1. Introduction
Despite many years of scientific research, the availability of better observations, and improvements made in numerical models to date there has been relatively little progress in the prediction of hurricane intensity. One of the major reasons for this deficiency arises from the fact that the fundamental physical processes that govern the intensity change are not fully understood (Wang and Wu 2004). There is indication that the inner-core asymmetric dynamics may play a key role in the hurricane structure and intensity changes and a better understanding of these processes may advance hurricane research and prediction.
Early studies of two-dimensional (2D) vortex fluids by Melander et al. (1987) recognized that any perturbed vortex will tend to relax back to symmetry (axisymmetrization) by generating vorticity filaments, even in the absence of dissipation. This phenomenon is referred to as “inviscid damping.” Building on these studies, Montgomery and Kallenbach (1997) developed an inviscid mechanistic model based on wave kinematics to study the underlying dynamics governing the outward-propagating spiral bands observed in hurricanes. They showed that vorticity perturbations on symmetric monopolar vortices (vorticity decreasing monotonically with radius) propagate outward as vortex Rossby waves (VRWs) throughout the region of nonzero vorticity gradient. As the VRWs move toward their critical radius, defined as the position where they corotate with the unperturbed fluid, vorticity is redistributed via wave–mean flow interaction to produce a reinforcement of the primary circulation. In the context of hurricane dynamics, the axisymmetrization process has also been studied by Montgomery and Enagonio (1998), Möller and Montgomery (1999), and Enagonio and Montgomery (2001). These studies found that axisymmetrization contributes to the intensification of an incipient hurricane. In a complementary study in plasma physics, Schecter et al. (2000) demonstrated that inviscid damping can go through two pathways: the initial perturbation can be weakly damped due to the decay of an excited quasimode (Briggs et al. 1970) or it can be strongly damped and decay through the process of global filamentation (spiral windup). In hurricanes, a quasimode appears as a slowly decaying inner-core vorticity perturbation that affects the outer-core dynamics.
The two different pathways of axisymmetrization for the case of hurricane-like vortices have been studied by Reasor and Montgomery (2001), Schecter et al. (2002), Reasor et al. (2004), Graves et al. (2006), and Martinez et al. (2010a). However, these studies were restricted to the class of monopolar vortices. In a mature hurricane, convection is organized in an annular ring (eyewall) near the radius of maximum wind. In terms of vorticity this can be pictured as an annular ring of uniformly high vorticity embedded in a low vorticity background (Yau et al. 2004). It remains to be determined how the axisymmetrization process will affect nonmonotonic vorticity profiles. It is one of the main objectives of the present work to fill this gap.
An intriguing part of hurricane research that may be linked to the dynamics of asymmetric disturbances is the secondary eyewall formation (SEF) frequently observed in intense hurricanes (Hawkins et al. 2006). Because secondary eyewalls are often associated with large intensity changes, a good understanding of the underlying dynamical processes is vital to improve hurricane intensity forecasting.
Several theories have been proposed for SEF. Nong and Emanuel (2003) suggested that external forcing may trigger a wind-induced surface heat exchange instability that results in SEF. Kuo et al. (2004) and Kuo et al. (2008) used a 2D barotropic vorticity framework to show that concentric rings can form as a result of the interaction of a strong core vortex with one or more weak vortices placed in its near environment. Their results emphasized the important role of the vorticity magnitude of the core vortex in straining out the weak vortices into a vorticity ring surrounding the former, due to the induced differential rotation. Terwey and Montgomery (2008) hypothesized that SEF may be induced by the anisotropic upscale energy cascade and axisymmetrization of convectively generated vorticity anomalies on a low-level radial potential vorticity gradient referred to as a beta skirt. They coined this the beta-skirt axisymmetrization (BSA) mechanism. Judt and Chen (2010) suggested that the generation and accumulation of potential vorticity in the rainband region was key for SEF in their simulated Hurricane Rita (2005). Huang et al. (2012) proposed an axisymmetric view for SEF, emphasizing the importance of unbalanced processes. In their simulation, the secondary eyewall formed after a sequence of structure changes, including the broadening of the tangential winds above the boundary layer, followed by an intensification of the radial inflow in the boundary layer underneath this region, and finally the generation of a secondary convergence zone owing to the development of supergradient winds. Rozoff et al. (2012) mentioned that SEF is likely a dynamical adjustment process to latent heating and emphasized the importance of the vortex inertial stability as a controlling factor.
Montgomery and Kallenbach (1997) were the first to hypothesize a possible connection between VRWs and secondary eyewall formation. In a 2D unforced barotropic framework Martinez et al. (2010b) explored further the ideas from Montgomery and Kallenbach (1997) and proposed that a wave–mean flow interaction mechanism is key to explaining enhancement of the primary circulation and the development of a secondary wind maximum. By perturbing their symmetric hurricane-like vortex with a wavenumber-4 asymmetry placed outside the high vorticity core, they showed that a secondary wind maximum arises because of the relaxation of this initial disturbance and a resonance interaction with the mean flow at the critical radius of the dominant wave modes. However, Moon et al. (2010) suggested that the asymmetry introduced by Martinez et al. (2010b) was too broad and too weak in magnitude and therefore may not be consistent with what is observed in real hurricanes. In addition, they argued that the axisymmetrization process in a 2D framework is an incomplete process and cannot describe the formation of the secondary eyewall.
In general, it is expected that the propagation and evolution of asymmetries are influenced by both friction in the boundary layer and diabatic processes. However, internal “dry” dynamics may reveal important mechanisms that could be overshadowed in a more complex framework. Nevertheless, Martinez et al. (2011) extended their idea to a three-dimensional (3D) framework and showed that SEF in their idealized full physics simulation was due to the outward propagation of wavenumber-1 VRWs that stagnate at the critical radius, leading to an acceleration of the primary circulation due to angular momentum redistribution. The results in Martinez et al. (2011) indicate that the contribution from eddies to SEF can be dominant, at least during the early stages of the SEF. In their idealized case, the net spinup produced by the eddies is on the same order of magnitude as the acceleration of the axisymmetric tangential wind inferred from their Fig. 4. The work by Abarca and Corbosiero (2011) also suggests that VRWs may have an important role in SEF. However, they did not study the wave–mean flow interactions. Finally, Menelaou et al. (2012) emphasized the important role played by VRWs and wave–mean flow interactions in the genesis of the secondary eyewall in their simulation of Hurricane Wilma (2005).
From the above discussion, it is clear that the dynamics of hurricane asymmetries are of great significance and need to be further explored. In addition, the large diversity and inconsistency among studies of the SEF to date indicate that this problem is not well understood and is far from being resolved. The intention of this paper is to better understand the underlying dynamics of the hurricane inner-core asymmetries and assess their impact on SEF. Our approach consists of using a simple 2D unforced barotropic model to simulate the evolution of 2D hurricane-like nonmonotonic vortices that mimic strong hurricanes in a mature stage, and study the formation of concentric rings of enhanced vorticity (secondary eyewalls) and a secondary wind maximum. Diagnostic studies will then be performed by applying the empirical normal mode (ENM) theory (Brunet 1994) to extract the dominant wave patterns from the dataset and to assess their role on intensity changes and SEF.
The remainder of this paper is as follows. Section 2 describes the model used in this study, the experimental setup, the initialization procedure, and some of the basic features and results obtained from the numerical experiments. Section 3 reviews briefly the eigenmode theory of small perturbations on 2D vortex flows, the generalized wave activity conservation laws, wave–mean flow interactions in a 2D framework, and the ENM method in a 2D framework. Diagnostic results from the ENM analysis are presented in section 4, and finally a summary and conclusions are discussed in section 5.
2. Model and initial vortex
a. Nondivergent barotropic spectral model
b. Setup of the experiments and initial conditions
In this work, two sets of experiments are conducted to study the dynamics of asymmetric disturbances in nonmonotonic vorticity profiles that mimic hurricane-like vortices during the mature stages of development. In the first experiment, by using a combination of exponential functions, two symmetric nonmonopolar stable vortices are prescribed and are perturbed with azimuthal wavenumber-2 asymmetries to mimic elliptical eyewalls. Elliptical eyewalls are not uncommon in hurricanes and have been previously documented by Kuo et al. (1999), Reasor et al. (2000), and Corbosiero et al. (2006) in Typhoon Herb (1996), Hurricane Olivia (1996), and Hurricane Elena (1985), respectively. Both Reasor et al. (2000) and Corbosiero et al. (2006) associated the appearance of the elliptical eyewall to a small-amplitude wavenumber-2 asymmetry. In addition, Corbosiero et al. (2006) pointed out the longevity of the elliptical eyewall in comparison with other short-living asymmetric structures (e.g., triangles and squares). The origin and persistence of an elliptical eyewall has been previously studied by Kossin et al. (2000). Specifically, they showed that elliptical eyewalls may arise as a result of an instability that occurs in the moat region due to the interaction of a central vortex and the inner edge of an outer vortex ring. Here, we will reveal another physical mechanism that may be responsible for the persistence of elliptical eyewalls.
The second experiment is initialized with an unstable symmetric annular vortex perturbed with a localized wavenumber-4 asymmetry placed slightly outside the primary eyewall. Sensitivity tests on the width of the wavenumber-4 asymmetry are performed. The two sets of experiments are designed to study the impact of the asymmetric disturbance dynamics on the vortex primary circulation and to investigate the mechanism leading to the formation of secondary wind maximum and concentric rings of enhanced vorticity (secondary eyewall formation) in hurricane-like vortices.
1) Experiment I
Basic state for the first (black solid line) and second (gray dashed line) simulation in expt I: (a) vorticity (s−1), (b) radial vorticity gradient (km−1 s−1), (c) tangential wind (m s−1), and (d) angular velocity (s−1).
Citation: Journal of the Atmospheric Sciences 70, 2; 10.1175/JAS-D-12-0192.1
Vorticity contour plots (×10−3 s−1) for the first simulation in expt I: simulation times (a) t = 0 h, (b) t = 2 h, (c) t = 4 h, and (d) t = 6 h. The model domain is 600 km × 600 km, but the results are presented in a subdomain of 233 km × 233 km.
Citation: Journal of the Atmospheric Sciences 70, 2; 10.1175/JAS-D-12-0192.1
As in Fig. 2, but for the second simulation in expt I.
Citation: Journal of the Atmospheric Sciences 70, 2; 10.1175/JAS-D-12-0192.1
Radius–time Hovmöller diagram of the asymmetric vorticity (×10−3 s−1) for expt I: (a) first and (b) second simulation. Thick (thin) black lines denote positive (negative) values (contours of ±0.2, 0.4, 0.6, 0.8, 1.0, 1.2, and 1.4 × 10−3 s−1).
Citation: Journal of the Atmospheric Sciences 70, 2; 10.1175/JAS-D-12-0192.1
Azimuth–time Hovmöller diagram of the asymmetric vorticity (×10−3 s−1) at 50-km radius for expt I: (a) first and (b) second simulation. Thick (thin) black lines denote positive (negative) values (contours of ±0.3, 0.6, 0.9, 1.2, and 1.5 × 10−3 s−1).
Citation: Journal of the Atmospheric Sciences 70, 2; 10.1175/JAS-D-12-0192.1
Vorticity contour plots (×10−3 s−1) for a new 12-h simulation for expt I: (a) first and (b) second simulation.
Citation: Journal of the Atmospheric Sciences 70, 2; 10.1175/JAS-D-12-0192.1
The results presented in this experiment suggest two different decay processes of the initial elliptical deformation. In the following sections we will present the physical mechanism that governs the evolution and lifetime of the asymmetries. In addition, the weak damping case bears a great resemblance with the results of Martinez et al. (2010a), who showed how an incipient storm (prescribed as a monopole) may intensify via the decay of an excited quasimode. However, the spatial structures obtained for this case are quite different than those in Martinez et al. (2010a), as will be shown in the following sections from the linear eigenmode analysis and the ENM analysis. This difference is attributed to the fact that we are dealing with a nonmonotonic equilibrium vortex.
2) Experiment II
Basic-state for expt II: (a) vorticity (s−1), (b) radial vorticity gradient (km−1 s−1), and (c) angular velocity (s−1).
Citation: Journal of the Atmospheric Sciences 70, 2; 10.1175/JAS-D-12-0192.1
Figure 8 shows the time evolution of the total vorticity in a subdomain of 294 km × 294 km. As expected, the annular ring deforms and breaks down into four distinct mesovortices early in the simulation. A linear stability analysis for a similar vorticity distribution carried out by Schubert et al. (1999) showed that wavenumber-4 disturbances are the fastest growing mode. Different from the experiment of Schubert et al. (1999), in our experiment the vorticity stripped off the outer edge of the high vorticity ring and organizes into a secondary ring of enhanced vorticity at ~85 km (Fig. 8f) that is associated with a secondary maximum in the tangential wind at the same radius (Fig. 9). Between 8 and 10 h the annular ring reforms at a smaller radius owing to the merging of mesovortices. This finding was also observed by Martinez et al. (2010b). Finally, by the end of the simulation a region of low vorticity (moat) can be clearly seen between 70-km and 80-km radius.
Vorticity contour plots (×10−3 s−1) for expt II: simulation times (a) t = 0 h, (b) t = 2 h, (c) t = 4 h, (d) t = 6 h, (e) t = 6 h, and (f) t = 10 h. The model domain is 600 km × 600 km, but the results are presented in a subdomain of 294 km × 294 km.
Citation: Journal of the Atmospheric Sciences 70, 2; 10.1175/JAS-D-12-0192.1
Azimuthally averaged tangential winds (m s−1) at t = 0, 5, and 10 h.
Citation: Journal of the Atmospheric Sciences 70, 2; 10.1175/JAS-D-12-0192.1
To further explore the impact of the spatial extent of the initial asymmetry on the vortex evolution and secondary eyewall formation, another experiment is carried out in which the spatial extent of the initial asymmetries is reduced considerably. For comparison, Fig. 10 shows the initial spatial structure of the imposed wavenumber-4 asymmetry for the two cases. Both asymmetries have similar amplitude that is one order of magnitude smaller than the basic-state vorticity. The major difference lies in their spatial extent. Figure 11 shows the time evolution of the total vorticity in a new 12-h simulation in which the width of the initial wavenumber-4 asymmetry is decreased. The results are in a good agreement with the previous simulation in which the secondary ring and the moat region can be clearly identified. The main difference lies in the time of occurrence of the secondary eyewall, which is delayed by 2 h.
Spatial structure of the wavenumber-4 asymmetry (×10−4 s−1) used to perturb the symmetric vortex in expt II: (a) 10-h simulation, and (b) 12-h simulation. Thick (thin) black lines denote positive (negative) values (contours of ±0.2, 0.4, 0.6, 0.8, and 1.0 × 10−4 s−1).
Citation: Journal of the Atmospheric Sciences 70, 2; 10.1175/JAS-D-12-0192.1
Vorticity contour plots (×10−3 s−1) for a new 12-h simulation for expt II in which the spatial extent of the initial asymmetry is decreased: simulation times (a) t = 0 h, (b) t = 4 h, (c) t = 8 h, and (d) t = 12 h.
Citation: Journal of the Atmospheric Sciences 70, 2; 10.1175/JAS-D-12-0192.1
3. Asymmetric dynamics in 2D vortex flows
To analyze the evolution of perturbations on 2D vortex flows a linear eigenmode analysis similar to Schecter et al. (2000), and the empirical normal mode (ENM) technique in the context of 2D Euler equations (Martinez et al. 2010a,b) are applied. In this section the eigenmode theory of small perturbations in 2D vortex flows, the generalized wave activity conservation laws, wave–mean flow interactions, and the 2D ENM method are briefly reviewed.
a. Linear eigenmode theory
b. Wave activity conservation laws and wave–mean flow interactions
c. 2D ENM method revisited
To extract and isolate the dominant wave modes from the dataset, the empirical normal mode method (Brunet 1994) is used. The ENM technique is similar to the empirical orthogonal functions (EOFs) in the sense that both are eigenvalue problems. Different to the EOF method, which is associated with a matrix whose elements are the covariances of one variable, the ENM method incorporates a self-adjoint matrix into the covariance matrix in such a way that each matrix element is in the form of wave activities. Therefore, an eigenvalue of the EOF matrix can only be interpreted as the variance, whereas an ENM eigenvalue represents the amount of wave activity carried by each ENM.
Finally, to study the contribution from individual ENM modes to wave–mean flow interactions, u′ and υ′ can be expanded into a set of basis functions similar to ζ′ [using (24); ζ′ replaced by u′ and υ′], and then replaced back into (20). Taking into account the biorthogonality condition of the PCs and ENMs, the time average of (20) can then be expressed as the summation of individual ENMs.
4. Diagnostic results
a. Experiment I
1) Linear eigenmode analysis and Landau pole calculation
In section 3a, we briefly reviewed the linear eigenmode theory (Schecter et al. 2000). In this section, (14) is solved to characterize the nature of the relaxation and maintenance of the elliptical eyewalls in experiment I. Figure 12 shows some of the eigenfunctions obtained from the linear analysis for the two cases. Figures 12a–d (Figs. 12e,f) correspond to the first (second) experiment in which the initial elliptical eyewall relaxes faster (slower) to a more axisymmetric eyewall. In the first experiment generic continuum modes were found with eigenfunctions that have a singular spike at their critical radius. This result suggests that the initial disturbance damps through the excitation of filaments distributed over the entire radial extent of the vortex domain (Martinez et al. 2010a). During the global filamentation the wave components of the packet that describe the initial disturbance interfere destructively, leading to fast decaying of the initial disturbance.
Linear eigenfuntion analysis for expt I: (a)–(d) the first simulation (global filamentation) and (e),(f) the second simulation (quasimode). The domain radius Rmax = 300 km and the critical radius r* for the quasimode case is ~54 km.
Citation: Journal of the Atmospheric Sciences 70, 2; 10.1175/JAS-D-12-0192.1
In the second experiment, continuum modes with spatial structures shown in Figs. 12e,f are obtained. The eigenfunctions of these modes are characterized by smooth spatial structures and a singular spike at the critical radius, which is found to be ~54 km. In the following section we show that the perturbation formed by the wave packet of these modes defines a quasimode. Note the differences in the spatial structures of these continuum modes from those obtained in Martinez et al. (2010a) (their Fig. 6b, bottom-right panel) in which the spatial eigenfunction is positive throughout the domain. In our case, however, it changes sign from negative to positive. This is attributed to the fact that the basic-state vorticity profile in this case is a nonmonotonic function of radius. In both cases the spatial patterns satisfies the relation 
2) Wave activity spectra
The wave activity spectra of the absolute values of pseudomomentum for the first few wavenumber asymmetries (1–6) for the two cases in experiment I are depicted in Figs. 13a,b. The ENMs here are sorted according to their pseudomomentum in descending order. The first mode has the largest and positive pseudomomentum and the last mode has the smallest and negative value. In general, the leading modes (1–10), which explain most of the total variance (95.6% and 97% in Figs. 13a and 13b, respectively) for wavenumber-2 asymmetries, are at least one order of magnitude larger than the remaining wavenumbers. For this reason we will consider only wavenumber-2 asymmetries in our analyses (Figs. 13c,d). Since pseudomomentum changes sign, it suggests the possible existence of a vanishing mode (depicted by the dip in the pseudomomentum) that separates the spectra into two regions. To the left (right) of this dip, modes have positive (negative) pseudomomentum and, according to the azimuthal phase speed formula (Chen et al. 2003), can be considered to be vortex Rossby waves that retrograde (prograde) with respect to the mean flow. Taking into account the last statement and the percentage of the total variance explained by the leading modes, the wave activity spectra in both cases are dominated by retrograde VRWs.
Experiment I wave activity spectra of pseudomomentum 
Citation: Journal of the Atmospheric Sciences 70, 2; 10.1175/JAS-D-12-0192.1
3) PCs and spatial patterns
One ENM mode alone can only act as a standing wave. To form a propagating wave, at least one pair of modes with similar contribution to the total variance, the same oscillation frequency, and high cross correlation among their spatial patterns (Zadra et al. 2002) is needed. Here, the existence of propagating waves in the two cases is verified and their main properties are studied. Figure 14 shows the time series for the first pair of ENMs (modes 1 and 2) of wavenumber-2 asymmetries and their corresponding power spectra for the first case. These modes contribute about 21.3% and 19% to the total wavenumber-2 asymmetry, respectively. In addition, the two modes have the same frequency of oscillation that is given by the pronounced peak in the power spectra (Fig. 14c). Finally, cross-correlation comparison between their complex vorticity spatial patterns (not shown) also indicate high values (−81.7% and 98.1%). As such, ENMs 1 and 2 form a retrograde VRW. Of interest is that the amplitude of these modes exhibits an overall damping behavior.
Time series and power spectra for ENM modes 1 and 2 of wavenumber-2 asymmetries for the first simulation in expt I. The time series are for 6 h.
Citation: Journal of the Atmospheric Sciences 70, 2; 10.1175/JAS-D-12-0192.1
Similarly, Fig. 15 shows the time series and power spectra for the first pair of ENMs for the second case. The variance explained by these modes is 42.2% and 36.3%, respectively. The amplitude of these modes indicate an early exponential decay (first 4 h) that becomes oscillatory as time evolves. This is in contrast to the previous ENMs and the observed overall damping. This behavior bears great similarity to the quasimodes found in Martinez et al. (2010a) (see their Fig. 8). As pointed out by Schecter et al. (2000), when a quasimode is excited, it exhibits an early exponential decay, but then bounces owing to the nonlinearities that arise from “trapping oscillations” in the cat’s eyes. Figure 16 shows the complex vorticity spatial patterns for these two ENMs. Cross-similarity comparison among the diagonal patterns indicates values of −99.96% and 99.94%, respectively. In addition, Fig. 16a resembles the discretelike continuum-mode eigenfunctions obtained from linear eigenmode analysis. In summary, as seen from the time series, the power spectra, the complex spatial patterns, and their high cross similarity, ENMs 1 and 2 indeed form a retrograte-propagating weakly damped quasimode. To the best of our knowledge, this is the first time that quasimodes are revealed in nonmonotonic vorticity profiles.
As in Fig. 14, but for the second simulation in expt I.
Citation: Journal of the Atmospheric Sciences 70, 2; 10.1175/JAS-D-12-0192.1
Complex spatial patterns of ENM modes 1 and 2 for the second simulation (quasimode) in expt I.
Citation: Journal of the Atmospheric Sciences 70, 2; 10.1175/JAS-D-12-0192.1
4) EP flux and its divergence
Eliassen–Palm flux can be interpreted as a flux of wave activity. Specifically, EP flux here is associated with the flux of eddy angular momentum. Figure 17 shows the EP flux for the two cases, namely the sheared VRW (Fig. 17a) and the quasimode (Fig. 17b). The critical radius for the sheared VRW is located at ~78 km and for the quasimode at 52 km. The resonance condition, ω = m0Ω(r*), is used to calculate the critical radius, where ω = 5.81 × 10−4 s−1 is the frequency of the sheared VRW mode and ω = 0.0010 s−1 is the frequency of the quasimode, and is given by the maximum peak in the power spectra. Here, m = 2 is the azimuthal wavenumber. In both cases a maximum in the EP flux is located close to the calculated critical radius and is consistent with the hypothesis of the VRWs propagating radially outward and redistributing angular momentum as they move toward their critical radius. After reaching the critical radius, the inward flux of eddy angular momentum starts to decrease. From the results here we can conclude that sheared VRWs impact the mean flow in a region located outside the eyewall, while a quasimode is primarily affecting the near-eyewall region. By calculating the EP flux divergence we can assess how sheared VRWs and quasimodes may affect the mean vortex [see (21)]. Figure 18 shows the EP flux divergence associated with modes 1 and 2 for both cases. For the case of a sheared VRW (Fig. 18a), the maximum in the EP flux divergence at a radius of ~80 km indicates an acceleration of the mean flow that occurs far outside the basic-state radius of maximum wind (RMW). In contrast, for the case of a quasimode (Fig. 18b) acceleration occurs close to the RMW (~40 km). Therefore, we may attribute the acceleration of the mean flow close to the RMW to the decay of an excited quasimode. This finding is consistent with Schecter (2008) [see his (19)] in which for γ0 = dζ0/dr < 0 (the case at RMW) the mean flow gains angular momentum as the quasimode decays.
Eliassen–Palm flux (m3 s−2) for (a) first (sheared vortex Rossby waves) simulation and (b) second (quasimode) simulation in expt I.
Citation: Journal of the Atmospheric Sciences 70, 2; 10.1175/JAS-D-12-0192.1
EP flux divergence (m2 s−2) for (a) first (sheared VRWs) simulation and (b) second (quasimode) simulation in expt I.
Citation: Journal of the Atmospheric Sciences 70, 2; 10.1175/JAS-D-12-0192.1
b. Experiment II
1) Wave activity spectra and principal components
Our second experiment is primarily designed to further explore the idea proposed by Martinez et al. (2010b) that secondary eyewall formation may be due to wave–mean flow interactions. For this reason the basic-state high vorticity core closely resembles their case. The main differences can be identified in the skirt region and the initial perturbation. In our case we perturb the basic-state vortex with a localized wavenumber-4 asymmetry. As shown previously, sensitivity tests to the spatial extent of the initial asymmetry resulted in similar results. In both cases a secondary ring of enhanced vorticity associated with a secondary maximum in the tangential winds can be identified by the end of the simulation period.
The ENM diagnostics will be restricted to wavenumber-4 asymmetries (the fastest growing modes) (Schubert et al. 1999; Martinez et al. 2010a). Figure 19 shows the wave activity spectra of the absolute values of pseudomomentum and pseudoenergy. The spectra again consist of prograde and retrograde VRWs and therefore the existence of barotropically unstable modes is possible. As previously mentioned, the empirical normal modes in this study are obtained by solving the eigenvalue problem of the time-covariance matrix. In this approach, the unstable modes are split into two ENMs that form a pair with the same total pseudomomentum, but opposite in sign. Therefore, when these two ENMs are recombined, the result is an unstable mode with zero total pseudomomentum.
Experiment II wave activity spectra of pesudomomentum 
Citation: Journal of the Atmospheric Sciences 70, 2; 10.1175/JAS-D-12-0192.1
We draw attention to the first few modes (1–6), which explain about 68% of the total variance for wavenumber-4 asymmetries. Martinez et al. (2010b) related the first pair of modes (1 and 2) to primary eyewall processes, such as contraction, and modes 5 and 6 to the secondary eyewall formation. Figures 20 and 21 show the time series and corresponding power spectra for modes 1 and 2 and modes 5 and 6. Modes 1 and 2 (Figs. 20a,b) grow in amplitude during the first 4 h, indicating that they are possible unstable modes. In later times, their amplitude decreases and start to oscillate. Modes 3 and 4 (not shown) have similar structures compared to the first pair. On the other hand, modes 5 and 6 exhibit a damping behavior in the first 6 h, consistent with the findings of Martinez et al. (2010b) (see their Fig. 7). In contrast, later on in time, their amplitude grows, a feature that may attributed to the rise of nonlinear processes. One of the spectral characteristics of this pair of modes is the nonmonochromaticity as depicted by the two peaks in the power spectra. Cross-correlation comparison among their complex spatial patterns (not shown) indicates high values of −99.9% and 99.9% for the first pair (modes 1 and 2) and −99.9% and 99.9% for modes 5 and 6. In addition, both pairs of modes have the same frequency of oscillation (Fig. 20c and Fig. 21c for modes 1 and 2 and modes 5 and 6). As such, ENMs 1 and 2 and ENMs 5 and 6 both form retrograde-propagating vortex Rossby waves.
Time series and power spectra for ENM modes 1 and 2 of wavenumber-4 asymmetries for expt II: the time series are for 10 h.
Citation: Journal of the Atmospheric Sciences 70, 2; 10.1175/JAS-D-12-0192.1
As in Fig. 20, but for ENM modes 5 and 6.
Citation: Journal of the Atmospheric Sciences 70, 2; 10.1175/JAS-D-12-0192.1
ENM modes at the other extrema of the wave activity spectra with small negative values of pseudomomentum may also be significant. Specifically, modes 298 and 299 (not shown) combine to form a prograde VRW that grows in amplitude in the first hours, similar to modes 1 and 2 (retrograde VRW). These counterpropagating VRWs possess similar observed periods (not shown) and suggest a possible phase locking that results in exponential instability.
2) EP flux divergence
Figure 22 shows the EP flux divergence from the contribution of the first two ENMs (Fig. 22a) and ENMs 3–6 (Fig. 22b). The EP flux divergence associated with ENMs 1 and 2 depicts a global maximum located at a radius of about 40 km, indicating an acceleration of the flow inside the initial RMW. This maximum coincides with the radius at which the annular primary eyewall reforms by the end of the simulation. Therefore, consistent with previous findings, modes 1 and 2 are associated with the contraction of the primary eyewall. A smaller maximum in the EP flux divergence can be seen also at ~60 km radius that coincides with the initial RMW. Modes 1 and 2 exhibit also a minimum in the EP flux divergence at ~70 km radius that may account for the spindown of the mean flow at this region. On the other hand, the EP flux divergence associated with the other two ENM pairs (3–6) indicates a maximum located close to the radius where the secondary eyewall develops. The critical radius for modes 5 and 6, given by the resonance condition ω = mΩ0, is found at a radius of ~75 km. For this experiment m = 4, denoting the wavenumber-4 asymmetries. It should be noted that the critical radius is located a few kilometers away from where the secondary eyewall develops. This discrepancy can be attributed to the polychromatic nature of these modes. It is significant to note that the maximum acceleration of the mean flow obtained from the higher order ENMs (pairs 3–6) is largely compensated by the spindown from ENM modes 1 and 2. The net effect is a spindown of the mean flow between ~60- and ~80-km radius. Therefore, the ENM analysis suggests that SEF occurs mainly from the spindown of the mean flow between ~60- and ~80-km radius. Note that these results are slightly different from those of Martinez et al. (2010b) in which the secondary eyewall formed mainly because of a strong acceleration of the mean flow in the region where the secondary eyewall eventually formed. In either case, SEF is explained through a wave–mean flow interaction mechanism, underlying therefore its crucial importance.
EP flux divergence for (a) ENM modes 1 and 2 and (b) ENM modes 3–6.
Citation: Journal of the Atmospheric Sciences 70, 2; 10.1175/JAS-D-12-0192.1
Further exploration of the results from the wave–mean flow interactions in the two experiments reveal physical mechanisms that deserve special attention. In experiment I, although a net spinup of the primary circulation outside the primary eyewall is obtained, there is no evidence of the formation of secondary wind maximum outside the primary eyewall, as revealed by a plot of the mean-tangential wind evolution (not shown). This particular result may explain why in some strong annular hurricanes SEF does not occur. However, one has to take into account that all our experiments use only one pulse of VRWs. Further testing using a forcing mechanism similar to the one used in Montgomery and Enagonio (1998) that allows the introduction of more pulses of VRWs may be helpful to clarify the results. In contrast, in experiment II a secondary vortex ring has been observed together with the formation of a secondary wind maximum. All of these results suggests that an unstable annular vortex may be more efficient for SEF than a stable annular vortex because the former is more prompted to shed higher vorticity filaments and generate secondary wind maximum outside the primary eyewall.
5. Summary and conclusions
The underlying dynamics associated with hurricane inner-core asymmetries and the axisymmetrization process can result in sudden changes in the intensity and structure of a hurricane. During axisymmetrization, vorticity filaments are excited and the initial asymmetry decays in time, even in the absence of dissipation. On some occasions the initial perturbation can be strongly damped through the process of spiral windup—that is, the excitation of filaments spiraling over the entire vortex domain—and the initial perturbed vortex returns to symmetry in rather a fast manner. In others, it can be weakly damped through the decay of an excited quasimode. The two pathways of inviscid damping have previously been studied in the context of monopolar vortices typical of incipient storms. Recently, Martinez et al. (2010a) suggested that quasimodes have a larger impact on the structure and intensity of a hurricane, and showed how quasimode mean flow interaction may intensify a weak storm leading to tropical cyclogenesis.
In this study, we take advantage of the simplicity of a 2D unforced barotropic nondivergent model to investigate the inviscid damping of strong nonmonotonic vorticity profiles that are typical of strong mature hurricanes. Although hurricanes are in general 3D nonconservative systems, 2D models can still reveal important dynamical mechanisms. Specifically, two experiments are configured to study, first, (experiment I) the relaxation and maintenance of elliptical eyewalls. Our second experiment (experiment II) aims to provide further insight on the fundamental processes that result in the formation of secondary eyewalls. The ENM method and EP flux formulation are then applied to extract the dominant wave patterns and assess their roles.
The results from experiment I show, first, how an elliptical eyewall may relax back to symmetry through global filamentation (sheared vortex Rossby waves) and, second, through the damping of an excited quasimode. In the case of sheared VRWs the initial ellipticity is strongly damped and the perturbed vortex returns back to symmetry in a few vortex rotation periods. In contrast, in the case of a quasimode, the initial perturbation is weakly damped and the axisymmetrization process eventually stops. Sensitivity tests were performed by doubling the initial time period of the simulation (12 h). It was found that, when a quasimode is excited, the initial perturbation remains undamped for a long time. Therefore a long-living elliptical eyewall may be associated with an excited quasimode. In the case of spiral windup, the dominant ENM modes (modes 1 and 2) for wavenumber-2 asymmetries combine to form a propagating VRW with an amplitude that is decaying throughout the whole simulation period. The critical radius for these modes is found to be well outside the inner-core vortex at a radius of ~78 km. The maximum acceleration of the flow is expected to occur close to the critical radius. Therefore, sheared VRWs may account for the formation of a secondary wind maximum and secondary eyewall formation. In contrast, the critical radius of the quasimode was found to be just outside the radius of maximum winds at ~52 km. Note that the critical radius obtained by linear eigenmode analysis (~54 km) and from the Landau pole calculation (~53 km) is in excellent agreement with that obtained from the empirical normal mode analysis. In addition, great similarity was observed between the radial eigenfunctions obtained from the eigenmode analysis and the complex spatial structures of ENMs 1 and 2. Of interest is that the EP flux calculations (interpreted as the flux of wave activity) indicate peaks that are collocated with the calculated critical radius for both cases. Finally the EP flux divergence map for the quasimode case indicates an acceleration of the flow close to the RMW. In summary, the long-lasting hurricane ellipticity can be associated with the excitation of a quasimode. Also, our result may provide a possible hint to the forecasting of the SEF. In the case of a long-lasting elliptical eyewall we should not expect a secondary eyewall to form in the near time period.
The results from experiment II help to provide a better understanding of the underlying processes leading to SEF. Recently, Martinez et al. (2010b) showed how concentric rings may arise owing to the axisymmetrization of asymmetric disturbances placed outside the primary vortex ring and wave–mean flow interaction. Here we further show that concentric rings may still arise even in the absence of large initial anomalies. Specifically, we found that the amplitude of the leading empirical normal modes (mode 1 and 2) for wavenumber-4 asymmetries exhibit exponential growth during the first hours of the simulation, and account for the instability and contraction of the initial vortex ring. Due to angular momentum conservation, when high vorticity is transferred into the center, another part of high vorticity must be transported outward at a greater radius (Schubert et al. 1999; Chen and Yau 2001).
In both of our simulations, regardless of the spatial extent of the initial perturbation, the outward-propagating filaments organize into a secondary ring of enhance vorticity. The results presented here thus emphasize the importance of VRWs in the SEF and, in contrast to Moon et al. (2010), show that a 2D model can be used to study SEF.
Acknowledgments
K. Menelaou wishes to gratefully thank Dr. David Schecter for the helpful discussions on quasimodes and for providing him with the algorithm of a quasimode solver. We are very grateful to the two anonymous reviewers for their constructive comments, which led to the improvement of this paper. The research reported here is supported by the Natural Science and Engineering Research Council of Canada and Hydro-Quebec through the IRC program.
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