On the Dynamics of Two-Dimensional Hurricane-Like Vortex ...

On the Dynamics of Two-Dimensional Hurricane-Like Vortex Symmetrization

Y. MARTINEZ AND G. BRUNET

Meteorological Research Division, Environment Canada, Dorval, Quebec, Canada

M. K. YAU

Department of Atmospheric and Oceanic Sciences, McGill University, Montreal, Quebec, Canada

(Manuscript received 12 March 2010, in final form 17 June 2010)

ABSTRACT

Despite the fact that asymmetries in hurricanes, such as spiral rainbands, polygonal eyewalls, and meso-

vortices, have long been observed in radar and satellite imagery, many aspects of their origin, space–time

structure, and dynamics still remain unsolved, particularly their role on the vortex intensification. The un-

derlying inner-core dynamics need to be better understood to improve the science of hurricane intensity

forecasting. To fill this gap, a simple 2D barotropic ‘‘dry’’ model is used to perform two experiments starting

respectively with a monopole and a ring of enhanced vorticity to mimic hurricane-like vortices during in-

cipient and mature stages of development. The empirical normal mode (ENM) technique, together with the

Eliassen–Palm (EP) flux calculations, are used to isolate wave modes from the model datasets to investigate

their space–time structure, kinematics, and the impact on the changes in the structure and intensity of the

simulated hurricane-like vortices.

From the ENM diagnostics, it is shown in the first experiment how an incipient storm described by a vortex

monopole intensifies by ‘‘inviscid damping’’ of a ‘‘discrete-like’’ vortex Rossby wave (VRW) or quasi mode.

The critical radius, the structure, and the propagating properties of the quasi mode are found to be consistent

with predictions of the linear eigenmode analysis of small perturbations. In the second experiment, the fastest

growing wavenumber-4 unstable VRW modes of a vortex ring reminiscent of a mature hurricane are

extracted, and their relation with the polygonal eyewalls, mesovortices, and the asymmetric eyewall con-

traction are established in consistency with results previously obtained from other authors.

1. Introduction

Although the circulation in a hurricane can be consid-

ered primarily axisymmetric, observations often reveal

asymmetric features in the form of outward propagat-

ing inner spiral rainbands and polygonal eyewalls (e.g.,

Lewis and Hawkins 1982; Jorgensen 1984). It is impor-

tant to understand the origin and dynamics of these

asymmetries, because they may be connected to sud-

den changes in the structure and intensity of hurri-

canes (Holland and Merrill 1984; Willoughby 1990a,b,c;

Montgomery and Kallenbach 1997, hereafter MK97;

Montgomery and Enagonio 1998; Challa et al. 1998;

Moller and Montgomery 1999; Reasor et al. 2000; Wang

2002a,b; Chen and Yau 2001; Chen et al. 2003).

To understand radar observations of outward propa-

gating spiral bands in hurricanes, MK97 developed an

inviscid mechanistic model based on wave kinematics

and wave-mean flow interaction. They started by inte-

grating exactly the linearized barotropic vorticity equa-

tion on an f plane following Smith and Rosenbluth (1990).

For the case of stable symmetric vortices with mono-

tonically decreasing vorticity profile (monopole vortices),

the solution was shown to contain low wavenumber ra-

dially propagating vorticity waves throughout the re-

gion of the vortex with nonzero vorticity gradients. The

solution for higher wavenumbers, although not exact,

also exhibited radially propagating waves. Extension

of the work to include the effect of divergence and a

variable deformation radius consistent with real hur-

ricanes was performed in the framework of a shallow-

water asymmetric-balance (AB) model (Shapiro and

Corresponding author address: Yosvany H. Martinez, Meteoro-

logical Research Division, Environment Canada, 2121 Transcanada

Highway, No. 453, Dorval QC H9P 1J3, Canada.

E-mail: [email protected]

NOVEMBER 2010 M A R T I N E Z E T A L . 3559

DOI: 10.1175/2010JAS3499.1

� 2010 American Meteorological SocietyUnauthenticated | Downloaded 10/08/21 09:09 AM UTC

Montgomery 1993). The results indicated the robustness

of the propagating vorticity waves as a result of the re-

laxation of the initial asymmetries. Because these waves

appear in both nondivergent and balance models, they

are not gravity waves. MK97 coined the term ‘‘vortex

Rossby waves’’ (VRWs) because the waves are disper-

sive and the restoring mechanism is based on the radial

gradient of background vorticity or more generally po-

tential vorticity (PV). As VRWs radiate outward in the

negative radial gradient of vorticity of the storm and

reach their critical radius, where they corotate with the

background flow, cyclonic (anticyclonic) eddy momen-

tum maximum is transported inward (outward) slightly

inside (outside) the critical radius. Therefore, the critical

radius provides a site for wave-mean flow interactions

and the wave-mean flow dynamics becomes a basic vor-

tex spinup mechanism that operates during the devel-

opment of the tropical cyclone.

The work of MK97 is rooted in the studies of vorticity

rearrangement or axisymmetrization on two-dimensional

(2D) vortex fluids and vortex merger of Melander et al.

(1987) that recognized axisymmetrization as a universal

process of vortex flows. During axisymmetrization, vor-

ticity filaments are shed and the perturbations that form

the initial deformation decay in time even in the absence

of dissipation. This phenomenon of decay during ax-

isymmetrization is also known as ‘‘inviscid damping,’’

and it has been successfully described by the 2D Euler

equations (Pillai and Gould 1994; Schecter et al. 2000).

The axisymmetrization problem for hurricane-like

vortex monopoles has been considered extensively.

Montgomery and Enagonio (1998) further clarify the

significance of the vortex axisymmetrization process for

the 3D problem of tropical cyclogenesis. In particular,

Montgomery and Enagonio (1998) examine the interac-

tion of small-scale convective disturbances with a larger-

scale vortex circulation in a nonlinear quasigeostrophic

balance model. The results in Montgomery and Enagonio

(1998) are later validated by Enagonio and Montgomery

(2001) in a shallow-water primitive equation framework.

Montgomery and Brunet (2002) conducted idealized lin-

ear and nonlinear numerical experiments for tropical

cyclones and polar vortex interiors to elucidate more

aspects of the vortex symmetrization problem and the

vortex Rossby wave/merger spinup mechanism pro-

posed by Brunet and Montgomery (2002). More re-

cently, McWilliams et al. (2003) developed a formal

theory for vortex Rossby waves and vortex evolution that

describes the balanced evolution of a small-amplitude,

small-scale wave field in the presence of an axisym-

metric vortex initially in gradient-wind balance and pro-

vides a new perspective on wave-mean flow interactions

in finite Rossby numbers regime.

The decay of the initial perturbations or inviscid

damping can go through two pathways. The perturbation

can be strongly damped and decay through a process of

global filamentation (spiral windup) or they resist the

spiral windup and are weakly damped due to the exci-

tation of a quasi mode. A quasi mode is a continuum

spectrum mode with a sharply peaked frequency spec-

trum and a smooth (delocalized) spatial distribution of

vorticity similar to that of a discrete mode. In this work,

we shall refer to this type of ‘‘special’’ continuum mode

as ‘‘discrete-like’’ VRWs. A quasi mode behaves like a

single azimuthally propagating wave (with a complex

angular frequency given by the ‘‘Landau pole’’) weakly

damped by critical layer stirring (Briggs et al. 1970;

Corngold 1995; Spencer and Rasband 1997; Schecter

et al. 2000, 2002; Schecter and Montgomery 2006). In

a hurricane, a quasi mode appears in the form of a slowly

decaying inner-core vorticity perturbation, but it may

affect outer-core dynamics.

The two pathways for vortex symmetrization (quasi-

modal versus spiral windup) in the context of non-

barotropic hurricane-like vortices were first studied by

Reasor and Montgomery (2001) in a study of 3D align-

ment and corotation of weak tropical cyclone-like vor-

tices in a quasigeostrophic framework. This work was

later extended, and a theory for the vertical alignment

of a quasigeostrophic vortex via quasimodal decay or

spiral windup is formally introduced by Schecter et al.

(2002). Reasor et al. (2004) and Graves et al. (2006) have

further clarified conditions for quasimodal or spiral

windup decay pathways for finite Rossby numbers re-

gime characteristics of real tropical cyclones and other

geophysical vortices.

Mature hurricanes have also been observed to be ac-

companied by other asymmetries such as polygonal eye-

walls and mesovortices (Black and Marks 1991; Lewis

and Hawkins 1982; Muramatsu 1986; Houze et al. 2006).

Schubert et al. (1999) used an unforced nondivergent

barotropic model to analyze the origin of polygonal

eyewalls and the breakdown of the eyewall in mature

hurricanes characterized by an annulus of elevated vor-

ticity (see also Kossin and Schubert 2001). This vorticity

profile satisfies Rayleigh’s necessary condition for in-

stability so that unstable waves may be generated. The

simulations in Schubert et al. (1999) display counteract-

ing VRWs that propagate on the inner and outer edges

of the eyewall, where the radial gradient of vorticity is

larger. These waves may phase lock and become baro-

tropically unstable (see also Terwey and Montgomery

2002) by extracting energy from the background flow and

reorganize the vorticity in the eyewall into mesovortices

and polygonal eyewall. Schubert et al. (1999) demon-

strated that during this process, the high vorticity in the

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eyewall mixes with the vorticity in the eye, leading to

eyewall contraction and the spinup of the eye.

From the above discussion, it is clear that asymmetric

features play an important role on the changes in the

structure and intensity of hurricanes. A better knowl-

edge of the dynamics of hurricane asymmetries is vital

to understand inner-core dynamics and ultimately im-

prove hurricane intensity forecasting systems. However,

many aspects related with the origin, propagation, space–

time structure, and dynamics of the hurricane asymme-

tries still remain unsolved. The goal of this series of

papers is therefore to better understand the kinematics

and dynamics of hurricane asymmetries, particularly in

the hurricane intensity changes. Our approach will be to

use both a simple 2D barotropic vorticity model and

a state-of-the-art full-physics model to simulate phe-

nomena related with the propagation and dynamics of

hurricane asymmetries. Diagnostic studies would then

be performed, including the application of the empirical

normal mode (ENM) method (Brunet 1994) and the

space–time empirical normal mode method to shed light

on how asymmetries and axisymmetrization affect the

hurricane structure and intensity through VRWs.

The specific objectives of this paper are to simulate

various asymmetric features in 2D hurricane-like vorti-

ces, such as elliptical and polygonal eyewall formation

and the formation of mesovortices in the eyewall, using

a simple 2D nondivergent barotropic model; to clarify

some aspects on the origin, structure, and dynamics of

the asymmetries; and to show the ability of the ENM

method to isolate the most diverse wave modes from the

datasets, including quasi modes and unstable modes. In

a second paper of the series (Martinez et al. 2010), we

used the same simple 2D nondivergent barotropic model

to simulate the process of secondary wind maximum

generation and investigate the role of asymmetries on

the formation of the secondary wind maximum that ac-

companies the secondary eyewall often observed in ma-

ture hurricanes. In the third paper of the series (Martinez

et al. 2010, manuscript submitted to J. Atmos. Sci.),

we simulated secondary eyewall formation in a realistic

hurricane environment using a high-resolution full-

physics numerical model to investigate important as-

pects on the dynamics of concentric eyewall genesis.

The organization of this paper is as follows: in sec-

tion 2, we discuss briefly the model and its initialization

and describe some of the most relevant features ob-

tained from the numerical experiments. In section 3, we

review the eigenmode theory of linear perturbations in

2D vortex fluids, the generalized wave activity conser-

vation laws, and the ENM method in a 2D nondivergent

barotropic vorticity equation framework. The ENM di-

agnostic results for the two experiments are presented

in section 4. Summary and conclusions are found in

section 5.

2. Numerical simulations

a. Nondivergent barotropic model

In general, it is expected that the evolution of asym-

metric disturbances and the propagation of VRWs in

hurricanes are influenced by boundary layer and moist

processes. However, the internal conservative ‘‘dry’’

dynamics could reveal important mechanisms that may

otherwise be overshadowed in a more complex frame-

work. We therefore employ a simple 2D nondivergent

barotropic model for our investigation (Bartello and Warn

1996).

In Cartesian coordinates the equation for the 2D non-

divergent barotropic unforced model on an f plane is

›j

›t1

›(c, j)

›(x, y)5 n=2j, (1)

where c is the streamfunction, j 5 =2c is the relative

vorticity, and ›(., .)/›(x, y) is the Jacobian operator in

Cartesian coordinates. The eastward u and northward y

components of the velocity can be expressed in terms

of c by u 5 �(›c/›y) and y 5 ›c/›x. A very small dif-

fusion coefficient n is chosen to control the spectral

blocking associated with enstrophy cascade to higher

wavenumbers. The model is solved using a doubly peri-

odic pseudospectral code that includes a leapfrog scheme

for the time integration.

b. Setup of the experiments

In total, two experiments are performed. Experiment I

is designed to study the different pathways (quasi modes

versus sheared VRWs) of the inviscid damping of asym-

metric disturbances, to verify the conditions under which

each pathway occurs, and to explore the relationship

between inviscid damping and the intensification of an

incipient storm (tropical cyclogenesis). To accomplish

our goal, we initialize experiment I with an equilibrium

vorticity profile that allows us to simulate the two path-

ways of inviscid damping. Our main focus, however, is on

the damping of the asymmetries via the excitation of a

quasi mode because of their large impact on the vortex

structure and intensity changes. Two basic-state vortic-

ity profiles are prescribed using a combination of ex-

ponential and polynomial functions. It will be shown

that one of the profiles supports quasi modes and that

the other does not. The symmetric vortex is also per-

turbed with azimuthal wavenumber-2 asymmetries. We

will compare the wave structures obtained from the ENM



diagnostics with those predicted from the eigenmode

theory of linear perturbations in 2D vortex fluids (see,

e.g., Sutyrin 1989; Montgomery and Lu 1997; Schecter

et al. 2000). Experiment II will be used to simulate the

evolution of perturbations in a mature annular hurri-

cane and the formation of polygonal eyewall and meso-

vortices and to investigate their relation with the rapid

vortex intensification. The mechanism of rapid hurri-

cane intensification via eyewall contraction resulting

from VRW instabilities (Schubert et al. 1999) is redis-

covered using the ENM perspective. The structure of

the most important unstable VRW modes will be revealed.

The initialization of experiment II follows Schubert et al.

(1999) with a symmetric annular vortex embedded in a

quiescent environment. The annular vortex is randomly

perturbed on the inner and outer edges.

1) EXPERIMENT I

The first simulation in experiment I is initialized with

the basic-state tangential wind y0(r), angular velocity

V0(r), vorticity j0(r), and its radial gradient g0(r) de-

picted in Fig. 1. In particular, the equilibrium vorticity

profile is given by

j0(r) 5 z

0[e�(r/r0)2

1 0.25(r/r0)2e0.5�0.1(r/r0)4

], (2)

where z0 5 0.0009 s21, r denotes radius or distance in

kilometers from the central axis, and r0 5 35 km. This

particular equilibrium profile describes a weak storm

with a radius of maximum wind (RMW) located at about

70 km and a maximum tangential wind of about 20 m s21

(Fig. 1a). Because the mean vorticity (Fig. 1b) is a mono-

tonic function of radius, the vortex satisfies Rayleigh’s

FIG. 1. Basic-state experiment I: (a) tangential wind (m s21), (b) vorticity (s21), (c) angular velocity (s21), and (d)

radial gradient of vorticity (km s21).



sufficient condition for linear stability (Gent and

McWilliams 1986). The angular velocity (Fig. 1c) decreases

monotonically with radius, and g0(r) is negative over the

entire domain with two local minima, one situated at

about 21 km and the other at about 75 km (Fig. 1d).

The equilibrium profile (2) is perturbed according to

Schecter and Montgomery (2006) to generate an ini-

tially (t 5 0) elliptical vortex with a total vorticity field

j(r, l, 0) given by

j(r, l, 0) 5 j0

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 0.5d2

p1 d cos(2l)

" #. (3)

Here, d 5 1.5 is a measure of the vortex ellipticity (ratio

between the semimajor axis and the semiminor axis) and

l is the azimuthal angle. For the first simulation, the

model integration is carried out using a double Fourier

pseudospectral code with 8002 grid points and a grid size

of 750 m, which corresponds to a domain of 600 km 3

600 km. Time differencing is accomplished using 6 s

as the time step. The integration time is 34 h, and the

output domain is 340 km 3 340 km. The time sampling

is every 4 min, resulting in 513 time samples.

Figure 2 shows the time evolution of the total vorticity

field. At t 5 0, the initial elliptical vortex is described by

(2) and (3). Subsequently it relaxes toward a more axi-

symmetric state. During the relaxation, spiral vorticity

filaments propagate outward until a certain radius where

they start to curl around to form stable configurations

known as ‘‘cat’s eyes’’ (Figs. 2e,f). Moreover, the vor-

ticity field contours become more axisymmetric in the

core of the vortex.

To determine the sensitivity to the initial vortex, Fig. 3

displays the evolution of the total vorticity field for a

different simulation initialized with the first term in (2)

only and also (3). In this case, the axisymmetrization

occurs at a faster rate, and the vorticity filaments dis-

tribute globally over the entire vortex domain. These

two simulations show that, although the ellipticities

of the vortices decay in time, the decay process goes

through different paths. In section 4a, we will elaborate

more on this issue.

2) EXPERIMENT II

In our second experiment, the expressions for the

vorticity equilibrium profile and the perturbations are

taken directly from Schubert et al. (1999). The basic-

state tangential wind, angular velocity, vorticity, and its

radial gradient are depicted in Fig. 4. The basic tan-

gential wind is weak inside 35 km but increases rapidly

between 40 and 50 km (Fig. 4a). Note that the maximum

tangential wind is approximately 54 m s21 and the RMW

is about 60 km. This vorticity profile is typical of a mature

hurricane with an annular ring of uniformly high vorticity

embedded in a low-vorticity background (Fig. 4b). Also

note that g0(r) changes sign inside the domain (Fig. 4d);

therefore, barotropically unstable VRWs may emerge

together with vorticity redistribution. The model is in-

tegrated in a domain with 5122 grid points (600 km 3

600 km). The grid size is approximately 1.17 km and the

time step 7.5 s. The integration time is 6 h. The time

sampling is every 2 min, giving a total of 145 time samples.

The evolution of the total vorticity field during the

first 6 h is depicted in Fig. 5. In general, we observe

waves that develop in the outer and inner edge of the

ring. As the two vorticity waves phase lock and grow,

mesovortices are generated and the high vorticity of the

ring tends to mix with the low vorticity of the eye, cre-

ating a polygonal eyewall appearance and the contrac-

tion of the ring. Although the results in Fig. 5 are similar

to those in Schubert et al. (1999), we will show how we

can apply the ENM technique to extract unstable VRWs,

study their kinematics, reveal their space–time struc-

ture, and evaluate their role in hurricane intensification

through wave-mean flow interaction computations. We

will also compare the results from the ENM diagnostics

with those reported in Schubert et al. (1999).

3. Methodology to study hurricane asymmetries

An overview of the methods to study hurricane asym-

metries is presented in this section. The eigenmode the-

ory of linear perturbations in 2D vortex flows is reviewed

and the ENM algorithm is presented in the context of

the 2D Euler equations. In this framework, wave activ-

ities, expressions of wave-mean flow interactions, and

formulation of the ENM adopt the simplest form.

a. Linear eigenmode analysis

A commonly used strategy to analyze wave processes

in fluid dynamics is to separate the flow variables into a

basic-state part that is a steady solution of the governing

equations and a disturbance part that is associated with

‘‘eddies’’ or ‘‘waves.’’ For example, assuming that the

primary circulation in a hurricane is axisymmetric, the

total vorticity field can be rewritten in cylindrical coor-

dinates and decomposed into contributions from a basic-

state or mean axisymmetric j0(r) term and a perturbation

or eddy j9(r, l, t) term in the form

j(r, l, t) 5 j0(r) 1 j9(r, l, t). (4)

To analyze the evolution of perturbations in 2D vortex

flows, we follow mostly the analysis in Schecter et al.

(2000). This formalism decomposes the perturbations

into independent eigenmodes or wave modes. For small



FIG. 2. Vorticity contour plots (31024 s21) for experiment I. The initialization uses (2) and (3). The

model domain is 600 km 3 600 km, but the results are presented in a subdomain of 340 km 3 340 km.

Dark red colors denote the maximum values of vorticity, dark blue represents small values of vorticity,

and white corresponds to zero values. Simulation times: (a) t 5 0 h, (b) t 5 6 h, (c) t 5 13 h, (d) t 5 20 h,

(e) t 5 26 h, and (f) t 5 33 h.



FIG. 3. As in Fig. 2, but the vortex is initialized using the first term in (2) and (3).



perturbation approximation, the inviscid version of (1)

can be linearized in polar coordinates

›

›t1 V

0(r)

›

›l

� �j9� 1

r

›c9

›lg

0(r) 5 0, =2c95 j9. (5)

Here, c9 denotes the perturbation streamfunction,

=2 5 (›2/›r2) 1 (1/r)(›/›r) 1 (1/r2)(›/›l2) is the cylin-

drical Laplacian, and V0(r) 5 1/r2

Ð r

0 dr1r

1j

0(r

1) is the

mean angular velocity of the vortex flow. Arbitrary small

perturbations can be represented as a superposition of

linear eigenmodes j9 5 h(r)ei(ml�vt), c9 5 C(r)ei(ml�vt),

where m indicates the azimuthal wavenumber. Substitut-

ing these expressions into (5), the following pair of equa-

tions for the radial eigenfunctions h(r) and C(r) are

obtained:

[v�mV0]h 1

m

rg

0(r)C 5 0, =2C 5 h. (6)

Equation (6) can be transformed into an integral ei-

genvalue equation for the vorticity eigenfunction h that,

when discretized, adopts the following form:

�N

j51L

ijh(r

j) 5 vh(r

i), (7)

where i 5 1, . . ., N; N is the total number of radial grid

points obtained from the domain discretization; and the

matrix elements Lij are real and are given by

Lij

5 mV0(r

i)d

ij�m

rj

ri

g0(r

i)G(m)(r

ijr

j)Dr. (8)

Here, Dr 5 N/Rmax

represents the radial grid size; dij is

the Kronecker delta; and G(m)(ri|rj) is the Green’s

function solution of the Poisson equation in (6) (ex-

pression on the left),

FIG. 4. As in Fig. 1, but for experiment II.



FIG. 5. As in Fig. 2, but for experiment II. Plots are for simulation times of (a) t 5 1 h, (b) t 5 2 h, (c) t 5 3 h,

(d) t 5 4 h, (e) t 5 5 h, and (f) t 5 6 h.



G(m)(rijr

j) 5� 1

2m

r,

r.

� �m

1� r.

Rmax

� �2m" #

, (9)

where r, (r.) is the smaller (larger) between ri and

rj. Equation (7) can be solved numerically following

Schecter et al. (2000). Nolan and Farrell (1999) solved

a similar equation, but they redefine the Green’s matrix

as rjG(m)(ri|rj). The solutions of (7) are discrete and con-

tinuum modes. The discrete modes are physical solution

to the linearized Euler equations and they have spatially

smooth eigenfunctions. On the other hand, the radial

eigenfunctions of a continuum mode have a singular

point where the fluid rotation is in resonance with that

mode (Corngold 1995; Spencer and Rasband 1997). For

the case of monotonic vorticity profiles, only neutrally

stable (real frequency) eigenmodes are supported.

Sometimes, for specific equilibrium vorticity profiles,

the continuum spectrum eigenmodes reveals, in addition

to the singular behavior at the critical radius, a non-

localized spatial distribution characteristic of discrete

eigenmodes. The singular behavior at the critical layer

is reflected by a finite peak when the eigenmodes are

found numerically. These modes tend, because of their

large multipole moment, to be excited easily by external

forcing or random perturbations. These discrete-like ei-

genmodes have a frequency spectrum distributed around

a certain value vRq

(where subindex Rq stands for the

real part of the frequency of this special class of modes

q), and when combined they form what is called a quasi

mode, which is a perturbation with a spatially smooth

structure and a frequency spectrum sharply peaked

around vRq. A quasi mode is not a solution of (7); in-

stead, it evolves like a single discrete retrograde wave

that is exponentially damped in the bulk of the system

during the early stage of evolution with a complex fre-

quency vq known as a Landau pole (Briggs et al. 1970;

Spencer and Rasband 1997; Schecter et al. 2000). How-

ever, its vorticity can grow in the vicinity of the critical

radius due to the dispersion of the continuum modes,

during which singular spikes unravel, forming a bump

across the critical layer. In Schecter et al. (2000), the

details of an algorithm to find the Landau poles is ana-

lyzed. In section 4a, we will discuss more on quasi modes.

In general, it is difficult to find an a priori result to

demonstrate the existence of quasi modes for a given

vorticity distribution. Although it seems that when g0(rC)

(where rC denotes the critical radius) is negative and

small, we can often find quasi modes. On the other hand,

it can be demonstrated that quasi modes becomes a

genuine discrete eigenmode by flattening the profile of

radial gradient of vorticity around the critical radius [i.e.,

by forcing g0(rC) 5 0]. A similar situation is analyzed in

Brunet and Haynes (1995) in the context of the evolution

of disturbances to a parabolic jet.

b. Wave activity conservation laws in 2D barotropicvortices

Equation (5) can be manipulated algebraically to ob-

tain a local conservation law of the form

›W

›t1 $ � F

W5 S

W, (10)

where W and FW

are quadratic forms of the disturbance

quantities and SW is the source/sink term. The quantity

W is called wave activity and the vector FW

represents

a flux of wave activity. Equation (10) has been shown to

be very useful for the case when SW is negligible. In this

case, (10) becomes a local conservation law that could be

used to diagnose wave processes.

To construct the small-amplitude wave activity con-

servation laws in our case, we assume that the symmetric

circulation in a hurricane is much larger than the asym-

metric one (Shapiro and Montgomery 1993). Our choice

of basic state assumes time invariance and azimuthal

invariance of the tangential wind and vorticity fields. If

the prognostic equation in (5) is multiplied by rj9/g0 and

the new expression is azimuthally averaged, a conservation

law for the azimuthal mean pseudomomentum density

(called simply pseudomomentum here)J follows from the

basic-state azimuthal invariance (Shepherd 1990),

›J›t

11

r

›

›r(�r2u9y9) 5 SJ , J 5�rj92

2g0

. (11)

Here, the overbar represents azimuthal average and SJare the azimuthally averaged sink/source term of pseu-

domomentum. From the time invariance of the basic

state, a conservation relation for the azimuthal mean

pseudoenergy density (called simply pseudoenergy here)

A will follow

›A›t

11

r

›

›r(ry

0u9y9) 5 SA, A5

y0

rJ 1

1

2(u92 1 y92).

(12)

The first term on the expression of pseudoenergy A is a

Doppler shift (DS) term associated with the background

wind y0 and the next two terms sum to the azimuthal

mean wave kinetic energy (K); SA is the azimuthally

averaged sink/source term of pseudoenergy.

c. Two-dimensional ENM method

Brunet (1994) developed the ENM decomposition

method by combining the EOF method (Lorenz 1956)



and the orthogonality properties of normal modes in the

context of wave activities. The ENM technique has been

applied in the past (Brunet and Vautard 1996; Charron

and Brunet 1999; Zadra et al. 2002; Chen et al. 2003).

Here, the ENM is cast in a 2D barotropic nondivergent

framework for the first time to describe several important

mechanisms based on VRW dynamics, such as inviscid

damping, asymmetric eyewall contraction, and polygonal

eyewalls.

The ENM algorithm starts by decomposing the asym-

metric disturbances into a set of modes or basis functions

that approach a set of true normal modes when the dis-

turbances are considered of sufficiently small amplitude.

For example, the vorticity disturbance field j9 can be

represented by the expansion

j9 5 �n,s

ans

(t)[j(1)ns (r) cos(sl) 1 j(2)

ns (r) sin(sl)], (13)

which includes a preliminary Fourier expansion in the

azimuthal direction indicated by the azimuthal wave-

number s, followed by a decomposition in ENMs in-

dicated by the integer mode number n. The term ans is

the time series and is also known as principal component

(PC) for wavenumber s, and j(1,2)ns is the azimuthal vor-

ticity cosine/sine component of the (ns)th ENM, respec-

tively. ENMs and PCs are found from an optimization

problem (see Zadra et al. 2002), and they are the eigen-

vectors of a space and a time-covariance operator, re-

spectively. The PCs are eigenvectors of the eigenproblem

�j

Tsijans

(tj) 5 l

nsa

ns(t

i),

1

T

ðT

0

ams

(t)ans

(t) dt 5 dmn

,

(14)

and

Tsij 5�

ðRmax

0

r2j9(ti)j9(t

j)

g0

dr. (15)

The operator Tij is the time-covariance matrix for wave-

number s, constructed with a metric defined by the

pseudomomentum J in (11). We prefer to work with the

pseudomomentum-based metric over the pseudoenergy

A–based metric given by (12) because the former contains

fewer terms, which makes the ENM diagnosis less sus-

ceptible to errors from numerical approximations. Note

that Tsij may be interpreted as the real part of a complex

time-covariance operator. It can be shown that both the

complex covariance and its real part can generate true

normal modes in the linear and conservative limit. Once

the PCs are found, the corresponding ENMs are obtained

using a projection formula. For example, the (ns)th nor-

mal mode of the vorticity for wavenumber s is given by

j(l)ns (r) 5

1

T

ðT

0

dt1j9(l)

s (r, t1)a

ns(t

1), (16)

where l 5 1, 2 indicates the cosine and sine components,

respectively. This strategy to find the ENM’s spatial–

temporal structures by solving first the eigenproblem for

the time-covariance operator (14) and (15) and then use

the projection equation (16) to find the spatial structures

is known as the snapshot method (Sirovich and Everson

1992).

The recognition of propagating modes in our system

happens by finding pairs of PCs with degenerate eigen-

values (wave activities) associated to the real and imagi-

nary part of a complex PC (Zadra et al. 2002). Mode

numbers [n, n 1 1] form a pair whose associated time

series is a complex PC An,s 5 an,s 1 ian11,s from which

the mode’s power spectrum, mean frequency, and phase

speed can be found. The theoretical values for every prop-

agating mode’s angular phase speed, frequency, and pe-

riod are computed using Held (1985): cn 5 �An/J n,

vth 5 s(An/J n), and T th 5 (2p/s)jJ n /Anj, respectively,

where the subscripts th stands for theoretical. More

details on these relations can be found in Brunet (1994),

Brunet and Vautard (1996), Charron and Brunet (1999),

and Zadra et al. (2002).

d. Vortex Rossby wave-mean flow interactions

Eliassen–Palm (EP) flux maps have been widely used

as a diagnostic tool in different contexts: for example, in

studies of baroclinic wave life cycles (e.g., Edmon et al.

1980; Thorncroft et al. 1993) and in hurricane distur-

bance analysis (Willoughby 1978a,b; Schubert 1985;

Molinari et al. 1995, 1998; Montgomery and Enagonio

1998; Enagonio and Montgomery 2001; Montgomery

and Brunet 2002; McWilliams et al. 2003; Chen et al.

2003). The EP fluxes are associated with flux of pseu-

domomentum, and its divergence can be interpreted as

an eddy-induced force per unit mass and therefore a

measure of the wave-mean flow interactions. In our case,

the EP theorem is applied to analyze the impact of prop-

agating VRWs on the mean vortex. Equation (11) can be

rewritten in a flux form

›J›t

1 $ � FJ 5 SJ , $ � FJ 51

r

›

›r(�r2u9y9), (17)

where $ � FJ is the divergence of the generalized azi-

muthal mean EP flux FJ 5 �ru9y9er (er is a unit vector

in the radial r direction). It is not difficult to connect the

azimuthal mean EP flux to the time variation of the

azimuthal mean tangential wind (angular momentum),

which is accomplished by means of the standard relation



›y0

›t5

1

r2

›

›r(�r2u9y9). (18)

Note that in the domain regions where $ � FJ . 0

($ � FJ , 0), VRWs lose (gain) pseudomomentum to

accelerate (decelerate) the mean tangential wind locally.

4. Diagnostic results

In this section, we present the results from the ap-

plication of the linear eigenmode analysis of section 3a,

the ENM diagnostics, the wave-mean flow interactions

computations, and EP flux calculations for the two

experiments.

a. Experiment I

1) QUASI-MODE VERSUS SHEARED VRWS

To study the linear excitation and evolution of per-

turbations on 2D vortices, in section 3a we reviewed the

algorithm described in Schecter et al. (2000). This for-

malism views any perturbation as a sum of independent

modes. It can be demonstrated that the eigenmodes of a

monotonic vortex are neutrally stable and form an or-

thogonal basis. However, a perturbation decays through

the dispersion of the wave packet formed by its con-

stituent modes. For example, in the simulations of ex-

periment I we observed how the perturbations that

describe the elliptical deformation of a monotonic vor-

tex decay in time. In this section, we study the exact na-

ture of this damping process.

As explained earlier, our choice of the basic-state vor-

ticity profiles for experiment I can be used to simulate

the two pathways of inviscid damping. Next we are going

to show that, in the first simulation where the symmetric

vortex is given by the entire expression (2), the inviscid

damping occurs via a decaying quasi mode. On the other

hand, for the second simulation where only the first term

of (2) is used in the basic state, the decay process occurs

FIG. 6. Linear eigenfunction analysis for experiment I: (a) case of global filamentation (strong damping), with

modes n 5 46, 110, and 210 and (b) case of decaying quasi mode (weak damping), with modes (top to bottom) n 5 46,

99, and 141. The mode n 5 141 corresponds to a discrete-like continuum. The domain radius Rmax is 300 km and the

critical radius rc for the (primary) quasi mode of the equilibrium vortex given by (2) is about 105 km.



via sheared VRWs. The first term in (2) describes a

Gaussian vortex basic state; when the second term in (2)

is included, however, the profile of vorticity is slightly

flattened around 48 km, creating the two local extrema

on the radial gradient of vorticity profile observed in

Fig. 1d. The profile in Fig. 1b is commonly found in

hurricanes (Mallen et al. 2005); furthermore, as will be

shown next, it supports asymmetric disturbances that

have a large impact on the parent vortex.

We will first verify the case of a decaying quasi mode.

For this reason, we compute the Landau poles of the

equilibrium vortex given by (2) to examine the possibility

of the existence of exponentially decaying perturbations.

The contribution of Landau poles to a perturbation be-

haves exactly like an exponentially damped mode with a

complex eigenfrequency vq 5 vRq1 vIq

i, where the real

(imaginary) part vRq

(vIq

) defines the angular frequency

(decay rate). The computation of the Landau poles for

the vortex represented by (2) gives the complex fre-

quency vq 5 0.0003 1 4.5 3 1026i and a critical radius rC

at about 105 km, where rC is found using the resonance

equation 2V0(rC) 5 vRq. The factor 2 in front of V0(r) is

due to the azimuthal wavenumber-2 disturbances. We

restricted the analysis of experiment I to the azimuthal

wavenumber-2 disturbances because they describe bet-

ter the elliptical deformation. This result suggests that

our equilibrium profile lies in the weak damping regime

in which the perturbation has a decay rate much smaller

than the rotation frequency (i.e. vIq/vR

q� 1). Moreover,

Schecter et al. (2000) demonstrated that the decay rate

vIqis proportional to the radial gradient of vorticity

evaluated at the critical radius rC [i.e. , vIq} g0(rC)]. This

implies that a weak damping occurs when the radial

gradient of vorticity at the critical radius g0(rC) is very

small. Then, the vorticity perturbation decays in time for

all the radii less than rC.

For a more complete characterization of the decaying

perturbation, we solve (7). Figure 6b displays selected

eigenfunctions obtained from solving (7) for the equi-

librium vorticity given by (2), and Rmax corresponds to

the radius of the entire computational domain which is

300 km. The eigenfunctions consist of positive and nega-

tive spikes localized about the mode’s critical radius.

However, several continuum modes, with real frequency

localized around vRq5 0.0003 s�1, are discrete-like,

with added spikes on either side of their resonant radius

at about 105 km. In the bottom panel of Fig. 6b, we plotted

one of the discrete-like continuum modes (n 5 141). As

was mentioned in section 3, the perturbation formed by

the wave packet of these neutrally stable discrete-like

continuum modes defines a quasi mode. Thus, the el-

liptical perturbation will excite a packet formed by the

discrete-like modes and the packet will eventually decay

due to destructive interference between the dispersive

modes. This result indicates that the observed (experi-

mental) damping of the ellipticity can be explained by

the exponential decay of a quasi mode.

Now, we will verify the decay of the perturbation via

sheared VRWs. When we assume that our symmetric

vortex is given by the first term in (2), then the results are

rather different. The computation of the Landau poles

for this special case gives a ratio vIq/vRq

5 0.3, which is

not much smaller than one. It indicates a case of strong

damping. We also observe that for this particular case

the perturbation does not fit the definition of a quasi

mode. Figure 6a shows some of the eigenfunctions ob-

tained from the linear analysis. No discrete-like patterns

were obtained, only filaments distributed over the entire

domain. The frequency spectrum of the perturbation

is broader compared to the sharply peaked frequency

FIG. 7. Experiment I wave activity spectra of (top) pseudomo-

mentum J (m2 s21) and total pseudoenergy A (m2 s22) and (bot-

tom) pseudomomentumJ and individual terms of the pseudoenergy

A, eddy kinetic energy K, and DS energy of the ENM azimuthal

wavenumber-2 disturbances.



spectrum that characterizes a quasi mode. It implies that

the decay of the initial deformation takes place via

global filamentation or sheared VRWs.

Quasi modes are more likely to impose a larger impact

on the structure and intensity of a hurricane. One reason

is that strongly damped non-quasi-modal perturbations

have a shorter life cycle than weakly damped quasi

modes that can resist the spiral windup. In addition,

a quasi mode possesses an exceptionally large multipole

moment, and it may exert the strongest influence on the

external flow. Also, in accordance with the reciprocity

argument (see Schecter 1999, section 3.4), a quasi mode

is the mode most easily excited by external forcing. For

all these reasons, our remaining diagnostics for experi-

ment I will focus on quasi modes that appear in the first

simulation.

2) WAVE ACTIVITY SPECTRA

The ENMs for pseudomomentum are sorted accord-

ing to their eigenvalues (i.e., their pseudomomentum),

in descending order. In general, ENMs with larger

pseudomomentums have longer time scales. Sorting ENMs

according to their eigenvalues is almost equivalent to

sorting the modes with different time scales. The first

mode has the largest and positive pseudomomentum

and the last has the smallest and negative value. The

modes on the extremes of the wave activity spectra have

the largest variance. The variance of a given mode is

defined here as the ratio between the absolute value of

the pseudomomentum of the mode and the total abso-

lute value of the pseudomomentum for a given wave-

number of the disturbances. The wave activity spectra

represented by the absolute values of pseudomomentum

and pseudoenergy of wavenumber-2 anomalies are depic-

ted in Fig. 7. We restricted our diagnostics to wave-

number 2, because they have the largest contribution to

the total variance. The absolute values of the pseudo-

momentum J is given in units of meters squared per

second and the pseudoenergy A in units of meters

squared per second squared. Valuable information on

the properties of the wave modes can be drawn from

these curves. A useful hint to analyze the spectra is to

locate first the mode with the absolute value of pseu-

domomentum closest to zero. This mode separates the

spectra into two regions. To the left of this mode, the

ENMs have positive pseudomomentum and therefore

(according to the phase-speed formula at the end of

section 3) a negative angular phase speed. Thus, these

modes retrograde relative to the mean tangential wind.

To the right, on the other hand, the ENMs have negative

pseudomomentum and form prograde modes. Figure 7

(top) shows the pseudomomentum and pseudoenergy

spectra for this experiment. In Fig. 7 (bottom), the pseu-

doenergy is split into the eddy kinetic energy component

(K) and the Doppler shift component (DS). The variance

FIG. 8. Experiment I time series and power spectra for ENM

modes 1 and 2 of wavenumber 2. The time series are for 34-h

simulation time, and the frequency in the power spectrum is

computed as follows: frequency (32p/34 h21).

FIG. 9. Evolution of the amplitude of the wave formed by ENM

modes 1 and 2 (solid line) and the evolution given solely by the

Landau pole (dashed line) of wavenumber-2 disturbances. The

horizontal t axis is labeled in terms of the model output time steps

(every 4 min). The total amount of time steps is 513, which cor-

responds to 34 h. Approximately 8 core rotations are elapsed in 200

units of time.



explained by the first two modes is about 86%. Note that

the wave activity spectra are dominated by retrograde

VRWs. The contribution from prograde modes is much

smaller.

3) ANALYSIS OF THE PCS AND THE SPATIAL

PATTERNS

Next, we are going to verify that the leading modes

in the diagnostics indeed form a propagating VRW. To

form a propagating wave, we need at least two modes

that have similar contributions to the total variance (i.e.,

degenerate eigenvalues), the same oscillation frequency,

and high cross correlations among their spatial patterns

(Zadra et al. 2002). A pair of modes that form propa-

gating waves is identified by comparing their time series

and the power spectra of the time series and by com-

puting the correlations among their complex spatial

patterns.

Figures 8a,b depict the time series for the first pair of

ENMs (modes 1 and 2) of the wavenumber-2 anomalies.

The amplitudes of these modes decay exponentially

during their early stage of evolution and become oscil-

latory in time later on (see also Fig. 9). This pattern of

evolution corresponds to the picture of a decaying quasi

mode. When a quasi mode is excited, its amplitude first

damps exponentially but then ‘‘bounces’’ due to the non-

linear effects generated from ‘‘trapping oscillations’’ in the

cat’s eyes, then asymptotes to a finite amplitude (see

Schecter et al. 2000). Inspection of Figs. 8a,b indicates

that these modes are in close quadrature, as verified

FIG. 10. Experiment I spatial patterns of ENM modes 1 and 2 of wavenumber 2.



from a lag computation (not shown). The power spectra

in Fig. 8c indicate that the first pair of modes have an

experimental period of roughly 6.3 h. The theoretical

periods of these modes are 6.2 h for mode 1 and 6.3 h for

mode 2. The period predicted by the Landau pole is

about 5.8 h (vRq’ 0.0003 s21), which is in good agree-

ment with the periods from the numerical experiment.

Figure 10 shows the vorticity ENM spatial patterns

corresponding to the first two modes of wavenumber-2

anomalies. The cosine and sine (real and imaginary)

components for mode 1 and mode 2 in (13) are shown

in Figs. 10a,b and Figs. 10c,d, respectively. The ENM

spatial patterns are smooth functions in the bulk of the

domain and they have two local extrema, one at about

23 km and the other at about 73 km. Note that the ENM

spatial patterns satisfy the relation j(2)n2(r) } rg0(r) for

n 5 1, 2 (Figs. 10b,d), which usually is a good approxi-

mation for the spatial patterns of a wavenumber-2 quasi

mode (Schecter et al. 2000). Note also that the spatial

structures of the leading ENMs (Fig. 10a) resemble the

eigenfunction of the discrete-like mode (n 5 141) de-

picted in Fig. 6b (bottom). The cross correlation be-

tween the pairs of diagonal patterns in Figs. 8a,d is

299.75% and between Figs. 8b,c is 99.9%. This large

cross similarity between the spatial patterns together

with the results from the wave activity spectra and the

time series indicates that the first pair of ENM indeed

forms a retrograde propagating VRW.

The above results leave little room for speculation on

the exact nature of the inviscid damping process in our

case. The damping is explained by the exponential decay

of a discrete-like VRW (quasi mode), which is well rep-

resented by the first pair of modes.

4) EP FLUX DIVERGENCE

Now we investigate the effects of radially propagating

VRWs on the mean vortex using the small-amplitude

approach of the EP flux theory. Figures 11a,b show

the contribution of the wavenumber-2 mode 112

anomalies to the numerator of the rhs of (18) (›/›r)

�r2u9y9� �

(EP flux divergence) and the EP flux, respec-

tively. It is evident from Fig. 11a that a dipole pattern

exists in the EP flux divergence map. The general picture

is maximum acceleration occurring slightly outside the

RMW (r 5 70 km) at 80 km, and maximum deceleration

occurring farther outside at 130 km. Using (18), we con-

clude that the total effect on the mean tangential wind is

FIG. 11. Experiment I: (a) plot of the term (›/›r)(�r2u9y9) in units

of m3 s22 (EP flux divergence) and (b) EP flux (m4 s22) for ENM

modes 1 and 2 of wavenumber 2.

FIG. 12. As in Fig. 7, but for the experiment II wavenumber-4

disturbances.



net spinup slightly outside the RMW. The critical radii

for modes 1 and 2 computed from the power spectra of

the ENM analysis and the resonance condition are lo-

cated at about 110 km. They match well the critical radius

of 105 km predicted by the Landau pole (see last panel

of Fig. 6b). The position of the zero local-mean flow

variation (or the location where the EP flux divergence

vanishes) is observed to be at about 103 km. The match

between the experimental critical radius and the position

of the zero local-mean flow variation is consistent with

the results in MK97.

Regarding the EP fluxes, Fig. 11b shows that fluxes are

positive throughout the domain and a maximum is lo-

cated slightly outside the critical radius. This indicates

that the inward flux of cyclonic eddy angular momentum

starts to decrease after reaching the critical radius.

b. Experiment II

1) WAVE ACTIVITY SPECTRA

It is easy to demonstrate, following the instability

analysis in Schubert et al. (1999) and Nolan and Farrell

(1999), that the wavenumber-4 disturbances are the fast-

est growing modes for the equilibrium profile used in this

experiment. For this reason, the ENM diagnostics will be

restricted to wavenumber-4 disturbances.

An inspection of the wave activity spectra (Fig. 12)

indicates that wave modes populate both regions of the

spectra, implying that both prograde and retrograde

VRWs are of importance. The change in sign in the

pseudomomentum spectrum suggests that barotropi-

cally unstable modes may be excited (see Held 1985).

There are formally (mathematically) two equivalent

approaches to extract the ENM modes: one is by solving

the eigenvalue problem of a space-covariance matrix

and the other is by solving the eigenvalue problem of

a time-covariance matrix (snapshot method). The two

approaches deal differently with the unstable modes.

The space-covariance matrix approach extracts the un-

stable modes directly, and it can be verified that these

modes carry zero total pseudomomentum. In this re-

search, however, we use the snapshot method by con-

venience because the dimension of the time-covariance

matrix is much smaller than the dimension of the space-

covariance matrix. In the space-covariance matrix ap-

proach, the unstable modes project on the kernel of the

eigenvalue problem. In the snapshot method the un-

stable mode is split into two EOFs that form a pair that

have the same but opposite-sign total pseudomomentum.

The unstable pairs are also easily recognized and matched

by their polarization signatures. So, when these two EOFs

are recombined, the result is an unstable mode with zero

total pseudomomentum.

FIG. 13. As in Fig. 8, but for the experiment II wavenumber-4

disturbances. The time series are for 6-h simulation time and the

frequency in the power spectrum is computed as follows: frequency

(32p/6 h21).

FIG. 14. As in Fig. 13, but for the last pair of ENM modes 144

and 145.



2) ANALYSIS OF THE PCS AND THE SPATIAL

PATTERNS

Figures 13a,b depict the time series for the first pair of

ENMs of wavenumber-4 anomalies. Figure 14 is as in Fig.

13, but for the last pair of ENMs (modes 144 and 145).

The power spectra for modes 1 and 2 (Fig. 13c) have their

maxima at 0.55 and 0.50 h, respectively, and for the

modes 144 and 145 (Fig. 14c) at 0.53 h. The correlation

between the two pairs of diagonal panels of vorticity

ENM space patterns of modes 1 and 2 in Figs. 15a,d and

Figs. 15b,c are 99.55% and 299.60%, respectively; be-

tween the two pairs of diagonal panels of vorticity, ENM

space patterns of modes 144 and 145 in Figs. 16a,d and

Figs. 16b,c are 99.64% and 299.52%, respectively. The

excellent match between the observed periods of the

first and last pair of modes together with the large values

of cross correlations among the complex spatial patterns

suggests a phase locking between counterpropagating

VRWs formed by the modes on the extrema of the

wave activity spectra. This phase locking could even-

tually result in barotropic instability.

3) EP FLUX DIVERGENCE

Figure 17 depicts the contribution from modes 1, 2,

144, and 145 to the EP flux divergence (Fig. 17a) and EP

flux (Fig. 17b) of the wavenumber-4 anomalies. Similar

to experiment I, a dipole structure is observed in the EP

flux divergence map, but in this case the whole pattern

is shifted toward the vortex center with maximum ac-

celeration located inside the RMW (r 5 60 km) and

maximum deceleration at/outside that radius. The result

is a ring that contracts. A mechanism based on eyewall

contraction has been proposed earlier to explain the in-

tensification of mature hurricanes (Schubert et al. 1999).

FIG. 15. As in Fig. 10, but for the experiment II wavenumber-4 disturbances.



However, it is noteworthy to mention that vorticity re-

arrangement is not the archetype for eyewall contraction

and hurricane intensification. The eyewall contraction is

now believed to occur primarily via convergence of ab-

solute angular momentum within the frictional boundary

layer (Smith et al. 2009). The model used in this study

cannot address these aspects. In fact, a less intense vortex

(with a weaker maximum wind) is obtained by the end

of the simulations. However, the vorticity rearrangement

process may still be considered a precursory signature for

rapid deepening because it may lead to rapid pressure

falls. In a real hurricane, friction and diabatic heating

forcings allow vorticity to eventually rebuild into an an-

nular ring surrounding the core of mixed vorticity. For an

ideal combination of dissipation and vorticity generation,

a vortex can eventually intensify because asymmetric mix-

ing contributes to an enhanced radial profile of vorticity

and pressure falls (Chen and Yau 2001; Rozoff et al. 2009).

To close this section, Table 1 summarizes some of the

main results obtained from the ENM diagnostics of the

two experiments.

5. Concluding remarks

There has been an increasing effort to understand the

role of vortex Rossby waves in hurricane structure and

intensity changes. The dynamical mechanism behind

processes such as tropical cyclogenesis, spiral rainbands,

polygonal eyewalls, and asymmetric eyewall contraction,

has been connected to the dynamics of VRWs. These

waves can participate actively in the control of the en-

ergy and momentum budgets in a hurricane (Guinn and

Schubert 1993; MK97; Montgomery and Enagonio 1998;

Moller and Montgomery 1999; Reasor et al. 2000; Wang

2002a,b; Chen and Yau 2001; Chen et al. 2003). The role

of VRWs in other processes such as concentric eyewall

FIG. 16. As in Fig. 15, but for the last pair of ENM modes 144 and 145.



genesis, however, is still poorly understood. Although a

hurricane is in general a three-dimensional nonconser-

vative system where moisture and boundary layer pro-

cesses are important, it is a common practice to use

simple two-dimensional conservative models to simplify

the physics to reveal important mechanisms that are not

overshadowed by the use of physics with higher com-

plexity. For example, VRW processes can be isolated

using filter models that allow only vorticity wave phe-

nomena to occur. In this study, we use a nondivergent

barotropic model (Bartello and Warn 1996) to carry out

two experiments, experiment I and experiment II, that

simulate the relaxation of asymmetric disturbances and

VRW propagation in 2D hurricane-like vortices during

the early and mature stages of the development of a

hurricane. The datasets of the two simulations are used

to diagnose VRW processes and assess their impact on

the vortex structure and primary circulation.

We take advantage of the ENM and the EP flux for-

mulation to decompose the asymmetric disturbances of

the system into wave modes to assess their role on in-

tensity change. The success of the ENM method lies in

the special way the mode’s orthogonality relation is

established (conserving wave activities) and on its ability

to manipulate large datasets. In contrast with other sta-

tistical flow decomposition techniques, the basis obtained

from the ENM method bear dynamical meaning, so they

are physically balanced. In this study, the ENM method

is used to study phenomenon related to VRW insta-

bilities and the evolution of ‘‘discrete-like’’ VRWs or

quasi modes in the context of hurricanes and validate

previous results on the inviscid damping of small per-

turbations in 2D vortex flows. Table 1 summarizes some

of the ENM diagnostic results.

In experiment I, a weak storm can intensify by a

wavenumber-2 quasi-mode-mean flow interaction mech-

anism, thus establishing the connection of inviscid damp-

ing and critical layer stirring in ‘‘tropical cyclogenesis.’’

The wavenumber-2 wave activity spectra are dominated

by continuum spectra retrograde VRWs. The first pair

of ENMs explained most of the variance, and the am-

plitude of their time series describes a VRW that decays

during the early and mature stages of the evolution. The

periods of the leading modes obtained from the com-

putation of the power spectra in the numerical experi-

ment match very well the theoretical results obtained

from the ratio of wave activities. Moreover, the periods

of the leading ENMs match those computed from linear

eigenmode analysis (Landau pole), and their spatial

patterns resemble the spatial structure of a quasi mode.

The EP flux divergence map indicates a dipole pattern

with acceleration and deceleration outside the RMW

and a net spinup on the primary circulation. The location

of the observed critical radius (where the frequency of

the ENM corotates with the background flow) well

matches the one computed from the Landau pole. In

summary the hurricane intensifies in association with

the damping of a discrete-like VRW or quasi mode

explained by a critical layer stirring mechanism.

The results from experiment II rediscover the mech-

anism of intensification of mature ring-like hurricanes

via VRWs instability and eyewall contraction (Schubert

et al. 1999). The wavenumber-4 wave activity spectra

derived from the ENM analysis indicate that both ret-

rograde and prograde waves were dynamically impor-

tant in our datasets. The time series of the wavenumber-4

leading (prograde and retrograde) modes exhibits an

exponential growing behavior during the first few hours

of the experiment. These modes form a discrete spec-

trum of unstable VRWs that counterpropagate and phase

lock as reflected from the match in frequencies between

FIG. 17. As in Fig. 11, but for experiment II for the sum of

contributions from modes 1, 2, 144, and 145 of wavenumber-4

disturbances.

TABLE 1. Summary of the ENM diagnostic results for the two

experiments. Table shows mode number, wavenumber, variance

explained var (%), theoretical periods Tth (h), observed periods

To (h), and the correlation among the spatial patterns Cor (%).

Expt Wavenumber mode var (%) Tth (h) To (h) Cor (%)

I 2 1 43.8 6 6.3 99.9

I 2 2 42.3 6.2 6.4 299.75

II 4 1 47 0.55 0.57 299.6

II 4 2 40 0.50 0.55 99.55

II 4 144 4 0.53 0.54 99.64

II 4 145 4 0.53 0.54 299.52



the leading retrograde VRWs and the leading prograde

VRWs. The spatial patterns for these VRW modes are

smooth, with maxima at locations where the radial gra-

dient of vorticity is larger (at the outer and inner edge

of the ring). The experimental and theoretical periods

matches very well for the leading modes. The EP flux

divergence map reveals a dipole structure with maxi-

mum acceleration inside the RMW and maximum de-

celeration at/outside the RMW. Net spinup maximum

occurs inside the RMW describing a mechanism of hur-

ricane intensification based on eyewall contraction.

Acknowledgments. The authors thank Dr. Peter

Bartello for all the help setting the model parameters.

The authors also thank Dr. David A. Schecter for pro-

vide us with the algorithm of a quasi-mode solver. Spe-

cial thanks go to Dr. Michael Montgomery and one

anonymous reviewer for their constructive comments on

an earlier version of this paper. This research is spon-

sored by the Natural Sciences and Engineering Research

Council and the Canadian Foundation for Climate and

Atmospheric Sciences.

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