Two-dimensional vortices with background vorticityThe two-dimensional character of geophysical flows...

Two-dimensional vortices with background vorticity

Citation for published version (APA):Velasco Fuentes, O. U. (1994). Two-dimensional vortices with background vorticity. Technische UniversiteitEindhoven. https://doi.org/10.6100/IR423643

DOI:10.6100/IR423643

Document status and date:Published: 01/01/1994

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https://doi.org/10.6100/IR423643

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Two-Dimensional Vortices with Background Vorticity

0 . U. Velasco Fuentes

Two-Dimensional Vortices

with Background Vorticity

by O.U. Velasco Fuentes

Oscar Velasco Fuentes Fluid Dynamics Laboratory Eindhoven University of Technology P.O. Box 513 5600 MB Eindhoven The Netherlands

Cover illustration: Sin t{tulo, Nor a Velasco Fuentes ( 1994).

CIP-GEGEVENS KONINKLIJKE BIBLIOTHEEK, DEN HAAG

Velasco Fuentes, Oscar Uriel

Two-dimensional vortices with background vorticity / Oscar Uriel Velasco Fuentes. - Eindhoven : Technische Universiteit Eindhoven. - lil. Proefschrift Eindhoven. - Met lit. opg. - Met samenvatting in het Nederlands en Spaans. ISBN 90-386-0134-4 Trefw.: geofysische stromingsleer

Two-Dimensional Vortices

with

Background Vorticity

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de Rector Magnificus, prof.dr J.H. van Lint, voor een commissie aangewezen door het College van Dekanen in het openbaar te verdedigen op

donderdag 27 oktober 1994 om 16.00 uur

door

OSCAR URIEL VELASCO FUENTES

geboren te México Stad

Dit proefschrift is goedgekeurd door de promotoren:

prof.dr ir G.J.F. van Heijst en prof.dr ir L. van Wijngaarden Universiteit Twente

This research was supported by the Netherlands Foundation for Fundamental Research on Matter (FOM) under grant SW-E-d 89.728.

A mispadres

Cosmos es caos pero DO Jo sabiamos o DO alcaDzamos a eDteDderlo.

J.E. Pacheco: Miro la tierra.

Preface

This thesis deals with the unsteady behaviour of quasi two-dimensional vortices in a rotating ftuid with gradients in the ambient vorticity. Since the first two chapters discuss the background of this research as well as the methods employed, here I shall confine myself to a few remarks.

A doctoral thesis is rarely the result of a single person's work, and this one is no exception. As a matter of fact, some chapters of this thesis have been ( or will he) published as joumal articles of which I am one of the authors. It is therefore inaccurate to place only one name on the cover of this thesis, but regulations do not allow otherwise. The small notes at the beginning of the aforementioned chapters are an unsuccessful attempt to correct that injustice. I have only slightly adapted the articles to include them here, together with some new chapters (which, in turn, will he submitted for publication later). Presenting the material in this way has some advantages, the main one being that a single chapter can he read as an independent work. A great disadvantage though, is that some degree of repetition occurs, which can become annoying. I ask the understanding of the reader in this respect.

I should like to thank the many people who have made contributions to this research. _ I have enjoyed working with my thesis advisor, GertJan van Heijst. The discussions I

had with him over the last four years have inspired most of the material presented here. Also the quality of this text has improved due to his carefut reading. I have benefited from conversations with Leen van Wijngaarden (University of Twente) and Anton van Steenhoven regarding the material discussed here. They also made important suggestions to improve the manuscript. The support and interest of Pedro Ripa (CICESE, México) on this research, as well as conversations about the Sixteenth Chapel are greatly appreciated. Conversations with Slava Meleshko (University of Kiev) about point vortices and dynamica! systems proved timely and valuable. Herman Clercx gave useful criticism about parts of the manuscript and was always willing to assist with various computer problems. Harm Jager and Eep van Voorthuisen provided much technica! assistance with the experimental equipment. I am especially thankful for the many times they solved problems on a short notice. Ion Barosan produced video animations of some of my numerical simulations, which have proved to he very useful in several occasions. Jan-Bert Flór and Casp-ar Williams, my roommates during the early years, provided an stimulating atmosphere and helped

with various experimental and numerical issues. Johan van de Konijnenberg and Menno Eisenga, my roommates during the last two years, were a great help during the writing process. They read parts of the manuscript and provided useful comments. Gert van der Plas was very helpfut with the image-analysis system. The MSc students Bart Cremers, Nicole van Lipzig and Joris Nuijten made important contributions to this research project. The help of the undergraduate students Elwin van den Bosch, Rob van Gansewinkel, Olaf Gielkens, Patriek Lemmens, and Roei Vanneer in performing laboratory or numerical experiments is greatly appreciated. I also want to thank the rest of the staff of the Fluid Dynamics Laboratory for making my stay in the Netherlands a pleasant one.

Finally, I want to thank Eva, Nora, Neil and Maurilio for their great support and the many experiences we have shared.

Contents

1 Introduetion

2 General theory and methods 2.1 Fluid motion in a rotating system ......... .

2.1.1 Approximations: j-, {3- and ')'-planes ... . 2.1.2 Topographie 'gradients' of ambient vortieity

2.2 Experimental methods ... . 2.2.1 Apparatus ...... . 2.2.2 Generation of vortiees 2.2.3 Flow visualization .. 2.2.4 Flow measurements .

2.3 N umerieal methods . . . . . 2.3.1 Point vortiees .... 2.3.2 Vortex-in-eell method .

2.4 Advective transport in two-dimensional flows . 2.4.1 The Poinearé map 2.4.2 Melnikov theory ........... .

3 Behaviour of a dipolar vortex on a {3-plane 3.1 Introduetion ............ . 3.2 The modulated point-vortex model 3.3 A meandering dipole ..... .

3.3.1 Qualitative observations .. 3.3.2 Flow measurements ..... 3.3.3 Trajeetory as a function of the tilting angle

3.4 Eastward versus westward travelling dipoles 3.5 ETD's for different values of j3 . 3.6 Conclusions . . . . . . . . . . . . . . . . . .

4 Adveetion by a dipolar vortex on a {3-plane 4.1 Introduetion ................. . 4.2 The physieal meehanism for transport ... . 4.3 Analysis of the modulated point-vortex model

4.3.1 Adveetion equations 4.3.2 Lobe dynamies .. 4.3.3 Melnikov function . 4.3.4 N umerieal results .

1

3

7 7 9

11 12 12 13 14 14 16 16 18 2:4 24 28

29 29 31 37 37 37 42 45 49 53

55 55 56 59 59 61 63 64

2

4.4 Size perturbations ..... 4.5 Experimental observations 4.6 Conclusions . . . . . . . .

5 Collision of dipolar vortices on a ,8-plane 5.1 Introduetion ................ . 5.2 Interaction of point-vortex dipoles .... .

5.2.1 Review of the non-modulated case. 5.2.2 Modulated coaxial couples . 5.2.3 Modulated parallel couples . 5.2.4 Mass exchange . . . . . .

5.3 Interaction of continuous dipoles . . 5.3.1 Experimental results .. . . 5.3.2 Numerical simulations using a vortex-in-cell method .

5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 A dipolar vortex on a ')'-plane 6.1 Introduetion ................... . 6.2 Propagation of a modulated point-vortex dipole 6.3 Transport by a meandering dipole .

6.3.1 Adveetion equations 6.3.2 Lobe dynamics . . 6.3.3 Melnikov fundion . . 6.3.4 Numerical results .. 6.3.5 Long time spread of particles

6.4 Conclusions . . . . . . . . . . . . . .

7 Unsteady behaviour of a tripolar vortex 7.1 Introduetion ............... . 7.2 Laboratory observations of an unsteady tripole . 7.3 Vortex motion ................. .

7.3.1 A non-modulated point-vortex tripole .. 7.3.2 A modulated point-vortex tripole .... 7.3.3 Comparison of experimental and numerical results .

7.4 Adveetion by an unsteady tripole .. . . ... .. . .. . 7.4.1 Adveetion by an asymmetrie point-vortex tripole 7.4.2 Adveetion by a modulated point-vortex tripole . 7.4.3 Experimental observations

7.5 Conclusions .... . ...... .

8 Conclusions

References

Samenvatting

Resumen

CONTENTS

70 72 74

77 77 78 78 81 84 88 92 92 97

100

101 101 102 107 107 109 110 111 117 118

121 121 122 124 124 130 134 135 136 146 148 150

153

155

159

161

Chapter 1

Introduetion

Like many other 'facts of nature', the essentially two-dimensional character of flow motion in the atmosphere and in the oceans is not evident in our daily life experience. lndeed, when one looks at breaking waves on a beach or at leaves being brought up and down in a windy day, one would not he ready to believe that 'the atmosphere and the oceans are close to a state of geostrophic equilibrium' . The latter is the name dynamica! meteorologists and physical oceanographers give toa state of slow, approximately two-dimensional motion. The cause of our 'misleading' observations is that we are not looking at the proper scales, neither in space nor in time. The two-dimensionality of atmospheric and oceanic flows (commonly referred to as geophysical fiows) is a feature of the large scales: it applies to the currents and wind systems extending over hundreds or thousands of kilometers on the Earth surface, and slowly evolving over periods of days or weeks.

The two-dimensional character of geophysical flows is mainly the consequence of two factors: (i) the Earth's rotation, which makes the fluid move in locally horizontal planes; and (ii) the geometry of the flow domain: the ocean and the atmosphere are thin layers of fluid of a few kilometers depth and thousands of kilometers in horizontal scale. The spherical shape of the flow domain has far reaching consequences. The horizontal plane in which the motion occurs is normalto theEarth's axis of rotation at the poles, and is parallel to it at the equator. In genera!, the horizontal plane and the axis of the Earth are located at an angle equal to the geographical latitude. As a consequence, the fluid effectively experiences a spatially varying rate of rotation. This effect is generally called the ,8-effect, and is the cause of many processes in the oceans and in the atmosphere, like planetary waves and the western intensification of the oceanic circulation (as apparent from, e.g., the Gulf Stream in the North-Atlantic and the Kuroshio Current in the North-Pacific). For phenomena occurring on latitudinal scales not larger than about thousand kilometers it is common to consider that the domain is flat and that the rate of rotation varies linearly in latitudinal direction (this is the so-called ,8-plane approximation). A similar approximation, but with a quadratic variation of the rotation rate is used for polar regions ( -y-plane approximation). If no variation is assumed, one speaks of an f-plane, which is a valid approximation for motions of latitudinal extent smaller than a few hundred kilometers.

A brief mention of two dynamica! equivalents of the ,8- and -y-plane approximations is in order. It can be shown, using conservation of potential vorticity in the shallow water model ( chapter 2), that a weakly varying topography causes an effective gradient of ambient vorticity. A flat sloping bottorn causes a uniform gradient and is thus equivalent to the ,8-plane; whereas a parabolic topography, like the free surface of a rotating fluid, is equivalent to the -y-plane.

3

4 Introduetion

"Shallow" in the topographic case is equivalent to "north" in the (3- or 1 -planes. A second equivalence appears in plasma physics. The dynamics in a magnetically-confined slab plasma is analogous to the (3-plane dynamics in geophysical flows, with the gradient of the plasma density playing the role of (3 . Similarly, the dynamics in a cylindrically confined plasma is analogous to that in a 1-plane, but a complete equivalence depends on the density dis tribution (Yabuki, Ueno & Kono 1993).

Two-dimensional flows are characterized by the emergence of coherent vortex structures (see, e.g., McWilliams 1984). Among these, the monopolar vortex, which consists of a patch of fluid rotating around a common centre, is the most frequently occurring vortex type. A more complex vortex structure is a dipole, consisting of two closely packed counter-rotating vortices. This vortex propagates in the direction defined by its symmetry line while carrying with it a fixed amount of mass. Therefore, the dipole can transport mass and momenturn over large distances compared with its diameter. The tripole completes the list of long-lived vortex structures. This vortex can be defined as a compact, symmetrie linear arrangement of three vortices, with the central vortex being flanked at its Jonger sicles by two weaker·vortices of oppositely signed vorticity. This symmetrie contiguration performs a steady rotation as a whole in the direction defined by the circulation of the central vortex.

The most spectacular example of a monopolar vortex is Jupiter's Great Red Spot , which still swirls strongly some 300 years after it was first observed. Similar long-lived structures can be found in the Earth's oceans -e.g., Gulf Stream rings. These rings can travel for hundreds of kilometers, while retaining their chemica) and biologica! water characteristics. Àlthough less abundant, dipoles have been observed in the oceans as 'mushroom-like' currents (Fedorov & Ginsburg 1989) and as 'blocking' events in the atmosphere (Haines & Marshall 1987). Recently, a tripolar vortex has been observedin the ocean (Pingree & LeCann 1992), viz. in the Gulf of Biscay.

Since the work of Stern (1975) much theoretica) work has been clone about the dipolar vortex both on a (3-plane and on a sphere. These dipolar solutions of the model equations are either stationary or translate steadily in zona! direction (see Flierl 1987) . In recent years , the motion of a dipole with an incidence angle with respect to the ambient vorticity isolines has received increasing attention. The unsteady motion of the dipolè has been stuclied using analytica) as well as numerical techniques, and point-vortex roodels as well as continuous distributions of vorticity (Kono & Yamagata 1977; Makino, Kamimura & Taniuti 1981; Zabusky & McWilliams 1982; Nycander & Isichenko 1990).

In contrast, few experimental studies on dipoles in a rotating fluid have been published. Until recent years the only studies devoted to the dynamics of dipoles in a homogeneous rotating fluid were those of Flierl, Stern & Whitehead (1983), and Fedorov, Ginsburg & Kostianoy (1989). Through the injectionof a turbulent jet (Flierl et al. 1983) , or the application of an air jet on the fluid surface (Fedorov et al. 1989) an asymmetrie dipole was generated, which subsequently moved in a circular path. The gradient of background vorticity caused by the parabolic free surface was estimated to be small in both studies. These authors therefore considered their experiments to be on dipoles on the f-plane.

It seemed thus relevant to study experimentaJly the behaviour of a single dipolar vortex on a (topographic) /3-plane. The experiments described in chapter 3 confirmed most of the analytica) and numerical results already reported in the literature. But new facts were also observed, like the growing size, the eventual break-up of an eastward travelling dipole (ETD), and the shrinking size of a westward travelling dipole (WTD). The hypothesis that

Introduetion 5

this asymmetrie behaviour is caused by generation of relative vorticity was confirmed using a vortex-in-cell method.

One of the most striking experimental observations, besides the meandering motion of the dipole, was the shedding of interior (dyed) fluid by the dipolar structure; this fluid then was stretched and formed thin, vertically-aligned bands. Similarly, ambient fluid was captured by the dipole and wrapped around the two vortex centres. These observations motivated the study of the adveetion of fluid particles in the velocity field of a meandering dipole ( chapter 4).

The study of the interaction of two zonally moving dipoles may contribute to an understanding of the dipole stability and of transport in a direction perpendicular to the isolines of ambient vorticity. The experimental observations described in chapter 5 verified some theoretica! results already reported in the literature (Kono & Yamagata 1977, Makino et al. 1981), like the partner exchange and the subsequent curved trajectories of the newly formed couples. However, a second partner exchange after the return of the couples totheir initia! latitude was not observed in the laboratory. Apparently, the realization of such an event requires a degree of symmetry that can only be achieved in numerical simulations.

Little work has been clone on the dynamics of the1-plane, which was introduced by LeBlond (1964) for the study of planetary waves in a polar basin. Nof (1990) used the same polar approximation to study monopolar and dipolar vortices. He found an unsteady propagation of the monopoles and obtained modon solutions equivalent to those on the ,8-plane; namely, isolated structures that propagate along !i nes of equal ambient vorticity ( circles in the 1-plane). The study of tilted dipoles on the 1-plane, described in chapter 6, seemed thus a worthwhile extension of our studies on ,8-plane dipoles.

Our interest in the 1-plane dynamics sterns from its topographic equivalent: the parabolic free surface of a rotating fluid. In particular, we were interested in the unsteady behaviour displayed by tripolar vortices generated off-centre as well as in the continuous stretching and folding of fluid patches produced by the unsteady vortex motion (van Heijst & Velasco Fuentes 1994). These features are in sharp contrast with previous experimental and numerical studies which showed that the tripole is a stabie and steady structure in a two-dimensional flow (e.g., van Heijst, Kloosterziel & Williams 1991, Orlandi & van Heijst 1992) . It is thus likely that the unsteady motion arises due to a three-dimensional effect; with the parabolic free surface of the fluid being the best candidate. In chapter 7 we use a modulated point-vortex tripole to support this hypothesis. The unsteady motion and the adveetion of fluid particles in this model are stuclied and compared with experimental observations.

We have used experimental, numerical and analytica! methods in studying vortices. The experimental set-up is simple: a tank filled with tap water mounted on a rotating table. Although the table can be controlled by a personal computer to have an angular velocity which is a non-trivia! function of time, the rotation period T was kept constant in the experiments described here. T had approximately the same value in most experiments (11 s) . The flow was visualized using dye or small particles floating on the surface of the fluid, and each experiment was recorded using a photographic or video camera. Measurements are also simple in essence: (i) the overall motion of the vortices is determined by following the motion of the patches of dyed fluid trapped by the vortices, and (ii) the velocity field, whether from streak photography or partiele tracking of video images, consists in determining the position of a partiele at a series of times and, from this series, the velocity at the desired time is computed by fini te differences and interpolation.

We have used a well known analytica! model: the point-vortex model introduced by

6 Introduetion

Helmholtz in the last century. For this research the strengths of the vortices are modulated on the basis of conservation of potential vorticity. Although in the literature only the ,8-plane modulation had been used, it was already mentioned by Zabusky & McWilliams (1982) that the modulation principle could be applied for different variations of the background vorticity.

For the study of adveetion we used the Lagrangian approach: This reduces the problem toa set of ordinary differential equations, i.e. toa finite-dimensional dynamica! system (Aref 1984). The powerful (geometrical) techniques developed for the study of such systems can be straightforwardly applied to the study of advection, with the advantage that the abstract phase space of a dynamica! system corresponds with the physical space where the flow takes place (see, e.g., Wiggins 1992). We have extensively used the 'lobe dynamics' technique to quantify the mass exchange between regions of the flow and to determine the location of the fluid patches that take part in that process (see chapters 4, 6 and 7). The Melnikov function is also used to evaluate the mass transport, but more importantly to show the existence of chaotic partiele trajectories. This has been clone using the temporal symmetries present in the advectio.n equations.

The set of 2N ordinary differential equations descrihing the motions of N point vortices has been integrated using a standard fourth order Runge-Kutta scheme (see, e.g., Press et al. 1986). The motion of tracers is computed in the same manner, but when they are assumed to define a contour in the flow, an algorithm must be implemènted to ensure that the contour is accurately defined during the whole flow evolution. The problem of defining a contour of varying length with a fini te set of points ( nodes) arises in various applications, like contour dynamics (see, e.g., PulJin 1992); consequently, various techniques have been proposed to cope with the problem. Here we use a cubic spline interpolation, which might not be the most efficient method to re-position the nodes, but it is simple and gives accurate results for the kind of problems discussed in this thesis.

The numerical integration of the equations of motion becomes extremely time consuming when the number of vortices becomes large. Therefore we turned into the vortex-in-cell (or cloud-in-cell) technique which was originally developed for the study of plasmas in the 1960s and later introduced in fluid mechanics by Christiansen (1973). We have added the modulation of the vortices' circulations in the method. The vortex-in-cell rnethod is a Lagrangian technique: it computes the evolution in time of the point-vortex positions. But, in order to compute the evolution in an efficient way, flow properties arealso computed on a regular grid fixed in space, i.e. the method gives also an Eulerian description of the flow (see chapter 2). These combined features make the vortex-in-cell method particularly suitable for comparison with our experimental observations: flow measurements are Eulerian, while flow visualizations are essentially Lagrangian.

In the following chapters we present the results of this combined study of unsteady dipolar and tripolar vortices. The analytica! and numerical computations of the vortex motion, as well as of the adveetion of fluid particles, are in agreement with experimental measurements. As mentioned in the final chapter and other places in this thesis, the point-vortex model captures the essential physical processes, whereas the vortex-in-cell method gives a more detailed description of th~ real flow.

Chapter 2

General theory and methods

2.1 Fluid motion in a rotating system

When the motion in a rotating fluid is referred to axes which rotate steadily with the fluid, Coriolis and centrifugal accelerations have to be added to the equations of motion. The centrifugal force (per unit mass) may be written as ~\7(0 x f')2 and this is equivalent, in a ftuid of uniform density, to a contribution to the pressure. With p now denoting a modified pressure which includes the effects of centrifugal forces as well as of gravity, the equation of motion with velocity ii relative to axes rotating with steady angular velocity 0 is

àil ~ n ~ 21\ ~ 1 n n2 ~ ""î"" + 1.1 • V U+ H X U= --V p +V V U, vt p

(2.1)

where the term 20 x ü is known as the Coriolis acceleration, although this term first appeared in LapJace's ti dal equations (Gil! 1982, p. 73). The degree to which the Co riolis acceleration inftuences the motion of ftuid elements evidently depends on the relative magnitude of Coriolis forces and other forces acting on the ftuid . In the present context, these other forces are inertia forces and viscous forces. If U is a representative velocity magnitude (relative to the rotating axes) and L is a measure of the distance over which U varies appreciably, the ratio of the magnitudes of the terms Ü· Vil and 20 x Ü is

u Ro = 2f2L"

The value of this ratio, known as the Rossby number but originally introduced by Kibel' (Gill 1982, p. 498), provides a measure of the relative importance of inertia and Coriolis accelerations. When Ro » 1, Coriolis accelerations are negligible; but when Ro « 1 the Coriolis accelerations are likely to be dominant.

The ratio of the magnitudes of the terms v\72ü and 20 x i1 is

V

E = 2flL2.

This ratio is known as the Ekman number and measures the relative importance of viscous forces with respect to Coriolis accelerations. For most geophysical ftows this number is very small (i.e., Coriolis forces domina te over viscous on es) and the ftuid can he considered as effectively inviscid. However, viscosity can play an important role through the (Ekman) boundary layers.

7

8 General theory and methods

The dominanee of the Coriolis accelerations at small Rossby numbers has remarkable consequences when the flow is also steady (relative to the rotating axes), as was first pointed out by Proudman (1916). Note that for 8üf8t = 0 and Ro -+ 0 the equation of motion becomes

~ 1 2n x a= --Vp.

p

This equation expresses a balance between Coriolis forces and forces due to pressure gradients. This balance is referred to as geostrophic balance in the geophysicalliterature. The pressure p is eliminated from this equation by taking the curl, and, as a consequence of conservation of mass one obtains

n. va= o. This relation , known as the Proudman theorem, holds approximately for Ro ~ 1 and states that 'slow' steady motion in a rotating fluid is uniform in the direction of the axis of rotation. For the motion to occur in a plane perpendicular to the axis of rotation it is necessary to have n. û = 0 sorriewhere in the fluid; for example, at a fixed boundary perpendicular to the rotation axis, as is the case in a rotating laboratory tank. However, if n · ü i- 0 somewhere on a line parallel to the axis of rotation, then the component of the velocity in the direction of fi is nonzero everywhere along that line. This theoretica! result was experimentally verified by Taylor (1923), who stuclied the flow that arises when a cylinder moves perpendicularly or parallel to the rotation axis of a rotating fluid.

A good description of important aspects of atmospheric and oceanic flows of large horizontal extent (say with linear dimensions of hundreds of kilometers) may be obtained from a simplified set of equations, based on the following idealizations:

(a) The ocean and the atmosphere are considered as layers of homogeneaus incompressible :fluid. Obviously, the density of the air varies as aresult of its compressibility; and the density of ocean water depends on its temperature and salinity. However , for many situations these effects may be neglected.

(b) The velocity of the fluid is considered to be the average velocity of water or air masses over the wholefluid layer. Although vertical currents do occur, this average motion is nearly horizontal and, in large scale flows, varies appreciably over horizontal distances larger than about 100 km . The effect of bottorn friction (Ekman layers) is also neglected.

( c) A variation in the fl uid depth h is allowed, but the variations of h over horizontal distances of order h are considered to be negligible (see figure 2.1). The only consequence of this slow variation of h is to impose on the fluid a non-zero rate of expansion in the horizontal plane. By consiclering conservation of mass of a vertical fluid column of small cross-section we find

"V· ü =.!._DH H Dt'

(2.2)

where D / Dt is the material derivative in the horizontal plane. For all other purposes the vertical component of the fluid's velocity may be neglected. This approximation, known as the 'shallow water' model, was introduced by Laplace for the study of tides.

The equations of fluid motion in the thin layer covering the planetary globe are better expressed in a spherical coordinate system (O,r/!,r), which rotates with the Earth, with the origin at the centre of the sphere and r ~ R in the whole layer (Ris the radius of the Earth) . The angle r/! is the latitude (i.e., rjJ = 0 at the equator and rjJ = 1r /2 at the north pole) , and the direction at which (} increases with rjJ and r constant is east (see figure 2.2) . The velocity is given by ü = (uo, u4>, u,) and the angular velocity by n = (O,nsin rjJ,ncos rfi) .

2.1 Fluid motion in a rotating system

h(x,y)

Figure 2.1: A sketch of a shallow layer of fluid.

Taking the curl of (2.1), with v = 0, leads to

aw _ "_ ( _ "') "_ Bt+u· vw= w+2H ·vu,

and substitution of (2.2) gives the relation

.!}_ (w + !) = 0 Dt h '

9

(2.3)

(2.4)

which expresses the conservalion of potential vorticity. Here f = 20 sin tfJ is the Coriolis parameter. On the (two-dimensional) spherical surface the material derivative takes the form

D 8 uo 8 uq, 8 -=-+---+-Dt 8t Rcos tjJ 80 R 8tjJ

and the relative vorticity in the radial (vertical) direction is

w = __ 1_ (8uq, _ Buo cos t/J) . R cos tfJ ao BtfJ

Equation (2.4) shows that the relative vorticity w of a fluid column can change as a consequence of the movement of the element to a place where the thickness of the fluid layer is different, or as the element movestoa different latitude. This important conservation property will be used throughout this thesis.

2.1.1 Approximations: f-, (3- and Î-planes

We consider motions that extend to a small range of latitudes centred at t/J = t/J0 and, for simplicity, we assume that the layer of fluid has a constant depth h0 • In this case, it is convenient to introduce the following cartesian coordinates

x = R(} cos t/Jo, Y = R(tfJ- tPo), z = r- R,


• y-plane •

Figure 2.2: Schematic representation of the /3- and ')'-plane approx.imations for the motion in a thin layer of fluid on a rotating sphere.

which forma right-handed system with z the upward vertical coordinate, and x and y pointing eastward and northward, respectively. With x and y as rectilinear coordinates and the Coriolis parameter having a constänt value Jo = 2!1 sin 1/>0 , the only explicit changes in (2.4) arising from this approximation occur in the expressions for the materiaJ derivative D / Dt and the relative vorticity

D a a a Dt = at + U x a x + Uy ay,

auy aux w = Tx -a;·

This approximation is known as J-plane approximation, and it was first mentioned by Kelvin in his workon waves in a rotating fluid, see Gill (1982).

Two improved approximations of (2.4) are obtained by allowing variations of the Coriolis parameter, while assuming that the motions still occur on a plane tangent to the Earth's surface (see figure 2.2). The variation of the Coriolis parameter J = 2!1 sin!/> is obtained by expanding it in a Taylor series around the reference latitude lj>0 :

J = 2!1(sinl/>o + ólj>cosl/>0 - (6.:)2

sinl/>0 + ... ).

Retaining the zeroth and first order terms leads to a linear variation of J

J =Jo+ /3y,

where

Jo = 2!1 sin 1/>o, f3 = 2fl cos 1/>o . R

This is known as the (3-plane approximation and leads to the conservation relation

D Dt (w + f3y) = 0. (2.5)

2.1 Fluid motion in a rotating system 11

Note that f3 decreases as 4> -+ 1r /2, and becomes exactly zero at the pole. Here, the second order term in the Taylor expansion of J becomes the leading term in the variation of the Coriolis parameter. A cartesian coordinate system is also used, but the orthogonal coordinates x and y are defined as

y = R(4>o- rf>).

With these definitions, the Coriolis parameter becomes

where

Jo= 2!1, n

"Y = m· This less common approximation is known as the -y-plane (Nof 1990) and leads to the conservation relatiori ·· ·

2.1.2 Topographic 'gradients' of ambient vorticity

Topographic /3-plane

(2.6)

We now consider the conservation of potential vorticity in the shallow water model (2.4) with a uniform Coriolis parameter J = Jo = 2!1. Let h have a small linear variation in some direction, say y, so that the fluid depth as a function of position is given by h(y) = h0(1- sy), with s a small parameter. Substituting this expression in (2.4) and expanding the result in a Taylor series one obtains:

D Dt (w + sJoy) = 0, (2.7)

where a small Rossby number (w/ Jo ~ 1) is assumed. This equation is equivalent to (2.5), showing that to this order of approximation the dynamics of a rotating layer of fluid with a linearly varying depth is equivalent to that of a fluid on a /3-plane. Obviously, the equivalent {3-value is f3 = sJo-

Topographic -y-plane

The free surface of a fluid rotating with an angular speed n acquires a parabalie shape h(r) = ho(1 + 0 2r2 /2gho) where ho is the fluid depth at the axis of rotation, r = (x 2 + y2 ) 112 is the distance to this axis, and g is the gravity acceleration. Substituting h(r) in (2.4) and expanding the result in a Taylor series one obtains

D F -(w - --r2) = 0,

Dt 8gho (2.8)

where a small Rossby number (w/ Jo ~ 1) is again assumed. This equation is equivalent to (2.6), showing that to this order of approximation the parabalie free surface of the fluid induces an "ambient vorticity" distribution equivalent to that in the polar region of a rotating sphere (the so-called -y-plane), with an equivalent -y-value of "Y = j3 /8gh0 .

12

D I


Figure 2.3: Schematic view of the experimental arrangement for the study of dipoles on a ,13-plane. The tank rotates with a constant angular velocity 11, and the ,13-effect is provided by a sloping bottorn (with shallow being equivalent to "north"). Dipoles are generated by dragging a small bottomless cylinder (8 cm in diameter) through the fluid while lifting it as indicated in the figure. The typical speed of the cylinder is 0.15 ms-1 •

2.2 Experimental methods

2.2.1 Apparatus

The experiments were carried out in a rectangular tank of horizontal dimensions 100 x 150 cm2 and 30 cm depth mounted on a rotating table. The angular speed of the system could he varied continuously and was in most experiments taken as n = 0.56 s-1 , so that the Coriolis parameterf = 1.12 s-1 • The working depthof the fiuid was varied from 15 to 20 cm.

The topographic /3-effect was provided by raising a false bottorn 4- 8 cm along one of the long sicles of the tank. With these parameter settings the equivalent value of f3 measured approximately 0.25 m -ls-1

• The topographic 1-plane is, obviously, always present in our free surface experiments; with the typical values of rotation rate and fluid depth mentioned above, 1 measured approximately 0.1 m-2s-1 • This topography generated a maximum gradient of ambient vorticity (2q) of about 0.05 m-1s-1 , which is smaller than the gradient generated by the sloping bottom. Therefore we may neglect the effect of the free surface in the experiments on a /3-plane.

This choice of parameter values was made in order to achieve two effects: (i) a dynamically relevant gradient of background vorticity without affecting the two-dimensionality of the motion; and (ii) a column of fl.uid long enough for the effect of the bottorn Ekman layers to he negligible on the time scale of the experiment, e.g., the time required for a dipole to move a distance equal to several dipole diameters or for a tripole to make a few rotations. These time scales are equivalent to 10-15 rotations of the table.

2.2 Experimenta.l methods 13

D

t

Figure 2.4: Schematic view of the experirnental arrangement for the study of tripolar vortices. The tank rotatea with a constant angular velocity 11, and the parabolle free surface of the fiuid provides an effect equivalent to that of a 1-plane at the pole of a rotating planet. As in figure 2.3 shallow is equivalent to "north", and thus the centre of the tank corresponds with the north pole. Monopolar vortices were generated by cyclonically stirring the fiuid within the small, bottornleas cylinder (::::: 20 cm in diameter), which then is quickly lifted. A tripoleis usually forrned as aresult of the instability of the rnonopole.

2.2.2 Generation of vortices

Each experiment was started by filling the tank to the desired height. Then the fluid was spun-up to an angular velocity of 0.56 s-1 , a process that takes typically a few minutes. In order to avoid even very weak background flows, however, in all experiments the fluid was allowed to spin-up for approximately 30 minutes.

Di pol es

A colurnnar dipole vortex was generaled by slowly moving a smal!, bottomleas cylinder of 8 cm diameter in a straight line relative to the rotating tank, while gradually lifting it (figure 2.3). By moving the cylinder very slowly (10-15 cms-1) and keeping its axis parallel to the axis of rotation, one guarantees that the forcing is almost two-dimensional. The vorticity generaled by the motion of the cylinder accumulates in a dipolar structure in the wake of the cylinder. This dipolar flow is confined in a vertically-aligned Taylor column, a feature well-known in rotating fluids. After typically 1-2 rotation periods the organization of the vortex flow is completed, and the mature, fairly symmetrie dipole travela through the fluid along an almost straight line.

The radius of deformation Rd = #/ f varied with the experimental configuration from 1 to 1.25 m. These values are much greater than the typical size of the dipole (0.1 m). A Rossby number for the generation process is defined as Ro = U /2f!R, with f! the rotation rate of the system and U and R the velocity and radius of the cylinder, respectively. According to the parameters given above this Rossby number is of order 2- 3. A Rossby number for the

14 General theory and meth~ds

resultant dipolar structure is defined with U as the maximal velocity and R as the distance between the points of the extrema! values of vorticity. This Rossby number is measured to be of order 0.1-0.2 when the dipale reaches a mature state, in all experiments discussed in this thesis.

The generation technique described above proved to be superior to the injection of a turbulent jet, since in that case the forcing is essentially three-dimensional. The jet is deflected by the Coriolis force and tends to move anticyclonically in a circular trajectory. One of the two-dimensional products of this process is an asymmetrie dipale (Fiierl et al. 1983)

Tripoles

A tripolar vortex was generated by creating an unstable manapolar vottex in the following way. A bottomless cylinder of about 20 cm in diameter is placed in the rotating tank and the fluid located within the small cylinder is stirred cyclonically, i.e., in the same sense as the rotation of the table. By quickly lifting the cylinder (figure 2.4), an isolated manapolar vortex is released in the uniformly rotating ambient fluid (N .B. isolated means that the total circulation of the monopole is zero). Under certain conditions, that are easily met, this vortex becomes unstable, resulting in the gradual formation of a tripolar vortex (van Heijst et al. 1991). When the formation process of the tripale is completed the three vortices are located on a straight line, with the satellites located at equal distances from the central vortex. This tripolar structure rotates as a solid body.

2.2.3 Flow visualization

For the vortex structures ( dipales and tripales) stuclied in this thesis a first series of experiments was done using dye (fluorescein or terasil blue) to visualize the flow. Such experiments provided important qualitative information, like the overall motion of the vortex structure and the adveetion of fluid parcels, and also allowed to verify the two-dimensionality of the motion.

The dye was added tothefluid within the small cylinder befare the vortex was generated. In each experiment the flow was recorded photographically (photographs were taken at intervals of typically 5-15 sec) or in videotape by a camera mounted in the rotating frame a bout 150 cm above the free surface of the fluid .

Dye experiments were done mainly to determine the trajectories of the individual vortices which form the dipolar or tripolar structures. Depending on the way an experiment was recorded the trajectories were obtained as follows: (a) by projecting the negative photographic film on a screen and by platting the vortex een tres; or (b) by capturing a series of video images in a personal computer and writing to a file the positions of the the vortex centres. In both cases, the vortex eentres were determined from the spiral structure of the dye within each vortex and from the shape of the whole vortex.

2.2.4 Flow measurements

Quantitative information was obtained by seeding the flow with small particles floating on the fluid's surface, which were assumed to follow fluid elements without affecting the flow itself. The partiele veloeities at a particular time were obtained by two methods: (a) streak photography and (b) partiele tracking on a video tape.

2.2 Experimental methods 15

In the experiments analysed with the streak photography method, pictures were taken every 5-15 s and the exposure time was varied typically from 1 s at the beginning of an experiment to 3-4 s at later stages. The typical duration of an experiment was 20-25 rotation periods. The velocity field was measured from the lengths and orientations of partiele streaks, which are approximated by straight line segments. With the help of a digitization table the coordinates of the start and end points of the streaks are ]oaded into a personal computer. The distance between these points divided by the exposure time gives the mean local velocity of the fiuid.

In the case of a video-recorded experiment, the velocity was obtained by using the partiele tracking feature of Digimage (Dalziel1992). Fora detailed description of the process the reader is referred to Dalziel (1992). Here we simply outline the main steps involved in obtaining the velocities. (a) Image capture: a series of video images (frames) is captured and digitized using a frame grabber; (b) partiele location: in each digitized image particles are identified using a number of attributes, such as intensity, size, and shape; and (c) partiele matching: it is determined which image particles in two successive frames represent the same physical particle; this is clone using spatial and temporal information in addition to the partiele charaderistics obtained in step (b ). The result of this process is a time-series (200-300 s) of the positions of individual particles. From this information the positions and veloeities of particles at a particular time can be obtained by interpolation.

Both methods yield, fora particular time, the velocity in a number of irregularly distributed points. These veloeities are subsequently interpolated onto a regular grid of 30 x 30 points by using cubic splines (for details of this technique, see Nguyen Duc & Sommeria 1988). The analytic functions that give the values of the velocity components u and v in each grid point canthen be differentiated to obtain the vertical component w of the vorticity in the grid points

àv àu w=---.

àx ày

The stream function "P is computed from the vorticity field by numerically solving the Poisson equation

\12"P = -w.

The boundary conditions for "P are obtained (within an arbitrary constant) from the line integral of the normal velocity on the edge.

The stream function "P' in a frame of reference moving with the vortex structure can be calculated by simpte transformation

"P' = "P- Ury + Uyx or "P' = "P + ~!1'[(x- xc? + (y- Yc) 2],

with Ur and Uy being the componentsof the vortex translation velocity, !1' the angular velocity of the structure with respect tosome point (xc, Yc)· These are corrections for motions relative to the rotating system, therefore the point ( Xc, Yc) is independent of the axis of rotation. In the case of pure .linear translation the vorticity remains unchanged, whereas a constant value (2!1') must be added in the case of a rotational motion.

The parameters needed to correct for rotation or linear translation are obtained from a short series of images including the one being analyzed. From this sequence, one can estimate the magnitude and direction of the velocity, or the angular speed and the apparent centre of rotation. Throughout the following chapters only the corrected stream function "P' wil! be used, but the prime wil! be omitted to simplify notation.

16 General theory and metbods

2.3 Numerical methods

2.3.1 Point vortices

One of the simplest theoretica! modelsof 2D vortical flows represents the vorticity distri bution as delta functions on the plane. These are known as "point vortices". A vortex with strength K.; located at point (x;, y;) moves with a velocity equal to the sum of the veloeities induced by the other N - 1 vortices in the system. The evolution of the set of vortices is governed by the set of 2N ordinary differential equations (see, e.g., Batchelor 1967)

dx;

dt

dy;

dt

1 N Yi- Yj --I: K.j--2-, 21r i=I rij

#i

1 N x·- x· - :L:K,j-·-2-', 21r j=I r;j

#i

for all values of i from 1 toN, where r;j is the distance between vortices i and j.

(2.9)

(2.10)

If we assume that a point vortex represents a small patch of vorticity, the circulation K.;

equals the (uniform) vorticity w; multiplied by the area of the patch a;, therefore K.; = w;a;. Conservation of potential vorticity implies that the relative vorticity w of a vortex tube moving in meridional direction changes as expressed by, e.g., (2.5)-(2.6). These equations, in addition to conservation of mass, yield the following modulation equations for the vortex circulation:

f-plane, ,8-plane, 1-plane.

(2.11)

The coordinates (x,y) are used as defined insection 2.1.1 for the different planes. Subindexes 'jo' denote the initia! value of some quantity associated to vortex j, while a single subindex denotes a quantity at any other time in the evolution. Several shortcomings of the model must he mentioned: (i) A potential vortex is nota solution of (2.5) or (2.6), nor of any similar equation having a non-uniform Coriolis parameter; (ii) a point vortex represents a finite area of the flow domain, but it is supposed that this area is not deformed during the evolution; (iii) equations (2.5)-(2.6) are valid for the whole flow field, and this representation applies them only to a finite number of patches of fluid, i.e., we are neglecting the generation of relative vorticity due to adveetion of initially passive fluid. This truncation error (Zabusky & McWilliams 1982) can he reduced by adding more point vortices, as will he discussed in the next section.

Motion of passive tracers

The ability of the point-vortex model to simulate real flows is best appreciated when, in addition to the motion of the vortices, the evolution of patches of passive fluid is also taken into account. Numerically this is clone by computing the evolution of a set of points that define the contour of the region of interest . If the range of i in equations (2.9)-(2.10) is taken to he 1- M, with M > N, then the particles i= N + 1, .. . , M represent passive tracers that

2.3 Numerical methods 17

(a) (b)

4,-----------------,

3

2 : ~

~ \

0 !\ I \ j \ : y . . .

\! \.! 10 20 30

s

Figure 2.5: Numerical computation of the adveetion of passive contours in the velocity field of a system of point vortices: (a) a typkal closed contour, (b) coordinates x and y of the points defining the contour shown in (a) as a function of the length s along the contour. New positions are computed by natura! cubic splines constructed between each smooth segment.

are advected by the velocity field created by the point vortices i = 1, ... , N. Obviously, the tracers do not affect the evolution of the point vortices .

In genera!, there is no way to define a priori an optima! number of points and their distri bution on the contour to achieve an accurate description of the curve in the time interval of interest. As the flow evolves, some segmentsof the line are subjected to a large stretching while others undergo little change or are contracted. As a consequence, any initia! distri bution of points becomes inadequate: in segments with large stretching the points move apart from one another, whereas many points accumulate in the contracted segments. It is clear that, as the distance between adjacent points exceeds a certain threshold value 8smax, new points must be introduced in between to guarantee an accurate description of the curve. On the other hand, as the distance between two consecutive points decreases below a value 8smin one of them can safely be discarded. However, instead of introducing and removing points, we have chosen to redistribute all points in a uniform way along the contour, after each time step. For this purpose the coordinates x and y of the points defining the contour are written as a function of the contour length s measured from some reference point (x0 , y0 ) (figure 2.5a), and a natura! cubic spline (Press et al. 1986) is computed between every smooth segment of the curve. The smoothness of the curve is monitored by computing the second difference (i. e. a discrete form ofthe second derivative). New positions are then computed at uniform intervals 8s along the contour. Figure 2.5a illustrates a closed contour defined by a large number of points and figure 2.5b gives the coordinates x and y of these points as a function of the distance s along the contour. The whole contour is composed of a number of smooth segments, one of which is indicated by the arrows in figure 2.5b.

The area enclosed by an arbitrary closed line is a conserved quantity in this two-dimensional

18

(a)

5 ,------- -------,

4

L 3 s

2

1 0 2 3 4 5

time


(b)

1.02 ,----- ---------,

0.98

0.96

0.94

0.92

0.9 0

&=0.01 d

'\ .. ""'.·· . ·-,;· ..

\ ·· .. \ ·· ..

\ ·· .. &~0.05d ~

\ \

\, -&=O.i d -- . .._., _.r·

2 3 4

time

5

Figure 2.6: Evolution of the contour length (a) and the conesponding enclosed area (b) for different 6s used in the computation of new points (figure 2.5). The test case is a dipole on a ,B-plane (see chapter 4); in one time unit the dipole advances a distance equal to the separation between the point vort i ces.

incompressible model, hence it provides a way to check the quality of the integration. This has been clone by monitoring the time evolution of the area for various 8s. A dipolar vortex on the ,13-plane was chosen as test case; more specifically, we follow the contour of the fiuid initially trapped by the structure (see chapter 4). As 8s = O.Old, with d the distance between the vortices, the area is conserved within 0.1 % (figure 2.6b), largervalues (8s = 0.05d,O.ld) produce larger errors and no significant improvement is achieved with smaller 6.s. Note the large error obtained for the stretching of the contour, approximately a factor 0.5, when a large 8s- is chosen (figure 2.6a).

2.3.2 Vortex-in-cell method

Discretization

Let us assume that we are given a vorticity distribution w(x , y). The first step in the numerical computation of the evolution of w(x,y) with the vortex-in-cell method is to represent this vorticity distribution by a finite set of point vor tices. In genera!, it is numerically convenient to assume that every point vortex represents an equal area of fluid a. The circulation of the vortex is thus determined by K-k = aw(xk, Yk), where (xk, Yk) is the vortex position. The determination of how many points are used and where they are initially located is strongly determined by the phenomenon we are interested in. For example, if no gradients of ambient vorticity are present, relative vorticity is conserved and the regions that contain vorticity at the initia! time are the only active regions at any time later in the evolution. In this case the point vortices are placed in the - usually finite- regions of fluid with a non-zero vorticity and, where possible, they are arranged along streamlines. This is easily clone if the structure


has circular symmetry or is elliptic, but for other cases (like the Lamb dipole) the non-trivia! functions descrihing the streamlines, in addition to the condition that point vortices represent equal areas of fluid, make this type of discretization difficult.

A similar situation occurs if gradients of ambient vorticity are present in the system. In this case relatiVè vorticity can be generated over the whole flow domain, even if only a smal! region contains vorticity at the beginning. Therefore, the whole flow domain is covered by a regular array of point vortices, each of them representing an equa] area of fluid. The vorticity of each vortex is determined from the function w(x,y) and wil! usually result in a large fraction of the vortices (90 %is a typical figure) having zero relative circulations initially. These vortices become active as they move to regions with different background vorticity, and for this reason they are sometimes called "ghost vortices".

Motion of the point vortices

Once we have the continuous vorticity field w(x,y) being represented by a finitesetof point vortices, the evolution may be computed by numerical integration of (2.9)-(2.10), with the possible addition of the point-vortex modulation (2.11). However, the integration of the equations of motion of this system becomes excessively time consuming when the number of vortices Np is higher than a few tens (the computation time is proportional to N;). An alternative approach, introduced in fluid mechanica! problems since the 70's (Christiansen 1973), makes it possible to compute the evolution of several thousands of points vortices in an efficient and accurate way. The essential step is to define a mesh where the group of vortices evo]ve. The vorticity in the grid points is then obtained by interpolation, and subsequently the stream function and the velocity are obtained as in any standard mesh technique. Finally, the veloeities of the point vortices are obtained by interpolation of the velocity field. In this case the computation time is proportional to NP if the mesh dimensions are kept constant (see Hockney & Eastwood 1981). Below we briefly describe the operations performed to move the set of point vortices one time step.

{i) Interpolation

Thesetof vortices with vorticity wk and coordinates (xk,Yk) are supposed to !ie inside a rectangular region covered by a Cartesian meshof dimensions Nx x Ny and cel! size b.x x b.y. Several methods have been used to evaluate the vorticity w;,j at the mesh point (i,j) (see, e.g., Hockney & Eastwood 1981). Here we use the four-point cloud-in-cell scheme (CIC4). To assign vorticity to the four surrounding mesh points it is assumed that the 'point' vortex possesses uniform vorticity Wk within a rectangular area A = b.x x b.y. Every surrounding point receives an amount of vorticity proportional to the area of the vortex that lies within that cel!, as illustrated in figure 2.7. If one writes the coordinates of a point vortex as Xk = ib.x+8x and Yk = jb.y + 8y, the CIC4 interpolating factors are given by

(b.x- 8x)(b.y- 8y),

8x(b.y- 8y),

8x8y,

(b.x- 8x)8y,

(2.12)

(2.13)

(2.14)

(2.15)


Sx

Ay

Figure 2.7: Cloud-in-cell (or 'area weighting') interpolation scheme. The 'point' vortex (solid circle) is assumed to represent an area equal to the rectangular cell t::J.xt::J.y. The intersection of this area with the areas corresponding to the ciosest four mesh points (solid squares) gives the fraction of the vorticity that is attributed toeach mesh point.

and vorticity is credited by

w· · 1,] (2.16)

(2.17)

(2.18)

(2.19)

where a is the area represented by each point vortex and A is the cell area. The distribution of vorticity on the mesh is obtained by applying these relations toeach vortex (k = 1, ... , NP) in the flow domain .

(ii) Stream function

The stream function is related to the vorticity by the Poisson equation '\1 2 '1/J = -w, which is discretized by the usual five point approximation

'1/Ji,j+l - 2'1/Ji,j + '1/Ji,j-1 + '1/Ji+l,j- 2'1/Ji,j + '1/Ji-l,j

(t::J.y)2 (~x)2 = Wi,j· (2.20)


This equation, with the addition of appropriate boundary conditions, is solved by the FACR (Fourier Analysis and Cyclic Reduction) method developed by Hockney (1970). This direct method is "the best way to solve (partial differential) equations" of elliptic kind and constant coefficients (Press et al. 1986). lt requires the dimensions of the grid to he a power of two (Nx = 2" and Ny = 2\ with pand q integer numbers), and allows for three types of boundary conditions in each direction. The conditions are (i) a prescribed function giving 'Ij; on the boundary, (ii) periodic and (iii) zero tangential velocity but non-zero normal velocity. The boundary conditions in one direction are independent of those in the other direction.

(iii} Veloeities

The velocity field, i.e. the velocity in the mesh points, is evaluated from the stream function by using centred differences

Ui,j = 1/;;,j+l - 1/;;,j-1

2/:).y 1/;i+l,j - 1/;i-l,j

2/:).x

(2.21)

(2.22)

The velocity of each point vortex is determined with the veloeities on the four dosest mesh points, which are the mesh points that define the cel! in which the point vortex is located. The same interpolation factors used to obtain the vorticity on the grid are applied to compute the velocity of a point

(iv) Time inlegration

fi u;,j + hui+l.i + hui+IJ+l + f4ui,j+l,

!I v;,j + hvi+I,i + hvi+l,i+l + f4viJ+i·

(2.23) (2.24)

The new positions of the point vortices are obtained by allowing the points to move a short time interval with the velocity computed above. For this purpose two sets of coordinates (xk, Yk)odd and ( Xk, Yk)even are introduced to express the vortex positions at alterna te times 2n/:).t and (2n + 1)/:).t. The new positions are computed by

x;;+ I = x;;- I + 2/:).tu};,

yi:+1 = Yi:- 1 + 2/:).tv;:,

(2.25)

(2.26)

where the superscript n denotes the time n/:).t. Note that the velocity field used to move the set n- 1 is computed by the vorticity distribution produced by the set n. Due to this feature the time integration scheme is called 'leapfrog'.

(v} New relative vorticity

As the new position of the point vortices is known, the computation of the new relative vorticity is made using (2.11).


{a) {b)

{c) {d)

{e) {f)

Figure 2.8: Evolution of an elliptical patch of uniform vorticity (Kirchhoff's vortex) computed using the vortex-in-cell method. Parameters of the simulation: 128x 128 mesh, 961 point vort i ces, ajb = 2, a = 8.ó.x, .ó.t = 0.1.

Tests

The quality of a numerical model is better tested with a simpte problem of which the analytica! solution is known. The Kirchhoff vortex, an elliptic patch of uniform vorticity in an otherwise quiescent fiuid, was chosen as test case. This vortex rotates without change of shape with a constant angular velocity n = abwf(a2 + b2

), where a and bare the lengtbs of the semiaxes and wis the vorticity within the ellipse. In addition to the usual conserved quantities, one has the angular velocity, the area and the shape of the vortex, as control parameters. The quality of the results produced by the model was tested as a function of two parameters, namely, the number of point vortices per cel! (Ne) and the time step (~t) .


(a) (b)

1.004 1.004

m m At K0.1

1.002 1.002 At =0.05 \

0.998 0.998 At =0.2

0.996 0.996

0 2 3 0 2 3

time time

Figure 2.9: Evolution of the momenturn M according to the vortex-in-cell method: (a) for different numbers of point vortices per cell; and (b) for different time steps. A Kirchhoff elliptical vortex with eccentricity 2 was used as a test case ( one rot at ion period is the unit time).

The discretization is clone in a circular vortex and this is transformed into an elliptical one. One point vortex is placed in the centre and more vortices are located on n concentric rings: 8j vortices (where j is the ring number) are uniformly distributed in each ring. This gives a total number of point vortices of Np = 4n(n + 1) + 1, each representing the same area of fiuid. The computations were clone using a 128 x 128 mesh with free-slip boundary conditions. The semiaxes of the ellipse were taken as a = 2b = 8~x and the vorticity w = 1; the angular velocity of the ellipseis thus n = 2/9. An example of the evolution is shown in figure 2.8.

Simulations were clone with n= 10, 15, 20, 25, 30 and 35; which resulted in Ne ~ 4, 10, 16, 26, 37, and 50. It was found that the error in the momenturn M decreases as the number of particles per cel! increases from 4 to 16 (figure 2.9a) butfora larger increase of points (16 to 26) the error does not decreases significantly. If the number of particles is kept constant ( n = 20) and the time step is varied, a decrease in the error is observed as the time step is changed from ~t = 0.2 to ~t = 0.05 (figure 2.9b), but a further reduction produces no impravement in the conservation of moment urn. N ote that M invariably displays a small-amplit u de oscillation with a frequency which is four times the rotation frequency of the elliptical vortex. This oscillation is caused by the discretization of the Poisson equation (see, e.g., Christiansen 1973). If the difference between the stream functions determined numerically and analytically is plotted, one obtains a pattem with an azimuthal mode 4. Therefore, the vortex experiences alternately outwarcis and inwards radial fiows as it rotates around its centre. These non-physical fiows generate the oscillations observed in figure 2.9.

We conclude this section by stressing that the vortex-in-cell method provides both a Lagrangian and an Eulerian description of the flow. That is to say, it gives at the sametime the distri bution of vorticity and velocity on a mesh fixed in space, and the positions and veloeities of a set of material particles.


2.4 Advective transport in two-dimensional flows

The general problem of adveetion can be stated, in the so-called Lagrangian representation, as follows (see, e.g., Aref 1984):

d~

x ~ dt = u(x,y,z,t). (2.27)

Here the advecting velocity field ü is a prescribed function of the spatial coordinates (x,y,z) and timet. This is a set of three ordinary differential equations, i.e. , it is a finite-dimensional dynamica! system. In this thesis we will be concerned with (2.27) when ü is two-dimensional, incompressible and unsteady, in which case (2.27) becomes

dx &IJl dt = ày'

(2.28)

where IJl is the strea.m function. Note that (2.28) defines a Ha.miltonian system of one degree of freedom if IJl is constant and of two degrees of freedom if IJl is time-dependent. The number of degrees of freedom (N) has strong consequences for the adveetion of particles. If N = 1 the streamlines are constant in time and ftuid particles move along them, i.e., partiele trajectories are integrable. If N > 1 ftuid particles can move along different streamlines at different times and, as a consequence, partiele trajectoriescan be chaotic.

2.4.1 The Poincaré map

Let us assume that the stream function IJl can be written as the sum of a steady component and a small, time periodic perturbation

IJI(x, y, t) = tPu(x, y) + f.tPp(x, y, t), (2.29)

where f is a small parameter and the perturbation t/;p has period Tp. In this periodic case a significant simplification of the description of partiele motion is achieved by using the Poincaré map: the map of the partiele location (x(t0 ), y(t0 )) to the location one period later (x(t0 + Tp), y(t0 + Tp)). Loosely speaking this corresponds to sampling the position of a partiele using the fixed time interval Tp. The fundamental step consist in consiclering only the sequences of sampled positions, and forgetting about the details of the trajectories between two samples. There are several justifications for doing so (Hénon 1983):

a. The essential properties of the differential system are reftected in equivalent properties of the mapping; for example, a partiele trajectory with period Tp gives rise to a fixed point in the Poincaré section, and the stability properties of the complete orbit and the fixed point are the same.

b. The new problem is much simpler: the number of dimensions is reduced by one. For example, in unsteady two-dimensional ftows the Poincaré map eliminates the temporal dimension. This reduction also makes the graphical representation of the results simpler.

c. The essential properties of the system, related to the long-term behaviour, are more clearly seen because the irrelevant details of the short term evolution have been eliminated. For example, the trajectory of a single partiele in an unsteady two-dimensional flow can be a complicated, self-intersecting curve: displaying the trajectories of a few particles produces a practically useless graph, due to its complexity.

2.4 Advective transport in two-dimensional flows

(a) (b)

I

\ \\ .. ···········

.. ~<· /

/: ··~---.... ...... .

i

(c) (d)

heteroclinic point

c

·. \ \

' '

25

Figure 2.10: (a) Streamline patterns of a steady two-dimensional flow. (b) Effect of a small perturbation on the closed streamlines: if the rotation number is rational a chain of islands appear in the Poincaré section ( see text ). ( c) Effect of a small perturbation on the separatrix: multiple trans versa! intersections of the stabie and unstable manifolds. ( d) Transport between different flow regions: the dotted area closetoA will be detrained in the next period (see text).

It is useful tomention some properties of the Poincaré map which, as stated in (a), stem from the properties of the flow field (Rom-Kedar, Leonard & Wiggins 1990):

a. Area conservation. Incompressibility implies conservation of area in a two-dimensional flow. Therefore the Poincaré map, constructed by samplingtrajectoriesof flow elements, also conserves area.

b. Orientation preservation. A consequence of area conservation is that fluid elements preservetheir orientation under the mapping. Geometrically, this means that a series of


points in a line are always located in the same order along the line; this implies that the interior of a closed contour is mapped to the interior of the mapping of the contour.

c. Variation of the section The shape of structures appearing in the Poincaré section depends on the chosen times at which particles trajectories are sampled. However, the topology is the ~ame in different Poincaré sections.

Islands of stability

For f = 0 in (2.29) the stream function W is time independent and a point and its mapping !ie on the same streamline. Let us assume that the streamline patterns of the steady flow have the form illustrated in figure 2.10a, which shows the case of a superposition of the flow due to a point vortex in the origin and a uniform shear flow in the y direction, used here only for the sake of illustration. There exist two fixed points P+ and P- of hyperbolic type, and there is additionally an elliptic fixed point in the origin. The collection of orbits forming a line that approaches P- as t -+ +oo, is called the stabie manifold; and the collection of orbits that emanates from P+ (i.e. approaches P+ ast-+ -oo), is called the unstable manifold. In the unperturbed case the unstable manifold of P+ and the stabie manifold of p_ coincide. This line is usually called separatrix because it di vides the flow field into qualitatively different regions: in the central region of figure 2.10a particles rotate periodically around the origin, whereas outside they have unbounded trajectories. A partiele can not leave the streamline along which it moves; for that reason a streamline is called an invariant curve of the steady flow.

Note that associated with every streamline in the central region there is a period T. which is the time needed for a partiele starting on the strearnline to make one complete circuit along the streamline. There are two types of streamlines depending on whether the number v = T,/TP (called the rotation number in the theory of dynamica! systems) is rational or irrational. Streamlines which have an irrational number vare covered densely by the sequence of positions of a particle; withno point being ever visited twice. On streamlines with a rational number v = i/ j (i and j being integers), the situation is completely different : a partiele located arbitrarily on the streamline comes back to its initia! position after i iterations, and the process repeats itself periodically. Obviously, each initiallocation in the same streamline leads to the same behaviour. There is thus an infinite number of i-cycles on this streamline. Under the perturbation, usually only two of these cycles survive, one is of hyperbolic type and the second one of elliptic type (figure 2.10b). Note that v varies continuously from zero at the origin to infinity at the separatrix. Hence, in the perturbed system there must exist an infinity of concentric chains of islands, present everywhere. This being also true for the perturbed flow fields stuclied in this thesis, we can ask ourselves why Poincaré sections like the ones shown in chapters 4, 6 and 7 do not display such chains of islands. The reason is that the islands rapidly decrease in size with i.

The behaviour, in the perturbed case, of streamlines with irrational rotation number v is predicted by the KAM theorem, suggested by Kolmogorov and proved independently by Arnol'd and Moser (see, e.g., Hénon 1983). This theorem establishes that an invariant curve persist under the perturbation if v is "sufficiently far from all rational numbers". Note that the theorem establishes only sufficient conditions for the persistence, i. e. there could exist streamlines that persist and simply do not satisfy those conditions. The converse, then, is necessary for destruction: a streamline that is destroyed under the perturbation must have a v which is "suffi.ciently close to a rational number" .

2.4 Advective transport in two-dimensional flows 27

Chaotic seas

For f =f:. 0 but sufficiently small the hyperbolic fixed points persist. If we follow the unstable manifold of P+ and the stabie manifold of P-, we find that they do not join smoothly as before, but interseet at an angle (see figure 2.10c), the intersection point is called a heteroclinic point (because the manifolds correspond to different fixed points). The existence of this point has far reaching consequences. All images of this point belong to the stabie manifold of P-, therefore they approach the fixed point P-. But they also belong to the unstable manifold of P+• therefore this manifold must oscillate to pass through all these points. Now the mapping is area preserving; therefore all successive lobes formed by the unstable manifold on one side of the stabie manifold must have the same area. But the base of these lobes tends to zero as the intersectiöns approach P-, so that their length must increase considerably. They become very thin and are elongated at an exponential rate in the direction of the unstable manifold of P-. The lobes follow the unstable manifold of P- in a parallel way and begin to oscillate as they approach the fixed point P+· The structure that results from the intersection of the manifolds of the two hyperbolic points is called a heteroclinic tangle (figure 2.10c).

In the previous section it was mentioned that for every rational value of v, there exists a chain of elliptic and hyperbolic points. The elliptic points give rise to islands and between each pair of hyperbolic points a new heteroclinic tangle is generated. The outer chaotic sea surrounding apparently a small number of islands which is seen in many systems, including the ones discussed in the following chapters, is not fundamentally different from other chaotic regionsin the hierarchy (Hénon 1983). Only it is larger, therefore we will confine our attention to these large islands and chaotic seas.

The intersecting manifolds expose the mechanism for transport of fluid between different flow regions in the following way. Note that the area enclosed by the segment AC along the unstable manifold of P+ and CA along the stabie manifold of P- maps to the area A'C' along the unstable manifold of P+ and C'A' along the stabie manifold of P- (figure 2.10d). If the trapped fluid is redefinedas the area enclosed by P+C on the unstable manifold of P+, Cp_ on the stabie manifold of P-, P+ c on the stabie manifold of P+, and cp_ on the unstable manifold of P- (figure 2.10d); then the area AC-CA represents the fluid that will be entrained in the next cycle, whereas the dotted area near A represents the fluid that wil! be detrained to the ambient fluid (indicated by the dotted area near A'). Since the flow is incompressible, the area entrained is equal to the area detrained in every cycle.

The unstable manifolds are constructed numerically in the following way. A line with length 8d, where d is a charaderistic length (e.g., the distance between the point vortices in chapters 4 and 6) and {j « 1, is located on the fixed point and its evolution in the perturbed velocity field is computed forward in time. The line will be stretched in the direction of the unstable manifold. The stabie manifold is constructed in a similar way, but the integration is now backwarcis in time. The exchange of mass can be evaluated directly from the discrete set of points defining the manifolds. Once a single lobe is identified the area follows from p. = f xdy along AC-CA. This method is valid for every amplitude of perturbation f.


2.4.2 Melnikov theory

Without explicitlysolving the adveetion equation (2.28), it is possible to predict the behaviour of the stabie and the unstable manifolds using the Melnikov function. This function is, up toa known normalization factor, the tirst-order term in the Taylor expansion about ~; = 0 of the distance between the stable and the unstable manifolds. The Melnikov function M(to) is defined as

M(to) = i: {h[i,.(t)]g2[i,.(t), t + t0]- h[i,.(t)]g1[i,.(t), t + t0]}dt, (2.30)

where i ... (t) = [x,.(t),y,.(t)] represents a partiele trajectory along the separatrix of the unperturbed flow, (ft, f 2) is the unperturbed velocity field, and (gt,g2 ) is the perturbation. These veloeities are given by

(a1f;,Jax, -a1f;,.fay), (a1f;pfax, -a1f;pfay).

Some properties of the Melnikov function are (see, e.g., Rom-Kedar et al. 1990):

(2.31)

(2.32)

a. A simple zero of the Melnikov function implies a transverse intersection of the stabie and unstable manifolds. If the Melnikov function has no zeros, there are no intersections of the manifolds (Melnikov theorem).

b. The Melnikov function is a signed distance measurement, it therefore contains information about the relative position of the manifolds.

c. The Melnikov function is a periodic function of t 0 having the same period as the perturbation flow.

Properties (i) and (iii) imply that if the Melnikov function has a simple zero, then the manifolds interseet an infinite number of times. Hence, a heteroclinic tangle is formed, where features as exponential stretching of fluid elements and horseshoe maps arise (Wiggins 1992). The Melnikov function provides thus a criterion for the existence of chaotic partiele trajectories in the perturbed flow.

One can also obtain an 0( e) approximation for the area of a lobe by using the Melnikov function (Rom-Kedar et al. 1990). The area of a lobe is given by

(2.33)

where to1 and t02 are two adjacentzerosof the Melnikov function M(t0 ) (i.e., they correspond to adjacent intersections of the unstable and stabie manifolds ).

The Melnikov function (2.30) and the lobe area (2.33) were computed numerically in the following way. (i) The trajectory along the unperturbed separatrix is computed by integration of (2.28), with ~; = 0, using a fourth-order Runge-Kutta scheme, (ii) the velocity fields J;(t) and g;(t) are calculated by simple substitution of iu(t) in (2.31)-(2.32); and (iii) the integrals were evaluated using the trapezoidal rule. The infinite limits of the integral in (2.30) were replaced by large but finite limits. By large we mean here that the use of an even larger value leaves the integral unaffected.

Chapter 3

Behaviour of a dipolar vortex on a ,8-plane

3.1 Introduetion

The two-dimensional character of large-scale motions in the oceans and in the atmosphere is due to both rotation, which makes the fluid move in (locally) horizontal planes, and geometry, since the ratio between horizontal and vertical scales is of order 0(103

). The dynamica of geophysical flows is further essentially influenced by the gradient of the Coriolis parameter in the latitudinal direction, usually referred to as the ;9-effect. The simplest model that includes these effects is the so-called equivalent barotropic vorticity equation, which expresses the conservation of potential vorticity in a homogeneous, incompressible two-dimensional flow on a rotating sphere (see Pedlosky 1979). In the context of plasma physics, this equation is known as Hasegawa-Mima equation, where a density gradient plays the same role as the gradient of the Coriolis acceleration (see, e.g., Makino et al. 1981). In this thesis the equation is used in the limit of infinite radius of deformation, but the results presented bere have relevanee to both geophysical flows and plasmas.

The equivalent barotropic equation bas a number of dipolar solutions, which arealso known as modons. The dipole, consisting of two closely packed counter-rotating vortices, has two remarka.ble properties: it possesses a separatrix and it has a non-zero linear momentum. This means that the dipole provides an efficient mechanism for transport of mass and momenturn over large distances compared to the dipole's radius. One partienlar charaderistic of modons is that they are either stationary or they translate perpendicularly to the gradient of ambient vorticity. An important question thus concerns the existence of similar structures that propagate transversally to those isolines.

Although a modon is not an exact solution when its symmetry line is not parallel to isolines of ambient vorticity, Makino et al. (1981) used "tilted"modons as initial conditions for numerical simulations. They found that the modons survived as coherent structures, moving along meandering trajectories in eastward direction or along cycloid-like paths in westward direction, depending on the tilting angle. Using a perturbation technique, Nycander & lsichenko (1990) also found these two regimes as well as the mode in between, in which the dipole perfarms a periadie looping excursion without any net displacement. They also calculated the decay rate based on vorticity generation in the wake of the dipole. The decay

This cha.pter wa.s origina.lly publisbed a.s a. journa.l artiele (Vela.sco Fuentes & va.n Heijst 1994).

29

30 Behaviour of a dipolar vortex on a (3-plane

was found to he negligible on the time scale of an oscillation of the dipole's motion.

Another approach to the problem has been the use of a. couple of point vortices of equal but opposite strengths and a.llowing varia.tions in rela.tive circula.tion to preserve potential vorticity during the evolution. Kono & Yama.ga.ta. (1977) first found the three "regimes" in the motion of the couples: (i) ea.stward meandering, (ii) westward cycloid-like trajectories, and (iii) the "non-propa.ga.ting" mode, in which the couple moves along an 8-shaped curve fixed in space. Za.busky & McWilliams (1982) presented calculations of a point-vortex couple and a dipole represented by two pairs of point-vortices. These point-vortex calculations showed good agreement with numerical simulations of a modon solution, at least for the first oscilla.tion. More recently, Kono & Horton (1991) presented an exact salution for the motion of the pointvortex dipole a.s well a.s numerical results of interactions of point-vortex couples. Hobson (1991) stuclied the problem in phase spa.ce terms and found that the dipole moving to the east has a. sta.ble trajectory, while that of a westward-travelling dipale is unstable. Ma.kino et al. (1981) found tl:ie same beha.viour in their numerical simulations of modons. However, they did not use the terms stabie or unstable trajectories, a.s pointed out by Nycander (1992), who refers to it as "tilt" instability.

A gradient of the a.mbient vorticity can also he caused by variations of the depth of a rotating fluid. When the variations are linear and small, the gradient of ambient vorticity is approxima.tely the same a.s that produced by the variation of the Coriolis parameter; this effect is sametimes referred to as "topographic /3-effect". Carnevale et al. (1988) numerically stuclied the motion of dipales over hills and ridges and found that under strong topographies the dipole hroke apart, the cyclonic half moving up-hili and the anticyclonic one down-hill. The evolution of the independent halves after the splitting corresponds with the general hehaviour of barotropic monopolar vortices on a {3-plane, namely: cyclones move to the (local) northwest and anticyclones move to the (loca.l) southwest. The problem of a single monopole on a /3-plane ha.s receivedconsiderable attention inthelast decades, both in theoretica.l studies (Adem 1956), in numerical and laboratory experiments (Carneva.le, Kloosterziel & van Heijst 1991), and in observational studies (see Robinson 1983).

Experimenta.l studies on dipolar vortices have concentrated ma.inly on situations without a gradient of ambient vorticity (/3-effect absent). Nguyen Duc & Sommeria (1988) stuclied vortex couples in a. thin layer of mercury, the flow being constrained to he two-dimensional by an externa.l magnetic field. They obtained steadily translating dipoles; the symmetrie dipoles (with zero net circula.tion) were seen totranslate along straight paths, whereas the trajectories of asymmetrie dipoles (with non-zero circulation) were found to he circular. Nguyen Duc & Sommeria also determined the relationship between the vorticity w and the stream function t/J of the lahoratory dipoles, and they found both linear and non-linear relations. Flór & van Heijst ( 1994) compared the Lamh model, which assumes a linear w-t/J relation in the interlor of the dipole, with their experimenta.l ohservations of dipales in a stratified fluid. The experimental results (vorticity distrihution, dipoles size and translation speed) show very good agreement with the Lamb dipole, both for dipales with a linear w-t/J relation and for dipales with a non-linear relation. In their rotating-fluid experiments, Flierl et al. (1983) stuclied dipolar vortices that contained non-zero circulation, and a.ccordingly travelled along a circular path. By taking into account the net circulation they obtained a modified Lamb dipole solution, which was found to agree well with their experimenta.l results.

The results to he discussed in this chapter concern the hehaviour of dipolar vortices when a weak gradient of amhient vorticity is present in the system. The following section reviews

3.2 The modulated point-vortex model 31

the main results of the modulated point-vortex dipoles. Experimental observations of dipole trajectories for different initia! directions, as well as flow measurements during one oscillation of the dipole, are presented in section 3.3. A comparison between west-travelling dipoles (WTD's) and east-travelling dipoles (ETD's) is made in section 3.4, on the basis of both experimental observations and numerical results obtained with a vortex-in-cell method. The behaviour of an ETD for different values of (3 is discussed insection 3.5. Finally, insection 3.6 we summarize the results and give some conclusions.

3.2 The modulated point-vortex model

For the sake of completeness, we first review the motion of two point vortices with arbitrary (initia!) circulations. The vortices' circulations are modulated on the basis of conservation of potential vorticity on a (3-plane. The case of a finite deCormation radius can he found in the refere_nces mentioned in the previous section.

In absence of background vorticity the evolution of a group of N point vortices is governed by a system of 2N ordinary differential equations (see, e.g., Batchelor 1967)

dxi 1 N Yi- Yi di= -271" L"r-~-,

J=l l]

dyi 1 N Xi- Xj di = 27r L.: "i~·

J=l l]

(3.1)

#i #i

where (xi, Yi) is the position of point-vortex i with circulation "i' and Tij is the distance between vort i ces i and j.

Large-scale motions on a rotating sphere ( the Earth) are essentially affected by the latitudinal variation of the Coriolis parameter J, which is defined as J = 2!1 sin 4>, with n the Earth's angular speed and 4> the geographic latitude. For motions occurring on scales smaller than a few degrees of latitude it is allowed to approximate the Coriolis parameter as a constant (local) value plus a linear variation in the meridional direction, i.e. J = Jo+ (3y, where Jo = 2!1 sin 4>0 and (3 == 2!1 cos 4>ol R, with R the Earth's radius. This approximation is known

· as the (3-plane model. Conservation of potential vorticity implies that the relative vorticity w of a vortex tube

moving in meridional direction changes as expressed by the following equation:

D Dt(w + f3y) = 0, (3.2)

where DI Dt = a I ot + uo I ox + vo I oy is the material derivative, u and V the veloeities in east (x) and north (y) direction, respectively.

Although a potential point vortex is not a solution of (3.2) for the case (3 =f. 0, some insight into the {3-plane dynamics can he gained by solving (3.1) with the circulations being modulated according to the principle of conservation of potential vorticity. For that purpose it is necessary to assign a certain area to the "point" vortex: under the assumption that a point vortex represents a small pat eh of vorticity, the circulation equals the (uniform) vortkity w multiplied by the area of the patch. If 1r L2 is the area associated to the point vortex, its circulation is then given by " = w0 1r U. Substitution of this expres si on in (3.2) yields

l3.3)


y

x

Figure 3.1: Schematic representation of the point-vortex model of a dipole.

where conservation of mass (area) has been used. Here y;0 represents the initia! latitude, at which the vortex has strength K.;0.

For a system of just two point vortices the distance d between them is a constant of motion. This fact makes it possible to describe the evolution of the pair with the position of the middle point and the direction of motion (figure 3.1). Substituting the new variables

XI + X2 t = YI + Y2, ( x 2 - XI) 'YJ = - -2-, ."

2 a= arctan ---- ,

Y2- Y1

in (3.1), one obtains

drt K.J - 1\.2 dt ~cos a,

dÇ /\.} - 1\.2 dt ~sina,

da "I + 1\.2 àt 27rd2 '

where the third expression is a general equation for the angular velocity of a pair of point vortices. Let K.o1 = K.o and K.02 = aK.0, so that the strength of each vortex is given by

K.I = K.o+1rL2(3(y10 - YI) ,

K.2 = aK.o + 7r L 2 f3(Y2o- Y2),

(3.4)

(3.5)

and one obtains the following system of ordinary differential equations governing the motion of the pair:

drt U cos a, (3.6) = dt

dÇ U sin a, (3 .7) = dt

da K.o (3L2 (3.8) = (l+a)27rd2 ---;p:-Ç, dt

3.2 The modulated point-vortex model 33

where Ko (1- a /31rL

2d ) U=- ------(cos a- cosa0 ) •

21rd 2 2Ko

Note that TJ does not appear on the right hand side of these equations. The evolution is thus governed by a system of two first-order ordinary differential equations. Therefore, the motion of two point vortices on the /3-plane is integrable for any values of the vortices' circulations (see, e.g., Hobson 1991). Two particular cases are analyzed below.

Vortex pair (K2o = KID}

Let us consider first the motion of two equal point vortices. Substitution of a = 1 in (3 .6)-(3.8) yields

dTJ (3L2 (3.9)

dt -4 cos a( cosa-cos a0 ),

df, (3L2 (3.10)

dt -4sina(cosa- cosa0 ),

da Ko /3L2 (3.11)

dt 7rd2- -;pE,·

Fora relatively weak /3-effect with respect to the vortex ei reulation (f3dL2 I Ko ~ 1) the angular velocity of the pair can be approximated by n ~ no = Kol7rd2. Then a= ao+f!t . From (3.10) it follows that there is an oscillation in meridional direction but nonet displacement in a period T = 21r Ir! . From (3.9) it follows that the pair moves with a mean velocity [! = - f3L 2 18, i.e. the pair moves westward both for positive and for negative vort i ces. Th is result resembles the continuous case, where both cyclonic and anticyclonic monopoles have a net westward drift on the /3-plane.

Vortex couple (K2o = -KID}

For the case of an initially symmetrie dipole a = -1 , (3.6)-(3.8) reduce to the following system of ordinary differential equations governing the motion of the pair:

dTJ = U cos a, ( 3.12) dt df,

U sin a, (3.13) = dt da (3L2

(3.14) = --;pf,, dt

where Ko ( /31r L

2d ) U=- 1- --(cos a- cosa0 ) .

21rd 2Ko

The weak- /3 assumption (f3dL2 I Ko ~ 1) is used again to obtain an approximation for the translation velocity of the dipole U~ U0 = ~~, 0 121rd. Here (3.13) and (3.14) can be combined to yield a single equation for the direction of propagation a

d2a L2

d(ï + d2 /3Uo sin a = 0. (3.15)

34 Behaviour of a dipolar vortex on a {3-plane

(a) N (b) N (c) N

s s s

Figure 3.2: Trajeetory modes of a. modula.ted point-vortex dipole on a. ,6-pla.ne: (a.) the mea.ndering mode, with eastwa.rd drift; (b) the "non-propa.ga.ting" mode, with the dipole moving a.long a.n 8-sha.ped páth fixed in spa.ce, a.nd ( c) the cycloid mode, with westwa.rd drift. The dipole was initia.lly symmetrie a.nd moved in the direction indica.ted by the a.rrow. The thick line represents the pa.th of the positive vortex, the thln line the pa.th of the nega.tive vortex. The upper a.nd lower pa.rts of the plots correspond wjth North a.nd South, respectively.

The solution should satisfy the initial conditions o:(O) = o:0 , which is the tilting angle, and o:'(O) = 0, since the couple is initially symmetrie. This is the non-linear simple pendulum equation. A few results can be drawn immediately from its linearized version: for small values of o:0 the oscillations have a. constant frequency (L/d)(f3U0 )

112 a.nd the wavelength ,\ a.nd the amplitude A of the oscillations are given by

,\ = ~ ( 1 - :5) 2~~' d{Uo

A == zo:oy 7f ·

(3 .16)

(3.17)

Proceeding with the analogy between the vortex pa.ir a.nd the pendulum, an ETD (o:0 == 0) dearly has a stabie trajectory: when a small perturbation is imposed, small oscillations around the equilibrium latitude a.rise. On the other hand, a. WTD (o:0 == ~) is unstable in the trajectory sense (see also Hobson 1991). This instability is also present in continuous modons and has important consequences for the stability of the structure itself (Makino et al. 1981, Nycander 1992). The effect on laboratory modons will be discussed insection 3.3.

Figure 3.2 shows numerical integrations of equations (3.1) and (3.3) with the corresponding regimes: (a) wave-like trajectory, (b) "8" -sha.ped trajectory and ( c) cycloid-like trajectory. These regimes were originally described by Kono & Yamagata (1977), although their paper seems to have escaped the attention of most of the investigators working in this field (most likely because the pa.per is written in Japa.nese).

For initially asymmetrie couples (a < 0, but a =f. -1) an approximate equation with the same form a.s (3.15) can also be obtained. But in this case U :::::: U0 = ~(1 - a)Ko/2~d and the initia! condition for the angula.r velocity changes to o:'(O) = f!0 = (1 + a)Ko/2~~, due to the asymmetry of the initial dipole. The same oscillatory behaviour arises for f!Ö < (U /~)f3U( cos o:0 + 1). However, in this case the equilibrium latitude is not the latitude of the initial position. The dipole meanders a.round a line loca.ted to the north of the initial position

3.2 Tbe modulated point-vortex model 35

(a) N (b) N

s s

(c) N (d) N

s s

Figure 3.3: Trajectories of initially non-symmetrie point-vortex dipoles on a ,8-plane. (a) Slightly negative net circulation causes meandering around a line southward of the equilibrium latitude, (b) slightly. positive net ei reulation causes meandering around a line northward of the equilibrium latitude, ( c) st rong negative circulation causes anticyclonic librations around a point that drifts to the west, and ( d) st rong positive net circulation produces cyclonic librations and westward drift.

if the positive vortex is initially stronger (figure 3.3b ). Likewise, if the negative vortex is stronger the equilibrium latitude is located to the south of the initia! position (figure 3.3a). If D~ >-(L2 /d?)f3U(coso.0 + 1) the dipole remains always asymmetrie and rotates around a point that moves westward (figures 3.3c,d).

It is appropriate to make some comments on the two length scales d and L present in the problem, being the distance between the two point-vortices and the radius of the 11-rea associated with each 'point' vortex, respectively. It is stressed that while being arbitrary, L must be much smaller than d, otherwise the discrete representation of the vortex patch is not accurate. Also, L must be finite, otherwise the modulation mechanism does not work.

The meandering motion obtained with this simple model does not depend on the special condition of having an initially symmetrie dipole. However, several effects are neglected in the model: (i) the deformation of the finite area represented by the point vortices, and most importantly, (ii) the generation of relative vorticity due to adveetion of ambient fluid . Naturally, the question arises of whether dipolar vortices in real flows would show phenomena similar to those described above. In order to provide an answer to that question, we have carried out a study of dipolar vortices in the well controlled Iaberatory environment.


Figure 3.4 : Sequence of plan-view photographs showing the meandering of a dipolar vortex on a (topographic) ,6-plane. The dipole was created as explained in chapter 2 and it travelled initially towards shallower fluid (top of the pictures) . The images were taken at times t = 0.45T (a), 2.25T (b), 4.05T (c), 5.4T (d), 9.45T (e), and 15.31T (f) after withdrawing the cylinder, with T = 11 .1 s the rotation period of the turntable. Experimental parameters: f = 1.13 s-1, h0 = 0.18 m, s = 0.22 m-1 , .6 = 0.25 m- 1s-1 .

3.3 A meandering dipale 37

3.3 A meandering dipole

3.3.1 Qualitative observations

The basic mechanism that determines the trajectory of a dipole on a ,8-plane has already been explained by Makino et al. (1981). In order to clarify some of the laboratory observations to be discussed later, the argument will be repeated here in termsof fluid-depth variations.

The dipolar vortex was generated by moving the cylinder along a straight line in "northern" direction, i.e. into shallower partsof the tank (figure 3.4a). Once the dipole has reached a mature state (figure 3.4b), it tends to move in a straight line to the northeast, where the fluid is shallower. Owing to the background rotation the two dipole halves experience asymmetrie effects, as can be understood from conservation of potential vorticity (2.4) the cyclonic vortex (w > 0) becomes weaker while the anticyclonic one (w < 0) becomes stronger, resulting in a trajectory curved in clockwise sense. The vertical squeezing also causes the dipole to widen, so that the vorticity extremes move slightly apart and the translation velocity decreases. The dipole reaches a maximum size and a maximum circulation asymmetry at its narthemmost position (figure 3.4c). The clockwise rotation brings the dipole southwards, and the asymmetry gradually diminishes (figure 3.4d). The dipole reeoversits zero net circulation when it crosses the equilibrium line, where the height of the fluid column is equal to its initia! value. Because of its own momenturn the dipole moves further into deeper regions ("south"), so that the column of fluid is stretched further. Owing to this stretching, the cyclonic half (w > 0) becomes stronger while the strength of the anticyclonic part of the dipole decreases in magnitude. As aresult the dipole's trajectory curves in anti-clockwise sense (figure 3.4e). The stretchingalso causes the vorticity extremes to get closer, and as a consequence the translation speed of the dipole increases at this stage. After reaching its southernmost position, the dipole moves back to its equilibrium depth (figure 3.4f). Experimentally, we have not been able to observe more than one cycle of the oscillatory path, because the dipole either reached the wall after one meandering or it was observed to split into two monopolar vortices. The latter feature will be discussed in more detail in section 3.5.

3.3.2 Flow measurements

Results equivalent to those described in the previous section are obtained when the dipole moves southward initially, as can be seen from the flow evolution (figures 3.5-3.8). When it reaches a mature state at timet= 1.77T, with T the rotation period of the table (T=11.3 s), the dipole moves in an approximate southeast direction. The cyclonic half is stronger, resulting in an anti-clockwise defiection of the di pole's trajectory. The density of the vorticity contours in figure 3.5a shows a clear asymmetry between the two dipole halves. The streamline pattem 1/J =const (see figure 3.5b) shows the same asymmetry; the centre of the rotational dipole motion at this stage is also visible in the upper right corner of the figure.

At t = 4.42T the dipole reaches its southernmost position; therefore a maximum asymmetry is observed in the contours of vorticity (figure 3.6a) and a decrease of size with respect to the previous stage (cf. figure 3.5a). The sameeffectscan be observed in the streamline pattem (figure 3.6b). Also, the curvature of the dipole trajectory has increased at this stage, as can be observed from the centre of rotation, which has moved closer to the dipole.

38

m[s-1} ~ ..

0.5

0 --0.5 . · ...

-1 0 10 20 30

-x (an}

(a)

10an f------i

(c)

. .....

40

Behaviour of a dipolar vortex on a /3-plane

m -[s-1] (d)

0.5

0

-0.5

-1 0 10 20 30 40

"' (an2s-1]

Figure 3.5: Measured flow characteristics of a dipolar vortex on a ,8-plane at t= L 77T, with T= 1 L3 s the rotation period of the tur.ntable. The dipole moved initially to the south and subsequently described a meandering path. The graphs represent (a) vorticity contours (contour interval CI=O.l s-1), (b) stream function contours (CI=2 cm2s-1 ) corrected for the observed motion of the dipole; ( c) distri bution of vorticity along a line crossing the points of extrema! vorticity val u es and ( d) w-1/J relatiort, obtained by plotting the vorticity value against the stream function value of every grid point in the interior of the dipole. Experimental parameters: f = Lll s-1

, ho = 0.17 m, s = 0.23 m-I, ,8 = 0.26 m-1s-1 .

As the dipole returns to its equilibrium depth (at t = 7.96T) the asymmetry is reduced, as can he observed from the v~rticity distribution (figure 3.7a). The dipole size increases and the centre of rotation moves farther away (figure 3.7b). However, the dipole has still positive circulation and its path is curved in anti-clockwise sense.

Finally, at t = 15.93T the dipole reaches its northernmost position, as shown in figure 3.8. The clockwise rotation and the vorticity contours reveal that the dipole has acquired a negative net circulation (figure 3.8a). The centre of rotation is now located to the south of the couple

3.3 A meanderïng dipale

1

m[s.1]

0.5

0

-0.5

-1

-... _.

(a)

0

0 """ 10an f------

. (c) ..

. ··-

0

. . ...

10 20 30

X [an]

40

m [s.1]

0.5

0

-0.5

Figure 3.6: As figure 3.5, but now fort= 4.42T.

39

(d)

(figure 3.8b ). The low density of contour lines in both the-vorticity and the stream function plot indicates the decay of the dipole, which is most likely produced by two mechanisms associated with the topographic ,6-effect. First, the generation of relative vorticity in the wake of the dipole, and second, the entrainment and detrainment of ambient fluid (or "breathing", Nycander & Isichenko 1990). The observed evolution of similar dipoles on the f-plane shows that other decay mechanisms like horizontal ditfusion or spin-down induced by the Ekman layersare negligible on this time scale (approximately 15 rotation periods).

The evolution of the total relative circulation within the dipole illustrates a few important characteristics. The circulation is obtained by a (discrete) surface integral of the vorticity values. The ratio E = - r+ ;r-' with r+ the positive circulation and r- the negative circulaÜon in the dipole, changes as the position of the dipole moves in meridional direction. At t = 1. 77T (figure 3.5) the ratio t measures 1.23, showing that there is an excessof positive vorticity and therefore an anticlockwise rotation. In the southernmost position (figure 3.6) the circulation reaches its most positive value and t =1.488. Afterwards the asymmetry starts to decrease;

40 Behaviour of a dipoJar vortex on a (3-plane

(a)

1

m [s"1) (c) m [s-1] (d)

·"· I 0.5 0.5

. 0 •. 0 .... -

-0.5 . ... -0.5

-1 -1 0 10 20 30 40 0 10 20 30

X [an] 'V [an2s.,]

Figure 3.7: As tigure 3.5, but now fort= 7.96T.

at t=7.96T (figure 3.7) € =1.16, thus the dipole still has positive net circulation. Finally, when the anticyclonic part becomes stronger (figure 3.8), the ratio € =0. 75. Naturally, the qualitative evolution of the ratio agrees with the observed trajectories. A comparison with the modulated point-vortex model is made using (3.4)-(3.5) and assuming that the dipole was initially symmetrie and moved in southward direction (as the generating cylinder). The initia! circulation of the point vortex is Ko = (r+- r-)/2 as computed at t = 1.77T and the gradient of ambient vorticity is the experimental value (3=0.26 m- 1s-1

• A free parameter used to "tune" the model is L, the radius of the area associated with the point vortices. The values obtained for €, being € = 1.2, 1.63, 1.14 and 0.74, respectively, campare very well with the corresponding values measured in the experiment (see above) . The value of L used to obtain this approximation is 9.44 cm, which is of the sameorder as the distance d between the points of maximal vorticity. Clearly, the condition L « d, which permits a discrete representation of the vorticity in the point-vortex model, is not satisfied. A smaller L (3 cm) generates val u es of the ratio € (1.02, 1.05, 1.01 and 0.97, respectively) closer to one than the values measured

3.3 A meandering dipole 41

1

I. m[s-1) (c) 0.4 (d)

m [s-1) 0.5 r.: 0.2 .......... ;: . . .· · ..

0 . . 0 !{ . . . . · ..... .)"

-0.2

~; -0.5

-0.4

-1 0 10 20 30 40 -20 -10 0 10

x [an) 'V [an2s·1]

Figure 3.8: As figure 3.5, but now fort= 15.93T.

in the experiment. However, the qualitative evolution of € is the same in the experiments and in the point-vortex model.

A remarkable feature in the vorticity distribution is the preserree of a small ring of oppositely-signed vorticity around the dipole. This ring is visible in the form of smal! humps in the vorticity cross sections, specially during the initia! stages (figures 3.5c and 3.6c). The shielding ring is caused by the adveetion of ambient fluid in meridional direction, and its influence on the dynamics of the dipole will be discussed in section 3.5.

A stationary solution of (3.2) satisfies q = F( 'Ij;) where q is the potential vorticity q = w + {3y, 'Ij; is the stream function in the system moving with the dipole and F is an arbitrary integrable function. Strictly, in the case of an oscillating dipole a stationary state is never established : the total circulation oscillates between positive and negative values and the translation velocity changes continuously in direction and in magnitude. In spite of this, values of vorticity and stream function were computed in a frame in which the dipole is "stationary", by correcting for the instantaneous displacement of the structure.


By plotting the vorticity against the stream function for every node of the interpolation grid, the resultant graph provides information about how close the dipole is to a stationary solution: 1f the points collapse onto a single line a well-defined functional relationship exists between vorticity and stream function. The scatter around the line can he due to experimental errors but more likely because the structure is not stationary. Furthermore, the form of the curve provides information about the nature of the solution.

The scatter plots presented hei:e are constructed only with points in the interior of the vortex structure; points from the exterior field all collapse onto the horizontal axis (w =0). Formally, one should plot the (corrected) potential vorticity q = w + {3y , rather than the corrected relative vorticity w , as a function of '1/J. In all experimental results reported here, however, the term {3d, with d the radius of the dipole, is at least ten times smaller than w ({3 = 0.3, d = 0.1, 0.2 < w < 1), so that w is a good approximation of q. The difference between scatter-plots w( '1/J) and q( '1/J) (not shown here) is in fact hardly noticeable.

A linear relation between w and '1/J was observed for the dipole at times t=l.77T and t=4.42T (figures 3.5d and 3.6d, respectively). The change in the slope is a consequence of the stretching of the dipole and can he, explained as follows. Most analytica! dipolar solutions of the equivalent barotropic equation (as well as the Lamb dipole) are based on an assumed linear relation between vorticity and stream function, i.e., w = k2 '1/J (see Flierl 1987). Under this assumption, the solutions must satisfy the dispersion relation ka = 3.83, where a is the radius of the dipole and 3.83 is the first zero of the first-order Bessel function of the first kind, which is one of the building blocks of the modon solutions.

For the laboratory dipoles the radius a is determined by using the closed streamline of the corrected stream fundion plot and then k is calculated using the dispersion relation. This result should agree with the value of k obtained from the scatter plot (the slopeis the square of k) . Experimental dipoles in absence of a sloping bottorn (J-plane) have been observed to satisfy these criteria. For the experiment being discussed here, it is observed that at t=l.77 the radius is a=11.04 cm, giving k=0.34. This corresponds very well with the value derived from the scatter plot (figure 3.5d), k=0.339. At t=4.42, the dipole radius a has decreased to 8.88 cm, giving k=0.43; this value is again in good agreement with the value k=0.447 ohtàined from the scatter plot (figure 3.6d). At the later stages, t=7.96 and t=l5.93, a weak nonlinearity in the w-'1/J relation is observed (figures 3.7d and 3.8d) . The fitted lines give the best approximation of the slope of the two branches. The value of k obtained from the scatter plots are approximately one half of those obtained from the observed radius of the dipole (using the dispersion relation ka=3.83). The disagreement is most likely associated with the nonlinearity of the w-'1/J relation.

3.3.3 Trajeetory as a function of the tilting angle

The squeezing and stretching mechanism described at the beginning of this section is active for every initia! orientation of the dipole axis. However, its effect is very different on dipoles moving initially at angles greater than 1r /2 (i.e., dipoles with a westward component intheir motion) and on dipoles moving at angles smaller than 7r/2 (dipoles with an eastward component).

When its initia! angle is less than 1r /2 the dipole acquires an asymmetry of the proper sign to pull it back to its equilibrium latitude. The smooth oscillation of a dipole with initia! northeast motion appears in figure 3.9a, and similar trajectories can he seen in figures 3.9b and 3.9f for dipoles released in northward and southward direction, respectively. In these three

3.3 A meandering dipole 43

(a) (b)

..... ! / I

. ··-' ~

. . . . 10an 10an

NE 1-------l N 1-------l

(c) (d)

: ~··· .. ' . . . . . . -· . . . . \

. ! . . -. . .. ,

10an 10an tiNJ 1-------l w 1-------l

(a) (I) .... /

. .. ..... . \

. . 10an 10an

sw 1-----1 s f-----1

Figure 3.9: Observed dipole trajectories for different initia! angles. The dots represent the position of the dipole centre at selected time intervals. Arrows indicate initia! and final directions of the observed dipole translation. Since the medium depth ho of the fluid in the dipole evolution varies from one experiment to another (in the range 15 to 18 cm) the ,8-effect changes accordingly in the range 0.25- 0.3 m-1s-1 .

examples the direction of the dipole changes in such a way as to make the structure return to its equilibrium latitude. Because of its own inertia the dipole overshoots and the process repeats.

Dipoles with tiltingangles greater than 1r /2 move initially away from the equilibrium latitude before finally returning to it. Figures 3.9c and 3.9e show examples of initially northwest and southwest moving dipoles. Figure 3.9d shows the trajectory instability of a dipole shot in

44

2.------------, (a)

1.5 0

0

aoooo 00

1 ;oo o • • • • .·· 0.5 . .

. NE

L~L·---~-----~---~ 0 0 10 20 30

(c)

1.5

.. / .. ···· 10

(e)

1.5

0 0

0.5

t/T

...... NW

20 1/T

.... . . . sw

t/T

30

Behaviour of a dipolar vortex on a {3-plane

2.-----------------, (b)

1.5

0.5 N

10 20 30 t/T

2.------------------. (d)

1.5

0.5 w

o~---~----~--~ 0

1.5

0.5

10

(f)

. .

oo

. .. ···· .

20 30 1/T

. . s

OOLZ•--~1~0---~2~0--~30 t/T

Figure 3.10: Sequence of measured linear displacement (solid dots) and size (open dots) as a function of time ( scaled by the rotation period T = 11.2 s ), for the di pol es shown in the conesponding frames in tigure 3.9. The dipole size is normalized with its size after reaching a mature state ( typically 3 to 4 revolution periods after withdrawal of the generating cylinder). The d.ipole displacement is normalized with the maximum displacement observed in a particular experiment.

westward direction. In the first two examples the dipoles make looping excursions in meridional direction before returning to their equilibrium latitude, while in the latter example ( cf. figure 3.9d) the dipole breaks up before being able to return to its initia! latitude.

These results confirm numerical and analytica! predictions by previous investigations (e.g., Makino et al. 1981, Hobson 1991, Nycander 1992) that an ETD corresponds toa stabie equilibrium while an WTD represents the unstable equilibrium.

3.4 Eastwa.rd versus westward travelling dipoles 45

The evolution of the velocity of the six experiments discussed above can be deduced from the linear distance travelled by the dipole as a function of time. For the individual experiments of which the observed trajectories are shown in figure 3.9, the linear displacement of the dipole between successive photographs was measured, and the results (represented by solid dots) are shown in figure 3.10. A common feature of all cases is that the velocity is approximately constant during the first stages of the evolution, but at later stages it decreases rapidly. Small perturbations can also be observed, which are likely to be produced by two effects. Firstly, the alternating squeezing and stretching of the vortex dipole moving into shallower and deeper areas, respectively, causes oscillations of the velocity magnitude with the same frequency as the meandering motion: the squeezed dipole moves slower than the stretched one. Secondly, the translation velocity of the symmetrie dipole is slightly larger than when the vorticity structure is asymmetrie; this perturbation cycle has a frequency twice that of the meandering cycle. A few attempts were made to extract these perturbation veloeities from the data sets shown in figure 3.10. Although insome cases one can easily observe the cyclic effects described above, most of the results are too noisy to be conclusive.

The alternating squeezing and stretching the dipole experiences when moving into shallower and deeper areas, respectively, causes also oscillations in the dipole 's size. Although the dipole shows a gradual overall increase in size, the oscillatory variations can be clearly recognized in the size measurements for the individual experiments of figure 3.9, which are represented by the open dots in figure 3.10. A comparison with the corresponding t rajectory plots (figure 3.9) confirms that the oscillations are the result of displacements in meridional direct ion. For example, the southward part of the di pole's trajectory in figure 3.9b corresponds with decreasing size (see figure 3.10b), whereas the dipole regains approximately its original size after returning to the initia! latitude. In figures 3.9a and 3.10a one observes a fast growth when the dipole moves northward, followed by a small decrease when it starts to move to the South.

3.4 Eastward versus westward travelling dipoles

An important result from both numerical and analytica! calculations of a tilted modon and its point-vortex counterparts is the dramatic difference between ETD's and WTD's (see, e.g., Makino et al. 1981, Nycander 1992). While dipoles of the former class are stabie and perform small oscillations when perturbed, the WTD's are unstable: even an extremely small perturbation causes them to travel over large distances in meridional direction. This difference was also observed in the laboratory. It was easy to obtain a dipole travelling to the east along an almost straight line. However, we never succeeded in getting a dipole moving to the west in a similar fashion. Very soon these dipoles deviated and drifted a long distance in northern or southern direction before breaking up or colliding with the tank walls (see, e.g., figures 3.9d and 3.9e). Note also that in (3.2) steady ETD solutions exist while WTD's always radiate Rossby waves, which were experimentally observed as very weak cells in the wake of the dipole. Pictures of streak-lines using exposure times of up to 30 s (not shown here) reveal the presence of weak cells of motion (Rossby waves) in the wake of a WTD, while no clear structure is observed in the wake of an ETD.

A few more differences were observed between eastward and westward travelling dipoles. The ETD's were always largerand slower than the WTD's, which were compact and travelled relatively fast. Figure 3.11a shows a typical example of the measured size and linear displace-


2 2

(a) (b)

1.5 1.5

oOO 0 oo 0

oooooooo 0

0ooo000oooo 0

ooooo ••••••• oo oo •••• ••• 0 oooooo .• •• • • •• • • •• • •• •

0.5 • • 0.5 • • • ETD • WTD • • • • • •

0 • 0 • 0 10 20 30 0 10 20 30

t!T t!T

Figure 3.11: Observed dipole displacement and dipole size as in figure 3.10, but now for: (a) East travelling dipole (ETD), experimental parameters are Jo= 1.12 s-1, h0 = 0.17 m, s = 0.23 m-1 , and {3 = 0.258 m-1s-1 , and (b) West travelling dipole (WTD), experimental parameters are Jo = 1.12 s-I, ho= 0.19 m, s = 0.21 m-1 , and {3 = 0.235 m-1s-1 .

ment of an ETD: an almost linear growth of the size is clearly observed while the velocity decreases monotonically in a gentie manner. In the evolution of a WTD (figure 3.llb) two stages can be distinguished: in the first stage the dipole experiences a decrease in size and has an almost constant (high) velocity, while in the subsequent stage the dipole grows and its drift velocity decreases. This last stage occurs when the dipole has moved to the South as a result of the instability of the westward trajectory. A WTD that drifts to the north shows the same decrease in size (see figures 3.9d and 3.10d) in spite of the squeezing that results from the motion towards shallow water.

The observed differences between ETD's and WTD's can be explained in termsof relative vorticity generation in the fluid exterior to the vortex dipole, due to adveetion of ambient fluid. If the dipole moves in zonal direction the sign of the generated vorticity is as follows: in front of the dipole fluid located above the symmetry line moves northward and acquires negative vorticity; on the other hand, below the symmetry line the fluid moves southward and acquires positive vorticity.

Downstreamof the dipole the effects are opposite: positive vorticity is generated above the symmetry line and negative vorticity is produced below this line. The schematic diagrams of figure 3.12 illustrate the process just described for both an ETD and a WTD. According to this mechanism the upstream side of the ETD becomes "shielded" by patches of opposite relative vorticity, whereas patches of matching vorticity can be expected to be formed in its wake. Besides decreasing the translation speed of the ETD, the secondary vorticity distribution has another important effect: the oppositely signed vorticity shield at the upstream side gives the flow -in combination with the primary vorticity distribution- a double-dipole character that is symmetrie with respect to the axis of the ETD. As indicated by the thick arrows, the upper dipole has a tendency to travel to the NW, whereas the lower dipole wil! tend to move to

3.4 Eastward versus westward travelling dipales 47

Figure 3.12: Schematic diagram showing the generation of relative vorticity by adveetion of ambient fluid. Tlfe open arrow indicates the translational motion of the dipolar vortex, the thin arrow shows the adveetion of ambient fl.uid induced by the moving dipole, and the solid arrow illustrates the tendendes of the dipole halves induced by the secondary vorticity field.

the SW. In other words: the secondary vorticity structure has a tendency (i) to decrease the translation speed, and (ii) to tear the ETD-halves apart. Fora WTD the adveetion of ambient fluid also results in bands of relative vorticity, as illustrated in figure 3.12b. In contrast to the ETD, however, the secondary vorticity upstream of the WTD is correlated with the primary vorticity distribution, whereas it is anti-correlated at the downstream side of the WTD. As indicated by the solid arrows, the secondary vorticity structure wil! most likely tend to enhance the compactness of the WTD and increase its translation speed. These are short term effects and should be present before a R.ossby wave field is established.

In order to verify this conjecture, a numerical simulation was carried out using the vortexin-cell technique described in chapter 2. In view of the continuous vorticity distributions observed in the Iabaratory experiments as discussed in section 3.3 (see also figures 3.5-3.8), the simulations were initialized by using Lamb's dipole solution. The flow domain was covered by a 33 x 33 mesh, with periodic boundary conditions at the meridional boundaries, and freeslip conditions at the latitudinal boundaries of the domain. A bout 10,000 point vortices were evenly distributed over the whole flow domain, although initially only a smal! number of them ( viz. only the ones lying inside the Lamb dipole) have non-zero relative vorticity. The others (lying outside the Lamb dipole) are so-called "ghost vortices" that become active as soon as they are advected in meridional direction, according to (3.2).

The results of the simulations are presented in figure 3.13. In the following discussion distances are given in units of the initia! diameter of the dipole and time in units of the time it takes to the Lamb dipole to travel that unit distance. The graphs on the left (frames a, band c) show the evolution of the flow field associated with the ETD. At time t=0.9 (figure 3.13b) the dipole has moved over a distance of 0.5 units to the east, and has acquired a band of anticorrelated vorticity at its sides; also, one observes the formation of tails at the downstream side, which denote the preserree of secondary vorticity that is correlated with the primary dipole. At t=l.8 (figure 3.13c) the asymmetrie structure of the secondary vorticity field is clearly visible. At this stage the ETD has only travelled over a relatively smal! distance, and shows a considerable widening (in comparison with the initia! state shown in figure 3.13a).


(a) (d)

(b)

(f)

Figure 3.13: Numerical simulations of the evolution of an ETD ( a-c) and a WTD ( d-!) using a vortexin-cell technique. The flow is initialized by taking a Lamb dipolar vortex; boundary conditions are free-slip at the latitudinal walls and periodic in meridional direction. Vorticity distributions are shown at t = 0, t = 0.9T and t = 1.8T, where T is the time required for the Lamb dipole to travel a distance equal to its diameter, in absence of the ;9-effect.

3.5 ETD's for different values of (:J 49

1.5 3

(a) (b)

oooooooooooooooo~ 2 • • • • • oo • 0 •

oooooooooooooooo •• 0.5 • • •••••• • • • •

• • • • I- plan a •• ~- plane • • • •

0 •• 0 •• 0 5 10 0 10 20 30

1/T 1/T

Figure 3.14: Observed dipale displacement and dipale si ze as in tigure 3.10, but now for (a) a dipale on an J-plane (ex perimental parameters Jo = 1.12 s-1 , ho = 0.2 m) and (b) an ETD on a moderate ,13-plane (,13 = 0.352 m-1s- 1, Jo= 1.1 s- 1 , ho= 0.17 m, s = 0.32 m-1 ).

Although the simulation was stopped at this stage, the flow behaviour thus found agrees very well with the conjectured ETD behaviour discussed above.

The graphs in the right-hand column of tigure 3.13 (frames d, e and f) show the numerical results for the evolving vorticity distribution associated with the WTD. It is easily observed that the WTD quickly acquires a very compact structure, with tails forming at its downstream side. The compactnessof the vorticity structure gives rise toa high translation speed, as can beseen by camparing the WTD graphs with the corresponding graphs of the ETD evolution. At t=0.9 (figure 3.13e) the WTD has shed two patches of oppositely-signed relative vorticity, which are the result of the adveetion of surrounding fluid, as schematically indicated in figure 3.12b. At t=1.8 (figure 3.13f), the WTD has re-entered the domain at the right (the domain is periadie in east-west direction), and has become even more compact. All these WTD features confirm the conjectured behaviour outlined before.

3.5 ETD's for different values of j3

The results in the previous section show that the value of (:J is of primary importance since the magnitude of the secondary vorticity is of order (:Jd, where d is the radius of the dipole. As (3 is increased the value of the generated vorticity gets closer to the values of the dipole's vorticity itself, thus producing a faster growth and eventually a breakup of the dipole. On the other hand, as (:J decreases the value of the vorticity generated decreases in camparisou with that of the dipole, and fora vanishing (3 this effect would not be present at all. Indeed, it was observed that on an f-plane the velocity of the dipale remained approximately constant until the dipale reached one of the walls after crossing the tank (figure 3.14a), whereas the size of the structure showed variations but remained very close to its initia! value.


3

(a) (b) 0

0

0 .. 0 .. 2 0 ... 0

. l 0 - 0 0

oOO oO

0

- 0 • • ·.:., •• 20c:m tlmi1g point • • . • ...... ~ .. · • •••

0 • 0 10 20 30

t!T

Figure 3.15: Laboratory observations of the break-up of an ETD: (a) Trajectodes of the dipole's vortex centres, the cyclonic half being represented by the upper dots and the anticyclonic half by the lower dots, and (b) the linear displacement (solid dots) and the size of the dipole (open dots) as a function of time. Experimental parameters are Jo = 1.12 s-1, ho = 0.16 m, s = 0.375 m-1 , and f3 = 0.42 m-1s-1 •

For a moderate (3 value (0.25 m-1s-1) the measured variation of size and velocity of an ETD is shown in figure 3.lla, while the same quantities are displayed in figure 3.14b for a somewhat greater value ((3 = 0.352 m-1ç 1 ). Obviously, the qualitative behaviour of the dipole structure is very much the same. In the case of a relatively st rong (3 (0.42 m -ls-1

)

the dipole is observed to split into two monopolar vortices (figure 3.15a). While gradually becoming axisymmetric, these monopolar vortices subsequently start to evolve independently, the cyclonic one (represented withsolid circles) moving to the northwest and the anticyclonic vortex ( solid squares) drifting to the southwest, in accordance with the observations reported by Carnevale et al. (1991). The evolution of the linear displacement (solid dots) and the distance between the vortex eentres (open dots) before, during and after the dipole's breakup is shown graphically in figure 3.15b. The horizontal part of the curve representing the linear displacement corresponds to the moment when the eastward motion is reversed into a westward one, and can be considered as the point of actual breakup. After the breakup both halves are observed to drift at a similar speed in westward diteetion with an additional small meridional velocity component (northward for the cyclonic vortex and southward for the anticyclonic one).

Flow measurements of a splitting ETD

In order to investigate the dipole-splitting process in more detail, an experiment was carried using astrong (3-effect ((3 = 0.52 m- 1s-1 ). Again, the initia! dipole was directed to the east . The formation of the dipolar structure was completed approximately 4-5 revolution periods after the lifting of the generating cylinder. The spatial distributions of the relative vorticity

3.5 ETD's for different values of fJ 51

(a)

20an

0.6 0.6 .. (c) (d) m[s-1] m [s-1]

0.4 0.4

0.2 0.2 ..

······. 0 .. · ... ···· 0 . -0.2

-0.2 ·. . -0.4 ·.· -0.4 ... -0.6

0 10 20 30 40 50 0 5 10 15 20

X [an] 'I' [an2s-1]

Figure 3.16: Measured flow characteristics of an ETD at an early stage of the splitting process. Time is t = 5.27T, with T = 11.4 s the rotation period of the turntable: (a) vorticity contours (Cl=0.06 s-1 ), (b) stream function contours (CI=l cm2s-1 ) corrected for the observed eastward motion of the dipole, (c) vorticity distribution along a line intersecting the points of extremal vorticity values, and ( d) the w-1/J plot obtained from the grid points in the interior of the dipole. Experimental parameters: Jo= 1.1 s-I, ho= 0.16m, s = 0.42 m- 1, /3 = 0.52 m-1s-1 .

and the (corrected) stream function of the flow at this stage (t = 5.27T) are presented in figures 3.16a,b. In comparison with the previously shown dipoles (see figures 3.5-3.8), the size of the dipolar vortex is now relatively large, and the two halves are not compactly attached as before. At this stage the structure moved slowly eastward, in a direction perpendicular to the line joining the two vorticity extremes. At a later stage ( t = 14.5T) the separation between the two halves has clearly increased (figures 3.17a,b ), and the di pole's break up was in fact completed. At this point a weak westward drift of the two halves was observed.


0.6 0.6

m[s.1) (c) m [s.1) d)

0.3 .·· . 0.3

0 .....

0 . ...... .. . . . . . ·0.3

.... ·0.3

·0.6 ·0.6 0 10 20 30 40 50 -5 0 5 10

X [an) "' [an2s-1)

Figure 3.17: Mea.sured flow characteristics of the ETD shown in figure 3.16 after the splitting ha.s been completed (at timet = 14.5T): (a) vorticity contours (CI=0.06 s-1), (b) stream function contours (CI=1 cm2s-1 ) corrected for the observed westward motion of the monopolar vortices, (c) vorticity distribution along a line intersecting the points of extrema! vorticity, and (d) the w-'lj; plot obtained from the grid points in the rectangular area indicated by the braken line in (a).

The vorticity dis tribution along the line joining the vortex eentres shows that at the earlier stage t == 5.27T (figure 3.16c) the splitting was already in progress: a srnall kink is clearly visible in the centre of the dipole, which is absent in the weak-{3 dipoles (see figures 3.5- 3.8). These observations agree with the numerical results of Couder & Ba.sdevant (1986) who found this result studying the formation of a vortex couple (in absence of {3-effects) frorn two initia! gaussian vortices of opposite sign. The corresponding sectionat timet== 14.5T (figure 3.17c) shows an alrnost complete breakup of the dipole, and the vorticity distribution resembles that of two isolated monopolar vortlees placed close together. Similar vorticity profiles in single rnonopolar vortlees were measured by Kloosterziel & van Heijst (1992) , see figures 3f and 4f of that paper.

3.6 Conclusions 53

The w-t/J plot of the dipole at the former stage (see figure 3.16d) contains considerable scatter (even after corrections), reflecting the non-steadiness of the structure, which was in fact about to undergo the splitting process. However, the points agglomerate around a non-linear curve. The w-t/J relation at t = 14.5T (figure 3.17d) shows two antisymmetrie branches which are reasonably approximated by a cubic polynomial. Comparison with the w-t/J relationship of an isolated monopolar vortex, for example as measured by van Heijst, Kloosterziel & Williams (1991) -see figure 11 of that paper-, confirms that the w-t/J plot of figure 3.17d indeed represents a combination of a cyclonic and an anticyclonic shielded monopolar vortex. The considerable scatter can be attributed to the different meridional drift components of the vortices (which can not be simultaneously corrected for), and also to the non-steadiness of the individual vortices: the shielded monopolar vortex on a ,8-plane is not quasi-stationary, and looses vorticity while drifting (see, e.g., Carnevale et al. 1991).

3.6 Conclusions

Modulated point-vortex couples with the axis tilted with respect of isolines of ambient vorticity perform oscillations around these lines (e.g., Kono & Yamagata 1977). In the linear approximation the amplitude and wavelengthof the path grow with increasing initia! velocity of the couple and decrease for increasing gradient of ambient vorticity (,8). A free parameter in the vortex model is the length scale associated with each "point" vortex. Slightly non-symmetrie dipoles show the same oscillatory behaviour. Strongly non-symmetrie couples rotate around a point that moves in west direction.

In a rotating laboratory tank the constant gradient of ambient vorticity is provided by the presence of a sloping bottom. In a series of dye experiments dipoles were initiated at different angles with respect of the isobaths. The trajectories of dipoles with an eastward component in their motion are in good (qualitative) agreement with the results obtained by the pointvortex modeland by direct numerical simulations of modons (Kono & Yamagata 1977, Makino et al. 1981). Dipoles with a westward component agree only in the perturbation enhancement mechanism but do not show the non-translating mode (8-shaped path) or the cycloid-Iike mode. Only fractions of the looping excursion can be observed. An explanation for this fact can be found in the large meridional displacements that the non-translating and cycloid-like modes imply. These displacements produce large asymmetries in the dipole leading to strong deformation of the weaker part both by the stronger partner and by the secondary vorticity field generated by adveetion of ambient fluid.

Detailed measurements of the velocity field and the vorticity distribution of the dipolar vortex show variations of the total relative circulation as the dipole moves into shallow and deep water, in qualitative agreement with the assumed conservation of potential vorticity. A functional relation is observed between vorticity and stream function, being linear at the first stages and becoming non-linearat later stages. The slope of the w-t/J line is observed to increase as the dipole moves into deep water and to decrease when the dipole climbs the topography. This effect is d ue to variations in the diameter of the vortex structure by alternating stretching and squeezing of the vortex column.

Generation of relative vorticity by adveetion of ambient fluid in meridional direction causes ETD's to grow and totranslate slower, while WTD's become compact and fast-moving during the first stages (before the development of the trajectory instability). The rate of growth of the ETD is determined by the magnitude of the secondary vorticity field, which depends on

54 Behaviour of a dipolar vortex on a /3-plane

the gradient of ambient vorticity (/3) and the radius of the structure. It is observed that a strong /3-effect leads to the breakup of the ETD. After the separation of the dipale into two manapolar vortices, each vortex drifts independently under the /3-plane dynamics, namely: the cyclonic half moves in northwestern direction and the anticyclonic vortex travels to the southwest. At an early stage the w-.,P plot of the ETD is weakly non-linear (sinh-like), whereas after the breakup the w-.,P plot has a local maximum and a local minimum, which is typical of two isolated manapoles of opposite sign located close together.

Chapter 4

Adveetion by a dipolar vortex on a ,8-plane

4.1 Introduetion

The results presented in chapter 3 verified the meandering motion of a single dipole on a (topographic) tJ-plane, a feature discovered in numerical and analytica! studies (e.g., Kono & Yamagata 1977, Makino et al. 1981). The visualization experiments also revealed the formation of a tail in the wake and spiral structures in the interior of the dipole; which indicates that detrainment and entrainment of fluid occurs as the dipole propagates. Kloosterziel, Carnevale & Philippe (1993) observed similar features both in numerical simulations and laboratory experiments. Furthermore, the flow measurements presented in section 3.3 confirm the alternating asymmetries in the stream function, which is the primary mechanism for entrainment and detrainment. A second mechanism for transport, only present on the topographic tJ-plane, was also observed in both visualizations and flow measurements. As the dipole moves uphill the distance between the two halves increases, causing entrainment of fluid, whereas when it moves downhili the dipole halves approach each other, resulting in detrainment of fluid.

In this particular case transport of !luid is dominated by adveetion so that the relative motionsof fluid pareels are all important. The study of these motions ('Lagrangian' view) can be pursued if the velocity field of the flow ('Eulerian' view) is known for all times. Using the point-vortex approach , and given a particular set of initia! conditions, the positions and strengtbs of the point vortices are known for all times, and consequently so is the stream function. The motion of individual particles, which is far more complex than the motion of the couple itself, can thus be extensively analysed using recent developments in the theory of nonlinear dynamica! systems (Rom-Kedar et al. 1990, Wiggins 1992). The stream function is further time periodic, and the problem can be reduced to the study of transport in a twodimensional map (the Poincaré map). First we identify the structures in the flow that are responsible for transport between different flow regions: the hyperbolic fixed points of the Poincaré map (stagnation points in the unperturbed flow field) and the associated invariant curves or manifolds passing through these points. The many intersections of these manifolds forma complicated structure ( heteroclinic tangle) that is responsible for the transport of fluid between different regionsof fluid. Knowledge of the dynamics of the heteroclinic tangle enables

This chapter has been submitted for publication as a joumal artiele (Velasco Fuentes, van Heijst & Cremers 1994).

55

56 Adveetion by a dipolar vortex on a {3-plane

us to calculate the amount of mass exchanged between the two dipole halvesas wellas between the dipole halves and the ambient fluid.

This "dynamical-systems theory" approach was used by Rom-Kedar et al. (1990) to study chaotic partiele motion due toa point-vortex dipole embedded in an asciilating external strainrate field. Their work should be consulted by readers interested in a more detailed discussion of the techniques used in the following sections. Hobson (1990) used the point-vortex model to study the behaviour of dipolar vortices on the (3-plane as wellas the adveetion of passive tracers. She showed the existence of transverse intersections of the unstable and stabie manifolds and therefore the chaotic character of partiele motion.

This chapter is organized as follows: in section 4.2 fluid exchange between the dipole and the ambient fluid in an experiment is illustrated and the physical mechanism for transport is discussed; in section 4.3 the adveetion of passive tracers in the velocity field of a modulated point-vortex dipole is analyzed and numerical computations of mass exchange in this model are presented; section 4.4 is devoted to a perturbation specific of the "topographic" (3-plane, namely, the variation of the distance between the dipole's halves as the vortex moves into shallower or deeper regions of the flow; in section 4.5 we make a comparison between the point-vortex model and experimental observations for a few points in the parameter space. Finally we summarize and give some conclusions in section 4.6.

4.2 The physical mechanism fortransport

It has been shown in chapter 2 that, for an inviscid homogeneaus fluid , there is a dynamica! equivalence between: (a) a layer of fluid having constant depth and a linearly varying rate of rotation (Coriolis parameter), on the one hand, and (b) a layer of fluid in uniform rotation and having a linearly varying depth, on the other hand. In both cases, the following expression for the conservation of potential vorticity can be obtained:

D Dt (w + (3y) = 0, ( 4.1)

where DfDt is the material derivative in the plane (x, y) and wis the vertical (z ) component of the relative vorticity. The form of the gradient of ambient vorticity (3 in y direction depends on the case being considered:

For case (a) (3 = (20.E/ R) cos tjJ0 , withOE and R the angular velocity and the mean radius of the Earth, respectively; and t/Jo a reference latitude (see chapter 2). In this approximation, known as the (3-plane model, all the effects of the Earth's curvature have been neglected, except in the gradient of the Coriolis parameter.

For case (b) (3 = sf0 , where Jo is the uniform Coriolis parameter, and s represents the smalllinear variation of the fluid depth, which is given by h(y) = h0 (1 - sy ).

The dynamica! equivalence between the bottorn topography and the gradient of the Coriolis parameter is used to study (3-plane dipoles in the laboratory. For the experiments reported in this chapter the angular speed of the table was 0.E=0.56 s-1 , so that the Coriolis parameter fo=20.s=1.12 s-1 . The false bottorn was raised 4 cm along one of the long sicles to provide the (topographic) (3-effect and the fluid depth at the centre of the tank was chosen to be 13 cm and 23 cm. With these parameter settings the equivalent value of (3 measured 0.194 and 0.344 m- 1s- 1 , respectively.

4.2 The physical mecbanism for transport

0 ~ UJ

(a)

(d)

(l) .

··--------

(b)

(a)

~ "\....u

57

(c)

(f)

Figure 4.1: Evolution of a dipolar vortex on the ,6-plane according to the modulated point-vortex model. The thick line is the boundary of the fluid that initially lies within the unperturbed separatrix. The broken line represents the instantaneous separatrix. Time increases from right to left and from top to bottorn and the interval between subsequent frames is approximately one eighth of the oscillation period.

Let us reeall the evolution of a point-vortex dipale on the (3-plane during one oscillation in order to clarify the mass exchange mechanism in the meandering dipole. In addition to the motion of tbe individual vertices, we fellow the evolution of the fluid carried by the unperturbed dipole, using the numerical technique described in chapter 2. The dipole is initially symmetrie and travels in northern direction carrying with it the fluid located within the unperturbed separatrix (figure 4.1a) . The trapped fluid moves with the dipole but as it moves northward, the negative vertex becomes strenger and the positive one weaker, leading to an asymmetrie separatrix ( tigure 4.1 b ): the area of fluid orbiting around the negative vortex becomes larger, implying that some ambient fluid bas been trapped, while the area of fluid orbiting around the positive vertex becomes smaller, which means that some of the interior fluid now lies outside the separatrix and wil! be detrained (figure 4.lc ). As the dipole returns to the initia! latitude (figure 4.le), the asymmetriesin the separatrix disappear. The dipole continues travelling southwards, however, and thus becomes asymmetrie again (figure 4.lf). Because of the strenger positive vertex, the area of fluid orbiting around it becomes larger and that of the weaker negative vertex smaller (figure 4.lg). The dipole returns to its initia! latitude (figure 4.li) and the process repeats itself.


(a) (b) (c)

{f)

~~

Figure 4.2: Sequence of experimental images showing the meandering of a dipolar vortex on a topographlc ,8-plane. The dipole travelled initially northward (i.e. towards shallower fluid). The images were taken at times (a) t=30 s, (b) t=45 s, (c) t=60 s, (d) t=75 s, (e) t=90 s, (f) t=l05 s, (g) t=120 s, (h) t=150 s, (i) t=l80 s, after withdrawing the cylinder. Experimental parameters: /=1.12 s-1 , ho=0.13 m, s=0.3 m-1 , .8=0.344 m-1s-1 •

Similar changes in the dipole's relative circulations and consequently in the shape of the separatrix were measured in laboratory dipoles {see figures 3.5- 3.8). The continuous changes in relative circulation are caused by the squeezing and stretching of the fiuid column in the topographic ,8-plane. However, the ftuid depth variations have further effects: as the dipole moves uphill the distance between the two halves increases, causing entrainment of fiuid, whereas when it moves downhili the dipole halves approach each other, resulting in detrainment of fiuid. Figure 4.2 shows the entrainment and detrainment of fiuid in an experimental dipole. The sequence shows qualitative similarities with the point-vortex model (figure 4.1), as well as some dilierences that will be discussed in section 4.5.

4.3 Analysis of the modulated point-vortex model 59

(a) x:, (b)

Figure 4.3: (a) Schematic representation of the point-vortex dipole on the ,8-plane. (b) Phase portnut a - ~; the thick line indicates the initia! conditions for an initially symmetrie dipole. All solutions are time periodic, and a is an even and ~ is an odd function of time.

4.3 Analysis of the modulated point-vortex model

4.3.1 Adveetion equations

It was shown in chapter 3 that the meandering trajectory of a dipolar vortex on the ,13-plane can be reasonably well simulated by a couple of point vortices with the circulations being modulated according to

d K 1 = Ko- .B.[Ç- 2(cosa0 - cos a)], (4.2)

d K2 = -Ko - .B.[Ç + 2(cosao- cos a)], ( 4.3)

where Ç is the latitudinal coordinate of the di pole's cent re and a is the direction of propagation measured from the east direction (see figure 4.3a). This modulation was first introduced by Kono & Yamagata (1977), and has been used by several authors since then (see chapter 3). It is also assumed that the coupleis initially symmetrie (with circulations Ko and -~~:0) and the vortices are separated by a distance d. (Note that ,13. is equivalent to 1r L2 ,8 in chapter 3, where 1r L2 is the area associated with the point vortex).

The path of the couple is described by a couple of ordinary differential equations for the latitudinal coordinate Ç of the dipole's centre and the direction of propagation a

where

dÇ u . - = sma dt ,

Ko .B. ( ) U = -2

d + - cos a 0 - cos a . 1T 41r

( 4.4)

62 Adveetion by a dipolar vortex on a f]-plane

(a) (b)

(c) (d)

B

p_

Figure 4.4: (a) Streamlines of an unperturbed point-vortex dipole. (b) The heteroclinic tangle in the perturbed case. The thick line is the unstable manifold and the thin line is the stabie manifold. ( c) The mechanism for transport between the cyclonic vortex and the ambient fluid. Region ABC D is mapped to A'B'C'D' (see text). (d) Transport between the two dipole halves. Region abcdis mapped to a'b'c'd' (see text).

perturbed case. Usually, the definition that best approximates the unperturbed case is chosen. However, for a dipoie on a f]-plane any choice greatiy differs from the symmetrie separatrix (compare figures 4.4a and 4.4c). Let A and C be two adjacent intersecÜons between the stabie and unstable manifolds, and B a point on the unstabie manifold and D a point of the stabie manifoid, as indicated in figure 4.4c. If one chooses the boundary between the cyclonic vortex and the ambient fluid as the iine formed by p+C on the unstabie manifoid of P+, and Cp_ on the stabie manifoid of P-, then a large fraction of fluid initially located out of the dipoie is included as interior fluid. On the other hand, a large fraction of fluid initially iocated in the dipole is excluded from the vortex if the boundary is chosen as the iine P+A on the unstabie manifoid of P+, and Ap_ on the stabie manifoid of P- - With the use of the former definition, the area ABCD in figure 4.4c, which maps to the area A'B'C'D', represents the fluid that will be detrained from the cyclonic vortex in the next cycle; whereas the dot ted area near A, which maps to the dotted area near A', represents the fluid that will be entrained. Since the flow is incompressibie, the area entrained is equai to the area detrained in every cycle. The overestimation of detrainment produced with the adopted definition of the boundary is


corrected with the use of an effectively detrained area, which is simply the intersection of the detrainment lobe with the area enclosed by the unperturbed separatrix.

Similarly, the tangle formed by the unstable manifold of P- and the stabie manifold of P+ gives rise to transport of fluid between the cyclonic and the anticyclonic halvesof the dipolar vortex (figure 4.4d) . In this case the area abcd is mapped to a'b'c'd', i.e. moves from the cyclonic half to the anticyclonic one; and the same amount of fluid (the dotted area close to a) leaves the anticyclonic vortex half and enters the cyclonic one.

4.3.3 Melnikov function

The Me/nikov function enables us to predict the behaviour of the stabie and the unstable manifolds without explicitlysolving the adveetion equations (4.8)-(4.9). This function is, up toa known normalization factor, the first-order term in the Taylor expansion about {3. = 0 of the distance between the stabie and the unstable manifolds. The Melnikov function M(to) is defined as

M(to) = J..: {!J[i"(t))g2[i,.(t), W + to; {3., ao), a(t + to; {3., ao)] -

h[i"(t)]gt[i"(t), Ç(t + to; {3., a 0 ), a(t + to; {3., a0 )]}dt, (4.20)

where i"(t) = (x"(t),y,.(t)) represents the partiele trajectory along the separatrix of the unperturbed dipole. The existence of a simple zero of the Melnikov function (M(to) = 0, oM(t0 )Jot -:f. 0) implies an infinite number of transverse intersections of the stabie and the unstable manifolds (see chapter 2 and references therein). Therefore the Melnikov function yields a specific criterion for (a) the exchange of mass between the dipole and the ambient fluid as wel! as between the two dipole halves, and (b) the existence of chaotic partiele trajectories, in termsof the system parameters a0 and {3 •.

One can also obtain an 0({3.) approximation for the area of a lobe by using the Melnikov function (Rom-Kedar et al. 1990). The area of a lobe is given by

l!o2

J.l = {3. M(to)dto + 0({3:), l o1

( 4.21)

where to1 and t02 are two adjacent zerosof the Melnikov function M(t0 ) (i.e. they correspond to adjacent intersections of the unstable and stabie manifolds ).

Some symmetries

Note that if the partiele trajectory along the separatrices is chosen in such a way that x"(t = 0) = 0 (i.e. t=O at the intersectien of the heteroclinic trajectory with the y axis ), the following symmetries hold for the time independent components of the velocity field:

fdx"(t)] h[i"(t))

Jdx"( - t)J - h[i"(-t))

Then the Melnikov function M(to) is equal to zero for all t0 = t0 such that the timeperiadie componentsof the velocity field have the same symmetry with respect to t0

gl[i"(t), t~ + t] = 91 [i,.( -t), t~- t]

92[i"(t), t~ + t] = -g2[i"( -t), t~- t]

64 Adveetion by a dipolar vortex on a (3-plane

These symmetries are satisfied if Ç(t~ + t) = Ç(t~ - t) and a(tö + t) = ±a(t0 - t), as can beseen in equations (4.14)-(4.19) . Thus t~ must correspond to an extreme latitudinal displacement in the motion of the dipole. In general t0 = (n + 1/2)T /2 for n an integer and T the period of the dipole's roeanclering motion. M(t0 ) has thus an infinite number of isolated zeros, two for every period of the perturbation.

4.3.4 Numerical results

Entraioment and detrainment

Figure 4.5 shows the detrainment lobe (of the cyclonic half) for constant a 0 = 1 and increasing (3. =(a) 0.04, (b) 0.08, (c) 0.12 and (d) 0.16. The lobeis thin and long for smal! f3. andreduces in lengthand increases in thickness with increasing f3 •. The area of the lobe increases with f3., but it occupies regions close to the unperturbed separatrix of the cyclonic half, indicating that a large portion of the core wil! remain trapped by the vortex. Similarly, figure 4.5 also shows the detrainment lobe for constant (3 = 0.1 and increasing initia! orientation angles a 0 = (e) 0.8, (f) 1.2, (g) 1.6 and (h) 2.0. The lobe increases in length and in thickness with increasing a 0 . Obviously the lobe area also increases. The lobe now "penetrates" closer to the positive point vortex, thus reducing the size of the positive core.

(a) (e)

n D (b) (I)~

~ (c)

:0 D (d)D (h)

~ Figure 4.5: The detrainment lobe of the cyclonic <lipole half for constant a 0 = 1 and f3.=( a) 0.04, (b) 0.08, (c) 0.12, and (d) 0.16. The detrainment lobe for constant {3. = 0.1 and ao= (e) 0.8, (f) 1.2, (g) 1.6 and (h) 2.0.


(a) (b)

0 ~~~~~~~~_L~~~~ 0.1 0.2 0.3 0 0.1 0.2 0.3

~· ~·

Figure 4.6: (a) Area (Jt) of the lobe detrained from the cyclonic half to the ambient fiuid computed using the Melnikov function. Contour interval 0.025, Jl is zero along bath axes. (b) Exchange ra te (Jt* = Jt/period). Contour interval 0.005.

Figure 4.6a shows the amount of fiuid exchanged between the cyclonic half and the ambient fiuid during one asciilation of the dipale computed using the Melnikov function. The initia! angle is varied from 0 to 11', and /3. is varied in the range 0-0.3. The area of the lobe increases with bath increasing /3. and o:o. The area is zero for o:o = 0: this initia! condition corresponds to the stable equilibrium (ETD) and no change of circulation occurs in the couple, and therefore no change of the farm of the separatrix. For o:0 -> 11' the area of the lobe does nat go to zero, since o:0 = 11' corresponds to the unstable equilibrium. The area of the lobe tends to a finite value which depends on /3.. For /3. = 0 the lobe area is also zero because then there is no variation of the circulation of the vortices, independently of the direction of propagation.

Figure 4.6b shows the exchange ra te (i.e. the amount of fiuid that is exchanged per unit time) between the cyclonic half and the ambient fiuid. This rate is obtained by dividing the lobe area (figure 4.6a) by the period of the di pole's asciilation (i.e. the perturbation period). The ra te is zero for /3=0 and for o:o = 0, where the lobe area is zero, but also for o:0 = 11', where the period of the asciilation goes to infinity. Therefore for every value of f3 -=/= 0 the exchange rate has a maximum and this occurs at the same value of o:0 ;:::: 1.9, within the resolution of our calculations.

The sections o:0 = 1 and /3. = 0.1 in the parameter plane have been chosen to compare the results given by the Melnikov function with those obtained by direct numerical integration of (4.8)-(4.9). The results agree very well, as should be, for small values of {3. and o:0 (figures 4.7a,c). For largervalues the Melnikov function underestimates the amount of fiuid that is detrained. The exchange rate increases in a linear manner with /3. (figure 4.7d) and shows a maximum for o:0 ;:::: 1.9 (figure 4. 7b ). Both methods show this maximum although they are slightly shifted, the one of direct numerical integration occurring for a greater value of o:0 •

66

0.4 ,------------...., (a)

0.3

l.I. 0.2 . . . 0.1

0 '--------'--~--'---~_) 0 2 3

0.4 ,------------...., (c)

0.3

l.I. 0.2

. . 0.1 ... ... 0 '--------'--~--'----_)

0 0.1 0.2 0.3

~·

Adveetion by a dipolar vortex on a {3-plane

0.03 .--------------, (b)

0.02

..... O.Q1

0.03 ,------------...., (d)

0.02

0.01

0 • 0 0.1 0.2 0.3

~·

Figure 4.7: Lobe area (J.L) and exchange rate (J.L*) computed using the Melnikov function (solid line), and direct numerical integration ofthe adveetion equations (asterisks) . .8=0.1 for (a) and (b); a 0 = 1 for (c) and (d). The squares indicate the "effectively detrained area" (see text).

The total exchange and the exchange rate between the negative vortex and the ambient fluid are exactly the same as those described above for the positive vortex. Because of the symmetries ( 4.5) the manifolds in the Poincaré section t = 0 are the mirror image with respect to the x axis of the manifolds in the Poincaré section t = T /2. The exchange of mass between the two dipole halves show the same qualitative behaviour as the exchange with the ambient fluid. The numerical calculations have revealed an approximate quantitative relation between them: 1.56J.LM < J.Lu < 1.61J.LM where J.L is the area of the lobe and the subiudexes U and M indicate exchange with the ambient fluid (through the upper separatrix) and between the two dipole halves (through the middle separatrix) , respectively.

Figure 4.7 also shows the effective area of fluid (J.LE) detrained during the first period of the dipole's meandering motion. This area is simply the intersection of the detrainment lobes (as illustrated in figure 4.5) and the unperturbed separatrix (figure 4.4a). Approximately 50 % of the detrainment lobe lies within the initia! separatrix. For the range of parameters stuclied here the following relation has been found: 0.48J.L < J.LE < 0.5J.L.

The amount of fluid detrained in one period does not depend on the period as it does


(a) (b) 0.3 .--------------,

0.2

s 0.1

o, 2

. .. . -::.·· '-s,

..

3 E

... ... ...

4

Figure 4.8: (a) S1, S2 and SJ are part of the open flow, the cyclonic core or the anticyclonic core, depending on whether the dipole is symmetrie (broken Line) or asymmetrie (solid line). (b) The markers represent the areas S1, S2 and SJ as a function of the asymmetry of the dipole ! = -K2/ K.J;

the line is an approx.imation given by S = 0.15(!- 1) - 0.04(!- 1 )2 + 0.005(!- 1?. The area of the unperturbed cyclonic half is taken as the unit area.

in the case of a vortex pair in an oscillating strain flow stuclied by Rom-Kedar et al. (1990). The most likely reason is that on the ,8-plane the perturbation period is, for the parameter range stuclied in this paper, much larger than the typical time scale of dipole propagation: the length of the dipole's trajectory during one oscillation is at least several times the distance between the point vortices.

Here the dominant factor is the amplitude of the perturbation, measured by the term ,B.A, where A is the amplitude of the radial displacement. The fluid area detrained in the first period is given by

0.31,B.A < !-tE < 0.33,B.A ( 4.22)

This result can be understood in the following way: the amount of detrained fluid should be related to the difference S1 between the area enclosed by the unperturbed separatrix and the area enclosed by the separatrix at the position of maximal asymmetry of the dipole (figure 4.8a). If the perturbation period is of the same order or greater than the orbit period "close" to the separatrix, then most of the fluid located outside the current separatrix will be advected to the dipole's wake. However, for a small perturbation period a significant portion of this fluid wil! be recaptured during the same asciilation of the dipole.

The shape of the separatrix depends on the ratio ~= -K.2 /K1 and the distance between the vortices d, which is constant. S1 has been computed numerically as a function of ~ and it has been found to be given by S1 ~ 0.15(!- 1) for 1 < ~ < 2 (figure 4.8b). On the ,8-plane the extreme values of ~ are reached at the position of maximum displacement in latitudinal direction, and the most important term in determining the amplitude of the strength's perturbation is ,B.A, as can be deduced from (4.2)-(4.3). The ratio between the circulations becomes € ~ 1 + 2,B.A, leading to S1 ~ 0.3,B.A. The latter relation compares very


well with ( 4.22). The same argument explains the almost constant ratio between the amount of fluid ex

changed by the two dipole halves and the fluid exchanged betweenone half and the ambient fluid (1.585±0.025 according to the Melnikov function calculations). Numerical computations using the asymmetrie separatrices give a ratio of SI/ S2 = 1.6 (see figure 4.8) .

Long time spread of particles

In the previous section we evaluated the amount of fluid (lobe area) that is exchanged between different regionsof the flow in one period of the meandering dipole. The specific f!uid area that undergoes such a process was also determined (i.e. the detrainment lobe) . In this section we explore the evolution of particles for Jonger periods. We discuss three cases: (a) a dipole with wave-like motion and net eastward drift (a0 = 0.5), (b) a dipole moving along an 8-shaped path and zero net drift (a0 = 2.241511), and (c) a dipole with cycloid-like motion and net westward drift (a0 = 2.5). The choice is motivated by one important difference between these cases: in (a) the dipole travels indefinitely in eastward direction without ever returning close to its initia! position, in (b) the dipole returns periodically to its initia! position and therefore fluid that was detrained can be recaptured by the dipole in a later passing, and in ( c) there is a series of points through which the dipole passes twice.

Particles were initially placed on a regular array within the detrainment lobe of the positive vortex ("bright" particles) and the negative vortex ("dark" particles), and their positions were sampledat times nT, where nis an integer and T is the period of the dipole's meandering.

In the wave-like mode (figure 4.9a) the dipole simply leaves a very thin and long tail along the meandering path. In case (a) one observes alternating bands of ambient and interior fluid, and the particles are spread in latitudinal and in westward direction over distances larger than the scale of the trajectory of the dipole itself (indicated by a white line in figure 4.9b ). There is a net westward transport of fluid in spite of the di pole's zero drift. This can be understood in the following way. As the dipole moves northward, the positive vbrtex (which occupies the west side of the couple) becomes weaker and detrains fluid, while the negative vortex (which occupies theeast side) becomes stronger and entrains fluid . On the other hand, as the dipole moves southward, the negative vortex (on the west side of the couple) becomes weaker and detrains fluid , while the positive vortex (on theeast side) becomes stronger and entrains fluid. In both cases the vortex located on theeast entrains fluid while the vortex located on the west detrains fluid, resulting therefore in a net mass transport in westward direction. Finally, in the cycloid-like mode, some particles are left behind in the form of large lobe structures, while other particles form a kind of front that ad vances in westward direction (figure 4.9c ). lt also can also be observed that the stabie cores ( the regions around the point vort i ces) decrease in size as the tilting angle increases.


Figure 4.9: Positions of fluid particles at t = nT, where T is the dipole's oscillation period and nis an integer number. The bright particles were originally located within the detrainment lobe of the cyclonic dipole half and the dark particles within that of the negative half. The white line represents the di pole's trajectory. (a) A dipole in the wave-like regime ( a 0 = 0.5); n = 0, 1, 2, 3. (b) The 8-shaped trajectory with zero net zonal drift (ao = 2.241511); n = 0,1 , ... ,6. (c) A dipole in the cycloid-like regime ( ao = 2.5); n = 0, 1, . .. , 10.

70 Adveetion by a. dipola.r vortex on a. (3-plane

4.4 Size perturbations

The meandering trajectory of a dipolar vortex on a topographic ,8-plane causes the water column to be alternatingly stretched and squeezed. As a result the distance between the vortex eentres will decrease as the dipole moves into deep water and increase as the dipole moves towa.rds shallower water (see figure 3.10). This process is periodic (with the same period as the variation of the vortices' strengths) and leads toentrainment of fluid during the "expanding" phase of the dipole and detrainment of fluid during the "shrinking" phase.

This effect is modelled using the point-vortex dipole with constant circulation ~e0 and -~e0 ,

respectively, but imposing on it a sine-like variation of the distance between the vortices

d = do(l + fd sinwt), (4.23)

where d0 is the initial distance between the vortices (as the dipole is located at its equilibrium latitude). The frequency w is the frequency of the roeanclering motion on the ,8-plane and for simplicity it will be taken as in the linear approximation used in chapter 3, see (3.15). The amplitude of the size perturbation td is related to the parameters of the topographic ,8-plane in the following manner. The depth of the fluid is given by h = h0 (1 - sy ), and by conservation of volume the radius of a column of fluid moving on this topography is given by r = r0(1 - s{t112 ~ r0 (1 + s{/2). This relation is assumed to hold for the variation of the distance between the two point vortices. In the linear approximation, the latitudinal position of the dipole's centreis given by { ;;:;: Asinwt, where A is the maximum latitudinal displacement. This leads to the following expression for the frequency w and the amplitude fd of the size perturbation:

w = fo/j, sA ,BA

fd = 2 = 4f2E'

Because of the symmetry about the x-axis we only need to examine the transport across the upper separatrix. Transport across the lower manifold is the same. The middle separatrix does not break up and therefore no exchange of fluid can occur between the two halves of the vortex dipole.

The adveetion equations for the point-vortex dipole with varying size can be written in the form of a periodically perturbed integrable Hamiltonian system

~; = h(x,y) + fd9t(x,y,t;e,w) + O(t:~), ~~ = h(x,y) + f'd92(x,y,t;t,w) + O(t:~),

by substituting the value of d given by (4.23) in (4.8)-(4.9) and makinga Taylor expansion of these equations around do. The functions /; are the same as in the ,8-plane perturbations, see (4.12)-(4.13), and g; are now given by

sinwt (x2 - y_: x2 - y; )

91= ]2 + J2 +2, - +

(4.24)

sînwt Y_ Y+) 92 ~ E +I~ x. (4.25)

4.4 Size perturbations 71

0.5 0.01

(a) 0.4 0.008

0.3 0.006

IL ll 02 0.004

0.1 0.002

0 0 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8

Jl. Jl.

Figure 4.10: Transport in the point-vortex dipole with oscillating size: (a) lobe area p, and (b) exchange rate p* as a function of (3,.. the Melnikov function , and * direct numerical integration of the adveetion equations.

Defining g; = gi sinwt, and using the symmetries present in the equations (/!, Y± and I: are even, and h and x are odd functions of t for the choice xu(t = 0) = 0), the Melnikov function becomes

M(to) = coswto J: {!I[x,.(t)Jg;[xu(t)]- h[xu(t)]g~[xu(t)]} sinwt dt. (4.26)

The integral is a function of /3,. through the dependenee on w and therefore will be denoted by F(/3.). F(/3.) is non-zero for the parameter region stuclied in this paper (0 < /3. < 0.3). M(t0 ) has therefore an infinite number of simple zeros [i.e. M(t0 ) = 0 and {JM(to)/&t0 = Oj for every point in the parameter space.

In figure 4.10 the solid line represents lobe area computed using the Melnikov function and the markers show the lobe areas obtained from direct numerical integration of the adveetion equations for t:d = 0.1. Different values of fd produce a similar curve, but with the lobe areas being some multiple of the ones shown here.

The lobe area decreases with increasing /3,.. This is because the period also decreases with /3,., and a shorter period allows for less fluid to he exchanged. However, this behaviour changes for the exchange rate (mass exchanged per unit of time). The exchangerateis maximal where wF(/3.)/21r is maximaL This maximum is reached for /3. = 1.8 ± 0.1.


(a) (b) ~·---"'!

. ' .· '

" ' ': ~ .t~

(d)

Figure 4.11: Experimental images showing the formation of lobe-like structures after half a period of the dipolar motion for different initia! direction of propagation and gradjent of ambient vorticity (a) ao = 1r /4, ,13=0.194 m- 1s- 1 , (b) a 0 = 1r /2, ,13=0.194 m- 1s- 1 , ( c) ao = 1r /4, ,13=0 .344 m-ls- 1 , ( d) ao = 1r /2, ,13=0.344 m- 1s-1 .

4.5 Experimental observations

In this section we present results of flow visualizations and compare them with numerical simulations. Four combinations of the parameter values f3 = 0.194, o1.344 m- 1s- 1 and a 0 = 7r/4,7r/2 were used.

In figure 4.11 the dipolar structures are shown after one half of the oscillation, when they have returned to their equilibrium latitude. Figure 4.1la,c show the dipoles started at an angle a 0 = 'lr/4: the structures are rather broad, with clear spiral patterns of undyed fiuid within bath dipale halves, indicating that entrainment has occurrep. For {3=0.344 m- 1s- 1

4.5 Experimental observations 73

(a) (b)

(C) (d)

Figure 4.12: The same as in figure 4.11 but now for numerical simulations using the modulated point-vortex model.

the dipole shows also a lobe structure that is being advected with the long tail (figure 4.11c). Figure 4.1lb,d show dipoles with initia! direction of propagation 1r /2. The dipoles are now more compact than those with initia! direction of propagation a0 = 1r /4. The spiral structure of ambient fluid is thinner, showing that little fluid has been entrained; and larger lobes of interior fluid are left behind, especially for ,8=0.344 m-1s- 1 (figure 4.11d).

In figure 4.12 the corresponding numerical simulations are shown. Both strength and size perturbations (4.2)-(4.3) and (4.23) are used in the adveetion equations. The parameter values and initia! conditions approximately match the values used in the experiments (figure 4.11). A fluid patch limited by the unperturbed separatrix has been foliowed during one half of the perturbation period.


Figure 4.12a shows two thin lobes, each one originating from a dipole half, and a second lobe is being formed in the anticyclonic vortex. Ambient fluid has been entrained but it is confined to a thin layer close to the original separatrix, indicating that large cores of interior fluid remain trapped by the dipole. (Note that on the rear as wellas on the front side of the dipole the "interior" and the "exterior" fluid are located in alternating bands, the ones formed by exterior fiuid are thicker ).

Figure 4.12c shows essentially the samefeatures as figure 4.12a but the lobes as wellas the spirals in the interior of the dipole are thicker. These structures are in an earlierstage of their evolution because now the period is shorter than in figure 4.12a. A comparison between the dipoles shown in figures 4.12b,d shows the same characteristic features and mutual differences as the dipoles shown in figures 4.12a,c. Note, however, that a third lobe-like structure appears in the anticyclonic vortex.

Qualitatively, similar features can be recognized such as entrainment and detrainment of fluid through the formation of lobes and spiral structures. Note that the long tail in the experiment is formed during the generation of the dipolar vortex (see chapters 2 and 3) and is notaresult of the perturbations in strengthor size. Another difference is that the ex perimental dipoles with a greater eastward component (a0 = 7r/4) have a net growth and therefore there is mainly en trainment of fluid and hardly any fluid is detrained. The growth of the eastward propagating dipoles is caused by the generation of relative vorticity in the dipoles exterior (see chapter 3); this secondary vorticity field gives rise to a shear that might be the cause of the large lobe in the cyclonic vortices (figures 4.1lb,d).

4.6 Conclusions

Mass transport during the meandering motion of a dipolar vortex on the {3-plane is investigated. The meandering motion occurs if the dipole travels transversally to lines of equal ambient vorticity. Due to conservation of potential vorticity the relative vorticity of the dipole changes asymmetrically, resulting in an alternatingly net positive and negative circulation. The stream function pattem changes accordingly, thus providing the possibility for fluid to escape or enter the dipolar structure.

The modulated point-vortex dipole models this basic mechanism appropriately (see chapter 3 and Kloosterziel et al. 1993). This model is therefore used to study the transport of mass between different regions of the flow during the meandering motion of the dipole. The equations of motion of passive tracers are shown to have the form of a periodically perturbed integrable Hamiltonian system. The time independent part of the stream function is that of the symmetrie vortex couple with circulations constant in time, and the time-periodic component corresponds to that of a couple of point vortices of the same sign, but with varying intensities.

Recent techniques in the theory of transport in dynamical systems (so-called lobe dynamics, see Rom-Kedar et al. 1990) are subsequently applied to calculate the amount of fluid that is entrained and detrained in every period of the dipole's meandering motion as well as the rate at which this interchange takes place. These quantities are computed as a function of the initial direction of propagation a 0 and the gradient of ambient vorticity {3. The amount of mass exchanged increases with increasing f3 and increasing a 0 and it is approximately proportional to the product f3A . The exchange rate has a maximum for a 0 ;:::: 1.9 for every value of {3 .

4.6 Conclusions 75

On a topographic ,8-plane the changes in relative vorticity are caused by squeezing of the fiuid column as the dipole moves uphill and stretching as the dipole moves downhill. These processes induce a periodic change in the distance between the points where the extreme values of vorticity occur, thus leading to detrainment as the points approach each other and entrainment of fiuid as the points move apart. The exchange of mass due to this mechanism is stuclied independently using a symmetrie point-vortex couple separated by a distance that changes periodically in time. For equal perturbation amplitudes (i.e. the same relative variation with respect to the unperturbed value), the size variation causes more exchange of fiuid than the variation of strength.

The point-vortex model is compared with experimental observations for a few points in the parameter space. The deformation of the initia! separatrix is computed numerically using both modulation of the vortices circulations and oscillation of the distance between the point vortices. The comparison is made after the dipole returns to its equilibrium latitude (isobath), i.e. after one half of the perturbation period. Good qualitative agreement exists in the main features: formation of spiral structures of ambient ( undyed) Huid in the interior of the dipole and Iobe-like (dyed) structures left behind in the wake of the dipole. The main differences are that the dipoles with a greater eastward component grow in the experiment (and therefore there is mainly entraioment of fiuid) and that larger lobes are formed in the experimental dipoles than in the simulations. The observed differences are most likely due to the absence of relative vorticity generation in the point-vortex model. As discussed in chapter 3 this secondary vorticity field is responsible for the expansion of dipolar vortices with an eastward component and also causes a shear that would produce the lobe.

Hobson (1990) did not explicitly compute the mass exchange, but she concluded from the structure of the heteroclinic tangles that the mass exchange should increase with the tilting angle. She later discussed the implications of her results for an atmospheric blocking - supposedly a WTD which remains stationary due to the eastward circulation- and concluded that this disturbance does notforma complete harrier for mass transport. This may he true, but it can not he concluded from the point-vortex model, which shows that the exchange of mass between a WTD and the ambient fluid is large because of the unstable trajectory. An atmospheric blocking, however, remains quasi-stationary for long periods. Therefore mass exchange should he small according to the results of the point vortex model. But the interpretation of an atmospheric blockingas a dipolar vortex has yet to overcome some difficulties, in the first place its surprising stability.

76 Adveetion by a dipolar vortex on a (3-plane

Chapter 5

Collision of dipolar vortices on a /3-plane

5.1 Introduetion

Various elements have stimulated the study of interactions between two dipolar vortices with continuous distribution of vorticity (like the Lamb dipole or ,8-plane modons). The most important role has probably been played by the stability issue, which has been stuclied mainly numerically due to the difficulties involved in an analytic approach. In most numerical studies the di pole's evolution is foliowed aftera small perturbation has been imposed on the structure, for example, by slightly changing the shape of the dipole or by the addition of a small random flow in the vicinity of the vortex (McWilliams et al. 1981). Another type of perturbation is the presence of a non-uniform external velocity field, like a strain (Kida, Takaoka & Hussain 1991) or a shear. These external fields have been thought of as an idealization of the infl.uence of other vortices acting at a large distance (compared to the size of the dipole). The character of these external infl.uences changes if the distance between the vortices is comparable to the di pole's diameter, which has led to the study of basic interactions such as the collision of two dipolar vortices (McWilliams & Zabusky 1982).

Another stimulus for the study of dipolar interactions is the fact that these structures tend to emerge spontaneously in two-dimensional turbulence. If the dipolar structures are abundant then some collisions must take place. In this context, dipolar interactions might play an important role in the redistribution of vorticity (e.g., Couder & Basdevant 1986).

In the presence of gradients of background vorticity (a ",8-effect") steadily translating dipolar vortices only exist if they propagate perpendicularly to those gradients. It has been suggested that collisions of zonally moving dipoles could give rise to transversally propagating ones (Horton 1989). The collision of dipoles would act thus as a triggering mechanism of the rich variety of phenomena displayed by a meandering dipole, as has been discussed in chapters 3 and 4. The argument, however, leaves unexplained why the processes that generate zonally travelling di po les ( whichever these processes are) would not be able to genera te transversally propagating ones.

Kono & Yamagata (1977) first introduced the modulated point-vortex model for the ,8-plane. They stuclied the dynamics of a single dipole and the head-on collision of dipoles of equal size but different circulations. In all these cases they observed the exchange of partners, a looping motion of the newly formed couples and a second interaction with partner exchange

77

78 Collision of dipolar vortices on a (3-plane

as the couples return to the system's symmetry line. After the second interaction the two couples reeover their initia! size, speed and direction of propagation. For this reason this is sometimes called a soliton-like or elastic interaction. Hobson (1992) showed that the motion of the individual point-vortices during the interaction of two coaxial couples is integrable. The symmetry present in this problem enabled her to find a Hamiltonian, and, by using this advantageous formalism, to study the regimes of motion that arise as a function of the system parameters. Makino et al. (1981) stuclied numerically the interaction of dipoles -with a continuous dis tribution of vorticity- of different sizes and speeds, and found the same fundamental behaviour as predicted by the point-vortex model of Kono & Yamagata (1977). However, a more detailed study by McWilliams & Zabusky (1982) showed that a soliton-like ooilision is a rare event among the variety of dipole interactions. An elastic collision arises only if, after the looping motion, the two dipoles re-approach the symmetry line perpendicularly. Th is is the case for modons of approximately the same si ze and speed, and for f3 smal! com pared with the dipole's vorticity.

In addition to the motion of the couples, the adveetion of fluid during the interaction process has received increasing attention in recent years. This has been mainly motivated by the interest in transport issues in plasmas. Horton (1989) stuclied numerically meridional transport due to the interaction of two equal modons travelling along parallel lines. He found that the net transport is maximal when the distance between the symmetry lines is of the same order as the modon radius, while it is minimal when the couples have a common symmetry line. Similar results were obtained for transport during the interaction of point-vortex couples by Kono & Horton (1991). In contrast, Nycander & Isichenko (1990), using theoretica! considerations about the oscillating motion of the resulting couples, found a larger net transport.

The main purpose of this chapter is to present experimental observations of the interaction of two zonally moving dipoles. In particular, we are interested in the motion of the vortex eentres and the adveetion of passive fluid . This issues are first discussed using modulated point vortices, which represent a simple model but still capture the essential physics of the interaction. Here mass transport is approached in the same spirit as the lobe-shedding by a dipole on a {3- or a {-plane (see chapters 4 and 6), where the aim was todetermine the region of fluid that is detrained or entrained and how much area undergoes such a process.

5.2 Interaction of point-vortex dipoles

5.2.1 Review of the non-modulated case

The collision of point-vortex couples (with constant circulations) has been stuclied by several authors since the last century (see Aref 1983). Here we present a brief review of the results that wil! be used in the discussion of interactions on a (3-plane. For a thorough study of the non-modulated case the reader is referred to the artiele by Eekhardt & Aref (1988).

It is assumed that both couples propagate perpendicularly to the gradient of background vorticity before the interaction. The effect of f3 thus beoomes important only after the partner exchange, as the new couples begin to move transversally to the lines of equal ambient vorticity. It will therefore be assumed that the motion of the vortices up to the partner exchange occurs as if the circulations were constant. The angle of propagation of the new couples (which will be called departure angle) is determined by the initia! configuration and it is independent of

5.2 Interaction of point-vortex dipoles

y (a)

d/2+~ + 0 3 \ x

i -d/2-~ t 0 4

-·-·1-·-·--·-·-·-·-·-·-·-·- - -·-·-·- - --- - - - -·--·--·-·-·-f·--··

-b/2 b/2

/~ \

1 0 \ \

2 0 / I

\ \

\

I I

I I

"'--...... 0 d~ 4

79

y (b)

' I \ I x

I \:JIXo

Figure 5.1: Schematic representation of the interaction of two coaxial couples: (a) before the partner exchange, and (b) after the partner exchange.

y

0 d/2-e

2 0 + -d/2-e

(a)

t 0 3

d/2+e I

-d/2+e + 0 4

-·-·+- -·-·-·- -·-·-·-·-·-·-·--·-·- -·-·-·-·-·- ---- -+- ---b/2 b/2

x

~3 x' 0 10\

\ \

\ \

' \ \

y

\ \

\ \

\ ' \

\ \

\ 0 4 \

~/.

(b)

x

Figure 5.2: Schematic representation of the interaction of two parallel coup les: (a) before the partner exchange, and (b) after the partner exchange.

(3. As has been discussed in chapters 3 and 4, the departure angle plays a fundamental role in determining the character of the subsequent motion of the dipole as wel! as the transport of mass between different regionsof the flow.

Among the large variety of possible interactions, we will confine our attention to the case of equal magnitude of the initia! circulation r;,0 (which is assumed to be positive), and with the two couples propagating in opposite directions along parallellines. These interactions lead to integrable motions of the vortices in the non-modulated case (Eckhardt & Aref 1988). Two types of interaction can be distinguished: (a) coaxial couples, when the couples have different

80 Collision of dipolar vortices on a {3-plane

si zes ( d- 2t1 and d + 2t2) but a common symmetry line (figure 5.1a); and (b) parallel couples, when the couples have equal size d but their symmetry lines are separated by a distance 2t, which is usually called the impact parameter (figure 5.2a). The values of t, t1 and t2 used here !ie within the range that generates a partner exchange. Although two coaxial couples are also parallel, we will use these terms as in the definitions above. Note that a head-on collision of two equal couples belongs to both categories, and will receive special attention in the discussion of mass transport .

lt is well known that the motion of a system of N point vortices is given by the system of Hamiltonian equations

dy; aH K,·-- --

1 dt - àx;' (5.1)

with

(5.2)

where r[i = (x;- Xj)2 + (y;- Yi) 2. The Hamiltonian H does not depend explicitly on time,

therefore it is an integral of motion; three more integrals are easily obtained from translational and rotational invariance (see, e.g. , Batchelor 1967)

N

Q LK.iXi, (5.3) i=l N

p = L"'iYi, (5.4) i=!

N

I = L K;(xf + y;). (5.5) i=!

Figure 5.1b shows the configuration after the partner exchange of two coaxial couples. The dependenee of the angle a 0 on the parameters t1 and t2 can easily be obtained from the conserved quantities (5.2)-(5.5), see Eekhardt & Aref (1988). Here Q = I = 0, whereas the other integrals of motion before ( subindex I) and after (subindex F) the interaction are given by the following expressions:

PI -2Ko(t1 + E2),

Hl ~ -K~ln[(d- 2t2)(d + 2t1)], Pp - 2Kod cos ao,

Hp ~ -K~ ln d2,

where, in the computation of the Hamiltonian, it has been assumed that the distance between the two couples is much larger than the distance between the vortices forming a couple (e.g., b ~ d in the initia! configuration). Conservation of Pand H during the whole evolution implies that

dt2

d- 2E2'

~ ( 1 + cos ao - J 1 + cos2 a 0 ) .

(5.6)

(5.7)

5.2 Interaction of point-vortex dipales 81

In principle, t 1 and t 2 can be chosen arbitrarily. However, the assumption that after the interaction process two new couples of size d emerge, uniquely determines the value of t1 once t 2 has been chosen, or vice versa, see (5.6) . The value of, say t 2 , then uniquely determines the departure angle a:0 in (5.7). The results presented in chapters 3 and 4 show the importance of the initia! direction of propagation a:0 in the evolution of a dipole on a ,8-plane; therefore, we wil! discuss the interaction process as a function of the departure angle a:o instead of t 2 •

Figure 5.2b shows the configuration after the collision of two parallel couples. In this case we are also interested in how the departure angle a:0 varies as a function of the impact parameter 2t. However, unlike the coaxial case, now the integrals of motion are not suftleient to obtain that relationship. Numerical experiments show that the departure angle is given approximately by

(5.8)

This empirica! relation agrees within 0.1% with the numerical and analytica! results of Eekhardt & Aref (1988), for the parameter region stuclied in this chapter. Therefore, in the discussion of parallel interactions on the ,8-plane, the angle a:0 given by (5.8) will be called the departure angle. It must be kept in mind, however, that this is not an exact expression.

5.2.2 Modulated coaxial couples

As illustrated in figure 5.1, in the non-modulated case the symmetry of the system with respect to the common axis of the couples is preserved during the interaction. On a ,8-plane this symmetry is preserved only if the symmetry line is parallel to the west-east axis (x). In that event the collision of coaxial couples is equivalent to the evolution of two vort ices on the half-plane y > 0 with a straight free-slip wall at y = 0. The evolution equations transform into

dx;

dt dy;

dt

_ _!__ [!2_ _ ,_3_; (y; ~ Y3-i _ Yi -~ Y3- i)] , 21r 2y; ri,3-i ri,3-i

= _ "-3-i (Xi- X3-i _ X;- X3-i) . 21r r l,3-i rl,3-i '

with the circulations being modulated according to (see chapter 3)

"-i = "-io + .B.(Yio- y;),

(5.9)

( 5.1 0)

(5.11)

for i = 1, 2, where r?,3-i = (x; - X3-i)2 + (y; + Y3-i)2. This symmetry has thus reduced the number of equations from eight (the evolution of N point vortices in the infinite planeis described by 2N equations) to four.

As in the case of a single dipole we will describe the evolution in termsof the coordinates (7J,Ç) of the middle point between the vortices in the upper half-plane and the direction of motion a (see figure 5.3). However, now the distance 8 between these two point vortices is time dependent. These new variables are given by

1] X1 + X 2

2 YI + Y2

2 (x2- Xt)

2 + (Y2 - yt)2,

arccos (YI ~ y2) .

82 Collision of dipolar vortices on a /3-plane

u

Figure 5.3: Schematic representation of the vortices in the upper half-plane for a coaxial interaction on the ,8-plane.

The time derivative of these equations and the use of (5.9)-(5.10) give, after cumhersome but straightforward algebra, the following equations of motion

diJ dt d~ dt d8

KJ K2 (KI - K2) COS a 2~(KI + K2) 2Ç + 8 cos a+ 2Ç- 8 cos a+ 8 + 82 sin2 a+ 4e'

8sina(!2- c2. 21 2)(KJ-K2), o o sm a+ 4~ . ( K1 K2 (K1+K2)cosa = sm a + - -'-------,-''---

dt 2Ç + 8 cos a 2Ç - 8 cos a 8

2~( K1 - K2) + ( K1 + K2)8 cos a) 82 sin2 a+4~2 '

= K 1 ; K2 + ( ;2 - 82 sin 2 ~ + 4Ç2 ) ( K 1 - K2), da

dt

(5.12)

(5.13)

( 5.14)

(5.15)

where a factor 1/47r has been removed from the right-hand side of equations (5.12)-(5.15). A further reduction can be achieved by using the conserved quantity (Zabusky & McWilliams 1982)

This leads to an expression that relates the latitudinal position ~ to the direction of propagation a

8( ) (Ko 8(cosa - cosa0 ) ) "2 cos a - cos ao (i - --'---2:------'-'- · (5.16)

5.2 Interaction of point-vortex dipoles 83

(a) (b) (c)

Figure 5.4: Trajectories of individual vortices in the course of the interaction of coaxial couples on the /3-plane. The thick lines show the trajectories of the positive vortices and the thin lines show those of the negative vortices. (a) A large ETD interacting with a small WTD produces a sine-like looping of the newly formed couples. (b) A large WTD collides with a small ETD and a closed loop is described by the new couples. ( c) A yet larger WTD against a smaller ETD gives rise to a self-intersecting loop.

Substitution of this result in (5.12)-(5.15) leaves a system of equations where Ti and ~ do oot appear in the right-hand side of any of them. Therefore two first order ordinary differential equations for 8 and a govern the motion of the vortices; furthermore, these equations are autonomous, which implies that the motion of two coaxial couples on the ,8-plane is integrable (Hobson 1992).

Before presenting the numerical results we makesome operational definitions. We consider the partner exchange to occur when 813 = 812 ( 8;j is the di stance between vortices i and j). The time elapsed between the two events 813 = 812 is called the scattering time T •. The loop length À is the distance in zona! direction between the points where the partner exchanges occur, and is considered to be positive when the second point lies to the east of the first one, and negative otherwise. The amplitude of the looping motion of the couples can be obtained by putting a=O in (5.16).

For smal! a 0 (i.e. the ETD larger than the WTD) the loop has a smal! amplitude and a sine-like shape, the newly formed couples reencounter eastward of the location of the first interaction (figure 5.4a). The loop-length decreases as the departure angle a 0 increases, and becomes zeroforsome critica! value Oe~ 2.26 (figure 5.4b). For largervalues of a 0 the second partner exchange occurs to the west of the first partner exchange, leading to a cycloid-like loop (figure 5.4c). In all cases the northern couple rotates clockwise and the southern couple anticlockwise. Note that the role of a 0 in determining the trajectory shape is the same as in the single-dipole case ( chapter 3).

The scattering timeT, decreases with .B. and increases with a 0 (figure 5.5a). The qualitative behaviour of the loop's amplitude A is the same as the amplitude of a meandering dipole, namely A increases with a 0 and decreases with .B. (figure 5.5b ). The loop length À decreases with increasing .B., as does the wave-length of a dipolar vortex, but it has a maximum for a particular value of a 0 (figure 5.5c) .

84 Collision of dipolar vortices on a (3-plane

(a) (b) (c)

Figure 5.5: (a) Scattering timeT., (b) amplitude A, and (c) length ..\of the loop described by the newly formed couples during a coaxial collision as a function of the departure angle eto and the grad.ient of ambient vorticity (J •.

5.2.3 Modulated parallel couples

Here the intervals 0 < a 0 < 1r /2 and 1r > a 0 > 1r /2 are equivalent. The labels ' northern' and 'southern' couples have to be interchanged as one describes the behaviour in one interval or the other.

For small values of a0 the new couple formed by the southern vortices performs small oscillations and propagates in eastward direction, while the new couple formed by the northern vortices propagates westward with a cycloid-like trajectory (figure 5.6a, where a 0 has a small negative value). The distance between the new couples increases continuously and their mutual infiuence decreases. Therefore, soon after the interaction each coup ie can al ready be considered as a single dipole moving in an otherwise quiescent fiuid (see chapter 3) . As the magnitude of a0 increases the net zona! velocity of the southern couple decreases while that of the northern couple increases (becomes less negative), for some value of a0 this couple shows no net zonaJ displacement, as is illustrated in figure 5.6b, which was computed using a vaJue of a 0 close to this critica! value. For slightly larger values of a 0 the two couples propagate eastward but their veloeities are different enough for the couples to behave independently a few oscillations after the partner exchange (figure 5.6c).

The departure angle a0 = 1r leads to a head-on collision of two identical couples, as discussed in the previous section. For small deviations from this valuè the newly formed couples reencounter almost at the initia! latitude but they are slightly misaligned -their trajectories are not parallel and they have a zona! shift. Owing to the small (but non-zero) impact parameter a new exchange of partners occurs, which re-establishes the original couples; however now the ETD propagates with a wavy trajectory, while the WTD moves along a cycloid-like path (figure 5.6d). The misalignment of the couples increases with both increasing a 0 and decreasing (3; and as a consequence the wavy trajectory of the re-established ETD increases in amplitude and decreases in wavelength, while the WTD trajectory has a decreasing amplitude and an increasing (less negative) wavelength.

5.2 Interaction of point-vortex dipales 85

(a) (b)

(c) (d)

Figure 5.6: As figure 5.4 but now for the interaction of two parallel equal couples. In (a), (b) and (c) a single exchange scattering occurs, and in ( d) two exchange scatterings take place. As the departure angle increases, the resultant state is: (a) two new couples propagate in opposite directions; (b) one new couple propagates in eastward direction while the other has nonet propagation; (c) two new couples propagate in eastward direction; and ( d) the two original couples propagate in approximately their original direction, with the corresponding stability properties.

For every value of f3. =I 0 there is a range of a 0 ( e.g., 1.1 < a 0 < 1.45 for (3. = 0.1), where the couples show multiple interactions, with partner exchange (exchange scat tering) as wel! as without partner exchange (direct scat tering). Figure 5.7a shows an example of this type of behaviour. After the first partner exchange, three interactions without exchange take place; a second exchange occurs later and after a new interaction without partner exchange, the two couples propagate in eastward direction with different zona! speeds. This example shows that it is impractical to study in detail the motion of the vortices in this parameter region. However, one would like to know, for example, if the dipole interactions continue indefinitely or eventually two independent couples emerge; and if the lat ter occurs, whether the couples are formed by the same vortices as initially. As a diagnostic parameter to explore these issues wedefine

( 5.17)

where li13 is the distance between vortices 1 and 3 (pertaining to different couples originally) and li12 the distance between vortices 1 and 2 (forming one of the original couples) . From this definition it is clear that a growing positive value of D indicates that the two original couples


(a)

3

2 (b)

D 0

-1

-2

-3 0 50 100 150

time

Figure 5.7: (a) An example trajectory in the multiple interaction regime (the angle used here is a0 = 1.405). (b) Evolution of D = log(ó13/ó12) for the numerical experiment shown in (a) (thick line); the typical evolution of D in the exchange scattering regime (upper line) and in the direct scattering regime (lower line) are also l5hown.

have recovered their identity and move away from each other, as illustrated by the (up per) thin line in figure 5.7b, which gives the evolution of D in the course of a head-on collision. If, on the contrary, D has an increasingly negative value then the two newly formed couples are moving apart from each other without interacting again, as is the case in the parallel collision with small departure angle represented by the (lower) thin line in tigure 5.7b. When multiple interactions occur the graph of D remains 'close' to the x-axis for a large period of the evolution. An intersection with the axis indicates an exchange scattering, whereas an approach to this axis without crossingit indicates a direct scattering interaction. An example of this is given by the thick line in figure 5.7b, which corresponds to the interaction shown in figure 5.7a.

The long-time average of D ( denoted by fJ) points out the final result of the interaction:


(a) (b)

1.4

a o 1.2 a o 1.2

0 0. 1 0.2 0.3 0 0.1 0.2 0.3

Figure 5.8: (a) Mean value of D = log(613 j612) [colour representation runs from blue (D = -5) to red (D = 5)]; and (b) number of partner exchanges (Ns) as a function of the departure angle ao and the gradient of ambient vorticity f3. [colour representation runs from blue (NE = 1) to red (NE = 6)] .

a large positive value indicates that in the final state the original couples have recovered their identity and behave almost independently, a large negative value indicates that two new couples propagate independently. A value of b close to zero indicates a more collective behaviour of the four vortices, which are generally arranged as two couples that at some st ages move away from each other and subsequently, due to the asymmetries induced by the ,8-plane, reverse their direction of propagation and approach each other to interact either with or without partner exchange. This process can take place many times.

The behaviour of D as a function of the departure angle a0 and the gradient of ambient vorticity ,8 is shown in figure 5.8a, and the number of partner exchanges (NE) that t ake place during the evolution is shown in figure 5.8b. The value of b depends evidently on the time interval used to make the average. On the other hand, numerical experiments with different time intervals have shown that Ns does notdepend on t he time interval if this is la rge enough. This result indicates that the final state for all initia! conditions (in the parameter region considered he re) is that of two independently propagating coup les .

For ,8. = 0.1 D has a negative value in the region given approximately by 0 < a 0 < 1.1 (see figure 5.8a), because the couples exchange partners only once (Ns = 1, figure 5.8b) . After that event the newly formed couples propagate independently of each other . On the contrary, D has a positive value in the up per region of the parameter plane ( 1.45 < a0 < 1r /2, for ,8. = 0.1 ), since the vortices exchange partners twice, which re-establishes the original couples. For initia! conditions located between these "regular" regions, the couples show a complicated beha~iour. In this region multiple interactions , of d irect and exchange scattering


type, take place; this is visible in the alternating bands of positive and negative lJ (figure 5.8a). For low values of /3 . the bands of positive and negative fJ correspond to NE being 1 or 2, but as /3. grows the number of partner exchanges increases up to several tens in small regions (figure 5.8b ). These results suggest that the fin al result of the interaction is sensitive to t he in i ti al con di ti ons.

5.2.4 Mass exchange

The motion of the vortices shows a major difference between the coaxial and the parallel cases. On the one hand, the callision of coaxial couples results in two exchange interactions, which strongly affect the vortex motion. However, this occurs only for a finite time, because the second interaction re-establishes the original couples, which subsequently travel perpendicularly to the gradient of ambient vorticity. For this reason the vortices have a constant circulation after the interaction; therefore neither interior fluid can escape, nor ambient !luid can be entrained, as discussed in chapter 4.

On the other hand, parallel couples never reeover their initia! zona! propagation. Both as independent couples and as mutually interacting ones , the couples meander around lines of constant ambient vorticity. The circulations are thus conlinuously changing and fluid masses can be entrained and detrained.

Coaxial couples

The exchange of mass during the interaction of two equal coaxia.l couples is asymmetrie. On the one hand , a large amount of fluid carried by the ETD is replaced by ambient fluid during the interaction. This is because the ETD's positive half moves northwards while its negative half moves southwards during the looping motion. Hence the two halves become weaker, see (5.11), and the area of trapped fluid decreases, which implies detrainment of fluid; but the vortices reeover their initia! circulat ion a.s they return to the symmetry line, t herefore they capture ambient fluid. On the other hand, the WTD exchanges a negligible amount of fluid , because after the first partner exchange its negative half moves northwards and its posi t ive half southwards. Hence the two halves acquire a. stronger circulation and the area of fluid trapped by each half increases, which implies entraioment of ambient fluid. However , most of this fluid is detrained as the temporary couples return to the symmetry line, where the vortices reeover their ini t ia! circulation ( and consequently lhe area of t rapped fluid reeovers its initia! size).

The mass exchanged by the ETD with the ambient fl uid was evaluated as a function of ,8 • . In figure 5.9, the broken lines represent the boundaries of t he areas of fluid initially trapped by the original couples, the thick line within lhe positive vortex of the ETD encloses the area of fluid that wil! be detrained and the thick line between the two couples indicates the ( equal) area of fluid that wil! be entrained into the positive vortex of the ETD. As /3. increases the detrainment lobe camprises regions closer to the point vort ices, a lt hough t he lobe areas do not change significantly in the range 0.1 < /3. < 0.3. The entraioment area has a larger latitudinal span for smal! /3. values , which is a consequence of the increasing ampli tude of the loop as /3. decreases. Corresponding areas in the lower half plane are not drawn in figure 5.9, sirree they are the mirror images, with respect to the x-axis, of the ones shown here. Moreover, if the figures are mirrored about a vertica.lline drawn through the middle point between the


(a)

0 -·.

-... -----~ -(b)

a -· ..

(c)

- ----···

Figure 5.9: Detrainment and en trainment regionsof the ETD (!eft-hand side) fora head-on callision with an equal WTD. The broken lines indicate the ftuid initially trapped by the couples. The results are plotted for {3. = 0.1 (a), 0.2 (b), and 0.3 (c) .

two couples, one obtains the position of the entrained fluid within the ETD and the tail of detrained fluid after the interaction. These locations are relative to the couples, in absolute space a shift in eastward direction -the loop length- has to be added. Figure 5.10 shows the area p of the detrainment lobe - scaled by the area of trapped fluid in the unperturbed case- as a function of the gradient of ambient vorticity f3. . The value of p increases with increasing (3., up toa maximum of 0.2 at f3. ~ 0.2, and beyond that value p decreases to 0.19 for (3 = 0.3.

When couples of different size collide the exchange of fluid is larger than in the case of two equal couples. This larger exchange is mainly the result of the size ditierences between the new couples and the original ones; variations of the vortex circulations as the couples move latitudinally play a secondary role here. It is helpfut to analyze the mass exchange due to this geometrical mechanism in the non-modulated case. The amount of mass trapped by

90 CoWsion of dipolar vortices on a {3-plane

0.2 . . 0.15

IJ. 0.1

0.05

0 0 0.1 0.2 0.3

Figure 5.10: Mass exchange during a head-on collision of two equal couples as a function of the gradient of ambient vorticity {3.: (a) total area p. exchanged during the process ( expressed as a fraction of the area trapped byeach vortex in the unperturbed case); and (b) exchange rate p.* = p.fT., where T. is the scattering time (see text ).

each half of a point-vortex dipole is proportional to the square of the distance d between the vortices; i.e. A = kál, where k ~ 1.4248. Therefore the fluid areas carried by each vortex before the interaction are k( d - EI} 2 and k( d + E2 ) 2 , for the vort i ces in the small and large couples, respectively. Note that here E1 and Ez are not independent of each other, they are related by the condition that the new couples have size d, see (5.6). After the partner exchange all vortices carry the same amount of fluid (kál). Therefore the vortices originating from the small coupleentraio an area of fluid Ae ~ 2kdE~, while an area Ad ~ 2kdE2 is detrained from the vortices originating from the large couple. In the modulated case the new couples return to the symmetry line of the system and, after a secoud interaction, the original couples are re-established. As a consequence, the small couple detrains the fluid captured during the first partner exchange and only retains its original fluid, while the large couple has to compensate the large portion of its original mass lost during the interaction by entraining ambient fluid.

Figure 5.lla illustrates a collision of a large ETD against a smal! WTD (the trajectory is shown in figure 5.4a). The broken lines represent the fluid trapped by the original couples, the thin lines an arbitrarily chosen streamline close to the separatrices. As the new couple reaches its northernmost position these patches are advected to the position indicated by the thick lines. The fluid patch trapped by the WTD has suffered little deformation, while the patch trapped by the original ETD is strongly deformed, and a large fraction has already been detrained. The broken line (hardly visible here) gives the current shape of the separatrix of the newly formed couple. Equivalent features are observed when a large WTD collides with a small ETD (figure 5.llb, the vortices' trajectories are shown in figure 5.4b) . But now it is the ETD which preserves most of its original fluid, while the WTD loses a large fraction of it and, consequently, entrains a corresponding area of ambient fluid as the couple is re-established.


(a)

' ' ~----------------------:

(b)

p 0------ ,

' ,'/ \ .. ' ' ' ' ' ' :-------------------:

Figure 5.11: Mass exchange during the callision of coaxial couples of different size. The broken lines indicate the initia] separatrices, the thin lines represent the streamlines arbitrarily chosen as initia! conditions, and the thick lines indicate the position of the fluid patches as the newly formed northern couple reaches its extreme Jatitudinal position. The cases plotted correspond with (a) a large ETD colliding with a smal] WTD, and (b) a small ETD colliding with a large WTD

In summary, when two equal coaxial couples collide the ETD exchanges fiuid with the surroundings while the WTD keeps its identity; and when the couples have a significant difference in size the larger couple exchanges large amounts of fiuid, while the smaller couple preserves most of its initia! mass .

Parallel couples

The exchange of mass caused by the interaction of parallel couples is larger than in the case of the head-on collision of equal couples. Here the process can be divided in different stages. The (first) partner exchange occurs very much like that in the non-modulated case and so does the exchange of mass, which implies that only masses of fl.uid close to the initia! separatrix change of region as aresult of the interaction (see also Meleshko et al. 1992). The


vort i ces that have the dosest approach exchange the largest amount of fiuid. Af ter this partner exchange the evolution strongly depends on the departure angle a 0 , as has been described in the previous section. In the range 0 < a 0 < 1.1 (a single partner exchange), the newly formed couples have an independent behaviour, moving periodically around the equilibrium latitude while propagating in zona! direction. Consequently, in this case mass transport is governed by the periodic perturbation of the vortex strengths, as discussed in chapter 4. On the other hand in the range 1.45 < a 0 < 1r /2, where the couples exchange partners twice, mass transport during the looping phase is similar to that due to a head-on collision, where the ETD exchanges significant amounts of mass while the WTD carries with it most of its original fl.uid (see section 5.2.4). In this case, however, each original couple has a periadie motion after the second partner exchange, therefore mass exchange occurs as in the single-dipole case (chapter 4).

In the range 1.1 < a0 < 1.45 (multiple interactions) the evolution of the fiuid initially trapped by the couples is more complicated. It is to be expected that, as a result of the various interactions -especially the direct scattering ones-, most :ftuid initially carried by the couples wil! be detrained. As a result fiuid wil! be most effectively stirred in this regime.

5.3 Interaction of continuons dipoles

5.3.1 Experimental results

A series of Iabaratory experiments was performed in order to compare with the results of the point-vortex model and of the numerical simulations using a vortex-in-cell method. The experimental set-up and the dipole generation are described in chapter 2, the only difference is that two dipoles moving along lines of equal depth were simultaneously generated. The objective was to produce a head-on callision of two equal dipoles moving perpendicularly to the gradient of ambient vorticity. However, from the knowledge about the behaviour of a single dipole ( chapter 3) it was expected that achieving such an interaction would be difficult. In the first place one has the different stability properties of the dipole trajectory: the pointvortex modeland Iabaratory experiments show that an ETD has a stabie trajectory, whereas that of an WTD is unstable; i.e. it is easy to produce a dipole travelling straight to the east, but a dipole initially travelling west acquires a latitudinal velocity component and makes a large looping motion (chapter 3). A second difference is that an ETD broadens and becomes slower whereas the WTD shrinks and moves faster - features that are a consequence of the generation of relative vorticity in the ambient fluid. Therefore, these features are not displayed by the point-vortex model but have been observed in experiments and numerical simulations (see chapter 3). The two asymmetries make it difficult to achieve a collision of two equal dipoles: in any real situation it is more likely that the collision wil! occur between unequal dipoles, propagating along non-parallel directions.

Even if two equal couples collide with zero impact parameter, there exists a further hindrance for the realization of a soliton-like collision on a topographic /3-plane (e .g., a laboratory tank with a sloping bottom). It consists in the following: the 'gradient of ambient vorticity' on a topographic /3-plane depends on 1/ H (see chapter 2); therefore the newly formed couples encounter different local gradients of ambient vorticity as they move uphill (north) or downhili (south). The northward moving couple experiences a larger f3 than the couple moving southwards, which leads to a northern loop with smaller amplitude and a shorter length than

5.3 Interaction of continuous dipales 93

the southern one. The couples return to their equilibrium latitude with a non-zero impact parameter, therefore after being re-established the original couples have a small perturbation with respect to their original direction of propagation. Note that this asymmetry also appears on a rotating planet, where the gradient of ambient vorticity is proportional to the eosine of the latitude. However small, these effects will prevent the realization of a soliton-like interaction, as we have confirmed during the numerous experiments performed, some of which are described below.

Visualization experiments

In a first series of experiments, the motion was visualized with dye, and the flow evolution was recorded with photo or video cameras. The positions of the vortex eentres were determined from the apparent centre of rotation of each vortex. This was done at fixed time intervals (5 sec) when a video camera was used, and at variabie times (10 or 15 sec) otherwise.

Figure 5.12 shows several stages of the interaction between an ETD and a WTD. At time t = 2.25T (with time scaled by the rotation period of the table T = 11.1 s) the couples are approaching each other. At this stage the ETD is slightly larger and better developed, while the WTD is relatively small and its anticyclonic half is not completely developed yet (figure 5.12a). At t = 4.5T the distances between the four vortex eentres are of similar magnitude; this can thus be considered as the moment of partner exchange (figure 5.12b). Note that the cyclonic halves are slightly closer to each other than the anticyclonic halves, which leads to larger deformation of the positive vortices. As the exchange process is almost completed (at t = 5.4T) large portions of dyed ftuid are exchanged between the positive halves; which alsoentrain ftuid from the new partner (figure 5.12c). In addition to producing large exchange of mass during the interaction, the misalignment also makes the northern couple move northwestward and the southern couple move southeastward (figure 5.12d). The northern couple performs therefore a large loop before the asymmetry induced by the ,8-plane is able to pull the vortex back to the equilibrium latitude, while the southern couple moves along a more sine-like trajectory (figure 5.12e). At this stage (t = 12.1T) two lobes are observed at the rear of the southern couple, in addition to the band of ftuid connecting it with the northern couple. The lobe that is being detrained by the cyclonic half contains undyed ftuid that was entrained during the interaction . In contrast, the northern couple shows large entrainment of undyed ftuid, which is located between the two halves and at the front side; and almost no detrainment. In the last stage shown here (at t = 16.21T) the vortices are approaching their equilibrium latitude (figure 5.12f), but the southern couple has advanced a larger distance in zona! direction and a second collision is not produced in this case.

Figure 5.13d shows the trajectories of the individual vortices during the interaction discussed above, and frames a-c show three examples of similar collisions between ETD's and WTD's with different departure angles. The initia! position of the vortices is indicated by dots, the thick lines represent the trajectories of the cyclonic vortices and the thin lines that of the anticyclonic partners. In tigure 5.13a the negative halves of both the ETD and the WTD are approximately aligned when the dipoles collide. After the partner exchange the northern couple has an almost eastward direction of propagation, therefore it is stabie and moves along a sine-like trajectory. The southern couple, on the other hand, has an almost westward direction of propagation, it is therefore unstable and makes a loop with a large latitudinal displacement. Figure 5.13b shows a similar case but with a smaller impact parameter. The newly formed couples have a larger latitudinal component than in case (a), but it is


Figure 5.12: Sequence of experimental images showing some stages of the callision between two zonally moving dipolar vortices on a topographic {J-plane. The dipoles were generated in zona! direction. The images were taken at times t = (a) 2.25T, (b) 4.5T, (c) 5.4T, (d) 8.1T, (e) 12.1T, and (f) 16.2T after withdrawing the cylinders, with T = 11.1 s the period of the rotating table. Experimental parameters: f = 1.12 s- 1 , gradient of the fluid depth 0.04, ho = 0.16 rn, and {3 = 0.344 m- l s-1.

5.3 Interaction of continuous dipoles

(a)

(c)

20cm t--------1

(b)

(d)

20cm ,_....

95

Figure 5.13: Experimental trajectoriesof the vortex eentres during dipolar callision on a topographic !3-plane. The period of the rotating table (T=11.1 s) and the gradient of the fluid depth (0 .04) were the same in all experiments shown; the mean depth ho (in cm) and the equivalent i3 value (in m- 1ç 1 ) are, respectively: (a) 18, 0.25; (b) 16, 0.28; (c) 18, 0.25; and (d) 16, 0.28.

still possible to observe a sine-like path of the northern couple and a cycloid-like path of the southern couple. Figure 5.13c shows a callision with almost zero impact parameter, although the WTD seemed to be larger than the ETD. The new couples return to their equilibrium latitude and a second callision takes place (this can be best observed in the video film of the experiment). The anticyclonic half of the original WTD was strongly deformed during this interaction, but a second partner exchange did not take place. This experiment is the best approximation of a soliton-like interaction we have observed.

Measurements

Experiments using fioating particles were clone in order to study the evolution of the flow field during the interaction process. The velocity field was measured and properties as vorticity and circulation were derived in the way described in chapter 2. In the experiment discussed below the mean depth of the water was h0 = 16 cm, and the sloping bottorn had a gradient of 0.04. The rotation period of the table was T = 11.1 s, therefore the Coriolis parameter has a value of f = 1.13 s- 1 • The equivalent value of /3 is 0.282 m- 1s- 1 •

96

(a}

(c)

• -

20an t) ,@

f---i

Collision of dipolar vortices on a (3-plane

(b)

20an 1----------1

(d)

20an f---i

[@)~ ~

Figure 5.14: Measured vorticity distributions at several stages in the evolution of the newly formed couples. Experimental parameters are: T = 11.1 s, f = 1.13 s-1 , ho = 16 cm, and {3 = 0.28 m-ls- 1

.

Graphs are shown for: t =(a) 2.5T; (b) 5.7T; (c) 7T; and (d) 10.2T.

Figure 5.14 shows the vorticity distribution at four stages in the interaction process. At time t = 2.5T (figure 5.14a) the exchange of partners has just taken place. The cyclonic vortices are stronger in both couples, and the vortices coming from the WTD are in turn stronger than those coming from the ETD. The ratio of the total positive circulation to the negative circulation t = -r+ ;r- within the dipoles are En = 0.95. and t 9 = 1.42, where the subscripts n, s denote the northern and the southern couples, respectively. At timet = 5.7T (figure 5.14b) the southern couple has passed its southernmost position, and it rotates anticlockwise, as can be expected from the stronger positive vortex (ts = 1.39). On the other hand, the northern couple is just approaching its extreme latitudinal position, as it rotates clockwise due to the stronger negative vortex (the extreme vorticity values have similar magnitude, but the net circulation is negative: tn = 0.41). At t = 7T (figure 5.14c) both couples are approaching the equilibrium latitude. The net circulation in the couples has the same sign as in the previous frame, but its magnitude gets closer to zero, i. e. the ratios


approach one: fn = 0.6 and ~:, = 1.26. The positive vortices have come very close to one another at timet = 10.2T (figure 5.14d), at this stage the negative halves, specially the one of the southern dipole, seem to have been decoupled from the positive halves, making possible a merger event between the two cyclonic vortices. After the merger event between the cyclonic vortices was completed, the negative vortices merged themselves; although in this final stage the flow was very weak.

5.3.2 Numerical simulations using a vortex-in-cell method

For comparison numerical simulations were made using the vortex-in-cell method described in chapter 2. The model is initialized using two equal Lamb dipoles separated by a distance d (the dipole's radius) . The impact parameter is 0.2d. The vorticity gradient f3 and the size d and maximum vorticity w of the dipole were chosen such that the parameter f3d/w = 0.046 is of the same order as in the experiment discussed in the previous section (0.03-0.05) . In the simulation discussed below time has been scaled, as in the experimental situation, by the rotation period of the table.

The grid has 128 x 128 points (with 20 grid points per dipole diameter) and the boundary conditions are free slip on the zona! walls and periodic in east-west direction. lnitially 250,000 point vortices were evenly distributed in the computational domain, although only the small fraction ( 4 %) that formed the di po les had net relative vorticity. The other 96 % had initially zero net circulation, i.e. they were so-called "ghost vortices" that became active as their latitudinal position changed.

Figure 5.15a shows the initia! distribution of relative vorticity. At time t = 3.6 the exchange has just taken place and the new couples move transversally to the lines of constant ambient vorticity. As the couples propagate in latitudinal direction they become asymmetrie and begin to rotate, the northern couple in clockwise sense and the southern one in the opposite sense (see figure 5.15c, which shows the situation at t = 7.2). Due to the presence of the couples, ambient fluid is advected and acquires relative vorticity when displaced in north-south direction; a feature that can be observed in the form of thin regions of oppositely signed vorticity at the flanks of the dipoles (figure 5.15d, t = 10.8). The southern dipole has a larger eastward velocity component than the northern couple (figure 5.15e), and as aresult the couples return misaligned to the initia! latitude (figure 5.15f) .

One of the main features of the method is that the positions of all points are foliowed in time (i.e. it is a Lagrangian technique) . This enables us to follow the location of species during the flow evolution. We will identify the species with the letters E or W, for point vortices initially located within the ETD or the WTD, respectively, and the subiudexes p or n to indicate the initia! sign of the vortex' circulation. For example Ep denotes the vortices located initially in the cyclonic (positive) half of the ETD.

The initia! conditions in this case approximate that of the experiment shown in figure 5.12. The boundary conditions, the number of vortices and the mesh size are equal to those in the numerical experiment discussed above, except for the impact parameter, which is 0.4d in this case. Figure 5.16a shows the distribution of species corresponding with the experimental situation shown in figure 5.12e. Species EP is represented by red dots, En by yellow dots, WP by blue dots and Wn by green dots. The largest mass exchange occurs between the two cyclonic vortices. A thin ring formed by species Ep (red) surrounds the core formed by species WP (blue), while a similar ring of species WP surrounds the core Ep . This is the result of their

98 Collision of dipolar vortices on a {:J-plane

(a) (b)

a a • ~V

" t (c) '

(d) 0

" ~ ~

~

• ~ • ()

(e) (f)

,~ ()

·~ (I =~ -~ u

Figure 5.15: Numerical simulation using a vortex-in-cell metbod (with vortex strength modulation) of an off-set collision of two Lamb dipoles. The parameters of the sim~lation are: 250,000 point vortices; a 128 x 128 mesh; periodic boundary conditions in east-west dir.eetion and free-slip on the zonal walls. The initia! conditions approximate the ones of the experiment shown in figure 5.14 (see text); and the sarne time scale is used to present the results. The distribJtion of relative vorticity is shownat t = (a) 0; (b) 3.6T; (c) 7.2T; (d) 10.8T; (e) 14.4T; and (f) 18T.


Figure 5.16: Numerical simulations of the experiment shown in figure 5.12e. (a) Vortex-in-eelt method : Position of point vortices initially located in the interior of the ETD (red: positive; yellow: negative) and the interior of a WTD (blue: positive; green: negative). The same numerical parameters used in figure 5.15 are used here, except that the impact parameter is 0.4d (where dis the dipole's radius). (b) Point vortex method : deformation of fluid patches initially trapped by the vortices. The colour representation is the sameasin (a) .

close approach during the partner exchange. The core Wn (green) shows a long tail , where one can also find species EP and WP. Similarly the core WP has a tail that also contains species EP. These long tails, as well as the the cusp observed where the two tails encounter each other, are partially the result of the shrinking process undergone by the WTD in the short time elapsed before the interaction. Ambient fluid is found in the interior of both newly formed couples, in the form of thin rings surrounding the vortex cores in the southern couple and as a relatively thicker intrusion of uncoloured fluid separating the two halves of the northern couple.

The same experiment was simulated using two parallel point-vortices couples. The parameter (3djw = 0.05 used here is a typical value in experimental situations, and the impact parameter 0.4d was chosen as in the vortex-in-cell simulation above. Two fluid contours per point-vortex were followed : one is initially located close to the separatrix and the second one is closer to the point vortex . This is done in order to show that the vortices retain an area of fluid that undergoes little deformations, whereas the regions that take part in mass exchange are thin bands of fluid located close to the separatrix. This simple model captures the main features observed in both the experiment and the vortex-in-cell simulation. Namely, vortex trajectories and the large exchange of mass between the cyclonic vort i ces (figure 5.16b ). However, in comparison with the vortex-in-cell simulation, the entrainment of ambient fluid is smallerand the tail of the anticyclonic half of the northern couple is almost absent in the point vortex model.


5.4 Conclusions

The collision of dipolar vortices on the /3-plane has been stuclied using both a point-vortex model and dipoles with more realistic vorticity distributions. The first model renders accurately the qualitative evolution observed in laboratory experiments, whereas the vortex-in-cell model reproduces the flow evolution in more detail, especially the adveetion properties.

A discussion of the collision of coaxial couples on the /3-plane, a problem known to be integrable (Hobson 1993) , has been presented. The emphasis is put on the dependenee of physical features like the amplitude, loop-length and scattering time on the relative sizes of the colliding dipoles (or equivalently the departure angle). In contrast, the interaction of two equal point-vortex couples with non-zero impact parameter is, in genera!, not integrable. However, for a large range of the departure angle o:0 and the gradient of ambient vorticity /3 the behaviour is regular: there is a single exchange scattering with the two newly formed couples having different net zona! speed. A smaller region seems to be sensitive to initia! conditions.

In the point-vortex model the head-on collision of two equal couples is not elastic when the trapped ftuid is also taken into account. The ETD detrains up to 20 %of the original ftuid -and consequently entrains the same amount of ambient ftuid- while the WTD preserves almost all its original fluid. The amount of mass exchanged by the ETD with the ambient fluid increases with /3 in the range 0 < /3 < 0.2 and then it slightly decreases. The lati tudinal displacement of the detrained fluid, however, decreases with increasing /3. In contrast, when the couples have different size large amounts of fluid is detrained from the larger couple, for a large ETD as well as for a large WTD. A fraction of this fluid is temporarily trapped by the vortices that initially formed the smaller couple, but it is finally detrained as the smalt couple is re-established. The numerical results suggest that the motion of a passive marker is regular in this case.

A large exchange of mass arises during the collision of parallel couples with non-zero impact parameter. The multiple interactions occurring for some initia[ conditions are very effective in destroying the cores of trapped fluid. In the final state at least one of the couples shows a meandering motion with large amplitude, resulting in large exchange of ma.ss a.nd chaotic partiele motion, according to results presented in chapter 4. The asymmetrie behaviour of an ETD and a WTD make the head-on collision an unusua.l event.

Chapter 6

A dipolar vortex on a ~-plane

6.1 Intrad uction

Many processes in the oceans and in the atmosphere owe their existence to the variation of the Coriolis parameter (!) with latitude. For mesoscale phenomena it is common to approximate f as a linear function with a gradient /3. Near the pole f3 varrishes and the quadratic term ( with coefficient 1) becomes dominant (Nof 1990) . The leading order term in the variation of the background vorticity produced by the parabolic free surface of fluid in a rotating (laboratory) tank has the same form of the 1-effect on a rotating sphere. Although one would not expect to find parabolic sea mountains, some isolated topographic features could create a distri bution of background vorticity similar to that of the 1-plane (namely closed contours of ambient vorticity). It is well known that there is an equivalence between the /3-plane dynamics in geophysical flows and the dynamics in a slab plasma, where the density gradient plays the role of /3. For a cylindrically confined plasma the dynamics is analogous to that of a 1-plane, but a complete equivalence depends on the density distribution (Yabuki et al. 1993) .

The dynamics of coherent vortices on the /3-plane have been stuclied intensively over the past decades. Initially studies were devoted to monopolar vortices and more recently the dipolar vortex, consisting of two closely packed regions of opposite vorticity, has been increasingly stuclied analytically, experimentally and numerically. Special attention has been given to the behaviour of the dipole when it propagates transversally to the lines of equal ambient vorticity. To study this problem Kono & Yamagata (1977) introduced a modulated point vortex model (see also Zabusky & McWilliams 1982 and Hobson 1991) and found that the couple meanders periodically around its equilibrium latitude. Numerical simulations and analytica! studies of continuous modons have shown the same behaviour (Makino et al. 1981, Nycander & Isichenko 1990). Experiments on dipolar vortices on a "topographic" /3-plane (Kloosterziel et al. 1993 and Velasco Fuentes & van Heijst 1994, see also chapter 3) have confirmed the meandering motion of a dipole and the stability properties of eastward and westward travelling dipoles (ETD's and WTD's for short). During the meandering motion of the dipole there is a continuous change of the streamline pattem and therefore ambient fluid can be trapped by the passing dipole, and interior fluid can be detrained as wel!. Using the point-vortex model Velasco Fuentes, van Heijst & Cremers (1994) (see also chapter 4) stuclied the exchange of mass between the dipole and the ambient fluid as a function of the gradient of ambient vorticity and the initia! direction of propagation of the couple. They obtained good

This chapter was originally published as a joumal artiele (Velasco Fuentes 1994).

101

"I

102 A dipolar vortex on a 1-plane

agreement between the analyticai-numerical results and experimental observations. Much less work has been clone on the 1-plane dynamics. Nof (1990) stuclied monopolar

and dipolar vortices on the 1-plane. He found exact analytica! solutions equivalent to the stationary barotropic modons on the ,8-plane. The modon's axis is perpendicular to the gradient of ambient vorticity, but the modonis asymmetrie (its total circulation is not zero). The asymmetry depends on the location of the stationary modon. Yabuki et al. (1993) used the modulated point-vortex model to study the propagation of a single dipole in a cylindrical plasma when the dipole's axis is not perpendicular to the gradient of ambient vorticity. The qualitative behaviour displayed by the dipole is the same as in the ,8-plane case.

This chapter is organized as follows. Section 6.2 deals with the propagation of the dipole, which is governed by a couple of ordinary differential equations; an approximate solution is obtained and compared with numerical integrations of the complete system. The adveetion of particles is analyzed in section 6.3; in particular, we present numerical results about entrainment-detrainment. Section 6.4 contains a summary and some final remarks.

6.2 Propagation of a modulated point-vortex dipole

Large-scale motions on a rotating sphere (the Earth) are essentially affected by the latitudinal variation of the Coriolis parameter J, which is defined as J = 2ns sin </1, with ns the Earth's angular speed and <P the geographic latitude. For motions occurring on scales smaller than a few degrees of latitude the Coriolis parameter can be approximated as a constant (local) value plus a linear variation in the meridional direction, i.e. J = Jo+ ,By, where Jo = 2ns sin <Po and ,8 = 2ns cos <Pol R, with R the Earth's radius. This approximation is known as the ,8-plane model. Close to the poles, however , ,8 goes to zero and the second order term in the Taylor expansion of J becomes more important, leading to the expression J = Jo -1r2 , where 1 = nsl R2 and r is the distance to the pole. This rarely used approximation is known as the 1-plane model (Nof 1990). In the context of large-scale geophysical fiows the 1-plane might seem rather limited in scope, but as mentioned in section 6.1, it may turn out to be useful in a wider range of situations, such as the "topographic" 1-plane and a cylindrical plasma.

Conservation of potential vorticity implies that the relative vorticity w of a vortex tube moving in meridional direction changes as expressed by the equation

(6.1)

where DI Dt = f) I ot + uo I ox + vo I oy is the material derivative, u and V the veloeities in the (x) and (y) directions, respectively, and r 2 = x2 + y2 is the square of the distance to the pole.

lnsight into the dynamics of flows with non-uniform background vorticity has been gained by studying the evolution of a few point vortices with a "modulated" circulation (Kono & Yamagata 1977, Zabusky & McWilliams 1982). In the same spirit, and knowing that a potential vortex is not a solution of (6.1) for 1 -::j:. 0, the evolution of a dipole on the 1-plane is stuclied using two point vortices modulated according to the principle of conservation of potential vorticity. For that purpose a certain area is assigned to the "point" vortex: under the assumption that a point vortex represents a small (undeformable) patch of vorticity, the circulation equals the uniform vorticity w multiplied by the area of the patch. If rr L 2 is the area associated to the point vortex, its circulation is then given by K = wrr L 2 .

6.2 Propagation of a modulated point-vortex dipo/e 103

Figure 6.1: Schematic representation of the point-vortex model of a dipole in polar coordinates.

Using (6.1) and conservation of mass one obtains

(6.2)

In this expression r;0 represents the initia\ distance of the point vortex to the pole, at which the vortex has strength ~~:;0 (N .B. Wh en two subindexes are used the fi.rst one denotes the point vortex and the 0 indicates that the initial value is taken). In what follows the short notation 1. = 1r L 21 will be used.

For a system of just two point vortices the distance d betweenthem is a constant of mot ion. The evolution of the pair is therefore completely described with the position of the middle point and the direction of propagation (see :figure 6.1). In view of the geometry of our domain it is convenient to use polar coordinates r and (} in radial and azimuthal direction, respectively. The path of the couple is given by the following set of ordinary differential equations for the radial and azimuthal coordinates (r, B) and the direction of propagation a:

dr U . dt =- sm a,

dB U - = -cosa dt r '

da =f! dt , (6.3)

where the magnitude of the velocity U and the rate of change of angular direction n are given by

u = K:j - /(2

~· K:j + /(2 u

n = 27rd2 - -;:- cos a.

This is a general set of equations descrihing the motion of t.wo point vortices. The second term in the expression of n is used for geometrical reasons: a is de:fined as the angle of the velocity


vector to the circle r = const (figure 6.1) at the position of the dipole's centre; therefore, a has a constant rate of change n = -(U fr) cos a fora dipole moving in a straight line. In the modulated case both n and u depend on the position of the dipole through the dependenee of ,çi on the radial coordinate.

One can define ,ç0 = (,ç10 - ,ç20)/2 and a= ,ç10 / ,ç20 so that the initia! velocity of the pair is equal to the velocity of a symmetrie couple with circulations ,ç0 and -,ç0. According to (6.2) and definitions given in figure 6.1 the circulation of each point vortex is given by

2,ço [ 2 2 J ,çl = ---1. r 0 -r +d(rcosa-r0 cosa0 ),

1-a (6.4)

2a,ço [ 2 2 ( )] ,ç2=---1.r0 -r -drcosa-rocosao. 1- a

(6.5)

As indicated in figure 6.1, ris the radial coordinate of the dipole's centre, and the subindex 0 denotes its initia! value. Substitution of these expressions in the definitions of n and U gives

1 U = -d[,ço- r.d(r cos a- r0 cos a 0)],

21f n 1 ( 1+a 2 2) u = 1rd2 ,ç01 _a +r.(r -r0) --;:-cosa.

(6.6)

(6.7)

By further imposing the condition a'(O) = 0, one obtains two equilibrium solutions: a(O) = 0 and a(O) = 1r, i.e. an ETD and a WTD, respectively. This sets the initia! asymmetry of the dipole

dcos a 0 - 2ro a= .

dcos ao + 2ro

As in Nof's (1990) study of continuous modons, the steady couples are asymmetrie and the asymmetry depends on the distance to the pole. In our case the initia! asymmetry depends also on the initia! direction of propagation: an ETD has positive net circulation, a WTD has negative net circulation and couples travelling straight poleward are initially symmetrie.

The equations (6.3)- (6. 7) describe the motion of the point-vortex dipole on the 1-plane. Note that the azimuthal coordinate does not appear on the right-hand side of any of the evolution equations; (6.3b) is therefore a subsidiary equation and the evolution is entirely determined by (6.3a) and (6.3c). Equivalently, in the ,8-plane case the east coordinate is also a subsidiary relation. This is obviously a consequence of the background vorticity being constant along the zona! coordinate (0 in the 1-plane and x in the ,8-plane).

Figure 6.2 shows numerical integrations of equations (6.3)-(6.7) . Three "regimes" can be distinguished: (a) westward cycloid-like, (b) non-propagating 8-shaped and (c) eastward wavelike. These regimes are also observed in the ,8-plane case, as has been discussed in chapter 3. Note that the maximum radial displacement is larger north of the equilibrium latitude than south of it. Similarly, the azimuthal displacement of the couple is shorter during the southern part of the trajectory than during the northern part. This is a consequence of the (local) gradient of background vorticity being stronger to the south of the equilibrium position. The same variation in circulation (asymmetry) is achieved with a smaller radial displacement, and then the couple returns to the equilibrium latitude more quickly. The asymmetry in the path becomes stronger with increasing values of 1 •.

6.2 Propagatîon of a modulated poînt-vortex dîpole 105

(a) (b)

···· ...

+

I

Figure 6.2: Trajectories of point-vortex dipoles on a 1-plane: (a) cycloid-like path with westward drift; (b) 8-shaped path without zona! drift; (c) sine-like path with eastward drift. The dipole moved initially in the direction indicated by the arrow. The thick line represents the pathof the positive vortex, the thin line the path of the negative vortex.

For the case of a small inhomogeneity (i.rod/"-o « 1) and a relatively large initia! radius (b.r jr0 « 1, where b.r = r- r 0 ), the di pole's speed is well approximated by U~ U0 = "-o/271'd and the angular velocity can be written as

These approximations make it possible to reduce (6.3) toa single equation for a:

a" + w2 sin a = 0, (6.8)

where w2 = 2"(.r0 U0 j7!'d2. A solution should also satisfy the initia! conditions a(O) = a0 and a'(O) = 0, where a 0 is the tilting angle. This equation is of the same form as the nonlinear simple pendulum equation and has therefore the same stability properties. This implies that an ETD ( ao = 0) corresponds to a stabie equilibrium: when a small perturbation is imposed, small oscillations around the equilibrium latitude arise. On the other hand, a WTD ( a 0 = 71') is an unstable equilibrium.

A few important results can be drawn immediately from the linearized version of (6.8): for smal! values of a 0 the oscillations have a constant frequency w and the azimuthal wavelength À and the radial amplitude A of the oscillations are given by

À Uo a~

(6.9) = 211'~(1- 4 ),

A Uo

(6.10) = ao-. w

Figure 6.3 shows a comparison between the period T, amplitude A and wavelength À for the linearized solution (of the approximate equation) and direct numerical computations of the complete set of equations (6.3)- (6.7). For constant a0 = 0.01 (figures 6.3a,b,c) the agreement


40"-----------, (a) (d)

30 0 · ----------------------- ---•----------

·20

10

oL---~--~----~ 0 0.01 0.02 0.03

Y·

10 (e)

8 ... A A

6

0.2

0.1

0.01 0.02 0.03 2 3 Y·

40..-------------, (c)

00 ,--- --------,, (f)

30 60

T 20 T 40

... 10 20

0 0'----0~.0-1 --0~.0-2---'0.03 oL----~--~--~ 0 3

Y·

Figure 6.3: Comparison between the approximate solution (line) and numerical calculations (squares): (a) azimuthal wavelength À, (b) amplitude A and (c) period Tas a function of 1. for ao = 0.01. The same for (d), (e) and (f) but now as a function of ao for 1. = 0.002.

between the linear approximation and the complete solution is very good for the range of 1 considered in this paper. For constant 1. = 0.002 (figures 6.3d,e,f) the agreement is only good for small values of ao. For the later analysis of transport in section (6.3) it is useful to note that the amplitude of the meandering path agrees within 5 % up toa value a 0 = 1.

The dipole's trajectories stuclied in previous paragraphs have all an oscillatory character: the orientation angle a varies periodically between the initia! direction of propagation a 0 and - a 0 . As in the ,8-plane case, depending on the parameter values and the initia! conditions, the dipole can enter the libration regime, where a increases or decreases indefinitely. To illustrate this behaviour a symmetrie dipole has been located initially on the pole (r0 = 0) . The dipole moves to a region of smaller ambient vorticity and acquires net positive relative vorticity,

6.3 Transport by a. mea.ndering dipole 107

12~-------------------,

(a) (b) 10

8

A 6

4

'Y·

Figure 6.4: The "libration regime" of the poip.t-vortex dipole: (a) an example of a trajectory (!. = 0.003 and r0 = 0); (b) amplitude A of the eycloid-type motion as a function of 1. (ro = 0).

which results in a cycloid-like path in anticlockwise sense, as can be seen in figure 6.4a. The radial amplitude of the motion decreases with increasing 1. as indicated in figure 6.4b.

6.3 Transport by a meandering dipale

A conple of oppositely signed point-vortices with constant circulations (i.e. not modulated) either move in a straight line or rotate with constant angular velocity. In both situations a constant area of fiuid is advected with the couple without change of shape. The shape and size of this region of fiuid depend solely on the ratio "-2/ "-I and the di stance d between the point-vortices. The latter is a constant of the motion with or without modulation of the vortex circulations. As described in the previous section, the roeanclering pathof the dipole is caused by the change of the vortex circulations: "-! decreases while " 2 increases (in absolute value) and vice versa. As a result the shape of the "trapped" area varies continuously, and due to conservation of mass, some fiuid masses are transported in turn from the interior to the exterior and reversely. This section addresses two main questions: how the amount of fluid entrained and detrained depends on the parameters of the problem; and where a fiuid partiele must be located to be entrained (or detrained) during the next period of the dipole's meandering path.

6.3.1 Adveetion equations

The stream function of the flow in a frame moving with the dipole is

(6.11)

..,

108 A dipolar vortex on a {-plane

where n· is equal to n except that it does not include the correction term (u /r) cos et; the equations descrihing the trajectories of fluid particles are

dx aw dt = 8y, (6.12)

This is a Hamiltonian system with IJ! playing the role of the Hamiltonian. In a steady flow, i.e. with a time independent 1l1, particles move along streamlines motions; therefore the partiele motion is integrable. Time dependent flows , however, can produce chaotic partiele trajectories, at least insome regionsof the flow. Using the stream function given by (6.11) the adveetion equations (6.12) become

dx

dt dy

dt

1 ("Iy- ~~:zY+] ,.... U -- --+-- +H y-27r L /+ '

-=-[~ + "zl- n·x 21r /_ I+ '

(6.13)

( 6.14)

where the definitions Y± = y±d/2 and I±= x 2 +Yl have been used. These equations, together with the equations descrihing the motion of the couple (6 .3)-(6.7), completely describe the motion of particles in the velocity field of a meandering dipole on the 1-plane. For most of the numerical calculations and the analysis that follows this form of the equations is suitable. However, for the computation of the Melnikov function, it is necessary to express the equations in the form of a periodically perturbed integrable Hamiltonian system:

dx dt = !1 (x, y) + {.gi[x, y, r(t; 7., eto) , et(t ; {., a0)],

dy dt = fz(x, Y) + 1.9z[x, y, r(t; 1., eto), et(t ; {., eto)].

The functions J; and g; are given by

!2

9I

where

Ko( 2 Y_ 2a Y+ 2(1+a)y 1) - --------+ --27r 1 - a L 1 - a I+ d2 ( 1 - a) d '

Kox ( 2 1 2a 1 2(1 +a)) 2;- l-aL +l-al+ -d2(1-a)'

1 ( ~~:1"Y_ ~~:2"Y+ ) - ------+ny-u 21r /_ I+ " " '

_.:_ ( 11:!-y ll:z-y - n ) 21r L + I+ " '

n" u"

r 2 - r~ - d(r cos Ct- ro cos eto),

r 2 - r~ + d(r cos Ct- r 0 cos eto),

2(r2 - r6)

d2 -(r cos Ct- r0 cos eto) .

This representation is exact.

(6 .15)

(6.16)

( 6.17)

(6.18)

(6.19)

(6.20)

( 6. 21)

(6.22)

(6.23)

(6.24)

6.3 Transport by a meandering dipale 109

(b)

(c) (d) B

p.

Figure 6.5: (a) Streamlines of an unperturbed point-vortex dipole. The ratio of the vortex circulations is "-!/"-2 = -1.1 and it corresponds to an ETD rotating steadily at a distance r0 = 10 to the pole. (b) The heteroclinic tangle in the perturbed case. The thick line is the unstable manifold (the observable structure in flow visualization) and the thin line is the stabie manifold. (c) The transport mechanism in the heteroclinic tangle (see text). Region ABCDis mapped to A'B'C'D'. (d) Region abcdis mapped to a'b'c'd'.

6.3.2 Lobe dynamics

A significant simplification of the description of partiele motion is achieved by using the Poincaré map -the map of the partiele location [x(t0 ),y(t0 )] to the location one period later [x(to + T), y(t0 + T)]. This map is constructed by sampling the position of a partiele, relative to the meandering dipole, every time the dipole returns to its initia! configuration, i.e. every time r = r 0 and a = a 0 .

The streamline patterns of the stationary flow (for 1. = 0) are illustrated in figure 6.5a. This field corresponds toa steady ETD (a0 = 0) for the case r 0 = IOd, where dis the distance between the point vortices. There exist two hyperbolic fixed points P+ and P- conesponding to the front and rear stagnation points of the dipole, respectively. The line that approaches p_ as t ~ +oo, is called the stabie manifold, and the line that emanates from P+ (i. e. approaches P+ ast~ - oo), is called the unstable manifold. In this unperturbed field the unstable manifold of P+ and the stabie manifold of P- coincide and correspond to the separatrix. There are

'!'


additionally three elliptic fixed points: two correspond to the positions of the point vortices and one correspond to the centre of rotation of the asymmetrie dipole. The last point is not shown here because it lies at a distance 10d from the dipole's centre, which is much larger than the size of the structure; therefore the flow around the dipole is almost parallel (note the small curvature of the boundary between the two dipole halves). The separatrix divides the flow in three regions: the free flow region, the positive-vortex core where particles rotate anticlockwise, and the negative-vortex core where particles rotate clockwise. The cores are trapped and travel permanently with the dipolar vortex in the unperturbed case. Note that the fluid areas trapped by the two vort i ces are not equal; the difference, however, is small ( the streamline patterns resembie the symmetrie case illustrated in figure 4.4a).

For 1. -:f 0, but sufficiently small, the fixed points persist and the unstable manifold of P+ smoothly emanates from P+ as before, but in this case undergoes strong oscillations as it approaches p_. The structure that results from the intersection of the manifo!ds of the two hyperbolic points is called a heteroclinic tang/e (figure 6.5b ). The intersecting manifolds ex pose the mechanism for transport of fluid between the interior and the exterior of the vortex dipole in the following way. Let A and C be two adjacent intersections between the stabie and unstable manifolds, and B a point of the stabie manifold and D a point on the unstable manifold. N ote that the area ABC D in figure 6.5c maps to the area A' B' C' D' . If the border between the fluid trapped by the cyclonic vortex and the ambient fluid is defined as P+C along the unstable manifold of P+ and Cp_ along the stabie manifold of P- then the area ABC D represents the fluid that will be entrained into the cyclonic vortex in the next cycle, whereas the dotted area near A represents the fluid that will be detrained. Since the flow is incompressible, the area entrained is equal to the area detrained in every cycle.

Similarly, the tangle formed by the unstable manifold of P- and the stabie manifold of P+ gives rise totransport of fluid between the cyclonic and the anticyclonic vortices (figure 6.5d), the border between these regions being defined as P+C along the stabie manifold and cp_ along the unstable manifold. In this case the area abcd is mapped to a'b'dd', i.e. moves from the cyclonic vortex to the anticyclonic one; and the same amount of fluid (the dotted area in figure 6.5d) leaves the anticyclonic vortex and enters the cyclonic one.

The exchange of mass can be evaluated directly from the discrete set of points defining the manifolds. Once a single lobe is identified the area follows from p = f xdy along e.g., ABCCDA. This method is valid for every amplitude of perturbation 1.·

6.3.3 Melnikov function

The behaviour of the stabie and the unstable manifolds can be predicted, without explicitly solving the adveetion equations (6.3)- (6.7), using the Melnikov function. This function is, up toa known normalization factor, the first-order term in the Taylor expansion about 1. = 0 of the distance between the stabie and the unstable manifolds. The Melnikov function M(t0 ) is defined as

M(to) = ;_: {fi[xu(t)]gz[xu(t), r(t + to; 1., ao), a(t + t 0 ; /. , a 0 )]

fz[xu(t)]gi[xu(t) , r(t + t 0 ; 1., a 0 ) , a(t + t 0 ; 1., a0 )]}dt , (6.25)

where xu(t) = [xu(t),Yu(t)] is the partiele trajectory along the separatrix of the unperturbed dipole. The Melnikov theorem shows that a simple zero of M(t0 ) implies a transverse intersection of the stabie and the unstable manifolds (see, e.g., Rom-Kedar et al. 1990), while one


intersection implies the existence of infinitely many intersections of the manifolds (i.e. a heteroclinic tangle). Therefore, the Melnikov function gives a specific criterion for the existence of (a) exchange of mass between different flow regions and (b) chaotic partiele trajectories, in termsof the system parameters a 0 and 1 •.

One can also obtain an 0(1.) approximation for the area of a lobe by using the Melnikov function (Rom-Kedar et al. 1990). The area of a lobe is given by

(6.26)

where t01 and t02 are two adjacentzerosof the Melnikov function M(t0 ) (i.e. they correspond to adjacent intersections of the unstable and stabie manifolds).

Some symmetries

Note that if the partiele trajectory along the separatrix is chosen in such a way that xu(t = 0) = 0, the following symmetries hold for the time independent components of the velocity field:

JI[iu(t)] = !2[xu(t)]

!I[xu( -t)], - h[iu( -t)].

Then the Melnikov function M(to) is equal to zero for all to = t~ such that the time-periodic components of the velocity field have the same symmetry

gt[iu(t), t~ + tj 92[iu(t), t~ + t]

gtfxu( -t), t~- t], = -g2[xu( -t), t~ - t].

These symmetries are satisfied if r( t0 + t) = r( t~- t) and a( t~ + t) = ±a( t0 - t), as can be seen in equations (6.19)-(6.24). Thus t0 must correspond to an extreme radial displacement in the motion of the dipole. In the linear approximation t~ = (n + 1/2)T /2 for n an integer and T the period of the dipole's meandering motion (see the appendix at the end of this chapter). M(t0 ) has thus an infinite number of isolated zeros, two for every period of the perturbation. Similarly, for the complete system of equations there exist an infinite number of t0 for which M(t0) = 0 and 8M(to)/8t0 =/:- 0 for all 1. =/:- 0 and a 0 =/:- 0,1r. The zeros correspond again to the maximum radial displacement in the dipole's motion.

6.3.4 Numerical results

Transport in the oscillatory regime

The heteroclinic tangle as wel! as the entraioment and detrainment lobes shown in figure 6.5 are schematic drawings used to illustrate the lobe dynamics. The structure of the intersecting manifolds for any value of 1. and a 0 in the parameter range used here is more complicated. The main differences are that (i) the manifolds are "Jonger" (measured along the unperturbed separatrix) due to the large perturbation periods and that (ii) any choice of the boundaries between different regions has little resemblance with those of the unperturbed case: they

1


(a) (e)

(b) (f)

(c) (g)

(d) (h)

Figure 6.6: The detrainment Jobe for constant ao = 1 and 1.= (a) 0.008, (b) 0.014, (c) 0.02 and (d) 0.026. The detrainment Jobe for constant 1. = 0.02 and ao= (e) 0.2, (f) 0.8, (g) 1.4 and (h) 2.0. The dashed area is the fraction of the lobe that lies within the unperturbed separatrix.

either leave outside a large region of the initially trapped ftuid or include a large region of ambient fluid. The latter definition has been chosen here and it has motivated the use of an "effectively" detrained area, which wil! be discussed later in this section.

Figure 6.6 shows the detrainment lobe for constant a 0 = 1 and increasing 1.= (a) 0.008, (b) 0.014, (c) 0.020, (d) 0.026. The lobeis thin and long for smal!/. andreduces in length and increases in thickness with increasing /.· The area of the lobe increases with 1., but it occupies rather "external" areas of the cyclonic vortex half, indicating that a large portion of the core wil! remain trapped by the couple. Similarly, figure 6.6 also shows the detrainment lobe for constant 1. = 0.02 and increasing initia! orientation angles a 0 = (e) 0.2, (f) 0.8, (g)

6.3 Transport by a meandering dipale

(a)

Y·

(b)

3 / 1

I 2

113

0.01 0.02 0.03

Y·

Figure 6.7: (a) Area (Ik) of the lobe detrained from the cyclonic half to the arnbient ftuid computed using the Melnikov function. Contour interval 0.025, Ik is zero along both axes. (b) Exchange ra te (Ik* = Jk/period). Contour interval 0.005.

1.4, (h) 2.0. The lobe increases in length and in thickness with increasing a 0 . Obviously the lobe area also increases. The lobe now "penetrates" closer to the positive point vortex, thus reducing the size of the positive core.

Figure 6. 7 a shows the amount of fluid exchanged between the cyclonic half and the ambient fluid during one oscillation of the dipole computed using the Melnikov function. The area is expressed as a fraction of the area trapped by each vortex in the unperturbed case (see, e.g., figure 6.5a). The initia! angle is varied from 0 to 71', and 1. is varied in the range 0-0.03. The area of the lobe increases with both increasing 1. and a 0 . The area is zero for a 0 = 0: this initia! condition corresponds to the stabie equilibrium (ETD) and no change of circulation occurs in the couple, and therefore no change of the form of the separatrix. For a 0 -+ 71' the area of the lobe does not go to zero, since a 0 = 71' corresponds to the unstable equilibrium. The area of the lobe tends toa finite value which depends on 1 •. For 1.=0 the lobe area is also zero because then there is no variation of the circulation of the vortices.

Figure 6.7b shows the exchange rate (i .e. the amount of fluid that is exchanged per unit time) between the cyclonic half and the ambient fluid. This rate is obtained by dividing the lobe area (figure 6.7a) by the period of the dipole's oscillation (i.e. the perturbation period). The rateis zero for 1.=0 and a 0 = 0, where the lobe area is zero, but also for a 0 = 71', where the period of the oscillation goes to infinity. Therefore for every value of 1. f:. 0 the exchange rate has a maximum and this occurs at the same value of a 0 ~ 1.9, within the resolution of our calculations.

The sections a 0 = 1 and 1. = 0.02 in the parameter plane have been chosen to compare the results of Melnikov calculations and the area of the lobe computed by direct numerical integration of (6 .13). Both methods agree very wel!, as should be, for smal! values of both 1. and a 0 (figures 6.8a,c). For largervalues the Melnikov function underestimates the amount of fluid that is detrained . The exchange rate increases in an approximately linear manner with

114 A dipolar vortex on a {-plane

0.8 0.08

(a) • • (b)

0.6 0.06

IL IL 0.4 0.04

0 0

0.2 0.02

0 0 2 3 2 3

IXo

0.8 0.08

(c) (d)

0.6 0.06

IL IL OA 0.04

0.2 0.02 0 0 0 0 0

0 0 0 0

0 0 0

0 0

0

0 0 0

0 O.Q1 0.02 0.03 0 0.01 0.02 0.03

'Y· 'Y·

Figure 6.8: Lobe area (p,) and exchange rate (p,*) computed using the Melnikov function (solid line) and direct numerical integration of the adveetion equations (squares). 1.=0.02 for (a) and (b ); a0 = 1 for (c) and (d). The dots represent the area of fiuid "effectively" detrained (see text).

/. (figure 6.8d) and shows a maximum for some a 0 (figure 6.8b). Both methods show this maximum although they are slightly shifted, the one of direct numerical integration occurring for a greater value of a 0 •

The mass exchange between the two dipole halves (ttM ), and that between the negat ive vortex and the ambient ftuid (pL) show the same behaviour as the one described above for the exchange between the cyclonic half and the ambient ftuid (ttu). It has been found numerically that these areas are approximately proportional in the parameter region considered here:

1.51ttM < /1U < 1.6lttM,

0.985ttL < /1U < l.015ttL·

Figure 6.8 also shows the effective area of ftuid (ttE) detrained during the first period of the dipole's meandering motion. This quantity is defined as the intersection between the detrainment lobe and the surface limited by the unperturbed separatrix (see the dashed areas in tigure 6.6). Approximately 40 % of the detrainment lobe lies within the initia! separatrix.

6.3 Transport by a meandering dipole 115

In the range of parameters stuclied here the effectively detrained area varies m the range: 0.35J.LM < J.LE < 0.47J.LM·

The amount of fluid detrained in one period does not depend on the period as it does in the case of a vortex pair in an oscillating strain flow (Rom-Kedar et al. 1990), or in the dipole with prescribed change of size stuclied by Cremers & Velasëo Fuentes (1994). The most likely reason is that on the 1-plane the perturbation period is, for the parameter range considered here, much larger than the typical time scale of dipole propagation: the length of the dipole's trajectory during one oscillation is at least several times the distance between the point vortices .

Here the dominant factor is the amplitude of the perturbation, measured by the term 1.A, where A is the amplitude of the radial displacement. The fluid area detrained in the first period varies in the range:

4.88/.A < J.LE < 5.44/.A (6.27)

This can be understood in the following way: the amount of detrained fluid should be related to the difference S between the area enclosed by the unperturbed separatrix and the area enclosed by the separatrix at the position of maximal asymmetry of the dipole. If the perturbation period is of the same order or greater than the orbit period "close" to the separatrix, then most of the fluid located outside the current separatrix wil! be advected to the dipole's wake. However, for a small perturbation period a significant portion of this fluid will be recaptured during the same oscillation of the dipole.

The shape of the separatrix depends on the ratio E = -/'i,2 / "' 1 and the distance between the vortices d, which is a constant. In chapter 4, S was shown to be given by S ~ 0.15(é-1) for 1 < E < 2. On the 1-plane the extreme val u es of é are reached at the pos i ti on of maximum displacement in latitudinal direction, and the most important term in determining the amplitude of the strength's perturbation is 2ro/.A, as can be deduced from (6.4)-(6.5) . The ratio between the circulations becomes E ~ 1 + 4ra/.A, teading to S ~ 0.6ra1.A . The latter relation compares well with (6.27), which was obtained using r0 = 10.

The same argument explains the almost constant ratio between the amount of fluid exchanged by the two dipole halves and the fluid exchanged betweenone half and the ambient fluid (approximately 1.56 according to the Melnikov function calculations). Numerical computations using the asymmetrie separatrices give a ratio of 1.6.

Transport in the libration regime

The same calculations have been performed using somewhat different vaJues of r0 (but all within the oscillating regime) and the overall behaviour is the same. The mass exchange slightly increases with growing r0 , specially for large values of both 1. and o-0 . The larger exchange of mass is aresult of the larger local gradient of ambient vorticity 2rol·

Let us briefly discuss how transport is affected when the dipole enters the "libration" regime, using as an example a symmetrie dipole initially located on the pole (r0 = 0 and o-0

not defined). Except for its periodic returns to the pole, the dipole is always located in areas where the background vorticity has a smaller value than at its initia! location (figure 6.4a), consequently the positive-vortex' circulation varies between its initia! value and some larger value, while the negative-vortex's varies between its initia! value and some smaller value (in magnitude).

116

(a)

(b)

(c)

' ' ' /

' /

1 A dipola.r vortex on a (-plane

Figure 6.9: (a) The entrainment lobe (solid line) for 1. = 0.03 and ro = 0, the broken line represents the initia! separatrix. (b) The entrainment lobe (see a) after half a period, the broken line shows the instantaneous separatrix. (c) The entrainment lobe (see a) after one period, the broken line shows that the instantaneous separatrix has returned to the initia! (symmetrie) shape.

The exchange of fluid therefore occurs as follows: (i) as the dipole moves equatorwarcis (during the first half of the period) fluid is entrained into the cyclonic half (mainly ambient fluid but also fluid from the anticyclonic half) and fluid is detrained from the anticyclonic half; (ii) as the dipole moves back to the pole (during the second half of the period) fluid is detrained from the cyclonic half, the larger fraction being ambient fluid captured during the previous stage; and fluid is entrained into the anticyclonic vortex. The core of fluid that remains trapped by the positive vortex is larger than the core of fluid carried by the negative vortex.

This process is illustrated in figure 6.9. The solid line in figure 6.9a shows the en trainment


0.2 .-------------------------------.

0.15

~ 0.1

. 0.05 ............. "·······

...............

0 0 0.01 0.02 0.03

'Y·

Figure 6.10: Lobe area J.L (thick line) cornputed by direct numerical integration of the adveetion equations (6.13)-(6.14) for dipoles in the "libration" regime. The broken line represents the fraction of J.L that enters the anticyclonic vortex and the thin solid line the fraction that returns to the arnbient fluid.

lobe ("the region of fluid that will be entrained in the next period") and the dotted line shows the shape of the initia! separatrix. The same lobe is shown after one half period in figure 9b, the whole lobeis contained in the new (asymmetrie) separatrix and it occupies the outermost regions of the cyclonic half. When the dipole returns to the pole only a smal! fraction of the "entrainment" lobe remains within the cyclonic half: most has been captured by the anticyclonic half and a fraction has returned to the free flow region (figure 6.9c).

The area of the entrainment lobe increases with /., as shown in figure 6.10 (thick solid line). The fraction of this lobe that is entrained into the anticyclonic vortex (broken line) shows the same behaviour, and the fraction that is returned to the ambient fluid remains approximately constant. A negligible amount of fluid is actually trapped by the cyclonic vortex, as can be seen in figure 6.9c.

6.3.5 Long time spread of particles

The previous section was devoted to the determination of the amount of fluid (lobe area) that is exchanged between different regions of the flow during a single oscillation of the meandering dipole. The evolution of particles for Jonger periods is explored in this section. The motivation is that, on the 1-plane, all dipole solutions return to regions of the plane they have visited before, and for some particular initia! conditions the dipole returns exactly to its initia! position. This has consequences for the spreading of particles. On the ,8-plane for example, once a patch of fluid has been detrained it remains in the wake of the dipole without undergoing much deformation anymore (see chapter 4). On the 1 ·plane, on the other hand, detrained

l 118 A dipolar vortex on a 1-plane

fluid can be recaptured again as the dipale returns to that region of the plane. Three particular situations are considered: (a) a dipale with an 8-shaped trajectory and

zero net zona! drift, (b) a dipale that returns exactly to its initia! position after one rotation around the pole and eight oscillations around its equilibrium latitude; and (c) a dipale that returns exactly to its initia! position after two rotations around tbe pole and seventeen oscillations around its equilibrium latitude. Particles were initially placed on a regular array within the detrainment lobe of the positive vortex ("dark" particles) and the negative vortex ("bright" particles), and their positions were sampledat times nT, where nis an integer and T is the period of the dipole's meandering.

In the case of an 8-shaped trajectory (a) one observes alternating bands of ambient and interior fluid, and the particles are spread in latitudinal and in westward direction over distances larger than the scale of the trajectory of the dipale itself (indicated by a white line in figure 6.11a). There is a net westward transport of fluid in spite of the dipole's zero drift. A net transport was also observed for the 8-shaped trajectory on the ,8-plane (see figure 4.9b), and this phenomenon was explained in the following way. As the dipale moves northward, the positive vortex (which accupies the west side of the couple) becomes weaker and detrains fluid, while the negative vortex (which accupies theeast side) becomes stronger and entrains fluid. On the other hand, as the dipale moves southward, the negative vortex (on the west side of the couple) becomes weaker and detrains fluid , while the positive vortex (on tbe east side) becomes stronger and entrains fluid . In both cases the vortex located on theeast entrains fluid while the vortex located on the west detrains fluid, resulting therefore in a net mass transport in westward direction.

Th ere is a major difference in the distri bution of particles between cases (b) and ( c). In (b), 16 braad islands of stability (larger than the dipole) arise: 8 correspond to the nartbernmost point and the rest to the 8 soutbernmost points of the dipole's trajectory, indicated by a white line in figure 6.11 b. The particles are distributed forming bands. Both dark and bright particles are well spread in radial direction.

For case (c) the islands have almast completely disappeared. Both species of particles occupy a band in radial direction with amplitude camparabie to the amplitude of the meandering motion (white line). The particles have a braader spreading in!atitudinal direction in case (b) but they have a better azimuthal spreading for case (c) .

6.4 Conclusions

A single dipale on the 1-plane perfarms a meandering motion around circles of equal ambient vorticity when its initia! direction of propagation is not parallel to these lines. The equations of motion of the point-vortex couple can be reduced to an autonomous system of two ordinary differential equations, and they are therefore integrable. In all solutions the dynamically relevant variables, namely the direction of propagation and the distance to the pole, are periodic. Th ere ex i st two trivia] solutions: (a) the eastward rotating dipale corresponds to a stabie equilibrium, small perturbation of the initia! condition leading to meandering motion with a small amplitude; and (b) the westward rotating dipale is an unstable equilibrium, and a small perturbation of the initia! conditions willlead to looping motions with large latitudinal displacements. Due to spatial variations of the gradient of ambient vorticity, the amplitude of the latitudinal displacement and the azimuthal drift are larger to the north than to the south of the equilibrium latitude.

6.4 Conclusions 119

Figure 6.11: Positions occupied by fluid particles after a series of iterations of the Poincaré map. The dark particles were originally located within the cyclonic dipale half and the bright ones within the negative half. (a) The 8-shaped trajectory without zona! drift (a0 = 2.257232), the positions of the particles after 5, 10, 15, and 20 periods are displayed. (b) A dipale that returns exactly to its initia! position after one rotation around the pole ( a0 = 1.021516), the particles' positions aft er 57-64 periods are displayed. ( c) A dipale that returns exactly to its initia! position after two rotations around the pole ( ao = 1.136182), the particles' positions after 57-64 periods are displayed. In all cases/. = 0.01 and the white line is the dipole's trajectory.


The adveetion equations for the dipole's velocity field can be exactly written in the form of a periodically-perturbed integrable Hamiltonian system. Relatively recent techniques for the study of transport in two-dimensional maps (lobe dynamics) are then used to study entrainment and detrainment of fl.uid during the dipole's meandering motion.

The amount of mass exchanged during one period of the meandering motion increases with both increasing 1. and o:0 • The lobe area is independent of the perturbation period, and it is approximately proportional to the product 1.A (A is the amplitude of the latitudinal displacement). The rate at which area is detrained has a maximum for some critica! value o:0 :::::: 1.9. This value is the same for the range of 1. considered in this work and within the resolution used to explore the parameter space.

The dipoles in the 1-plane propagate in a limited region of the plane and periodically return (close) to areas they have occupied before. As a consequence; they are able to stir the fl.uid more efficiently than for example dipoles on the ;1-plane. A major difference in spread of particles exists between dipoles which are periodic in the azimuthal variabie and dipoles which are not. In the former case the Poincaré map shows broad areas of unstirred fl.uid coinciding with the maximum-radial displacementsof the di pole's meandering trajectory.

Appendix: Melnikov function for the linear system

Using the same approximations leading to (6.9)-(6.10), the perturbation to the circulations and the angular velocity are given by

1t2-y = 2ro6.r, 4ro

-df6.r,

where 6-r = -(Uoo:o/w)sinwt, and the perturbation to the linear velocity is zero. These relations lead to the following periodic componentsof the velocity field (g1):

1 ( y_ y+ ) - -2ro(- + -) + y 6-r , 21r L I+

92 = x ( 1 1 4ro) - 2ro(- + -)-- 6-r . 211" L I+ d2

Defining g; = g'; 6-r, and using the symmetries present in the equations [!I, Y± and I± are even and h and x are odd functions of t for the choice xu(t = 0) = 0] , the Melnikov function becomes

M(to) = -(Uoo:o/w)coswto 1: {fdxu(t)]g;(xu(t)]- h[xu(t)]g~[xu(t)]} sinwt dt.

The integral is nonzero for all values of o:0 . M(t0 ) has therefore an infinite number of simple zeros [i.e. M(to) = 0 and 8M(t0 )/at0 i= 0] in every point of the parameter region stuclied here (0 < 1. < 0.03 and 0 < o: < 1r).

Chapter 7

Unsteady behaviour of a tripolar vort ex

7.1 Introduetion

In the realm of coherent vortices the tripole follows the monopolar and the dipolar vortices in increasing order of complexity. This relatively new vortex structure can be defined as a compact, symmetrie linear arrangement of three patches of continuously distributed vorticity, with the central vortex being flanked at its Jonger sicles by two weaker vortices of oppositely signed vorticity. This symmetrie configuration performs a steady rotation as a whole in the direction defined by the circulation of the central vortex.

The emergence of a symmetrie tripole from an unstable monopolar vortex was originally mentioned by Leith (1984), who subjected a minimum enstrophy vortex to a random asymmetrie perturbation. This transition from a monopolar toa tripolar vortex has been observed later both in laboratory experiments (van Heijst & Kloosterziel 1989; van Heijst et al. 1991; Flór et al. 1993) and in numerical simulations (Carton, Flierl & Polvani 1989; Orlandi & van Heijst 1992). Tripoles have also been observed to arise spontaneously in two-dimensional turbulent flows (Legras, Santangelo & Benzi 1988) and by a collision of two misaligned dipolar vortices (Orlandi & van Heijst 1992). Recently, a tripolar vortex has been observed in the ocean ( Pingree & LeCann 1992), viz. in the Gulf of Biscay. In this case the tripolar structure consistedof an elongated anticyclonic central vortex and two smaller cyclonic satellites; it had a horizontal dimension of 50-60 km and rotated with a period of eighteen days.

Van Heijst et al. (1991) stuclied the generation processof the tripolar vortex in a rotating homogeneaus fiuid from the instability of a cyclonic vortex, and the further evolution of the symmetrie tripole. They compared flow measurements (stream function and vorticity distribution) with a point-vortex tripole and found that the main features shown by the laboratory tripole, namely, the overall rotation of the structure and the topology of the streamlines in the co-rotating frame, are wel! described by the model. Flór et al. (1993) observed the emergence of a tripolar vortex from the instability of a monopolar vortex generated in a non-rotating linearly stratified fluid. This structure, however, existed only for a short period before the destruction of the satellites and the subsequent transformation into a new monopolar vortex.

Carton et al. (1989) numerically stuclied the instability of isolated monopolar vortices (i.e. monopoles with zero net vorticity) subjected to a mode 2 perturbation and found that the evolution depends on the steepness of the vorticity profile. For vorticity gradients in

121

122 Unsteady behaviour of a tripolar vortex

an intermediate range the monopolar instability leads to the formation of a tripolar vortex. Similarly, Orlandi & van Heijst (1992) numerically stuclied the instability processof an isolated monopolar vortex with a similar smooth vorticity distribution. They subjected the monopole toa random perturbation, and the mode 2 component was observed to dominate the evolution . This scenario is in agreement with the experimental situation, where the vorticity profile is smooth (van Heijst et al. 1991) and it is to be expected that the perturbations do not contain any preferred mode, but the most -unstable one finally dominates.

The tripole has been observed to he a stabie and steady structure both experimentally and numerically: it can exist for many rotation periods without noticeable change of shape or size. However, it has been recently observed (van Heijst & Velasco Fuentes 1994) that when the tripoleis generated off-axis in a rotating fiuid, the structure rapidly loses its symmetry and the individual vortices show a complicated quasi-periodic motion. This behaviour produces a continuous stretching and folding of (dyed) ftuid patches which were initially located within one of the vortices; a feature that suggest the existence of chaotic motion of ftuid particles. A similar unsteady motion of the vortex eentres has been observed in numerical simulations by Carton & Legras (1994), although in their case the unsteady motion and the asymmetry appear gradually after several rotations of the structure.

The observation that quasi-periodic motion rapidly arises when the tripole is generated off-axis, as well as the numerical experiments showing the long steadiness of the tripole in the two-dimensional case, suggest that the parabolic free surface of the rotating ftuid is the cause of the unsteady behaviour. In this chapter we use a modulated point-vortex model to support this hypothesis. A non-modulated asymmetrie point-vortex tripole, which displays some of the features observed in the experiments and in the modulated model, is used as a paradigm for the study of the motion of the vortices and the adveetion of ftuid particles.

7.2 Laboratory observations of an unsteady tripole

In the experiment described here the tank was filled up to a height ho = 17 cm, and the angular speed of the table was n = 0.566 s-1 ; therefore the difference in water depth between the axis of rotation and the ciosest walls (in a 100 x 150 cm rectangular tank) was 4 mm. This parabolic free-surface topography produced a value of 1 = S13 jgh0 = 0.109 m-2s- 1 (see chapter 2).

A tripolar vortex was generated by stirring the ftuid cyclonically in a bottomless cylinder of 19 cm in diameter, placed in the rotating ftuid at 30 cm from the rotation axis , then quickly lifting this cylinder. The flow was visualized using ftuorescein, which was added to the ftuid af ter stirring it and before the generating cylinder was removed. An isolated monopole (i.e. a vortex with a single rotation centre and zero net vorticity) was thus produced in the uniformly rotating ambient fl.uid. This vortex became unstable, resulting in the gradual formation of a tripolar vortex. When the formation process of the tripole is completed (figure 7.1a) the two satellites are located at equal distances from the central vortex and the three vortices are located on a straight line. The structure rotates as a solid body and, due to the offcentre location of the tripole, the satellites are alternately squeezed and stretched as they rotate around the central vortex. Note that, as a satellite rotates around the central vortex , it encounters a difference of about 1.5 mm in fiuid depth between its locations of minimum (20 cm) and maximum ( 40 cm) separation form the tank centre, in this particular experiment. This seemingly small value turns out to have a major effect on the evolution of the structure. After

7.2 Labora.tory observa.tions of a.n unstea.dy tripole 123

Figure 7.1: Plan-view photographs showing the evolution of a tripolar vortex in a rotating free surface fiuid ( experimental parameters ho = 17 cm, n = 0.566 s-1 , R = 30 cm). Pictures were taken at time t = 0.9 (a), 5.4T (b), 7.2T (c), 9T (d), 10.8T (e), 12.6T (f), 14.9T (g), 18.46T (h), and 22.1T (i); with the rotation period of the tableT = 11.1 s.

l 124 Unsteady behaNiour of a tripolar vortex

about one rotation of the satellites, a small asymmetry begins to develop and the vortices lose their linear arrangement (figure 7.1b). After this instant the central vortex moves away with one of the satellites (figure 7.1c), but this newly formed coupleis asymmetrie (the circulation of the central vortex equals the sum of the circulations of the satellites, in absolute value) and moves along a curved trajectory (figure 7.1d). The collision of the couple with the satellite left behind (figure 7.1e) results in an exchange of partners and the process repeats itself, although now the looping excursion of the new coupleis larger (compare figures 7.1g and 7.1e); and, as the couple re-encounters the satellite left behind, the vortices are arranged in an almost rectangular triangle (figure 7.1i).

Another remarkable feature observed in this experiment is the complicated distri bution of dye. Each time the central vortex pairs with one of the satellites, the new couple leaves behind a tail of dye that joins it with the remaining satellite (figure 7.1c). As the couple returns, the tri po lar structure traps the ambient ( undyed) fiuid located between the couple, the satellite and the tail of dyed fiuid (figure 7.1d). Subsequently, as the central vortex changes partner, this ambient fiuid is wrapped around the new couple (figure 7.1e). This process occurs every time the central vortex changes partner and leads to the formation of alternating filaments of dyed and undyed fiuid, as is revealed by a close examination of the tail joining the couple and the monopole in figure 7.1h,i. It is clear that a patch of fluid that at some stage enters the tail, is later subjected to repeated stretching and folding; this is an indication that the adveetion of fiuid pareels in some flow regions is chaotic.

The following sections are devoted to a detailed study of the unsteady motion of the vortices and the adveetion of fluid parcels. This is clone using point vortices, which give a good agreement with experimental observations in the steady case (van Heijst et al. 1991).

7.3 Vortex motion

7.3.1 A non-modulated point-vortex tripole

Three point-vortices display a large variety of regimes of motion as the vortices' strengths and relative positions are changed . It therefore can not be a surprise that the three vortex problem has attracted the attention of scientists for more than a century. In fact, it was already known to Poincaré ( 1893) that the motion of three vort i ces with arbitrary circulations is integrable. However, the actual reduction of the motion to quadratures has been the subject of numerous works (Gröbli 1877, Synge 1949, Novikov 1976, Aref 1979).

Our interest in the motion of three point vortices sterns from laboratory observations of the tripolar vortex. Therefore the range of parameters reduces considerably: the set of vortices is assumed to have total zero circulation, with the strengths of two of them ( the so-called 'satellites') being equal ( -K0 ); as has have been measured van Heijst et al. (1991) in laboratory experiments in a rotating tank. The motion of a set of point vortices with these charaderistics has been stuclied by Rott (1989), who showed that it can be effectively reduced to an adveetion problem in a two-dimensional steady field (the phase space), whereas Aref (1989) showed that this reduction is possible for all sets of three point vortices with vanishing total circulation.

7.3 Vortex motion 125

(a) (b)

81 c d c 82

a E 9

d 81

·- ---··-··-~·-- - 82

Figure 7.2: (a) Initia! conliguration of the point-vortices in the asymmetrie tripole with E respresenting the asymmetry; (b) general contiguration of the system: a is the distance between satellite 1 (SI) and the central vortex ( C), b is the distance between satellite 2 (S2) and C, and () is the angle between the lines joining S1 and S2 with C (measured in anticlockwise sense from S1 to S2 ).

The following is detined as the "standard" contiguration at time t = 0. The three point vortices are located on the y axis (tigure 7.2); the vortices with circulations -K0 are called satellites, the one located at y = d being denoted by 5 1 and the one at y = -d by 5 2 • The central vortex ( wi th circulation 2K0 ) is located at y = t: and will be denoted by C. Th is choice does not represent a loss of generality, since more general initia! configurations reach this particular one in the course of their evolution. Those configurations can thus be transformed into the "standard tripole" after an appropriate rotation of the coordinate system and by rescaling of time and length. The triangular contiguration that translates without change of shape represents an asymptotic state for some critica! value of E, as will be discussed below. The motion of this system of point vortices on an infinite domain is given by the Hamiltonian equations

dy; aH K·----

' dt - OX;, (7.1)

with

for N = 3, where r[1 =(x;- Xj)2 + (y;- Y1) 2

• The first integral of motion is the Hamiltonian itself, which does not depend explicitly on time. Three more integrals can easily be obtained (see, e.g., Batchelor 1967) from invariance with respect to translation and rotation

N N N

Q = LK;x;, p = LK;y;, I= L K;(x7 + yf). i=l i=l i=l

1 126 Unsteady behaviour of a tripolar vortex

(a) (b)

9 57tl4 -------_;-'0..--------------------- -----------

b 0'-------'----~--'---------'

0 0.5 1.5

Figure 7.3: Behaviour of the asymmetrie point-vortex tripole: (a) trajectoriesof the point vortices C (thick line), St (thïn line) and S2 (dotted line); (b) evolutïon of the <listances a (thick line) and b (thin line) and the angle (} (dotted line) . The calculations were performed for f = 0.4.

These conserved quantities have the following values for the initia! configuration defined above:

Q = 0, (7.2)

PI = 2~~:ot, (7.3)

Ir -2K0 (J2- t2), (7.4) K2 2d

H1 - ___Q_ In . (7.5) 21!" (d-t)2(d+t)2

In the non-modulated case the vortex circulations are constant, therefore the absolute positions of the vortices are not important, the distances between the vortices are thus suftkient to describe the state of the system (Aref 1979). It is more convenient though, to choose the distance a between C and St, the distance b between the C and S2 , and the angle 0 between the line joining the central vortex with the satellites (figure 7.2b). This choice allows a distinction between two configurations where the point vortices are located at the vertices of similar triangles but in different order.

An example of the vortex trajectoriesof an initially asymmetrie point-vortex tripole ( E = 0.4) is shown in figure 7.3a, where the vortices' initia! and final positions are indicated by filled and open dots, respectively. The structure has a net translation in the positive x direction, which is the case for allE > 0 (for E < 0 the translation occurs in negative x direction) and the satellites rotate periodically around the central vortex C. Figure 7.3b shows the time evolution of the distances a (thick line) and b (thin line), and the angle (} (dotted line); the time is scaled by the rotation period Tof the symmetrie tripole; i.e. T = 81r2J2 /3~~:0 . The distance a(t) has a minimum when b(t) has a maximum, and vice versa, i.e. they have a phase shift of half a period. At these instants the vortices are arranged in a straight line [B(t) = 1r], as is indicated by the two aligned arrows in figure 7.3b. When the vortices are located at the vertices of an


(a) (b)

6

• 1cY u/U

(1)/(l 4

101

2

10°

0 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8

E E

Figure 7.4: (a) Frequency wfrJ and (b) mean propagation speed u/U of the tripole contiguration shown in figure 7.1a as a function of the initial perturbation e of the position of C. These quantities are scaled by the frequency n = 3/î,o/ 47rd2 of the symmetrie tripole ( € = 0), and the velocity u = "'o/ 41rd of the symmetrie dipole ( e = 1 ).

isosceles triangle (a = b) the angle reaches an extreme value (maximum or minimum), as is indicated by the single arrow in figure 7.3b. The amplitude of the perturbation of both a(t) and b(t) is c, as will be proved below.

As shown by Rott (1989), three regimes of motion may arise as a function of the initia! asymmetry c (Rott uses the more general definition b = P/2,..o, which is equal to c for our particular choice of initia! configuration): (i) for c --+ 0 the three vortices are separated by distances of the same order during the whole evolution; (ii) for c --+ 1 the central vortex is close to S1 while S2 permanently remains at a much larger distance; and (iii) for c » 1 the two satellites are close together while the central vortex remains far from them. We consider here how the frequency, the mean speed and the perturbation amplitude of a, b and () vary in the range 0 < c < 1, i.e. only in the first two regimes of motion. The critica! value of é

at which the change form regime (i) to regime (ii) takes place is also briefly discussed. In the discussion below, frequencies have been scaled by the angular frequency n = 3"'o/47rcf of the symmetrie tripole (c = 0), veloeities by the velocity U = "'o/47rd of a symmetrie dipole ( c = 1 ), and lengths have been scaled by d.

Tripole regime

The tripole regime occurs in the range 0 < é < 0.544. For all c -:f 0 the vortices C and S 1 move initially in the same direction. These can be considered as forming a "couple", with S 1 rotating around C relatively fast in comparison with the rotation of vortex S2 • Due to its asymmetry, the couple C-S1 moves towards S2 , and an exchange of partner takes place. The new couple


(a) (b) (c)

~IÄ s,

1• ·I s2

2EJ3112

Figure 7.5: lntermediate stages for different regimes of motion: (a) tripole regime: at timet= T/4 the vortices form an isosceles-triangle configuration; at this point c pairs with s2, leaving SI behind; (b) for the critica! value f = 0.544 the vortices move asymptotically towards an equilateral-triangle configuration; and ( c) dipole-monopole regime: for f > 0.544 the three vortices reach a new linear configuration at t = T /2, but now C occupies an extreme position. T is the period of the motion.

C-S2 moves initially away from Slt but it is also asymmetrie and moves back to re-encounter the monopole left behind. A periadie motion, with C pairing in turn with SJ and S2 , arises. The angular frequency of this motion increases slightly from 1, and reaches its maximum value at f ~ 0.49, see figure 7.4a. A further increase of f leads to a large decrease of the frequency. The tripole's mean translational speed increases continuously from zero at f = 0; for small f

the increase is slowandit is faster as € approaches its critica! value (figure 7.4b). In this regime the vortices reach an isosceles-triangle configuration after one quarter of

a period (figure 7.5a). The vortices SJ and S2 are located at the ends of the triangle's base, which has a lengthof 2À. The vortex Cis located at (x, y) , at a distance {j from the base of the triangle. The integrals of motion uniquely determine the size and shape of the configuration and the value of the coordinate y, whereas x remains undetermined. The integrals of motion are given at this stage by

Py = 2Koa,

]y -2Ko(b2- 2ay + a2

),

11:2 2b Hy = _ _QJn .

27r ( a2 + b2)2

Conservation of P, I and H during the whole evolution implies that

(7.6)

(7 .7)

(7.8)

where d and € are known. The height of the triangle is thus equal to f and the length 2À of its base can be obtained from the third equation. Once À is known, the position y of the central vortex follows from the second equation.


Dipole-monopole regime

Beyond the critica! point (0.544 < t < 1), one finds the dipole-monopole regime. C and 5 1

can be considered as being a couple since the distance between them is much smaller than that between C and 52 • The couple thus rotates due to its own asymmetry and propagates due to the presence of the third vortex (52 ), which in turn shows an average translation with a wobbling perturbation superimposed. The frequency in this range is again a growing function of f (thick line in figure 7.4a). An empirica! approximation of it is given by the rotation frequency of an asymmetrie dipole formed by vortices C and 51 (assuming that 52

has no infl.uence on this couple). The angular frequency of this asymmetrie dipole is given by f2D = "-o/27r(1- t)2 , which approximates the frequency of the three vortex system very wel! for f--+ 1 (the broken line in figure 7.4a) .

The mean translation speed of the three vortices decreases with growing t (thick line in figure 7.4b ). An empirica! approximation for this speed is obtained by defining a virtual dipole, which is formed by vortex 52 and a vortex with the same total circulation as the asymmetrie couple C-51 located at the centre of rotation of this couple. The virtual dipole so defined moves with a velocity given by UD = "-o/47rt (represented by the broken line in figure 7.4b) , which approximates the velocity of the three vortices very wel! as t --+ 1.

Since the central vortex C does not change partner in this regime, the vortices never reach an isosceles-triangle configuration. Instead, they take positions on a straight line after one half period (figure 7.5c), but now the central vortex C occupies one end of the linear configuration. Satellite 51 is separated by a distance TJ from C, and 52 by a distance Ç . The integrals of motion at this stage are given by the following expressions:

which lead to

h = "-o(TJ + Ç), h = -K-o( -2y(ry + Ç) + T)

2 + e), "5 ç -7]

HL - 47r ln (712 + Ç2)2,

y =

2t , e - 2tÇ - d2 + 3t2

2t

As before, f and d are known and these three equations are sufficient to determine the configuration of the vortices. The value of x can not be determined from the integrals of mot ion.

Equilateral triangle

At the critica! value Ec ~ 0.544, the initiallinear configuration transforms asymptotically into an equilateral triangle with sicles 2tc/V3 (see tripale regime). The two negative vortices are located on the base of the triangle and the positive vortex on the top (see figure 7.5b). This triangle translates at a constant speed Uc = 3K-0 /47rE (indicated by a square in figure 7.4b) without change of size or shape. This configuration is unstable, i.e. corresponds to a fixed

l


point of hyperbolic type in the phase space (Rott 1989). Since it takes an infinite time to reach a hyperbolic fixed point ( or to escape from i t) the frequency of the system is zero. The frequency is larger than zero (and the velocity smaller than Uc) in figure 7.4 because calculations have been done close, but not at the critica! point.

The asymmetrie contiguration discussed in this section reproduces some of the features observed in the experiments; in particular, it renders the essentially time dependent behaviour of the tripole, which goes in succession through linear and isosceles-triangle configurations. However, it fails to reproduce the growing amplitude of the asymmetry; which is in fact approximately zero as the tripoleis formed. In other words, this model describes the evolution of an asymmetrie tripole, but it does not explain how the asymmetry arises. The following section intends to provide an answer to the latter issue.

7.3.2 A modulated point-vortex tripole

Laboratory experiments like the one discussed in section 7.2 have shown that the evolution of the tripoleis essentially influenced by its initia! separation from the axis of rotation . This suggest that the unsteadiness of the tripole is caused by the parabolic free surface of the rotating fluid in which the vortex is generated. As the satellites rotate around the central vortex the nonuniformity of the fluid depth produces a continuous stretching and squeezing of fluid columns; which, having to preserve potential vorticity, alternately gain and lose relative vorticity. These effects can be introduced in the point-vortex model by modulating their circulation on the basis of conservation of mass and potential vorticity on this topographic 1-plane (see chapter 2):

(7.9)

where 1. = 1rL21 (with L the radius of the area associated with the 'point'-vortex), and r;o is the initia! distance of the point vortex to the tank centre, at which the vortex has strength ~C;0 . Due to this modulation, the absolute positions of the vortices essentially determine the evolution; the three parameters used in the previous section are thus not enough to uniquely describe the state of the system. Six parameters determine the absolute positions of three vortices, but due to the rotational symmetry of the 1-plane, only five are needed to describe the state of the system. These are chosen to be the distances a and band the angle 0, which define the relative positions of the vortices, and the distance c from the pole to vortex s2 and the angle a between this line and the line joining S2 and C (figure 7.6b). The evolution of the point-vortex tripole on the 1-plane is stuclied as a fundion of two parameters: the value of 1. (which is equivalent to varying the rotation rate of the tank, and therefore the curvature of the free surface) and the initia! position of the tripole, expressed by the distance R of the central vortex to the pole (i .e. the rotation axis) . Initially the vortices are located on a straight line and the satellites are separated by the same distance d from C (figure 7.6a). In all cases discussed here the linear arrangement is tangent to the circle r= R at the position of C (figure 7.6a).

The evolution of the point-vortex tripole can be described as follows . As the satellites rotate around C, S1 moves towards a region of larger ambient vorticity, this negative vortex thus has to become stronger (more negative), while S2 moves towards areasof smaller ambient vorticity, becoming weaker (less negative). The veloeities induced by the satellites on the central vortex do not cancel each other anymore and C begins to move in the direction of the stronger satellite (St) . Although initially small, this asymmetry is enough to give a net


(a) (b)

c a

d a

c

R d

Figure 7.6: (a) Initial conditions of a symmetrie point-vortex tripole on a 1-plane. (b) General contiguration of three point vortices on a 1-plane. The parameters a, b and 8 are detined as in tigure 7.2, cis the distance from s2 to the pole and a is the angle between the lines joining s2 to the pole and to C.

drift to the tripole, which shifts polewards. This shift increases the asymmetry, which in turn strengthens the tripole's drift (as has been shown insection 7.3.1 for the non-modulated case). However, because of the circular shape of the ambient-vorticity isolines, the northwestward drifting tripole reaches a maximum northward displacement and then continues its drift but now in southward direction. The asymmetries decrease and the tripole approximately reaches the initia\ symmetrie state as it approaches its initia\ latitude. A typical set of trajectories is shown in figure 7.7a, where the initia! northwestward motion can be clearly observed as well as the growth of the asymmetry. However, as the structure moves away from the pole the asymmetries decrease. The evolution of each of the parameters a, b and (} show the same behaviour (figure 7.7b). Two characteristic periods can be defined in these t ime series: (i) the short period is defined as the time elapsed between two adjacent local maximum (or minimum) values, this period depends on /. , R and the pair of extreme values under consideration (i.e. the rotation period of the satellites around C changes during the evolution of the tripole); and (ii) the long period is the time elapsed between two returns of the tripole to its initia! latitude (e.g., figure 7. 7a shows vortex trajectories during approximately one long period), this period depends only on the values of 1. and R.

The high frequencies (i.e. the inverse of the short period) displayed by the tripole for different initia! positions are shown in figure 7.8a. Every rotation of the sate llites around the central vortex gives rise to one frequency, a number of which appear during each long period; for example, in the numerical experiment shown in figure 7.7a sl and s2 rotate approximately eight times around C : there are thus eight points on the vertical line R = 3 in figure 7.8a. Here, the solid line represents the frequency of the unperturbed tripole, which is taken as unity. It is obvious that for all initia! conditions the first rotations of the satelli tes around C occur at a frequency close to the frequency of the unperturbed tripole. As the tripole moves poleward the frequencies are lower due to the combined effects of a larger separation

132 Unsteady bebaviour of a tripolar vortex

(a) (b) 3n12 ~---------------,

(\ 71 /\ (\

'J ; :

V V 2

0

0 2 4 6 8 10

Figure 7.7: Modulated point-vortex tripoleon the 1-plane: (a) typical trajectoriesof the individual vortices c (thick line), SI (thin line) and s2 (dotted line); (b) evolution of the parameters a (thick line), b (thin line) and (} (dotted line). The calculations were performed for 1. = 0.01 and R = 3.

of the satellites from the central vortex, and the lower strength of C and the higher strengths of S1 and S2 • As R increases the frequency band broadens; but the higher limit remains approximately constant, while the lower decreases in a linear manner.

In addition, the extreme values of the perturbation of the distance a(t) between S1 and C, as a function of the initia! position R, are shown in figure 7.8b. The filled dots represent tÎmax = ( amax- d)/ d and the open ei rel es represent tÎmin = ( amin - d)/ d, where amax and a min

are the maximum and minimum values of a(t) during the tripole's evolution, respectively. The graph shows that, as R increases, tÎmax grows linearly; while ÎZmin slowly decreases as R becomes non-zero but then remains approximately constant in the range of R considered here.

Similar effects are observed if the initia! position of the tripole is kept constant and the value of 1. is varied. For 1. = 0 the unperturbed symmetrie tripole is recovered; but with increasing 1. the band of frequencies broadens (figure 7.9a) . The broadening occurs towards the lower frequencies, as in the case of varying initia! position. The value of tÎmax grows in a nonlinear manner with 1. ( see figure 7 .9b ), whereas tÎmin is approximately constant (but smaller than the unperturbed value) for the range of parameters stuclied here.

We conclude this section with some remarks about the integrability of the motion of a modulated point-vortex tripole. If the numerical experiments are allowed to continue for several long periods then more high frequencies appear, although all of them lye within the same band displayed during the first long period (e.g., figures 7.8a and 7.9a). This results suggests that the motion of the modulated tripoleis integrable in the parameter region ex plored in this chapter (0 < R < 4 and 0 < 1. < 0.04) . However, for largervalues of R the three point vortices have been observed to enter the dipole-monopole regime for one period (i.e. at least one linear-configuration event occurred with C not located between the satellites) and then they went back to the tripole regime. These regime changes might suggest the emergence of chaotic behaviour of the system.

7.3 Vortex motion

(a)

lllij ~<:<~::=.::.:~~::::':::": • I" • ,• "·:" "•:

::.:.··· •• 0

0.8 ··;.: ...

0.5

.. . ·:::•''

.. .. ·····. ··.::

R

. ... '

...

a

(b) 2,.---------------,

.·· .·

Of----------------1 ... ···································

R

133

Figure 7.8: (a) Frequencies exhibited by a(t) during the motion ofthe modulated point-vortex tripale on the 1-plane and (b) Ûmax (filled dots) and Ûmin (open dots), as a function of the initial position (R) of the tripale (and fora constant 1. = 0.01). The solid lines give the conesponding values for the unperturbed tripole.

(a)

1

III!H1 111·:· · i!:: ,; : 0.8 'i

0.5

: : : : 1 : : J : ~ ! • • •

·:.·: .. · .. a ••• • ••• 0

• •• 0

••• • 0

•••• •• 0 0

• I !! : I::: • • • o • •

••• J ... ······ ·

'Y·

(b)

2r-------------------------

... ... .. .·

... ... . ... ... .....

01--------------------------1

······································

-1 0 0.01 0.02 0.03 0.04

'Y·

Figure 7.9: The sarne as in figure 7.8, but now as a function of 1. and for constant distance to the centre R = l.S.


+ (a) of=······· ... (b)

10an

Figure 7.10: (a) Observed trajectoriesof the vortex eentres of the tripolar vortex shown in figure 7.1 and (b) numerical simulation of case (a) using three modulated point vortices. The central vortex C is represented by a thick line, s1 by a thin line and s2 by a dotted line.

7 .3.3 Comparison of experiment al and numerical results

The trajectory described byeach vortex in the laboratory experiment discussed insection 7.2 is displayed in fi.gure 7.10a. The thick line represents the central vortex, the thin line represents S~> and the dotted line represents S2 . Two features are clearly seen in this figure: (i) the increasing amplitude of the cycloid-like motion of the central vortex, and (ii) the westward drift of the structure as a whole (the cross indicates the tank's centre, which is the pole of the topographic 1-plane ). Both effects can also be seen in the evolution of the parameters a, b, and () (figure 7.1la) . In this figure the two aligned arrows indicate that extreme values of a and b occur at a linear configuration ( () = 1r), whereas the single arrow indicates that an isosceles-triangle configuration (a= b) is reached when the angle ()takes alocal extreme value, as in the point-vortex models described in previous sections.

A numerical simulation was performed to compare with these experimental observations. The value of 1 and the initia! positions of the point vortices were chosen to correspond to that of the experimental situation. The circulation of the vortices was estimated in the following way. A mean angular speed of the tripole was computed using

where f!max is the observed initia! anguJar speed (0.33 S-1 ), T is the duration of the exper

iment (300 s), and the value of er (0.013 s-1) was computed using the results of van Heijst

et al. (1991), who measured an exponential decay of the angular speed of the steady tripolar vortices. This decay is most likely due to the Ekman bottorn layers. It was further assumed that initially the tripole had zero net circulation and the circulation of the two satellites was equal (see van Heijst et al. 1991). The radius of the circular area associated to the point

7.4 Adveetion by an unsteady tripale

(a)

31t/2 31t/2

... ",.........._ ......... 8 1t ... (_·~ ___ _;:_·._ ...... ... .· - - - -,:- - - - - - -

··· .... .,···· · ...... ··· ...

1t

·-... 2 ·-. 2

0

0 5 10 15 20 25

(b)

0 5 10 15

8

' ' ' ---'-----

'

20

' ' ' '

b

25

135

Figure 7.11: Evolution of the parameters a(t), b(t) and 9(t) for: (a) the Iabaratory experiment shown in ligure 7.10a; and (b) the numerical simulation shown in ligure 7.10b.

vortices was d0 /2, where do = 11 cm is the initia! distance from the satellites to the central vortex.

The resultant vortex trajeetori es (figure 7.10b) show growing cyeloids and a net northwestward drift of the structure, as observed in the experiment. Figure 7.llb shows the evolution of the parameters a, b and 8 in this simulation, which shows a good agreement with the observations (figure 7.lla), as can be expected from the good agreement between the observed and the computed vortex trajectories. There is, however, a time shift between the two time series: the numerical tripole evolves more slowly. This is probably due to the use of a large a

for the decay of the angular speed; this a was chosen as an average of the values measured by van Heijst et al. (1991) .

7.4 Adveetion by an unsteady tripole

The complicated dye patterns observed in figure 7.1 are the result of tbe essentially timedependent motion of the tripole. This behaviour is also displayed in the two models described in sections 7.3.1 and 7.3.2, namely: (i) an asymmetriclinear arrangement without modulation, (ii) an initially symmetrie tripole with modulation. The non-modulated point-vortex tripole is time periodic and the techniques used to study partiele motion by a dipolar vortex on the {3- and 1-planes can be straightforwardly applied. The modulated tripole, on the other hand, is quasi-periodic and more general definitions of the Poincaré map and the Melnikov fundion are needed for the study of partiele motion and mass transport; however, only the Poincaré map will be used here.

Although the non-modulated model does not reproduce the growing asymmetry obse~ved in the laboratory situation, we believe that a good understanding of the partiele motion

i


(a) (b)

d

d

2e

Figure 7.12: (a) Trajectoriesof the point vortices in a frame fixed in space (E=0.4); (b) the same trajectoriesin a frame moving with the central vortex and rotating with the frequency of the satellites ( see text ).

according to this model helps addressing the more complicated quasi-periodic case. For this reason, we will first discuss the transport charaderistics of the non-modulated tripole model, before consiclering the effects of the modulation.

7.4.1 Adveetion by an asymmetrie point-vortex tripole

We first consider the motion of fiuid particles in the velocity field of an asymmetrie tripole. This is most conveniently clone by choosing a reference system in which the point vortices are either at rest or move periodically around an unperturbed position. The choice madehereis a system that moves with the time-dependent velocity ü(t) of the central vortex C, and rotates with the average angular velocity w(t) of the satellites around C. It will prove convenient to write the velocity and the angular speed in the following way

ü = (t:U',t:V'), w = f!+ t:f!',

(7.10)

(7.11)

where t: is the initia! shift of the central vortex C, and nis the angular speed of the symmetrie tripole. In these equations the primed variables are essentially time dependent, and the factor t: stresses the fact that they appear only if the structure is asymmetrie (t: -1- 0) . Note that w is given (in nondimensional form) in figure 7.4a, whereas the velocity given in figure 7.4b is the average value of U' over one period of the tripole's motion. Figure 7.12a shows the trajectories of the point vortices in a frame fixed in space for the case € = 0.4, while the same trajectories in the co-moving frame are shown in figure 7.12b. In this moving system the satellites rotate around the unperturbed position (0, ±d) in clockwise sense. Note that the paths are circumscribed by a square of sicles €.

7.4 Adveetion by an unsteady tripale

The equations of motion of a fluid partiele in the moving frame are given by

dx

di dy dt

137

(7.12)

(7.13)

where the definitions X; =x- x;, Y; = y- y; and I;= Xl + Y/ have been used. The positions of the vortices are given by (x;,y;), with i= 0 for the central vortex and i= 1,2 for satellites 51 and 52 , respectively.

These equations, together with the evolution equations (7.1), govern the motion of particles in the velocity field of an asymmetrie tripole. This form of the equations is suitable for most of the calculations; however, for the computation of the Melnikov function the equations must be written in the form of a periodically perturbed integrable Hamiltonian system:

~; = f 1(x, y) + t9I[x, y, i 1(t; t ), i 2(t; t)],

~~ = h(x, y) + t92[x, y, x1(t; t), i 2(t; t)],

(7.14)

(7.15)

where i; = (x;, y;) is the position of satellite i, which can be written in the following form:

x· . = Y1

Y2 =

tÇ;,

d + fTJJ, -d- fTJ2·

(7.16)

(7.17)

(7.18)

The perturbed form (7.14)-(7.15) is obtained by substituting (7.16)-(7.18) in (7.12)-(7.13) and expanding these equations in a Taylor series around t = 0. The zeroth and first order terms give the functions f; and 9;:

!I = K-o ( y_ y+ 2y) n --+---+y 2n I_ I+ k ' (7.19)

h - K-oX ( J_ + J_ - ! ) - f!x 2n I _ I+ I0 '

(7.20)

91 = - LTJI + 2L(xÇI + LTJI) I+TJ2 + 2Y+(x6- Y+TJ2) f!' U' r: + n + y - , (7.21)

92 LÇl- 2x(xÇI + y_TJJ) I+6- 2x(x6- Y+TJ2) n' I

E + n - x-V, (7.22)

where the new definitions Y± = y ± d and h = x 2 + Y,f have been introduced, and Io is defined as before.


(b)

(c) B (d)

·····--........... .

Figure 7.13: (a) Stream function of the unperturbed point-vortex tripole in a frame rotating with the structure; (b) heteroclinic tangles of the perturbed tripole; (c) fluid exchanged between 51 and the ambient flow region; and ( d) fluid exchanged between S2 and C.

Lobe dynamics

In this time-periodic case a significant simplification of the description of partiele motion is achieved by using the Poincaré map -the map of the partiele location [x(t0 ),y(t0 )] to the location one period later [x(t0 + T), y(t0 + T)]. Loosely speaking this corresponds to sampling the position of a partiele every time the tripole returns to its initia! contiguration (i.e. (} = 1r,

a= d- f. and b = d + f.).

The streamline patterns of the stationary tripole ( f. = 0) are illustrated in figure 7.13a. There exist two hyperbolic fixed points P+ and P- connected by a collection of orbits ( partiele trajectories) that approach P- as t ----+ +oo, called the stabie manifold, and a collection of orbits that emanate from P+ (i.e. approach P+ ast----+ -oo), called the unstable manifold. In this unperturbed case the unstable manifold of P+ and the stabie manifold of P- coincide and correspond to the separatrices. There are additionally three elliptic fixed points corresponding

7.4 Adveetion by an unsteady tripole 139

to the positions of the point vortices. The separatrices divide the flow in four regions: the free flow region, where particles simply rotate in clockwise sense around the tripole; the satellite cores, where particles rotate anticlockwise around one of the satellites; and the positive core where the particles rotate clockwise. The cores are the areas of trapped fluid that rotate as solid boclies with the tripole.

For E =f 0, but sufficiently small, the fixed points persist and the unstable manifold of P+ smoothly emanates from P+ as before, but in this case undergoes strong oscillations as it approaches P-· Similarly, but for t --> -oo, the stabie manifold smoothly emanates from P- and undergoes strong oscillations as it approaches P+ . The structure that results from the intersection of the manifolds of the two hyperbolic points is called a heteroclinic tangle (figure 7.13b ), and indicates how transport of fluid between different flow regions occurs. The points A and B, which are two adjacent intersections between the stabie and unstable manifolds, are mapped to the points A' and B'. Correspondingly, the dotted area close to B in figure 7.13c maps to the dotted area close to B'. If the boundary between the fl.uid trapped by satellite S1 and the ambient fluid is redefined as P+B along the stabie manifold and Bp_ along the unstable manifold then the area bounded by segments of the stabie and unstable manifolds between A and B represents the fluid that will be entrained into S1 in the next cycle. Since the flow is incompressible, an equal area is detrained in the same period. Similarly, the tangle formed by the stabie manifold of P- and the unstable manifold of P+ gives rise to transport of fluid between S1 and the central vortex C (figure 7.13d).

Finally, the exchange of mass can be evaluated directly from the discrete set of points defining the manifolds. Once a singlelobeis identified the area follows from J.l = f xdy along, e.g., AB-BA.

Melnikov function

Without explicitlysolving the adveetion equations (7.1), it is possible to predict the behaviour of the stabie and the unstable manifolds using the Melnikov function. The Melnikov function M(t0 ) is defined as

M(to) i: {JI[i,.(t)]g2[x,.(t),i1(t + t0 ; E) , X2(t + t0 ; E)]

h[i,.(t)]gl[i,.(t), i1(t + to; t), i 2(t + to; t)]}dt, (7.23)

where i,.(t) = [x,.(t),y,.(t)] is the partiele trajectory along the separatrix of the unperturbed tripole.

The Melnikov theorem shows that a simple zero of M(to) implies a transverse intersection of the stabie and the unstable manifolds (see, e.g. , Rom-Kedar et al. 1990), while one intersection implies the existence of infinitely many intersections of the manifolds (i.e. a heteroclinic tangle).

One can also obtain an O(t ) approximation for the area of a lobe by using the Melnikov function , which is a first-order approximation of the distance between the stabie and unstable manifolds. The area of alobeis given by the following relation (Rom-Kedar et al. 1990):

(7.24)

where t01 and to2 are two adjacent zerosof the Melnikov function M(t0 ) (i. e. they correspond to adjacent intersections of the unstable and stabie manifolds ).


Some symmetries

Note that if the partiele trajectory along the separatrix is chosen in such a way that x,.(t = 0) = 0, the following symmetries hold for the time independent components of the velocity field:

h[x,.(t)J h[xu(t)]

Jdx,.( -t)J, - h[xu( -t)].

Then the Melnikov fundion M(t0 ) is equal to zero for aJI t 0 = t~ such that the time-periodic components of the velocity field have the same symmetry:

gl[xu(t), t~ + t] 92[xu(t), t~ + t]

These symmetries are satisfied if

Ç;(t~ + t) = V'(t~ + t) = TJ;(t~ + t) U'(t~ + t) =

91[x,.( -t), t~- t], -g2[xu( -t), t~- t].

-Ç;(t~- t), - V'(t~ - t), TJ;(t~- t), U'(t~- t),

which implies that t0 must correspond toa linear configuration of the three point vortices (see figure 7.12b). M(t0 ) has thus an infinite number of isolated zeros, two for every period of the perturbation.

Numerical results

Some of the lobes that change of region in the next period are given in figure 7.14a, which represents the case € = 0.1. The thick lines represent entrainment lobes of satellite 1 and the central vortex, while the thin lines represent detrainment lobes of the central vortex and satellite 2. The letters in figure 7.14a give more precisely the location of the lobe at time t = 0 and at t = T (with T the period of the tripole). For example, the lobe denoted by A-S1

is located in the ambient fluid and will be transferred to satellite S1 after one period. The corresponding lobes for the cases € = 0.2 and € = 0.3 are shown in figures 7.14b and 7.14c, respectively; although the notation has been omitted in these frames. It is obvious that the lobe area increases with increasing €.

Note that every lobe is to be entrained intoor detrained from a satellite, i.e. a satellite is involved in any mass exchange event, either with the central vortex or with the ambient ftuid. For this reason it is natura! to choose the area of a satellite in the unperturbed case (2.054d2

) as the unit area. In what follows all areas are given as a fraction of this unit area; for example, the ftuid area trapped by the central vortex, which is 58.5 % of that trapped by a satellite, is written simply as 0.585. In figure 7.15 the areas of the different entrainment lobes are given as a fundion of the initia! asymmetry €. The lines represent calculations using the Melnikov function, while the markers give the result of direct numerical computations. Note that the ambient fluid exchanges equal amounts of mass with S1 (squares) and with S2 (crosses). Similarly, the central vortex exchanges equal amounts with S1 ( circles) and S2

(triangles). This apparently surprising result can be explained by two facts: (i) the shape


(b) (c)

Figure 7.14: Entrainment and detrainment lobes of the perturbed point-vortex tripole for different asymmetries: € = (a) 0.1; (b) 0.2; and (c) 0.3.

of the manifolds (and therefore the lobe shape) depends on the time at which the Poincaré section is constructed; at the same time the Poincaré map is area preserving, due to the incompressibility of the flow field. In other words: the shape of the lobe changes from one Poincaré section to the other but its area is conserved; and (ii) t he position of St with respect to C at t == nT (with T the period of the perturbation and n an integer number) is the same as the relative position of S2 (with respect to C) at t == (n + 1/2)T. Since a lobe of S2 at t == 0 bas the samearea at t == T/2 (or at any other time), it has also the samearea as the entrainment lobe of St at t == 0.

The lobe area for large t is of order I (figure 7.15a). This does not mean that the whole core of St wil! he substituted by new fluid coming from the exterior region or the central vortex C. A large fraction of the lobe will in fact he found within C and S2 after one period. The mechanism for that is the intersection of the lobes corresponding to different regions.

Long time evolution of fluid particles

The evolution of particles after several iterations of the Poincaré map is invest igated in tbis section. The knowledge of the structure of the tangle helps to efficiently compute the evolution of particles. For smal! values of t particles were placed in the lobe S2-C , and every period it is determined in which region each individual partiele is located. The fraction of particles that are located in the different regions approaches a constant value (different in every region) as the number of periods grows (figure 7.16). In the range 0 < t < 0.25 this effect is more rapidly achieved with growing E, and in all cases it is completed by period 15. Similar results are observed if the particles are located initially in another lobe; with the same percentage of particles as in figure 7.16 (within an accuracy of 3 % in our calculations) being located in different regions after 15- 20 periods.


(a) (b)

0.8 0.8

... 0.6 0.6

.. ······ .......... .. ····

0.4

0.2

+ 0.4

0.2 0.4 0.6 0.2 0.4 0.6

Figure 7.15: (a) Lobe area I" as a function of € ( expressed as a fraction of the area trapped by a satellite in the unperturbed case). The lobe areasof S1 and S2 are indicated by asolid line, those of C are indicated by a braken line. The lines represent Melnikov calculations and the markers represent direct computations using the adveetion equations. (b) The corresponding exchange rate I"• =I" fT, according to the Melnikov calculations.

(c) 100~---------------.

80

80

%N

20

ij~=O~~i~OO~~~~i~oOO 0

0 5 10 15 20

n

80

80

%N

20

n

(a) 100~---------------.

80

80

%N

20 o0

o0

o 00 oo0°ooo0 0 oo

a 6 aaA6

aaAA6666 66 A.Qoo 9 0 oa o o 00 ooo8oo

0 0 10 15 20

n

Figure 7.16: Percentage of particles (N) located in the ambient fluid (crosses), C (squares), S1 ( triangles ), and s2 (ei rel es) after n periods, for different initia! asymmetries: € = (a) 0.05; (b) 0.15; and ( c) 0.25. A regular array ~f 3000 particles were initially located within the lobe that moves from S2 to C4 during the first period.

For € close to the critica! point (for both lower and larger values), the tangle has a complicated structure and the identification of the lobes becomes a laborious task. Therefore we turned to brute force calculations for the study of partiele motion in this range of the parameter €. Now we are interested in the general structures that arise in the Poincaré section rather than in an accurate description of the motion of species through the different flow regions. For this purpose particles were located in rectangular arrays close to the point vortices (fig-

7.4 Adveetion by an unsteady tripole

(a)

/=• particles - = e

\ =•

(b)

particles

143

(c)

.;?

. . 1 !} 3

Figure 7.17: Initia! location of tracers for brute force calculations: (a) in the unmodulated -but asymmetrie- case 1100 particles were located within each rectangle; (b) in the modulated -but initially symmetrie- point-vortex tripole N particles were placed within each region determined by an arbitrarily chosen streamline and the separatrix (N = 2000 within each satellite and N = 1400 in the central vortex); and (c) linear-configuration events of a point vortex tripoleon a 1-plane (the number indicates the rotations of the satellites around the central vortex, i.e. the period).

ure 7.17a) and their positions sampled after every quarter period (i.e. as the vortices are in a linear arrangement or as they form an isosceles triangle). However, we show here only the Poincaré sections constructed at a linear configuration of the vortices.

In the range 0 < t < 0.5 the structures governing the transport between the different vortex regions are qualitatively similar, but the si ze of the fiuid permanently trapped by the individual vortices ( cores) decreases with increasing c A larger area, which surrounds the three vortices, is also permanently trapped by the tripole, and the fiuid inside this region (but outside the cores) is strongly stirred. This extended region of trapped fiuid has an almost circular shape for small values of t, but it becomes more oval and decreases in size with growing t. It reaches its minimum size for t:::::: 0.49 (figure 7.18a) . Here, a new closed region of unstirred fiuid appears between the central vortex C and satellite S2 • A further increase of t leads to the formation of small lobe-like structures at the rear side of the oval region. These lobes greatly increase in size in the range 0.5 < t < 0.54, resulting in an opening and destruction of the oval region (figure 7.18b). Beyond the critica! point te = 0.544 the qualitative structure of the Poincaré section does not change, in spite of the change of regime (we refer here to the Poincaré section constructed at t = nT; in contrast, sections constructed at t = (n + 1/2)T change strongly). Notwithstanding the qualitative agreement, a remarkable difference does exist; namely, the almost complete destruction of the regions of permanently trapped fiuid around the satellites (figure 7.18c), although a large area of fiuid trapped by C still persists.


Figure 7.18: Positions of particles aftera series of iterations of the Poincaré map for: f = (a) 0.49; (b) 0.54; (b) 0.56; and ( c) 0.65. Th ree regular arrays of particles (1000 each) were placed within the reetangles indicated in figure 7.17a. "Bright" particles were initially located in the central vortex and "dark" particles were located in the satellites

For even larger values of € a new oval region of trapped fluid begins to form around the three vortices, and the area of permanently trapped fluid around the point vortices increases. This is illustrated in figure 7.18d for the case € = 0.65, where it can be seen that the core region of the 'independent' satellite s2 has greatly increased, and a new region of unstirred fluid appears between the couple C-S1 and satellite S2 • Lobe-like structures are present in the upper half plane, indicating that fiuid escapes from this region; whereas the lower half plane shows no detrainment of ftuid. It has been shown insection 7.3.1 that for € -+ 1, the three vortices translate like a dipolar vortex, with the couple C-S1 effectively being substituted by a virtual vortex having the same net circulation of this asymmetrie couple and located at its


--o :' . .. .... ,

····•····

Figure 7.19: Adveetion of tracers in the dipole-monopole regime (here E = 0.8). The contours of the patches at t = 0 (broken lines) coincide with streamlines of the virtual dipole ( see text ). The shape of the contours after two periods is shown by solid lines.

centre of rotation. The validity of this analogy for the adveetion of passive tracers has been also tested. Several fiuid regions that would be carried by the virtual dipole without change of size or shape have been placed in the velocity field of the three point vortices. For large values of the initia! asymmetry (t > 0.75) the samearea of fluid is carried by the tripole withalmost no change of shape, a feature that is illustrated in figure 7.19 for the case f = 0.8. The fiuid patches initially coincided with the (steady) streamlines of the virtual dipole, as indicated by the broken lines in figure 7.19. They were allowed to evolve with the three vortices during two periods, and the final positions of the curves are indicated by solid lines. lt is clear that the separatrix has not been substantially deformed, and the lines surrounding satellite 2 have not changed either. Only lines close to the couple C-51 show a strong deformation. This result indicates that there is a fixed volume of fiuid trapped by the tripole and that noticeable stirring takes place only in the vicinity of the couple C-Sh which could be considered as a periodically perturbed dipole. Partiele motion in this region can therefore be analyzed as it has been clone in previous chapters for the dipole on the {3- and 1-planes.

The break-up of the oval region of trapped fiuid around the tripole, as one approaches the critica! value fc ~ 0.544 from smaller or larger values of t, is most likely the result of the velocity reaching its maximum value at fc. Fort: close to fc the vortices translate almost as an equilateral triangle during a large fraction of each period. The fiuid lost by any of the vortices during this phase is rapidly advected to the wake of the structure and can not be recaptured when the central vortex makes a cycloid-like loop with one of the satel!ites.


7.4.2 Adveetion by a modulated point-vortex tripole

In sectioo 7.3.2 we showed that there are some parallels between the motion of an asymmetrie tripole (with the point-vortices haviog constant circulatioos) and a modulated tripole (being initially symmetrie but having circulations that are functions of position) . Both structures translate in the direction perpendicular to the line at which the vortices are ioitially positioned. The asymmetrie tripole moves in a fixed direction; the modulated tripole, on the other hand, has a direction of propagation that changes cootinuously. In both cases the mean speed during one period increases with the amplitude of the perturbation (see section 7.3.1); but this amplitude is constant in time for the asymmetrie tripole, whereas in the modulated case the asymmetry changes continuously.

The aoalogies exhibited by the motion of the vortices in these models might suggest that the adveetion of passive tracers also shows significant similitudes. This was verified by numerical experimeots performed in a few points of the parameter plane (1., R), an example of which is discussed below (for the case Î• = 0.01 aod R = 3). Fluid particles were located initially between the separatrix and the streamline at distance 0.2d from the separatrix in the satellites, aod the streamline at distance 0.125d from the separatrix in the central vortex (as indicated in figure 7.17b). Partiele positions are then plotted every period; i.e. every second lioear-coofiguration event. In the Poincaré sections the positions are translated and rotated in such a way that the vortex C lies in the origin of the coordinate system and the satellites S1 aod S2 are located at some distance on the y axis. Figure 7.17c shows the real point-vortex positions aftereach period (and the complete trajectories are shown in figure 7.7a).

As cao be seeo in figure 7.7b, the tripole undergoes a growing phase between period 1 and 4. After each period the distance between the satellites increases while the central vortex C pairs with S2 • Due to its increasing size, the tripole entrains fiuid at the rear side and advects it to the front side (see figure 7.20a, which shows the particles at period 2). The asymmetry iocreases until period 4 and in every period en trainment of fiuid occurs (figures 7 .20b,c). After period 4 the asymmetry begins to decrease and lobe-like structures are formed at the rear side; which indicates the detraioment of fiuid, in this case not from the vortices themselves but from the oval region of stirred fiuid that surrounds the tripole. At period 8 the tripole reaches its initia! latitude and becomes (approximately) symmetrie again. A new growing phase occurs between periods 8 aod 12, and the process repeats itself.

From the satellites themselves interior Huid is detrained during the first growing phase and the core of trapped fiuid reduces in size, but after the first growing phase (periods 1- 4) the core sizes do not change sensibly any further. The fiuid patches detrained from the satellites are well stirred, remaining in the surroundings of the tripole for at least one long period.

The tripole's asymmetry changes continuously in the course of the evolution; this implies that the distance between the satellites in two subsequent Poincaré sections is, in genera!, not the same. However, the difference between those distances is small if the 1-effect is small (i.e. if Rd{./"o ~ 1 and R = O(d)); this implies that the vortices have approximately their unperturbed circulation and the evolution duriog one period is determined by the contiguration at the beginning of that period. At the end of the period the separations have slightly increased though, and this new contiguration is the main factor in determining the evolution of the tripole in the new period. This argument suggests that the series of Poincaré maps generated during the evolution of a tripolar vortex for a single initia! condition in the parameter plane (1., R), approximately correspond with the Poincaré sections described in the asymmetrie (but non-modulated) case as a function of the parameter t.

7.1 Advection by all unstea.d_y tz"ipole 147

Figure 7.20: Adv~(:tion by the roo.-lulat~d point·vortex tripol~ on the ')'·plane: the graphs show the positions of particle~ iu different Poim3.rii sH:tiollS in a single numerical experiment. The calculations

were performd for R = 3, 1. = 0.01. The vorteiC tmjectories are show11 in figure 7.7a and th~ IH>sitions od the vortices at period 2··7 <'u'c shown in figur~ 7.17t.. "Bright" pi;!,rtitks were initially [of.atcd in the centra.! vortex while "dark" pa,·ticles were located in the satellites


(a)

(c)

Figure 7.21: Plan-view of the experimental tripolar vortex shown in tigure 7.1. Linear or isoscelestriangle configurations are observed to occur at t = 0 (a), 3.2T (b), 5T (c), 7.7T (d) and 10.4T (e) (with T = 11.1 s the rotation period of the table).

7 .4.3 Experimental observations

For the analysis of transport in the experimental situation we return to the experiment shown in figure 7.1. The distribution of dye at some particular instants in the evolution is analysed. These instants are chosen as those at which linear (0 = 1r) or isosceles-triangle (a = b) configurations occur.

7.4 Adveetion by an unsteady tripale 149

(a)

+ 20cm

(b) (c)

+ +

(d) (e)

+

Figure 7.22: Numerical simulation of the evolution of fluid initially trapped by the modulated pointvertex tripole. Calculations were performed using the same initia! conditions as those in the experiment shown in ligure 7.21 and results are shown as the point vortices reach the conesponding conliguration. The contours of the patches initially trapped by the symmetrie tripole are shown at t = 0 (a), 5.8T (b), 8.5T (c), 12.6T (d) and 15.8T (e); with T the timescale used in ligure 7.21.

The dye used for visualization was placed originally in the outer ring of the monopolar vortex. Therefore, as the tripole is formed, most of the dye becomes concentrated in the satellites, while the central vortex consists of undyed ftuid. This was done so because one satellite is always involved in the exchange of ftuid, either with the central vortex or with the


ambient fluid. Fluid can pass from one satellite to the other through the central vortex, and from the latter to the ambient fluid through one of the satellites.

Time t = 0 is detined as the instant at which the generation of the tripole was completed, and the rota.tion period of the table (T = 11.1 s) is taken as the unit of time. In tigure 7.21a one can see a fairly symmetrie structure, with St and S2 closely attached to C and the three vortices arranged in a linear manner. However, the tripole soon becomes asymmetrie and although at t = 3.2T the three vortices are again linearly arranged (tigure 7.21b), S2 has separated from C, while St remains attached to the central vortex, forming a dipole-like structure. Bands of fluid (both dyed and undyed) conneet this couple with the monopole at the front si de of the tripole ( which moves to the left in the pictures ), while fluid is being entrained between the couple and the monopole S2 at the rear side of the tripole. Due to its asymmetry the couple C-S1 moves back and collides with S2 at t 5T (:figure 7.21c). A large ring of undyed fluid surrounds satellite S2 at this stage. The central vortex C changes partner and moves away with vortex S2 , forming a compact couple. As the structure reaches a new linear contiguration at t = 7.7T the amplitude of the asymmetry has grown noticeably (tigure 7.21d). The band of fluid connecting the couple with the monopole is thicker and contains larger lobes of ambient fluid, and a large amount of fluid is being entrained at the rear side of the tripole. The couple C-S2 moves back and collides with vortex St at t = 10.4T ( tigure 7.21e). The kink of the tripoleis larger than in frame (c), and the ring of ambient fluid surrounding St is also larger than the ring around S2 in (c).

The same modulated point-vortex model used to simulate the motion of the vortex eentres (:figure 7.10b), is used now to simulate the adveetion of fluid patches. The initia} contours of these patches were chosen to he the separatrices of the unperturbed tripole. As in the experiment described above the contours are plotted when a linear or an isosceles-triangle contiguration occurs. In that sense the series correspond with one another, although not exactly in time (see section 7.3.3). All the main features observed in the laboratory experiment can be observed in the simulation. Namely: formation of bands of interior fluid at the front side and entrainment of ambient fluid at the rear side, with the amount of entrained fluid increasing as time progresses (compare tigure 7.22b with 7.22d or tigure 7.22d with 7.22e).

7.5 Conclusions

In this chapter we have stuclied the unsteady motion of a tripolar vortex on a rotating freesurface fluid. It has been observed in the laboratory that a tripole generated off-centre is initially symmetrie ( the three vortices are located on a straight line and the satellite vortices are equally separated from the central vortex) but it soon beoomes asymmetrie, which results in the emergence of a kink in the arrangement and the pairing of the central vortex with a satellite. The tripole then shows an essentially time dependent behaviour, with the central vortex pairing in turn with one of the satellites. A (non-modulated) point-vortex tripole shows the sametime dependent behaviour. As in the experiments this asymmetrie structure bas net linear momentum, but in contrast with the observations the motion is periodic and the asymmetries have a constant amplitude.

The squeezing and stretching of fluid columns due to the parabolic free-surface of the fluid (the so called 1-plane) is proposedas a mechanism for the enhancement of the asymmetry and the propagation of the tripole. The hypothesis is tested using a point-vortex tripole modulated on the basis of conservation of potential vorticity on the ')'-plane. This structure is initially

7.5 Conclusions 151

symmetrie, but due to the modulation it slowly becomes asymmetrie and, in addition to the rotational motion of the satellites around the central vortex, the structure as a whole begins to propagate towards the local northwest. The rotation period of the satellites around the central vortex is not constant, and the perturbation amplitudes increase as the tripole shifts polewards. The amplitude of the perturba.tion and the frequencies of the motion have been stuclied as a function of the initial position of the tripole and of the value of 1 •. The maximum amplitude increases with increasing Rand 1., and the range of frequencies also increases, the highest frequencies being in all cases approximately tha.t of an unperturbed tripole.

The motion of particles in the velocity field of the non-modulated point-vortex tripole has been stuclied using the "dynamica.l-systems theory" approach, and it was found tha.t for all valnes of f f 0 chaotic partiele motions appear and fluid is exchanged between the different flow regions. The amount of mass exchanged during one period increases with increasing e. Transport of mass in the modulated case is more complicated in the sense that it eau not be stuclied in a single two-dimensional map but with a series of two-dimensional maps. With the proper sealing of the lengths, the structure of every Poincaré map in this series agrees qualitatively with a particular e in the periodic case.

Chapter 8

Conclusions

In this the~is we have studied the bcha.viotll' of dipolar and tripolar vortices in the presence of gradients of ambient vorticity (;3- and ')'-ph\ncs). In particular, we have analysed the motion of the vortex centrt~s a.nd the chaotic advection arising from the unsteady vortex motion. The main resul l.s a.u~ enumerated at the end of chapters 3 to 7. Here we want to make some general rema1·ks.

The results about the dipole on a {3- and on a 1-plane suggest that, for any we<tkly varying awbit~nt vorticity, the dipok meanders around isolines of a.rnblent vorticity. A dipole travelling to the local east has a st;thle trajectory, whereas a. dipole moving in loca.l westward directioll has an unstable Lrajcdoty. As a result of the meandering motion fluid masses are exc.hangcd between the dipole and the ambient flow, the amount of which is proportional to the gradient of ambient vorticity and Lo the amplitude of the meandering path.

The vortex structut·es generated in the laboratory have an essentially continuous distribution of vorticity; furthermore, due to gradients of ambient vorticity (;3-elTed ), relative vorticity i:; generated all over the till¥ In spite of this, it lws been shown that a few (modulated) point vortic.es can simulate tbt; ob~erved flows very well. For example, there is good agreement between the motion of the vortex centres in experiments and simulations. But even more striking is the a.grct;ment between the advedion of patches of active fluid in the experiments and passive fluid in th~ modeL The good agreement a.rises probably because mainly low·valued vorticity i~ involved in detrainment and entrainment in the experimental situation (recall that the vorticity is zero along the separatrix, both in theoretical models like the Lamb dipole and in tl!!~ cxpe1·iments). Naturally, there exist some disagreements between the point-vortex model and experimental observation~; in those cases the vortex-in-cell method, which takes into account IJ1(~ continuous distribution of vod.icit.y as well as the generation of ambient vorti<:ity by using a large amount of point vortices, provides a bdkr agreement. This particular aspect of the numerical results can be interpreted as follows: "If the objed i~ to make the best possible simulation of a real flow, a large munbcr of degrees of freedom in the form of point-vortices are required. If, on the other hand, the object is to arrive at physical under· standing, it is desirabk to rep1·esent the motion with as few dcgtee~ of freedom as poo~ible." Where the plH·ases in italics substitute "forecast of a meteorological event" and "grid points'', respectively, in the original text by Charney (1963).

We conclude by mentioning some direct extensions of the work presented in this thesis. [n all cases discussed here, the lobe-~hedding mechanism was studied using point vortices; a. similar study should be done using vortices with continuously distributed vorticity. For example, in contrast with the point-vortex case, the motion of a medon on the #-plane is

15:3

r

154 Conclusions

not periodic (e.g., Zabusky & McWilliams 1982). This non-periodicity, in ad dition to the non-zero relative vorticity present in the whole field, makes an analytica! approach difficult. But a numerical study could be undertaken.

Our results about dipolar vortices on the Î-plane concern exclusively point-vortex dipoles. A study of the ETD and the WTD in this case might be worthwhile. In contrast with the /3-plane case, it is likely that the Rossby-wave field will be asym111etric in the north-south direction. Therefore the zona! motion of the ETD and WTD would be perturbed, and a meandering motion may arise.

The results of chapter 4 and 6 indicate that, in genera!, the motion of three point vortices with modulated strength is chaotic. This was shown , using two pah icular modulations (.Band Î-planes), for the restricted problem of two vortices and a passive particle. However, the modulated point-vortex tripole discussed in chapter 7 showed a regular behaviour in the parameter region explored here. The conditions fora possible transition to chaotic behaviour deserve further attention.

155

References

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Samenvatting 159

Samenvatting

Het onderzoek dat in dit proefschrift is beschreven is gericht op de dynamica van tweedimensionale wervelstrukturen in een omgeving met gradiënten in de achtergrondvorticiteit, en het advectieve transport dat een gevolg is van de beweging van deze wervels . De motivatie van het onderzoek komt voort uit de geofysische stromingsleer. In eerste benadering zijn atmosferische en oceanische stromingen tweedimensionaal. Dit komt voornamelijk door twee factoren: de rotatie van de aarde, en de geometrie van het domein waarin de stromingen plaatsvinden. De bolvorm van de aarde veroorzaakt een variatie van de achtergrondvorticiteit met de geografische breedte. Als de bestudeerde verschijnselen in een klein gebied plaatsvinden (ongeveer 100 km in noord-zuid-richting) kan de achtergrondvorticiteit als uniform worden beschouwd (f-vlak-benadering). Is de afmeting in noord-zuid-richting tussen de 100 en 1000 km, dan is de achtergrondvorticiteit te beschouwen als een lineaire functie van de geografische breedte ((3-vlak-benadering) . In de nabijheid van de polen is de achtergrondvorticiteit een kwadratische functie van de afstand tot de pool ( 1-vlak benadering).

De wervels die in dit proefschrift aan de orde komen zijn de dipool en de tripooL De dipool bestaat uit twee compacte gebieden met tegengestelde vorticiteit, en beweegt in de richting gedefinieerd door zijn eigen as. Als de twee dipoolhelften een gelijke sterkte hebben bewwegt de dipoollangs een rechte lijn; is een van beide sterker dan de andere, dan beschrijft de dipool een cirkelbaan. De tripool wordt gevormd door drie gebieden met vorticiteit van alternerend teken. In het symmetrische geval zitten de drie wervels op een rechte lijn met de centrale wervel exact in het midden en is de centrale wervel twee maal zo sterk als de satellieten.

De gebruikte methoden zijn zowel experimenteel, theoretisch als numeriek. De experimenten zijn gedaan met een tank gevuld met kraanwater, geplaatst op een roterende tafel. Gradiënten in achtergrondvorticiteit zijn bewerkstelligd door variaties in de diepte van de waterlaag. Visualisatie en metingen zijn uitgevoerd om de banen en circulaties van individuele wervels, alsmede de uitwisseling van massa, te bepalen. De theoretische analyse is gedaan met behulp van het gemoduleerde-puntwervelmodel; de modulatie van de puntwervelsterkte is gebaseerd op behoud van potentiële vorticiteit. Puntwervels zijn ook gebruikt in de numerieke simulaties. Twee soorten numerieke simulaties zijn uitgevoerd. (i) Contourkinematica: Met deze methode worden materiële lijnen, gedefinieerd door honderden passieve deeltjes, geadvecteerd door een paar puntwervels (maximaal vier), die de enige actieve deeltjes in de stroming zijn. (ii) Wervel-in-cel. Deze methode gebruikt duizenden actieve deeltjes om de stroming te beschrijven.

Voor de meanderende baan van een dipool op een (3- of 1-vlak zijn analytische benaderingen gevonden. De beweging van een passief deeltje in het stroomingsveld van een meanderende dipool is bepaald door een stelsel vergelijkingen dat de vorm van een periodiek verstoord integreerbaar Hamilton-systeem heeft. Daarom is advectief transport door een meanderende dipool ook analytisch bestudeerd met behulp van dynamische-systeemtheorie. Het is aangetoond dat zowel op een (3- als op een 1-vlak vloeistof wordt uitgewisseld tussen de beide dipoolhelften, alsmede tussen de dipool en zijn omgeving. De hoeveelheid massa die uitgewisseld wordt is evenredig met de gradiënt van achtergrondvorticiteit en de amplitude van de

160 Samenvatting

meanderende baan van de dipool. De tripool vertoont een ingewikkelde quasi-periodieke beweging op een 1-vlak. Deze instationaire beweging veroorzaakt ook uitwisseling van massa tussen de drie wervels die de tripool bevat, alsmede tussen de vloeistof in de nabijheid van de tripool en de tripool zelf.

Resumen 161

Resumen

Esta disertación trata del movimiento de remolinos bidimensionales en fluidos en rotación y del transporte de masa originado por estos remolinos. Las investigaciones acerca de fluidos bidimensionales en genera!, y de remolinos en particular, han sido estimuladas en gran medida por la relevancia de estos fenónemos en la dinámica de la atmósfera y los océanos. El carácter bidimensional de las corrientes atmosféricas y oceánicas ( usualmen te llamadas corrientes geofisicas) es un ras go de las escalas grandes: se aplica a las corrientes y vientos que se extienden horizontalmente por cientos o miles de kilómetros y que evolucionan lentamente en perioclos de dias o semanas.

La bidimensionalidad de las corrientes geofisicas es una consecuencia de dos factores principales: (1) la rotación terrestre, que obliga al fluido a moverse en planos localmen te horizontales, y (2) la geometria de la región donde occurren las corrientes, ya que el océano y la atmósfera son capas delgadas de fluido que tienen una profundidad de unos cuantos kilómetros y se extienden horizontalmente por miles de kilómetros. La forma esférica de la Tierra tiene otras consecuencias importantes. El plano horizontal en que el movimiento ocurre es perpendicular al eje de rotación terrestre en los polos, y es paralelo a ese eje en el ecuador. En genera!, el eje de rotación y el plano de movimiento forman un ángulo igual a la latitud geográfica. Consecuentemente, el ftuido experimenta efectivamente una rotación no uniforme. Este fenómeno es usualmen tellamadoel efecto (:J, y es la causa de muchos procesos dinámicos en la atmósfera y el océano, como las ondas planetarias y la intensificación occidental de la circulación oceánica (como se puede observar, por ejemplo, en la Corriente del Golfo en el Atlántico y la Corriente de Kuroshio en el Pacifico). Cuando el fenómeno que se estudia no se extiende latitudinalmente más de unos cien kilómetros, es válido considerar que el dominio es plano y la rotación es uniforme (esta idealización se conoce como aproximación del plano!) . Si el fenómeno tiene una extensión latitudinal mayor, pero no superior a unos mil kilómetros, es usual considerar que la rotación varia linealmente con la latitud (aproximación del plano fJ). Para fenómenos que ocurren en escalas similares, pero cerca de los polos, la variación de la rotación es aproximada como u na función cuadrática de la distancia al polo ( aproximación del plano 1).

Es conveniente mencionar la existencia de equivalentes dinámicos de los planos fJ y I· Con el uso de la conservación de la vorticidad potencial puede demostrarse que una variación pequefia de la profundidad del fluido produce un gradiente de vorticidad ambiental. Un plano inclinado causa un gradiente uniforme y es por Jo tanto equivalente al plano (:J, en tanto que una topografia parabólica, como la superficie libre de un fluido en rotación, es equivalente al plano I· "Somero" en el caso topográfico es equivalente a "norte" en los planos fJ y 1'· Una segunda equivalencia aparece en la dinámica deun plasma magneticamente confinado. En este caso el gradiente de la densidad del plasma tiene un efecto equivalente al del gradiente de la vorticidad ambiental en un ftuido geofisico. La dinámica de un plasma confinado planamente es equivalente a la de un fluido en el plano (:J, en tanto que en un plasma cilindricamente confinado la dinámica es análoga a la de un ftuido en el plano 1'· U na equivalencia completa depende de la distribución de densidad del plasma (Yabuki et al. 1993).

T

162 Resumen

El movimiento de fl.uidos bidimensionales se caracteriza por la aparición de remolinos coherentes (vea, por ejemplo, McWilliams 1984). Entreestos remolinos, el que ocurre con más frecuencia es el remolino monopolar o monopolo, que consiste en una región de fiuido que rota alrededor de un punto comun. U na estructura más compleja es el remolino dipolar o dipolo, formado por dos remolinos monopolares estrechamente acoplados que rotan en direcciones opuestas. El dipoio se desplaza en la dirección definida por su eje de sirnetria y acarrea consigo una cantidad fija de masa; por lo tanto, el dipoio puede transpotar masa y momento a través de distancias grandes comparadas con su diámetro. El tripolo completa la lista de remolinos estables conocidos. Esta estructura puede definirse como un arreglo linea! y compacto de tres remolinos, con el remolino central fl.anqueado en sus lados largos por dos remolinos más débiles donde el fl.uido rota en dirección contaria a la del remolino centra!. Esta configuración simétrica rota uniformemeute en la dirección definida por el remolino centra!.

El ejemplo más espectacular deun remolino monopolar es la Gran Mancha Roja de Jupiter, que continua girando 300 ai'ios después de haber sido observado por primera vez. Estructuras similares existen en los océanos terrestres. Por ejemplo, los remolinos de la Corriente del Golfo pueden viajar cientos de kilómetros, mientras retienen las propiedades qufmicas y biológicas del agua que contienen. Aunque menos abundantes, los remolinos dipolares también ocurren en la naturaleza, en los océanos como "corrientes en forma de hongo" (Fedorov y Ginsburg 1989) y como "bloqueos" en la atmósfera (Haines y Marshall 1987). Recientemente un remolino tripolar fue observado en el Golfo de Biscaya (Pingree y LeCann 1992).

A partirdel trabajo de Stern (1975) diversas investigaciones teóricas han sido dedicadas a la dinámica de remolinos dipolares en el plano {3 y en la esfera. Invariablemente, los remolinos dipolares que satisfacen las ecuaciones de movimiento son estacionarios o se transladan con velocidad constante en dirección zona! (vea, por ejemplo, Flierl 1987). En ai'ios recientes se ha estudiado con atención creciente el movimiento de dipolos que tienen un ángulo de incidencia con respecto a las lineas de isovorticidad ambiental (es decir, respecto de los paralelos geográficos). El movimento del dipoio ha sido estudiado mediante técnicas analfticas asf como numéricas, y usando modelos de vórtices puntuales asi como remolinos con una distribución continua de la vorticidad (vea, por ejemplo, Kono y Yamagata 1977, Makino et al. 1981, Zabusky y McWilliams 1982, Nycander e Isichenko 1990).

En con traste, pocos estudios experimentales acerca de dipolos en fl.uidos en rotación habfan sido publicados cuando se inicio esta investigación. Probablemente los unicos estudios dedicados a la dinámica de los dipolos en un fiuido homogéneo en rotación eran entonces los de Flierl et al. (1983) y Fedorov et al. (1989). La inyección deun chorro turbulentoen el interior dellfquido (Flierl et al. 1989), o la aplicación de un chorro de aire en la superficie (Fedorov et al. 1989) crean un dipoio asimétrico, que se mueve en una trayectoria circular. En ambos estudios se estimó que el gradiente de vorticidad ambiental eausado por la superficie parabólica es pequeîio. Por lo tanto, estos autores consideron que sus experimentos son relevantes para la dinámica del plano f.

Por ello, nos parecfa importante estudiar experimentalmente el comportamiento de un dipoio en el plano {3 topográfico, asi como la interacción de dipolos que se mueven zonalmente en dirección contraria. Los experimentos descritos en el capftulo 3 confirmaron la mayorfa de los resultados analfticos y numéricos publicados con anterioridad por otros investigadores. Nuevos hechos fueron también observados, como el tamai'io creciente y el rompimiento del dipoio que se mueve en dirección oriental, asf como la reducción de tamai'io del dipoio que se mueve en dirección occidental. La hipótesis de que este comportamiento asimétrico es el

Resumen 163

resultado de la generación de vorticidad relativa fue confirmada usando el método de vórticeen-celda.

Una de las observaciones experimentales más sorprendentes, aparte del movimiento serpenteante del di polo, es la expulsión de fluido inicialmente acarreado por el di polo. El fluido expulsado es a continuación estirado hasta formar bandas muy delgadas, de tal manera que el dipoio deja a su paso una estela de fluido entintado (es decir, inicialmente localizado en el interior del remolino). Similarmente, otras rnasas de fluido son capturadas por el dipoio y posteriormente son enrolladas alrededor de cada una de sus mitades. Estas observaciones motivaron el estudio de la advección de partfculas en el campode velocidad del dipoio (capftulo 4). Los objetivos principales al investigar este prohierna son (1) calcular la cantidad de masa que es intercambiada entreel dipoio y sus alrededores y (2) determinar las regiones del fluido que son atrapadas o expulsadas por el remolino.

El choque de dos remolinos dipolares es un proceso que ha recibido gran atención, entre otros motivos porque contribuye a dilucidar la cuestión de la estabilidad del remolino y por sus posibles implicaciones en el transporte de masa. Los investigadores mencionados arriba en relación con el movimiento del dipoio trataron también algunos aspectos de la colisión de dos dipolos que se mueven en dirección zona!. Las observaciones experimentales que se presentan en el capitulo 5 verifican algunos de los resultados publicados con anterioridad, como el intercambio de pareja entre los dos dipolos y la trayectoria curvada de los nuevos remolinos dipolares. U na segundo intercambio de pareja (que, por Jo tanto, re-establece a los dipolos originales) no pudo ser observado en ellaboratorio. Al parecer, tal evento requiere de un grado de sirnetria que sólo puede ser obtenido teóricamente.

Escasos trabajos han si do publicados acerca de la dinámica del plano"'(, que fue introducido por LeBlood (1964) para el estudio de ondas planetarias en una cuenca polar. Nof (1990) estudió el comportamiento de monopolos y dipolos en este plano. Encontró que los monopolos no son estacionarios y obtuvo soluciones dipolares (modones) equivalentesalas del plano (J, es decir, estructuras aisJaclas que se propagan a Jo largo de los paralelos de latitud geográfica. El estudio de dipolos en el plano"'(, que se presenta en el capftulo 6, es una extensión de los estudios acerca de la dinámica del dipoio en el plano (J (capftulos 3 y 4).

Nuestro interés en la dinámica del plano 1 tiene sus orfgenes en su equivalente topográfico: la superficie parabólica en un fluido en rotación. En especial, estábamos interesados en el movimiento inestacionario mostrado por los remolinos tripolares generados a cierta distancia del eje de rotación, asf como en el continuo estiramiento y plegamiento de regiones de fluido a que da origen ese movimiento (van Heijst y Velasco Fuentes 1994). Estos fenómenos contrastan notablemente con estudios experimentales y numéricos publicados con anterioridad en los que se muestra que el tripolo es una estuctura estacionaria y estable en un fluido bidimensional (vea, por ejemplo, van Heijst et al. 1991 , Orlandi y van Heijst 1992). Esto sugiere que el movimiento inestacionario es el resultado de un proceso tridimensional. El mejor candidato parece ser el alargamiento y compresión de columnas de agua que resultan de la superficie parabólica del fluido. El movimiento inestacionario y la advección de particulas son descritos en el capftulo 7.

Métodos experimentales, numéricos y analfticos fueron usados para estudiar el movimiento de los remolinos y el intercambio de masa. El arreglo experimentales sencillo: un tanque lleno de agua de la llave montado en una mesa que rota a una velocidad constante. Las corrientes fueron visualizadas mediante tinta o pequeîias partfculas que flotan en la superficie del agua, y cada experimento fue registrado con cámaras fotográficas o de video. Las mediciones son

164 Resumen

también simples en prinCJplO: (a) el movimiento de los remolinos se determina siguiendo el movimiento de las regiones de fluido entintado atrapadas por los remolinos y (b) el campo de velocidades, ya sea a partir de fotografîa de larga exposición o seguimiento de partkulas en video, se obtiene por diferencias finitas e interpolación a partir de las posiciones de las partkulas en una serie de tiempos.

Los métodos analfticos son bien conocidos: el modelo de vórtices puntuales introducido por Helmholtz en el siglo pasado. Aquf las intensidades de los vórtices son moduladas con base en la conservación de vorticidad potencial. Aunque en la literatura sólo la modulación en el plano /3 habfa sido usada, ya Zabusky & McWilliams (1982) mencionaron que el mismo principio podria ser aplicado para diferentes variaciones de la vorticidad ambiental.

Para el estudio de la advección se aplica la descripción Lagrangiana de la corriente. Esto reduce el prohierna a un conjijnto de ecuaciones diferenciales ordinarias, es decir, a un sistema dinámico de dimensiones finitas (Aref 1984). Las poderosas técnicas geométricas desarrolladas para el estudio de esos sistemas pueden ser directamente aplicadas al estudio de la advección, con la venta ja de que el abstracto espacio de fasedeun sistema dinámico corresponde al espacio fisico donde el flujo tiene lugar. La técnica de la 'dinámica de lobulos' (vea Wiggins 1992) es usada extensamente para cuantificar el intercambio de masa y para determinar la localización de las rnasas de fluido que toman parteen ese proceso (capitulos 4, 6 y 7). También la función de Melnikov es usada para evaluar el transporte de masa, pero, aun mas importante, esta función se usa para probar la existencia de trayectorias caóticas. Esto se hace con ayuda de las simetrias temporales presentes en las ecuaciones de advección.

El conjunto de 2N ecuaciones diferenciales ordinarias que describen el movimiento de N vórtices puntuales es integrado usando un esquemade Runge-Kutta de cuarto orden (vea, por ejemplo, Press et al. 1986). El movimiento de partfculas es calculado de manera similar. Pero cuando las partfculas definen un contorno en el fluido, se usa un algoritmo para asegurar que el contorno es definido adecuadamente durante la evolución de la corriente. El prohierna de definir un contornode longitud creciente con un numero finito de puntos (nodos) aparece en diferentes aplicaciones, como la dinámica de contornos (vea PulJin 1992). Consecuentemente, varias técnicas han sido propuestas para tratar con este problema. Aqui se usa un interpolador de trazador cubico natura!, el cual probablemente no es el método más eficiente para reposicionar los nodos, pero es simple y proporciona resultados adecuados en la clase de problemas tratados en esta disertación.

La integración numérica de las ecuaciones de movimiento se vuelve excesivamente larga cuando el nûmero de vórtices se vuelve muy grande. Por lo tanto, en esos casos usamos la técnica del vórtice-en-celda, que fue originalmente desarrollada para el estudio de plasmas en los aiios 60 y que fue introducida en la mecánica de fluidos por Christiansen (1973). El método del vórtice-en-celda es una técnica lagrangiana: calcula la evolución temporal de la posiciones y velociclades del conjunto de vórtices puntuales. Pero para hacerlo de una manera eficiente, las propiedades de la corriente son calculadas en u namalla fija en el espacio, es decir, el método proporciona también una descripción euleriana de la corriente (vea el capitulo 2). Debido a estas caracteristicas el método del vórtice-en-celda produce resultados especialmente adecuados para ser camparados con nuestras observaciones experimentales: las mediciones son eulerianas, mientras que la visualización es esencialmente lagrangiana.

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Curriculum Vitae

Oscar Uriel Velasco Fuentes was born in México City on June 22, 1965. From 1980 to 1983 he received pre-university education at the Centro de Estudios Cientfficos y Tecnológicos Cuauhtémoc.

In 1983 he started a B.Sc. program in Geophysical Engineering at the lnstituto Politécnico Nacional in México City. He graduated cum laude in 1988 with a research performed at the lnstituto de Ciencias del Mar y Limnologfa, under the direction of Dr. Adela Monreal Gómez. From 1989 to 1990 he was enrolled in an M.Sc. program in Physical Oceanography at the Centro de lnvestigación Cientifica y Educación Superior de Ensenada (CICESE), in Baja California, México.

He left CICESE without obtaining the M.Sc. degree and moved to the Netherlands in October 1990; where he became a Junior Scientist (Onderzoeker in Opleiding) at the Fluid Dynamics Laboratory of the Eindhoven University of Technology. The research described in this dissertation is the result of four years of workunder the supervision of Professor GertJan van Heijst.

Stellingen behorende bij het proefschrift van O.U. Velasco Fuentes

27 oktober 1994

1. De beweging van drie puntwervels met een gemoduleerde sterkte is, in het algemeen, niet integreerbaar. Dit proefschrift, hoofdstukken 4, 6 & 7.

2. Dipolaire wervels in. een vloeistof met gradiënten in de achtergrond-vorticiteit meanderen rond lijnen van constante achtergrond-vorticiteit. De oostwaarts bewegende dipool heeft een stabiele baan terwijl die van de westwaarts bewegende dipool instàbiel is. Dit proefschrift, hoofdstukken 3 & 6.

3. De hoeveelheid massa die wordt uitgewisseld tussen een dipoolwervel en zijn omgeving is evenredig met de gradiënt van de achtergrond-vorticiteit en met de amplitude van de meanderende baan. Dit proefschrift, hoofdstukken 4 & 6.

4. Het geheugen bevat geen herinneringen, het is een fabriek van herinneringen met als grondstoffen het verleden en het heden. P. Vroon, Kop-Zorgen.

J.E. Pacheco, Ciudad de la Memoria.

5. De vooroordelen van gewone westerse mensen jegens niet-westerse groepen en individuen zijn gecreëerd door de politieke en intellectuele leiders die vorm hebben gegeven aan de westerse cultuur, welke gekenmerkt zou zijn door tolerantie ten opzichte van de ander. Edward Said, Culture and Imperialism.

Antonello Gerbi, The Dispute of the New World: the Bistory of a Polemic, 1750-1900.

6. Beschaafde landen garanderen dat alle mensen die zich in dat land bevinden, onder gelijke omstandigheden, op dezelfde manier zullen worden behandeld. Diezelfde beschaafde landen zorgen ervoor dat de omstandigheden van sommige van hun burgers gelijker zijn dan die van andere burgers.

7. De geïndividualiseerde samenleving is de samenleving van de eenzame massa.

8. Ons Arcadia is het anti-Utopiavan onze voorouders.

9. Steden, net als bloemen, groeien en bloeien in het platteland.

Theses belonging to the doctoral dissertation of O.U. Velasco Fuentes

October 27, 1994

1. The motion of three point vortices with modulated circulation is not, in genera!, integrable.

This dissertation, chapters 4, 6 & 7.

2. Dipolar vortices in a fl.uid with gradients of ambient vorticity meander around the lines of equal ambient vorticity. The eastward rnaving dipale has a stabie trajectory, whereas that of a westward rnaving dipale is unstable. This dissertation, chapters 3 & 6.

3. The amount of fl.uid exchanged between a dipolar vortex and its surroundings is proportional to the gradient of ambient vorticity and the amplitude of the dipole's meandering motion. This dissertation, chapters 4 & 6.

4. The memory does not contain remembrances, it is a Jactory of remembrances with the past and the present as raw materials. P. Vroon, Kop-Zorgen.


5. The prejudices of ordinary western people with respect to non-western groups and individuals were created by the politica! and intellectual leaders who shaped the western culture, which would be characterized by toleranee with respect to the other. Edward Said, Culture and lmperialism.

Antonello Gerbi, The Dispute of the New World: the Bistory of a Polemic, 1750-1900.

6. Civilized countries guarantee that, under equal circumstances, all people in that country will be treated in the same way. The same civilized countries ensure that the circumstances of some of their citizens are more equal than those of other citizens.

7. The individualized society is the society of the lonely mass.

8. Our Arcadia is the anti-Utopia of our ancestors.

9. Cities, just like fl.owers, grow and fl.ourish in the countryside.

Postuiados pertenecientes a la disertación doctoral de O.U. Velasco Fuentes

27 de octubre de 1994

1. El movimiento de tres remolinos puntuales con intensidad modulada no es, en genera!, integrable. Esta disertación, capftulos 4, 6 y 7.

2. Los remolinos dipolares en un fluido con gradientes en la vorticidad ambiental se propagan oscilando alrededor de las lineas de isovorticidad ambiental. El dipoio que se mueve en dirección oriental tiene una trayeetaria estable, mientras que la trayeetaria del dipoio que se desplaza hacia el oeste es inestable. Esta disertación, cap[tulos 3 y 6.

3. La cantidad de masa intercambiada entre un remolino dipolar y sus alrededores es directamente proporcional al gradiente de vorticidad ambiental y a la amplitud de la trayeetaria oscilante del dipolo. Esta disertación, capftulos 4 y 6.

4. La memoria no contiene recuerdos, es una fábrica de recuerdos que tiene como materias primas al pasado y al presente. P. Vroon, Kop-Zorgen.


5. Los prejuicios de las personas occidentales comunes respecto a grupos o personas no occidentales fueron creados por los lideres politicos e intelectuales que le dieron forma a la cultura occidental, la cual estaria caracterizada por la tolerancia respecto al otro. Edward Said, Culture and Imperialism.

Antonello Gerbi, La Disputadel Nuevo Mundo: Historia de una Polémica, 1750-1900.

6. Los paises civilizados garantizan que, bajo iguales circunstancias, todas las personas que se enenentren en ese pais serán tratados de la misma manera. Esos mismos paises civilizados procuran que las circunstancias de algunos de sus ciudadanos sean más iguales que las de otros.

7. La sociedad individualizada es la sociedad de la masa solitaria.

8. Nuestra Arcadia es la anti-Utopia de nuestros antepasados.

9. Las ciudades, como las flores, ereeen y florecen en el campo.

Two-dimensional vortices with background vorticityThe two-dimensional character of geophysical flows...

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