Jennifer Verlaine Lukovich

A thesis submitted in conformity with the requirements

for the degree of Doctor of Philosophy

Gaduate Department of Physics

University of Toronto

@ Copyright by Jennifer Verlaine Lukovich 2001

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Large-Sede Mixmg hr the Middle Atmosphme

Doctor of Philosophy, 2001 Jennifer Verlaine Lukovich, Depart ment of Physics, University of Toronto

bstract

Large-scale mixing in the Earth's rniddle atmosphere (stratosphere and mesosphere) is

governed by various fluid-dynarnical phenornena. Sought in the present investigation is a

quantification of the notion that the stratosphere, dominated by planetary-scde Rossby

waves, is characterized by ustirring", while the mesosp here, dominated by interna gravity

waves, is characterized by "rnixing". The concepts of balance versus imbalance and spec-

t r d nonlocality versus locality are used to aosess results from two shallow-water numerical

experiments representative of stratospheric and mesospheric dynamics on a quasi-horizontal

isent ropic surface.

Topologicai properties of the fiow are exarnined using various geometrical diagnostics. As-

sessrnent of the OkubeWeiss and Hua-Klein criteria, which characterize strain- and vorticity-

dominated regions, reveals that velocity gradients are strongly influenced by the presence of

an unbalanced component to the flow. Patchiness, wherein the average Lagrangian velocity

is monitored, dernonstrates persistence of coherent spatial structure in the zonal direction,

owing to the presence of a zonal sheax Bow, and the destruction of spatial structure in

the meridional direction. Spatial distributions of Liapunov exponents, which monitor the

stretching properties of the Bow, illustrate the erosion of largescale strain-dominated regions

(characteristic of stratospheric dynamics) with the inclusion of an unbalanced component.

The results from each of these diagnostics suggest that stirring provides an appropriate de-

scription for stratospheric dynamics, and mixing, as characterized by turbulent diffusion, for

meaos pheric dynamics.

S tat ist ical analyses of absolute (single- particle) and relative ( two-particle) dispersion fur-

ther elucidate these connections. Single-particle statistics are governed by the Lagrangian

velocity, end dative dispersion by the LagMgian vcfociSr gradients. The former b e h v b m

is demonstrated in the non-Fickian ( t 2 ) zona1 dispersion evident for both the stratospheric

and mesospheric cases. Differences between the stratospheric and mesospheric regimes are

however manifested in distinctive meridional dispersion statistics a t long times. By contrast,

relative dispersion is shown to effectiveiy capture the distinction between spectrally nonlocal

and local dynamics, and in so doing, to reflect the unbaianceci dynamics characterized by

shallow dopes of the kinetic energy spectra. Explored also in the present work is the determi-

nation of an appropriate description for passive tracer dynamics in spatially and tempordly

irregular flow.

iii

1 would like to thank Professor Shepherd for guidance and support provided. Technical

assistance provided by S. Gravel, K. Ngan, K. Semeniuk and J. Koshyk is also appreciated.

In addition, 1 would like to thank Professors Sinervo and Strong for their contributions as

cornmittee memben during my years as a doctoral student at the University of Toronto, and

Professor Desai for helpful suggestions.

Financial support was provided by the Natural Sciences and Engineering Research Coun-

cil of Canada, OGSST, the Department of Physics a t the University of Toronto, and the

Sumner foundat ion,

Contents

1 Introduction 1

. . . . . . . . . . . . . . 1.1 The Stratosphere and Mesosphere: Salient Features 1

. . . . . . . . . . . 1.2 Stirring vs Mixing: Kinematic and Dynamical Differences 3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Thesis Structure 6

2 The Shallow-Water Mode1 8

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Equations of Motion 8

. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 WhytheSWModel? 12

. . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Properties of the SW Mode1 12

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Balance and Imbalance 14

. . . . . . . . . . . 2.3 S hallow- Water Mode1 Implementat ion: Numerical Mode1 17

. . . . . . . . . . 2.4 Forcing Mechanism: Planetary- and Gravity-Wave Forcing 19

. . . . . . . . . . . . . . . . . . 2.5 Kinetic Energy Spectra and Spectral Slopes 25

. . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Energy Spectra and Diagnostics 26

. . . . . . . . . . . . . . . 2.6.1 Spherical Harmonic Decomposition Scheme 26

. . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Polarization Relations 28

. . . . . . 2.6.3 Spatial and Temporal Autocorrelations and Power Spectra 28

3 Flow Visualisation 34

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Stirring and Mixing 35

. . . . . . . . . 3.1.1 Stirring and Chaotic Advection. Mixing and Diffusion 38

. . . . . . . . . . . 3.1.2 Atmospheric Considerations and Expected Results 40

. . . . . . . . . . . . . . 3.1.3 Stirring and Bdance; Mixing and Imbaiance

3.2 The OkubeWeiss Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Okub* Weiss Criterion Results . . . . . . . . . . . . . . . . . . . . . 3.2.2 Hua-Klein Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Hua-Klein Criterion for Divergent Flow . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Hua-Klein Criterion Results

3.2.5 Time Evolution of Velocity Gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Patchiness

3.3.1 Patchiness Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Liapunov Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . 3.4.1 Finite-Time Liapunov Exponents

3.4.2 Relationship between OkubeWeiss Criterion and Liapunov Exponents

3.4.3 Maximum Liapunov Exponent Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Cu mdat ive Assessrnent of Geometrical Properties

4 Statistical Diagnostics 73

4.1 Absolute/Single Particle Dispersion . . . . . . . . . . . . . . . . . . . . . . . 74

4.1.1 Spatial Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.1.2 Non-Fickian Diffusion and the Advection-Diffusion Equation . . . . . 75

4.1.3 Absolute Dispersion and the Lagrangian Correlation Coefficient . . . 77

4.1.4 Temporal Considerations . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.1.5 Absolute Dispersion and Balanced Dynamics . . . . . . . . . . . . . . 80

4.2 Absolute Dispersion and Patchiness . . . . . . . . . . . . . . . . . . . . . . . 82

4.2.1 Absolute Dispersion and Stirring . . . . . . . . . . . . . . . . . . . . 83

4.3 Absolute Dispersion Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.3.1 Probability Distribution Functions for Single-Particle Displacements . 91

. . . . . . . . . 4.3.2 Higher-Order Moments and LargeScale Intermittency 93

. . . . . . . . . . . . . 4.3.3 PDF Resdts for Singleparticle Displacement 93

4.3.4 Asrcmnerrtof AbsotateDispers iorrhIts . . . . . . . . . . . . . . . 99

4.4 Relative/Tw*Part icle Dispersion . . . . . . . . . . . . . . . . . . . . . . . . 100

4.4.1 Relative Dispersion and Separation between a Pair of Particles . . . . 100

4.4.2 Relative Dispersion and the Advection-Diffusion Equation . . . . . . 101

4.4.3 Relative Dispersion and the Lagrangian Correlation Function for Rel-

at ive Veloci t ies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.4.4 Temporal Considerations . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.4.5 Relative Dispersion and Mixing . . . . . . . . . . . . . . . . . . . . . 107

1.4.6 Relative Dispersion Results: Temporal Scaling Laws . . . . . . . . . . 108

4.4.7 Local and Nonlocal Dynamics: Kinetic Energy Spectra and Scaling Lawsll6

4.4.8 Structure Functions and Spatial Scaling Laws . . . . . . . . . . . . . 118

4.4.9 Structure Function Resuits . . . . . . . . . . . . . . . . . . . . . . . . 121

4.4.10 PDFs and the Distance Neighbour Function . . . . . . . . . . . . . . 123

4.4.1 1 Higher-Order Moments . . . . . . . . . . . . . . . . . . . . . . . . . . 125

4.4.12 PDFs for Relative Separation: Results . . . . . . . . . . . . . . . . . 125

4.5 Additional 1)iagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

4.5.1 The Two-Particle Correlation Function and Correlation Dimension . 131

4.5.2 Correlation Dimension Results . . . . . . . . . . . . . . . . . . . . . . 132

. . . . . . . . . . . . . . . . . . . . 4.5.3 FiniteTime Liapunov Exponents 134

. . . . . . . . . . . . . 4.5.4 PDF for the Finite-TimeLiapunov Exponent 135

5 Conclusions 137

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Summary 137

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Future Directions 141

A Description of GEM Shallow-Water Mode1 143

. . . . . . . . . . . . . . . . . . . . . . A . 1 Changes to the Governing Equations 144

Bibliography 145

vii

List of Figures

1.1 Kinetic energy spectra calculated from Canadian Middle Atmosphere Mode1

data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1 Geopotential height field used for PW forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Geopotential height field used for GW forcing

. . . . . . . . . . . . . . . 2.3 Geopotential height field used for PW/GW forcing

. . . . . . . . . . . . . . . . . 2.4 Zond winds for PW and PW/GW experiments

. . . . . . . . . . . . . . 2.5 Meridional winds for PW and PW/GW experiments

. . . . . . . . . . . . . . . . . . . . . . . 2.6 Timeaveraged kinetic energy spectra

2.7 Spatial longitudinal autocorrelations for P W. P W/G W and G W experiments . 2.8 Temporal autocorrelation of the totd geopotential for the PW and PW/GW

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . experiments

. . . . . . . . . . . . . 2.9 Power spectra of the divergence and potential vorticity

2.10 Time mean of the zonally-averaged divergence variance for various experiments . 2 . t 1 Time mean of the zonally-averaged potential vorticity variance for various

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . experiments

. . . . . . . . . . . 3.1 Quadrants of the Re(A)-lm(A) and Re(X2)-lrn(X2) plane 46

3.2 Contour plots determined from the Weiss criterion Qw and the OkubeWeiss

. . . . . . . . . . . . . . criterion QOw for the PW and PW/GW experiments 47

. . . . . . . . . . . . . . . 3.3 Divergence field for PW and P W/GW experiments 48

. . . . . . . . . . . 3.4 Potential vorticity field for PW and PW/GW experiments 49

viii

Contou~ pl&§ debennid Erom ek Hus-Klein mterioo QHK and the Hua-

Klein criterion with divergent component for the PW and PW/GW experiments . 54

Contour plots of the Okubo-Weiss criterion for the P W experiment at various

instants in time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Patchiness plots for zonai winds for PW and PW/GW experiments

Patchiness plots for meridional winds for PW and PW/GW experiments . . . . . . . . . . . . . . . Tracer fields for PW aod PW/GW forcing at t = 50 days

Spatial distribution of finitetirne Liapunov expoaents at Mnous instants in

time for the PW and PW/GW experiments . . . . . . . . . . . . . . . . . . .

Ensemble-averaged Lagrangian autocorrelations of the zona1 and meridional

wind components for the PW. PW/GW and GWRT experiments . . . . . . . Trajectory of a particle for the PW . PW/GW and GWRT experiments . . . . Ensemble-averaged zonal mean single- part icle di placements for P W. P W /G W

and G W KI' experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ensemble-averaged absolute zona1 and meridional dispersion . . . . . . . . . . PDFs for zonal and meridionai absolute displacements for the PW. PW/CW

and GWRT experirnents . . . . . . . . . . . . . . . . . . . . . . . . . . . PDFs for zona1 absolute displacement at different instants in time . . . . . . . PDFs for meridional absolute displacements at different instants in time . . . Ensemble-averaged Lagrangian autocorrelat ions of the veloci ty gradients for

the PW. PW/GW and GWRT experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trajectories of neighbouring particles

Ensemble-averaged relative zona1 and meridionai dispersion . . . . . . . . . . Short- and intermediate t ime regimes for ensemble-averaged relative zonal dis-

persion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.12 Tempordy-averaged structure function and corresponding kinetic energy spec-

tra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . 4.13 PDFs of zond and meridionai separation 127

. . . . . . . . . . . . . . 4.14 PDFs of zona1 separation at different instants in time 128

. . . . . . . . . . . 4.15 PDFs of meridional separation at different instants in time 129

. . . . . . . . . . 4.16 Correlation function for the PW and P W/G W experiments 133

. . . . . . . . . . . . . . . . . . . . . . 4.17 PDFs of finite-time Liapunov exponent 135

Chapter 1

Introduction

In order to understand the manner in which chernicd species interact in the Earth's atm*

sphere, it is necessary to examine the combined effect of dynamical, radiative and chemical

processes. The current investigation seeks to address one aspect of the dynamical prob-

lem, namely, how a temporally evolving flow transports Buid elements on quasi-horizontal

isentropic surfaces. In particular, the focus will be on mizing in the stratosphere and mes*

sphere. Previous st udies have shown that ait hough chaotic advection provides an apprw

priate description for stratospheric dynamics (Ngan 1997), increased spatial and temporal

irregularity in the mesosphere. introduced with the enhanced presence of interna1 gravity

waves, demands an alternat ive interpretat ion (Shepherd et al. 2000). The characterization

of large-scale mixing in the rnesosphere is hence the motivation for the present work.

1.1 The Stratosphere and Mesosphere: Salient Fea-

Transport in the middle atmosphere is a combination of slow zonal-mean vertical and rnerid-

ional motion (diabatic circulation) and rapid quasi-horizontal motion (ou isentropic sur-

faces). Vertical transport of chemical species in the stratosphere primarily proceeds via

the Brewer-Dobson circulation. which refers to the large-scale ascent of air in the tropics

and the large-scale descent of air in the extratropics. The vertical transport of tracers is

modified, however, by regions of strong quasi-horizontal mixing. It is the quasi-horizontal

CHAPTER 1 , INTRODUCTION 2

modion on isentmpic surfaces, that k, the natme of stratospheric and mesospheric mixing,

that comprises the essence of the present investigation.

Fluid motion in the stratosphere, located between the tropopause at - 10-15km and the

stratopause at - 50km, is dominated by a largescale, low-frequency, quasi-horizontal ve-

loci ty field (Juckes and McIntyre 1987). A phenornenon known as Charney-Drain filtering

(Charney and Drazin 196 1 ) removes the synopt ic-scale Rossby waves result ing from t r o p

spheric baroclinic instability, and permits propagation of only the planetary-scale Rossby

waves ( forced by cont inentd-scde topographic feat ures and land-sea thermal contrasts) into

the wintertime stratosphere. Because the forcing mechanisms are stationary the planetary

waves tend to be stationary (zero frequency) as well. Erosion of the polar vortex when these

planetary waves break results in the development of filamentation and srnall scales of motion

as material is ejected from the polar vortex into the secdled "surf zone", a region of strong

mixing in midlatitudes. Such an "erosion-entrainment" process sets up conditions for chemi-

cal ozone depletion in the sprinotime polar vortex. This proces bas hwn the oubject of much

study with regard to the dynamics of a perturbed vortex (e.g. Polvani and Plumb 1992). A s

will be described below, i t is the filamentation process that enables stratospheric dynamics

to be described in terms of chaotic advection.

By contrast, the mesosphere, occupying the region between the stratopause and mesopause

at .- 85km, is distinguished by ubiquitous gravity-wave activity. Due to decreasing density

with increasing altitude, internd inertia-gravity waves (geoerated by topography, convection

and other tropospheric processes) grow in amplitude as they propagate upwards into, and

thus domiaate, the mesospheric region. (They axe present in the stratosphere too, of course,

but with much smailer amplitudes. Roughly speaking the waves grow by an order of mag-

nitude every 30km.) Although adequate global observational data is not available for the

mesosp here, due to the st ringent spat i d and temporal resolut ion requirements necessary to

accurately capture rnesospheric winds, wind fields generated from General Circulation Mod-

els (GCMs) suggest that the mesosphere is characterized by increasing spatial inegularity

CHAPTER 1. INTRODUCTION 3

witk altitude (Kostiyk et d 1999). As demonstrateci by Shepherd e t aI. (20001, there is

also an increased temporal variability relative to the stratosphere. This is manifested, for

exarnple, in the sensitivity of contour advection experiments to temporal t runcation of the

winds.

1.2 Stirring vs Mixing: Kinematic and Dynamical Dif- ferences

Qualitative differences between stratospheric and mesospheric dynamics may be quantified

through consideration of balanced and unbalanced Bow. Balanced dynamics is defined here

as t hat for which the (slow) vortical motion dominates, whereas unbalanced dynamics de-

scribes a situation wherein the (fast) inertia-gravity waves dominate. To undentand this

distinction it is useful to consider the ratio of timescales for the vortical and inertia-gravity

wave cornponents of the flow. The vortical frequency is VIL. where L and V denote the char-

acteristic horizont ai lengt h scde and vclocit y assuciatcd ni t b a given BOW. Shallow-water

inertia-gravity waves, as are examined in the present investigation, have a frequency

where f is the Coriolis parameter, g is the gravitational acceleration, H is the mean depth

of the shallow layer of Buid, and k is the total horizontal wavenumber. The ratio of the slow

and fast frequencies E is thus

CI' - F r - Ro = L , / f 2 + g H k & -

for Ro = V/ f L the Rossby number and Fr = the Froude number, with k = 1/L.

The separation in timescales is evident as Fr -t O (for arbitrary Ro) and as Ro + O (for

arbitrary Fr). Both kinds of balance, low Ro and low Fr, yield a divergent component

of the flow that is much smaller than the rotational component. In the present investiga-

tion, unbalanced dynamics therefore refers to the case where the rotational and divergent

components are comparable.

CHAPTER 1. 1NTRODUCTlON 4

Pisnetary rives h the stratosphere constitute bdanced plienomena. Stratospheric flow is

dominated by coherent vortical structures (Ngan and Shepherd 1999a) induced by breaking

planetary waves, also known as cat's eyes, which are characteristic of the surf zone. Small-

scale, high-frequency gravity waves in the mesosp here contri bu te a divergent cornponent to

the flow. Spatial wavenumber kinetic energy spectra and their slopes, as shown in Figure

1.1, provide a means by which the rotational and divergent components may be compared

(Koshyk et al. 1999). In terms of an energy spectrum with power-law scaling E ( k ) - k-", cornparison of Bow fields generated from different GCMs demons t rates t hat the stratosphere

is characterized by a steep spectrum of balanced motion with n > 3, whereas the rnesosphere

is distinguished by a much shallower spectrum of unbalanced motion a t smaller spatial

(higher wavenumber) scales, with n < 3. Although these are only GCM results, they are

consistent with aircraft measurements of shallow kinetic energy spectra a t smaller spatial

scales in the lower stratosphere (Bacmeister et al. 1996).

Notnworthy, and kpy to the present study, are analogous spectral slope properties for

what are referred to as (spectrally) nonlocal and local dynarnics. For nonlocal dynarnics,

Tota I

m1000K ' - - \ f

a m K - - \ 1

Wovo Numbar n Wave Nurnbor n Wavo Numbw n

Figure 1.1 : Mont hly averaged total (left panel), rotational (rniddle panel), and divergent (right panel) kinetic energy spectra venus total horizontal spatial wavenumber calculated from January Canadian Middle At mosphere Modei data, at several isentropic levels. 320 K is the tropopause (- lokrn), 1000K is the rniddle stratosphere (- 35km), and 4000K is the rniddle rnesosphere (- 70km). Frorn Shepherd e t al. (2000).

CHA PTER 1. INTRODUCTION 5

kracei. evdution ab a given length s d e is g o v d by vefocity featmes at m c k Iarger terrgth

scales, whereas tracer evolution is determined by comparable length scales for local dynamics.

A s outlined by Bennett (1984), for a kinetic energy spectmm described by E(k) - k-", nonlocal dynamics are distinguished by slopes steeper than n = 3, and local dynamics by

shallower spect rai slopes. The relationship between the kinet ic energy spectmm and locality

is manifested in the expression for the nonlinear (advective)

time, associated with a given wavenumber k (Babiano et al.

for ko the low-wavenumber cut-off. For n > 3, the integral

and tracer evolution is characterized by a single timescale,

timescale, or eddy t urnaround

1985):

(1.3)

converges strongly as k + ca set by the large scales of the

flow (nonlocal dynamics). When 1 < n < 3, in contrast, rE(k) is dominated by contribu-

tions from wavenumbers close to k, correspondhg to small spatial scales (local dynamics).

No single timescale then exists as the flow is characterized by a wide range of eddies with

a correspondingly wide range of eddy tumaround times. For n c 1, the system is con-

sidered diffusive. This argument indicates that a possible interpretation for mesospheric

dynamics is a spectrally local description. Furthermore, it has been shown that s t r a t e

spheric dynamics are consistent with the concept of chaotic advection (Pienehumbert 1991,

Ngan and Shepherd lggga), which is a special case of nonlocal dynamics: the flow is char-

acterized by a quasi-regular velwity field, with largesede spatial structure and with ta-

grangian t rajectories following chaotic pat hs. Central to the present investigation is the es-

tablishment of diagnostics to confirm the distinction between stratospheric and mesospheric

dynamics in terms of nonlocal and local dynamics, respectively.

A pertinent question in the study of local and noniocal behaviour concerns the separation

of particles in a turbulent fluid. As noted by Bennett (1984), absolute and relative disper-

sion provide a means by which this question may be addressed. While absolute dispersion

monitors the motion of a single rnarked particle, and its translation relative to a h e d l+

cation, relative dispersion monitors the spread of a cluster of particles, thus depicting the

CHA PTER 1. INTROD UCTlON 6

eReet OF eddies smdk than tke cluster s k . The retevance of buth concepts to middre at mc+

spheric dynamical considerations is illustrated by the aforementioned filamentation process.

As materid is ejected from the polar vortex, Iarge-scale tracer gradients in the flow field

are generated by the differential translational motion of the advected particles, reflecting

nonlocal dy narnics. In the mesosphere, however, increased small-scale variability gives rise

to local dynamics, and the generation of small-scale structure, so that mixing becomes the

operat ive mechanism of transport.

1.3 Thesis Structure

The goal of the current study is to distinguish stratospheric from mesospheric tracer dynamics

using a variety of diagnostics t hat demonstrate local or nonlocal properties. The structure of

the presentation used to achieve this end is as follows. Chapter 2 describes the shailow-water

model, a singlelayer model containing both rotational and divergent components. Since

large-scdc mixing is the emphasis of this investigatioa, wherr 'largesçde" riieaus flow for

which the horizont al scales great ly exceed vert icd scales suppressed by vert ical stratification,

thus rendering d i d a tw*dimensional interpretation of the flow, middle atmospheric dy-

namics of a single isentropic surface are mimicked through forcing of the geopotential height

field in the shallow-water model. This approach is justifiable for timescales over which the

motion is adiabatic, typically a week or so. The forcing of the bottom "topographyn in the

shallow-water model represents forcing of the isentropic layer by waves propagating up from

below. Considered in particular is the question of slow (balanced) and fast (unbalanced) dy-

namics, both of which are represented in the shallow-water model. Findly, the nature of the

forcing used to implement the largescale planetary waves (Ngan and Shepherd 1999a) and

the small-scale, high-frequency gravity waves is discussed in detail, and diagnostics such as

power spect ra, kinetic energy spectra, and spatial and temporal autocorrelat ions presented

in order to verify an appropriate flow field for the mesospheric regime.

In order to understand how particles disperse in a fluid, it is instructive to examine the

CHAPTER 1. lNTRODUCT1OiV 7

topologid strnetnres associateci with a g i v e ~ ~ dynamicat regime. Chapter 3 seeks to address

the question of flow structure, and to examine the erosion of coherent structures present

in the stratospheric dynamicd regime (as a consequence of the planetary waves) wi th the

introduction of gravity waves. Such a geometrical interpretation is achieved with the aid

of a diagnostic known as the OkubeWeiss criterion, wherein the flow is partitioned into

strain-dominated and vorticity-dominated regions. This chapter thus provides a means of

flow visualization in order that the statistical diagnostics used in Chapter 4 to assess the

local/nonlocal featurev prevalent in the mesospheric/st ratospheric regions be fortified w i t h

knowledge of topological structures that enhance either stirring or mixing.

W hile the geometrical interpretation of Chapter 3 provides a qualitative perspective on

how particles disperse in balanced and unbaianced Bow, a statistical assessrnent of local

and nonlocal dynamics provides a quantitative aoalysis of the nature of mixing and stirring.

The essence of Chapter 4 consists of the examination of single (absolute) and two-particle

(relative) dispersion ta idrntify 1 0 4 and nonlocal dynamics. Pursued is the idea that abse

lute dispersion addresses the phenornenon of stirring, as is evident in the stratospheric surf

zone. By contrat , relative dispersion is thought to demonstrate the relative contributions

between strain and vorticity, thus providing evidence for mixing within a given dynamical

regime. Verification of scaling laws for the structure function as derived by Bennett (1984)

and Babiano et al. (1985) is sought, as is the investigation of additional diagnostics such as

finite-time Liapunov exponents and the correlation dimension. Chapter 4 seeks to demon-

strate the distinguishing mixing propert ies between the two dynamical regions resulting from

the unbalanced component ubiquitous in mesospheric dynamics.

Chapter 2

The Shallow- Water Model

Although stratospheric dynamics c m be described in terms of a stably stratified, essen-

tially non-divergent system, the ubiquitous presence of gravity waves in the mesosphere

does not permit such a description. A dynamical system of equations that incorporates

the slowly-varying vortical mot ion in addition to the rapidly-varying gravi ty waves is t here-

fore necessary. The shallow-water ( S W) equat ions provide such a frarnework. This chap ter

presents the salient features of the shallow-water model that render it an appropriate tool

for the study of the dynamics of quasi-horizontal isentropic layers in the middle atmosphere.

Discussed briefly is the numerical model used to represent the shallow-water model, in addi-

tion to the stratospheric forcing mechanism previously implemented (Ngan 1997). Outlined

in greater detail is the forcing scheme adapted to generate flow reminiscent of mesospheric

dynarnics, as observed frorn Generd Circulation Model simulations. Explored also is the

concept of batmced/unbdanced dynamics. The SW eqnations are nsed to demonstrate that

the separation of the flow into rotational and divergent components serves as a first approxi-

mation to balance and imbalance. No particularly new results are presented in this chapter.

It serves only as a presentation of the conceptual and experimental set-up.

2.1 Equations of Motion

The Navier-Stokes (NS) equations provide the foundation upon which the study of 0uid

mechanical phenornena is built. The fluid mechanical form of Newton's second law of motion,

CHAPTER 2. T H E SHALLOW-WATER MODEL 9

the NS equations are centrat to the study of tmbalence. The NS equations are expressed

according t o the mornentum equation

and the cont inuity equation

for Ü = (24, V , W ) the velocity field, p ( 5 , t ) the fluid density, p(2,t) the pressure field, u

the (kinematic) viscosity and P the external stirring force applied to the systern. The

notation & = & + 5 . V denotes the material derivative, or the d e r i ~ t i v e following a fluid

particle. Evident in the NS equations is the nonlinear advective t e r n t in addition to the

rnolecular viscosity term, viscosity being the constant of proport ionality between the stress

(representative of the momentum Aux (Frisch 1995)) and the symmetric part of the velocity

gradient tensor. Thus, the effects of both large- and smdl-scale properties of the flow are

depic ted in these equat ions of mot ion.

Fluid motion in atmospheric and oceanic systems necessitates consideration of a rotating,

rat her t han inertial, frame of reference. Wit hin the context of geophysical fluid dynamics,

the mornentum equation (2.1) in a rotating reference frame and in the absence of viscous

effects is expressed as

where 0 = 7.292 x I O - ~ S - ' is the Earth's rotation rate. In addition, 6 = c # ~ + & is the

gmpotentid. with & = -GM/r the gravitational potential, for G the gravi tational constant,

M the mass of the Earth, and t the distance from the Earth's centre, and #c = rL2R2/2 the

potent id associated wit h the cent rifugal force, where rl is the perpendicular distance to the

Earth's axis of rotation. The continuity equation is expressed as for the inertial reference

frame.

In order to derive the shallow-water equations from the equations representative of Buid

motion in the rniddle atmosphere, it is instructive to analyse the primitive equations of

CHAPTER 2. T H E SHALLOW- WATER MODEL 10

motion (definecl befow) nsing isentropic rat her t han Cmtesian coordinates. Isent ropic IeveIs

correspond to surfaces of constant entropy, and represent different quasi-horizontal surfaces of

the atmosphere. In isentropic coordinates, the equations of mot ion for cont inuously stratified

Bow are expressed in terms of the potential temperature @ as (Salmon 1998)

for the momentum equations, where

and

for the continuity equation, with f = 2R sin 6 the Coriolis parameter (here 0 denotes lati-

tude), k the unit normal vector to the x-y plane, and hdO, with h G &/a@ G le, denoting

the vertical separation between isentropic surfaces. Also, û = (u, v ) is the horizontal veloc-

ity, and V is taken to be a horizontal operator. Inherent in atrnospheric considerations of

largescale motion is the assumption known as the hydrostatic approximation, which gives

rise to an additional equation in the isentropic primitive equations formalism,

This relation is assumed valid for srnall vertical relative accelerations, and for regimes in

which the horizontal length scale of motion greatly exceeds the vertical lengt h scale. The

primitive equations of motion (2.4)-(2.7) are the hydrostatic version of the NS equations

(wit hout viscosity or forcing).

The shallow-water mode1 consists of a single fluid layer with a varying surface height q ,

and bottom topography Iocated at depth 2 = -K. Topography mimics forcing from below.

In Cartesian coordinates, the SW equations may be derived from the equations of motion

(2.3) with the imposition of the hydratatic approximation in the form

CHAPTER 2. THE SHALLO W- WATER MODEL

Entegratiou of (2.8) mrniug h o p e o n s ttrrid ( p = constant) yields for the pressure

The geopotential 4 is defined such that 4 i gz. For homogeneous fluid of variable depth,

the mass continuity equation is derived From the non-divergent condition V Ü = O. The

rotating SW equations are thus given by

and, for the continuity equation,

where again Ü = (u, v ) , V = (a,, O,) and h = + H denotes the vertical thickness of the

fluid layer.

Noteworthy is the similarity in form between the isentropic, hydrostatic equations for

a single layer and the shallow-water equations. In particular, the continuity equation in

isentropic coordinates (2.6) reduces to that for the shallow-water mode1 (2.11) with the

already established equivalence h = az/%. The horizontal momentum equations (2.1) and

(2.10) are analogous, the only difference coming from the pressure-gradient force which, in

the primitive equations, is determined by interactions with other isentropic layers.

There is also a direct mat hematical correspondence between the primitive equat ions of

motion in 2D and the SW equations. The horizontat wtocity is divergent, as shown by

the continuity equation (2. II). The SW fluid is in fact incompressible in 3D, with a free

surface, but the SW equations look the same as those for a compressible 2D fluid, with a

special choice of equation of state. According to this analogy, surface gravity waves in the

SW equations correspond to sound waves in the 2D compressible equations. Although the

S W modei neglects coupling between different iseot ropic layers in the Eart h's at mosphere,

its ability to capture the effects of rotational and divergent dynamics renders it a feasible

mode1 from which at least a partial understanding of middle atmospheric dynamics m- be

buitt.

CHAPTER 2. THE SHALLOW- WATER MODEL

2.2.2 Why the SW Mode??

As mentioned in the Introduction, the SW model incorporates both balanced and unbal-

anced dynamics. In particular, the solutions to the SW equations consist of (unbalanced)

fast modes, in the form of inertia-gravity waves, and (balanced) slow modes, in the form of

planetary waves and vortical dynamics. It is the inclusion of both types of motion which

renders the SW mode1 an appropriate setting for the study of planetary-wave ( P W) and

gravity-wave (GW) activity. Alternative models such as the NS equations with the Boussi-

nesq approximation may also be used to replicate salient features of middle atmospheric

dynamics, but at the cost of making the system three-dimensional. For the purposes of the

present study, the SW model provides a sufficient framework for the study of balanced and

unbalanced dynarnics on a single isentropic surface.

In the above derivations, the effects of viscosity are neglected. This is thought to be an

accurate assumption for the middle atmosphere. Molecular viscosity, as ascertained from ki-

netic theory, possesses an approximate value of v . 165cm2/s, corresponding to a molecular

mean free path of A 5 6.5 x 10-'crn (Salmon 1998). Estimates for eddy diffusivity induced

by 3D effects, as inferred from lower stratospheric balloon data ( Alisse et al. 2000), show it

to be negligible for 6s > IOkm, and the saine estimate would then apply for eddy viscosity

assuming an eddy Prandtl nurnber of order unity. Since the spatial resolution in the present

study is on the order of 300km, it can be safely assumed that viscous effects within the

stratosphere and mesosphere are smdl.

2.1.2 Properties of the S W Model

Fast and slow dynamics, which as discussed below are central to the characterization of

balance and irnbalance, can be identified in terms of the properties of the SW model. Of

particular interest is the dispersion relation for the linearized system of equations. Lineariza-

tion of the SW equations proceeds with expansion of the velocity about its rest state Û = O,

with the assumption that the surface height is close to being constant. It is instructive

CHAPTER 2. THE SHA L LO W- WATER MODEL 13

to +te the SW eqtcations in t m s of the vorticity and divergence, and the corresponding

streamfunction + and velocity potential X, to explore the distinctive features of the vortical

and divergent components of the Bow. For + and x defined by the Helmholtz decomposition

the relative vorticity given by C &u - 8,u and the divergence given by 6 &u + a,v, it follows that C = V2$ and 6 = V2y. Treating j as constant for now (the f-plane

approximation), the resulting linearized SW equations are therefore expressed as (in the

notation of Salmon 1998)

and

while the continuity equation takes the form

with 9 = gq. These Iinearized equations of motion give rise to plane-wave solutions the

form $' = $oexp(i(k=x + k,y - ut)) , y = y0 exp( i (k ,z + k,y - ut)) and r$ = &exp(i(krs + kvy - wt )), with the dispersion relation

where w denotes the frequency, and k = d k r 2 + kY2 the horizontal wavenurnber. The solution

w = O corresponds to balanced dynamics, and is associated with the vortical motion. (There

are no Rossby waves on the f-plane.) The solution w = IJP + gHk2 denotes the frequency

associated wit h inertia-gravity waves propagating in opposite directions. In the absence of

rotation the gravity waves travel with a critical phase speed equal to the group velocity,

given by cg = awlak = @. Gravity waves become dispersive only in the presence

of rotation. The gravity-wave behaviour is characterized by two limiting cases: that in

CHAPTER 2. THE SHALLOW-WATER MODEL 14

wkickr the h d n m t a t wavenumber is much smaikr than the inverse of the Rmsby radius of

deforrnation LR = m/ f , which is the defining horizontal length scale in this problem,

for which the waves are inertial, and that in which the horizontal wavenumber is much

greater than L~-', for which the waves are pure (non-rotating) gravity waves. In the middle

atrnosphere, as previously argued, the horizontal length scale greatly exceeds the vertical

scales of motion, as is implicit in the SW mode1 considerations.

2.2 Balance and Imbalance

In the Introduction, balanced flow was defined in terms of potent ial-vorticity dominatrd

flow with weak divergence, and unbalanced flow in terms of flow wherein the rotational

and divergent components are comparable. Such a distinction is easily illustrated by the

SW equations. From the diopenion relation (2. i6), the concept of fast and slow dynamics

may be identified (Nore and Shepherd 1997) and balanced/unbalanced dynamics accordingly

defiued. As uulliiitd in the Introduction, the ratio E of vorticd frequencies V I L to gravity-

wave frequencies d m (as found in the previous section) is given by the expression

When E < 1, there is a timescale separation and the concept of fast/sIow dynamics is

meaningful. Within this context, balanced dynamics refers to the situation where the slow

frequencies dominate. It is evident from (2.17) that E can be small for either Ro < 1 (quasi-

geostrophic (QG) dynamics) or Fr << 1 (barotropic dynamics). The condition Ro « 1

yields geostrophic balance, but is not valid near the equator. Hence Fr « 1 is in some sense

preferable for global Bows. Both limits give rise to a situation wherein vorticity greatly ex-

ceeds divergence, which is the basis for the current operational distinction between balaoced

and unbalanced flow.

In order to elucidate such a distinction, it is helpful to examine the leading-order be-

haviour of the Iinearized SW equations (2.13)-(2.15) in the context of the potentiai vorticity,

CHAPTER 2. THE SHALLO W- WA'rER MODEL

defined as

and the divergence 6. The potential vorticity q (which is materially conserved by the S W

equations) is slow to leading order, since it evolves on the advective timescale L/V. This

can be seen explicitly as follows. For small departures from the surface height , r ) < H, the

linearized potential vorticity is, from (2.18),

This expression is valid for both balanced and unbalanced dynamics. For linear gravity

waves, the time derivative of (2.19) becomes

for 4 = g? the geopotential height. Using (2.13) and (2.15), it follows that aplat = O for

linearized dynamics; q is "frozen" on the fast timescale.

In contrast, the divergence is fast to leading order and mishes for slow dynamics. Con-

sidering the slow behaviour (which is steady on the fast timescale and thus represents the

zerdrequency mode) as determined by quasigeostrophic dynamics (Ro « l), the second

term of (2.13) cannot be balanced by any other term. Balance then requires that 8 = V2x = O

for slow motion (to leading order in Ro). whence 6 « <. Similarly for balance as determined

by barotropic dynamics (Fr < 1): from (2.151, it is shown that 6 = O is necessary (to

leading order in Fr) to compensate for the dominant 11 Fr2 tenn.

The relative vorticity C is both fast and slow at leading order (this is apparent in the

coupling between geopotential height and relative vorticity in (2.14) and (2.20)). As noted

above, q = f / H for a gravity wave for al1 time. Equation (2.20) then implies that i # 0,

and we may expect that 6 -- < for unbalanced flow.

Separation of balanced and unbdanced Bow according to the relative magnitude of vor-

ticity and divergence is o d y a first approximation; the fast and slow modes are distorted

at higher orden in a. Namely, the potential vorticity is coupled to fast dynamics, and the

CHAPTER 2. T H E SHALLOW- WATER MODEL 16

divergence to stow dynamics, at &km d e r . k ft the coaphg of slow and fast dynamics at

higher orders in the small parameter E which renden the definit ion of balance and imbalance

somewhat fuzzy.

Balanced (slow) quasigeostrophic dynamics rely on the assumpt ion t hat contribut ions

from the acceleration term in the momentum equation (2.10) are small relative to contribu-

tions from the Coriolis parameter f. (This can be seen also in the definition for Ro, which

is the ratio of the acceleration to the Coriolis terms: Ro cg 1 implies that the acceleration

terms are srnall.) Also noteworthy is the neglect of acceleration terms for balanced dynam-

ics associated with barotropic modes. Relaxation of the balance assumption â/at .- VIL

in (2.13) shows that the time rate of change of the relative vorticity i = V2+ and of the

divergence 6 = V2y are comparable. Thus for unbalanced (fast) dynarnics, there is no

longer a separation of timescales: the vorticity gradients evolve on timescales comparable

to those associated with the divergence. The geometric equivalent for the concepts of ad-

vective timescales [slow motion) and fast rlynnrnirs will he explored in the analysis of the

Okub+Weiss/Hua-Klein criteria in Chapter 3, and in the concepts of nonlocality and locality

addresszd using relative dispersion in Chapter 4.

I t is usefui to recall previous shallow-water analyses which have investigated the role

of divergence in altering the descript ion of twedimensional turbulence. Potent ial vort ici ty

is conserved in tw-dimensional turbulence: in the absence of an extra dimension, vortex

stretching and tilt ing are not perrnitted. The direct enstrophy cascade refers to the transfer

of potential enstrophy (square of potential vorticity) from large to small scales of motion. By

contrast, the inverse energy cascade refers to the transfer of energy from small to large scales

of motion. Farge and Sadourny (1989) found the inertia-gravity-wave and vorticai nonlinear

interactions to be dynamically distinct. However, it was shown that the inertia-gavity waves

reduce bot h the direct enst rophy cascade, responsi ble for füamentation in 2D turbulence, as

well as the inverse energy cascade associated with the development of coherent structures.

Unbalanced motion itself has a direct cascade, as shown by Faxge and Sadourny (1989) and

CHA PTER 2. THE SHA LLO W- WATER MODEL Lï

argua+ t heore t idS . by Warn (1986).

In contrast with the findings of Farge and Sadourny (1989), the statistical equilibrium

arguments of Warn (1986) irnply that energy is extracted from the rotational modes and

supplied to the gravitational modes during relaxation to equilibrium. The balanced compo-

nent of the flow is thereby eroded until most energy is found a t small scdes, suggesting the

presence of a direct energy cascade. Both investigations demonstrate, however, that it is the

gravity waves, uninhibited by the potential enstrophy constraint, which are responsible for

energy transfer from large to small scales. -4 direct energy cascade is therefore possible for

unbalanced dynarnics, and impossible for bdanced dynamics. The destabilizing influence

of the gravity waves upon the more coherent vortical structures prevalent in the balanced

component of the flow, and the associated erosion of the large-scale straining mechanisms

that give rise to the development of steep gradients or filaments in tracer fields, suggests

an association of the unbalanced component of the flow with mixing, or the homogeniza-

tion of srnall-sali. ~triirture. We oeek to determine, in this study, the nzturc of such a

correspondence.

It should be noted that the aforement ioned analyses were performed for low Ro. In

the present investigation, the influence of the unbalanced component upon the balanced is

sought for the complementary scenario: that is, for Fr *: 1 and arbitrary Ro. The condition

Fr « 1 includes the possibility of investigation of leogth scales much smaller than the

Ross by deformat ion radius.

2.3 Shallow- Water Model Implementat ion: Numerical Model

The numerical model used to implement the SW model equations of motion (2.10), (2.1 1)

is a semi-Lagrangian, finite-element scheme. The semi-Lagrangian scheme incorporates fea-

t ures from bot h Eulerian advect ion schemes, wherein the advect ion of particles is determined

hom a fixed geographical location, and a fully Lagrangian advection scheme, where instead

CHAPTER 2. THE SHALLOW-WATER MODEL 18

one httows the ftnid particte trajectories (Staniforfii and (=ôté 1991 ). Provided by the Me-

teorological Service of Canada (fomerly known as the Atmospheric Environment Service),

the shallow-water numericd model is spherical and is known as the Global Environmental

Multiscale (GEM) model. In order to demonstrate how the goveming equations for the

shallow-water model are solved nurnerically, it is instructive to rewrite the S W equations in

the symbolic form (Côté et al. 1990)

for F the dependent quantity (the spherical wind images Lr =- u cos(8) and V = vcos(8)

where 6 denotes the latitude, for the momentum equations, and the geopotential height field

4 for the continuity equation), and G the linear and nonlineu terms found in the equations

of mot ion. Semi- Lagrangian discret izatioo is t hen manifested in the equations

where F+ = F(X,O,t + At) , G+ denote the parameters e d u a t e d at the time level of the

forecast, while Fo = F(X - a, 0 - 0, t ) and G" are evaluated upwind, at the previous time

level. The parameters a and ,B denote the distance traveled in X and 0 in time At. In

addition, a, b are numerically determined constants subject to the constraints Ci ai = O and

More usefat is the expticit representation for the continuity and moment um equations

which govern the shallow-water model. Included in Appendix A is a description of the

numerical implementation of the shallow-water model equations.

Previous studies (Bartello and Thomas 1996) have shown t hat the semi-implicit , semi-

Lagrangian model seems to damp gavity waves strongly. This feature is overcome here by

forcing the gavi ty waves with suficient amplitude. While gravity-wave darnping would be

problematic in studying shallow-water turbulence, it is not a problem for the examination

of a mesospheric regime where the gravity waves axe still linear (Le. not breaking), as is

considered in the present investigation.

CHAPTER 2. THE

2.4 Forcing Forcing

This section ou tlines

SHALLOW-WATER MODEL 19

Mecharrisrn: Pianetary- and Gravity- Wave

the forcing mechanism implemented in the shallow-water mode1 to

mimic both stratospheric and mesospheric dynamics. This includes a basic state, chaxac-

teristic of the westerly jet found in the wintertime hemisphere (taken to be the Northern

Hemisphere), and adapted from the jet proposed by Juckes and McIntyre (1987), in addition

to planetary-wave forcing and gravity-wave forcing. The remainder of this thesis is dedicated

to determining how the planetary- and gravity-wave forcing, superimposed on the mean flow,

influence the kinematics of the flow field in terms of mixing and transport.

The basic-state zonal jet is given by the expression (Ngan 1997)

for

A = 55, B = 13, al = 1.1, a2 = 0.3, bl = 0.5. (2.24 )

This westerly jet possesses a maximum at 50°N, and a zerewind line at 23ON. The cor-

responding height field, derived from the gradient-wind balance relations, constitutes the

initial geopotential perturbation + I l g , or the geopotential result ing from the basic state.

Superimposed on the basic state is the planetary-wave (PW) forcing, implemented as flow

over topography. This fming is intendecl to generate the dynamics resniting from Rossby

waves in the wintertime stratosphere, which are normally of zonal wave-1 or wave-2, and thus

to reflect mixing properties of a balaoced flow field, where motion is governed by potential

vorticity. The stationary-wave forcing is implemented via the expression (Ngan 1997)

where & = (1/2)hpw is the forcing amplitude, A(t ) = 1/2(1 + tanh((t - 5)/2.5)) is the

tirne envelope (with t in days) which switches on the forcing over t = 10 days, B(0) = 2

Bo cos(0) exp [- ( ( 9 - t lO) /O, ) ] is the latitudinal envelope and C(X) = 2, cos(X), praviding

CHAPTER 2. THE SH-4LL0 W- WATER MODEL 20

a waue-l forcing in longitude A. The pmmtefs & = ? 5 O , B, = 30Q, & = 2.6, ,; = f .O

and hPW = 9.8 x 10~rn~/s* are used, as in the study by Ngan (1997), which give a realistic

amplitude for planetary waves in the middle stratosphere. The peak of the latitudinal

envelope occurs at about 60°iV, a feature which will appear in the divergence and potential

vorticity fields examined in Chapter 3. As will be discussed in Chapter 3, the dynamics

established by the P W forcing scheme are characterized by a coherent "st irring" mechanism,

known as chaotic advection. Figure 2.1 shows the structure of the geopotential height field

providing the applied wave l forcing. This forcing mechaoisrn generates planetary waves

which break in a critical layer.

Figare 2.1: Geopotentid height fieM rrsed for PW forcing, with contour levet spacing of 900 m2/s2, a t t = 20 days. Dashed lines denote negative contours.

For the gravi ty-wave (G W) forcing mechanism, bot h spatial and temporal aspects of the

forcing are constructed to mimic a small-scale, Buctuating field characteristic of the mese

spheric regime (as inferred from GCMs ) . Mat hemat ically, the forcing funct ion in spectral

space is described by the equation

&w(w)116 exp i19(n, rn, t), if n' < n < N k ~ ( n , m ) = IO, ot herwise.

where n is the total wavenumber, rn is the zona1 wavenumber, n' = 3 is the minimum

CHAPTER 2. THE SHALLO W- WATER MODEL 21

wavenumber fomd f o r the €Mi case an& = 1.5 x 10m'f s2 is the amplitude of the

gravity-wave forcing. The choice of amplitude is to some extent arbitrary but is chosen to

yield a kinetic energy spectrum representative of the middle mesosphere. The exponent 116

in (2.26) was determined empirically to obtain a realistic spectral dope of the kinetic energy.

The phase 3(n , m, t ) is modified ternporally via a recursive filter scheme

where 7 defines the temporal autocorrelation T of the forcing according to the relation r /At =

1/(1 - y), which rneasures the memory time in timesteps. Here y = 0.92, corresponding, for

mode1 time steps of At = 15 minutes, to an e-foiding time r of approximately three hours,

and r is a random number between O and 2r. The perturbation field is then transformed to

a regular latit ude-longit ude grid using spherical harmonies such t hat

This is the form of the gravity-wave forcing that is incorporated into the GEM shallow-water

model.

The geopotential field generated from the GW forcing scheme at a particular tirne is

depicted in Figure 2.2. Noteworthy is the spatial variability in the geopotential, in contrast

to that defined for the PW forcing. As exhibited in the figure, an addit ional tapering function

is impfemented to avoid spurious effects a t the poks where the spatial discretization has

singular behaviour. The tapering function is given by

I z ~ e - (1 + tanh (*) ) , 9 -

where the 7 case refers to tapering in the Northern and Southern Hemispheres, respectively.

Numerical parameters are selected empirically to ensure that the gravity-wave forcing en-

cornpasses the midlatitudes and surf zone, while avoiding regions near the poles.

Also included in the model are iinear relaxation terms for the geopotential and velocity

field: the system is slowly relaxed to its basic state on a timescale of 10 days, namely, to

CHAPTER 2. THE SHALLO W-WATER MODEL

Figure 2.2: Geopotential height field used for GW forcing, with contour level spacing of 900 nz2/s2, at t = 20 days. Dashed lines denote negative contours.

the westerly jet, in order to balance the forcing and reach a statistically steady state. The

modified moment um and continui ty equat ions are presented in Appendix A. 1.

Three experiments are considered for the present investigation. The superposition of the

stationary-wave forcing and the basic-state shear is hereafter recognized as the planetary-

wave (PW) experiment, and is used to mimic stratospheric dynamics. The addition of gravity

waves through the gravity-wave forcing scheme allows the interaction between planetary-wave

and gravity-wave activity, and is denoted for the remainder of the study as the PW/GW

experiment. The PW/GW experiment is thought to reflect mesospheric dynamics due to the

presence of a small-scale, high-frequency component to the flow. In the absence of stationary-

wave forcing, the experiment is referred to as the GW experiment, and redects behaviour

resulting from the combined effects of the mean flow and gravity waves. The present study is

prirnarily based on cornparison of results from the P W and P W/G W experiments. Reference

is made to the G W experiment in some instances to illustrate the effect of the stationary-wave

forcing on teniporal properties of the system fields (such as the geopotent i d or veloci ty field).

In Chapter 1, a fourth experiment, similar to the GW experiment but with a temporally-

CHAPTER 2. THE SHA LLOW- WATER MODEL

Figure 2.3: Geopotential height field used for PW/GW forcing, with contour level spacing 2 2 of 900 m /s , at t = 20 days. Dashed lines denote negative contours.

decorrelated flow field (i.e. the recursive filter scheme is replaced by a random temporal

forcing), is used as a benchmark for Brownian motion. and is referred to as the GWRT

(gravity wave, random in time) experimeat . For the current investigation, the model was run wi th longitude-lati tude spatial resolut ion

of 145 x 72 g i d points, and sampled every three hours (except where otherwise stated). It

should be noted that for the given spatial resolution and model timestep, there is a minimum

spatial resolution of A r 175krn at midlatitudes, and a minimum temporal resolution of

At = ISmin. The runs extended for a mode1 time of 70 days, in order to ensure that

statistical equilibrium was attained. Subsequent analyses neglect the first 20 days of the

run, unless otherwise noted, in order to avoid spurious effects (evident in time series (not

shown) of various physical quantities) resulting from the initial implementation of the forcing

mechanisms.

Figure 2.3 is a soapshot of the geopotential height field used for the PW/GW forcing.

Notewort hy is the dominance of small-scale features south of about 70" N, in cont rast to the

coherent structures prevalent for the P W case.

CHAPTER 2. THE SHALLOW-WATER MODEL

Figure 2.4: Zona1 wind fields at t = 70 days for the a ) PW experiment and b) PW/GW experiment, with contour intervals of 10 m / s . Dashed lines denote westward-directed winds.

The corresponding zona1 wind fields for both the PW and PW/GW experiments are

presented in Figure 2.4 ab t = 70 days. Apparent in the PW experiment is the existence of

coherent feat ures depicting the cat's eye structure characteristic of stratospheric winds in the

wintertime extratropical stratosphere. By contrast, increased spatial variability is evident

for the PW/GW case.

The enhancement of fluctuations is even more pronounced for the meridional winds, as

demonstrated in Figure 2.5. Some structure exists in the meridional component of the wind

field over the poles and in the surf zone for the PW case. However, gravity waves appear to

destroy this structure in the PW/GW experiment. The implications of this will be studied

in Chapters 3 and 4.

CHAPTER 2. THE SHALLOW-WATER MODEL

Figure 2.5: Meridional wind fields at t = 70 days for a) PW experiment with contour intervals of 1 m / s , and b) PW/GW experiment with contour intervals of 3 m/s . Dashed lines denote sout hward-directed winds.

2.5 Kinetic Energy Spectra and Spectral Slopes

The present study, as emphasized in Chapter 1, is motivated by the desire to quantify mixing

in the middle atmosphere arising from balanced and unbalanced dynamics. The irnpetus for

this characterization cesides in the spectral slopes obtained for stratospheric and mesospheric

spatial wavenumber spectra of kinetic energy examineci in an intercornparison study of Mid-

dle Atmosphere Generai Circulation Models (MAGCMs) (Koshyk et al. 1999). i t was shown

there that the model kinetic energy spectra exhibit a steep siope in the stratospheric region.

relative to a significantly shallower dope apparent at higher altitudes in the mesospheric re-

CHAPTER 2. THE SH.4LLO W- WATER MODEL 26

giou. The steep s p d siope for the stratospheric case is a signature of baianced dynarnics:

upward-propagating, breaking planetary waves in the wintertime stratosphere give rise to

large-scale, coherent features in the Bow. The shallow spectral slope demonstrated at the

small scales of the mesospheric kinetic energy spectra is thought to be a signature of unbai-

anced dynarnics: decreasing densi ty wit h altitude allows gravity waves to grow wit h altitude

as they propagate upwards into the mesospheric region. (The gravity-wave spectrum is a p

parent in aircraft measurements in the lower stratosphere (Bacmeister et al. 1996) but only

at smaller spatial scales.) Bdanced dynarnics are distinguished by the rotational component

overwhelming the divergent component. This may be understood in terms of the vortical

Bow induced by the planetary waves in the stratosphere. By contrast, unbalanced dynam-

ics are active when the rotational and divergent components are comparable. The physical

process responsible for this behaviour in the real atmosphere is the enhanced gravity-wave

activity at higher altitudes. A central question to emerge from kinetic energy spectra and

spectral sloprs is: How will the ~~nbalanced component of the Bcw, reflected in the shallow

kinetic energy spectra, rnodify the properties of rnixing found for non-divergent flow? This

is the starting point of the present analysis.

2.6 Energy Spectra and Diagnostics

This section presents the diagnostics used to demonstrate that the dynarnical fields obtained

do possess properties characteristic of what is argued to be a rnesospheric wind field. Such

diagnostics further emphasize the distinguishing characteristics of the simulated stratospheric

and mesospheric regimes, as detemined from the PW and PW/GW experiments.

2.6.1 S pherical Harmonic Decomposition Scheme

Since the horizontal velocity field is composed of both a rotational and divergent component.

as depicted by the Helmholtz decomposition (2.12), while the kinetic energy is expressed as

CHAPTER 2. THE SHALLOW- WATER MODEL '27

(hem t h e ande brackets denote aPeragingm att tatitndes and fongit udes) the kinet ic energy

spectrum may be decomposed into a rotational and divergent component such that

where n and m denote the total and zond wavenumbers, respectively. The spectra were

computed numerically using Canadian Centre for Climate modelling and anaiysis (CCCma)

subroutines.

Figure 2.6: Time-averaged (20 to 70 days) kinetic energy spectra for a) totd, b) rotational, and c) divergent flow components.

Figure 2.6 depicts the ternpordly-awraged kinet ie energy spectra for the total, rot at ionai

and divergent components of the winds, where temporal averaging was implemented from

day 20 to day 70 to ensure that the spectra were determined from an equilibrated flow.

The PW experiment bas a steep spectral dope and the rotational component dominates

the divergent cornponent. For the PW/GW experiment, the steep spectral dope at large

spatial scales (small wavenurnber) gives way to a shallower dope at smaller scales, and the

divergent component is not small compared to the rotationai component at the smailer scales.

So t hese experiments do mimic the characteristics of strat ospheric and mesospheric kinet ic

energy spectra found by Koshyk et al. (1999). In order to f u t her examine the partitioning

CHAPTER 2. THE SHALLO W- WATER MODEL 28

between roiat i ond and divergent kinetic energp in the gravity-wave part of the spect ra, it is

useful to examine polarization relations.

2.6.2 PoIarizat ion Relations

The purpose of this section is to derive rui expression for the expected partitioning between

KErot and KATdiv as a function of n, based on the polarization relations for gravity waves.

Polarization relations indicate how various fields are related in a wave structure (Gill 1982).

and are obtained by substituting plane-wave solutions into the linearized equations of motion.

From (2.13), it follows that C = -if d/w, so that

where the last equality is obtained from the dispersion relation (1 .1 ) with k replace4 by

n / a . Relation (2.32) implies that h'Erot/KEdi, + 1 as n + O, while KErot/f<Ediv -t O

as n -t oa (which is the Fr < 1, finite Ro regime). Noteworthy also is the connection of

both limiting cases to LR. Inertial waves are characterized by the former ( ~ ~ ~ k ~ « 1) and

gravity waves by the latter ( ~ ~ ~ k ~ > 1). In this cootext, the ratio of the Rossby radius of

deformation to the length scale of interest ( L = k-') governs the nature of the unbalanced

motion, that is, the relative magnitude between the rotational and divergent components

of the flow. At moderate scales, up to n - 35, it is true that h*ErOt < KEdiv and that

K Erot/KEdiu decreases with increasing n, as expected from (2.32). The d u e of n for which

g H n 2 / a 2 f 2 = 1 is n - 2, taking f = 10-*s-'. The sharp increase in h' Erot/KEdiu observed

for the very smallest scales might be an artifact of the semi-implicit , semi- lagrangian t ime-

stepping scheme employed by the GEM shallow-water model, which distorts the behaviour

of the highest-frequency waves. However, this is only speculation.

2.6.3 Spatial and Temporal Autocorrelations and Power Spectra

Spatial and temporal autocorrelations provide an indication of the spatial and temporal

scales a t which the fields of interest decorrelate. Longitudinal spatial autocorrelations are

CHAPTER 2. THE SHA LLO W- WATER MODEL

dcrrfated arrordmg to the expression

where indices i and j denote the longitudinal and latitudinal grid points at which 4 is

positioned, k denotes the index corresponding to the longitudinal spatial lag, and LC; is the

number of longitudinal grid points. The overbar indicates temporal averaging and (9) here

denotes the zona1 mean of 6. Shown in Figure 2.7 is the longitudinal spatial correlation

associated with the geopotential for the PW, PW/GW and GW experirnents at 47.5OiV as

a function of the spatial lag p. The lack of much distinction between the PW and PW/GW

experiments may be attributed to the latge-scale features set by the planetary-wave forcing

mechanism. There is no decorrelation at al1 in the PW case; the oscillatory behaviour with

strong negat ive correlations merely reûects the spatial periodicity of the wave- 1 P W held.

Some evidence of the G W field can be seen in the rapid partial decorrelation at short lags

in the PW/GW case. For the CW experiment, a much more random regime is established

so that the geopotential field decorrelates at much smaller scales.

Figure 2.7: Spatial longitudinal autocorrelations of 4 for PW, P W/G W and G W experiments at latitude 47.5ON.

CHAPTER 2. THE SHALLOW-WATER MODEL 30

The temporat autocorretation is expresseci, in a manner andogous to the spatid auto-

correlat ion, as

for a tempord lag r , where here the angle brackets denote averaging over al1 longitudes

and the range of latitudes from 20°N to 60° N, and over tirne. Figure 2.8 demonstrates the

temporal autocorrelations established for the PW and PW/GW regimes: once again, the

geopotentid decorrelates more rapidly in the P W/GW case t han in the P W experiment due

to the presence of the high-frequency component to the flow. The geopotential does not

appear to decorrelate at d l for the PW experiment, over the time interval shown; again, the

negative correlations reflect the periodicity (here in time) of the P W field.

Figure 2.8: Temporal autocorrelation of the total geopotential # for PW (red) and PW/GW experiments (blue). Shown in the inset is the temporal autocorrelation for the PW/GW experiment for the total geopotential (solid) and the perturbation geopotential #jGw (dashed). sampled every quarter hour from days 60 to 65. The time lag r is in units of days for the lower panel and hours for the inset.

Sbown in the inset of Figure 2.8 are the tempord autocorrelations for the total geopoten-

tial 4 and perturbation geopotential 4Gw for the PW/GW experiment. The perturbation

CHAPTER2. THESHALLOW-WATER MODEL 31

geopotentid, tkat is, the forcing gwpotentiat that is read into the mode!, has an e-foîding

time of approximately 3 hours, which is consistent with the pararneter specification outlined

in Section 2.4. The total geopotential, however, which characterizes the actual dynarnical

field, decorrelates much more rapidly, as is evidenced by a decorrelation time on the order

of an hour. This more rapid timescale is the timescale of the gravity waves themselves.

Figure 2.9: Power spectra of a) the divergence 6 (in s-*) and b) the potential vorticity q = g(i + /)/(cf + cb') (in r n - ' ~ - ~ ) for the PW (red) and PW/GW (blue) experiments.

Shown in Figure 2.9 are power spectra for the divergence and potential vorticity q =

g ( c + J ) / ( V +dl) for both the PW and PW/GW experiments. Observed in the power spectra

for the divergence is a slight accumulation at low frequencies for the PW case (presumably

slaved to the slow dynamics), and over a range of higher frequencies for the PW/GW case.

In the latter case most divergent power is subinertial (with period less than a day), hence

presumably due t o gravity waves. The power spectrum for the P W experiment appean to

mimic the high-frequency behaviour eshibited for the P W/G W experirnent . suggesting sonie

weak spontaneous generation of GWs in the PW case. The peak found at a frequency of

24d-' appears for the divergence sarnpled every quarter and every half h o u , for a mode1

time step of a quarter hour. It is therefore not a Nyquist frequency effect, but instead may be

CHAPTER 2. THE SHALLOW- WATER MODEL

Figure 2.10: Time mean of the zonally-averaged divergence variance for the different exper- iments. The zona1 average is cornputed over all longitudes. Shown in b) is the divergence variance for the PW experiment; note the difference in scale.

Figure 2.11: Time mean of the zonally-averaged potential vorticity variance for the different experiments. The zona1 average is computed over al1 longitudes. Shown in b) are ( q 2 ) and (g)2 for the P W/GW experirnent; note the difference in scale.

CHAPTER 2. T H E SHALLO W- WATER MODEL 33

abtributecl to the spat id resototion of the modet If we assnme the 2Az wave to be s t rongly

damped, then a minimum wavelength of 4Ax - 1000km, with H = (& + 4' ) lg - lOkm,

gives a period of At 4 A z / m - lhr corresponding to a frequency of 24d-'. (The

Coriolis term is i r r e l e ~ n t for such high frequencies.) The sarne analysis for an experiment

run at h d f the spatial resolution (72 x 36 grid points) yields At - 2hr, and indeed results

in a peak at a frequency of 12d-l. So the peak corresponds to the highest gravity-wave

frequency resolved by the mode1 at the given spatial resolution.

The power spectra for the potential vorticity indicate that the vortical cornponent of the

flow is controlled by rnuch slower time scales for both the P W and PW/GW experirnents. The

potential vorticity associated with the PW/GW experiment possesses some power at higher

frequencies, demonstrating a fast component to the potential vorticity. These diagnostics,

taken together with the spatial spectra, suggest that the P W experiment is characterized by

balanced dynamics at al1 scales, while the PW/GW experirnent is characterized by balanced

dynamics at large scales and by tinhalanc~d rl ynamifs at small rcalee.

The time mean zonally-averaged divergence and potential vorticity variance is repre-

sented in Figures 2.10 and 2.1 1. respectively, for the PW, PW/GW and GW experirnents.

Cornparison of the PW/GW and GW experiments demonstrates that the planetary-wave

forcing appears to suppress the divergence variance. The time average of the potential vor-

ticity variance demonstrates greatly enhanced variance for those experiments with a large

balanced component to the Bow: the divergent component does not significantly induce fluc-

tuations in the potential vorticity, as expected. (The high d u e s a t the North Pole are an

art ifact of numerical discret izat ion.)

In summary, the results conveyed by the kinetic energy spectra, autocorrelations and

power spectra confirm that the PW/GW forcing mechanism has indeed achieved what it

was supposed to do, and serves to mimic mesospheric dynamics.

Chapter 3

Flow Visualizat ion

In order to understand the nature of mixing in the middle atmosphere, it is instructive

to examine geometric features which characterize the velocity field. This chapter seeks

to present an interpretation of the kinematic properties evident in the idealized middle

atmosphere described in Chapter 2. In particular, stirring and mixing are defined, and their

applicability to balmced and unbalanced flow, respectiveiy, explored. The behaviour of

passive tracer motion in baianced and unbalanced flow is examined t hrough the parti tioning

of the flow into ellipt ic and hyperbolic components according to the O kubdVeiss criterion.

This Eulerian diagnostic is supplemented by the Lagrangian diagnostic of patchiness, wherein

the dynamical properties of the velocity field may be andysed. Liapunov exponents are

further used as an aid to understanding what the flow field looks like: that is, where one

might find regions of strong mixing, or barrien to transport.

Of central interest in this chapter is the influence of the unbalanced component of the

velocity field upon the coherent structures prevalent in a regime described by chaotic advec-

tioa (wherein a regular velocity field gives rise to irregular particle paths). The pertinent

question to be asked is, how is baianced flow modified with the presence of an unbalanced

component? How are velocity gradients, or the strain field, d e c t e d with the introduction of

a srnall-scale. high-frequency component to the flow? In particular, what are the implications

of this modified flow topology for mixing?

CHAPTER 3. FLOW VlSUA LIZATION 35

This section seeks to present concrete definitions for stirring and mixing, which will later

assist in the identification and quantification of transport in balanced and unbalanced flow.

Stirring is characterized by the development of increased spatial gradients in a tracer

field (Garrett 1983), while mixing refers to the hornogenization of small-scale structure. The

distinction between stirring and mixing in the context of geophysicd fluid dynarnics was first

examined by Eckart (1948). Here it was shown that the competing effects of stirring and

mixing are manifested in an Eulerian representation of passive tracer advection.

In this representation, the advect ion of passive t racers in 2D is described by the advection-

diffusion equation

for 7 the passive tracer field and

equations excep t for the fact that

K the diffusivity. This is

in this instance, with the

rerniniscent of the Navier-Stokes

prescri bed Eulerian veloci ty field

Y, the differential equation is linear in 7. Expressed in terms of the passive tracer gradient,

the advection-diffusion equat ion yields (see e.g. Salmon 1998)

w here

denotes the mean-square gradient of, or spatial variability in, the tracer field, and ri is the

outward normal. Significant in this expression is the sign and magnitude of the stirring

t e m (U V7)V2T relative to the diffusive term i((V2n2. Mixing associated with the

diffusive term acts to decrease the spatial gradients in the tracer fields. By contrast, the

stirring associated with the advective term generally increases the variability in the tracer

field (Eckart K M ) , alt hough ei t her an increase or a decrease in the mean-square gradient

can occur, depending upon the orientation of the Bow relative to the tracer gradient. Cocke

(1969) presented a general argument that line elements are in general stretched, and that,

CHAPTER 3. FLOW VlSUALIZATION 36

comeqnentb, tracer gradients witt increase on average as a resuIt of stirring. Certainly in

statistical equilibrium, (3.1) implies that the stirring term must act to increase the tracer

gradients (apart from boundary effects). The role of the advection-diffusion equation in

describing tracer transport will become clear in Chapter 4 wit h a statistical interpretation

of transport, and is mentioned here only in contrast to the Lagrangian interpretation upon

which a dynamical systems and thus geometrical analysis is built.

The essence of stirring is captured in the seminal paper by Aref (1984), who studied tracer

evolution using a Lagrangian approach, governed by the differentid equation for particle

positions

A dynamical systems interpretation of transport identifies stirring according to the stretching

and folding of material lines defined by continuous sets of fluid particle trajectories. Mixing

follows as a consequence of diffusion or coarse-grain averaging. Stirring and mixing may be

identifieci by exarnining relative motion about fixed points. This is best demonstrated for a

Lagrangian interpretation of the non-divergent , twedimensionai, steady equations of motion

(Ottino 1989). Within such a system, a stagnation or fixed point is a coordinate position

for which Y ( & ) = O. Thus, for an arbitrary point in space, t = Zo + 2, linearization about

the point of zero flow yields for the dynamical equations of motion

where arrows denoting vectors have been (and are hereinafter) dropped, is the strain

or velocity gradient matrix and X = (rt,y'). Since this equation can be appiied to the

separation AX between two fluid parcels such that d A X / d t = aV/ax(AX), such a char-

acterization provides a means by which local dynarnics may be monitored, as discussed in

Chapter 4. The eigenvdues of (3.5), as will be outlined in the section describing the deriva-

tion of the OkubeWeiss criterion, determine the nature of the stagnation point, and thus the

Buid particle trajectory behaviour near this point. Imaginary eigenvaiues give rise to elliptic

CHAPTER 3. FLOW VlSCrALlZATlON 37

points, and r ed eigendues to hyperbolic points. Stabfe and unstable manifolds of hyper-

bolic fixed points, perceived intuitively as one-dimensionai surfaces in a twedimensional

phase space, define the topological structure of the flow field. Stable and unstable manifolds

of a stagnation point correspond to the set of initial conditions that approach it as t + oo and t -t -00, respectively. Furthemore, the intersection of stable and unstable manifolds at

two different stagnation points gives rise to a heteroclinic structure, known as a separatrix.

What happens when the extra dimension of time is introduced, so that V = V(x, t )? It

is such a situation that is of interest for many fluid dynamical systems, and for geophysical

flows in pasticular. In time-periodic or quasi-periodic flows, Buid particle trajectories are

allowed to evolve in time. An extended phase space exists, comprised of the original spatial

degrees of freedom (x, y) in addition to time. Instantaneous stagnation points are t hen, in

the hyperbolic case, replaced by hyperbolic trajectories. The linearized equation of motion

in this case is t hus

where y ( t ) denotes the hyperbolic trajectory, where here the term "hyperbolic t rajectory"

refers to a Ruid particle trajectory in extended phase space with saddle-type structure. or a

moving saddle-point. Central to the concepts of transport and rnixing is the phenomenon

wherein transverse intersections of stable and unstable manifolds emerge as a consequence

of an unsteady flow field. For an incompressible flow with periodic or quasi-periodic time

dependence, these invariant regions of transport, knowo as lobes, are bounded by stable and

unstable manifolds. The essence of the dynamical systems description of transport in this

case resides in lobe dynamics, which serve as mediators of transport. For Bows with general

time dependence, the concept of a hyperbolic trajectory becornes ambiguous and templates

for transport and mixing are determined using techniques t hat detect Lagrangian coherent

structures over finite time intervals, such as finite-time stable and unstable manifolds. Phe-

nomena such as filamentation are described by the flux into and out of these regions. Near

hyperbolic fixed points, geometric patterns referred to as *tendrilsn characterize the devel-

CHAPTER 3. FLOW VlSUALlZATlON 38

opmenb of filaments generahed by t h e interseetion of stable and anstable manifolds. M m

importantly, Ytendrils" are a signature of straining mechanisms which give rise to stirring.

By contrast, trajectories near elliptic fixed points are identified by "whorls", as the trajec-

tory encircles the elliptic point. It is the generation and interaction of these features which

provides a visual interpretation of stirring and mixing in the fiow.

3.1.1 Stirring and Chaotic Advection, MWng and Diffusion

In order to distinguish between stirring and mixing, it is useful to examine the concepts of

chaotic advection and turbulent diffusion in the context of the differential equations which

govern the Lagrangian representat ion. For a steady or t ime-independent, non-divergent,

twedimensional Bow field, pat hlines correspond to streamlines associated wi t h the well-

defined streamfunction @(x, y). Phase space trajectories and transport are t hen descri bed

by Hamilton's equat ions (e.g. Wiggins 1988)

In this representation, phase space is identical to physical space. Moreover, the systern is in-

tegrable for steady flow. This has important consequences for the particle trajectories: t hcir

paths are determined completely and follow the streamlines. Non-integrability results when

time-dependence is introduced. In t his case, a spatially-smooth, t ime-periodic veloci ty field

in an Eulerien representation can give p i s e te ehaotic particle trajectories in the Lagrangian

representation, a phenornenon known as chaotic advection. Studies by Aref (1984) regard-

ing the passive tracer field produced by two "stirrers", and investigations of stratospheric

dynarnics (Pierrehurnbert 1991, Ngan and Shepherd 1999a) have shown that filamentation

is indeed a characteristic feature of chaotic advection. A quantitative validation of chaotic

advection exists in the measurernent of the exponent i d lengt hening of material contours.

Liapunov exponents, as will be discussed later in the chapter, by monitoring the separation

between neighbouring trajectories, provide the means by which the stretching of material

contours may be ascertained.

CHAPTER 3. FLOW VISUALEATION 39

Mixing ensues when t h d d patMmeP eirctosed by the separatrk structure estabtished

by stable and unstable manifolds are perturbed temporally: after successive iterations, or as

time progresses, the filamentation process produces tendrils of ever-smaller scales as t + a?. Mixing can be identified when regular particle trajectories are replaced by irregular ones.

Turbulent advection presumes a stochasticaily-defined Bow field. A probabilist ic interpre-

tation, such as that explored in Chapter 4: enables passive tracer behaviour to be investigated

in the presence of a spatially and temporally irregular flow field. As noted by Aref (1984),

it is the relationship between diffusion and the andom-walk problem (see e.g. Vergassola

1999) which permits the Lagrangian stochastic representation to be regarded as analogous

to the advect ion-diffusion equation.

Thus, mixing in the Lagrangian problem can be unambiguously identified with diffusion

when particle paths follow independent random walks; in the Eulerian representation. the

governing equation of motion for the tracer dynamics is then given by

where K is the diffusivity. The diffusion approximation is a consequence of the property

that a well-defined decorrelation time exists for the veloci ty field (Vergassola 1999); any two

particles thereafter follow independent randorn waiks. The presence of Lagrangian autocor-

relations dong the particle trajectories, however, as are apparent for flow fields characterized

by coherent at ruet ures f Garret t et al. 1985), renders the diffttsion approximation mmlid. For

well-defined coherent structures, as in chaotic advection, the advective term in the Eulerian

description dominates, and the Lagrangian interpretation provides a suitable framework from

which the fixed points may be discerned. and the paxticle trajectories oear such fixed points

determined. When no decorrelation time is present, anomalous or non-Fickian diffusion gov-

ems tracer particle dynamics, as will be explored in Chapter 4. Moreover, it is necessary

to determine a diffusion coefficient appropriate to al1 and not just the smallest scales of

motion, as in (3.8). This is the subject of turbulent diffusion theory, whose applicability

to mixing in the current study resides in the investigation of anomalous diffusion proper-

CHA PTER 3. FLOW VlSUA LIZATlON 30

tm in the -ce of arr mrbatanced component to the ftow. White it has been shown that

chaotic advection, and hence stirring, describes stratospheric dynamics (Pierrehumbert 1991.

Ngan and Shepherd 1999a), it is anticipated that mesospheric dynamics will be distinguished

by mixing characterized by the properties of turbulent diffusion.

3.1.2 At mospheric Considerations and Expected Results

Within the stratospheric context, stirring is associated with filamentation as air is stripped

from the edge of the polar vortex. Tracer fields subjected to stratospheric winds, generated

both numerically and from observations, demonstrate the presence of a heteroclinic structure

in the velocity field (Ngan and Shepherd l999a). The "cat's eye" wave-1 or wave-2 planetary-

wave structure can be described by the intersection of stable and unstable manifolds. The

surf zone, located at mid-latitudes in the stratosphere, depicts a region of strong mixing and

is a consequence of breaking planetary waves (McIntyre and Palmer 1983, Shepherd 1998).

Mixing in this region may be attributed to the repeated stretching and folding of material

lines associated with the breaking planetary waves. In contrast, mixing in the mesosphere is

thought to be governed by the significant unbalanced component to the flow resulting from

the gravity waves. The extrerne case of mixing, normal (in the Gaussian sense) diffusion, is

never fully at tained for the timescales of interest in the stratosphere.

Previous studies (Ngan 1997, Shepherd et al. 2000) have shown that stirring, manifested

in the dynamicat phenornenon of chaotic achrection, and the physicd phenornenon of erosion

of the polar vortex within the wintertime stratosphere, is the predominant transport mecha-

nism in the stratosphere. By contrast, and based upon numerical simulations of mesospheric

dynarnics using a general circulation mode1 (Shepherd et al. 2000). it is anticipated that the

dominant small-scale, high- frequency gravi ty waves, super-imposed on the zonal fiow and the

planetary waves, will introduce a coarse-graining mechanism over many spatial scales to give

rise to a variant of diffusion, and thereby demonstrate enhaoced mixing. The plausibility

of this hypothesis is suggested by the very nature of the forcing mechanisms irnplernented

to mimic stratospheric and mesospheric dynamics in this study. The flow field for the PW

CHAPTER 3. FLOW VISUALIZATlON 41

experiment is driven by a determrmstic ge~potentid fretd. The resuhing Bow is batanced

and quasi-stat ionary. Tracer fields advected by wind fields generated from the shallow- water

mode1 nevertheiess follow chaotic particle paths in the surf zone (Ngan 1997). In contrast,

gravity-wave forcing by the geopotential height field for the PW/G W experiment is random

in space and Markovian in time. The stochastic nature of the forcing is projected ont0 the

velocity field through the shallow-water equations. In fact, as the inset to Figure 2.8 makes

clear, the Bow is more random than the forcing itself. That is, the randomness introduced

via the small-scale, high-frequency gravity waves gives rise to stochastic equations of me

tion for the tracer problem, which, in turn, should demonstrate sustained chaotic dynamics

for most particle paths, and thus enhanced rnixing. It should be noted that in the current

study, the velocity field to which the tracer field is subjected for the gravity-wave forcing

consists of a wide range of eddy sizes contributing to the dispersion of particles, but is not

turbulent. In fact, the PW/GW case used to simulate mesospheric dynarnics is comprised of

forwd-dissipative gav i ty waves, rather than gravity-wave turbulence. This is believed to be

an appropriate way to mimic a randorn field of gravity waves propagating vertically through

a given isentropic layer. However the efiect on tracer behaviour may nevertheiess be similar

to turbulent diffusion.

As mentioned in Chapter 2, gravity waves have bot h a rotational and a divergent compo-

nent although, as shown in the kinetic energy spectra of Figure 2.6, the divergent component

dominates. For the PW/GW case, one can therefore imagine the Buctuating field as being

comprised of a system of divergent "st irrers* wit hin the non-divergent field generated from

the planetary-wave forcing, each stirrer evolving randomly in time, in a manner analogous

to the point-vortex stirring rnechanism described by Aref (1984). One possible question to

be addressed from such a configuration is whether spiral structures, resulting from motion

around an elliptic fixed point, emerge or if. instead, filaments depicting advection near hy-

perbolic fixed points are evident, with the establishment of straining regions between elliptic

fixed points. Furthemore, how are the topological structures evident in the balanced (vorti-

CHA PTER 3. FLOW VISUALIZATION 42

cat) flow field modified by the prese~~ce of a dioergent cornponent? This is the centrai question

to be addressed in the present chapter. Are elliptic or hyperboiic components enhanced or

eroded with the presence of divergence? More generally, is the kinematic description of

balanced dynamics significantly altered by the unbalanced component? If so, what are the

implications for mixing? The punuit of possible explanat ions to t hese questions comprises

the essence and remainder of this chapter.

3.1.3 Stirring and Balance; M k h g and Imbalance

Central to the characterization of balance is the description of slow dynamics, as determined

by a dominant rotational component to the flow. This is manifested in the quasigeostrophic

(QG) approximation through the condition ~b = O (where the subscript b refers to balanced

dynamics), which is tantamount to the assumption that the QG and gravity-wave dynamics

are uncoupled. Consequently, the flow is determined entirely by the strearnfunction t,bb for

balanced dynamics. From the Helmholtz decomposition (2.12) it follows that

in a manner reminiscent of the Hamiltonian formulation (3.7). The existence of a balanced

streamfunction irnplies that chaotic advection, founded on streamfunction topology and re-

quiring spatial coherence in addition to a definit ive timescale, provides an appropriate inter-

pretation for tracer advection by bsfanced dynamics. This has indeed been found to be t h e

case for stratospheric dynamics, as demonstrated by numerous authors (Pierrehumbert 1990,

Ngan 1997, Shepherd et al. 2000). Significant also is the fact that for 0ow described by bal-

anced dynamics and chaotic advection, acceleration terms resulting from temporal variations

in the velocity field are small.

By contrast, the unbalanced dynamics are determined not by a prescribed velocity field.

but rather by a stochastic process. A probabilistic approach, as explored in Chapter 4, is

t herefore necessary to describe the propert ies associated wit h enhanced gravi ty-wave act ivi ty.

Furthemore, acceleration terms in the momentum equation can no longer be disregarded.

CHAPTER 3. FLOW VISUALlZATlON 43

&mebrie diagnostics ddesa-ibed in s t r b t sections are nsed to test t he nature of

the correspondence between stirring and balance (as depicted by the PW experiment). and

mixing and imbalance (as depicted by the P W/G W experiment ).

3.2 The Okubo-Weiss Criterion

In order to understand the nature of mixing in the middle atmosphere, it is useful to ex-

amine the geometric features which characterize the velocity field. Information regarding

the flow topology provides insight regarding the passive tracer behaviour as particle tra-

jectories sarnple different regions of space. The behaviour of passive tracer motion in t w e

dimensional. non-divergent flow may be dernonstrated by a partitioning of the flow into

vorticity-domiaated (elliptic) and strain-dominated (hyperbolic) components, known as the

O kubo-Weiss criterion (Okubo 1970, Weiss 1991 ). This is achieved through considerat ion

of the eigenvalues for the velocity gradient tensor, â V / a X , determined from the linearized

equaticns of relativc mot ion,

as addressed in Section 3.1. The square of the eigenvalues XI of the velocity gradient mat rix

yield the Weiss criterion for non-divergent flow

w here

for Sn = &U - a,w the stretching rate, S, = &v + a,u the shear component of the strain,

and w = &v - a,u the relative vorticity ( P r o v e n d e 1999). Strain-dominated regions are

represented by the condition

Qw > 0, (3.13)

whereas regions identified by

QW < Q

CHAPTER 3. FLOW VISUALIZATION 44

c o r r e ~ p ~ ~ ~ d to vorticity-dominatecf (ettiptic) regions. For Qw > O the eigenvdues for the

strain matrix are real, so that stretching of materiai lines occurs in the direction of one

eigenvector, and compression of material lines occurs in the direction of the second eigen-

vector. Particles in strain-dominated regions thus experience significant dispersion and net

displacement. For Qw < O the eigenvalues are imaginary, so that the particle-trajectory

mot ion wi t hin elli pt ic regions is characterized by trapping. Weiss ( 199 1) furt her demon-

strated that for non-divergent flow, where the velocity field can he characterized in terms of

the streamfunction $J alone (as shown by ( 3 3 1 , particle trajectories are determined by the

strearnfunction topoiogy.

Central to the derivation formulated by Weiss is the assumption that the velocity gradi-

ents are slowly varying with respect to the vorticity gradients. (Since vorticity is materially

advected, vorticity gradients correspond to tracer gradients.) As demonstated by variouc

authors (Basdevant and Philipovitch 1994, Hua and Klein L998), such an assumption is re-

strictive, and the Weiss criterion has been shown ta hold anly at the ccntrc of vortcx cores

and near stagnation (singular) points. (Lagrangian versions of the O kubo- Weiss criterion,

which moni tor the finite-time hyperbolici ty of a trajectory, generalize the conditions of t his

assumption, while also providing a rneans by which invariant regions may be ascertained

(Haller 2000, Lapeyre et al. 2001).) However, in spite of this restrictive assurnption, the

Weiss criterion has been frequently used to assess the dynamical properties of a turbu-

lent flow (Provenzale 1995, Elhmaidi et al. 1993). Turbulent flow can often be described

in terms of coherent structures ( Lesieur 1997); tw-dirnensional turbulence in part icular is

distinguished by the presence of coherent structures immersed in a background turbulent

flow (McWilliarns 1984, Bartello and Warn 1996, Weiss 1998). The enstrophy cascade, or

vorticity variance cascade from large to small scales of motion as a consequence of the con-

servation of vorticity in 2D turbulence, is identified with vortex filamentation, whereas the

merging of vortices is thought to reflect the inverse energy cascade from small to large scdes

of motion. The Weiss criterion, initidly proposed to explain the phenornenon of enstr*

CHAPTER 3. FLOW VISUA LIZATION 45

phy transfer in M hydrodynamicd sysfems constrained by the conservation of vorticity (as

there is no stretching permitteci in 2D Bow, in contrast to 3D flow), provides a geometrical

means by which the link between the physical and spectral interpretations may be estab-

lished. Although the detection of Lagrangian coherent structures provides a more rigorous

account of geometrical templates that govern turbulent mixing (Haller 2001), the Eulerian

diagnostic of the OkubctWeiss criterion serves as a qualitative assessrnent of strain- and

vort ici ty-dominated regions.

For the purposes of the present investigation, it is instructive to examine the topoIogica1

features of the flow in the presence of gravity waves, as incorporated by the shallow-water

model. Okubo (1970), in addressing the study of singularities in oceanic flow, established a

condition analogous to the Weiss criterion for a divergent velocity field. In this case, four

motions are ascribed to the velocity gradient field: the stretching rate Sn, the shear rate of

strain S., the vorticity w and the divergence 6. The eigenvalues for the strain matrix are

expressed, in contrast to the non-ditvergent case, as

A* = +* d m ) * 2

It is evident from this relation that the competition between the deformation

(3.15)

and rotation-

dominated regions is dec t ed by the divergence. Consequently, a much larger class of dynam-

ical regimes exists with the presence of a divergent component to the flow. For both eigen-

values real and either positive or negative, in contrast to the hyperbolic regime established

for A- < O < A+, there may exist either unstable or stable motion. For eigenvalues which

are cornplex conjugates of one another, trajectories follow spirals around stable (Re(A) < 0)

or unstable (Re(X) > 0) points. In this case, as 6 + O one recovers the elliptic regime of the

Weiss criterion, where the motion is neutrally stable.

The criterion used in the present investigation to

divergent flow is given as

assess stirring and mixing properties in

2161 oF2)) ?

CHAPTER 3. FLOW VlSUALIZATlON

Figure 3.1: Quadrants of the a) Re(X)-lm(X) plane and b ) Re(X2)-im(A2) plane. Vertical and horizontal hatching depict hyperbolic and elliptic regions, respect ively.

which is, as for the Weiss criterion, the real component of the square of the largest eigen-

value. Positive values correspond to regions of strong divergence or deformation, since

the real part of AI is then larger than its imaginary part. Values with Qow < O indi-

cate the dominance of elliptic fixed points, since the real part of At is then smaller than

i ts irnaginary part. (The correspondence between hyperbolic and elliptic regions in the

Re(A)-h(X) and Re(X2)-im(X2) planes is depicted in Figure 3.1 .) One shortcoming of

t his parameterizat ion is its failure to detect regions of negat ive divergence, or convergence.

As will be shown later in the discussion pertaining to the Hua-Klein criterion, however, this

characterization provides the moet efficient means by which the Weiss triterion, its divergent

counterpart, and the corresponding Hua-Klein criterion may be compared.

3.2.1 Okubo-Weiss Criterion Results

Figure 3.2 shows both the original Qw and the proposed Qow field at t = 45 days, where here

and for the remainder of Chapter 3,t = O corresponds to day 20 of the iû-day experiments.

in order to be consistent with the temporal characterizations of Chapter 4. Regular latitude-

longitude plots, as opposed to stereographic plots, are shown, since the focus of the present

investigation is on the surf zone, rather than the polar vortex. As expected, littie distinction

CHAPTER 3. FLOW ViSUALIZATlON

Figure 3.2: Contour plots determined from the Weiss criterion Qw for a ) PW and c) PW/GW experiments, and from the OkubeWeiss criterion with divergent cornponent Qow for b) PW and d) PW/GW experiments, al1 at t = 45 days. Units are in s-*.

is apparent between Qw and Qow for the PW experiment, as indicated in panels a) and b).

This is consistent wit h the kinetic energy spectra shown in Figure 2.6, where the magnitude

of the divergence is seen to be less than that of the vorticity in the PW experiment. Recall

that Qw and Qow are equivalent in the limit of vanishing divergence. Figure 3 . 3 ~ ~ ) shows

that the divergence that is present in the PW erperiment is a slaved component of 6 linked

t o the planetary wave, not a gravity wave. Evident in Figure 3.2 is the erosion of large-

scaie spatial gradients in the Bow with the presence of the unbalanced component in the

PW/GW experiment. Whereas distinctive rotation-dominated regions are prevdent in both

CHAPTER 3. FLOW VlSUA LIZATION

Figure 3.3: Divergence field (in s-') a t t = 45 days for a) PW and b) PW/GW experiments.

Qw and Qow for the PW experiment, with an apparent well-defined distinction between

regions of stretching (red) and rotation (blue), such a segregation disappears for the P W/G W

case. .4lthough one might find this behaviour to be intuitively plausible, the validation for

this phenomenon, as will be shown from additional flow visualization diagnostics, requires

an understanding of the interplay between advection in the zona1 direction resulting from

the basic-state shear, and transport in the meridional direction resulting from the combined

effects of the planetary wave and the gravi ty waves. The erosion of largescale structure in the

spatial gradients suggests t hat a diffusive mechanism arises resulting from the stochasticity

associated wi t h the gavity-wave act ivity. However. t his is only an instantaneous pict ure, and

CHAPTER 3. FLOW VlSUALIZATION

Figure 3.4: Potential vorticity field (in r n - l s - l ) at t = 45 days for a) PW and b) PW/GW experiments.

the question arises whether there is sufficient temporal persistence to lead to real diffusion.

Also notewort hy is the development of elongated features in the meridional direct ion

in the PW/GW experiment (Figure 3.2d). A more isotropie spatial distribution rnight be

expected in this case, due to the small-scale variability associated with the gravity waves.

In contrast, however. streaks of higher strain or divergence-dominated regions are evident

t hroughout the Qw, and more markedly the Qow field. Evident in panels a) and b) for the

PW experiment is the existence of a rotation-dominated region at midlatitudes. As will later

be discussed, this is a signature of the surf zone and in particular of the core of the cat's eye

CHAPTER 3. FLOW VISUALMATION 50

stmeburn. Pawk a) and b} in Figmç 3.2 kdicztte tht particies iocated withm this region

may remain trapped within the surf zone. Panels c) and d) demonstrate a partial survival of

the rotation-dominated region wit h the presence of gravity waves. By contrast, the coherence

associated with the straining mechanisms for the PW case is lost in the PW/GW case.

Shown in Figure 3.4 is the potentiai vorticity (PV) defined by (2.18) for the PW and

PW/GW experiments at f = 45 days. The most striking feature is the insensitivity of the PV

field to the presence of gravity waves. Linear gravity waves have zero PV signature, but it is

st il1 notable t hat the slow component of the dynamics remains essentially intact. Thus we can

think of the gravity waves as being superposed on the PW structure, to a first approximation.

It is interesting to note, however, that the PV contours surrounding the critical layer at which

the planetary waves break are somewhat modified in the PW/GW experiment. Although

in general the potential vorticity is an active tracer, it is approximately passive in a critical

layer (Rossby-wave critical-layer theory, e.g. Killworth and McIntyre 1985). Under these

conditions. the enhanced variability in the PV field also suggests that mixing is e n h a n 4

by the gravity waves, across the entire criticai layer. This is consistent with the picture

presented by the Okubo- Weiss criterion, which suggests small-scale regions of stretching

interspersed t hrough the larger-scale structures [panels c) and d) of Figure 3-21.

It is important to note that the Q field derived from the Okubo-Weiss criterion is deter-

mined a t a single instant in time: no information is provided regarding its time evolution.

A true evaluation of the changes in the dynarnical properties of the flow that aise with

the presence of an unbalanced (fast) component rnay therefore only be achieved when the

time evolution of the velocity gradients is considered. The latter feature is the premise upon

which the Hua-Klein criterion is founded.

3.2.2 Hua-Klein Crit erion

Although the Weiss criterion is applicable within the cores of elliptical regions, and in strongly

hyperbolic regions, the relaxation of the assumption that the velocity gradients evolve more

CHAPTER3. FLOW VlSUALlZATlON 51

slow'y than the vortieity gradients (Basdevant arrà Phitipovitch 1994, Hua and KIein 1998)

provides an alternative met hod for parti tioning the veloci ty field into distinct dynarnical

regimes. Weiss ( 199 1 ) addressed the subtleties associated wit h t his approximation and pur-

ported its validi ty based on a direct relationship between the t imescales associated wi t h

vorticity gradients, wherein only smdl scales of motion are considered, and with the gra-

dients of the strain. Weiss furt her argued that t his approximation should be applicable

for the purpose of investigating the direct cascade of enstrophy from large to small scales

(Weiss 1991). However, the OkubeWeiss assumption breaks down for a fluid with complex

time dependence such that the velocity gradients dong a particle path are rapidly changing.

The debate regarding tbis issue is manifested in the more general problem in turbulence

in which no separation of scales exists: that is, ail scales of motion, not only the smallest

scales, must be exarnined. The Hua-Klein criterion is based on the 2D non-divergent Euler

equations in a Lagrangian frame. Particle motion is described in terms of the Lagrangian

acceleration

7~ = X, (3.17)

for .Y the (Eulerian) position vector, as opposed to the velocity field. As a result, the square

of the eigenvalues for the acceleration gradient tensor, instead of the square of the eigenvalues

for the velocity gradient tensor, provides the cri terion by w hich the flow is partit ioned into

dynamically distinct regimes. An equivalent representation for (3.17) is given by

The eigenvaiues of the acceleration gradient matrix then assume the form

where it is observed that the radicand, contaioing the total time derivatives of the velocity

gradients and vorticity, distinguishes the Hua-Klein criterion from that proposed by Weiss.

The criterion used in the present anaiysis to interpret the Hua-Klein criterion is the real

CHAPTER3. FLOW VlSUALIZATIOlV

component of the k q e d eigeavairre af the acceteration gradient matrix:

The interpretation of the Hua-tilein criterion is similar to that for the OkubeWeiss

criterion. In particular, Q H K > O corresponds to strain-dominated regions (except in the

case where the st rain-dominated regions are more rapidly varying in time than the vortici ty-

dominated regions), and Q H R < O corresponds to vorticity-dominated regions. Central

to the distinction between the OkubeWeiss and Hua-Klein criteria is the inclusion of the

acceleration gradient t e m s in the latter. For unbdanced flow, where the timescales are

rapid, one might therefore expect the QsK criterion to be more appropriate than Qw. As

is explored later in this chapter, the analysis of unbalanced flow however requires that QHK

be generalized to divergent flow.

Although both the OkubeWeiss and Hua-Klein criteria are based on the premise of

spectral nonlocality, it is the QHK fi44 that, in monitoring the tirne-rate-of-change in the

velocity gradients, permits evaluation of the small scales of motion resulting from the un-

balanced component to the flow. Similarity in the Qw and QHK fields indicates that the

velocity gradients vary more slowly than the vorticity gradients, which is to Say that the

large scdes vary more slowly than the srnall scales. In this case dynamics are governed by

chaotic advection and tracer evolution by nonlocality. Dissimilarity in the two fields sug-

gests that spectrally local dynamics, activated by the unbalanced dynamics, is captured by

QHK The connections between locality versus nonlocality, and balance versus imbalance

will be addressed in Chapter 4. wit h the st udy between local and nonlocal dynarnics explored

t hrough single- and twepart icle statistical diagnostics.

3.2.3 Hua-Klein Criterion for Divergent Flow

As was noted in Section 3.2.2, the role of the Hua-Klein criterion in assessing imbalance

requires the generalization of QwK to divergent flow. In order to address dispersion in

divergent flow it is necessary to determine the eigenvalues for the acceleration gradient

CHAPTER 3. FLOW VISUALIZATION 53

tmsor for a ftow consisting of the normat Sn and shear S., strain rates and the vort icity w,

in addition to the divergence 6. From (3.18), the acceleration gradient tensor is determined

from the expression ab- av '

= (& + (a3) x, where the 1 s t t e m follows from (3.10). The eigenvalues for the acceleration gradient tensor

in the divergent case are expressed as

(3.22)

The criterion implemented to investigate the evolution in the velocity gradients is defined by

the real component of the largest eigenvalue, which we cal1 QL. It is evident t hat in the limit

6 -+ 0, QL reduces to Q H K . With the additioiial assumption of slowly-evolving flow fields, QL

becomes Qw. The expression (3.22) also validates the choice for the Okubo criterion: in the

limit of slorvly-evolving fields and non-vanishing divergence, QL and Qo are equivalent . As for the Qw, QOw and QwK fields, large positive values of QL correspond to strain-

dominated regions, and negat ive values to vorticity-dominated regions. It is necessary to

employ this interpretation with caution, however, since the divergence and its time derivat ive

may contribute to large positive values or small negative d u e s . That is, regions with positive

values may reflect a large divergent component, and significant alterations in the divergent

component w-ith time, rather than regions with enhanced filamentation as for the spectrally

nonlocal, non-di vergent case. Negat ive values may also reflect regions of st rong convergence.

The interpretation in this instance, however, might not be as misleadiag as for the enhanced

maxima case, since regions of strong convergence would plausibly be associateci with regions

of strong trapping.

3.2.4 Hua-Klein Criterion Results

Shown in Figure 3.5 are the Q fields derived from the Hua-Klein criterion for the PW and

PW/GW experiments. On the lefi is the Hua-Klein criterion derived for non-divergent flow,

CHAPTER 3. FLOW VlSUALlZATlON

a) b)

Figure 3.5: Contour plots determined from the Hua-Klein criterion for the a) PW and c) PW/GW experiments. and from the Hua-Klein criterion with divergent component for the b) PW and d) PW/GW experiments, al1 a t t = 45 days. Units are in s - ~ . Note the difference in scde with respect to paneh c) and d) in Figure 3.2.

QHK, and on the right is the field resulting from the modified criterion which incorporates

divergence, QL. Noteworthy is the fact that for the PW experiment the Hua-Klein criterion

generates fields that are similar to those of the Okubo-Weiss form, particularly at the cores

of eiliptic regions, and wit hin hyperboiic regions. This suggests t hat the velocity gradients

are slowly varying with respect to the vorticity gradients at midlatitudes, which in turn

indicates that the changes in the rotational component of the flow, associated with slow

dynamics, dominate those of the midlatitude region. Trapping regions, indicated by negative

CHAPTER 3. FLOW VEUALIZATION 55

Q values for bobb criteris, m a i n intact 6th the hctnsiorr of the time mtntion terms in

the veloci ty gradients. The longer-range straining and shearing influences of the coherent

elliptic region on the surrounding flow field resulting from a dominant advective term in the

Hua-Klein criterion are evidenced in the enhancement of the strain-dominated regions in

panels a) and b) of Figure 3.5. This is a consequence of the fact that the Hua-Klein criterion

represents a second-order correction to the Okube Weiss criterion, and thus serves as a more

accurate criterion for detecting actual Lagrangian behaviour. Nevertheless, the similarity

between Qow and QL, mitnifested in the survival of cotierent features in the flow, namely

in the elliptic regions, and enhanced representation of strain-dominated regions, implies the

existence of a distinctive separat ion in t imescales, and hence nonlocal dynamics.

A much more marked contrast between the Okub*Weiss and Hua-k'lein criteria is demon-

strated in the Q fields for the PW/GW experiment. The QHa field is increased by two

orders of magnitude relative to the Qw field. This may be explained in terms of the

dominance associated with the strain and vorticity tirni. d ~ r i v a t i v ~ r for the PW!CW ex-

periment. For a tirnestep of At = 9009, and for S - I O - ~ S - ' , from (3.20) it follows that

QHC - @ - ISI 5 10-8û-2, and sirnilarly for elliptic regions, in contrast to S2 5 I O - ~ O S - ~ .

The straining region present around latitude 70" N and seen in Qw in Figure 3.2, panel c),

broadens to overwhelm the region north of 50°N, as seen in panel c) of Figure 3.5. The

difference between Qw and QHK is to be expected for the PW/GW experiment, due to the

large unbdanced component set by the high-frequency gravity waves. Less obvious, however,

is the cause for the homogenization of the enhanced spatial structure, and disappearance of

the eLlipt ic region found at 40° N in the Q field. This behaviour would appear to suggest

that the fast component of the dynamics induced by the gravity waves significantly affects

the time evolution of the normal and shear components of the strain tensor.

Significant changes are also seen between QOw and QL for the PW/CW experiment.

Whereas the Qo field is controlled by the divergent, strain and vorticity components, the

QL field is controlled by the time derivative associated with each of these components. The

CHAPTER 3. FLOW VISUALIZATION 56

two orders of magnitude increase in the QL fierd may therefore be exptained as for QHK,

but applied to the divergent cornponent and its time derivatives. Exhibited in the QL field

are negative regions ( blue) interspersed amongst regions of st rong deformat ion ( red). Since

such negative regions axe not found in the QHK field, the negative values can be attributed

only to the time rate of change in the divergent component.

In general, the significant differences found between the Okubo- Weiss and Hua-Klein

diagnostics for the PW/GW experiment underline the fact that in the presence of an un-

balanced cornponent to the flow the evolution of the velocity gradients is not slower than

the evolution of the vorticity gradients. In fact, the strain and divergent components are if

anything evolving more rapidly in the presence of an unbalanced component to the flow. Io

particular, a term such as s will typically be larger in magnitude than S2 since the gravity-

wave timescale is much shorter than the advective timescale S-'. The significant gradients

in vorticity found in the PW experiment are replaced, with the inclusion of gravity waves,

hy romparably and more rapidly-evolving gradients in the strain and divergent components.

The Iack of a separation in timescales between the changes in velocity and vorticity gradients

for the PW/G W experiment is suggestive of the fact that tracer evolution is described by

local dynarnics. In contrat , the similarity in the Okubo-Weiss and Hua-Klein fields for the

P W experiment indicates t hat the large scales are slowly evolving with respect to the small

scales of motion, so that a distinct separation of timescales does exist. Thus for the P W

experiment, evolution of the passive tracer field ought to be govemed by nonlocal dynarnics,

namely, by the large-scale straining mechanisms observed in both Qow and QL.

3.2.5 Time Evolution of Velocity Gradients

The time evolution in the Qo field for the PW experiment is depicted in Figure 3.6. While

the Okubo-Weiss criterion indicates strain, vorticity and divergence-dominated regions at

a given instant in time, rather than the stretching and folding of material line elements,

examination of the time evolution of the QOw field provides some sense of the emergence of

coherent features in the flow. The vorticity-dominated region found at midlatit udes (depicted

CHAPTER 3. FLOW VlSUALIZATlON

Figure 3.6: Contour plots of the OkubeWeiss criterion Qow for the PW experiment at a) t = 0.5 days, b) t = 5 days, c) t = 30 days. and d) t = 45 days.

in blue in Figure 3.6) retains its structure until t = 30 days, after which the region appears to

elongate and after t = 45 days breaks apart into elongated features (panel d). This reflects

the time dependence of the cat's eye structure itself, presumably associated with breakup

of a PV filament. Regions of s t rong deformat ion emerge surrounding t h e cat 's eye structure

after thirty days. Evolution in the QL field is essentially identical t o Figure 3.6.

The evolution in Qow for the PW/GW experiment (not shown) exhibits no apparent

alteration over time in strain and vort icity dominated regions.

CHAPTER 3. FLOW VISUALlZATlON

3.3 Patchiness

Another geometric diagnostic used to assess the nature of dispersion is patchiness, or the tep

resentation of distinct average velocities in the flow (Pasmanter 1988, Malhotra et al. 1998).

While the Weiss cri terion examines the kinematic features associated wi th t h e geometric

properties of the flow, patchiness may be used to analyse its dynamical properties, through

investigation of the evolution of the Lagrangian average velocity. As a template for the role of

geometrical properties in governing transport and mixing, that is, the manner in which fluid

particles traverse the domain, or homogenize smail-scale structure, patchiness provides one

link to the probabilistic interpretation of how particles separate in a fluid. More concisely,

in tracking the evolution of regions with distinctive average velocities, patchiness monitors

the intersection of transport regions, w hile also providing statist ical informat ion about the

transport processes. The usefulness of patchiness for the present investigation in assessing

the nature of dispersion is twofold: the evolution in regions wi th distinct average velocities,

or patches, reflects the advection of a single tracer particle, while the degradation of such

patches should help in the interpretation of relative separation with respect to ballistic versus

diffusive behaviour, a concept which will be addressed in Chapter 4.

The mathematical definit ion of patchiness is the finite-t ime average Lagrangian veiocity.

Presented here is a synopsis of the mathematicd formalism relating patchiness to single-

particle dispersion, as outlined in Malhdra and Wiggios (1998). For a given Buid particle

t rajectory, the Lagrangian-mean zona1 veloci ty measured dong the part icle ~ a t h is expressed

(in the notation of Malhotra et al. 1998) as

for an interval of time t. The absolute, or singleparticle displacement, describing the trans-

lat ion of a particle and reflecting the sweeping motion of largescale eddies, is defined by

CHAPTER 3. FLOW VlSUA LlZATION 59

w h e ~ the angle backets specify spatiat averaging over ztO). The variance of the particle's

displacement is then characterized by

Equident expressions exist for meridional transport. Bot h (3.10) and (3.2 1) describe the

instantaneous evolut ion of neighbouring tracer part ides. Pat chiness, however, invest igates

the time evolution of the average Lagrangian velocity associated with a given region of

phase space. In particular, the collective time rate of change associated with fluid particle

trajectories is recorded by regions of distinct average velocity. Expressed in terms of the

average velocity, the single- part icle displacement becomes

where rn, = (v,) is the spatial average of the Lagrangian-mean velocity. Similady, the evolu-

tion of patchiness with time determines the dispersion of the passive tracer, as demonstrated

by the relation

~ * ( t ) = Dp(t)t2, (3.27)

for D, = ( (v , - rnJ2). It is this relation which defines patchiness dispersion. Intuitively, it

can be thought of as a measure of the spatial inhomogeneity in the Bow, and as a means of

determining how the fluid particles sarnple the phase space defined by the fluid flow.

tndeed, it kas been shom (Mdhotra et al. 1999) that patchiness is usefut for the de-

tection of luge-scale inhomogeneities in the Bow, and as a benchmark for nonergodicity, or

spatial and temporal nonuniformity. This can be seen in the relation between patchiness

dispersion and anomalous dispersion. Malhot ra et al. ( 1998) demonstrated t hat D, ( 1 ) goes

to zero eit her when the flow has zero temporal mean, or when ergodicity holds: i.e. when the

time average is equal to the spatial average. Birkhoff's ergodic theorem, which purports the

equivalence between time and ensemble averages, was used by Me& and Wiggins (1994) to

determine necessary and sufficient conditions for t2 dispersion. In particular, if Dp( t ) does

not go to zero as t + os, e.g. because of isolated regions, then A2(t) - t a and the motion

CHAPTER 3. FLOW VISUALlZATlON 60

is battistic. I t is pmposed that the patchiness diagnostic shodd emphasize the tendency for

the unbalanced component of the flow to induce mixing by prornoting ergodicity.

As demonstrated by Pasmanter (l988), patchiness depicts the interplay between regular

and chaotic orbits: stretching is prevalent in flow dynamics controlled by a basic-state shear;

patches representing chaotic orbits with the same drift velocity are elongated in the direction

of the mean flow. As will be discussed below, this phenomenon is observed for the zonal

winds in the PW experiment. The same study found that the presence of turbulent eddies,

analogous in some respects to the gav i ty waves in the PW/G W experiment , gives rise to a

degradation of patches.

3.3.1 Pat chiness Result s

Indicated in Figure 3.7 are the patchiness plots for the zona1 winds in the PW and PW/GW

experiments at t = 0, 10 and 20 days. The numerical code used to compute the patchiness

plots was provided by the Contml and Dynamical Systems division of the California Inst i tute

of Technology at the "Lagrangian Transport, Stirring, and Mixing in Geophysical Flows"

summer school, held in August , 1999.

The initial distribution [panel a), Figure 3.71 corresponds to the velocity field as observed

in the Eulerian representation and as shown in Figure 2.4. Evident in both Figures 2.4 and

3.7 is the wave-1 structure resulting from the PW forcing. Regions of maximal patchiness

at t = O coincide with maximum instantaneoos zond velocities. For t > O, since patchiness

monitors the average velocity along fluid particle trajectories, the translation in a part ide

cluster is recorded. Hence patchiness is connected with the concept of absolute dispersion.

For regions with steep spatial gradients, as is found between regions with distinct average

velocity, or distinct patches, the dispersion of individual particles wit h respect to t heir initial

position is increased due t o the prevalence of strong shear established by the mnal jet.

Maximal values initially found at the core of the jet and centred at 180°E, are, by t = 10

days, found over al1 longitudes. This indicates that all particles in the central latitude band

sample the strongest zonal velocities over a 10-day period. It is interesting to note that the

CHAPTER3. FLOW VISUALIZATION

Figure 3.7: Patchiness plots (as a function of Z(0)) for zona1 winds (normalized and in m/s ) for the PW experiment at a) 1 = O days. b) t = 10 days, and c) t = 20 days, and for the PW/GW experiment at d) t = O days, e ) t = 10 days, and f ) t = 20 days.

CHAPTER 3. FLOW VISUA LIZATION 62

distinction betweeu patches f m d south of the zond jet in panet c) of Figure 3.7 is apparent

in the strain-dominated regions surrounding the elliptic region to the east of 240° E and to

the west of 140°E of Figure 3.2 for the PW experiment. The cat's eye, Iocated around the

zero wind line (- 20°N) is depicted by patchiness close to zero, and characterizes the surf

zone associated wi t h breaking planet ary waves.

Noteworthy is the slight erosion of the coherent structure with the presence of the unbal-

anced component for the P W/GW experiment [panels d) , e) and f)]. The variability found

at the edge of the eastward jet suggests that some mixing occurs at the periphery of the

basic-state shear, because particles with different Lagrangian histories are contributing to

the structure. Enhanced transport into and out of the cat's eye structure is also apparent.

Even at t = O the cat's eye takes on a more diffusive character for the PW/GW experiment.

The erosion of patches is consistent with the behaviour predicted by Pasmanter (1988) in

considering the effects of turbulent diffusion, and with the inclusion of noise as examined

hy Malhotra et al. (1998). Larger patches are relztively insensitive to the spatially randam

and Markovian-in-time fluctuations: it is only the smaller patches which disappear witb the

presence of a rapidly-varying component to the flow. While the variability in patchiness ge-

ometry with the gravity waves might be thought to imply increased transport and perhaps

mixing, the penistence of the coherent features discards such a proposition. Dispersion in

the direction of the basic-state flow would appear to be largely unaffectecf by the presence

of gravity waves.

Also noteworthy is the fact that the patches have not yet spread over the globe after 20

days, for either the P W or PW/GW experiments. Ergodicity, or the uniform sampling of

phase space, has thus not been achieved, as is expected for atmospheric regimes. In fact it

is a very long way from being achieved. This would once again exclude the possibility of a

well-mixeci region for the Bow, since mixing is a signature of ergodicity.

Examination of the patchiness plots for the meridional winds, shown in Figure 3.8, reveds

greatly enhanced erosion with tirne of the coherent structures in the meridional direction

CHAPTER 3. FLOW VlSUALIZATlON

Figure 3.8: Patchiness plots (as a function of i(0)) for meridional winds (normdized and in m / s ) generated for the PW experiment at a) t = O days, b) t = 10 days, and c ) t = 20 days, and for the PW/GW experiment at d) t = O days, e) t = 10 days. and f) t = 20 days.

CHAPTER 3. FLOW VlSUALIZATION 64

relative bo bhst found for the zoxrat flow. Fbrtkmrrmq regions of maximum patchhess for

the PW case become fragmented with the presence of the unbalanced component, as is

demonstrated with the movement of regions with lower average velocity into the core of the

high-velocity region. In this respect patchiness cao be construed as a diagnostic for the

detection of mixing, whereas Liapunov exponents, as described below, serve as a diagnostic

of stirring.

It is interesting to note from Figure 3.8 that the initial patchiness plot for the PW/GW

experiment is more diffusive in character than at t = 10 days. Srnaller patches merge to pro-

duce the larger patch shown in blue at midlatitudes, which coincides with the elliptic region

found in the surf zone from the Okubo-Weiss criterion. The patch begins to disintegrate

at t = 20 days. One possible explanation for this is that the introduction of an initially

random field to the meridional flow generates motion &in to Brownian motion. As tirne

progresses, the passive tracer trajectories sample regions defined by the slow dynamics of

the fiow field. and subsequently traverse neighbouring domains to allow transport wi thin the

surf zone in the meridionai direction. If the gravity waves give a random walk on top of the

(spatially smooth) large-scale velocity field, then their effect on v, rapidly goes like d, and

hence becomes negligible compared to the contribution to v, from the largescale motion.

which goes like t . Consequently, the patchiness "organizes" until the large-scale motion itself

breaks up (after 20 days).

Pertinent to the present consideration is the difference between the zonal and meridional

patchiness plots. Panels c) and f ) of Figure 3.7 indicate that very little transport occurs

across the jet for either the PW or PW/GW experiments. Panels c) and f ) of Figure 3.8,

by contrast demonstrate t hat the gravity waves contribute to significant erosion of regions

with distinct meridional velocity. The distinction between the Okubc+Weiss and patchiness

diagnostics, narnely the significant alterations in the velocity gradient field with the presence

of gravity waves, and a Iack of alteration in the average zona1 Lagrangian velocity, may there-

fore be ascribed to the aforementioned cornpetition between zonal advect ion and enhanced

are considered separately, whereas the velocity gradient analysis for the O ku bo- Weiss and

Hua-Klein criteria involves the gradients of both the meridional and zona1 winds. The corn-

bined effect from both for ffow with a large unbalanced component (as demonstrated in the

PW/GW case) appears to result in the destruction of large-scale structure to the instan-

taneous velocity gradients. However the gravity-wave effects saturate over time, leading to

more coherence in the Lagrangian patchiness diagnostics.

3.4 Liapunov Exponents

Liapunov exponeots describe the rate at which neighbouring particle trajectories diverge,

while dso portraying how fluid interfaces stretch. Considered in the Liapunov analysis is the

evolution of the tangent vector rü(t ) = AZ(Zo, t ), for a trajectory initially located at position

4 and its neighbour found a t position Z0 + A&. Linearizing the Lagrangian equation of

motion (3.4) once again, but expressed in tems of thc .tangent vector Iü' ratber than the

velocity field, yields (Lichtenberg and Liebecman 1992)

where

is the strain tensor. The average exponential rate of divergence, or maximum Liapunov

exponent, is then given by

where d(&, t ) denotes the Euclidean n o m of the tangent vector

or the separation between the neighbouring particte trajectories. 4 d(&.O), and N is

the dimension of the system. The equivalence between (3.28) and (3.6) at t = O suggests a

CHAPTER 3. FLOW VlSUALlZATION 66

connectiorr betweerr tire Okuba-Weiss critakm and Liaponov exponents. In particuiar, A+

defined in (3.15) is the finite-time Liapunov exponent at t = 0, and the Liapunov exponent

is the average of A+ along particle paths. However, as is later noted, the correspondence

between the two diagnostics holds only for flow governed by strain, rather thon vorticity.

Intuitively, Liapunov exponents are a measure of the degree of chaos inherent in the

phase space since they indicate a sensitive dependence on initiai conditions. Positive values

are indicative of regions wi t h st rong exponential divergence, and t hus chaot ic mixing, while

Liapunov exponents w i s h in regions of regular transport wbere, for example, separation is

only algebraic in time (as in a shear flow). Moreover, different regions of physical space are

delineated by di fferent values of the Liapunov exponent . Geometrically, one may interpret Liapunov exponents as a measure of the distortion

inherent in a disk of passive tracer particles. For non-divergent, and thus area-preserving

flow, the area of this disk does not change with time. Thus, for a the semimajor axis and b

the semiminor axis of the elongated dink, or ellipse, their product c6 = c2 is constant, with f

the radius of the initial disk of tracer. As will be shown in Chapter 4, the Liapunov andysis is

reminiscent of relative dispersion which moni tors the separat ion between a pair of particles

within an ensemble. Indeed, as has been noted by Provenzale (1999), for infinitesimally

close neighbouring trajectories, the ensemble average of the maximum Liapunov exponents

is reflected in the relative dispersion attained.

3.4.1 Finite-Time Liapunov Exponents

Since, for most practical purposes. it is not possible to compute the infinite-time limit for the

Liapunov exponent, a finite-time variant is introduced (e.g. Pierrehumbert 1991 ). The above

equation is therefore modified su that the Liapunov exponent is averaged over a sequence of

finite tirne intervals. Moreover, the initial separation vector &(O) for a given tirne interval

r is determined according to the relation (Lichtenberg and Lieberman 1992)

CHA PTER 3. FLOW VISUALIZATlON

is the separation after the kth of n time intervals, and E CC I to ensure an initial small-

amplitude perturbation. The fini te-tirne approximation to the maximum Liapunov exponent

is t hen

w here

denotes the time average over the separations obtained for each interval. The limit of van-

ishing separation between neighbouring trajectories is replaced by the reinitiaiization proce-

dure; the sequence of finite steps serves to replace the infinite time limit. It is the finitetirne

Liapunov exponent which will be considered in the present investigation.

Relationship between Okubo-Weiss Criterion and Liapunov Exponents

In considering both the OkubeWeiss criterion (an Eulerian diagnostic) and Liapunov ex-

ponents (a Lagrangian diagnostic), a natural question which a ises is, how are the straining

mechanisrns indicated by the two diagnostics related? Are the localized regions of stretching

derived from the eigenvalues of the s train tensor suggestive of positive maximum Liapunov

exponents, and hence of chaotic regions? What role do vortical structures play in the Lia-

punov andysis?

It has b e n shown (Pierrehumbert 1992) that while Liapunov exponents indicate the rate

of stretching, the eigenvalues of the strain tensor do not. The two quantities arc equivalent

only when the rotational component of the flow is negligible. Even for such a situation, the

loog-time average for the stretching rate of an infinitesimal element is a spatially nonlocal

quantity, whereas the eigenvalues for the strain tensor are s p a t i d y local quantities. In ad-

dition, the Okubo- Weiss criterion is fiamedependent; e.g. a hyperbolic region can become

CHAPTER 3. FLOW VlSUA LIZATION 68

elliptic un de^ rapid rotation (Tabor end Kllrpper 1995). An mstantaneous Enlerian diagmx-

tic using the eigenvectors of the st min matrix, rather than the velocity gradient matrix, seeks

to eliminate elliptic regions resulting from rotation of the strain wes. Recent studies have

shown that the developrnent of a frame-independent diagnostic (Haller 2001) establishes

a greater correspondence between Eulerian and Lagrangian descriptions of transport and

mixing.

3.4.3 Maximum Liapunov Exponent Results

Presented in Figure 3.10 is the spatial distribution of the finitetirne Liapunov exponents for

the two experiments. Shown are the distributions for days 10, 30 and 50. Liapunov expo-

nents were computed using the particle advection code developed by Ngan (1997); numerical

methods used to compute Liapunov exponents are included therein.

Included as reference are the tracer fields a t t = 50 days for the PW and PW/GW

experiments. Evident in the PW experiment in Figure 3.9 is the cat's eye structure. By

contrast the PWjGW experiment yieids a passive tracer field which is characterized by

more diffusive behaviour. Such features are observed also for the Liapunov exponents, as is

shown below.

Prevalent in the P W case of Figure 3.10 are regions with large Liapunov exponents at the

edge of the cat's eye structure emanating from the hyperbolic points, indicative of particles

that have experienced strong stretching. Stirring is manifested in the developrnent of fila-

ments indicating strong expooential divergence, illustrating the chaotic advect ion description

of the stratospheric regime. For the P W/G W experiment , there is widespread stretching, but

with a remnant of the elliptic region at the centre of the cat's eye. The latter feature suggests

trapping in this region. Alt hough regions indicat ing strong stretching do not coincide wi t h

those found for the Eulerian interpretation at a given instant in tirne, this difference can be

attributed to the spatially noniocal nature of the Liapunov analysis and the spatially local

nature of the OkubeWeiss criterion. That is, the maximum Liapunov exponent monitors

the evolution in the separat ion between particle trajectories over a given time interval, t hus

CHAPTER 3. FLOW VISUA LIZATION

Figure 3.9: Tracer fields for PW (upper) and PW/GW (lower) forcing at t = 50 days. The init i d distri but ion consists of 21824 particles distributed in an area-weighted manner with a latitudinal spacing of l0 between 15.5" N and 52.5" N, and a longitudinal spacing of 0.5" x cos(@).

CHAPTER 3. FLOW VISUALIZATION

Figure 3.10: Spatial distri but ion of fini te-t ime Liapunov exponents (uni ts of daYs-') for the PW experiment at a) t = 10 days, b) t = ?O days, and c) t = 50 days, and for the PW/GW experiment at d) t = 10 days, e) t = 30 days, and f) t = 50 days.

CHAPTER 3. FLOW VISUALIZATION 71

providing a sense of the tracer ctynamics. By contrast t h OkubWeiss criterion determines

fixed points at a specific time, irrespective of prior flow properties, providing only a kine-

matic description. One rnight expect the nonlocal/locaJ discrepancy to be remedied to some

extent by the Hua-Klein criterion, yet cornparison of Figures 3.10 and 3.5 demonstrates that

the spatial distribution of maximal Liapunov exponents is not in very good agreement with

the Q H K field either. Of course, the Hua-Klein criterion only takes account of instantaneous

time tendencies, which rnay not accurately reflect the time evolution over a period as long

as 10 days. Thus it is still essentially a local diagnostic. Moreover, particle dispersion is not

governed entirely by straining mechanism.

3.5 Cumulative Assessment of Geometrical Propert ies

The effect of the unbalanced component of the flow upon the balanced component has been

shown, through the O kubo- Weiss/Hua-Klein, patchiness, and Liapunov-exponent diagnos-

tics, to be manifested largely in the spatial gradients of the flow field, and to a lesser extent in

the average meridional Lagrangian velocity. The observed qualitative similarity between the

Qow and QL fields for the PW experirnent and their dissimilarity for the PW/GW experi-

ment indicates that the velocity and vorticity gradients are evolving on different timescales

in the former case, and on timescales of comparable magnitude in the latter case. This

is consistent with the notion of spectrally nonlocal dynamics in the PW experiment and

spectratty tocd dyuarnics in the PW/O W experiment. Patchiness indicates a persistace of

coherent spatial structure for the zonal winds with the introduction of the gravity waves,

and a partial destruction of spatial structure in the meridional winds.

The geometrical diagnostics presented in this chapter aiso convey the influence of small-

scale, high-frequency gravity waves upon the velocity gradients. The Okubo- Weiss and Hua-

Klein criteria demonst rate the erosion of smoot h velocity gradients that occurs with the

introduction of a gravity-wave component to the flow. In contrast, the rotation-dominated

region characteristic of the surf zone appears to survive the introduction of the gravity-wave

component [panels c) and d) of Figure 3.21. Evident in the Liapunov spatial distribution

CHAPTER 3. FLOW VlSUALIZATlON 72

andpis is the correspondhg erosiorr of spatidy coherent strain-dominated regions, ait hough

the weak-stretching region within the cat's eye appears to persist, implying trapping.

These results suggest that the divergent component of the flow acts to erode the gra-

dients of the velocity field while preserving the coherent stmctures prevalent in the zonal

component, and homogenizing coherent structures in the meridionai Bow field. This may be

interpreted in terms of the interplay between advective and diffusive processes. In the zonal

direction, the gravity waves are forced to compete with both the basic-state shear and the

planetary waves, and are overwhelmed by the advection induced by the basic-state shear.

By contrast, in the meridional direction the gravity waves compete only with the planetary

waves, and overwhelrn the planetary waves, t hus giving rise to diffusive behaviour.

Also emphasized in the analysis of the geometric diagnostics explored in this chapter

are the implications of the modified flow topology for mixing. The destruction of strain-

dominated regions and partial survivd of the elliptic regions (in particular, the cat's eye)

as evidanced in the Okiih+Weirr/Hua-Klein criteria appearo to suggst that !arge particle

displacements dong the direction of maximal strain are suppressed, while mixing proceeds

and may be enhanced by the presence of gravity waves. The degradation of patches in the

meridional patchiness plots for the P W/G W experirnent implies the dominance of mixing

in the meridional direction. In contrast, in the zonal direction the gravity-wave effects on

patchiness rapidly approach the diffusive limit and are ovenvhelrned by the ballistic compo-

nent arising from the basic zonal shear flow. Liapunov exponent values showing maximal

straining regions surrounding the cat's eye structure exhibited in Figure 3.10 for the PW

experiment are suggestive of stirring. This is further manifested in the lack of sensitivity of

the PW experiment to the inclusion of time dependence in the velocity gradients (as noted

in the comparison of the coherent features for Qow and QL ), which in turn provides a sig-

nature of chaotic advection. The nature of stirring and mixing may be quantified through

the examination of spectrally nonlocal and local dynarnics, the essence of which is explored

in the following chapter using the concepts of single- and two-particle statistics.

Chapter 4

Statistical Diagnostics

While the geometric interpretation of Chapter 3 provides a qualitative perspective on how

particles disperse in balanced and unbalanced flow , a stat ist ical assessrnent of spect rd ly

local and nonlocal dynarnics, as defined in the Introduction, provides a quantitative aodysis

of the nature of mixing and stirring. The essence of this chapter therefore consists of the

examination of single- and tweparticle dispersion to identify spectrally nonlocal dynarnics,

wherein the evolution of a given length scale is dominated by much larger length scales,

versus local dynamics, where a given length scale is influenced by comparable length scales.

in part icular, the question of the relationship between balance vs. imbalance and spectrally

local vs. nonlocd interactions will be addressed.

A quantitative understanding of stirring and mixing may be achieved with the analysis

of single-particle (absolute) and two-particle (relative) statistics. Absolute dispersion, in

monitoring the displacement of partictes from their initial location due to advect ion, captures

stirring, and transport more generally. The translation of a cluster of marked particles,

which tends to reflect sweeping by largescale motions, is thus addressed by single-particle

statistics. By contrast, relative dispersion, in tracking the separatioo between a pair of

particles due to contributions from eddies of size comparable to their interparticle distance,

captures mixing. The spread in the cluster size is monitored by tweparticle statistics. While

absolute dispersion indicates the presence of coherent structures, relative dispersion describes

the relative contributions between strain and vorticity.

Considered in the present chapter are three experiments. In addition to the planetary-

CHAPTER 4. STATISTICA L DIAGNOSTICS 74

wave f PW) iuid ptanetérry/gratrity-waw (PWJGW) experiments investigated in Chapter 3,

a third experiment consisting of a gravity-wave forcing of the geopotentid height field that is

random in space and time, known as the GWRT (gravity-wave, random in time) experiment,

is studied. According to this analysis, the effect of the unbalanced dynarnics on the balanced

flow may be discerned according to its characteristics relative to two extremes: the PW case.

and the Fickian diffusion GWRT case, where Gaussian statistics are expected to dominate.

Stat istical diagnostics such as absolute and relative dispersion, the Probability Distribut ion

Functions ( PDFs) of these properties, and structure functions will be used to confirm estab

lished scaling laws, and hence to diagnose local and nonlocal behaviour. Turbulence theory

suggests (Frisch 1995) that higher-order moments are necessary to test the vaiidity of the

scaling laws derived from dimensional arguments. In light of this consideration, higher-order

moments of the PDFs of the relative separation will be investigated. Furthemore, signa-

tures of stirring and mixing will be quantified through such diagnostics as the correlation

dimension (a-sru~iatrrl with rnultifractal anaiysis) and PDFs of finite-time Liapunov expo-

nents, in addition to the previously encountered diagnostic of pat chiness. Key questions in

this particular investigation are: to what extent does absolute dispersion quantify transport

and stirring? To what extent does relative dispersion quantify mixing?

4.1 Absolute/Single Particle Dispersion

4 . 1 . Spat id Considerations

Taylor ( 1921 ) proposed that transport in incompressible turbulent fluid be described ac-

cording to the mean-square displacement of individual particles. This concept of absolute

dispersion is defined as (in the notation of Provenzale 1999)

where x i ( t ) denotes the position of the i th particle a t the time t, and N is the number of

particles in the ensemble. Angle brackets denote averaging over the particle ensemble, as

CHAPTER 4. STATlSTlCA L DIAGNOSTICS 75

opposed to a spat id and temporal average wtn& is msend for aiternative diagnostics such as

the Liapunov exponent, discussed below. Ensemble averages are assumed to be independent

of the initial time, although this is probably not strictly true in these experiments.

A bsolute dispersion captures the sweeping effects of large-scale eddies on passive t racers.

and in doing so, indicates the presence of large-scale (organized) structure in the flow. Due

to the wide assortment of eddy scales present in a turbulent fïow, or, in this case, a random

spectrum of gravity waves, no scaling laws can be derived as a function of particle displace-

ment. Particle motion is determined by the geometric properties of the physical space, in

addition to the stirring mechanism driving the flow (Majda and Kramer 1999). The concept

of absolute dispersion is useful when considering how the particle trajectories evolve with

time: spatial characteristics of the flow field are manifested in the temporal scaling relations.

4.1.2 Non-Ficlcian Diffusion and the Advect ion-Diffusion Equa- tion

Passive tracer behaviour, as noted at the beginning of Chapter 3, is govemed by the

advection-diffusion equation (3.1). Since it is the behaviour of a collection of particles rather

than a single particle which is of interest in ascertaining passive tracer transport, it is conve-

nient to consider the advection-diffusion equation for the mean passive tracer concentration

where here K refers to an effective eddy diffusivity, or the sum of the molecular diffu-

sivity K and a turbulent eddy diffusivity hPT resulting from fluctuations in the advecting

velocity Ü = + i?, for Y the mean advective velocity and the randorn, turbulent fluc-

tuations. Contributions from molecular difision will be ignored for the purposes of the

present investigation, for reasons similar to those presented in Chapter 2 to justify the ne-

glect of eddy viscosity in the governing equations of motion. It should be noted that (4.2)

is not a closed equation: that is, the process of averaging introduces a new variable f i , via

(i? .- V7) = -V * ( KT * V(7)) (Majda and Kramer 1999). It is the closure problem which is

CHAPTER 4. STATISTICA L DIAGNOSTICS 76

respomibte for the ômbigtxity hhereut in demnng a gendized advection-diffusion equation,

and as a consequence, an eddy difisivity.

Pertinent to the present discussion is the relationship between the eddy diffusivity and

single-part icle dispersion (Batchelor 1949). The absolute eddy diffusivi ty is defined such t hat

(Babiano et al. 1990)

As will be shown in subsequent sections, the eddy diffusivity is not defined for al1 flows: in

particulat, for those exhibiting anomalous, or non-Fickian diffusion.

Non-Fickian diffusion. or non-diffusive dispersion, describes a situation where particle

transport is not governed by a constant eddy diffusivity. In particular, the diffusion equa-

tion used to describe Brownian motion of molecuIes is not necessarily appropriate for the

description of large-scale motion. Rather, eddy diffusivity may be dependent on space and

time. In such a case several timescales are present, as opposed to a single timescale. An

additionai statist ical diagnostic which detects the presence or absence of a single rms veloc-

ity for the advecting flow is anomalous advection, depicted in the scaling law for the mean

singleparticle displacement

M = ( d t ) - xW),

as shown by del-Castillo-Negrete (1998). Both non-Fickian diffusion and anomalous advec-

tion demonstrate the shortcomings of the advection-diffusion equation in depicting transport

in 2D turbulence and its variants. In contrast to the kinetic theory of gases, where molecular

colIisions occur at length scales much shorter than the length scale over which the velocity

field varies, no such separation of length scales exists in turbulence. The andogous "mean

free pathn in fluids, known as the "mixing length" (Salmon 1998) or the distance over which

momentum transfer occun between fluid particles. is not constrained to be much smaller

than the largescale inhomogeneities in the flow itself. Fluid particles may move distances

comparable to the variations in the velocity field before exchanging momentum with other

fluid particles. This was noted by Richardson (1926), and is widely evidenced in atmospheric

CHAPTER 4. STATISTICAL DIAGNOSTICS

and oceimic phenmena.

4.1.3 Absolut e Dispersion and the Lagrangian Correlation Coef- ficient

The relationship between the Lagrangian correlation coefficient and absolute dispersion was

emphasized by Taylor (1921) in his assessrnent of dispersion in turbulent flow. Considered

was isotropic and homogeneous turbulence in a onedimensiond flow with continuous particle

motion. Correlations between the velocity associated wit h a fluid particle at a given time t ,

and the sarne fluid particle a t a later time t + r , provide a means by which the displacement

of a single particle rnay be quantified. The correlation coefficient is defined such that

for V the departure of the Lagrangian speed from the ensemble average associated with a

given particle, and where angle brackets denote, once again, the average over particle trajec-

tories. In considering the time average of the correlation between the velocity of the particle

a t an initial tirne, and its velocity at a later time, one is led to the following relationship

between the correlat ion coefficient and absolute dispersion (Taylor 192 1, Provende 1999.

assuming that the Bow is statisticdfy stationary so that (P(t)) is independent of time.

However, since the displacement .Y

encountered time rate of change in

may be expressed as

= x(t ) - x ( 0 ) - ( x ( t ) - r ( 0 ) ) = /O V ( r ) d r , the previously

the absolute dispersion, or the absolute eddy diffusivity,

&ter using symmetry in time. The absolute dispersion A2 = (X2) is t hen of the form

CHA PTER 4. STATlSTlCA L DIAGNOSTICS

An equivafent representation for c4.8) is

for the mean-square distance traveled by a particle in time T (relative to the ensemble

average). It is relationship (4.9) which enables the role of absolute dispersion in identifying

flow inhomogeneities to be demonst rated. As particle t rajectories sample different regions

of space, the memory R(r ) of the initial velocity (or the correlation between the initial and

final velocity as a function of time) will change accordingly, thereby modifying the absolute

dispenion A2.

4.1.4 Temporal Considerations

Universality implies an independence of 0ow statistics upon the forcing mechanism used

to perturb the flow (Frisch 1995). Although universal scaling laws for the displacement of

individual Buid particles cannot be established as a result of the sensitivity of singleparticle

statistics to the large-scale properties of the Bow, it hss been shown (Elhmaidi et al. 1993)

that temporal scaling laws can be constructed based on the topological structure of the

Bow. In this sense, absolute dispenion provides a benchmark from which ideas in dynamical

systems theory may be interpreted: barriers to transport, Kolmogorov- Arnold-Moser ( KAM )

tori (or invariant phase-space surfaces that survive temporal perturbations to the flow),

separatrices and their influence on the dispersion of passive tracers are aJl evidenced in

temporal scaling laws for single-particle dispersion. This idea was noted briefly in Chapter 3,

and will be addressed further in the evaluation of the relationship between single-particle

dispersion and pat chiness.

The evolution of a passive tracer in homogeneous, isotropie turbulence can be sepa-

rated into three temporal regimes, specified according to the Lagrangian integrai timescale

(Taylor 1921 )

CHAPTER 4. STATISTICA L DIAGNOSTICS 79

For suffieienbly short Limes, 7' « Tc, the particks h e a memory of their initid v e h -

ity. Thus, the correlation coefficient is close to unity, and the temporal dispersion relation

becornes

A* = (v*)T*. (4.11)

for T the interval of elapsed tirne. This is known as the ballistic regime, because the rms

displacement is linear in tirne. By contrast, for very long times, T > TL (assuming that

TL c oc), the particles lose memory of their initial velocities and

for K = (V2)Tt the dispenion coefficient (Provenzale 1999). In t his case, the particles follow

independent random walks as in Brownian motion.

Of particular interest in describing the properties of a flow using single-part ide dispersion

is the scaiing behaviour

r12 - T C ) (4.13 j

when a # 1 and a # 2. Anomalous diffusion refers to the phenornenon wherein long-

time, large-scale tracer transport exhibits behaviour that departs from the expected diffusive

a = 1 limit. Central to the established link between scaling Iaws and flow topology is the

aforernent ioned Lagrangian integral timescale ( Majda and Kramer 1999). The existence of

a finite decorrelation time, as shown above, eventually gives rise to diffusion. By contrast,

superdiffusion (a > 1) is distinguished by the absence of a definitive Lagrangian timescale.

such t hat

Responsible for superdiffusive behaviour are long-range correlations in the Lagrangian veloc-

ity, corresponding t o large-scale inhomogeneities in the Eulerian field. Oscillatory behaviour

in the Lagrangian coefficient R(T) (i.e. positive and negative values) leads to subdiffusive

behaviour. In this case

CHAPTER 4. STATISTICAL DIAGNOSTICS 80

SubdiRusiotl is characteristic of troppmg events, whick m i d occur if particies fotiowed

regu1a.r orbits and experienced a coherent, ternporally-periodic velocity field.

While Taylor ( 1921 ) and Bat chelor ( 1949) found that absolute dispersion increases less

rapidly than ballistic scaling and more rapidly than diffusive scaling at intermediate times,

that is, for times comparable to TL, it is not obvious that any scaling will hold for in-

termediate times. There is also the possibility that TL does not exist, as for Lévy Rights

(Shlesinger et al. 1987, Weeks 1996), which give rise to anomalous scaling even as T + oo. The precise nature of the scaling exponent a is dependent on the fiow topology. Previous

studies (MeziC and Wiggins 1994) have shown that if dispersion is considered over two top*

logically distinct regions, then T2 scaling is always attained in the long-time limit. The

essence of this argument lies in the concept of ergodicity, or the uniform sampling of phase

(in this case physical) space by a passive tracer trajectory, or orbit; when the dynarnics

are not ergodic, Ta scaling is inevitable. In considering absolute dispersion properties over

strongly elliptic and hyperbolic regions nrparatrly, howewr, as defined by the Weiss crite-

rion, Elhmaidi et al. (1993) found from numerical experiments that dispersion in elliptic

regions is characterized by a T ~ / ~ scaling law, while hyperbolic domains are characterized by

a shallower T'/' scaling law.

4.1.5 Absolute Dispersion and Balanced Dynamics

Et has b e n shown (Batchetor 1949) that singleparticte statistics capture the dispersion

properties resulting from low-frequency mot ions in the flow. Absolute dispersion may t hus

be construed as an appropriate description for balanced dynamics, and an inappropriate

description for unbalanced dynarnics. The justification for this statement is demonstrated

in the following derivat ion from Batchelor ( 1949).

An alternative representation for absolute dispersion (Babiano et al. 1987, Provenzale 1999)

resides in (4.9) and its reiation to the Lagrangian energy frequency spectrum L(n), where

n/2n denotes the frequency of oscillation (in time). The correlation coefficient R ( r ) is the

CHA PTER 4. STATISTICA L DlAGNOSTlCS 81

00

R ( r ) = L(n) exp(inr)dn, (4.16)

where L(-n) = L(n) so that R( r ) is real. Absolute dispersion is, from (4.9), related to the

Lagrangian energy frequency spectrum t hrough the expression (Batchelor 1949)

A ~ ( T ) = 4(v2) /- L(n) (1 - cos(nT))

dn. O n*

Noteworthy in the above analysis is the asymptotic temporal behaviour of dispersion. For

short t imes, (1 -cos(nT))/n2 - T2 and the resulting ballistic behaviour reflects contributions

from motions of al1 frequencies, as particles undergo coherent and unidirectional motion.

However, for long times the factor (1 -cos(nT))/n2 is controlled by the low-frequency modes:

linear dispersion is thus a reflection of the significant contributions from the slow modes. It

Follows that absolute dispersion, in failing to capture the dispersion properties associated with

the high-frequency component to the flow, is an inappropriate tool with which to examine

unbalanced dynamics. This concept will be demonstrated and emphasized repeatedly in

the analysis of the dispersion statistics obtained for the PW, PW/GW and purely diffusive

(GWRT) experiments.

For a Lagrangian energy spectrum L(n) - n-', absolute dispersion is found to obey the

following scding laws (Babiano et al. 1987, Elhmaidi et al. 1993):

for p > 1, or

.A2 ry p+=

for -1 < p c 1. From (4.9), such considerations also give rise to the power law

for the correlation coefficient for - 1 < p < 1. The significance of these relations r a t s in the

insensitivity of the asymptotic scaling laws (4.1 1) and (4.12) to the nature of the flow. In the

CHA PTER 4. STATISTICA L DIAGNOSTICS $2

case of hhomogeneom flow, as consictmd h m , the possibitity exists of anomdous diffusion

a t intermediate and late times for Lagrangian energy spectra which decrease with increasing

Lagrangian frequency, wherein contributions from the lowest-frequency modes will domi-

nate. Thus while absolute dispersion can depict the prevaience of spatial inhomogeneities,

as evidenced in superdiffusive behaviour, its ability to distinguish between slow and fast

dynarnics is limited by the dominance of the low-frequency motions. It has also been shown

that nonlocal dynamics, a concept investigated later in the chapter in the discussion of rela-

tive dispersion, yields indistinguishable high-frequency spectral dopes (Babiano et al. 1987).

This appears to suggest that while the nature of low-frequency motions is captured by single-

particle statistics, rapid oscillations are not. Such behaviour may dso be understood in terms

of the long-time lirnit for large and small scales of motion: A2 + T rapidly for small scales,

whereas A2 + T2 t'or large scales, so it is the long-time scaling for the large scales t hat dom-

inates. This once again supports the notion that absolute dispersion will serve as a useful

4.2 Absolute Dispersion and Patchiness

As dernonstrated in Chapter 3, patchiness is closely related to single-particle dispersion, as

evidenced by the relation

for D, the patchiness dispersion. Thns the ttreory presented above accounts for the &or-

ganization" of the patchiness in time, e.g. between Figures 3.8d) and M e ) , as the effects

of the gravity waves average out. An understanding of the relationship between spatial

inhomogeneities in the 0ow field, depicted by regions of distinctive average velocity, and

anomalous diffusion rnay be achieved by examining the patchiness properties of Figures 3.7

and 3.8 and their manifestation in the temporal scaling regimes established for zona1 and

meridiond absolute dispersion. In addition, the phenomena of ergodicity and mixing in the

surrogate stratosphere (the P W experirnent) and mesosphere (the P W/G W experiment ) are

elucidated.

Enhanced variability in a passive tracer field arises from stirring: increased spatial gradi-

ents reflect barriers to transport which subsequently define topologically distinct dynarnical

regions. Temporal scaling laws for absolute dispersion monitor the effect of geometry upon

dispersion. Thus, to what extent is stirring depicted by absolute dispersion? What effect

does imbalance have on the stirring properties? The statistical diagnostics of variance and

PDFs for the particle displacement seek to answer these queries.

4.3 Absolute Dispersion Results

Statistical analyses performed in this and subsequent sections are based on an ensemble of

21824 particles, initidly aligned dong latitudes separated by 1 .O0 and with a longitudinal

spacing of 0.5' x cos(@), giving an area weighting to the distribution to account for variations

in the longitudinal distances. The particles are initially confined to the domain bounded by

longitudes 0.2' and 358' and latitudes 15.5* and 52.5". Tracer trajectories and calculations

of the zonal dispersion allow for Uwrappingn past 0° or 360'. Moreover, flow statistics are

calculated only after day 20 in the 70-day experiment, in order that statistical stationarity

is ensured and that transient dynarnical effects are rninimized. Each particle time step

corresponds to a duration of 15 minutes. The passive tracer field is generated using a particle

advection scheme developed by Ngan (1997). Significant alterations to the code include the

incorporation of relative dispersion diagnostics, explored later in the chapter, in addition to

rninor additions such as PDF computations for single- and two-particle statistics.

Shown in Figure 4.1 are the ensemble-averaged Lagrangian autocorrelat ions for the

zonal and meridional winds, computed for the planetary-wave (PW), planetary/gravity-

wave (PW/GW), and gravity-wave with random temporal forcing (GWRT) experirnents.

It should be noted that an ensemble of - 700 particles was used in contrast to the 21824

particles used in o t her statistical analyses, due to cornputer mernory constraints. Evident in

Figure 4.1 is the absence of a decorrelation time for the zonal winds: the presence of the

CHAPTER 4. STATISTICA L DIAGNOSTICS

- PN - PWm

GWRT

Figure 4.1: Ensemble-averaged Lagrangian autocorrelations of the a) zona1 and b) meridional wind components for the PW, PW/GW and GWRT experiments.

CHAPTER 4. STATlSTlCA L DIAGNOSTICS 85

zond jet in the form of a bmic-state shear estabhttes hg-mnge corretations in the tow.

Moreover, such long-range correlations are apparent not only for the PW experiment, but

also for the PW/GW and GWRT experiments. Hence it is the zonal shear rather than the

PW forcing that is responsible for the absence of a definitive Lagrangian integral timescale

in the direction of the zonal flow. According to (4.10), one would therefore expect that zonal

dispersion of individual particles within the ensemble be characterized by superdiffusion.

It should be noted that the system is not in an exact statisticaily stationary state, so

that a nonzero limit as t -t oo may reflect transient effects.

Noteworthy also in Figure 4.la) is the relative insensitivity of the autocorrelation to the

forcing mechanism employed in each of the experiments, and thus to the detailed structure

of the flow itself. That is, the high degree of variability in the zonal direction, as manifested

in the smaller scales of the zonal winds and the corresponding tracer fields, is lost in this

diagnostic.

Rather different bohaviotir is exhi hi t d in the Lagrangian correlat ion function for the

meridiond winds, shown in Figure 4.lb). The meridional winds do not appear to decorrelate

for the PW experiment. For the PW/GW and GWRT experiments, however, the Lagrangian

correlation function decays rapidly. For the PW/GW experiment, the correlation coefficient

decays to zero after approximately 10 days, with an e-folding time of several hours, as

demonstrated in the inset of Figure 4.1 b) . The decay in the G WRT experirnent is even more

abrupt - the correlation coefficient decays to zero after several hours, with an e-folding tirne

comparable, it would appear, to the temporal resolution of 15 minutes used in the shallow-

water model. Such behaviour is to be expected in the absence of a meridional shearing

mechanism: the lack of a definitive decay tirne for the PW experiment is a manifestation

of the planetary-wave forcing in the meridional direction. The rapid falloff for the GWRT

experiment impiies that a purely diffusive regime is rapidly established for the meridional

winds. The existence of a finite Lagrangian timescale implies diffusive behaviour in the

meridional direction in the long-time limit for the PW/GW and GWRT experirnents, with

CHAPTER 4. STATISTICA L DIAGNOSTICS 86

the approach to the diffusive regime being more rapid for the GWRT experiment than for

the PW/GW experiment.

Figure 4.2: Trajectory of a single maxked particle in the PW (red). PW/GW (blue), and G WRT experiment (green).

It is instructive to examine the trajectory followed by an individual particle in each of the

three experiments. shown in Figure 4.2. Of interest are the eddy turnaround times apparent

in the figure. The red loop in the PW case corresponds to an eddy turnaround time of

approximately 38 days. No such eddy turnaround times are discernible from the PW/GW

experiment for the particle initiated at this particular location. It is interesting to note,

upon cornparison with the passive tracer field for the PW experiment, Figure 3.9, t hat the

region traversed by the particle is the surf zone. The presence of gravity waves results in

enhanced variability in the pat h followed by the part icle. The overall displacement, however.

is comparable in the two cases. Singleparticle dispersion is thus controlled by the sweeping

effects exerted by the balanced compooent when tracking the motion of a single particle.

Small-scale structure in the velocity field associated with the presence of the unbalanced

component is unimportant when considering the advection of individual particles.

Depicted in Figure 4.3 is the ensemble-averaged mean single part icle displacement M

-- PW PWIGW

- - GWRT

Figure 4.3: Ensemble-averaged absolute a) zona1 rnean (log-log scale) and b) meridional mean (linear scaies) single-part ide displacements for the P W (red), P W/G W (blue). and GWRT (green) experiments.

PW PW/GW GWRT

üme (deys)

Figure 4.4: Ensemble-averaged absolute a) zonal dispersion and b) meridional dispersion for the PW (red), PW/GW (blue). and GWRT (green) experiments.

CHAPTER 4. STATlSTICAL DIAGNOSTICS 89

[see (4.4}} for the PW, PW/GW an& GWRT experiments. The siight offset at the origin in

Figure 4.3a) is due to the fact that M (O) = O for each experiment, so that for the log-log

plot the initial mean shown is at t = 3hr. As expected a mean transport velocity is defined

for transport in the zonal direction by the basic-state shear flow. This is apparent also in the

zonal patchiness (Figure 3.7). A well-defined mean meridional velocity is not a . apparent, of

course, although there is a weak meridional drift towards the North Pole as one expects for

breaking planetary waves, analogous to the stratospheric Brewer-Dobson circulation (e.g.

Shepherd 2000). From t hese results, i t is apparent t hat zonal single-part icle advection is

govemed by the Northern Hemisphere jet: particles are swept along the streamlines sur-

rounding the cat's eye structure. More interesting behaviour should however be apparent in

the meridional direction where it is the nature of the forcing mechanism employed, rather

t han the basic-state shear, t hat is responsi ble for single-particle dispersion.

Figure 4.4 depicts the ensemble-averaged absolute dispersion of individual particles for

the PW, PW!C.W and GWRT experiments in the zonai and meridional directions. Evident

in Figure M a ) is the ballistic scaling for al1 experiments, anticipated with the presence of

coherent structure and the absence of a finite Lagrangian correlation timescde. Figure 4.4a)

is basically just the square of Figure 4.3a). The coherent structure in this case is the zonal

shear flow, which gives rise to linear growth in the r m s displacement through zonal advec-

tion. The similarity in the behaviour of the three experiments suggests that single-particle

dispersion is insensitive to the behaviour of small-scale eddies. This is consistent with the

previous argument that single-particle dispersion is a measure of the influence of large-scde

inhomogenei t ies on part icle advect ion. Neit her the smaller-scale gravi ty waves result ing from

the forcing at higher wavenumbers, or even the planetary wave, are reflected in the advection

of a single particle in the zonal direction. It is interesting t o note that Malhotra et al. (1998)

found the same scaiing behaviour for horizontal transport in a meandering jet, used to mirnic

transport in the Gulf Stream.

As demonstrated in the snapshot of the meridional winds (Figure 2.5) at t = 50 days and

CHAPTER 4. STATISTlCAL DIAGNOSTICS 90

in Che pabchiness plok, presented in Ghapter 3, some structure is apparent for the meridional

component of the flow. The coherent features are, however, much less dominant than for the

zonal component. The structure is even less distinctive for the PW/GW experiment. One

would therefore anticipate more diffusive-like behaviour for the ensemble-averaged merid-

ional dispersion than for the zonal dispersion. One can imagine, in a manner analogous to

the physicd picture presented by Pasmanter (1988), that both chaotic and regular parti-

cle trajectories exist in the meridional direction, so t hat subdiffusive rather t han baIIistic

behaviour might be expected. This is indeed found to be the case for single-particle disper-

sion. Two scaling regimes are established for the P W and P W/G W experiments. A ballistic

regime extends to t - 3 days, indicating that the particles experience unidirectional motion

until this time. Subdiffusive behaviour is apparent after t - 4 days, when the absolute

dispenion has a value of - 3 x 1o5krn2, corresponding to an rms absolute displacement of

approximately 500km. Although meridional dispenion, unlike zonal dispersion, is ultimately

limited by the spherical geometry. this upper bound is not yet apparent here: the largest

rms displacements attained by the end of the cdculation are only about 1000km. Differ-

ences between the P W and PW/GW experiments become apparent only after t - 20 days.

While the P W experiment shows trapping, dispersion in the P W/G W experiment continues

to evolve subdiffusively. More marked is the contrat with the GWRT experiment: in that

case, diffusive behaviour is exhibited for most of the temporal range considered. This is

a consequence of the absence of a planetary wave: no coherent features are present in the

meridional direction for this spatially and tempordly decorrelated flow.

The role of the temporal autocorrelation in determining temporal scaling behaviour for

single-particle statistics is t herefore supported in the present analysis for bot h zonal and

meridional dispersion. The lack of a definitive decorrelation time for the zonal Lagrangian

winds is manifested in the ballistic scding, or t2 dispersion, for each of the PW: PW/GW and

GWRT experiments. For the meridional winds, the existence of a finite decorrelation time

for the GWRT experiment is maaifested in diffusive behaviour. Sirnilarly, diffusive (actually,

CHA PTER 4. STATISTICAL DIAGNOSTICS 91

suMiKwive) behaviow appears alter irpprmtirnateky 4 days for the PW/GW experiment in

accordance with the longer decorrelation tirne in that experiment. For the PW experiment,

the strongly subdiffusive behaviour seen after 20 days, characteristic of trapping phenornena

(Weeks 1996), is reflected in oscillatory behaviour in the correlation coefficient presumably

associated with periodic behaviour as particles undergo regular orbits within the cat 's eye

structure. A run longer than 50 days may be necessary to determine a more accurate

representat ion of the long-time behaviour; in particular, to deduce whet her the correlat ion

coefficient is indeed oscillatory.

Cornparison of the absolute dispersion statistics with the patchiness plots for the zonal

and meridiond average velocity further elucidates the scaling laws obtained for the PW and

PW/GW experirnents. Evident in Figure 3.7 is the persistence of coherent structure, even

with the inclusion of the unbalanced component to the flow. Variations can be observed

only at the edge of the surf zone, indicating mixing into and out of the surf zone. Further

supported is the concept of nonergodicity (Ma ié and Wiggins 1994). The preaanre of the

zonai jet establishes barriers to transport which prevent a uniform sampling of phase space.

The patchiness plots for the meridional average velocity indicate the persistence of coherent

features at the North Pole and within the sud zone, after t = 10 days, as shown in panel

b). After t = 20 days, however, these coherent features are eroded, particularly for the

PW/GW experirnent. This is manifested in the scaling regimes shown in Figure 4.4: absolute

meridiond dispersion is described by ballistic scaling for about 3 days, after which diffusive

(or subdi ffusive) behaviour is apparent.

4.3.1 Probability Distribut ion Functions for Single-Particle Dis- placements

The probability distribution function (PDF) of particle displacements provides a measure of

the evolution of the mean passive scaiar concentrat ion ( Bat chelor 1949,

Majda and Kramer 1999). As the second moment of the PDF for single-particle displace-

ment X is the variance, or dispersion, of the passive tracer, considerations relating passive

CHAPTER 4. STATlSTICA L DIAGNOSTICS 92

t r i r m transport to tagrangiair vefocity adocot~efations are equatly appiicabte to PDF anal-

yses. However, furt ber informat ion is available from the PDFs, since higher-order statistics

of the motion cm be cdculated according to the relation

(X") = 1- XnP(X, t ) d X . (4.22) O

Of particular interest is the long-time behaviour of the statistics, namely, the asymptotic

form for the PDF as the diffusive Iimit is attained. The Central Limit Theorem predicts

that the PDF approach a normal, or Gaussian, distribution in the long-tirne limit, that is,

for t > TL (Monin and Yaglom 19777, with the requirement that the random variables of

the process, namely the singleparticle displacements, are either statistically independent or

are weakly dependent and belong to distributions having finite moments. Tracer behaviour

is governed in that case by the diffusion equation

for the diffusion coefficient. Conversely, a non-Gaussian limit is attained with the pres-

ence of long-range correlations (or the absence of a definitive decorrelation t ime). Trans-

port governed by (4.23) is referred to as Fickian diffusion. For a non-zero ensemble mean

advective velocity (q, transport is characterized by the advection-diffusion equat ion (e.g.

Majda and Kramer 1999)

A Gaussian distribution is characterized completely by its first and second moments, namely

the mean M and variance a* = A*.

Closely related to the Central Limit Theorern is the previously rnentioned concept of

ergodicity (Frisch 1995). In the limit t -t oo, the deviation between the time average and

the ensemble average for a diffusive system scdes as t - I I 2 , and differences between the

time and ensemble averages are negligible: fluid particle trajectories sample phase space

u n i f d y . However, wben long-range correlations are present in the Lagrangian velocity

CHA PTER 4. STATISTICAL DIA GlVOSTlCS 93

Md, the diffusive description for transport is mr fonger TT&& h this instance, a nodocal

advection-diffusion equation, and in particular a nonlocal effective diffusivity, is required.

4.3.2 Higher-Order Moments and Large-Scale

The existence of long-range correlat ions in Buid part icle transport

s kewness

and flatness

where o = is the standard deviation of the single particle

Int ermit t ency

is rnanifested in the PDF

(4.26)

displacement. Skewness

is a measure of asymmetry in the PDF, and is zero in the Gaussian limit; flatness moni-

tors distri bu tion bmadness, and F = 3 for Gaussian PDFs. Broader-than-Gaussian tails

with F > 3 are suggestive of long-range correlations in the Lagrangian velocity which, in

turn. give rise to large particlc diaplac~rn~nts - h ~ n w the term large-scale intermittency

(Majda and Kramer MN). As the advected fluid particles encounter coherent structures,

advection by Brownian motion, expected for homogeneous turbulence, is replaced by anorna-

lous transport evidenced in non-Gaussian PDFs. This is because individual particles are not

influenced by irregularities at the smallest scaies, but rather by irregularities at much larger

scales. Pertinent questions in this and many investigations are: Under what conditions will a

Gaussiaa PDF be achieved? When are non-Gaussian statistics prevalent? What topological

features in the flow are responsible for the long-range velocity correlations which give rise to

such non-Gaussian statistics? As such, the PDF of the particle displacement provides the

means by which departures from Gaussianity may be ascertained, and more importantly,

highlights deviat ions from the advect ion-diffusion description of transport .

4.3.3 PDF Results for Single-Part ide Displacement

Presented in this section is an analysis of the PDFs obtained for the single-particle dis-

placement bx = x( t ) - x(0), measured in kilometers. The bin sizes differ for the zona1 and

CHA PTER 4. STATISTICAL DIAGNOSTlCS 94

m e r i d i d displacement in accotbce wibh the maximum displacernent acliievd in the t w ~

directions. A bin size of - 2200km was used in the case of zond dispersion (dlowing for

wrapping in latitude) and - 200km in the case of meridiond dispersion.

Shown in Figure 4.5 axe the PDFs for zonal and meridional single-particle displacement

associated with the PW, PW/GW and GWRT experiments at t = 50 days. Included as ref-

erence are the Gaussian fits to the distributions, to emphasize the non-Gaussian behaviour

of the zonal dispersion statist ics relative to the meridional dispersion statistics. The skewed

behaviour evident in the zonai distri bution for each of the experiments indicates the dis place-

ment of particles eastward by the zonal flow. (This does not necessarily create skewness S,

since X is the deviation from the mean.) The shape of the zonal PDF is similar for each

of the three experiments, in spite of the differences in forcing mechanism. This reflects the

dominant role of the zonal flow; a maximum jet speed of roughly 50rns-1 times 50 days

yields a maximum zonal displacement of about 2.5 x 105km, as seen in Figure 4.5a).

In contrast to the PDFs for zond displacement, a more Gaussian distribution is attained

for the meridional-displacement PDFs, particularly for those experiments including gravity

waves. In the case of the PW/GW experiment, this may be related to the patchiness plots

of Chapter 3: with the introduction of a gravity-wave component to the flow, the erosion of

regions with distinct ive meridional velocity wit hin the surf zone gives rise to a homogenization

of the small-scale structure. In the case of the G WRT experiment, the only coherent feature

in the fiow is the zonal jet. A s shown in the trajectory of a single particle for this case

(Figure 4.2), particles are rneridionally const rained by the partial barrier to transport t hat is

established by the basic-state shear. Flow dong separatrices is replaced by a random walk in

the meridional direction. Nevert heless, the maximum nort hward single- particle displacement

is achieved for the GWRT experiment.

Shown in Figure 4.6 is the time evolution of the zonal-displacement PDFs. In al1 experi-

ments, the PDFs broaden with time. Particles are translated from their origin by advection

by the largescale structures of the ~ O W , regardless of the presence or absence of gravi-

CHAPTER 4. STATlSTlCAL DIAGNOSTICS

Figure 1.5: Probability distribution function for a) zonai and b) meridional absolute dis- placement for the PW, PW/GW and GWRT experiments at t = 50 days. Gaussian fits to the distributions are shown in each case by a dotted line.

CHA PTER 4. STATlSTlCA L DIAGNOSTICS 96

w a v a Furthemore, the PDFs for b h e zond disptafcment are non-Oaassian, emphasizing

the anticipated problems discussed in Section 4.2.2 regarding the use of a constant diffusivity

to describe dispersion in a generalized flow. For the case of shear-parallel displacement, it

h a b e n shown that Gaussian behaviour is indeed an inappropriate description.

The skewness and flatness, computed from (4.25) and (4.26) respectively, monitor de-

partures from Gaussianity. The asymmetry in the flow resulting from the basic-state shear

is represented by the skewness in each of the three experiments. A nonzero value indicates

non-Gaussian behaviour, as is to be expected for shear-parallel transport. W hile ail ex-

periments axe somewhat skewed, the GWRT experiment has twice the skewness of the two

othen. Differences between the GWRT experiment and the two others are also exhibited in

the flatness, which monitors, as was previously discussed, the probabili ty of large paxticle

displacements. The increase in the flatness for the GWRT experiment arises from a more

compressed PDF and suggests that the gravity waves, in the absence of coherent features

established by the PW forcing. inhibit the likelihood of a wide range of sweping v ~ l o d i e s

in the zond direction.

Very broad, truncated PDFs like those shown in Figures 4.6a) and 4.6b) have flatness

factors that are essentially independent of time, as is observed in panels d) and e) . It

is easy to see that a "box-car* PDF bounded by maximum and minimum values yields

F = 915, which is consistent with the values found for the PW and PW/GW experiments.

Enhanced Batness for the GWRT experiment reflects the slaated nature of the PDF for

positive displacements. For any symmetric distribution, S = 0: the larger values for the

GWRT experiment are a manifestation of the peak for small displacernents which shifts the

mean to lower values. An important feature to be noted from Figure 4.6 is that no particular

distinction is seen between the P W and PW/GW experiments. It appears tme that for the

most part, absolute dispersion is more sensitive to the largescale than to the smd-scale

flow, but there are exceptions such as the avoidance of trapping that seems to corne from

the small-scale gravity waves ( t > 20 in Figure 4.4).

CHA PTER 4. STATISTICAL DIAGNOSTlCS

Figure 4.6: Probability distribution function for zona1 absolute displacement at different instants in time for the a) PW, b) PW/GW, and c) GWRT experiments. Also shown are the skewness S in c i ) , and flatness F in e) .

CHA PTER 4. STATISTICAL DIAGNOSTICS

Figure 4.7: Probability distribution function for meridional absolute displacement at different instants in time for the a) PW, b) PW/GW, and c ) GWRT experiments. Skewness and flatness are depicted in d) and e), respectively.

CHAPTER 4. STATlSTlCA L DIAGNOSTICS 99

Depicteci iin Figure 4.7 is the bime evduhion of the PDFs for t h e mtklio~ai disptacement.

Here, the role of the gravity waves in establishing a more diffusive regime is evidenced in the

more Gaussian PDFs. The skewness factor is essentiaily zero for the PW/GW experiment

- the flow is symrnetric, i.e. there is no preferred direction of displacement - while the

Batness is close to F = 3, as it is for the G WRT experiment as well. While the planetary-

wave forcing creates some organized structure in the meridional component of the Bow, the

gravity waves act to erode any coherent features, even after 10 days [se, for example, panel

f ) in Figure 3.81. In the absence of any barriers to transport, the particles are allowed to

disperse freely throughout the domain, as shown in the increased breadth of the PDF at

t = 50 days for the GWRT experiment. The more compact nature of the PDF for the PW

experiment may be attributed to the subdifhsive or "trapping" behaviour exhibited in the

dispersion analysis (see Figure 4.4).

4.3.4 Assessrnent of Absolute Dispersion Results

The results found for zonal absolute dispersion (Figure 4.4) in the presence of a zonal flow

in the present investigation are consistent with results found for horizontal transport in a

meandering jet (Samelson 1992). The presence of coherent structure in the flow and the sub-

sequent lack of a definit ive Lagrangian t imescale renders the assumpt ion of diffusive transport

invalid for total and zonal dispersion. It has been shown both in the dispersion statistics

and in the PDF results that a constant eddy diffusivity does noh acturately represent zonal

dispersion for the flows considered in the present investigation. By contrast , meridional

dispersion demonstrates the effect of temporal fluctuations on dispersion statistics.

The PDF results further demonstrate that transport and stirring do appear to be c a p

tured by absolute dispersion: the sweeping effects evident for both the PW and PW/GW

experiments (see Figure 4.2) are depicted by non-Fickian diffusion in the zonal direction,

manifested in the large particle displacements in the zonal PDFs. Mixing appears to be the

dominant mechanism of transport in the meridional direction, as demonstrated in the disper-

sion statistics and Gaussian-like PDFs. As will be discussed for the remainder of this thesis,

CHAPTER 4. STATiSTlCA L DIAGNOSTlCS 100

an understanding of the relative separstion betmerr a pair of partides is n e e s a q to deci-

pher passive tracer dynamics within Bow governed by large-scale balanced and small-scaie

unbalanced dynamics.

4.4 Relat ive/Two-Part icle Dispersion

While absolute dispersion rnonitors the translation of a single particle, relative dispersion

monitors the separation between a pair of particles. This section seeks to define relative

dispersion and subsequently demonstrate the usefulness of this diagnostic in addressing the

distinction between dynarnics in Bow governed by large-scde components, as is evidenced in

the stratosphere with planetary-wave activity, and dynamics in Row governed by small-scale,

high-frequency components, as is thought to be the case in the mesosphere where gravity-

wave activity dominates. Furthemore, relative dispenion is compared with other mixing

diagnostics in order that a quantitative assessrnent of mixing in the middle atmosphere may

be established.

4.4.1 Relative Dispersion and Separation between a Pair of Par- t ides

The concept of relative dispersion was introduced by Richardson in the early twentieth

century (Richardson, 1926), and was motivated in part by the question: Does the wind

possess a velocity? For Ar the particle displacement over an interval of rime At, the

quanti ty

need not approach a finite value as At + O. Moreover, if the time average of the individually-

marked particles is considered, an increase in the timeof averaging wilI not generate a limiting

value: energy-containing eddies disperse t h e single particle to ever-larger scdes. Richardson

noted that the phenornenon responsible for the lack of a definitive velocity is the prevalance

CHAPTER 4. STATISTICAL DIAGNOSTICS

of a wide range of eddies in t h e EartVs atmmpbere. R~ktive dispersion is defmed as

where here N is the number of particle pairs with initial separation Do, and the angle

brackets denote an ensemble average. In monitoring the relative motion between a pair of

particles, relative dispersion incorporates the effects of wying-sized eddies upon a cluster of

particles. Both particles experience comparable velocities from larger-scale eddies; differences

are manifested in velocities associated wit h eddies smaller t han the interparticle separat ion.

As the range of eddy sizes to which the particle pair is subjected increases, an increasingly

wider spect rurn of eddies contri butes to relative dispersion.

4.4.2 Relative Dispersion and the Advection-Diffusion Equatioii

Yet anot her, and perhaps more fundamental, question addressed in Richardson's develop

ment of relative dispersion is the presence of a wide range of eddy diffusivities from at-

mospheric measurements. Such an analysis requires investigation of turbulent diffusion, or

diffusion in the presence of eddies. Richardson's study was motivated by a desire to determine

a "generalized" eddy diffusivity, independent of particle location, but dependent on particle

separation. As has been emphasized t hroughout t his investigation, diffusion in the absence

of eddies, also known as Fickian diffusion. is described by the standard advection-diffusion

equation (4.2) with constant diffusivity K. Richardson (1926) noted that tbis expression

provides an incomplete description of atmospheric dispersion in the presence of eddies. To

remedy the shortcomings evident in' the Fickian equation expressed in terms of concentra-

t ion, Richardson proposed an alternat ive description for the dispersion of passive tracers,

expressed in terms of the mean number of neighbours per unit length, q. The quantity q

monitors the spread in a cluster of particles. The modifieci expression monitoring the evolu-

tion of the separation vector d with magnitude 161 = V between a pair of particles is thus

given as

CHAPTER 4. STATISTICAL DIAGNOSTICS 1 02

Here, the dat ive diffoskity F ( @ , or diffusivity of neighbours, reflects f he variabie eddy

diffusivity values that are evident from atmospheric measurements. Empirical arguments

and observations yield (Richardson 1926)

for the relative diffusivity, where c is a constant indepeodent of relative separation, and is

determined from observations.

The analytic expression for the reiative diffusivity, or time evolution in the separation

variance, underlines the role of relative dispersion in addressing the range of eddy diffusivity

values in the atmosphere. The relative diffusivity is defined, in a manner &in to the definition

for the absolute eddy diffusivity (4.3), as (Babiano et al. 1990)

for D2 = (6 fi), and where the arguments & and t have b e n introduced tc cmphn-

size the dependence of the eddy dihsivity on the initial interparticle separation & and on time t. Thus, in contrast to the single-particle diffusivity K, the relative diffusivity

need not be constant in time. In fact, the significance of the relative diffusivity relies on

its ability to demonstrate that the rate of diffusion increases with interparticle separation

(Richardson 1926).

4.4.3 Relative Dispersion and the Lagrangian Correlat ion Func- tion for Relative Velocit ies

In order to demonstrate t he role of relative diffusivity in identifying the distinct temporal d

regimes for relative dispersion, it is useful to express the relative separation vector D as (see

for example McComb 1990, Babiano et al. 1990)

CHAPTER 4. STATISTICAL DIAGNOSTICS 1 03

where Ptt, Di) = &/dt is the retative tagrangian velocity. The rnean-squared separation

or relative dispersion is then given as

~ ~ ( t , Bo) = (fi 6 ) = ~o~ + 2/t@0 o IV(T, &))dr + 2 j t / ' (w$, 6) - W(T, &)drdtt, O O

(4.33)

where angle brackets denote an ensemble average over particle pairs wi th init i d separation

Do. The arrow denoting vectors will hereinafter be dropped. Although analogous to the

expression (4.3) obtained for absolute dispersion in Section 4.1.3, the non-stationarity of the

relative velocity vectors W prevents the use of a temporal autocorrelation and Lagrangian

correlation coefficient for the derivat ion of temporal scaling iaws for tweparticle stat istics.

That is, since the relative velocity grows with tirne as the particles separate, the character-

istics oi relative dispersion are not influenced solely by the correlations between initial and

final velocity states, as with one-particle statistics. Rather, the particles are subjected to

eddies of varying size as time progresses. A tweparticle integral timescale may nevertheless

be introduced in a rnanner similar to that for the single-particle case. Its re lewce in deter-

rnining the distinctive dynamical regimes according to which the temporal relative statistics

may be quantified is, however, limited by the fact that this entity is a function of the initial

particle separation.

Investigation of the evolution in the separation variance, or relative diffusivity, further

demonst rates the effect of nomtationarity upon relative dispersion statistics. From the ana-

lytic expression for the relative diffusivity given by (4.31), and (4.32), the relative diffusivity

is expressed as

The relative diffusivity is therefore dependent on the duration of diffusion and on the initial

separation. Thus, in the infinite-time limit, a definitive value for the diffusivity may oot be

at tained frorn t his relation, highlighting once more the underlying problem upon w hich the

concept of relative dispersion was built : t hat of descri bing the dispersion of part ides wi t hin

CHAPTER 4. STATISTICA L DlA GNOSTICS 104

a coostiao6ky thanging, notiaiffusive regime, wkick is not governeci by a singfe advective

timescde, but rather by a multitude of timescales associated with a multitude of eddies.

Relative dispersion monitors the straining rnechanisms acting on scales comparable to

the interparticle separation, while absolute dispersion monitors the translation of individual

particles (Batcheior 1931). This suggests t hat the correlations of the Lagrangian velocity

gradients, instead of the Lagrangian velocities thernselves, provide a link between relative

dispersion and temporal autocorrelations, in a manner andogous to the Lagrangian integral

t imescale descri bed in Section 4.1.3. Central to the analysis in C hapter 3 was the examination

of geometric properties of t he flow according to the linearized equations of motion. The

aoalogue of (3.5) for tw-particle statistics is

where X ( t ) = ( x ( t ) , y(t)) and with

axp) = n,

the initial particle separation. The relative dispersion integral may thus, from (4.33), be

written as

Hence it is the Lagrangian velocity spatial derivatives, rather t han the Lagrangian velocities,

t hat control the long-t ime behaviour of the relative dispersion statist ics. In addition, the

distinct ion bet ween spatial and temporal considerations is les well-defined for the t wo-

particle analysis t h a n for single-particle statistics. This is again a manifestation of the

monitoring of the spread in a cluster.

4.4.4 Temporal Considerations

An alternative derivation to (4.33), presented by Babiano et ai. (1990), is an anaiysis based

on the differentid equation for the separation between a pair of particles. Presented below

CHAPTER 4. STATlSTlCAL DlA GNOSTlCS 105

is e brief summay of hheir work in lur sttempt to motivate the statistiegf lutakysis perfarmed

on the P W and PW/GW experiments.

Introduced as a means of emphasizing the accelerating process of relative dispersion, the

pair is described by the equation time evolution of the separation of a particle

(Babiano et al. 1990)

for the magnitude ID - Do J2 of the separation variance. Equation (4.38) can be derived from

(4.32) by different iat ion, resu bst itut ion, and integrat ion by parts. The ensemble-averaged

solution [where the ensemble average is taken to ensure consistency with (4.33)] of this

differential equation is given as

where

contains the dynamical informat ion of relative dispersion. It is t his expression t hat enables

the asymptotic relative dispersion behaviour to be determined. The inclusion of the acceler-

ation term in this equation is reminiscent of the Hua-Klein criterion presented in Chapter 3.

and suggests a greater sensitivity to high-frequency (and hence to unbalanced) motion.

For the short-time limit t -t 0, the first term in (4.40) dominates and the relative

dispersion assumes the form

where

is the Eulerian second-order (one-dimensionai) structure function which in (4.41) is applied

at t = O. The relative acceleration is constant for short times as the particles undergo

CHA PTER 4. STATISTICA L DIAGNOSTlCS 106

lin- disptacement. The s a m e expression c m be derivecf f m (4.33)' since W(tt ) - W(r j

as t', r + O. According to Batchelor (1951), the quasi-ballistic behaviour is expected for

homogeneous turbulence (so that ( W ( t ) ) = O ) when D - Do GK Do, or when fluctuations in

the part ide separation are significantly Iess than the initial interpart icle separation. A more

quantitative evaluation is suggested by Babiano et al. (1990). In that study, asyrnptotic

behaviour for short times in homogeneous turbulence is expected for t « Tz, where a n

equivalent representation to the definition presented in (1.3) is given by

where Z is the enstrophy of the flow. This quantity is sometimes called the eddy turnaround

time. The characteristic timescale associated with

(Babiano e t al. 1985)

m

a given length scale D is defined as

(4.44)

w here

ia the longitudinal structure function, and t his provides an alternative definit ion for the

Lagrangian integral timescale. Fur homogeneous flow.

where it is assumed that contributions from strain and relative vorticity axe comparable (i.e.

t hat the Q field as determined from the O k u b e Weiss criterion does not exhi bi t topologically

distinct st min or vort ici ty-dominated regions ) . However , for inhomogeneous Bow , or for flow

characterized by large-scde structure, the characteristic timescale Tc is defined such that

Contributions from regions of the flow governeci by strain, vorticity and divergence are

included in t his definition. Again, the characterist ic tirnescale for inhomogeneous fl ow acts

as the two-particle counterpart to the single-particle Lagrangian integral timescale TL.

CHAPTER 4. STATlSTlCA L DlAGNOSTICS 107

In the long-t ime tirnit t + ou, (4.40) sattrrates and relative dispersion is characterized by

the scaling law

D2 - t . (3.48)

Particle pairs lose memory of their time origin, so that ( ~ ( t ' ) w(T)) + 2(v(t1)v(r)). Also,

the relative and absolute displacement are related by the expression D = Do + ( X ( a l ) -

X ( a 2 ) ) , where a 1 and a* are Lagrangian coordinates, so that D2 - D,-,' - 2A2. Equation

(4.48) then follows from the diffusive limit attained for absolute dispersion (4.12). The

asymptotic result (4.48) is perhaps more readily obtained from (4.33), using the property

that particles wander independently and the definition (.Y) = J,' V ( r ) d r .

For intermediate times, Richardson's formulation (1926) purports, from observations, the

relation for a separation-dependent eddy diffusivity

from which the famous t3 latv rcsults:

Babiano et al. (1990) rederived Richardson's law with the assumption of statistical station-

arity in the Lagrangian relative accelerations, so that (dW(t ) / d t d W ( r ) / d r ) = R( t - r).

and with the condition that the correlation between the initial velocities and those at a

later time vanish. Moreover, it was demonstrated that the t3 law holds for TL < t < TE,

where TE denotes the time necessary for the scales associated with the most energetic eddies

to be reached. Richardson's Iaw (4.50) may also be derived from (4.33) using the assump

tion of statisticd stationarity in the Lagrangian acceleration and the definition W ( t ) =

4.4.5 Relative Dispersion and M U n g

Significant in the relative dispersion andysis is the role of velocity gradients, rather than

the velocity field itself, for interparticle separation. This has important implications for the

CHAPTER 4. STATISTICAL DIAGNOSTICS 108

mixing properkies of the ffow, thst is, for t h e homogenization of smaitscate structure. The

primary question to be addressed in the twc+particle statistical analyses that follow is, to

what extent does relative dispersion reflect the effect of the unbalanced component upon

balanced dynarnics? In addition, does relative dispersion describe the onset of mixing?

4.4.6 Relative Dispersion Results: Temporal Scaling Laws

Relative dispersion statistics are computed as for the single-particle stat istics. The initial

particle spacing in the zonal direction is an area-weighted Ar = 4' x cos(O), and in the

meridional direction is Ay = 3O, unless otherwise noted. The ensemble again consists of

21824 particles, advected for an interval of 50 days. The magnitude of the separation between

pairs of particles is monitored so that ( x i ( t ) - ( t )) = O, in order to guarantee symmetry

in labeling.

In order to predict the relative dispersion of passive tracers, it is useful to examine the

Lagrangian correlat ion coefficients associated wit h the gradients in the Lagrangian veloci ty.

Shown in Figure 4.8 are the Lagrangian autocorrelations for the latitudinal and longitudinal

gradients in the zonal (u, and u, respectively) and meridional (v, and v, respectively) winds.

The largest spatial gradients in the Lagrangian velocity are found at the edge of the zonal

jet (see for example Figures 3.7, 3.8), hence in u,. Thus the establishment of a definitive

decorrelation time for the dispersion of a pair of particles depends in large part on the

correlation coefkient associated witk u,. ft is not possible to distinguish the characteristic

timescale for the zonal and meridional components of the flow separately, since the gradients

of u and v combined are evident in the definition for Tc. Rather, Tc depicts the integral

timescale for total dispersion 'D1 = D , ~ + D,'. Figure 4.8 demonstrates t hat the veloci ty gradients associated wi t h the divergence d =

u, + v,, namely u, and u,, decorrelate only for the P W/GW and GWRT experiments, as

is to be expected with the presence of the gravity-wave forcing. The decorrelation time

is on the order of r = 4 days for R(u,, r ) and r = 3 days for R(v,, T) for the PWiGW

experiment. In contra t , the decorrelation times are on the order of hours in the GWRT

CHAPTER 4. STATISTlCAL DIAGNOSTlCS

----- - -at -.. _.- -- --4

/

t . . . . l . .

O IO 20 me m t (diVI

Figure 4.8: Ensembleaveraged Lagrangian autocorrelations of the lat i t udinal and longit u- dinal gradients for the zonal a ) u, and b) u, and meridional c) v, and ci) v, winds, for the PW (red), PW/G W (blue) and G WRT (green) experiments.

experiment. The velocity gradients in the PW experiment do not appear to decorrelate at

all. In particular, the lack of a definitive integrai timescde for the latitudinal gradient of the

zonal winds u,, which is the dominant term for the vorticity w = v, - u,, suggests that the

PW experiment evolves on very slow timescales. By contrast, the rapid decorrelation in both

u, and v, for the PW/GW experiment suggests that the unbalanced component dominates

for t 3 days. That is, dispersion for short times in the PW/GW experiment is governed

by the unbalanced dynamics. One would therefore expect t3 dispersion to be achieved more

rapidly in the presence of gavity waves. The autocorrelations for the Lagrangian spatial

CHA PTER 4. STATISTICAL DIAGNOSTICS 110

derivatives further reflet the ambiguiky i n k t in the f f i i t i o n (4.47) of an appropriate

characterist ic timescale Tc for flow dominated by large-scale spatial inhomogenei ties: the

inclusion of divergent and straining effects modifies the value for Tc- Investigation of the

relative dispersion temporal scaling laws, as will later be discussed. will demonstrate the

influence of the fast dynamics associated witb gravity waves upon the slow dynamics linked

to pianetary-wave activity.

" f -- PW.l

PW. 2

GWRT, 1

GWRT, 2

l : . t n j 200 250

A

Figure 4.9: Trajectories of neighbouring particles for the a ) P W and PW/G W, and b) G WRT experiments, with initial separation Ar - 4'.

Depicted in Figure 4.9 are the particle trajectories followed by neighbouring particles for

the PW, PW/GW and GWRT experiments, initially separated by Ax - 4'. The presence

of the gravity waves introduces a secondary length scale. Al1 particles in the PW and

PW/GW experiments follow trajectories governed by the large-scde properties of the flow:

this is apparent in the sweeping arc traversed by each of the four particles. The spatial

and tempord Buct uations in the gravity-wave case, however. induce a meandering from

the smooth trajectories seen in the P W experiment. In this sense, a secondary and much

smder length scale is established, which is superimposed upon the large-scde particle path.

Significant is the variability introduced by the unbdanced component to the Bow. While

CHAPTER 4. STATISTICAL DlAGNOSTlCS 111

the turo particles f o b v simikw ares for Ihe Pw expriment and hardb separate at dl after

50 days, the introduction of gravity waves in the PW/GW experiment leads to a far greater

particle separation. S hown in panel b) of Figure 4.9 are trajectories followed by two particles

subjected to velocity fields generated from the G WRT experirnent. The particles follow a

random wdk in 8.

Figure 4.10 depicts the ensemble-averaged zonai and meridional relative dispersion D2

as a function of tinie. Although, as previously mentioned, the relationship between relative

dispersion and velocity gradients is meaningful for the consideration of the total dispersion,

exarninat ion of the coatri butions from the zonal and meridional components of relative dis-

persion helps to clarify the nature of dong-shear and cross-shear transport. Since the results

for total and zonal relative dispersion are similar, the latter serves as a surrogate for the

former. Apparent in the upper panel of Figure 4.10 is the short-time asymptotic behaviour

(4.41 ) for the relative separation between a pair of particles in the zonal direction, evidenced

in the plateau established for the PW. PW/GW and GWRT experiments. How~ver, the h r a -

tion of this behaviour differs between experiments. In particular, Tc,,, < Tc,,,,, < Tc,,

where Tc denotes the integral timescale. A plot of D2 - Do2 versus tirne, shown in Figure

4.11a), dernonstrates that short-time behaviour is characterized by linear, rather than ballis-

tic dispersion in each of the three experiments. This may be attributed, from (4.41), to the

non-zero mean advective t e m ( W(t )). Whether or not t or t2 dispersion is attained depends

on the magnitude of the initial separation Do. A smaller initial zonal separation is necessary

for the t2 terrn to dominate in (4.41), and thus for t2 scaling to be achieved. Figure 4.11a)

also shows that the t scaling lasts longer for the PW experiment, relative to the PW/GW

and GWRT experiments, for the given initial zonal separation. For the PW experiment,

Tc -- 1 day which is in agreement with the integral timescale Tz defined by (4.43) predicted

for flow governed by vortical dynarnics. For the PW/GW and GWRT experiments, however.

Tc < Tz. The presence of an unbalancecl component to the flow renders invalid the derivation

of an integral timescale bas& on the enstrophy (the usual eddy tuniaround timescale). The

CHA PTER 4. STATISTICA L DlAGNOSTICS

id

I

1 OT A

time (days)

I 10 '

L 1 ,

1 OT tirne (days)

Figure 4.10: Ensemble-averaged relative a) zona1 and b) meridional dispersion for the PW (red). PW/GW (blue), and GWRT (green) experiments.

CHAPTER 4. STATISTICA L DIAGNOSTICS

. - -. .- - PW

- - - - ewmw GWRT

Figure 4.11: Ensemble-averaged relative zonai dispersion a) D2 - D ~ * for aa initial rrns

separation of 4 0 = 44jkm, and b) LI2 on a log-linear plot frorn day 'L to day 20, for the PW (red), PW/CW (blue), and G WRT (green) experinients.

CHAPTER 4. STATISTICA L DIAGNOSTICS 114

temporal vwiability associatel wibb thegravity m v e s mkmces sepmation between particle

pairs, thus destroying memory of initial particle separation more rapidly than for the PW

experiment (as illustrated in Figure 4.9). Noteworthy also in Figure 4.10 is the fact that

Richardson's t3 Iaw emerges much more rapidly from the short-time asymptotic behaviour

for the PW/GW and GWRT experiments, in contrast to the PW case.

A regime intermediate t o the short-time and t3 regime rnay exist for the PW case from

t = 5 to 15 days. Shown in Figure 4.11b) is the dispersion from day 2 to day 20 on a

log-linear plot; more linear behaviour is dernonstrated for the PW experiment than for the

P W/GW and G WRT experiments. This behaviour might be explained by the exponential or

Kraichnan-Lin law (Babiano et al. 1985, 1990). This law may be derived upon integration

of (4.31), in accordance with the assuniption of a single timescale Tz, where F .- LI2, as will

be discussed in the following section. Thus

so t hat

The exponential law is found to be dependent on initial interparticle separation, and in-

dicative of the eficiency of turbulent diffusion. Such exponential behaviour occun only for

timescdes on the order of Tz. It is appropriate that this behaviour should hold for the PW

experiment, where balaaced dynamics yeti a definitive tirnesede, rsther thon for the PW/GW

and GWRT experiments, where no such single timescale exists.

Also of interest in the upper panel of Figure 4.10 is the enhanced relative dispersion for

the PW/GW and GWRT experiments, as exhibited in the values attained at t = 50 days

relative to t hose of the PW experiment. This behaviour may be undentood in t e m s of the

role of temporal variability in relative separation: a pair of particles initially trapped within

a given region rnay be allowed to escape. This has important implications for regions such as

the surf zone and the edge of the polar vortex in the middle atmosphere: the introduction of

gravity waves d o w s particles trapped within the core of the cat's eye structure to permeate

CHA PTER 4. STATISTICAL DIAGNOSTICS 115

surrounding regiow, whih pardieles f o d near theedgeof the p o k vortex may wander into

midlat i tude regions. Geornetrical considerations have shown t hat gravi ty waves dest roy the

coherent straining mechanisms inherent in the P W case (see for example Figures 3.2 and 3.5).

As such, gravity waves may be viewed as a catalyst for mixing, and relative dispersion the

tool with which the nature of this mixing is observed.

Meridionai dispersion [part b) of Figure 4.101 is characterized by a quadratic ( t 2 ) regime

for short times (on the order of hours) and diffusive behaviour at intermediate times, for

each of the three experiments. The fact that long-time asyrnptotic diffusive behaviour is

achieved at such an early stage is consistent with the meridional absolute dispersion results.

Diffusion is the dominant mechanismof transport in the meridional direction. In the absence

of planetary waves such diffusive behaviour persists for long times, as is demonstrated in

the GWRT experiment. However, the introduction of organized structure to the flow with

the planetary-wave forcing and the suppression of the gravity waves results in trapping

phenornena in the PW experiment. as shown in the plateau cormsponding ro a rnwidional

trapping region of width - lOOOkm or - 10". This value is comparable to the meridional

width of the region of strong vorticity at midlatitudes shown by the Okubo-Weiss and Hua-

Klein criteria in Chapter 3. The combined effect of planetary and gravity waves erodes this

inhomogeneity, as shown in the continued increase in D2 after t - 20 days for the PW/GW

experiment, yet the effect of the planetary waves is still reflected in the subdiffusive behaviour

for t > 4 days, in striking contrast to the GWRT experiment.

The differences apparent in the time evolution of the Qow and QL fields (i.e. coherent

structures for the PW case in both the Okubo-Weiss and Hua-Klein criteria, and homog-

enized regions resulting from spatial and temporal stochasticity in the P W/G W case) are

manifested in the temporal scaling laws seen in Figure 1.10. Rapid decorrelation in the Ion-

gitudind zond wind gradients u, and latitudinal meridional wind gradients v, (Figure 4.8)

gives rise to a short ballistic regime and rapid difision for the PW/GW case. The subranges

observed in the mnal direction correspond to those for which t < Tc, as demonstrated in the

CHAPTER 4. STATISTlCA L DIAGNOST!CS 116

Bat cmve for sbrt t h e s , and for which Tc c t < TI, for TI an intermediate time f evident ly

longer than the integration time of the experiment), wherein Richardson's t3 law holds. As

was demonstrated by the Okubo-Weiss criterion and the Liapunov exponent assessrnent of

Chapter 3, the influence of the unbalanced component of the flow upon the balanced corn-

ponent is manifested in the erosion of veloci ty gradients, and barriers to transport, rather

t han expressly in the velocity field. Hence it is the tweparticle statistics, rather than single-

particle statistics, that elucidate the distinguishing features of bdanced and unbalanced

dynamics for transport .

4.4.7 Local and Nonlocal Dynarnics: Kinetic Energy Spectra and Scaluig Laws

Built into the concept of relative dispersion is the distinction between spectrally locai and

nonlocal dynamics. This distinction may be quantified using scaling laws based on two-

paxticle statistics, namely in the scaling laws derived from the dynamical and measurable

quantit ies of the structure func tion and dispersion. This considerat ioi establishes the essence

of the present section, and is one of the pillars upon which the present investigation is based:

that of linking the geornetrical properties of the Bow to the dynamical properties of passive

tracer behaviour, as established using statistical diagnostics.

Central to the determination of spectral locality or nonlocality is the absence or presence,

respectively, of a single timescale. For a kinetic energy spectnun scaluig as E(k) - k-", where k is the horizontal spatial wavenumber, the distinction between local and nonlocal

dynamics caa be quantified according to the spectral dope on a log-log plot for the kinetic

energy spectrum. A recapitulation of the distinct dynamical regimes is presented here to

provide a reference from which dynamical scaling laws for the spatial aspects of particle

dispersion may be derived. Tracer evolution is governed by nonlocal dynamics for n > 3.

Nonlocal dynamics are characterized by a single timescale Tz, set by the large scales of

the Bow. The steep spectrum is associated with balanced dynamics and hence T, = Tz.

Dispersion within this regime is described by relative statistics. Most strain cornes from smail

CHAPTER 4. STATISTICAL DIAGNOSTICS 117

wauenurnbers, Le. lwge =des, md the defonntttion rate is ind-t of wavemtrnber.

When 1 < n c 3, dispersion of neighbouring particles is still relative, but the system is

described by spectrdly local dynarnics. Within such a regime, most strain cornes from

large wavenumbers or small-scale eddies. No single timescale exists, rather a multitude of

timescales associated with the wide range of eddies contribute to particle dispersion. Thus

the deformation rate increases with wavenumber. For n < 1, the system is characterized

by a diffusive regime. Particles follow dynamics akin to Brownian motion and dispersion is

described by absolute statistics.

Balance and imbalance are inextricably linked to the concept of local and nonlocal dy-

namics, in spite of the distinction apparent in the respective references to temporal and

spatial dynamical properties. The connection comes through the aforementioned role of the

integral timescale. The integral timescale Te is determined by the scales of motion prevalent

in the Bow, and in turn depends on the spectral slope of kinetic energy. It is worthwhile

to note that the integral timescale Tz as defined by ! 1.3) or (4.43) assumes that the flow

is governed by the enstrophy and thus the vorticity, so principally by the slow dynamics.

In fact, Tz is controlled by al1 velocity gradients, i.e. the strain, vorticity and divergence,

so that (4.47) provides a more reasonable description of the flow dynarnics. The latter is

nevertheless captured by the total energy spectrum.

Although the above categorization of spectral slopes serves as a means by which the

nature of mixing within a given regime may be established, rneasurements from atmospheric

observations or laboratory experiments tend to be instead of structure functions and diffusiv-

ities. The kinetic energy spectrum is then derived from these measured quantities. Babiano

et al. (1985) and Bennett ( 1984) have explored the role of the structure function and relative

eddy diffusivity in assessing the nature of local and nonlocal dynamics in twdirnensional

turbulence. Presented in the following section is a synopsis of their work against which the

numericai experiments of the present investigation are tested.

CHA PTER 4. STATlSTlCA L DIAGNOSTICS 118

4.4.8 Structure hn&o'lls ancf Sp& S c a h g Laws

The structure function is as before defined for V ( X ) the Lagrangian velocity in physical

space according to (Babiano et al. 1985)

for X = ( x , y) the position vector and D the separation vector, and w here averaging occurs

over al1 Eulerian coordinates. However, for the spectral space representation of the separa-

t ion veloci ty V ( k ) , the one-dimensional structure funct ion is related to the one-dimensional

energy spect rum t hrough the expression

It is t his relation t hat gives rise to the nonlocal/local distinct ion for the stmct ure funct ion

scaling laws. Following Babiano et al. (1985), the structure function can be interpreted

in cerms of spatial scaies larger jk'D < 1 j and smaller (ka » 1) than the interparticle

separation Z). In particular for E(k) - k-", the previous expression becomes

for a .- D-'. The first term on the right-hand side of this expression depicts the enstro-

phy Z = J' kZ E(k)dk, and the second, associated with scdes smaller than the interparticle

separation, the spectral energy. For n > 3, the enstrophy integral converges, and passive

tracer behaviour is govemed by the largest spatial scales. Nonlocal dynamics are therefore

depicted by a structure function scaling law of the form

Local dynarnics, for which 1 < n < 3, involve length scdes cornpaxable to the interpart icle

separation, so t hat k - V- ' . The structure funct ion scaling Iaw is t hen of the form

CHA PTER 4. STATISTICA L DIAGNOSTICS

Sdhimihriby hdds in this regime.

For the diffusive regime, where n < 1, the structure function is determined by the energy

at the smailest spatial scales. The structure function is then constant with respect to the

separat ion vector.

It is iuterest ing to note t hat (4.55) illust rates the previously encountered correspondence

between relative dispersion and velocity gradients, and in some sense sumrnarizes the rea-

sons for this correspondence. In particular, (4.55) shows that the relative dispersion of two

particles is controlled by velocity gradients on scales larger than the interparticle separation

D, provided the energy spectrum is steeper than k-' (Bennett 1984). Largescale straining

mechanisms, or shear, then give rise to either local or nonlocal dynamics. For a spectrum

shallower t h a n k-', the dispersion is controlled by the velocity field rather than by its gra-

dients on scales smaller than D, and in this case dispersion is absolute.

The structure function plays a role analogous to the Lagrangian au tocorrelat ion coeffi-

cient. as is evidenced in the expression

where

and E is the total energy.

single-particle Lagrangian

R(V) = ( V ( X ) V ( X + D))

( V 2 ( X ) )

The integral timescale Tc can be viewed as complementary to the

timescale TL. More precisely, the structure function is the spatial

counterpart of R ( r ) defined in Section 1.1.3. This parallel is manifested in the alternative

definition for the Lagrangiaa integral timescale (4.44) (Babiano et al. 1985). In this way.

largescale structure is reflected in the presence of a single timescale determined by the

energy-containing eddies, and reflected in scaling laws indicative of noniocal dynamics.

Babiano et al. (1985) asserted that the structure function is a useful diagnostic in veri-

fying the nonlocal nature of a dynamical regime. However, dynamical information may be

lost concerning the spectral slopes of kinetic energy spectra inferred from stmct ure funct ion

CHAPTER 4. STATISTICA L DIAGNOSTICS 120

meastuementsr when S ( D ) - D2, =y kinekir ewrgy s p s k ~ u r n a& Least acl sbeep as k3 is

permitted. This insensitivity to the nature of the largescale straining mechanism acting on

the particle pair is andogous to the situation found for single-particle dispersion. Of course,

wi t h numerical experiments one can compute bot h quantities explicit ly. Examination of the

dopes generated for bot h the structure function and the kinet ic eaergy spectra nevertheles

provide a means by which the mixing properties of the flow may be identified according

to spectrally local or nonlocai dynamics. Structure function scaling laws distinguish be-

tween local and nonlocal dynamics; kinetic energy spectra are used to distinguish between

unbdanced and bdanced components of the flow.

The instantanmus relative eddy diffusivity is defined, in contrast to (4.31), as

and is related to the longitudinal structure function SII(V) by (Babiano et al. 1985)

Alt hough this expression is not used in the explicit calculation of the eddy diffusivity, the

scaling relations derived for the structure function can be used to derive the appropriate

relative dispenion scaling laws within spatial rather than temporal regimes. For long times,

where the neighbouring particles are decorrelated in both space and time, the relative dif-

fusivity, as shown by Richardson, is twice the value of the constant diffusivity obtained for

single part icle dispersion. That is,

F, -+ 2h' (4.62)

for a spatially and tempordly decorrelated Bow field. The factor of two is a consequence

of the fact that particle pairs rather than individual particles are used to monitor particle

dispersion. This is in contrast to the laws predicted from the longitudinal structure function

for n < 1. While the particles are in this case decorrelated in space, it is not necessarily true

that they are decorrelated in time.

CHAPTER 4. STATlSTlCAL DIAGNOSTICS 131

Noteworthy also from sucb relations ie &e emeEgenee of Richardson's empiricat pu3

scaling law. Richardson considered atmospheric observations available in the early twentieth

century, and it is now known that such observations give n = 513 up to synoptic scdes of

motion (Nastrom et al. 1984). This regime emerges at even larger scales in the mesosphere

(Koshyk et al. 1999). Central to the investigation of this relation is the dependence of the

diffusivity on separation, and its independence on position. This assumption is used to derive

an appropriate PDF, or distance-neighbour funct ion, from which the statist ical properties

of dispersion resulting from both nonlocal and local dynamics may be derived. The essence

of this derivation is presented in Section 4.4.10.

4.4.9 Structure Function Results

The structure functions presented in this section are computed from the zonal and rnerid-

ional winds according to (4.53), for displacements in the zonal direction only. Shown in

Figure 4.12 are the structure functions associated with the PW, PW/GW and GWRT ex-

periments cornputed at 45ON, and averaged over 50 days. The structure functions a t other

latitudes are similar. The scaling laws appropriate for nonlocal and local dynamics as derived

for twedimensional turbulence (Bennett 1984, Babiano et al. 1985) appear to be obeyed for

each of the t hree experiments. That is, For the case of P W forcing where nonlocal interac-

tions are expected to dominate, a 'DZ scaling law is obtained at al1 spatial scales. However,

with the ptesence of the smaller-sc& eddies ~esulting from the GW forcing in the P W/G W

experiment, the structure function approaches an intermediate scaling law of S ( V ) -- v2I3,

followed by S ( D ) - b at the largest scales where nonlocal effects become important. By

contrast, the structure function approaches a constant for the GWRT experiment, as is to

be expected for the diffusive regime established by the spatially random forcing.

The results indicated in Figure 4.1%) coincide with the scaling laws established for the

kinetic energy spectra as presented in panel Figure 4.12b). The unbalanced component to

the flow in the PW/GW experiment is rnanifested in the D2I3 structure function scaling laws

CHAPTER 4. STATrSTICAL DIAGNOSTlCS

Figure 4.12: Temporally-averaged structure function at 45"N for the PW (red), PW/GW (blue), and GWRT (green) experiments. lndicated also are the slopes corresponding to n = 2 and n = 213. Shown in b) are the corresponding kinetic energy spectra, with reference slopes indicated.

CHAPTER 4. STATISTICA L DIAGNOSTICS 113

at intemediete spatid seales. This correspo~tds to Etk) - k-q3, which approxirnates the

kinetic energy spectrum achieved in this experiment. Two distinct scaling regimes are evident

for both the kinetic energy spectra and the structure function in the PW/GW experiment.

Figure 4.12 suggests that the scaling laws derived for 2D turbulence do indeed hold for

unbalanced Bow. That is, the structure function scaling laws appear to be insensit ive to the

temporal variability introduced by the gravity-wave forcing, or to the presence of a strong

divergent cornponent to the Bow.

4.4.10 PDFs and the Distance Neighbour Function

Closely related to the PDF of relative separat ion is Richardson's distance-neighbour funct ion

q, defined for one dimension as the mean number of neighbours per unit length. With the

assumption that the eddy diffusivity depends only on separat ion, based on the empirical law

F - c D " / ~ , relative diffusion can be described by the differential equation (Richardson 1926)

known as Richardson's non-Fickian diffusion equation. Noting the resemblance of this equa-

tion to the heat equation. its solution is expressed as

for a = z>$ and A a constant indepmdent of tirne ar the sep-tion. This relation is

applicable wi thin the self-similar regime characterist ic of local dynamics. Richardson further

demonstrated that the second moment of the distance-neighbour function represents the

standard deviation of the cluster of particles from its mean.

Bat chelor ( 1951 ) proposed an alternative representation for turbulent diffusion and the

PDF according to the assurnption that the diffusivity depends on time rather t han separation.

This suggestion takes into account the variability in the diffusivity for the duration over which

diffusion occurs. Batchelor furt her proposed that the diffusivity depend on the particle

CHAPTER 4. STATISTICAL DIAGNOSTICS

sepanrtion variance 02 = (6 6) = (W), mch that

rather than on the relative separation D alone. The differential equation describing relative

diffusion is in this case (Batchelor 1951 )

where F ( t ) is the eddy diffusivity, the solution of which is the normal distribution

for

D2 OC t3

and A a constant. While both conjectures yield distinct PDFs, Richardson's t3 law is seen

to hold for an eddy diffusivity dependent either on sepmation or time.

A natural question to emerge from both investigations is the nature of the PDF resulting

from an eddy diffusivity dependent on both separation and time. Hentschel and Procaccia

(1984) have explored this phenomenon and discovered, using arguments from fractal analysis,

that the diffusivity describing relative dispersion for local dynamics should be of the form

where A is a constant, independent of particle separation or the duration of diffusion, while

a and b are exponents determined from fractal statistics. The principles upon which such

assumptions are based reside in the examination of the fractal nature of clouds: diffusion

does vary with t ime, t hus rendering Richardson's proposal inappropriate, while fractal st ruc-

tures distinguish the changing cloud shape with time, rendering Batchelor's proposal of a

separation-independent diffusivity invalid. The corresponding diffusion equation is expressed

CHA PTER 4. STATISTICA L DIAGNOSTICS

the ~01ution of which is

for E and ~ ( b ) constants in time and separation. For a = O and b = 413, correspond-

ing to Richardson's hypothesis of a time-independent diffusivity, the PDF scales as the

distance-neighbour function, whereas for b = 0, the PDF is consistent with that proposed

by Batchelor.

Each of the PDFs derived above is applicable for local dynamics. However, Lundgren

( 1981 ) found nonlocal dynamics to be characterized by a lognormal PD F. Self-similarity is

no longer an appropriate description in this case.

Of interest in the present investigation is the nature of the relative-separation PDFs

derived for the PW, PW/GW and GWRT experiments in both the zonal and meridional

directions. Examined in particular is the nature of the distribution, that is, whether it is of

the shape purported by an assumption of the dependence of diffusivity on separation. or timr.

It is hoped that analysis of the PDF sbape might provide some insight regarding necessary

conditions for the prediction of an appropriate eddy diffusivity for a given atmospheric

regime.

4.4.11 Higher-Order Moments

Consideration of the higher-orde~ moments of the PDF provides an additional means by

which nonlocal dynamics can be distinguished from local dynamics. Constant Batness, de-

fined in Section 4.3.2, is expected for local, self-similar dynamics, whereas exponentially

increasing flstness reflects nonlocal dynamics (Bennett 1984).

4.4.12 PDFs for Relative Separation: Results

Figure 4.13 depicts the PDFs of zonal and meridional separation for the PW, PW/GW and

GWRT experiments at t = 50 days. Differences are manifested both in the cores and the

tails of the PDFs for each of the experiments. The finite width of the peak in the zonal

CHAPTER 4. STATISTICAL DIAGNOSTICS 126

separabion P DF f o ~ the P Ur expetiment saggests that s o w particles remain trapped witbin

a region of spatial extent AD, - 7000km or -. 60'. This is comparable to the horizontal

width of the recirculating region found at mid-latitudes (see Figure 3.2). By contrast, the

peak of the PDF is narrower for the PW/G W experiment. This would suggest that particles

have been transported out of the confines established by the spatial inhomogeneities. For the

GWRT experirnent, in which no coherent features are present, the peak is narrower. Barriers

to transport set the maximum separation between a pair of particles, and are reflected in

the PDF tails. For the zonal dispersion, the tails of the PDFs are not suited to a Gaussian

description. Moreover, the PDF exhibits behaviour similar to that dernonstrated in the PDF

of zonal absolute displacement shown in Figure 4.5a). Trapping in the meridional direct ion

leads to a limit in the zond dispersion for the PW case. This is in agreement with Taylor's

(1953) theory of sheiu-induced dispersion, which demonstrates the influence of meridional

dispersion on zonal dispersion in the presence of a zonal shear flow.

The lower panel of Figure 4.13 exhibits. as for the single-particle ~ t a t ~ i s t i r s , Gaiissian

behaviour for the meridional separation for each of the experiments. The widt h of the PDF

is confined to an interval roughly corresponding to the meridionai extent of the hemisphere.

The large-scale straining mechanisms exerted by the planetary waves appear to inhi bit the

relative separation between particles, confirming the trapping alluded to above.

The zond and meridional PDF results obtained for the PW experiment are consistent

with those found by Er-EI and Peskin (1981) in their examination of balloon dispersion in

the atmosphere. It is interesting to note that the stratospheric measurements analysed in

that study also obeyed the exponential scaling law found between t = 5 and 15 days for the

PW experiment shown in Figure 4.10 of the present investigation.

Depicted in Figure 4.14 is the time evolution of the PDFs for the zonal separation.

Separation is evidently much more rapid in the presence of gravity waves. Noteworthy is

the difFerence in the behaviour of the tails of the PDFs. The stretched tails evident in the

PW/GW case are suggestive of large particle displacements induced by the advection by

CHA PTER 4. STATISTICA L DIAGNOSTICS

P W PW IGW GWRT

Figure 4.13: PDF of a) zooal separation and b) meridional separation for P W. P W/G W. and GWRT experiments at t = 50 days. Dotted lines depict the corresponding Gaussian distributions.

CHAPTER 4. STATISTlCAL DlAGNOSTlCS

a)

Figure 4.14: periment, b) 4-

g lu'

O D, ( h l

PDF of the zona1 separation at different instants in time for a) the PW ex- the PW/GW experiment, and c) the GWRT experiment. Flatness is shown in

CHAPTER 4. STATISTICAL DIAGNOSTICS

Figure 4.15: As in Figure 4.14, except for the meridional separation.

CHAPTER 4. STATISTICA L DlAGNOSTICS 130

the basic-state shea and plaaetery waw, c o m b i d with enkrmced locai sepmation remking

from diffusive processes. The more rapid decay in the tails for the GWRT case relative to

the PW/G W experiment indicates that separation between particle pairs is, for the G WRT

case, governed by standard diffusion in the rneridiond direction: the non-Gaussianity is a

consequence of the zonal shear. As demonst rated by the O k u b e Weiss criterion and Liapunov

exponent diagnostics of Chapter 3, the unbalanced cornponent acts t o homogenize or erase

the gradients of the flow prevalent for nonlocal dynarnics. Such a phenomenon appears to

be reflected in the distinct ive PD Fs of relative separat ion obtained for each experiment . Local behaviour is depicted in the equilibration of the flatness of the zonai separation

PDF for long tirnes in the PW/GW and GWRT experiments. In contrast, spectral non-

locality is evidenced in the rapidly increasing flatness of the zonal separation PDF in the

P W experiment . Large Ratness values reveal the combined effects of large-scale transport

and small-scale fluctuations ( Er-El and Peskin 1981). Such behaviour is not observed for

the meridional separation (Figure 4.15): the signature of Gauss ian i t~ F = 3: is apparent in

al1 experiments. As shown in Figure 4.15, the meridional PDFs for the PW experirnent vary

little in time between 30 and 50 days. The PDFs for the PW/GW and GWRT experiments

reveal that diffusion as incurred by the gravity waves results in enhanced separation between

particle pairs.

The results from t his section demonstrate that relative dispersion does convey the differ-

ences apparent in nonIocd and local dynarnics, and in so doing, demonstrates the effect of

the unbalanced motion on passive tracer transport. Two-particle statistics moni tor the s e p

aration bet ween pairs of particles result ing not only from fluctuations of comparable lengt h

scales in the velocity field, but also from largescale straining mechanisms. Cornparison of

the PDFs with the O kub- Weiss and Hua-Klein diagnostics demonstrates t hat the existence

of topologically distinct features (i.e. strain and vorticity dominated regions) gives rise to an

inhibition of particlepair separation: a large number of part icles remain trapped within the

cat's eye structure. Barriers t o transport are manifested in the finite breadth of the PDFs.

CHAPTER 4. STATISTICAL DIAGNOSTICS

Nevertheles, gravity waves enhance p M e septmthr.

Richardson's form for the distance neighbou: function is found to be an accurate repre-

sentat ion for zona1 relative dispersion. This suggests that a separat ion-dependent diffusivity

is valid when considering the net dispersion between a pair of particles. A Caussian distri-

bution is appropriate for meridional considerations, in agreement with Batchelor's proposal

(1951) that the diffusivity be dependent only on time. This is a consequence of the compar-

ative homogeneity of the meridional flow, and the consequent random-wdk-like behaviour

experienced by passive tracen as they move in the meridional direction.

These results are consistent wi th the turbulent diffusion studies of Boffetta et al. ( 1999).

where it was shown that relative pair statistics described by Richardson's t3 scaling iaw

and the distance neighbour function provide an appropriate description of non-intermittent

turbulence. The attainment of similar results for the PW/GW experiment in the present

case suggests that turbulent diffusion provides an appropriate description foc Bow that is

strongly influenced by unbalanced dynamics. in contrast to the chaotic advection description

of transport used to describe flow in the presence of largescale spatial inhomogeneities.

4.5 Additional Diagnostics

4.5.1 The Two-Particle Correlation Function and Correlation Di- mension

It has been proposed that absolute dispersion Iends itself to a multifractal interpretation

of particle dispersion (Provenzde 1995). A diagnostic used to determine whether or not

fractal behaviour is an appropriate description for singleparticle statistics is the two-particle

correlation function, which measures the number of particles N separated by a distance

las than a specified distance r . The correlation function is defined as (Provenzale 1995.

Pierrehumbert 1990) I N

where f&, denotes the distance between particles i and j and 0 is the Heaviside step function.

CHAPTER 4. STATISTICAL DlAGNOSTlCS 1 3'2

Qualitat.iveLy, the co~dat ion lunetion monitors the mk of partictes within a cluster

found to be less than a distance r apart, and so rnay be constnied as a two-particle diagnostic.

In contrast to relative dispersion, however, which monitors the spread in a particle cluster, the

correlation function tracks the spatial extent of a cloud of particles. The use of the correlation

function to explore passive tracer dynamics in surface Boatable experiments similar to the

P W/G W experiment considered here (see for example Sommerer and Ott 1993), suggests

that such a diagnostic might be useful in edua t ing the meridional structures evident in the

velocity gradient fields with the presence of a divergent cornponent [as seen in Figures 3 .54

and 3.10f)J.

The scaling law for the correlation function is given by

where D2 is the correlation dimension. It is the correlation dimension that quantifies mixing:

a fluid is said to be completely mixed if 4 = 2. Filaments are depicted by dimension 4 = 1,

and clusters of points are depicted by D2 = O. The correlation dimension therefore serves as

another tool with which the distinction between stirring, as reflected in filarnentation, and

mixing, distinguished by homogenization, may be quantified.

4.5.2 Correlation Dimension Results

Depicted in Figure 4.16 is thecorrelation function for both the PW and PWIGW experiments

at t = 10 and t = 50 days. Included also in Figure 4.16 is a superposition of the PW and GW

experiments a t 10 and 50 days, in order to demonstrate distinctive regions of mixing and

filarnentation for each scenario. The correlation function is cornputed according to (4.72):

separations between each of the 21824 particles are calculated, and those with separation

less than a prescribed radius r ,

recordeci. For simplicity we take x = aXcos(8) and y = a$, where a denotes the Earth's

radius, although this becomes inaccurate at large scales.

CHAPTER 4. STATISTICAL DIAGNOSTICS

Figure 4.16: Correlation function for the a) PW and b) PW/GW experiments at t = 10 and t = 50 days. Shown in c) is the superposition of each experiment a t both instants in time. Indicated also in each panel is the slope corresponding to Dz = 2.

The contrast between the PW and PW/GW experiments is depicted in t.he shallow

spectral slope at srndl distances that emerges in the latter case with the progression of time,

contrary to the anticipated behaviour at the smailest spatial scales (Pierrehumbert 1990).

For the PW experiment, the interval over which homogenization of srnall-scale structure is

evident, manifested in a dope of 2, extends to smaller scales with time. For the PW/GW

experiment, however, the opposite phenomenon is observed. Although one would expect

homogeneity at the smallest scdes for the PW/GCV experiment, the passive tracer cluster is

characterized by filarnentary features. That is, the homogenization of srnall-scale structure

exhibited at t = 10 days is replaced by a shallower slope at smdl spatial scales at t = 50

days. The correlation funct ion sat urates at a separat ion of approximately 180' ( not shown),

CHAPTER 4. STATISTICAL DIAGNOSTICS

whkh is the rnrutimam seperotiorr of two partides on the sphere.

4.5.3 Finite-Time Liapunov Exponents

Revisited in this section is the concept of the Liapunov exponent from a statistical penpec-

the . Introduced in Chapter 3 as a means of monitoring the separation between a pair of

part icles wi t h infinitesimai initial separation, the PD Fs for the Liapunov exponents provide

a quantitative means of depicting regions of enhaaced mixing. It is instructive to relate

the PDFs of the finitetime Liapunov exponents to the ranges prevalent in the correlation

function, in order to establish a connection between the mixing properties dist inguished by

the PDFs and the geometric regions characteristic of distinctive values of the correlat ion

dimension. Numerous authors (Pierrehumbert 1991, Provenzale 1999) have explored the

connection between fractai geometric analysis, upon which the concept of the correlation

dimension is founded, and the PDF for the Liapunov exponent. In particular, it has been

shown (Pierrehumbert 1991 ) t hat the phase transition or "elbow" in the correlation function

H ( r ) can be determined from the peak in the Liapunov exponent. This is best demonstrated

through the spectral interpretation of the Liapunov exponent definition given by (3.34),

with ko the characteristic wavenumber for the ini t i d distribution. The purpose of the present

section is to ascertain the mean exponentid growth rate associated with stretching of material

lines in the PW and PW/GW experiments.

Does the evolution of the separation PDFs correspond to that of the largest Liapunov

exponent? Moreover, since Liapunov exponents are a measure of exponential divergence be-

tween neighbouring particles, and hence of chaos and the nonlocal properties of the flow field,

will the behaviour for the Liapunov exponents be comparable o d y for the PW experiment?

CHA PTER 4. STATlSTlCA L D IAGNOSTlCS

4.5.4 PDF fm the Pinite--Tirne Liapmrov Exponent

In Chapter 3 the spatial distribution of finitetime Liapunov exponents was presented in

order to depict regions of strong exponential stretching of material contours, and hence

exponential divergence of neighbouring particle trajectories. Shown in Figure 4.17 are the

corresponding PDFs for the PW and PW/GW experiments at different instants in time.

Figure 4.17: PDF of finite-time Liapunov exponent for the a ) P W and b) P W/GW experi- ments at different instants in time.

The PDFs narrow with tirne. A more well-defined peak is evident for the PW than for the

PW/GW experiment. This c m be explained, once more. in terms of the nonlocal dynamics

associated with the PW Bow. The presence of a single timescale, set by the large-scale,

CHAPTER 4. STATlSTlCAL DIAGNOSTICS 136

low-f~equency, or balancd eompomnt of the flow, gives rise to e*pondd separation of

neighbouring particles, and regions of strong chaotic mixing. The mean exponential growth

rate, as determined from the PDF peak, corresponds to t - 20 days, which is roughly half the

eddy tumaround tirne found for the trajectory of a single particle in the surf zone, as shown

in Figure 4.2. A much shorter Liapunov exponent corresponding to t - 5 days was found by

Pienehumbert (199 1) for mixing in stratospheric Bow . The reason for this discrepancy may

be due to the inclusion of particles outside the surf zone in the computation of the Liapunov

exponent. Those particles whose advection is governed by the zona1 mean flow will give rise

to algebraic, rather than exponential stretching. Such a feature is exhibited in Figure 3.10,

north of - 40'. In such a case the Liapunov exponent will tend to zero as t + m.

The finite-time Liapunov exponent PDF attained for the PW/GW experiment [Fig-

ure 4.17b)l has converged to a distribution that is less sharp than that found for the P W

experiment, owing to the enhanced spatial and temporal variability introduced with the sig-

nificant unbalaaced component to the Row. A mean exponential growth rate rorr-ponding

approximately to t .- 8 days is found for the PW/GW experiment. Such may reflect the

"sloshing" nature of the gravity-wave activity: the stretching associated wit h trajectories

governed by large-scale straining mechanisms may be in tempted with the presence of Spa-

tial and temporal fluctuations. The existence of larger Liapunov exponents for the PW/GW

experiment suggests greater mixing.

Chapter 5

Conclusions

5.1 Summary

Middle atmospheric dynamics are governed by numerous transport rnechanisms, manifested

in the phenomena of stirring and mixing. The present investigation has sought to eluci-

date the association of stratospheric dynamics with stirring, as determined from a chaotic

advection description of transport, and of mesospheric dynamics with mixing, as deter-

mincd from a turbulent diffusion description of transport. lu particulor, the concepts of

balance vs. imbalance and spectral nonlocality vs. locality have been used to assess re-

sults from the planetary-wave ( P W) experiment representative of st ratospheric dynamics

and the planetary-wavelgravity-wave ( P W/GW) experiment thought to depict mesospheric

dynamics. Geometrical and statistical analyses have shown that the introduction of a large

unbalanced component to the flow, as manifested in shallow slopes of kinetic energy spectra,

does indeed have important implications for transport and mixing.

Assessment of the geometrical properties of the flow in Chapter 3 reveals that velocity

gradients are strongly influenced by the presence of an unbalanced cornponent to the Bow.

This is exhibited by the O k u b Weiss and Hua-Klein criteria as an erosion of velocity gra-

dients in the PW/GW as compared to the PW experiment. A new diagnostic is developed

which generalizes the Hua-Klein criterion to divergent flaw, which is essential for the present

study. Spat i d distribut ions of finite-time Liapunov exponents demonstrate t hat coherent

st rain-dominated regions characterist ic of a chaotic advection descript ion of transport are

CHA PTER 5. CONCL üS1ONS 138

ako eroded. Patcbess, which mmonitms the mean tagrangian vetocity, demonstrates trans-

port into and out of the zonal jet. While a slight degradation of patches rnay be found at the

edge of the basic-state jet, the patchiness plots reveal the persistence of a coherent structure

for the zonal wind and the destruction of spatial structure in the meridional direction.

The techniques adopted in C hapter 3 furt her provide a visual interpret at ion of nonlocal

and local behaviour, but now understood in the spatial sense. Liapunov exponent analysis

is spatially nonlocal; the OkubeWeiss analysis local. A more nonlocal interpretation of

the velocity gradient field resides in the Hua-Klein criterion wherein the eigenvalues for

the acceleratioo gradient tensor are computed. Spectraily nonlocal behaviour for the P W

experiment is manifested in regions of st rong exponentid divergence along the unstable

manifold bounding the cat's eye structure, as shown in the spatial distribution of the finite-

time Liapunov exponents. Such regions disappear for the PW/GW experiment, and the

emergence of spect rally local behaviour is captured by the OkubeWeiss criterion in the

rlifferenrcs evident in the Q fields for the PW and PW/GW experirnents. The results from

the Liapunov, OW and HK analyses suggest, from a qualitative perspective, that balanced

dynamics rnay be described by stirring, and unbalanced dynamics by mixing.

Statistical analyses performed in Chapter 4 furt her support the connection between stir-

ring and mixing phenomena on the one hand, and balanced and unbalanced dynarnics on the

other. Results from this chapter also emphasize the role of single-particle statistics in de-

scribing spectrally nonlocal phenomena as determined by the large-scale eddies, and relative

dispersion in describing bot h nonlocal and local behaviour. As for patchiness, unbalanced

dynarnics has Little effect on the singleparticle analysis for zona1 flow. Dispersion is gov-

erned by the Lagrangian correlation coefficient which is found to be comparable in the zonal

direction for the PW and PW/GW experiments, owing to the presence of the basic-state

shear. Consequently, non-Fickian diffusion is evident in the dispersion statistics for both

experirnents. In addition, little distinction is to be found in the PDFs of zond displacement

for the PW and PW/GW experiments. The inclusion of a flow generated from a spatially

CHAPTER 5. CONCL USIONS 139

and tempord1y-decorrelated forcing mechaniSm, as chzmtcterized by tire G W W e x p h e n t ,

provides additional illustration of the role of absolute dispersion in capturing large particle

displacements due to advection by the basic-state shear. W hile the zona1 dispersion stat is-

tics can be used to confirm theoretical arguments, the rneridional dispersion statistics, in

the absence of the zonai jet and asy unredistic -wrapping3 mechanism in the rneridional

direction, are of genuine physical significance. In particular, the Gaussianity of the statistics

for the PW/GW and GWRT experiments in the meridional direction suggests that diffusion

does provide an appropriate description for transport in the mesosphere.

A shal!ow spectral slope of kinetic energy associated with a large unbalanced cornponent

to the flow is, however, reflected in relative dispersion statistics. Relative dispersion is

shown to capture the existence of numerous timescdes induced by the temporal variability

inherent in the gravity waves, in contrast to the single advective timescale associated wi th

the balanced dynarnics. Rapid decorrelation of the Lagrangian derivatives ot her t han u,

demonstrates that the separation between a pair of particles will initially hr i n f l w n r ~ d hy

the unbalanced component of the Bow. Richardson's t3 law is thus achieved more rapidly

for the PW/GW and GWRT experiments than for the PW experiment. The existence of

stretched tails in the PDFs of zona1 relative separation for the PW/GW case, in contrast to

the finite width of the PDF achieved for the PW experiment, further dernonstrates the role

of the unbalanced component in enhancing separation between particle pain. Large values

of Batness suggest that the relative separation is determined by the combined influence of

large-scale straining mechanisms induced by the basic-state shear, and fluctuations of size

comparable to the interparticle separation. Therefore relative rather than absolute dispersion

captures the differences between the Ptratospheric" and "mesospheric" dpamical regirnes

resulting from the stronger gravity-wave activity in the latter.

The results from Chapter 4 rely on the premise that displacement is controiled by the

Lagrangian velocities, and separat ion by the Lagrangian velocity gradients. Exponential

stretching of line elements, as reflected in large Liapunov exponents near the edge of the

CHAPTER 5. CONCL USIONS 140

surf zone for the P W case, is repiwenbd by ~ ~ e - ~ c k statistics, as is the presence of

orgmized structures resulting from a stat ionaxy perturbation to the basic-state shear. This

suggests a correspondence between absolute dispersion and stirring. The homogenization

of distinct topological regions in the PW/G W experiment , exhibited by the OkubeWeiss

and Hua-Klein criteria in particular, is manifested in the notably different behaviour of the

relative dispersion statistics for the PW and PW/GW experiments. The finite width of

the PDF of zoo$ separation in the PW experiment coincides with the spatial extent of the

mixing region, with D 2 150' being the zonal width of the cat's eye. Relative dispersion

statistics, in capturing stretching and diffusion processes, c m t herefore be construed as a

signature of mixing.

Also to be gathered from the statistical analysis of Chapter 4 is the determination of

an appropriate description for passive tracer dynarnics in spatially and temporally irregu-

lar flow. Superdiffusive scaling obtained for single-particle dispersion in each of the PW,

PW/GW and GWRT experiments suogests that t h e advection-diffusion rquation with con-

stant eddy diffusivity provides an incomplete description of transport for inhomogeneous flow

(in t his case, flow controlled by a basic-state s heu) . Moreover, relative dispersion stat istics

demonstrate that an equation describing evolution in the distance neighbour function wi t h a

separation-dependent diffusivity, as proposed by Richardson, may provide a reasonable esti-

mate of transport in the direction of the zonal flow. The present results further demonstrate

that a time dependent relative diffusivity may be a suitable approximation for regions where

diffusion is the governing mechanism of transport. Such an implementation would however

be appropriate only for transient problems, such as those associated with pollutant release.

Although results at spatial and temporal scales comparable t o or smaller than the spatial

and temporal resolution used in the present shallow-water scheme may vary with the use of

alternative choices for grid-scale truncation, the results at larger spatial and temporal scales,

such as are considered here, should be independent of such mode1 choices. Alternative in-

tegration and time-stepping schemes that implement spectral, rather than semi-Lagrangian,

CHAPTER 5. CONCL USIONS 141

techniques nay dsv yieid d 8 m t resutts at the d k s t spatid scdes and highest fre-

quences. However, as indicated in the experiment characterization of Chapter 2, the results

conveyed by the kinetic energy spectra, spatial and temporal autocorrelations, and power

spectra indicate t hat the essent i d physicd feat ures of stratospheric and mesospheric dynam-

ics are indeed c a ~ t u r e d by the PW and PW/GW experiments, suggesting that the results

are independent of specific model choices.

5.2 Future Directions

One possibility for future work resides in the cornparison of the present results with mid-

dle atmosphere GCMs such as the Canadian Middle Atmosphere Mode1 (CMAM). The

shallow-water model results have shown that diffusion provides an appropriate description

for meridional transport in the presence of gravity waves. These results are, however, some-

what limited by issues of convergence (a run longer than 70 days may be necessary) and by

error introduccd t tirough the semi-imp lici t , senii- Lagriuigiau shdlow-water scheme. Anai-

ysis of a more realistic situation, such as is presented by GCMs, would verify whether or

not shallow spectral slopes are indeed rnanifested in Caussian statistics in the meridional

direction.

Given the existence of Gaussian behaviour for meridional transport, the role of relative

dispersion in detecting unbalanced (fast) dynamics could also be used in the development of

a first-principles diffusion parameterization for CMAM in the mesosphere. The existence of a

shallow energy spectrum in the mesosphere seems like a chailenge for parameterization, due

to the lack of a separation in scales between the resolved and unresolved motions. However.

the existence of diffusive behaviour provides a justification to implement a diffusivity whose

coefficient is that appropriate for the grid-scale truncat ion.

In addit ion, the present twedimensional flow could be extended to include considerat ion

of three-dimensional (diabatic) Bow, which is relevant in the rnesosphere for timescales of a

week or more.

CHAPTER 5. CONCL USIONS 142

The gmvity-wave forcing s c ~ i m p t e m m t e d kere gives rise tu a situation wherein grav-

ity waves are superposed on a planetary wave characteristic of 2D stratospheric turbulence.

Previous studies of 2D turbulence have focused on the existence of coherent structures.

There has been comparatively little work on turbulent diffusion in systems with fast and

slow degrees of freedom. The present analysis, however, as is demonstrated by the relative

dispersion statistics, explores the combined effects of coherent structures and waves. and

in particular of the role of fast oscillatory dynarnics. The structure-function results indi-

cate that the scaling laws derived for 2D turbulence are insensitive to temporal dependence.

Since shallow-water gravity waves play a role analogous to sound waves in 2D compressible

turbulence, one possible avenue for future reseaxh may be the investigation of mixing in 2D

compressible turbulence.

Appendix A

Description of GEM S hallow-Water

Included in this appendix is a description of the numerical implementation of the shallow-

water equations described in Chapter 2. The horizontal momentum equations are expressed

fur the zuiial wiiicl coiiipouetiL, and for the meridionai wind component

The continuity equation in discretized form is

Furtherrnore. thegeopotentid tieight field #is dehedsuch that + = tp*++t+bs+~PW+@Gw,

where fl = 9.8 x 8 x 103 m2s-* is the unperturbed geopotential with H = 8.0km the

equivalent fluid depth, 4' is the perturbation geopotentid, not to be confused with dpw and

q5Gw described in Section 2.4, and 4s = 0.0 is the lower surface geopotentid. Common to

each of these expressions is the evaiuation of the forecast term based upon the parameters

obtained upwind from previous time levels. By eliminating the divergence bet ween the

continuity and divergence equat ions, the geopotent ial perturbation 4' is obtained from a

Helmholtz equation. The winds are then obtained frorn the divergence equation via back-

substitution.

APPENDlX A. DESCRIPTION OF GEM SHALLOW- WATER MODEL

A. 1 Changes to the Goveming Equations

The modified momentum and continuity equations, in the notation used for the GEM for-

mulation, are expressed as

Du' -+fa x Ü+V+ -,P(Ü- Dt .O ) ( A 4

for the momentum equations, with üo the basic-state velocity, and n = 8.64 x 105s, corre-

sponding to a darnping timescale of about 10 days. Similady for the continuity equation,

where &' is the basic-state geopotential. Furt hermore the discretized momentum equat ions

yield for the zona1 wind component

and

for the meridional component. The discretized version of the continuity equation is expressed

It is the relaxation terms in these equations that enable the velocity and geopotential fields

to approach statistical equilibrium.

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