1. Introduction
It is well known that secondary eyewall formation (SEF) is a common feature occurring during the mature stage of a tropical cyclone (TC). In SEF, a new eyewall forms outside the existing eyewall and an eyewall replacement cycle (ERC) often follows thereafter. During an ERC the secondary (or outer) eyewall gradually contracts while the inner eyewall weakens and eventually dissipates. Over the years, with the advances in satellite and radar observations as well as numerical modeling, double eyewalls and the corresponding ERCs have received significant attention (e.g., Willoughby et al. 1982; Black and Willoughby 1992; Houze et al. 2006, 2007; Sitkowski et al. 2011; Yang et al. 2013). In general, it has been revealed that an ERC usually halts the intensification of the TC, broadens the area of damaging gale-force winds, and even leads to TC weakening followed possibly by a period of reintensification. Subsequent to the replacement of the inner eyewall by the outer eyewall, another SEF and ERC could occur (Hawkins and Helveston 2008). Furthermore, the phenomenon of tertiary eyewall (although more rarely) has also been reported in the literature (e.g., McNoldy 2004; Zhao et al. 2016).
Despite the large number of papers on both SEF and ERCs in the past few decades, most of the published work focused on the fundamental mechanisms responsible for their initial formation. The proposed SEF mechanisms include vortex Rossby wave–mean flow interaction (e.g., Montgomery and Kallenbach 1997; Martinez et al. 2011; Menelaou et al. 2012, 2013), barotropic vorticity dynamics (Kuo et al. 2004, 2008), β-skirt axisymmetrization (Terwey and Montgomery 2008), dynamics within the atmospheric boundary layer (ABL) (e.g., Huang et al. 2012; Abarca and Montgomery 2013, 2014; Kepert 2013; Montgomery et al. 2014; Kepert and Nolan 2014; Wang et al. 2016), and the dynamics of mesoscale and convective-scale rainbands (e.g., Judt and Chen 2010; Rozoff et al. 2012; Zhu and Zhu 2014).
On the other hand, there are relatively few studies on the mechanism responsible for the dissipation of the inner eyewall (the completion of an ERC) and the subject remains not well understood. In some early studies, Willoughby et al. (1982) and Willoughby (1988) suggested that the outflow associated with the inner eyewall is counteracted by the forced upper-level inflow and descent from the outer eyewall leading to a disruption of the secondary circulation. However, this hypothesis was not supported by the findings of Rozoff et al. (2008) as the latter results (obtained by using the transverse circulation equation associated with a balanced vortex model) indicated that the decay of deep convection in the inner eyewall arises from a rapid increase of inner-core static stability resulting from the strengthening outer eyewall convection.
In a different context, another dissipation mechanism suggests that the development of an outer eyewall actively contributes to the inner eyewall demise by interrupting its low-level inflow (e.g., Samsury and Zipser 1995; Houze et al. 2007). This imposed barrier can have a twofold effect. First, it acts to reduce the boundary layer inflow and convergence underneath and just outside the inner eyewall. Such reduction decreases the inward transfer of angular momentum, which is a necessary ingredient for the maintenance of the inner eyewall against surface friction (Huang et al. 2012).1 Second, by suppressing the moist entropy-rich inflowing air, the barrier reduces the supply of moist entropy to the inner eyewall. As a result, the warm core of the inner eyewall cannot be maintained. According to the wind-induced surface heat exchange mechanism (WISHE; Emanuel 1986), a weaker warm core leads to weaker inflow, which in turn further weakens the warm core and so on. Zhou and Wang (2011) performed an axisymmetric equivalent potential temperature 
Besides the direct suppression of moist entropy supply due to the barrier effect, the moist entropy supply can also be indirectly reduced by downdrafts (linked to latent heat release in secondary eyewalls) within the moat region. These downdrafts can transport low 
It should be noted, that the mechanisms described above are all axisymmetric. In the most recent decade, more attention has been directed to the asymmetric aspect of ERCs. Didlake et al. (2017) analyzed observation obtained during the ERCs of Hurricane Gonzalo and found evidence of descent from the outer eyewall overriding the updraft of the decaying inner eyewall in the downshear quadrants. The inference is that the aforementioned suppression mechanism of inner eyewall outflow suggested by Willoughby et al. (1982) and Willoughby (1988) could work in concert with the decreasing moist entropy supply to weaken the inner eyewall. However, Tsujino et al. (2017) showed from their simulation of Typhoon Bolaven (2012) that the asymmetric structure of the outer eyewall had a significant impact on the contraction rate of the outer eyewall. Moreover, their moist entropy budget and backward trajectory analyses indicated that the inward moist entropy supply within the boundary layer was still sufficient to maintain the long-lived inner eyewall (maintained for more than 20 h; defined by Yang et al. 2013) of the simulated Bolaven. This indicates that boundary layer effects could play an important role in the inner eyewall dissipation. A similar finding based on the observational analysis of Hurricane Ike (2008) was reported by Zhang and Perrie (2018). Ike’s double eyewalls were maintained for more than 30 h.
Barotropic instabilities have been found to be associated with the double-eyewall phenomenon. Kossin et al. (2000, hereafter KSM00) identified two types of barotropic instability in the vorticity field of TCs with a double-eyewall structure. The first is the instability across the outer eyewall (referred to as type-1 instability) and the second is the instability across the moat (referred to as type-2 instability). These instabilities can yield asymmetric features in the evolution of the vorticity field. For example, in a composite study of TCs having multiple eyewalls2 in the western North Pacific basin from 1997 to 2011, Yang et al. (2013) conjectured that barotropic instabilities might play a role in determining the maintenance time of the multiple-eyewall structures. However, their conjecture is based on the statistics of the observed size of the outer eyewall and the width of the moat. They provided no direct evidence for the presence of the instabilities nor a detailed study of the underlying dynamics. Nevertheless, radar imagery of some TCs with double eyewalls (McNoldy 2012) did show asymmetric features in the inner eyewalls (especially elliptical shapes) for a period of time prior to their dissipation [e.g., Hurricane Earl (2010)3 and Hurricane Maria (2017)4]. Since the elliptical evolution could in principle be explained by the type-2 barotropic instability proposed by KSM00, the observed features from the radar imagery may imply a possible role played by the instability in inner eyewall dissipation. It is one of the objectives of the present study to validate or refute such a hypothesis.
Specifically, the aim of our study is to provide insight on the fundamental mechanisms responsible for inner eyewall dissipation. We attempt to determine whether barotropic instability that operates across the moat region is a suitable candidate that can partly explain the inner eyewall dissipation. In this first part of the study, the analysis will be based on a numerical investigation of Hurricane Wilma (2005). Wilma was the most intense hurricane recorded in the Atlantic basin, and it had two ERCs with the first one lasting for about 24 h (Chen et al. 2011). Our focus will be on the first ERC of Wilma.
The next section presents a brief overview of Hurricane Wilma. Section 3 provides the configurations of the numerical model and the simulation. Section 4 contains a description of some simulated inner-core structures and their evolution during Wilma’s first ERC. Section 5 discusses these results with particular focus on the relation between barotropic instability across the moat and the dissipation of the inner eyewall using the tools of linear stability analysis and nonlinear numerical experiments with a 2D nondivergent barotropic vorticity model. The conclusions are given in section 6.
2. Brief overview of Hurricane Wilma
Hurricane Wilma (2005), which formed over the Caribbean Sea, was the most intense Atlantic hurricane and the second-most intense TC recorded in the Western Hemisphere [after Hurricane Patricia (2015; Kimberlain et al. 2016)]. Its origin could be traced back to an unusually broad monsoon-like lower-tropospheric trough over much of the Caribbean Sea as early as 0000 UTC 11 October (Pasch et al. 2006; Chen et al. 2011). This trough was then split into two parts, with the southern one developing into a tropical depression (TD) near Jamaica by 1800 UTC 15 October when the surface circulation was well defined and the deep convection was sufficiently organized. In the subsequent few days, this TD was embedded in a region between a subtropical high over Mexico and the Gulf of Mexico, and another subtropical high over the North Atlantic Ocean. This synoptic configuration produced a weak steering flow, which drove the TD westward to west–southwestward for 1 day and then south–southwestward to southward for another 2 days. At 0600 UTC 17 October, the TD was upgraded to a tropical storm (TS) and named Wilma.
By early 18 October, Wilma’s strengthening intensity reached hurricane level as it turned west–northwestward. Starting from 1800 UTC on the same day, Wilma underwent a remarkably rapid intensification (RI) process for a 12-h period until 0600 UTC 19 October. During this RI period, Wilma’s central sea level pressure (SLP) dropped by 29 hPa in the first 6 h and by 54 hPa in the second 6 h (Fig. 1b). The minimum central SLP and the 1-min maximum sustained surface winds were 892 hPa and 150 kt (1 kt ≈ 0.51 m s−1; category 5 on the Saffir–Simpson hurricane wind scale), respectively, just after RI. During the RI stage, the U.S. Air Force reconnaissance observations indicated that Wilma’s eye contracted to a diameter of 2 n mi (~3.7 km), which is the smallest eye known to the National Hurricane Center staff (Pasch et al. 2006). The intensification continued for another 6 h. Wilma reached its peak intensity at 1200 UTC 19 October, with the 1-min maximum sustained surface winds of 160 kt, and the minimum central SLP of 882 hPa (Fig. 1b) which broke the Atlantic record of 888 hPa set by Hurricane Gilbert in 1988 (Lawrence and Gross 1989; Willoughby et al. 1989).
Comparison between the WRF-simulated and observed Hurricane Wilma (2005) during the 72-h period of 0000 UTC 18 Oct–0000 UTC 21 Oct 2005. The observation refers to the NHC best track data. (a) The 6-h observation (green circles) and the simulated track (blue stars). (b) Time series of observed TC central sea level pressure (green circles) compared with the simulation values (blue solid curve), and time series of observed TC maximum surface wind speed (red circles) compared with the simulation values (blue dotted curve). The plotting frequency for the simulation output is 30 min.
Citation: Journal of the Atmospheric Sciences 76, 4; 10.1175/JAS-D-18-0191.1
During the subsequent 24 h, Wilma weakened from 882 to 910 hPa because of an ERC. Then Wilma underwent another ERC before it made landfall on Cozumel Island (2145 UTC 21 October) and then the northeastern Yucatan Peninsula. Consequently, Wilma weakened further to 960 hPa over a 60-h period from the completion of the first ERC, with 1-min maximum sustained surface winds of 85 kt (0000 UTC 23 October). Around that time, Wilma turned north and arrived at the southern Gulf of Mexico. Subsequently, Wilma reintensified over the southeastern Gulf of Mexico when it turned to move northeastward, and then made a third landfall over southwestern Florida near Cape Romano around 1030 UTC 24 October with 1-min maximum sustained surface winds of 105 kt. For more details, interested readers are referred to Pasch et al. (2006) and Chen et al. (2011).
3. The numerical model and the simulation
The simulation of Hurricane Wilma is performed using the Weather Research and Forecasting (WRF) Model, version 3.8.1 (Skamarock et al. 2008). WRF is a state-of-the-art three-dimensional full-physics compressible primitive equation system that has been widely used in the literature. The model is configured on a quadruply nested grid with two-way interaction. Only the innermost domain follows the TC vortex. The outermost domain consists of 140 × 140 horizontal grid points with a grid spacing of 27 km. The nesting ratio is 3:1, such that the innermost domain has a grid spacing of 1 km. For the 9-, 3-, and 1-km domains, the corresponding numbers of grid points are 286 × 286, 484 × 484, and 346 × 346, respectively. In the vertical direction, 30 σ levels are configured. The major model physics include the Thompson scheme for microphysics (Thompson et al. 2008), the Betts–Miller–Janjić cumulus parameterization scheme for deep convection (in the two outer domains only; Janjić 1994, 2000), the Mellor–Yamada–Janjić (Eta) TKE scheme for the planetary boundary layer (Janjić 1990, 1996, 2002), the RRTM scheme for longwave radiation (Mlawer et al. 1997), and the GSFC scheme for shortwave radiation (Chou and Suarez 1994). The initial and boundary conditions are supplied by the Geophysical Fluid Dynamics Laboratory (GFDL) hurricane model forecast data, and the surface conditions by the National Centers for Environmental Prediction (NCEP) Global Forecast System (GFS) final (FNL) analysis data. The constant sea surface temperature (SST) is given by the NCEP real-time global SST data with 0.5° resolution at 0000 UTC 18 October 2005. The simulation is initialized at 0000 UTC 18 October 2005, and runs for 72 h. It is worth mentioning that Wilma has been previously simulated by Chen et al. (2011), Menelaou et al. (2012), and Gadoury (2012).
4. Simulation results
Figure 1 displays the model-predicted storm track and intensity of Wilma together with the 6-hourly best track and the observed intensity issued by the National Hurricane Center (NHC; Pasch et al. 2006). It is clear that there is a northeastward bias in the simulated track in the late stages (approximately 185 km too far to the north–northeast of the best track at the end of the simulation; see Fig. 1a). However, the model reproduces the observed northwestward movement quite well. Note that this kind of track bias was also present in the simulations of Chen et al. (2011), Menelaou et al. (2012), and Gadoury (2012). Worthy of note is that the predicted and the observed evolution of central SLP show very good agreement, both in the trend and in the peak values (Fig. 1b). The discrepancy in maximum wind speed between the observation and simulation is larger, but the predicted maximum wind speed still captures the time of peak intensity. More specifically, the initial spinup, RI, and subsequent weakening stages are all captured by the model. Although the simulated RI started about 6 h earlier than the observation, and with a less intense deepening rate, the predicted peak intensity and time of occurrence are consistent with the best track.
Perhaps the most significant aspect of this simulation is that the model successfully captures the first SEF and the subsequent ERC, leading to the weakening of Wilma (see Fig. 2 for the radar reflectivity evolution at z = 2 km). The simulated ERC lasted about 27 h (from 1800 UTC 19 October to 2100 UTC 20 October).
The radar reflectivity (dBZ) of the simulated Wilma at z = 2 km at t = (a) 42, (b) 59.5, (c) 62, (d) 64, (e) 66, and (f) 69 h. The distance is with respect to the center of the innermost simulation domain.
Citation: Journal of the Atmospheric Sciences 76, 4; 10.1175/JAS-D-18-0191.1
The outer eyewall of the actual Wilma (inferred from satellite microwave imagery)5 formed around 1200 UTC 19 October and the inner eyewall dissipated around 1100 UTC 20 October.6 Note that during this time period, no land-based or airborne radar observations were available. Thus, relying solely on the relatively low-resolution satellite microwave imagery, we were unable to determine the degree of similarity between the simulated ERC and that of the real Wilma.
To set the stage for our analyses, it is appropriate to first examine the evolution of Wilma’s secondary eyewall after it has been established. It should be noted that an eyewall is associated not only with the radar reflectivity maximum, but also the relative vorticity ζ maximum. As such, the evolution of both fields at three levels will be examined: the “lower levels,” “middle levels,” and “upper levels” refer respectively to the ranges from z = 800 m to 2.5 km, from z = 2.5 to 7.5 km, and from z = 7.5 to 11 km. These ranges are determined subjectively from the evolution patterns of ζ, with layers having analogous patterns categorized into the same group (not shown).7 Within the lower levels, the evolution and orientation of the elliptic inner eyewall is very similar. Moving up into the middle levels, this elliptical pattern gradually (but steadily) weakens and disappears. From the bottom layer of the upper levels, the trochoidal oscillation of the inner eyewall becomes much more obvious. Note that at this stage, the analyses exclude any contribution from the boundary layer [z < 800 m; in terms of the inflow layer depth as defined in Zhang et al. (2011)]. That being said, the extent to which the outcome of our current examination remains significant (or is modified) in the presence of a boundary layer requires further investigation. This additional investigation will be explored and reported in a separate paper.
Figure 3 shows the vorticity field averaged over the lower levels at six times starting from the time of SEF (t ≈ 42.0 h; Fig. 3a). This SEF time is subjectively determined as the start time from which a complete outer ring of radar reflectivity emerges along with a clear and complete secondary ring of vorticity. It can be seen that after SEF, the outer eyewall begins to contract gradually (see also Fig. 2). Of interest is that by t = 59.5 h (Fig. 3c), the inner eyewall has deformed into a distinct elliptical structure. Subsequently, the deformation continues to amplify, resulting in an inner eyewall with a pronounced elliptical vorticity distribution. Eventually, the elongated inner eyewall vorticity patch touches the outer eyewall vorticity ring from t ≈ 63.5 h (e.g., Fig. 3e). Note that, during this period of elliptical elongation, the magnitude of the inner eyewall vorticity decreases (i.e., the color becomes paler) and the size of the inner eyewall vorticity patch largely decreases. (Figure 13, which will be discussed in section 5, gives a clearer picture of the vorticity evolution after SEF.) Another salient feature is the oscillating wobbles exhibited by the inner eyewall. This is the primary reason as to why the location of the eye appears to deviate significantly from the geometric center of the outer eyewall [Figs. 2 and 3; readers are referred to Menelaou et al. (2018) for a discussion of a possible mechanism for oscillating wobbles].
The relative vorticity (×10−3 s−1) of the simulated Wilma averaged over the lower levels 
Citation: Journal of the Atmospheric Sciences 76, 4; 10.1175/JAS-D-18-0191.1
Evolution of radial flow and a possible dilution effect
As mentioned, the period of elliptical deformation coincides with a simultaneous reduction of the inner eyewall vorticity. Based on this, it is reasonable to hypothesize a possible connection between the decay of the inner eyewall and this dominant asymmetry. To elaborate on the process of inner eyewall decay, the evolution of the radial flow at lower levels (based on Earth-relative winds) is shown in Fig. 4. After SEF, the radial velocity exhibits an azimuthal wavenumber-1 pattern. With time, and as the moat width decreases to a certain extent, the dominant pattern of the radial flow switches to a wavenumber-2 structure (at t = 59.5 h; Fig. 4c). This wavenumber-2 mode appears to be important as it represents the normal strain flow pattern which could lead to the dilution of the vorticity within the eye and the inner eyewall (the inner vorticity patch): there are a pair of radial outflow regions and a pair of radial inflow regions (e.g., Fig. 4d). By looking at Figs. 3 and 4 together, it is clear that the radial outflow pair acts to elongate the inner vorticity patch thereby inserting high-ζ air of the elongated part to the moat. Given that the elliptically elongated inner vorticity patch has significant cyclonic rotating motion relative to the moat, the ζ at the two ends of the inner vorticity patch can filament rapidly into small radial scales as they become axisymmetrized. The reason is that a moat is generally a region of active potential enstrophy cascade to small scales because of the presence of severe differential rotation (KSM00, and references therein). Meanwhile, the radial inflow pair acts to advect the low-ζ air from the moat into the region of the eye and the inner eyewall. This inward advection, if not short lived, can also greatly dilute the inner vorticity patch by mixing with the low-ζ air from the moat. Since the dominant wavenumber-2 pattern is retained for a long time, the pairs of radial inflow and outflow result in substantial dilution of the inner vorticity patch (i.e., the outer part of the inner vorticity patch largely mixes with the moat air, and therefore the area of inner vorticity patch decreases significantly). (For a more direct illustration of this dilution mechanism, readers are referred to section 5c and the schematic diagram in Fig. 20.) The discussion of the dynamics behind the wavenumber-2 pattern is presented in the next section.
The low-level radial velocity (m s−1) of the simulated Wilma averaged over the lower levels 
Citation: Journal of the Atmospheric Sciences 76, 4; 10.1175/JAS-D-18-0191.1
5. Analyses and discussion
a. Linear stability analysis
As previously mentioned, KSM00 performed a linear stability analysis for a TC-like vortex with a double-eyewall structure. For their purpose, the system was idealized as a simple piecewise constant four-region model with a monopolar inner eyewall (region 1), the moat region (region 2), an annular secondary eyewall (region 3), and a far-field region with zero vorticity (region 4). We extended KSM00’s model into a five-region model by replacing the monopolar inner eyewall by an annular inner eyewall similar to the simulated Wilma (see Figs. 2 and 3). Since a necessary condition for barotropic instability (Rayleigh’s inflection point criterion) is that the basic-state vorticity gradient changes sign somewhere in the domain, it is clear that barotropic instability can be potentially excited in three regions throughout the vortex: one across the inner ring, one across the moat, and one across the outer ring.
The profile used for the linear stability analysis. The red curve represents the axisymmetric profile of simulated Wilma’s relative vorticity at t = 59.0 h, which is approximated and simplified by the corresponding piecewise constant function (gray shading). The values used in this five-region approximation are 
Citation: Journal of the Atmospheric Sciences 76, 4; 10.1175/JAS-D-18-0191.1
The dimensionless vortex parameters of the simulated Wilma at t = 59.0 h.
The corresponding solution for the case described in Fig. 5 is demonstrated in Figs. 6 and 7 in terms of the maximum value of the dimensionless growth rate 
Isolines of the maximum value of the dimensionless growth rate 
Citation: Journal of the Atmospheric Sciences 76, 4; 10.1175/JAS-D-18-0191.1
Isolines of the maximum value of the dimensionless growth rate 
Citation: Journal of the Atmospheric Sciences 76, 4; 10.1175/JAS-D-18-0191.1
Isolines of the maximum value of the dimensionless growth rate 
Citation: Journal of the Atmospheric Sciences 76, 4; 10.1175/JAS-D-18-0191.1
Isolines of the maximum value of the dimensionless growth rate 
Citation: Journal of the Atmospheric Sciences 76, 4; 10.1175/JAS-D-18-0191.1
To summarize, at the time just before the low-level radial flow switches from an m = 1 to an m = 2 pattern, the simulated Wilma’s vortex structure is most unstable to an m = 2 type-2 instability.
Before advancing, it is emphasized that the above analyses are linear and incorporate simplified step functions. In the following subsection, the dominance of type-2 instability in the simulated Wilma is further verified with nonlinear nondivergent barotropic vorticity experiments which consider continuous vorticity profiles.
b. Nonlinear nondivergent barotropic experiments
1) The model
2) The experiments
Before proceeding, recall that after Wilma’s secondary eyewall has been established at t ≈ 42.0 h, the dominant low-level radial flow pattern switches from m = 1 to m = 2 at t = 59.5 h.8 The transition implies that during the time just before the transition, Wilma’s low-level ζ structure might have been subjected to the m = 2 barotropic instability excited across the moat region. To verify whether this is a reasonable hypothesis, a group of two experiments is conducted using the NDBV model. In the first experiment (denoted by experiment A1), the model is initialized with Wilma’s axisymmetric low-level vorticity profile (following the abovementioned process) 6 h before the m = 2 transition (at t = 53.5 h). On the other hand, in the second experiment (denoted by experiment A2) the model is initialized with Wilma’s vorticity profile only 30 min before the transition. The initial axisymmetric vorticity profiles of all NDBV experiments are shown in Fig. 10.
The initial axisymmetric ζ profiles of all NDBV experiments: A1 (solid blue), A2 (dotted blue), B1 (solid red), B2 (dotted red), C1 (solid green), and C2 (dotted green).
Citation: Journal of the Atmospheric Sciences 76, 4; 10.1175/JAS-D-18-0191.1
Figure 11 depicts the results of experiment A1 in the form of total vorticity maps. From Fig. 11a, it can be seen that the m = 2 perturbation across the moat becomes obvious at tNDBV = 9.0 h. The inner core (inner vorticity patch) becomes slightly elliptic at tNDBV = 12.0 h. As per Fig. 11b, there is only weakly exponential growth with growth rate 
NDBV experiment A1. (a) Relative vorticity ζ (shading) and its perturbation 
Citation: Journal of the Atmospheric Sciences 76, 4; 10.1175/JAS-D-18-0191.1
As in Fig. 11, but for NDBV experiment A2. (a) Note that black contours are not plotted in the bottom panels since 
Citation: Journal of the Atmospheric Sciences 76, 4; 10.1175/JAS-D-18-0191.1
c. Evolution of the strength of the inner vorticity patch
Although we have established that the m = 2 type-2 barotropic instability was indeed excited in the simulated Hurricane Wilma, it remains to be determined whether its long-term nonlinear effect contributed to the weakening (and eventually the dissipation) of the inner vorticity patch (or the inner eyewall). To this end, we examine the time series (after SEF) of the circulation of the inner vorticity patch 
The calculation of 
Figure 13 shows the time evolution of 
Time series of the circulation (km2 s−1) of the inner vorticity patch 
Citation: Journal of the Atmospheric Sciences 76, 4; 10.1175/JAS-D-18-0191.1
d. Evolution of the inner vorticity patch at the middle and upper levels
Returning to Fig. 13, it is evident that in contrast to the lower levels, the vorticity at the middle and upper levels did not experience a second period of faster weakening. To investigate this difference, we examine the radial flows at the middle and upper levels in Figs. 14 and 15 (the corresponding low-level field is shown in Fig. 4). At the middle levels at t = 59.5 h (Fig. 14a), the radial flow is characterized by an m = 2 structure. However, this feature appears short lived as by t = 60.5 h, the dominant wavenumber pattern switches to m = 1 which remains the significant asymmetry until the end of the simulation. At the upper levels, the flow is characterized by an m = 1 pattern all the time.
The radial velocity (m s−1) of the simulated Wilma averaged over the middle levels 
Citation: Journal of the Atmospheric Sciences 76, 4; 10.1175/JAS-D-18-0191.1
As in Fig. 14, but for the upper levels 
Citation: Journal of the Atmospheric Sciences 76, 4; 10.1175/JAS-D-18-0191.1
To explain why the m = 2 pattern fails to be dominant at the middle and upper levels, two groups of additional experiments will be conducted with the NDBV model initialized with a number of vorticity profiles extracted from the mid- (group B experiments) and upper levels (group C experiments) of the WRF simulation. Each group consists of two independent experiments. Within group B, experiment B1 is initialized with an axisymmetric vorticity profile when the dominant radial flow structure transitions from m = 1 to m = 2 (at t = 59.0 h). Experiment B2 is initialized with a vorticity profile right after the flow transitions back to m = 1 (at t = 60.0 h).
Figures 16a and 17a depict respectively the B1 and B2 total and perturbation vorticity at selected times. In B1, the vortex retains its axisymmetric structure for around 7 h. By tNDBV = 9.0 h, a clear m = 2 asymmetry has been excited (Fig. 16a). This characteristic vortex deformation implies that the WRF vorticity profile at the middle levels is susceptible to a type-2 instability. Despite this, there is a large difference in the growth rate between B1 and A2 (in which an m = 2 asymmetry becomes discernible by tNDBV = 3.0 h). The estimated e-folding time from the fitted exponential curve for A2 (the red curve in Fig. 12b) is about 0.7 h while that for B1 (the red curve in Fig. 16b) is much longer—around 6.3 h. This different growth rate indicates that the vortex at the middle levels is significantly more stable relative to its low-level counterpart. Additionally, the vortex in B2 becomes slightly asymmetric at tNDBV = 12.0 h with a slower growth rate (longer e-folding time of 9.0 h). Taken together, the group B experiments indicate that the midlevel vorticity profile of the simulated Wilma is marginally stable and hence a dominant m = 2 radial flow pattern cannot be sustained.
NDBV experiment B1. (a) Relative vorticity ζ (shading) and its perturbation 
Citation: Journal of the Atmospheric Sciences 76, 4; 10.1175/JAS-D-18-0191.1
As in Fig. 16, but for NDBV experiment B2.
Citation: Journal of the Atmospheric Sciences 76, 4; 10.1175/JAS-D-18-0191.1
Regarding group C (figure not shown), experiment C1 is initialized with an upper-level vorticity profile extracted from the WRF simulation at t = 59.0 h (the selected time is chosen to match the one in B1). Experiment C2 is initialized with a vorticity profile extracted a few hours later (t = 63.5 h). In C1, until tNDBV = 12.0 h the flow does not develop obvious m = 2 asymmetry (larger e-folding time relative to experiment B2). Comparing this outcome with C2, it appears that an m = 2 asymmetry develops much faster in the latter experiment (e-folding time of 4.1 h). Nevertheless, here the instability remains much slower than that in experiment A2 (e-folding time of 0.7 h). It is therefore suggested that at upper levels, the m = 2 instability may not be sufficiently intense to develop an equivalent asymmetric radial flow pattern. Note that the NDBV framework does not support the excitation of an exponential m = 1 instability. As a result, the m = 1 pattern (see Fig. 15) does not emerge in these experiments.
In summary, the above results suggest that the absence of a second (and faster) period of 
e. Thermodynamic effect
Another effect due to the mixing of air induced by barotropic instability of the vortex is the modification of its thermodynamic structure. In a TC with a single eyewall, Persing and Montgomery (2003) argued that mixing between the eye and the eyewall, perhaps induced by the nonlinear stirring from VRWs at the inner edge of the eyewall, can stir higher 
Figure 18 presents the evolution of low-level equivalent potential temperature 
Equivalent potential temperature 
Citation: Journal of the Atmospheric Sciences 76, 4; 10.1175/JAS-D-18-0191.1
Time series of the sum of equivalent potential temperature (K km2) of the inner vorticity patch 
Citation: Journal of the Atmospheric Sciences 76, 4; 10.1175/JAS-D-18-0191.1
In passing, we mention that similar to the evolution of low-level 
It should be cautioned that the thermodynamic effect described above is incomplete because the changes in vertical motion or convection, which are out of the scope of this paper, can have substantial effects on the inner eyewall strength. For instance, it is possible that the low-level radial outflow pair during the occurrence of type-2 instability induces nonnegligible downdrafts (if the divergence of outflow pair exceeds the convergence of inflow pair), which counteract the convective updrafts within the inner eyewall. The resultant weaker updrafts may yield a weaker latent heat release and a weaker warm core, unfavorable for the maintenance of the strength of the TC vortex. In addition, both ABL physics and the effect of diabatic heating are excluded in the theoretical framework of the five-region model developed from the work of KSM00. Therefore, the effects of the moist entropy distribution and convection in a TC with a double eyewall on type-2 instability could not be identified in the current paper. Further work would be required to shed light on the interaction between the type-2 instability, convection, and the thermodynamic structure of a TC with a double eyewall.
6. Conclusions
In TCs with multiple eyewalls, the problem of how the inner eyewall dissipates remains an open question. Some recent radar observations indicate that the inner eyewalls become elliptical prior to their dissipation. It is conjectured that barotropic instability across the moat (see KSM00; aka type-2 barotropic instability) may play a role in this process. To investigate this problem, a 72-h high-resolution WRF simulation of Hurricane Wilma (2005) was performed. The simulation reproduced reasonably well the track, initial spinup, RI, peak intensity, and subsequent weakening of Wilma. However, the simulated track is about 185 km too far to the north–northeast of the best track at the end of the 72-h simulation. Of importance is that the simulation captures Wilma’s first ERC, with similar duration.
The simulation results indicate that the low-level eyewall became more and more elliptical for a 10-h period prior to its dissipation. During this period, the radial flow changed from a dominant pattern with wavenumber m = 1 to m = 2, which is suggestive of the emergence of a type-2 barotropic instability mode with m = 2. To ascertain this process in a TC with a double eyewall, a linear stability analysis in the context of a 2D nondivergent barotropic vorticity model was performed. Specifically, the four-region model of KSM00 was extended to a five-region model and a type-2 barotropic instability was obtained. The linear results are supported by nonlinear computations using a nonlinear nondivergent barotropic vorticity (NDBV) model. By initializing the NDBV model with the axisymmetric vorticity profiles from the WRF simulation before the wavenumber changed from m = 1 to m = 2, it was confirmed that the initialized vorticity profiles were susceptible to the m = 2 type-2 barotropic instability.
Of particular importance is the finding of how this instability results in the dissipation of the inner eyewall. The proposed mechanism is presented in a schematic diagram (Fig. 20). The radial flow associated with the dominant m = 2 mode of a type-2 barotropic instability has a normal strain pattern. If this pattern lasts for a long time, then the outer part of the inner vorticity patch would be greatly diluted (mainly in the form of shrinkage of the inner vorticity patch)—the inner eyewall significantly decays—because of the rapid filamentation at the two ends of the elliptically elongated inner vorticity patch associated with the cyclonic differential rotation (with respect to the moat), as well as the intrusion of low-ζ air from the moat which is associated with the radial inflow pair. To provide more direct evidence, the time series of 
A schematic diagram of the dilution of vorticity of the inner eyewall and the eye by mixing with the lower-vorticity air in the moat, due to the wavenumber-2 radial flow pattern associated with type-2 barotropic instability. Red arrows indicate the radial outflow pair elliptically elongating the inner vorticity patch while blue arrows indicate the radial inflow pair transporting low ζ into the inner vorticity patch. The dark gray curved arrow indicates the cyclonic rotation (relative to the moat) of the inner vorticity patch.
Citation: Journal of the Atmospheric Sciences 76, 4; 10.1175/JAS-D-18-0191.1
To find out the reason why there was no such second period of significant decrease in 
Another finding is that the type-2 instability can also have detrimental effects on the intensity of the simulated Wilma by modifying the low-level thermodynamic structure of the storm through the mixing of lower 
In closing, it is cautioned that the presented analyses focused on exploring a possible pathway to the dissipation of the inner eyewall without taking the frictional boundary layer into consideration. However, boundary layer effects could play an important role in the inner eyewall dissipation (Huang et al. 2012; Kepert 2013; Montgomery et al. 2014; Kepert and Nolan 2014; Tsujino et al. 2017; Zhang and Perrie 2018). The extent to which the possible contribution from type-2 instability remains significant (or is modified) in the presence of a boundary layer will be explored and reported in a follow-up paper.
The GFDL hurricane model data used for the simulation were kindly provided by Timothy Marchok at NOAA/GFDL. We thank Peter Bartello for providing the nondivergent barotropic spectral model. We also thank Michael T. Montgomery and two anonymous reviewers for their constructive comments that improved the manuscript. This research reported here is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC)/Hydro-Québec Industrial Research Chair (IRC) program. The simulation was performed on the supercomputer Graham supported by Compute Canada (www.computecanada.ca).
APPENDIX
The Matrix 
This appendix provides the details of solving the eigenvalue problem (9) and of the associated matrix operator 
REFERENCES
Abarca, S. F., and M. T. Montgomery, 2013: Essential dynamics of secondary eyewall formation. J. Atmos. Sci., 70, 3216–3230, https://doi.org/10.1175/JAS-D-12-0318.1.
Abarca, S. F., and M. T. Montgomery, 2014: Departures from axisymmetric balance dynamics during secondary eyewall formation. J. Atmos. Sci., 71, 3723–3738, https://doi.org/10.1175/JAS-D-14-0018.1.
Barnes, G. M., E. J. Zipser, D. Jorgensen, and F. Marks Jr., 1983: Mesoscale and convective structure of a hurricane rainband. J. Atmos. Sci., 40, 2125–2137, https://doi.org/10.1175/1520-0469(1983)040<2125:MACSOA>2.0.CO;2.
Bartello, P., and T. Warn, 1996: Self-similarity of decaying two-dimensional turbulence. J. Fluid Mech., 326, 357–372, https://doi.org/10.1017/S002211209600835X.
Bister, M., and K. A. Emanuel, 1998: Dissipative heating and hurricane intensity. Meteor. Atmos. Phys., 65, 233–240, https://doi.org/10.1007/BF01030791.
Black, M. L., and H. E. Willoughby, 1992: The concentric eyewall cycle of Hurricane Gilbert. Mon. Wea. Rev., 120, 947–957, https://doi.org/10.1175/1520-0493(1992)120<0947:TCECOH>2.0.CO;2.
Bryan, G. H., and R. Rotunno, 2009: The influence of near-surface, high-entropy air in hurricane eyes on maximum hurricane intensity. J. Atmos. Sci., 66, 148–158, https://doi.org/10.1175/2008JAS2707.1.
Chen, H., D.-L. Zhang, and J. Carton, 2011: On the rapid intensification of Hurricane Wilma (2005). Part I: Model prediction and structural changes. Wea. Forecasting, 26, 885–901, https://doi.org/10.1175/WAF-D-11-00001.1.
Chou, M.-D., and M. J. Suarez, 1994: An efficient thermal infrared radiation parameterization for use in general circulation models. NASA Tech. Memo. 104606, Vol. 3, 85 pp.
Didlake, A. C., Jr., G. M. Heymsfield, P. D. Reasor, and S. R. Guimond, 2017: Concentric eyewall asymmetries in Hurricane Gonzalo (2014) observed by airborne radar. Mon. Wea. Rev., 145, 729–749, https://doi.org/10.1175/MWR-D-16-0175.1.
Eastin, M. D., P. G. Black, and W. M. Gray, 2002a: Flight-level thermodynamic instrument wetting errors in hurricanes. Part I: Observations. Mon. Wea. Rev., 130, 825–841, https://doi.org/10.1175/1520-0493(2002)130<0825:FLTIWE>2.0.CO;2.
Eastin, M. D., P. G. Black, and W. M. Gray, 2002b: Flight-level thermodynamic instrument wetting errors in hurricanes. Part II: Implications. Mon. Wea. Rev., 130, 842–851, https://doi.org/10.1175/1520-0493(2002)130<0842:FLTIWE>2.0.CO;2.
Emanuel, K. A., 1986: An air–sea interaction theory for tropical cyclones. Part I: Steady-state maintenance. J. Atmos. Sci., 43, 585–604, https://doi.org/10.1175/1520-0469(1986)043<0585:AASITF>2.0.CO;2.
Emanuel, K. A., 1988: The maximum intensity of hurricanes. J. Atmos. Sci., 45, 1143–1155, https://doi.org/10.1175/1520-0469(1988)045<1143:TMIOH>2.0.CO;2.
Emanuel, K. A., 1995: Sensitivity of tropical cyclones to surface exchange coefficients and a revised steady-state model incorporating eye dynamics. J. Atmos. Sci., 52, 3969–3976, https://doi.org/10.1175/1520-0469(1995)052<3969:SOTCTS>2.0.CO;2.
Gadoury, J., 2012: Impact of numerical grid spacing and time step size on vortex Rossby waves in secondary eyewall formation in a simulation of hurricane Wilma (2005). M.S. thesis, Dept. of Atmospheric and Oceanic Sciences, McGill University, 115 pp.
Hawkins, J. D., and M. Helveston, 2008: Tropical cyclone multiple eyewall characteristics. 28th Conf. on Hurricanes and Tropical Meteorology, Orlando, FL, Amer. Meteor. Soc., 14B.1, https://ams.confex.com/ams/28Hurricanes/webprogram/Paper138300.html.
Hazelton, A. T., and R. E. Hart, 2013: Hurricane eyewall slope as determined from airborne radar reflectivity data: Composites and case studies. Wea. Forecasting, 28, 368–386, https://doi.org/10.1175/WAF-D-12-00037.1.
Hendricks, E. A., and W. H. Schubert, 2009: Transport and mixing in idealized barotropic hurricane-like vortices. Quart. J. Roy. Meteor. Soc., 135, 1456–1470, https://doi.org/10.1002/qj.467.
Hoskins, B. J., M. E. McIntyre, and A. W. Robertson, 1985: On the use and significance of isentropic potential vorticity maps. Quart. J. Roy. Meteor. Soc., 111, 877–946, https://doi.org/10.1002/qj.49711147002.
Houze, R. A., Jr., and Coauthors, 2006: The Hurricane Rainband and Intensity Change Experiment: Observations and modeling of Hurricanes Katrina, Ophelia, and Rita. Bull. Amer. Meteor. Soc., 87, 1503–1521, https://doi.org/10.1175/BAMS-87-11-1503.
Houze, R. A., Jr., S. S. Chen, B. F. Smull, W.-C. Lee, and M. M. Bell, 2007: Hurricane intensity and eyewall replacement. Science, 315, 1235–1239, https://doi.org/10.1126/science.1135650.
Huang, Y.-H., M. T. Montgomery, and C.-C. Wu, 2012: Concentric eyewall formation in Typhoon Sinlaku (2008). Part II: Axisymmetric dynamical processes. J. Atmos. Sci., 69, 662–674, https://doi.org/10.1175/JAS-D-11-0114.1.
Janjić, Z. I., 1990: The step-mountain coordinate: Physical package. Mon. Wea. Rev., 118, 1429–1443, https://doi.org/10.1175/1520-0493(1990)118<1429:TSMCPP>2.0.CO;2.
Janjić, Z. I., 1994: The step-mountain eta coordinate model: Further developments of the convection, viscous sublayer, and turbulence closure schemes. Mon. Wea. Rev., 122, 927–945, https://doi.org/10.1175/1520-0493(1994)122<0927:TSMECM>2.0.CO;2.
Janjić, Z. I., 1996: The surface layer in the NCEP Eta Model. Preprints, 11th Conf. on Numerical Weather Prediction, Norfolk, VA, Amer. Meteor. Soc., 354–355.
Janjić, Z. I., 2000: Comments on “Development and evaluation of a convection scheme for use in climate models.” J. Atmos. Sci., 57, 3686, https://doi.org/10.1175/1520-0469(2000)057<3686:CODAEO>2.0.CO;2.
Janjić, Z. I., 2002: Nonsingular implementation of the Mellor–Yamada level 2.5 scheme in the NCEP Meso Model. NCEP Office Note 437, 61 pp.
Judt, F., and S. S. Chen, 2010: Convectively generated potential vorticity in rainbands and formation of the secondary eyewall in Hurricane Rita of 2005. J. Atmos. Sci., 67, 3581–3599, https://doi.org/10.1175/2010JAS3471.1.
Kepert, J. D., 2013: How does the boundary layer contribute to eyewall replacement cycles in axisymmetric tropical cyclones? J. Atmos. Sci., 70, 2808–2830, https://doi.org/10.1175/JAS-D-13-046.1.
Kepert, J. D., and D. S. Nolan, 2014: Reply to “Comments on ‘How does the boundary layer contribute to eyewall replacement cycles in axisymmetric tropical cyclones?’” J. Atmos. Sci., 71, 4692–4704, https://doi.org/10.1175/JAS-D-14-0014.1.
Kimberlain, T. B., E. S. Blake, and J. P. Cangialosi, 2016: Hurricane Patricia (EP202015) 20-24 October 2015. National Hurricane Center Rep., 32 pp., https://www.nhc.noaa.gov/data/tcr/EP202015_Patricia.pdf.
Kossin, J. P., W. H. Schubert, and M. T. Montgomery, 2000: Unstable interactions between a hurricane’s primary eyewall and a secondary ring of enhanced vorticity. J. Atmos. Sci., 57, 3893–3917, https://doi.org/10.1175/1520-0469(2001)058<3893:UIBAHS>2.0.CO;2.
Kuo, H.-C., L.-Y. Lin, C.-P. Chang, and R. T. Williams, 2004: The formation of concentric vorticity structures in typhoons. J. Atmos. Sci., 61, 2722–2734, https://doi.org/10.1175/JAS3286.1.
Kuo, H.-C., W. H. Schubert, C.-L. Tsai, and Y.-F. Kuo, 2008: Vortex interactions and barotropic aspects of concentric eyewall formation. Mon. Wea. Rev., 136, 5183–5198, https://doi.org/10.1175/2008MWR2378.1.
Lawrence, M. B., and J. M. Gross, 1989: Atlantic hurricane season of 1988. Mon. Wea. Rev., 117, 2248–2259, https://doi.org/10.1175/1520-0493(1989)117<2248:AHSO>2.0.CO;2.
Martinez, Y., 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, https://doi.org/10.1175/2010JAS3501.1.
McNoldy, B. D., 2004: Triple eyewall in Hurricane Juliette. Bull. Amer. Meteor. Soc., 85, 1663–1666, https://doi.org/10.1175/BAMS-85-11-1663.
McNoldy, B. D., 2012: Tropical cyclone radar loops. University of Miami, http://andrew.rsmas.miami.edu/bmcnoldy/tropics/radar/.
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, https://doi.org/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, https://doi.org/10.1175/JAS-D-12-0192.1.
Menelaou, K., M. K. Yau, and T.-K. Lai, 2018: A possible three-dimensional mechanism for oscillating wobbles in tropical cyclone-like vortices with concentric eyewalls. J. Atmos. Sci., 75, 2157–2174, https://doi.org/10.1175/JAS-D-18-0005.1.
Mlawer, E. J., S. J. Taubman, P. D. Brown, M. J. Iacono, and S. A. Clough, 1997: Radiative transfer for inhomogeneous atmosphere: RRTM, a validated correlated-k model for the longwave. J. Geophys. Res., 102, 16 663–16 682, https://doi.org/10.1029/97JD00237.
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, https://doi.org/10.1002/qj.49712353810.
Montgomery, M. T., S. F. Abarca, R. K. Smith, C.-C. Wu, and Y.-H. Huang, 2014: Comments on “How does the boundary layer contribute to eyewall replacement cycles in axisymmetric tropical cyclones?” J. Atmos. Sci., 71, 4682–4691, https://doi.org/10.1175/JAS-D-13-0286.1.
Nolan, D. S., and M. T. Montgomery, 2000: The algebraic growth of wavenumber one disturbances in hurricane-like vortices. J. Atmos. Sci., 57, 3514–3538, https://doi.org/10.1175/1520-0469(2000)057<3514:TAGOWO>2.0.CO;2.
Nolan, D. S., M. T. Montgomery, and L. D. Grasso, 2001: The wavenumber-one instability and trochoidal motion of hurricane-like vortices. J. Atmos. Sci., 58, 3243–3270, https://doi.org/10.1175/1520-0469(2001)058<3243:TWOIAT>2.0.CO;2.
Oda, M., M. Nakanishi, and G. Naito, 2006: Interaction of an asymmetric double vortex and trochoidal motion of a tropical cyclone with the concentric eyewall structure. J. Atmos. Sci., 63, 1069–1081, https://doi.org/10.1175/JAS3670.1.
Pasch, R. J., E. S. Blake, H. D. Cobb III, and D. P. Roberts, 2006: Hurricane Wilma 15-25 October 2005. National Hurricane Center Rep., 27 pp., https://www.nhc.noaa.gov/data/tcr/AL252005_Wilma.pdf.
Persing, J., and M. T. Montgomery, 2003: Hurricane superintensity. J. Atmos. Sci., 60, 2349–2371, https://doi.org/10.1175/1520-0469(2003)060<2349:HS>2.0.CO;2.
Rozoff, C. M., W. H. Schubert, and J. P. Kossin, 2008: Some dynamical aspects of tropical cyclone concentric eyewalls. Quart. J. Roy. Meteor. Soc., 134, 583–593, https://doi.org/10.1002/qj.237.
Rozoff, C. M., D. S. Nolan, J. P. Kossin, F. Zhang, and J. Fang, 2012: The roles of an expanding wind field and inertial stability in tropical cyclone secondary eyewall formation. J. Atmos. Sci., 69, 2621–2643, https://doi.org/10.1175/JAS-D-11-0326.1.
Ryglicki, D. R., and R. E. Hart, 2015: An investigation of center-finding techniques for tropical cyclones in mesoscale models. J. Appl. Meteor. Climatol., 54, 825–846, https://doi.org/10.1175/JAMC-D-14-0106.1.
Samsury, C. E., and E. J. Zipser, 1995: Secondary wind maxima in hurricanes: Airflow and relationship to rainband. Mon. Wea. Rev., 123, 3502–3517, https://doi.org/10.1175/1520-0493(1995)123<3502:SWMIHA>2.0.CO;2.
Schecter, D. A., and M. T. Montgomery, 2007: Waves in a cloudy vortex. J. Atmos. Sci., 64, 314–337, https://doi.org/10.1175/JAS3849.1.
Schubert, W. H., M. T. Montgomery, R. K. Taft, T. A. Gunn, 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, https://doi.org/10.1175/1520-0469(1999)056<1197:PEAECA>2.0.CO;2.
Sitkowski, M., J. P. Kossin, and C. M. Rozoff, 2011: Intensity and structure changes during hurricane eyewall replacement cycles. Mon. Wea. Rev., 139, 3829–3847, https://doi.org/10.1175/MWR-D-11-00034.1.
Skamarock, W. C., and Coauthors, 2008: A description of the Advanced Research WRF version 3. NCAR Tech. Note NCAR/TN-475+STR, 113 pp., https://doi.org/10.5065/D68S4MVH.
Stern, D. P., and D. S. Nolan, 2009: Reexamining the vertical structure of tangential winds in tropical cyclones: Observations and theory. J. Atmos. Sci., 66, 3579–3600, https://doi.org/10.1175/2009JAS2916.1.
Stern, D. P., J. R. Brisbois, and D. S. Nolan, 2014: An expanded dataset of hurricane eyewall sizes and slopes. J. Atmos. Sci., 71, 2747–2762, https://doi.org/10.1175/JAS-D-13-0302.1.
Terwey, W. D., and M. T. Montgomery, 2008: Secondary eyewall formation in two idealized, full-physics modeled hurricanes. J. Geophys. Res., 113, D12112, https://doi.org/10.1029/2007JD008897.
Thompson, G., P. R. Field, R. M. Rasmussen, and W. D. Hall, 2008: Explicit forecasts of winter precipitation using an improved bulk microphysics scheme. Part II: Implementation of a new snow parameterization. Mon. Wea. Rev., 136, 5095–5115, https://doi.org/10.1175/2008MWR2387.1.
Tsujino, S., K. Tsuboki, and H.-C. Kuo, 2017: Structure and maintenance mechanism of long-lived concentric eyewalls associated with simulated Typhoon Bolaven (2012). J. Atmos. Sci., 74, 3609–3634, https://doi.org/10.1175/JAS-D-16-0236.1.
Wang, H., C.-C. Wu, and Y. Wang, 2016: Secondary eyewall formation in an idealized tropical cyclone simulation: Balanced and unbalanced dynamics. J. Atmos. Sci., 73, 3911–3930, https://doi.org/10.1175/JAS-D-15-0146.1.
Willoughby, H. E., 1988: The dynamics of the tropical cyclone core. Aust. Meteor. Mag., 36, 183–191.
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, https://doi.org/10.1175/1520-0469(1982)039<0395:CEWSWM>2.0.CO;2.
Willoughby, H. E., J. M. Masters, and C. W. Landsea, 1989: A record minimum sea level pressure observed in Hurricane Gilbert. Mon. Wea. Rev., 117, 2824–2828, https://doi.org/10.1175/1520-0493(1989)117<2824:ARMSLP>2.0.CO;2.
Yang, Y.-T., H.-C. Kuo, E. A. Hendricks, and M. S. Peng, 2013: Structural and intensity changes of concentric eyewall typhoons in the western North Pacific basin. Mon. Wea. Rev., 141, 2632–2648, https://doi.org/10.1175/MWR-D-12-00251.1.
Zhang, F., D. Tao, Y. Q. Sun, and J. D. Kepert, 2017: Dynamics and predictability of secondary eyewall formation in sheared tropical cyclones. J. Adv. Model. Earth Syst., 9, 89–112, https://doi.org/10.1002/2016MS000729.
Zhang, G., and W. Perrie, 2018: Effects of asymmetric secondary eyewall on tropical cyclone evolution in Hurricane Ike (2008). Geophys. Res. Lett., 45, 1676–1683, https://doi.org/10.1002/2017GL076988.
Zhang, J. A., R. F. Rogers, D. S. Nolan, and F. D. Marks Jr., 2011: On the characteristic height scales of the hurricane boundary layer. Mon. Wea. Rev., 139, 2523–2535, https://doi.org/10.1175/MWR-D-10-05017.1.
Zhao, K., Q. Lin, W.-C. Lee, Y. Q. Sun, and F. Zhang, 2016: Doppler radar analysis of triple eyewalls in Typhoon Usagi (2013). Bull. Amer. Meteor. Soc., 97, 25–30, https://doi.org/10.1175/BAMS-D-15-00029.1.
Zhou, X., and B. Wang, 2011: Mechanism of concentric eyewall replacement cycles and associated intensity change. J. Atmos. Sci., 68, 972–988, https://doi.org/10.1175/2011JAS3575.1.
Zhu, Z., and P. Zhu, 2014: The role of outer rainband convection in governing the eyewall replacement cycle in numerical simulations of tropical cyclones. J. Geophys. Res. Atmos., 119, 8049–8072, https://doi.org/10.1002/2014JD021899.
Here, it should be noted that different to Huang et al. (2012), Kepert (2013) argued that SEF tends to increase the frictionally induced secondary circulation. If the moat separating the two eyewalls is characterized by a local vorticity maximum, then sufficient subsidence may occur there to maintain the inner eyewall’s updraft. Readers with interest on this debate are also referred to Montgomery et al. (2014) and Kepert and Nolan (2014) for further details.
Although “concentric eyewalls” is widely used in the literature, it is believed that this terminology can be rather misleading especially when the inner eyewall is in trochoidal oscillation. Therefore, the phrase “multiple eyewalls” is preferred in this paper. It refers to any two or more coexisting eyewalls of a TC without the restriction that the geometric centers of all the inner and outer eyewalls must be the same.
The animation can be accessed online (http://andrew.rsmas.miami.edu/bmcnoldy/tropics/earl10/Earl_30-31Aug10_SanJuan.gif).
The animation can be accessed online (http://andrew.rsmas.miami.edu/bmcnoldy/tropics/maria17/Maria_19-20Sep17_TJUA.gif).
This satellite microwave imagery can be accessed online (https://tropic.ssec.wisc.edu/real-time/mimic-tc/2005_TWENTY-FOUR/webManager/mainpage.html).
On the other hand, according to the weather reconnaissance aircraft vortex data messages, the observed first ERC of Wilma began at a time between 1822 and 2017 UTC 19 Oct, and ended at a time between 1046 and 1910 UTC 20 Oct. These vortex data messages can be obtained online (http://hurricanes.ral.ucar.edu/structure/vortex/vdm_data/2005/vortex_AL252005_WILMA.txt).
The corresponding center of each group is determined by the potential vorticity (PV) centroid (e.g., Ryglicki and Hart 2015).
Radial flow pattern of m = 2 associated with type-2 instability emerges and develops from t = 57.0 h (not shown). It becomes dominant at t = 59.5 h.
One may also study the strength evolution of the inner vorticity patch in terms of its spatial mean 
The decline rate of 
The m = 1 instability, which manifests itself as the trochoidal oscillation of the eye of a single-eyewall TC, is reported in Nolan and Montgomery (2000, and references therein), Nolan et al. (2001, and references therein), and Schecter and Montgomery (2007). As a contrast, the m = 1 pattern in our WRF simulation is due to the m = 1 instability between the inner and outer eyewalls leading to the trochoidal oscillation of the inner eyewall as a whole within the secondary eyewall (Oda et al. 2006; Menelaou et al. 2018).
From the thermodynamics viewpoint, a strengthening outer eyewall will lead to an enhanced warm core, and thereby increasing potential temperature. Assuming the water vapor mixing ratio does not decrease substantially in the moat during this time, it is expected that 
Although the temporary change of dominant radial flow pattern is a qualitatively reasonable explanation for these pauses, the definite underlying mechanism requires further investigation.
