## Abstract

Despite the fact that asymmetries in hurricanes (e.g., spiral rainbands, polygonal eyewalls, and mesovortices) have long been observed in radar and satellite imagery, many aspects of their dynamics remain unsolved, particularly in the formation of the secondary eyewall. The underlying associated dynamical processes need to be better understood to advance the science of hurricane intensity forecasting. To fill this gap, a simple 2D barotropic “dry” model is used to simulate a hurricane-like concentric rings vortex. The empirical normal mode (ENM) technique, together with Eliassen–Palm (EP) flux calculations, are used to isolate wave modes from the model datasets to investigate their impact on the changes in the structure and intensity of the simulated hurricane-like vortex.

From the ENM diagnostics, it is shown that asymmetric disturbances outside a strong vortex ring with a large vorticity skirt may relax to form concentric rings of enhanced vorticity that contain a secondary wind maximum. The fact that the critical radius for some of the leading modes is close to the location where the secondary ring of enhanced vorticity develops suggests that a wave–mean flow interaction mechanism based on vortex Rossby wave (VRW) dynamics may explain important dynamical aspects of concentric eyewall genesis (CEG).

## 1. Introduction

Although the circulation in a hurricane can be considered primarily axisymmetric, observations often reveal asymmetric features in the form of outward-propagating inner spiral rainbands and polygonal eyewalls (e.g., Lewis and Hawkins 1982; Jorgensen 1984). It is important to understand the origin and dynamics of these asymmetries because they may be connected to sudden changes in the structure and intensity of hurricanes (Holland and Merrill 1984; Willoughby 1990c,a,b; Challa et al. 1998).

Earlier theories explained the physics of outward-propagating hurricane rainbands as inertia–gravity waves (Abdullah 1966; Kurihara 1976; Willoughby 1978a,b; Lewis and Hawkins 1982). In contrast, MacDonald (1968) suggested that spiral bands could be described as “Rossby-type” waves. Both inertia–gravity waves and Rossby-type waves may coexist in hurricanes. However, only recently has it been recognized that they may play important roles in affecting the structural and intensity changes of hurricanes by controlling the energy and momentum budgets (Guinn and Schubert 1993; Montgomery and Kallenbach 1997, hereafter MK97; Montgomery and Enagonio 1998; Möller and Montgomery 1999; Reasor et al. 2000; Chen and Yau 2001; Wang 2002a,b; Chen et al. 2003).

To understand radar observations of outward-propagating spiral bands in hurricanes, MK97 developed an inviscid mechanistic model based on wave kinematics and wave–mean flow interaction. They started by integrating exactly the linearized barotropic vorticity equation on an *f* plane following Smith and Rosenbluth (1990). For the case of stable symmetric vortices with monotonically decreasing vorticity profile (monopole vortices), the solution was shown to contain low-wavenumber radially propagating vorticity waves throughout the region 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 hurricanes was performed in the framework of a shallow-water asymmetric balance (AB) model (Shapiro and Montgomery 1993). The results indicated the robustness of the propagating vorticity waves. 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 dispersive and the restoring mechanism is based on the radial gradient of background vorticity or more generally potential vorticity (PV). Individual VRW packets propagate outward up to the stagnation radius where the radial propagation ceases. It is noteworthy to mention that for the inviscid free-wave problem, Tung (1983) and M. Montgomery (2006, personal communication) showed that the stagnation radius is collocated with the so-called “critical radius” where the phase speed of the waves matches the mean flow angular velocity. Relative strong shears confine the wave packets to the vortex core, while relatively strong vorticity gradients allow the wave energy to disperse more radially outward.

The studies of MK97 reveal the important result that as VRWs radiate outward in the negative gradient of radial vorticity of the storm and reach their critical radius, cyclonic (anticyclonic) eddy momentum 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 vortex spinup mechanism that operates during the development of the tropical cyclone.

Möller and Montgomery (1999) further examined the kinematics and wave–mean flow interaction of VRWs using a barotropic nonlinear AB model and diagnose the evolution of various initial asymmetric PV disturbances with different wavenumbers (1, 2, and 3) and strengths, as well as their effect on the symmetric vortex. The computations of the mean-flow variation reveal patterns similar to those reported in MK97. Furthermore, these results were in good agreement with some experiments from a primitive equation model for both weak and strong asymmetric disturbances.

It should be pointed out that all these previous calculations were restricted to the class of monopole vortices that is typical of incipient storms or hurricanes in the early stage of formation. In a mature hurricane, however, organized convection forces an annular ring (eyewall) with uniformly high vorticity embedded in a low-vorticity background (Chen and Yau 2001; Yau et al. 2004). Despite their limitations, some of these earlier results have been verified in the framework of 3D high-resolution full-physics simulations of real and idealized hurricanes (Chen and Yau 2001; Wang 2002a,b; Chen et al. 2003).

A still unsolved problem in hurricane asymmetries is the mechanism for the formation of secondary eyewalls. In general, the eyewalls and rainbands in hurricanes are in constant evolution. During the stage of tropical cyclogenesis, only a few disorganized spiral rainbands are observed to propagate. When the storm develops further, some spiral rainbands organize to form a primary eyewall that contracts with time (if conditions are favorable), leading to a stage of rapid intensification. As the hurricane becomes mature, it may enter another stage with convection outside the primary eyewall organizing into a secondary ring of towering thunderstorms containing a secondary tangential wind maximum (usually called the second/secondary eyewall) encircling the inner eyewall. If conditions are favorable, the outer eyewall intensifies and starts to propagate inward. The inner eyewall then weakens and is eventually replaced by the outer eyewall in a process known as the eyewall replacement cycle (ERC). Concentric eyewall and the ERC are usually associated with intensity change. Observational studies of this phenomenon include the work of Willoughby et al. (1982), Willoughby (1990c,a,b), Black and Willoughby (1992), and Kossin and Sitkowski (2009). With the introduction of passive microwave techniques, observations of concentric eyewalls and ERC became more frequent. Hawkins et al. (2006) estimated that about 70% of all the major hurricanes in the Atlantic Ocean have undergone ERC.

A better knowledge of concentric eyewall and ERC is vital to understand inner-core dynamics, short-term intensity trends, and upgrading of landfall warnings (Willoughby 1990c,a,b; Black and Willoughby 1992; Samsury and Zipser 1995).

Four main theories have been advanced to explain the genesis of concentric eyewalls. The first hypothesizes the important role played by symmetric instability (Willoughby et al. 1984; Willoughby 1988); the second, proposed by MK97, is that the genesis of concentric eyewalls may take place via vortex axisymmetrization; the third is rooted in the work of Nong and Emanuel (2003) on a finite-amplitude wind-induced surface heat exchange (WISHE) instability mechanism; and the fourth is the beta-skirt axisymmetrization (BSA) hypothesis of Terwey and Montgomery (2008). The MK97 theory hypothesizes that VRWs propagate radially outward from the original eyewall region as spiral bands embedded in the radial gradient of PV that serve as a waveguide. As they propagate, they transfer angular momentum to a critical radius where their phase velocity matches the mean flow angular velocity. Through the inviscid relaxation of asymmetric disturbances, the primary circulation outside the primary eyewall may be reinforced. The work by Kuo et al. (2008) almost qualifies as a fifth hypothesis for secondary eyewall formation. Kuo et al. (2008) studied two-dimensional binary vortex interactions and the evolution of some configurations into concentric rings as a result of a strong nonlinear process. Secondary eyewall formation, however, can occur regardless of binary vortex interactions as revealed by observations and numerical model experiments.

From the above discussion, it is clear that asymmetries in hurricanes are important in hurricane dynamics. In particular, the role of asymmetries in secondary eyewall formation and the ERC is not well understood. The goal of this paper is therefore to better understand the kinematics and dynamics of hurricane asymmetries, particularly in the formation of concentric eyewalls. Our approach would be to use a simple 2D barotropic vorticity model to simulate secondary eyewall formation. Diagnostic studies would then be performed, including the application of the empirical normal mode (ENM) method to shed light on how asymmetries and axisymmetrization affect secondary eyewall formation through VRWs.

Our specific objectives are to use a simple 2D nondivergent barotropic model to simulate secondary wind maximum formation, to investigate the role of asymmetries in the formation of the secondary wind maximum that accompanies the secondary eyewall, and to explore the original ideas of MK97 on the impact of VRWs and a mechanism based on wave–mean flow interactions on secondary eyewall formation.

In Martinez (2008) and Martinez et al. (2010a) the ENM method is used to isolate from the dataset the most diverse asymmetric features, including quasimodes and unstable modes, and then to assess their role in the hurricane structure and intensity changes. In Martinez et al. (2010b, manuscript submitted to *J. Atmos. Sci.*) a variation of the ENM method, called the space–time empirical normal modes technique, is used to study the role of hurricane asymmetries on the secondary eyewall formation in a hurricane simulated by a full-physics model.

The organization of this paper is as follows. In section 2 we discuss briefly the model and its initialization and describe some of the most relevant features obtained from the numerical experiments. In section 3 we review the generalized wave activity conservation laws and the ENM method in a 2D nondivergent barotropic vorticity equation framework. The ENM diagnostic results for the experiment are presented in section 4. A summary and conclusions are found in section 5.

## 2. Numerical simulations

### a. Nondivergent barotropic model

In general, it is expected that the evolution of asymmetric 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 might otherwise be overshadowed in a more complex framework. We therefore employ a simple 2D nondivergent barotropic model for our investigation (Bartello and Warn 1996).

In Cartesian coordinates the equation for the 2D nondivergent barotropic unforced model on an *f* plane is

where *ψ* is the streamfunction, *ξ* = ∇^{2}*ψ* is the relative vorticity, and ∂(·, ·)/∂(*x*, *y*) is the Jacobian operator in Cartesian coordinates. The eastward *u* and northward *υ* components of the velocity can be expressed in terms of *ψ* by *u* = −∂*ψ*/∂*y* and *υ* = ∂*ψ*/∂*x*. A very small diffusion coefficient *ν* is chosen to control the spectral blocking associated with enstrophy cascade to higher wavenumbers. The model is solved using a doubly periodic pseudospectral code that includes a leapfrog scheme for the time integration.

### b. Setup of the experiments

Our experiment is designed to study how the relaxation of asymmetric disturbances outside an intense hurricane would lead to the formation of stable configurations such as concentric rings of enhanced vorticity (eyewalls) that contain secondary wind maxima. The initialization of this experiment begins by defining a symmetric annular vortex with asymmetric disturbances similar to the one used in Schubert et al. (1999). However, a vorticity skirt represented by a region of significant vorticity gradient is added outside the ring so that it serves as a waveguide for the propagation of VRWs far from the eyewall. Also, wavenumber-4 asymmetric disturbances are placed outside the primary eyewall. Terwey and Montgomery (2008) point out that very mature hurricanes have the tendency to develop vorticity skirts, which may be an important ingredient for the formation of secondary eyewalls. The results from our experiment will shed light on a mechanism of secondary eyewall formation based on the internal hurricane dynamics.

The basic-state tangential wind, vorticity, and angular velocity are depicted in Fig. 1. Figure 1d shows the skirt of the vorticity profile. The basic tangential wind is weak inside 35 km but increases rapidly between 40 and 50 km (Fig. 1a). Note that the maximum tangential wind is approximately 54 m s^{−1} and the radius of maximum wind (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. 1b). The model is integrated in a domain with 512 grid points × 512 grid points (600 km × 600 km). The grid size is approximately 1.17 km and the time step 7.5 s.

Specifically, the symmetric vortex in this case is given by

The symmetric vortex is perturbed by placing random perturbations in the inner and outer edges of the high-vorticity ring in a manner similar to Schubert et al. (1999) and also by placing wavenumber-4 asymmetric disturbances (following MK97’s section 2e) in the region of the vorticity skirt. We chose wavenumber-4 disturbances because they dominate the wave activity (pseudomomentum and pseudoenergy) and because they represent the fastest-growing modes for this particular symmetric vortex (Martinez 2008). The expression for the initial vortex perturbation *ξ*′(*r*, *λ*, 0) is given as

where *ξ*_{0}^{(skirt)}(*r*) and *ξ*′^{(skirt)}(*r*, *λ*, 0) denote the equilibrium and the perturbation vorticities in the skirt, respectively, with the form

The values of various quantities are *r*_{0} = 20 km, *r*_{1} = 30 km, *r*_{2} = 45 km, *r*_{3} = 50 km, *r*_{4} = 65.5 km, *d*_{1} = 7.5 km, *d*_{2} = 8 km, *α*_{1} = 1.731 663, *α*_{2} = 0.567 921 8, *ζ*_{1} = 3.575 970 × 10^{−4} s^{−1}, *ζ*_{2} = 3.285 34 × 10^{−3} s^{−1}, and *ζ*_{3} = *ξ*_{0}^{(skirt)}(*r*_{3}). Note that *ζ*_{amp} = 10^{−5} s^{−1} represents the amplitude of the random perturbations. The vorticity disturbances in the skirt are placed outside the RMW at about 78 km and their initial amplitude is about 15% of the vorticity of the ring. Also, *S*(*s*) = 1 − 3*s*^{2} + 2*s*^{3} is the basic cubic Hermite shape function satisfying *S*(0) = 1, *S*(1) = 0, and *S*′(0) = *S*′(1) = 0. In this experiment the simulation time is 9 h and the time sampling is every 2 min, giving a total of 271 time samples.

Figure 2 shows the evolution of the total vorticity field in a subdomain of 360 km × 360 km. Polygonal eyewalls and mesovortices are observed during the first hours of the simulation (Fig. 2b), which is in good agreement with the results from Schubert et al. (1999). However, a new element observed in our simulation is the feature of the eyewall persisting for longer times because it sort of rebuilds from the ongoing merging of the mesovortices in the inner core region. The positive perturbations outside the ring axisymmetrize quickly, creating outward-moving vorticity filaments. The negative perturbations, however, axisymmetrize at a slower rate. We observe a small amount of negative vorticity being pulled out as satellites around the primary ring (Figs. 2c,d). The vorticity filaments evolve in different ways. Some filaments close to the vortex ring merge rapidly with it (Fig. 2c), while others far from the vortex ring persist for longer times (Fig. 2d). Dritschel (1989) and Kevlahan and Farge (1997) provide some clues on the evolution of the vorticity filaments. They suggested that coherent vortices may have a stabilizing effect on the vorticity filaments located not too far or not too close from the vortex, such as the case of the outer filaments in Fig. 2d. The stability of the inner filaments close to the vortex can be explained by the fact that these filaments may first be strained and then become thin enough to be ultimately suppressed by viscosity (Figs. 2c,d). Details on this “thinning” process are provided in Rozoff et al. (2006), which explains the origin of rapid filamentation zones with very low vorticity values known in radar terminology as a free echo region or “moat.” Note the configuration of the outer filaments forming a secondary ring of enhanced vorticity that resembles the secondary eyewall of a mature hurricane. The work of Kossin et al. (2000) supports the results of Dritschel (1989) and Kevlahan and Farge (1997) and explains that under certain circumstances the secondary eyewall may become barotropically stable because of the stabilizing effect of the adverse shear generated by the inner vortex.

The formation of a moat is evident in Fig. 2d. Note the low-vorticity area between the primary eyewall and the outer filaments. There are two ways to explain the origin of these features in a hurricane. One way is to consider that they form as a result of subsidence that inhibits convection to create a cloud-free region. The other explanation of their origin comes from a 2D nondivergent barotropic prospective (Guinn and Schubert 1993; Shapiro and Montgomery 1993; Kossin et al. 2000; Rozoff et al. 2006) that considers the moat as a rapid filamentation or strained/dominated region that forms as the result of ongoing vorticity wave breaking caused by the differential rotation in that area. The repeating wave breaking generates “cat’s eyes” structures that thin out with time until they are diffused away by viscosity. Rozoff et al. (2006) discussed the formation of the moat in weak and strong hurricanes. Their results indicate that the moats are more evident in very strong hurricanes.

Figure 3 depicts the evolution of the axisymmetric tangential wind at the selected times 0, 3, 4.5, 6, and 9 h. A secondary wind maximum starts developing outside the primary eyewall after 3.5 h. It reaches maximum strength at 4.5 h and then gradually weakens and finally disappears after 7 h.

Our end-state configuration (Fig. 2d) exhibits similar features to those explained in Kossin et al. (2000) and Rozoff et al. (2006), including the development of a rapid filamentation zone or moat and a stable secondary ring of enhanced vorticity. Once the secondary ring is established, its maintenance can be explaining by using the results in Kossin et al. (2000) on the stability of secondary eyewalls. However, after 12 h a period of intense mixing between the primary eyewall and the eye begins, leading to the gradual disappearance of the primary eyewall (Schubert et al. 1999) and the outer filaments. The end state is a monopole vortex.

## 3. Methodology to study hurricane asymmetries

An overview of the methods to study hurricane asymmetries is presented in this section. The ENM method of Brunet (1994) is presented in the context of the 2D Euler equations (Martinez 2008). In this framework, wave activities, expressions of wave–mean flow interactions, and formulation of the ENM adopt the simplest form.

### a. Wave activity conservation laws in 2D barotropic vortices

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 coordinates and decomposed into contributions from a basic-state or mean axisymmetric *ξ*_{0}(*r*) term and a perturbation or eddy *ξ*′(*r*, *λ*, *t*) term in the form

For small perturbation approximation, the inviscid version of (1) can be linearized in polar coordinates:

Here *ψ*′ denotes the perturbation streamfunction, ∇^{2} [=(∂^{2}/∂*r* ^{2}) + (1/*r*)(∂/∂*r*) + (1/*r* ^{2})(∂^{2}/∂*λ*^{2})] is the cylindrical Laplacian, *γ*_{0}(*r*) is the radial gradient of the mean vorticity, and Ω_{0}(*r*) is the mean angular velocity of the vortex flow:

Arbitrary small perturbations can be represented as a superposition of linear eigenmodes of the form

where *m* indicates the azimuthal wavenumber. The linear equations system (5) and (6) can be manipulated algebraically to obtain an equation of the form

where *W* and **F** are quadratic forms of the disturbance quantities, and *S _{W}* is the source/sink term (Martinez 2008). The quantity

*W*is called wave activity and the vector

**F**represents a flux of wave activity

*W*. Equation (10) has been shown to be very useful for the case when

*S*is negligible. In this case, (10) becomes a local conservation law that could be used to diagnose wave processes. It is well known that in general wave energy or enstrophy defined in the usual sense do not satisfy (10) and cannot be considered as candidates to define wave activity quantities (Held 1985). This is simply because both wave enstrophy and energy are not conserved by the linear dynamics. In general,

_{W}*S*contains terms that may be associated with forcing or dissipation but would also contain terms of third or higher order in the wave amplitude. For the case of small-amplitude conservative waves

_{W}*S*is exactly zero. When finite-amplitude waves are considered, however, the nonlinear terms of the governing equations introduce a source or sink even in a conservative framework. To overcome this difficulty, it is useful to work in an Eulerian framework to express the equations in term of the wave activity density and its flux. McIntyre and Shepherd (1987) applied the energy-Casimir method of Arnol’d (1966) to construct wave activity theorems in an Eulerian framework in which the term

_{W}*S*vanishes for all conservative motions. Haynes (1988) extended these finite-amplitude conservation relations to the forced and dissipative primitive equations on spherical coordinates.

_{W}Wave activities and their corresponding relations have been used in past studies of the dynamics of large-scale Rossby waves to clarify the development and propagation of wave disturbances and their interactions with the mean flow. To construct the small-amplitude wave activity conservation laws in our case we assume that the symmetric circulation is much larger than the asymmetric one (Shapiro and Montgomery 1993). Our choice of basic state assumes time invariance and azimuthal invariance of the tangential wind and vorticity fields. The azimuthal invariance of the basic state will lead to the conservation of angular pseudomomentum density and the time invariance to the conservation of pseudoenergy density (hereafter, pseudomomentum and pseudoenergy). If (5) is multiplied by *rξ*′/*γ*_{0}, and the new expression is azimuthally averaged, a conservation law for the azimuthal mean pseudomomentum 𝒥 follows from the basic-state azimuthal invariance

where

Here the overbar represents azimuthal average and *S*_{𝒥} is the azimuthally averaged sink/source term of pseudomomentum. Equation (11) can be multiplied by −*υ*_{0}/*r* and added to the azimuthally averaged eddy kinetic energy [Wang 2002b, see his (2.1) and (2.2)], and then from the time invariance of the basic state a conservation relation for the azimuthal mean pseudoenergy 𝒜 will follow

where

The first term on the right-hand side of (14) is the azimuthal mean Doppler shift (DS) term associated with the background wind *υ*_{0} and the next two terms sum to the azimuthal mean wave kinetic energy *K*; *S*_{𝒜} is the azimuthally averaged sink/source term of pseudoenergy.

### b. Vortex Rossby wave–mean flow interactions

Eliassen–Palm (EP) flux maps have been widely used as a diagnostic tool in all kind of contexts, such as in studies of baroclinic wave life cycles (e.g., Edmon et al. 1980; Thorncroft et al. 1993) and in hurricane disturbance analysis (Willoughby 1978a,b; Schubert 1985; Molinari et al. 1995, 1998; Chen et al. 2003). The EP fluxes are associated with flux of pseudomomentum 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 propagating VRWs on the mean vortex. Equation (11) can be rewritten in a flux form:

where

is the divergence of the generalized azimuthal mean EP flux **F**. It is not difficult to connect the azimuthal mean EP flux to the time variation of the azimuthal mean tangential wind (angular momentum). When the different quantities in the tangential wind momentum equation (Wang 2002a) are decomposed into an azimuthal mean part and a disturbance part and then azimuthally averaged, it becomes

By noting that the term on the right-hand side of (17) is the azimuthal mean EP flux divergence term in (16) divided by *r*, the connection between the azimuthal mean EP flux divergence and eddy forcing on the mean flow is now established. Note that in the domain regions where **∇** · **F**_{𝒥} > 0 (**∇** · **F**_{𝒥} < 0), VRWs lose (gain) pseudomomentum to accelerate (decelerate) the mean tangential wind locally.

### c. Two-dimensional ENM method

We have explained how it is possible to use the wave activity conservation relations to study VRWs. However, the wave activity concept is also very useful when working with modal decompositions (Held 1985). Since wave activity involves globally conserved quadratic quantities associated with asymmetric disturbances, their conservation laws lead to a metric appropriate to define orthogonality conditions for the wave modes. It means that, for example, the wave pseudomomentum and pseudoenergy can be decomposed into contributions from individual modes or VRW modes because the mode orthogonality is defined in the appropriate sense from the wave activity conservation laws. Wave energy and enstrophy as usually defined, on the other hand, are not conserved quantities, and therefore they are a poor choice of metric for flow decomposition. The energy of a disturbance cannot be separated into contributions of the energy from individual modes. Modal decompositions in the context of wave activities have been used in the past. Ripa (1981) utilizes pseudomomentum orthogonality in a study of wave–wave interactions. Held (1985) discussed how the orthogonality of modes in shear flows can be defined from the concept of pseudomomentum. He applied this result to planetary waves in horizontal and vertical shear. Brunet (1994) developed the ENM decomposition method by combining the EOF method (Sirovich and Everson 1992) and the orthogonality properties of normal modes in the context of wave activities. The ENM technique is similar to the EOF in that they are both defined from eigenproblems involving a “covariance” matrix. In the traditional EOF method, the eigenvalues and eigenvectors are derived from a covariance matrix whose elements are simply the covariance of a disturbance field. This choice, however, lacks physical meaning when one interprets the spatial–temporal structures of the wave modes. The covariance matrix in the EOF analysis is introduced in such form that its elements define nonconservative quantities such as wave enstrophy or energy. In contrast, the ENM method includes a self-adjoint matrix in the definition of covariance matrix, such that each of its elements adopts the form of wave activities.

The ENM technique has been applied in all kind of diagnostic studies, including to diagnose weakly nonlinear Rossby waves within the framework of a shallow water numerical model (Brunet and Vautard 1996), to isolate inertia–gravity waves from the Geophysical Fluid Dynamics Laboratory (GFDL) SKYHI general circulation model dataset to assess their impact on the zonal wind in the context of 3D primitive equations in spherical coordinates (Charron and Brunet 1999), and to analyze the variability of the historical National Centers for Environmental Prediction (NCEP) global analysis dataset and identify large-scale Rossby wave quasimodes (Zadra et al. 2002). Chen et al. (2003) applied this technique in the context of a full-physics model-simulated hurricane and were able to isolate VRWs and connect them to spiral bands. In a recent study, Martinez (2008) cast the ENM in a 2D barotropic nondivergent framework to describe several important mechanisms based on VRWs dynamics, such as elliptical eyewalls, inviscid damping, asymmetric eyewall contraction, and polygonal eyewalls. Here, the ENM is cast in a 2D barotropic nondivergent framework to describe several important mechanisms based on VRWs dynamics such as the genesis of secondary wind maximum in barotropic vortices.

The ENM algorithm starts by decomposing the asymmetric disturbances into a set of modes or basis functions that approach a set of true normal modes when the disturbances are considered of sufficiently small amplitude. For example, *ξ*′ can be represented by the expansion

which includes a preliminary Fourier expansion in the azimuthal direction indicated by the azimuthal wavenumber *s*, followed by a decomposition in ENMs indicated by the integer mode number *n*. The variable *a _{ns}* represents the time series, also known as principal components (PCs) for

*s*, and are the azimuthal vorticity cosine/sine components of the (

*ns*)th ENM, respectively. ENMs and PCs are found from an optimization problem (see Zadra et al. 2002) and they are the eigenvectors of a space and a time covariance operator, respectively. The PCs are eigenvectors of the eigenproblem

where

The operator *T _{ij}* is the time covariance matrix for

*s*, constructed with a metric defined by the pseudomomentum equation (12). Following Zadra et al. (2002), 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 normal mode of the vorticity for

*s*is given by

where *l* = 1, 2 indicate the cosine and sine components, respectively. This strategy to find the ENM’s spatial–temporal structures by first solving the eigenproblem for the time covariance operator (19) and (20) and then using the projection equation (21) to find the spatial structures is known as the snapshot method (Sirovich and Everson 1992).

### d. Power spectra, mean frequencies, and phase speeds

The recognition of propagating modes in our system happens by finding pairs of PCs with degenerate eigenvalues (wave activities) associated with the real and imaginary part of a complex PC. Zadra et al. (2002) studied the statistical issue behind this recognition and proposed three strategies to overcome the problem. We are going to use their approach in which the propagating pairs would tend to be sorted as pairs when the PCs are organized in decreasing order of eigenvalues as an outcome from the snapshot method. Then the mode numbers [*n*, *n* + 1] form a pair whose associated time series is a complex PC *A*_{n,s} = *a*_{n,s} + *ia*_{n+1,s} from which the mode’s power spectrum, mean frequency, and phase speed can be found. The theoretical values for every propagating mode’s intrinsic phase speed, frequency, and period are computed using Held (1985), yielding *c _{n}* = −𝒜

_{n}/𝒥

_{n},

*ω*

_{th}=

*s*𝒜

_{n}/𝒥

_{n}, and

*T*

_{th}= (2

*π*/

*s*). |𝒥

_{n}/𝒜

_{n}|, respectively, where subscript 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).

## 4. Diagnostic results

In this section we present the results from the application of the ENM diagnostics, the wave–mean flow interactions computations, and EP flux calculations for our experiment.

### a. Analysis of the PCs and the spatial patterns

In analogy with the ENM analysis for the vortex ring in experiment II in Martinez et al. (2010a), we will analyze only the wavenumber-4 disturbances. The wave activities (pseudomomentum and pseudoenergy) spectra for this case (not shown) are similar to those depicted in experiment II in Martinez et al. (2010a, their Fig. 12). Retrograde and prograde waves dominate the spectrum. Figure 4 shows the vorticity ENM spatial patterns for the first pair (modes 1 and 2) and Fig. 5 for the third pair (modes 5 and 6). The ENM spatial patterns for the second pair (modes 3 and 4) (not shown) and the last pair (modes 270 and 271) (not shown) are similar to those of the first pair. We will show that the contribution to the local mean-flow variations from the first two pairs and the last pair of ENMs could explain only processes related to the primary eyewall such as the eyewall contraction mechanism described in Martinez et al.’s experiment II. When modes 5 and 6 are included in the analysis, however, we observe a significant development of wave activities outside the primary ring close to the radius where the secondary ring of vorticity filaments develops.

The time series for the first pair of ENMs is represented in Figs. 6a and 6b and their corresponding power spectra in Fig. 6c. Figure 6c indicates that the period of these modes is about 0.75 h. These modes grow during the first 3 h, in agreement with experiment II in Martinez et al. (2010a). However, their amplitude decay later on in time and starts to oscillate. The ENM spatial patterns of these modes exhibit similarities with the spatial patterns discussed in Martinez et al.’s experiment II and may be related to the unstable modes. Figures 7a and 7b show the time series of modes 5 and 6, respectively, which depict an overall damping behavior. The observed period of modes 5 and 6 is about 1.5 h as indicated by the power spectra in Fig. 7c.

Next we are going to verify that the leading modes in the diagnostics indeed form propagating VRWs. 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 a propagating wave is identified by comparing their time series and the power spectra of the time series and computing the correlations among their complex spatial patterns.

For the first pair (modes 1 and 2) the cross-correlation analysis between the pair of diagonal patterns in Figs. 4a and 4d is −99.75% and between Figs. 4b and 4c is 99.9%. For the pair of modes 5 and 6 the cross correlation between the patterns in Figs. 5a and 5d is −98.73% and in Figs. 5b and 5c is 96.7%. This large cross similarity between the spatial patterns together with the results from the wave activity spectra and the time series indicate that the first pair of ENMs indeed form a retrograde propagating VRW. The observed and predicted periods match well. Although we only focus on the leading ENM modes of wavenumber-4 disturbances, they represent about 56% of the variance. Table 1 contains a summary of some of the ENM results.

### b. EP flux divergence

Figure 8 shows the map of the EP flux (Fig. 8c) and its divergence (Fig. 8a) computed from the contribution of the first three pairs (modes 1–6) and the last pair (modes 270 and 271) of ENMs. A magnified view of the EP flux divergence in the region of the skirt that is represented by the dashed box in Fig. 8a is plotted in Fig. 8b. The dipole formed by the plus and minus signs indicates tangential mean-flow acceleration and deceleration respectively. Note that a very distinctive feature not reported in previous studies characterizes the mean-flow variation, such as the signature of a quadrupole structure represented by the inner dipole around 50 km (Fig. 8a) and the outer dipole around 100 km (Fig. 8b). Inside the primary ring a maximum tangential wind acceleration develops mainly from the action of ENM modes on the extreme of the wave activity spectra (1, 2, 270, and 271), similar to that observed in experiment II in Martinez et al. (2010a). However, outside the primary ring the action of modes 5 and 6 is to induce a region of secondary maximum tangential wind acceleration at about 100 km.

The critical radius *r _{c}* for modes 5 and 6 can be found using the resonance condition 4Ω

_{0}(

*r*) =

_{c}*ω*and the power spectra in Fig. 7c. The factor 4 in the formula denotes the wavenumber-4 disturbances. According to the power spectra in Fig. 7c, these modes have a frequency

_{Rq}*ω*= 1.163 × 10

^{−3}s

^{−1}. Their critical radius is located at about 103 km. Note that the critical radius of modes 5 and 6 is close to the position of the zero local-mean flow variation associated with the outer dipole as indicated in Fig. 8b.

A secondary wind maximum forms outside the primary ring and is contained inside the region formed by the secondary vorticity ring observed during the simulations. The critical radii of some of the leading ENMs (such as modes 5 and 6) are found well outside the primary ring. The evolution of the initial disturbance can be described by a packet formed by sheared VRW modes that interact resonantly with the mean flow at their critical radius. The relaxation of the asymmetric features external to the primary eyewall would enhance the primary circulation outside the primary eyewall.

## 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 mechanisms behind processes such as tropical cyclogenesis, spiral rainbands, polygonal eyewalls, and asymmetric eyewall contraction have been connected to the dynamics of VRWs. These waves can participate actively in the control of the energy and momentum budgets in a hurricane (Guinn and Schubert 1993; MK97; Montgomery and Enagonio 1998; Möller 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 genesis, however, is still poorly understood. Although a hurricane is in general a three-dimensional nonconservative system where moisture and boundary layer processes 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 complexity. For example, VRW processes can be isolated using filter models that allow only vorticity wave phenomena to occur. In this study, we use a nondivergent barotropic model (Bartello and Warn 1996) to carry out an experiment that simulates the relaxation of asymmetric disturbances and VRW propagation in 2D hurricane-like vortices during the mature stage of the development of a hurricane. The datasets of the simulation are used to diagnose VRW processes and assess their impact on the vortex structure and primary circulation.

We use the ENM and the EP flux formulation to decompose the asymmetric disturbances of the system into wave modes to assess their role on intensity change. The usefulness of the ENM method lies in the special way the mode’s orthogonality relation is established (conserving wave activities) and its ability to manipulate large datasets. In contrast with other statistical flow decomposition techniques, the basis obtained from the ENM method bears dynamical meaning, so it is physically balanced.

Our experiment aims to shed some light on the process of formation of secondary eyewalls containing secondary wind maxima. The results showed how wavenumber-4 asymmetric disturbances placed outside a primary ring relax to reinforce the primary circulation outside and create a secondary ring of vorticity filaments. Our stable configuration consists of a primary ring, surrounded by a region of low vorticity or “moat” and then the secondary ring. The stability of the secondary ring can be explained using the stability analysis in Kossin et al. (2000). The leading ENM spatial patterns for the wavenumber-4 disturbances reveal peaks on the edges of the primary ring, consistent with the results in experiment II in Martinez (2008). However, a secondary peak develops outside the rapid filamentation zone. The EP flux divergence map obtained from the contribution of the leading VRW modes depicts a quadrupole pattern, with a maximum acceleration inside the RMW and a second maximum outside the primary ring close to the place where the ring of vorticity filaments establishes. We found that the locations of the observed critical radius for some of the leading modes were close to where the secondary peaks in the ENM spatial pattern are revealed, suggesting that the circulation may be reinforced in that area because of a VRW–mean flow interaction mechanism. In a companion paper (Martinez et al. 2010b, manuscript submitted to *J. Atmos. Sci.*) we will show using a full-physics model how a similar mechanism assisted by Ekman pumping may explain important dynamical aspects of secondary eyewall formation.

## Acknowledgments

We like to thank Dr. Peter Bartello for provide us with the barotropic model and for all his guidance to set the model parameters. This research is sponsored by the Natural Sciences and Engineering Research Council and the Canadian Foundation for Climate and Atmospheric Sciences.

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## Footnotes

*Corresponding author address:* Yosvany H. Martinez, Meteorological Research Division, Environment Canada, 2121 Transcanada Highway, No. 453, Dorval QC H9P 1J3, Canada. Email: yosvany.martinez@ec.gc.ca