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