1. Introduction
Accurate forecasting of hurricane intensity remains an open and challenging problem. While track forecasts have improved steadily over the past years (e.g., McAdie and Lawrence 2000), intensity forecasts have not. One of the main reasons for this deficiency originates from the fact that the internal asymmetric dynamical processes, and their connection to structural and intensity changes, are not fully understood (Wang and Wu 2004).
Although the hurricane is thought to be primarily an axisymmetric vortex (Ooyama 1982), recent observations suggest that this is true only for very strong storms, and that the hurricane vortex is largely an asymmetric system. Such asymmetric structures accompanying the symmetric circulation are categorized as (i) inner rainbands (e.g., Jorgensen 1984); (ii) outer rainbands; (iii) polygonal eyewalls (Lewis and Hawkins 1982; Muramatsu 1986) and mesovortices (Black and Marks 1991); and (iv) deep rotating convective clouds referred to as “vortical hot towers” (Hendricks et al. 2004; Montgomery et al. 2006; Nguyen et al. 2008). Both inner and outer rainbands are spiral-shaped banded features that possess, however, distinct dynamics (Wang 2009; Li and Wang 2012). Apart from the aforementioned asymmetries, other frequently observed features, such as secondary eyewalls (e.g., Willoughby et al. 1982; Black and Willoughby 1992) that can be thought to be axisymmetric structures (to the first order), may originate from asymmetric dynamical processes (Abarca and Corbosiero 2011; Martinez et al. 2011; Menelaou et al. 2012). A better understanding of the essential dynamics behind such asymmetric structures is thought to be crucial for the improvement of hurricane prediction.
In a recent observation study of Hurricane Dolly (2008), Hendricks et al. (2012) observed that, while rapidly intensifying, the primary eyewall of Dolly exhibited a highly asymmetric structure with pronounced polygonal shapes and embedded mesovortices. Based on this, it is reasonable to ask what mechanism generated these asymmetries and what is the connection, if any, to the rapid intensification of Dolly? This question can be further generalized to include any rapidly intensifying asymmetric hurricanes, and it is one of the main objectives of this study to further elaborate on the current understanding. In an effort to explain the origin of Dolly's polygonal eyewall, Hendricks et al. made use of a very simple unforced shallow-water model initialized with a thin vorticity ring constructed in such a way that the maximum tangential wind resembled that of Dolly. They showed how similar asymmetric structures can form simply owing to barotropic instability and the breakdown of the vortex ring. To better understand this process one has to consider that in terms of vorticity the eyewall can be pictured as an annular ring of elevated vorticity (Yau et al. 2004), with large vorticity gradients at its edges. Vorticity is increasing (decreasing) with radius on the inner (outer) edge. From the vortex Rossby wave (VRW) theory, such change in the sign of the radial vorticity gradient (from positive to negative) will result in the formation of two counterpropagating waves. If these waves become phase locked so as to intensify each other, dynamic instability and the breakdown of the annular vortex ring may result (Schubert et al. 1999). In terms of balanced dynamics, barotropic instability can also explain part of the reduction of the minimum surface pressure in Dolly (Hendricks et al. 2012). This can be understood if one considers that the inward vorticity mixing that follows the eyewall breakdown will increase the centrifugal term. However, such simple dry dynamics cannot explain the increase of the low-level winds observed in Dolly.
The connection among barotropic or combined barotropic–baroclinic instability of vortex rings, polygonal eyewalls, mesovortices, and vorticity mixing has been the subject of numerous numerical studies. However, most of these studies were cast in highly idealized frameworks, ranging from unforced two-dimensional (2D) (Schubert et al. 1999; Kossin and Schubert 2001; Kossin et al. 2002; Hendricks et al. 2009), forced 2D (Rozoff et al. 2009), linear dry 3D (Nolan and Montgomery 2002; Nolan and Grasso 2003), nonlinear dry 3D (Hendricks and Schubert 2010), to moist 3D (Schecter and Montgomery 2007). The majority of the aforementioned studies complement the hypothesis of barotropic instability as the major cause that leads to the destabilization of the vortex. However, the inclusion of moisture in the study by Schecter and Montgomery (for the purpose of a more realistic simulation) showed that the inclusion of clouds may have an important influence on eyewall instability. In addition and in contrast to barotropic instability, Hodyss and Nolan (2008) emphasize the potential importance of another destabilization mechanism: the Rossby–inertia–buoyancy instability. Using the 3D linear, nonhydrostatic model of Hodyss and Nolan (2007), they showed how a monopolar vortex (vorticity decreasing monotonically with radius) that precludes barotropic instability by coupled VRWs can be rendered unstable by the coupling between a VRW and an inertia–buyoyancy (IB) wave. Their results were validated in a 3D nonlinear model with an idealized environment in which all model physics (moisture, surface friction, and radiation) were excluded. In real hurricanes, friction and diabatic processes are also important and may further complicate eyewall instabilities. Thus, it remains to be determined to what extent barotropic instability can explain asymmetric polygonal eyewall structures in realistically simulated hurricanes. As such, investigating this question is another objective of this study.
In addition to the question related to the origin of eyewall asymmetries, there is also disagreement in the literature in terms of the impact of such asymmetries on hurricane intensity. For example, Schubert et al. (1999) showed in an unforced 2D framework that barotropic instability, and the inward mixing of vorticity from the eyewall into the eye, decreased the maximum tangential wind as vorticity was transported away from the eyewall. Using a 3D hydrostatic primitive equations model initialized with an idealized environment, Yang et al. (2007) also found that inner-core asymmetries reduced hurricane intensity. In their simulation, the inward potential vorticity (PV) mixing resulted in a less tilted eyewall, which in turn suppressed the cooling and drying effects of downdrafts, leading to reduced surface fluxes underneath the eyewall. In contrast, Rozoff et al. (2009) using a forced 2D model pointed out the potential importance of vorticity mixing in intensifying the vortex. Similar conclusions were also drawn in Martinez et al. (2010) in the context of an unforced inviscid 2D model. Chen and Yau (2001) found that the inward PV transport by the VRWs resulted in the intensification of their hurricane inside the radius of maximum wind (RMW). Finally, the recent work of Persing et al. (2013) showed that rotating deep convective plume structures act to intensify the hurricane vortex by upgradient radial momentum fluxes.
From the above discussion it is clear that the inner-core dynamics of hurricane asymmetries are of great significance and need to be further explored. The intention of this paper is, first, to better understand the origin of asymmetric eyewall structures in a realistically simulated real hurricane that undergoes rapid intensification and, second, to assess the impact of these asymmetries on the rapid intensification of the hurricane. Our results aim to provide a bridge between simpler numerical frameworks, theories, and observations.
Our analysis employs the dataset obtained from the simulation of Hurricane Wilma (2005) reported in Menelaou et al. (2012). Using the Weather Research and Forecasting model (WRF), Menelaou et al. were able to reproduce Hurricane Wilma reasonably well in terms of track and intensity evolution. Their simulation captured the remarkable rapid deepening of Wilma as well as the subsequent eyewall replacement cycle that led to the weakening of the storm. Diagnostic studies are then performed by applying the empirical normal mode (ENM) (Brunet 1994) method to extract the dominant wave modes from the dataset, to study their space–time structure and kinematics. The ENM technique is similar to empirical orthogonal functions (EOFs) in the sense that both are eigenvalue problems. However, unlike EOFs, the ENM method takes advantage of the conservation law of wave activities and can decompose fields into a set of basis functions that approach a set of true normal modes for sufficiently small amplitude disturbances. The wave activity spectra can be constructed in terms of the ENMs, and the properties of the dominant modes can be further studied. The ENM method has been previously used successfully to separate rotational-dominated modes from gravity modes (Chen et al. 2003; Martinez et al. 2011; Menelaou et al. 2012). Modes with combined rotational and gravitational properties also can be identified using the ENM technique. As such, the Rossby–inertia–buoyancy instability mechanism proposed by Hodyss and Nolan (2008) can be tested as a possible mechanism for the formation of the polygonal eyewall observed in the simulated Wilma. Recently, Menelaou et al. (2013) in a study of 2D inviscid asymmetric annular vortex flows applied the ENM technique to identify the existence of a weakly damped quasimode. The quasimode is of physical importance, as it can be associated with the lifetime of an asymmetric structure. For details of the ENM technique, interested readers are referred to Charron and Brunet (1999), Zadra et al. (2002), Chen et al. (2003), and Martinez et al. (2011). Finally, the Eliassen–Palm (EP) flux theorem is used to assess the impact of wave modes on the structure and intensity change of the simulated hurricane.
The remainder of this paper is as follows. Section 2 describes briefly the model setup and the period of analysis; section 3 revisits the governing equations that form the basis for the derivation of two wave activity conservation laws. The EP flux theorem in isentropic coordinates and the wave activity conservation laws are summarized. Diagnostic results from the ENM analysis and the EP flux calculations are presented in section 4. Finally, a summary and conclusions are discussed in section 5.
2. Model description and period of analysis
Figure 1a shows the model domain configuration, simulated track, and the track determined from the best-track dataset obtained from the National Hurricane Center (NHC). These are 6-hourly postanalyses of hurricane position and intensity based on a posthurricane assessment of all available data. For the setup of the experiment, we designed a quadruply nested simulation that was initialized at 0000 UTC 18 October 2005 and run for 72 h. The three stationary outer domains α, β, and γ have horizontal grid spacings of 27, 9, and 3 km and horizontal grid meshes of 140 × 140, 286 × 286, and 484 × 484, respectively. The innermost domain δ follows the hurricane vortex (δs and δf being the starting and final positions, respectively) and has a horizontal resolution and grid mesh of 1 km and 346 × 346. The model is integrated forward in time with a very small adaptive time step for the highest resolution domain, and the dataset are saved every 30 s. Such high temporal resolution allows for a detailed investigation of the evolution of the hurricane. For further details on the model physics and initialization, the readers are referred to Menelaou et al. (2012).
(a) Model domain configurations, superimposed with the simulated 10-m wind speed (shaded) at 0000 UTC 18 Oct 2005. The horizontal resolution of the stationary domains α, β, and γ is 27, 9, and 3 km, respectively. The horizontal resolution of domain δ that follows the hurricane vortex is 1 km; δs and δf denote the initial and final positions of the 1-km-resolution domain. The black curve with crosses indicates the NHC best track and the black curve with open circles indicates the simulated track. (b) Time series of hurricane central pressure (hPa) for the NHC (big black open circles) compared with simulated values from the simulation (thick gray line). Small black filled circles indicate the present 6 h of analysis.
Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-13-091.1
Figure 1b shows the NHC best track (open black circles) and predicted (thick gray line) central pressure of Wilma. The small filled black circles indicate the present 6-h period of analysis. The analysis will focus on this time framework (with 0 and 6 h being the beginning and end, respectively), unless otherwise noted. Throughout this time, Wilma exhibited distinct inner-core structural variability with the primary eyewall being highly asymmetric while rapidly intensifying. This sequence of events is not uncommon in real hurricanes and has been previously documented by Corbosiero et al. (2006) and Hendricks et al. (2012). Specifically, Hendricks et al. (2012) reported prominent polygonal eyewall signatures and distinct mesovortices during a 6-h rapid intensification of Hurricane Dolly (2008). They attributed these asymmetries to the result of a convectively modified barotropic instability. One has to note that, although barotropic instability in the context of dry dynamics offers an elegant explanation for the formation of polygonal eyewalls (Schubert et al. 1999), moist convective processes present in real hurricanes are expected to affect this process. While the end state of a dry barotropically unstable annular vortex is most likely a monopolar structure (Schubert et al. 1999; Kossin and Schubert 2001), the continuous PV production through latent heat release can maintain the annular shape of the PV ring (Chen and Yau 2001). In addition, the secondary circulation induced by the diabatic process can locally increase the PV where there is vertical mass divergence. That convection occurs on short time scales of the order of tens of minutes, while the e-folding time of adiabatic barotropic instability is of the order of several hours (Schubert et al. 1999), underlines their significant differences. Finally, apart from the diabatic processes, surface friction is also expected to influence barotropic instability. Some questions related to the impact of surface friction and more precisely the impact of subgrid vertical turbulent mixing on barotropic instability were recently addressed by Zhu et al. (2013). Figure 2 shows the hourly evolution of the azimuthally averaged tangential winds at ~700-m height. The initial (0 h) maximum tangential wind is approximately 74 m s−1 and occurs at a radius of about 20 km. By 6 h, the RMW has decreased to nearly 16.5 km, while the tangential wind speed has increased by about 16 m s−1.
Azimuthally averaged tangential winds (m s−1) at ~700 m height from 0 to 6 h at a 1-h interval.
Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-13-091.1
Figures 3 and 4 provide a detailed view of Wilma's inner-core variability as captured by snapshots of simulated reflectivity (shaded) and potential vorticity (gray contours) at ~1.8-km height and at 15-min time intervals. An azimuthal wavenumber, m = 4, is evident from the reflectivity plots early in the evolution. At this time the primary eyewall, envisioned as a hollow tower of PV (Yau et al. 2004), breaks down into distinct mesovortices, as can be seen from the discrete pools of high cyclonic PV (e.g., Fig. 3d). The PV then starts to get mixed into the eye, while the eyewall asymmetries evolve from higher to lower azimuthal wavenumbers (m = 3, Fig. 4x). Of interest is that a transition from high to low azimuthal wavenumbers was also observed by Hendricks et al. (2012). Accompanying the process of inward PV mixing, some high PV from the eyewall is also drawn outward and takes the form of spiral filaments that resemble inner spiral bands. We remark that in Figs. 3 and 4, only high values of PV are shown (50 PVU and above) since we want to emphasize the eyewall breakdown. Additional plots of simulated reflectivity and PV with a lower contour interval (not shown) indicate a positive correlation between spiral-shaped reflectivity bands and cyclonic PV, consistent with the idea that these inner spiral bands are convectively coupled vortex Rossby waves (Guinn and Schubert 1993; Chen and Yau 2001; Wang 2002a,b). We point out that the coupling between the PV and reflectivity bands is by itself not sufficient to infer that these features are indeed VRWs, although we will verify the existence of VRWs from the ENM analysis. Figure 5 shows the simulated cloud water mixing ratio at ~2.7-km height (shaded), superimposed with the asymmetric PV (black contours). The PV asymmetry is defined here as the deviation from the azimuthal wavenumber-0 component of PV in the Fourier series (which is identical to the symmetric field). The thick dashed orange lines mark the location of inner spiral bands, and the red arrow follows the evolution of one. One can clearly see that cyclonic PV asymmetry and cloud water mixing ratio are in phase, again indicating that these inner bands are most likely convectively coupled VRWs. While these results complement the aforementioned studies, they contradict the recent findings of Moon and Nolan (2012), where outward-propagating spiral bands did not possess VRW characteristics. We point out that, even though the period of analysis focuses on the aforementioned 6-h period, the inner core of the simulated Wilma remains highly asymmetric (not shown) throughout the remaining period of rapid intensification. Thus, the dynamical processes to be discussed in the following sections are expected to transpire throughout this time.
Simulated reflectivity (dBZ) (shaded) superimposed with PV (PVU) (thick gray contours of 50–100 PVU, with interval of 10 PVU) at ~1.8-km height. The time period is from (a) 0 to (l) 3 h at 15-min intervals.
Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-13-091.1
As in Fig. 3, but for the time period from (m) 3 to– (x) 6 h.
Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-13-091.1
Simulated cloud water mixing ratio (shaded) (kg kg−1) at ~2.7-km height and simulated PV asymmetry (PVU) (black contours of 4–20 PVU with 2-PVU intervals). The time period is from 0 h 41 min to 0 h 51 min at 2-min intervals.
Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-13-091.1
The presence of distinct azimuthal asymmetries and the eyewall breakdown bear a great similarity to the barotropic or combined barotropic–baroclinic instabilities of hurricane-like vortices, seen in previous idealized studies. Figure 6 shows the vertical structure in terms of the instantaneous simulated radar reflectivity. There are two important features to note. First, the prominent wavenumber-4 pattern observed at this time is more pronounced in the lower levels (Figs. 6a–c), while the eyewall becomes more axisymmetric at upper levels (Fig. 6d). This indicates that the eyewall asymmetries are mainly confined to the lower levels during the release of the instability, consistent with the findings of Hendricks and Schubert (2010), and Hendricks et al. (2012). Second, the asymmetric pattern appears to have a similar phase from 930 to 3400 m with no profound tilt in the vertical. This suggests that baroclinic instability may not be the dominant cause of these asymmetries but barotropic instability is the key mechanism, in agreement with the conclusions of Hendricks et al. (2012).
Simulated reflectivity (dBZ) at height (a) 930, (b) 2700, (c) 3400, and (d) 10 200 m.
Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-13-091.1
3. Methodology
The generalized wave activity conservation laws are applied in the context of hurricane asymmetries to study the properties of wave disturbances and their interaction with the mean flow. In this section the primitive equations that form the basis of the conservation of two wave activities, namely, pseudomomentum density and pseudoenergy density (hereafter pseudomomentum and pseudoenergy) are revisited together with the EP flux formulation. Finally, the flux conservative equations for pseudomomentum and pseudoenergy are presented.
a. The hydrostatic primitive equations and the Eliassen–Palm flux in cylindrical and isentropic coordinates
In the context of hurricane studies, the EP flux theorem has previously been used by several authors. More specifically, Willoughby (1978a) applied the EP relation to analyze linear waves on a barotropic mean vortex in an effort to explain hurricane spiral rainbands as inward-propagating inertia–buoyancy waves. In a companion paper, Willoughby (1978b) extended the EP derivation for a baroclinic mean vortex and verified his original findings. Molinari et al. (1995) made use of this theorem to study the interaction of Hurricane Elena (1985) with an upper-level trough and showed how such an interaction may have affected the intensification of Elena. Chen et al. (2003) used these expressions to study wave–mean flow interactions associated with inner spiral rainbands that possess characteristics of sheared VRWs. Finally, Martinez et al. (2011), and Menelaou et al. (2012) showed that the EP flux theorem can be used to study the impact of outward-propagating waves on the formation of secondary eyewalls. In the present study, EP calculations are performed to provide further insight on the potential impact of unstable wave modes on the rapid intensification of Hurricane Wilma.
b. Wave activity conservation laws in isentropic coordinates
4. Dataset
As previously mentioned, the dataset to be analyzed is taken from the simulation described in Menelaou et al. (2012). To obtain a detailed temporal evolution of the flow, the simulation results are output every 30 s over the 6-h period of interest, giving a total of 721 time samples. The storm motion is calculated following the method introduced by Liu et al. (1999) and is then removed from the total horizontal wind to obtain the relative winds. The model dataset are interpolated from sigma to isentropic coordinates. There are 14 isentropic levels ranging from 310 to 362 K, at 4-K intervals. The lowest isentropic level excludes the boundary layer, while the highest isentropic level extents up to about 14.5-km height. The basic state, taken to be the time and azimuthal mean state, is extracted from the total dataset, and the perturbations are decomposed into different azimuthal wavenumbers. In the present diagnostic study, wavenumber-3 and -4 anomalies are analyzed because they are the most profound in the selected time window.
Figure 7 shows the basic-state tangential wind, PV, angular velocity, radial gradient of PV, pressure, and isentropic density: the RMW is about 20 km and tilts slightly outward with height (Fig. 7a); the basic-state PV exhibits the characteristic bowl-shaped structure (Fig. 7b); the maximum angular velocity occurs at a radius of about 16 km (Fig. 7c); and the elevated pressure surfaces in the storm center suggest a warm-core structure (Fig. 7e). Moving radially outward, the basic-state radial PV gradient changes sign from positive to negative at a radius of about 12 km at the lower isentropic levels (Fig. 7d). As such, the hurricane vortex satisfies the Charney–Stern necessary condition for combined barotropic–baroclinic instability (Montgomery and Shapiro 1995) and may become unstable. Schubert et al. (1999) showed how polygonal eyewalls and mesovortices can form as the result of unstable discrete VRWs. In a diagnostic study using the ENM technique in a 2D framework, Martinez et al. (2010) further explored the ideas of Schubert et al. (1999) and found that the amplitude of their dominant wave modes exhibited early growth, supporting the idea that unstable growing modes were the cause for the breakdown of their hurricane-like vortex. Figure 8 shows a snapshot of the asymmetries under current investigation (m = 4, 3) in terms of PV, taken at 1 h (m = 4) and 4.5 h (m = 3). It is of interest to note that in both cases the anomaly is made up of two concentric sets of PV perturbations that occur close to the maximum positive and negative γ0. These perturbations may be interpreted as counterpropagating VRWs that can help each other to grow if they become phase locked, leading to barotropic instability (e.g., Schubert et al. 1999; Nolan and Farrell 1999; Nolan and Montgomery 2002; Schecter and Montgomery 2007). In the next hours, the (m = 4) asymmetry goes through an axisymmetrization process and is almost completely damped (not shown). In contrast, the double concentric structure of m = 3 asymmetry remains well defined while amplifying (not shown). In the following sections we show that the structure and the time evolution of the wave modes extracted in the present diagnostic study further complement these concepts. It will be shown that the spatial patterns of the dominant modes resemble the double PV perturbation, indicating that this is the dominant pattern of variability.
Basic-state (a) tangential wind (m s−1), (b) potential vorticity (PVU), (c) angular velocity (s−1), (d) radial potential vorticity gradient (PVU km−1), (e) pressure (hPa), and (f) isentropic density (kg K−1 m−2).
Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-13-091.1
Potential vorticity perturbations PV′ on the 322-K isentrope of (a) azimuthal wavenumber 4 (at time 1 h, contour interval ± 2.5 PVU) and (b) azimuthal wavenumber 3 (at time 4.5 h, contour interval ± 2 PVU).
Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-13-091.1
5. Diagnostic results
a. Wave activity spectra
The wave activity spectra of the absolute values of pseudomomentum and pseudoenergy for wavenumber-4 and -3 asymmetries are depicted in Fig. 9. The ENMs here are sorted according to their total pseudomomentum in descending order (Figs. 9a,c). The first mode has the largest and positive pseudomomentum, and the last mode has the smallest and negative pseudomomentum. Both wavenumbers have a similar distribution, although the leading modes for m = 4 asymmetries appear to have slightly larger values of pseudomomentum. This result is expected, if one recalls the time evolution of Hurricane Wilma as seen in Figs. 3 and 4. Since pseudomometum changes sign, it suggests the existence of a vanishing mode (the dip in pseudomomentum) that separates the spectra into two regions. According to the angular phase speed formula Cn =−
(a),(c) Wave activity spectra total pseudomomentum 
Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-13-091.1
b. Principal components and ENM mode characteristics
Figures 10 and 11 show the time series and power spectra for the first two modes for m = 4 and 3. We draw attention to the first two modes since they explain a significant part of the total variance (31.5% and 19% contribution to the total m = 4 and 3 asymmetry, respectively). Modes 1 and 2 for m = 4 (Figs. 10a,b) grow in amplitude during the first 1.5 h, indicating that they are possibly unstable modes. At later times, their amplitude exhibits a damping behavior. In contrast, modes 1 and 2 for m = 3 (Figs. 11a,b) do not indicate a clear oscillating signal during the first two hours. However, starting from 3.5 h their amplitude shows overall growth, again suggesting that they may be unstable modes. Considered together Figs. 9 and 10 are consistent with the time evolution of Hurricane Wilma (Figs. 3 and 4) in which the primary eyewall exhibited an early m = 4 asymmetry, followed by a transition at later times to a lower-wavenumber asymmetry m = 3.
Time series and power spectra for ENM modes 1 and 2 of wavenumber-4 asymmetries from 0 to 6 h.
Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-13-091.1
As in Fig. 9, but for wavenumber-3 asymmetries.
Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-13-091.1
We point out that alone, one ENM mode can only act as a standing wave. To form a propagating wave, at least two modes that have the same oscillation frequency, similar contribution to the total variance, and high cross correlation among their complex spatial patterns are required (Zadra et al. 2002; Chen et al. 2003). In both cases, modes 1 and 2 have the same oscillation frequency that is given by the peak in the power spectra (Figs. 9c and 10c). Their contribution to the total m = 4 (m = 3) asymmetry is 15.8% and 15.7% (9.7% and 9.3%), respectively. Figure 12 shows the real and imaginary (cosine and sine, respectively) components of the complex PV normalized by the maximum PV amplitude for ENM modes 1 and 2 of m = 4. As expected, the PV perturbation is mostly confined in the primary eyewall region. The cross-correlation comparison between the two pairs of diagonal panels are 99.9% and −99.8%, respectively. High cross-correlation values (99.9% and −99.6%) are also found among the complex PV spatial patterns (not shown) of modes 1 and 2 for m = 3. As such, modes 1 and 2 for both m = 4 and 3 asymmetries form a retrograde vortex Rossby wave.
(a),(c) Real (cos) and (b),(d) imaginary (sin) parts of the complex spatial patterns of wavenumber-4 PV disturbances normalized by the maximum PV amplitude for ENM mode (left) 1 and (right) 2.
Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-13-091.1
Empirical normal modes at the other extreme of the wave activity spectra with small negative values may also be significant. As an example, modes 720 and 721 for m = 4 asymmetries (not shown) combine to form a prograde VRW with a similar observed period to modes 1 and 2. This suggests a possible phase locking of these counterpropagating waves that may result in exponential instability. This scenario can be seen by examining their structure. Figure 13a shows the PV amplitude of modes 1 and 2 
PV amplitude 
Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-13-091.1
c. Eliassen–Palm flux and its divergence
The EP flux can be interpreted as a flux of wave activity [see (16)]. The horizontal component of the EP flux vector is related to the radial eddy angular momentum flux (barotropic process), and the vertical component is related to the radial eddy heat flux (baroclinic process). The divergence of the EP flux is interpreted as an effective eddy forcing on the mean flow. It is therefore interesting to calculate the EP flux vector and its divergence associated with the unstable wave modes extracted from the ENM analysis. Such calculations further clarify the relative importance of barotropic and baroclinic processes, as well as the impact of unstable modes in accelerating/decelerating the mean flow.
Figure 14 shows the horizontal component of the EP flux vector (Figs. 14a,c) and the EP flux divergence (Figs. 14b,d) for ENM mode 1 and 2 for m = 4 (Figs. 14a,b) and m = 3 (Figs. 14c,d) asymmetries. The vertical component is at least one order of magnitude smaller than the horizontal component of the EP flux, and therefore not shown. As such, the dominant process here is primarily barotropic in nature. These results further complement the findings of Hendricks and Schubert (2010). Moving outward from the storm center, the horizontal component of the EP flux for mode 1 and 2 for m = 4 asymmetries changes sign from positive to negative; that is, the waves transfer angular momentum radially inward and outward, respectively. The change of sign in the EP flux occurs close to the initial radius of maximum wind. This indicates that the VRWs act to decelerate the mean flow at this location, since momentum is transported away, while intensifying the mean flow inward and outward of this region (Fig. 14b) since momentum is redistributed in these regions. Taking into account that the RMW contracts while intensifying indicates that unstable VRWs may have a positive impact on the intensification of the hurricane vortex. In addition, the acceleration of the flow outside the RMW further suggests a possible connection of VRWs with secondary wind maxima (secondary eyewalls). Such a connection between VRWs and secondary eyewalls in mesoscale numerical simulations has been previously made by Abarca and Corbosiero (2011), Martinez et al. (2011), and Menelaou et al. (2012). In a recent study on a highly idealized 2D framework, Menelaou et al. (2013) showed how secondary rings of enhanced vorticity (secondary eyewalls) may arise owing to barotropic instability and the outward propagation of VRWs. The extent to which these results apply in real hurricanes needs to be examined further and will be the focus of future studies. Deceleration (acceleration) of the mean flow at the RMW (inside and outside the RMW) is also suggested from the EP flux divergence associated with mode 1 and 2 for m = 3 (Fig. 14d) asymmetries.
(a),(c) The horizontal component of the EP flux vector (×109 kg m K−1 s−2) and (b),(d) the divergence of the EP flux (×106 kg K−1 s−2) for ENM modes (top) 1 and (bottom) 2 of wavenumber-4 and -3 asymmetries.
Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-13-091.1
The aforementioned results obtained from the ENM method and the calculations of EP flux divergence (in which an acceleration of the mean flow is suggested to occur radially inside the RMW), along with the coexistence of the polygonal eyewall with the contraction and intensification of the mean flow, suggest that the polygonal eyewall and the mesovortices, resulted from the release of barotropic instability by counterpropagating unstable VRWs, can assist in the intensification process. To further corroborate that, indeed, wave activity is transferring momentum to the mean flow, the EP flux divergence calculated directly from the model output and without any modal decomposition is shown in Fig. 15. The plots are made at two instants of time when the m = 4 (Fig. 15a) and m = 3 (Fig. 15b) asymmetries were more pronounced. We mention that the EP flux divergence is weighted by 
Divergence of the EP flux, ρ−1σ0−1∇ · F (m s−1 h−1), for the total asymmetry taken at time (a) 2.5 and (b) 3.6 h.
Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-13-091.1
6. Summary and conclusions
Observations indicate that hurricanes are highly asymmetric systems during their intensification stage. A striking recent example of this is the pronounced polygonal eyewall structure and the distinct mesovortices that were observed during the rapid intensification of Hurricane Dolly (Hendricks et al. 2012). The coexistence of such asymmetries with the rapid intensification suggests that internal vortex asymmetric dynamics may be an important element of the intensification. However, the underlying dynamics associated with the formation of such asymmetric structures, and their interconnection to the intensification, are not well understood. Although numerous studies have focused on the origin of polygonal eyewalls and mesovortices, the majority of these studies are limited to simple numerical frameworks that deviate from actual hurricanes, therefore leaving a gap between such studies and observations. The intention of the present paper is to fill this gap and to bridge theoretical and simple numerical studies with the observations. This is done using the dataset of a realistic simulation of Hurricane Wilma (2005) reported in Menelaou et al. (2012).
To a certain extent, the observed behavior of Hurricane Dolly during the rapid intensification period matched the simulated behavior of Wilma. Specifically, in a 6-h period during the remarkable rapid intensification of Wilma, the primary eyewall exhibited a distinct azimuthal wavenumber-4 (m = 4) asymmetry and embedded mesovortices, followed by a transition to a lower-wavenumber (m = 3) asymmetry. Of interest is that a similar transition from higher- to lower-wavenumber asymmetries was also observed in Hurricane Dolly (Hendricks et al. 2012). The vertical structure of Wilma showed that the asymmetry is more pronounced in the lower levels with higher levels being more axisymmetric. The phases of the asymmetry are also found to be similar at lower levels with no apparent tilt in the vertical. Taken together, these results suggest that the origin of these asymmetries is most likely barotropic instability that grows most rapidly at the lower levels, which complements the recent findings of Hendricks and Schubert (2010) and Hendricks et al. (2012).
The importance of barotropic instability being the major driving mechanism for the asymmetric structures is also supported from the spatial structure of the m = 4 and m = 3 PV anomalies. The PV anomalies consist of two concentric rings located close to the maximum and minimum γ0, suggesting that these may be counterpropagating waves that, if phase locked, can lead to barotropic instability. From the empirical normal mode analysis, the extracted dominant modes possess characteristics of unstable vortex Rossby waves, again supporting the possible importance of barotropic instability. Specifically, modes 1 and 2 for m = 4 showed an early amplitude growth followed by a damping behavior, while modes 1 and 2 for m = 3 showed growth only at later times. The concept obtained from the time evolution of the dominant modes is consistent with the evolution of the asymmetries seen in Wilma, and the transition from higher to lower azimuthal wavenumbers. Finally, from the EP flux diagnostics it is found that these unstable VRWs act to decelerate the mean flow at the initial RMW by transporting cyclonic angular momentum both radially inward and outward, therefore accelerating the mean flow in these two locations. Taking into account that during the rapid intensification of Wilma, the radius of maximum wind contracted while intensifying, suggests that the VRW may have a positive impact on the intensification. In addition, the acceleration seen outside the RMW wind also suggests the possible connection of these waves in the initiation of secondary wind maxima. Further analysis is needed to make a conclusive statement.
Acknowledgments
K. Menelaou gratefully thanks Ariaan Purich, a former colleague at McGill University, for her assistance in proofreading an earlier version of the manuscript. We are grateful to the three anonymous reviewers for their constructive remarks, which led to the improvement of this paper. The reported research is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and Hydro-Québec through the IRC program. Computations were performed on the CLUMEQ supercomputer through Compute Canada.
REFERENCES
Abarca, S. F., and K. L. Corbosiero, 2011: Secondary eyewall formation in WRF simulations of Hurricanes Rita and Katrina (2005). Geophys. Res. Lett.,38, L07802, doi:10.1029/2011GL047015.
Black, M. L., and H. E. Willoughby, 1992: The concentric eyewall cycle of Hurricane Gilbert. Mon. Wea. Rev., 120, 947–957.
Black, P. G., and F. D. Marks, 1991: The structure of an eyewall meso-vortex in Hurricane Hugo (1989). Preprints, 19th Conf. on Hurricanes and Tropical Meteorology, Miami, FL, Amer. Meteor. Soc., 579–582.
Brunet, G., 1994: Empirical normal mode analysis of atmospheric data. J. Atmos. Sci., 51, 932–952.
Charron, M., and G. Brunet, 1999: Gravity wave diagnosis using empirical normal modes. J. Atmos. Sci., 56, 2706–2727.
Chen, Y., and M. K. Yau, 2001: Spiral bands in a simulated hurricane. Part I: Vortex Rossby wave verification. J. Atmos. Sci., 58, 2128–2145.
Chen, Y., G. Brunet, and M. K. Yau, 2003: Spiral bands in a simulated hurricane. Part II: Wave activity diagnostics. J. Atmos. Sci., 60, 1239–1256.
Corbosiero, K. L., J. Molinari, A. R. Aiyyer, and M. L. Black, 2006: The structure and evolution of Hurricane Elena (1985). Part II: Convective asymmetries and evidence for vortex Rossby waves. Mon. Wea. Rev., 134, 3073–3091.
Guinn, T. A., and W. H. Schubert, 1993: Hurricane spiral bands. J. Atmos. Sci., 50, 3380–3403.
Held, I. M., 1985: Pseudomomentum and the orthogonality of modes in shear flows. J. Atmos. Sci., 42, 2280–2288.
Hendricks, E. A., and W. H. Schubert, 2010: Adiabatic rearrangement of hollow PV towers. J. Adv. Model. Earth Syst.,2 (4), doi:10.3894/JAMES.2010.2.8.
Hendricks, E. A., M. T. Montgomery, and C. A. Davis, 2004: The role of “vortical” hot towers in the formation of Tropical Cyclone Diana (1984). J. Atmos. Sci., 61, 1209–1232.
Hendricks, E. A., W. H. Schubert, R. K. Taft, H. Wang, and J. P. Kossin, 2009: Life cycles of hurricane-like vorticity rings. J. Atmos. Sci., 66, 705–722.
Hendricks, E. A., B. D. McNoldy, and W. H. Schubert, 2012: Observed inner-core structural variability in Hurricane Dolly (2008). Mon. Wea. Rev., 140, 4066–4077.
Hodyss, D., and D. S. Nolan, 2007: Linear anelastic equations for atmospheric vortices. J. Atmos. Sci., 64, 2947–2959.
Hodyss, D., and D. S. Nolan, 2008: The Rossby-inertia-buoyancy instability in baroclinic vortices. Phys. Fluids,20, 096602, doi:10.1063/1.2980354.
Jorgensen, D. P., 1984: Mesoscale and convective-scale characteristics of mature hurricanes. Part II: Inner core structure of Hurricane Allen (1980). J. Atmos. Sci., 41, 1287–1311.
Kossin, J. P., and W. H. Schubert, 2001: Mesovortices, polygonal flow patterns, and rapid pressure falls in hurricane-like vortices. J. Atmos. Sci., 58, 2196–2209.
Kossin, J. P., B. D. McNoldy, and W. H. Schubert, 2002: Vortical swirls in hurricane eye clouds. Mon. Wea. Rev., 130, 3144–3149.
Lewis, B. M., and H. F. Hawkins, 1982: Polygonal eye walls and rainbands in hurricanes. Bull. Amer. Meteor. Soc., 63, 1294–1300.
Li, Q., and Y. Wang, 2012: A comparison of inner and outer spiral rainbands in a numerically simulated tropical cyclone. Mon. Wea. Rev., 140, 2782–2805.
Liu, Y., D.-L. Zhang, and M. K. Yau, 1999: A multiscale numerical study of Hurricane Andrew (1992). Part II: Kinematics and inner-core structures. Mon. Wea. Rev., 127, 2597–2616.
Martinez, Y. H., G. Brunet, and M. K. Yau, 2010: On the dynamics of two-dimensional hurricane-like vortex symmetrization. J. Atmos. Sci., 67, 3559–3580.
Martinez, Y. H., G. Brunet, M. K. Yau, and X. Wang, 2011: On the dynamics of concentric eyewall genesis: Space–time empirical normal modes diagnosis. J. Atmos. Sci., 68, 457–476.
McAdie, C., and M. B. Lawrence, 2000: Improvements in tropical cyclone track forecasting in the Atlantic basin, 1970–98. Bull. Amer. Meteor. Soc., 81, 989–998.
Menelaou, K., M. K. Yau, and Y. Martinez, 2012: On the dynamics of the secondary eyewall genesis in Hurricane Wilma (2005). Geophys. Res. Lett.,39, L04801, doi:10.1029/2011GL050699.
Menelaou, K., M. K. Yau, and Y. Martinez, 2013: Impact of asymmetric dynamical processes on the structure and intensity change of two-dimensional hurricane-like annular vortices. J. Atmos. Sci., 70, 559–582.
Molinari, J. S., S. Skubis, and D. Vollaro, 1995: External influences on hurricane intensity. Part III: Potential vorticity evolution. J. Atmos. Sci., 52, 3593–3606.
Montgomery, M. T., and L. J. Shapiro, 1995: Generalized Charney–Stern and Fjortoft theorems for rapidly rotating vortices. J. Atmos. Sci., 52, 1829–1833.
Montgomery, M. T., and R. J. Kallenbach, 1997: A theory for vortex Rossby-waves and its application to spiral bands and intensity changes in hurricanes. Quart. J. Roy. Meteor. Soc., 123, 435–465.
Montgomery, M. T., M. E. Nicholls, T. A. Cram, and A. B. Saunders, 2006: A vortical hot tower route to tropical cyclogenesis. J. Atmos. Sci., 63, 355–386.
Moon, Y., and D. S. Nolan, 2012: Spiral rainband propagation: Vortex-Rossby waves vs. tropical squall lines. Preprints, 30th Conf. on Hurricanes and Tropical Meteorology, Ponte Vedra Beach, FL, Amer. Meteor. Soc., 8C.8. [Available online at https://ams.confex.com/ams/30Hurricane/webprogram/Paper205457.html.]
Muramatsu, T., 1986: The structure of polygonal eye of a typhoon. J. Meteor. Soc. Japan, 64, 913–921.
Nguyen, V. S., R. K. Smith, and M. T. Montgomery, 2008: Tropical-cyclone intensification and predictability in three dimensions. Quart. J. Roy. Meteor. Soc., 134, 563–582.
Nolan, D. S., and B. F. Farrell, 1999: Generalized stability analyses of asymmetric disturbances in one- and two-celled vortices maintained by radial inflow. J. Atmos. Sci., 56, 1282–1307.
Nolan, D. S., and M. T. Montgomery, 2002: Nonhydrostatic, three-dimensional perturbations to balanced, hurricane-like vortices. Part I: Linearized formulation, stability, and evolution. J. Atmos. Sci., 59, 2989–3020.
Nolan, D. S., and L. D. Grasso, 2003: Nonhydrostatic, three-dimensional perturbations to balanced, hurricane-like vortices. Part II: Symmetric response and nonlinear simulations. J. Atmos. Sci., 60, 2712–2745.
Ooyama, K. V., 1982: Conceptual evolution of the theory and modeling of the tropical cyclone. J. Meteor. Soc. Japan, 60, 369–379.
Persing, J., M. T. Montgomery, J. C. McWilliams, and R. K. Smith, 2013: Asymmetric and axisymmetric dynamics of tropical cyclones. Atmos. Chem. Phys. Discuss., 13, 13 323–13 438, doi:10.5194/acpd-13-13323-2013.
Rozoff, C. M., J. P. Kossin, W. H. Schubert, and P. J. Mulero, 2009: Internal control of hurricane intensity variability: The dual nature of potential vorticity mixing. J. Atmos. Sci., 66, 133–147.
Schecter, D. A., and M. T. Montgomery, 2007: Waves in a cloudy vortex. J. Atmos. Sci., 64, 314–337.
Schubert, W. H., 1985: Wave, mean-flow interactions, and hurricane development. Preprints, 16th Conf. on Hurricanes and Tropical Meteorology, Houston, TX, Amer. Meteor. Soc., 140–141.
Schubert, W. H., M. T. Montgomery, R. K. Taft, T. A. Guinn, S. R. Fulton, J. P. Kossin, and J. P. Edwards, 1999: Polygonal eyewalls, asymmetric eye contraction and potential vorticity mixing in hurricanes. J. Atmos. Sci., 56, 1197–1223.
Wang, Y., 2002a: Vortex Rossby waves in a numerically simulated tropical cyclone. Part I: Overall structure, potential vorticity, and kinetic energy budgets. J. Atmos. Sci., 59, 1213–1238.
Wang, Y., 2002b: Vortex Rossby waves in a numerically simulated tropical cyclone. Part II: The role in tropical cyclone structure and intensity changes. J. Atmos. Sci., 59, 1239–1262.
Wang, Y., 2009: How do outer spiral rainbands affect tropical cyclone structure and intensity? J. Atmos. Sci., 66, 1250–1273.
Wang, Y., and C.-C. Wu, 2004: Current understanding of tropical cyclone structure and intensity changes—A review. Meteor. Atmos. Phys., 87, 257–278.
Willoughby, H. E., 1978a: A possible mechanism for the formation of hurricane rainbands. J. Atmos. Sci., 35, 838–848.
Willoughby, H. E., 1978b: The vertical structure of hurricane rainbands and their interaction with the mean vortex. J. Atmos. Sci., 35, 849–858.
Willoughby, H. E., J. A. Clos, and M. G. Shoreibah, 1982: Concentric eye walls, secondary wind maxima, and the evolution of the hurricane vortex. J. Atmos. Sci., 39, 395–411.
Yang, B., Y. Wang, and B. Wang, 2007: The effect of internally generated inner-core asymmetries on tropical cyclone potential intensity. J. Atmos. Sci., 64, 1165–1188.
Yau, M. K., Y. Liu, D.-L. Zhang, and Y. Chen, 2004: A multiscale numerical study of Hurricane Andrew (1992). Part VI: Small-scale inner-core structures and wind streaks. Mon. Wea. Rev., 132, 1410–1433.
Zadra, A., G. Brunet, and J. Derome, 2002: An empirical normal mode diagnostic algorithm applied to NCEP reanalysis. J. Atmos. Sci., 59, 2811–2829.
Zhu, P., K. Menelaou, and Z. Zhenduo, 2013: Impact of sub-grid scale vertical turbulent mixing on eyewall asymmetric structures and mesovortices of hurricanes. Quart. J. Roy. Meteor. Soc., doi:10.1002/qj.2147, in press.