1. Introduction
Secondary eyewall formation (SEF) in tropical cyclone (TC) eyewall replacement cycles (ERCs) (e.g., Willoughby et al. 1982; Sitkowski et al. 2011, 2012) has sustained attention in the TC community for decades because of its elusive predictability and significant impact on TC structure and intensity change. Beyond the sensitivity of numerical TC simulations to convective-scale processes, part of the challenge in operational forecasting is that the exact physical factors responsible for SEF have not been discerned from a list of plausible mechanisms. In recent years, improved resolution in full-physics TC simulations has resulted in numerous studies of SEF and ERCs (e.g., Zhang et al. 2005; Terwey and Montgomery 2008; Wang 2008a,b, 2009; Hill and Lackmann 2009; Hogsett and Zhang 2009; Wu et al. 2009; Judt and Chen 2010; Martinez et al. 2011; Zhou and Wang 2011; Abarca and Corbosiero 2011; Chen et al. 2011; Wu et al. 2012; Huang et al. 2012, hereafter HMW12; Fang and Zhang 2012, hereafter FZ12). This suggests that the understanding and predictability of SEF will continue to improve.
Early theory on SEF that is still relevant today focused on idealized, axisymmetric models in thermal wind balance. Generally, the idea was to understand the TC system-scale response to adding a heat source outside of the radius of maximum wind (RMW) of a TC-like vortex. Using a variation of the Eliassen (1951) transverse circulation equation for axisymmetric, balanced flow, Shapiro and Willoughby (1982, hereafter SW82) studied the impacts of adding point sources of diabatic heating to the region outside of the RMW for a variety of tangential wind profiles. They specifically found that when the tangential wind decays more slowly with radius, the enhanced inertial stability reduces the low-level radial inflow. However, the larger angular momentum associated with the broader wind field more than compensates for the reduced inflow and thereby increases the tangential wind tendency near the heat source. Using a similar model, Schubert and Hack (1982, hereafter SH82) and Hack and Schubert (1986, hereafter HS86) elucidated the existence of a nonlinear instability in which diabatic heating can increasingly induce larger tangential wind tendencies as inertial stability increases in the region of heating.
More recent literature has established the importance of asymmetric perturbations in vortex evolution. One of the simplest models used in studying the dynamics of vortex asymmetries and their role in SEF is the two-dimensional (2D) nondivergent barotropic model. In the unforced, barotropic framework, Kuo et al. (2004, 2008) proposed that concentric vorticity rings can evolve from certain initial conditions consisting of one or more patches of weaker vorticity in the vicinity of a strong TC-like vortex. However, Moon et al. (2010) showed that small-scale, high-amplitude vorticity perturbations arising from processes such as rainband convection are more typical outside of the eyewall of a mature TC and that these perturbations do not axisymmetrize into concentric vorticity rings.
The barotropic framework has also been used to develop theory for vortex Rossby waves (VRWs) and their connections to vortex axisymmetrization (e.g., Guinn and Schubert 1993; Montgomery and Kallenbach 1997). Montgomery and Kallenbach hypothesized that VRW interactions with the mean flow may contribute to SEF. In particular, a negative radial vorticity gradient of finite width outside of the RMW supports the outward propagation of VRWs until the waves slow down toward the stagnation radius where the VRW group velocity goes to zero. Inside this stagnation radius, cyclonic eddy momentum is transported inward and may contribute to SEF in this region. In this same framework, Martinez et al. (2010) analyzed the nonlinear evolution of asymmetric vorticity perturbations produced by barotropic instability in the symmetric mean vortex. Their analysis diagnosed wave–mean flow interactions that produced a secondary wind maximum near the critical radius of leading modes in their numerical experiment. The neglect of the three-dimensional (3D) dynamics, friction in the boundary layer, and convective processes is a serious limitation of any barotropic study. Overall, the role of axisymmetrization in SEF requires further examination in 3D modeling frameworks.
Given the 3D nature of SEF, Terwey and Montgomery (2008) proposed a β-skirt axisymmetrization (BSA) mechanism incorporating both barotropic dynamics and 3D convective processes. In this mechanism, SEF is initiated by the anisotropic upscale cascade of convectively generated potential vorticity (PV) anomalies on a low-level, negative PV gradient (i.e., β skirt) outside of the RMW. The collocation of convection and the β skirt are necessary conditions for SEF in this hypothesis. Qiu et al. (2010) find that SEF occurs near the VRW stagnation radius and in a convectively active region of their simulated TC’s β skirt. They further revealed that the eddy momentum flux convergence near the stagnation radius led to the outward expansion of the β skirt. The full-physics TC simulations of Hogsett and Zhang (2009), Martinez et al. (2011), and Abarca and Corbosiero (2011) also emphasize the potential role for VRWs in SEF.
Regardless of the mechanisms that may spin up the wind field and/or force latent heating outside of the primary eyewall, a number of studies stress the importance of dynamical adjustment to latent heating in SEF and size evolution. Using an idealized model, Moon and Nolan (2010) showed that realistic distributions of asymmetric diabatic heating from stratiform and convective precipitation in outer rainbands can induce secondary wind maxima and other circulation features often observed in rainbands. Some studies (Wu et al. 2009; Judt and Chen 2010) have carried out PV budgets in TC simulations to show that the PV generation from concentrated rainband activity in the incipient outer eyewall region is key to SEF. Wang (2009) found that artificially increasing latent heating in the region of outer rainbands of simulated TCs led to SEF and an outward expansion of winds. Hill and Lackmann (2009) showed that enhancing the environmental relative humidity facilitates SEF and larger simulated TCs. This latter result is consistent with idealized balance model results described in Nong and Emanuel (2003), who showed that, along with wind-induced surface heat exchange (WISHE) and finite-amplitude perturbations, relative humidity may be an important factor in SEF. The importance of relative humidity in SEF has been supported by the observational study of Kossin and Sitkowski (2009). Similarly, a series of papers by Xu and Wang (2010a,b) suggests that the distribution of low-level moist entropy and initial vortex size impact the size of a TC. Finally, an idealized full-physics TC simulation by FZ12 on the β plane showed that β shear in a motionless environment can impose a significant heating asymmetry that promotes the generation of an outer eyewall. This β shear results from the height variation of the TC circulation’s advection of planetary vorticity (Ritchie and Frank 2007).
Along with the developing body of work on dynamical adjustment, recent work suggests that unbalanced processes are relevant to SEF as well. HMW12 used the full-physics TC simulation of Wu et al. (2012) to understand symmetric aspects of SEF. Consistent with the studies of Smith et al. (2008, 2009) and Smith and Montgomery (2010), HMW12 suggest that an expanding wind field in and above the boundary layer leads to increased boundary layer inflow due to an inward pressure gradient force that exceeds the counteracting but frictionally reduced centrifugal force. Because of the net increase in absolute angular momentum in a region of strong inflow, supergradient winds can develop in and just above the boundary layer. An adverse pressure gradient force associated with the supergradient winds may enhance low-level convergence and facilitate SEF. Using airborne Doppler radar and dropsonde data from Hurricane Rita (2005), Didlake and Houze (2011) found supergradient forces exist in the boundary layer near Rita’s outer eyewall.
Taken together, the above studies suggest that SEF is likely a dynamical adjustment process to heating that may be augmented by unbalanced processes in the boundary layer. From this point of view, a more complete theory for SEF requires fully resolving where, when, and how quickly rainband activity will transform into an outer eyewall. With evidence that a broader vortex structure increases the local production of kinetic energy in a region of enhanced latent heating (e.g., SW82; HS86; Xu and Wang 2010b), we propose that the vortex inertial stability structure helps control the likelihood and speed of SEF as well. Using data from a realistic simulation of a TC, we now have the ability to evaluate pioneering theoretical predictions (e.g., SW82) with realistic time-evolving fields of vortex structure and diabatic and frictional forcing. Here, we use a high-resolution numerical simulation of SEF in a TC to examine the TC response to diabatic heating and frictional forcing and the role of the TC’s response in SEF and structure change. Section 2 briefly describes the full-physics model setup. Basic aspects of the TC simulation and the SEF event in the simulation are described in section 3. Section 4 describes processes governing TC structure change and their impact on kinetic energy efficiency. Section 5 examines the role of inertial stability in SEF from the perspective of a balanced vortex model. Finally, discussion and conclusions are provided in section 6.
2. Model setup
This study utilizes the Advanced Research Weather Research and Forecast Model (WRF) version 2.2 (Skamarock et al. 2005) to simulate an intense TC in an idealized setting. The simulation is carried out on the β plane with three domains containing horizontal grid spacing of 9, 3, and 1 km and domain sizes of 10 800 × 10 800 km2, 1620 × 1620 km2, and 1000 × 1000 km2, respectively. Two-way interactive nesting is used with vortex-following nested domains. As a result, the model TC is always situated in the center of the finest domain. The model has 35 vertical levels with the model top near 28 km. The vertical grid spacing is stretched from 0.6-km spacing near the surface to 3.6-km spacing near the top of the model domain. Radiation is neglected as in Nolan (2007) to suppress convection on the outer grids. The WRF Single-Moment (WSM) six-class scheme with graupel (Hong et al. 2004) is used. This simulation contains no large-scale environmental flow and the outer spiral rainbands are nearly contained within the finest domain. As such, cumulus parameterization is not included in any grid. Turbulent fluxes are parameterized using the Yonsei University (YSU) planetary boundary layer scheme (Noh et al. 2003).
The simulation is initialized with an axisymmetric cyclonic vortex similar to that used by Rotunno and Emanuel (1987) in a quiescent atmosphere and over the ocean with a constant sea surface temperature of 29.15°C. The initial unperturbed atmospheric temperatures are obtained from the Jordan (1958) mean hurricane season sounding for the Caribbean. The initial baroclinic vortex is centered at 20°N and in gradient-wind balance with the maximum azimuthal wind speed of 16 m s−1 at a radius of about 102 km, equivalent to a strong tropical depression.
3. Simulation overview
The TC simulation studied here is similar to the lower resolution (3-km horizontal grid spacing) simulation examined in FZ12. In FZ12, a wavenumber-1 pattern of stratiform rainband precipitation resulting from β shear led to evaporative cooling and a subsequent sharpening of the low-level equivalent potential temperature θe gradient outside the primary eyewall. This gradient coincided with a region of convective initiation preceding SEF. The SEF event analyzed in the current simulation also emerged as the β shear weakened and the rainband activity symmetrized. We focus on a time frame corresponding to SEF that covers the 72–144-h period of simulation. A brief overview of the general evolution during this time frame is now presented.
The low-level (z = 1 km) synthetic radar reflectivity in Fig. 1 provides an overview of the precipitation during the simulated ERC. During the 102–127-h time frame, asymmetric rainband activity 50–100 km from the center organizes into a more symmetric outer eyewall. Also, moat formation appears early. At 102 h (Fig. 1a), a weak radar echo region materializes on the southern and eastern sides of the eyewall while more vigorous rainbands are found farther out from center. Pronounced clearing around the outer edge of the eyewall was evident even earlier than 102 h (not shown). In the next 25 h (Figs. 1b–f), a wavenumber-1 asymmetry persists in the moat, with weaker radar echoes appearing on the southern side of the primary eyewall. Beyond the region of SEF, there is a broad wavenumber-1 pattern of concentrated rainfall that rotates from south to north during this period. By 127 h (Fig. 1g), a concentric eyewall configuration exists. Eyewall replacement follows over the next 10 h and results in a larger eye (Figs. 1h,i).
Low-level (z = 1 km) synthetic radar reflectivity (dBZ) at (a) 102, (b) 107, (c) 112, (d) 117, (e) 122, (f) 127, (g) 132, (h) 137, and (i) 142 h.
Citation: Journal of the Atmospheric Sciences 69, 9; 10.1175/JAS-D-11-0326.1
Hovmöller diagrams highlight basic kinematic and thermodynamic changes throughout the period preceding and accompanying the ERC. Figure 2a shows the low-level (z = 1 km) azimuthal-mean synthetic radar reflectivity. A moat of low precipitation clearly emerges in the azimuthal-mean state before SEF occurs. This low-precipitation moat moves inward with time. At about 95 h, the intensity of radar reflectivity in the eyewall increases as an earlier, minor ERC-like process completes. The outer eyewall precipitation starts out diffuse between 60- and 140-km radius (95–115 h), but it eventually solidifies into a more coherent structure (115–125 h). Consistent with most ERCs, the outer eyewall contracts inward with time.
Hovmöller diagrams of the azimuthal-mean (a) synthetic radar reflectivity (dBZ) at z = 1 km, (b) vertical velocity (m s−1; the zero contour is thick black) at z = 5 km, (c) tangential wind (m s−1) at z = 0.5 km, (d) radial wind (m s−1; the zero contour is in thick black) at z = 0.5 km, (e) temperature (°C) at z = 0.5 km, and (f) equivalent potential temperature (°C) at z = 0.5 km.
Citation: Journal of the Atmospheric Sciences 69, 9; 10.1175/JAS-D-11-0326.1
The midlevel (z = 5 km) azimuthal-mean vertical wind (Fig. 2b) progresses similarly to the radar reflectivity. Weak subsidence begins to appear outside of the primary eyewall around 95 h as the primary eyewall intensifies. Not until the outer eyewall becomes more symmetric does the Hovmöller diagram indicate substantial upward motion outside of the moat (~125 h). The radial structure of tangential wind evolves rapidly during this time as well. Figure 2c shows the tangential wind at z = 0.5 km. Even before the ERC takes place, the tangential wind generally expands outward with time. However, there is a local weakening and contraction of tangential winds just outside the RMW after the earlier ERC-like process completes around 95 h. A sharp and rapid expansion of the tangential winds occurs after 105 h in association with SEF. A striking secondary tangential wind maximum in the azimuthal mean is never indicated near the time of SEF. Individual cross sections, however, indicate that clear secondary wind maxima appear in just some sectors of the TC (e.g., Fig. 3). Also, consistent with the flight-level data studied in Sitkowski et al. (2011), these secondary wind maxima are evident before concentric eyewalls have clearly emerged in the precipitation fields.
South-to-north cross section of the zonal wind (gray) and west-to-east cross section of the meridional wind (black) at z = 0.5 km and 122 h.
Citation: Journal of the Atmospheric Sciences 69, 9; 10.1175/JAS-D-11-0326.1
Figure 2d shows the low-level radial flow evolution during an ERC. Some weak outward flow is seen in the eye throughout the entire time period, whereas the flow is generally inward outside of the primary eyewall. Low-level inflow increases significantly as the primary eyewall’s convection intensifies at 95 h, but thereafter it progressively weakens as the moat and outer eyewall develop. Stronger inflow develops outside of the outer eyewall by 125 h. It is worth noting that outside of 80–100 km, the low-level inflow intensifies well before SEF.
Azimuthal-mean low-level (z = 0.5 km) temperature and θe fields are provided in Figs. 2e and 2f. Between 75 and 100 h, air cooler than 22°C exists in the region of precipitation shown in Fig. 2a due to cool downdrafts. As the moat matures, warmer temperatures emerge in this region, likely tied to subsidence warming, downward mixing into the boundary layer, and the relatively warm sea surface temperatures below. This pattern is also noticeable in the θe field as well. However, in response to the expanding low-level wind field and increased latent heat flux at the ocean surface, the θe field still generally expands outward. The rapid outward increase in θe at around 100 h is similar to an increase seen in FZ12. Moreover, the radial gradient of θe sharpens around 60–100 km from center at this time. As indicated in Fig. 2a, this region experiences invigorated precipitation at this time. In comparison to FZ12, the spatial scales of the inner and outer eyewalls are smaller in this higher-resolution simulation, but otherwise both simulations are similar. As discussed in FZ12, this frontlike structure may expedite SEF.
4. Symmetric aspects of secondary eyewall formation
a. Absolute angular momentum budget
To assess symmetric aspects of SEF, we now focus our attention more closely on the 48-h period following 96 h, which captures SEF and the subsequent ERC. A rapid expansion of tangential winds occurs during this time frame. An absolute angular momentum budget is carried out to illustrate the leading contributions to the expansion of the winds.
Figure 4 shows an example of the 
The (a) actual and (b) diagnosed changes in 
Citation: Journal of the Atmospheric Sciences 69, 9; 10.1175/JAS-D-11-0326.1
b. On the kinetic energy efficiency of secondary eyewall formation
Figure 5a provides a snapshot of I at 97 h, which is prior to the SEF event. To emphasize the low-level skirt of I extending outward from the primary eyewall, values of I exceeding 3 × 10−3 s−1 are not contoured in the eye–eyewall region. Much larger values of I (exceeding 7 × 10−3 s−1) are found in the vicinity of the primary eyewall, which coincide with the strongest 
The parameters (a) I (10−3 s−1) and (b) 
Citation: Journal of the Atmospheric Sciences 69, 9; 10.1175/JAS-D-11-0326.1
Over the next 15 h (97–112 h), I increases throughout the troposphere outside of the primary eyewall (Fig. 5c). In the region defined by 40 ≤ r ≤ 60 km and z ≤ 5 km, the average value of I increases by 25% (0.17 × 10−3 s−1). There is also a substantial reduction of I in the core of the TC coinciding with decreased 
Figure 6a shows the midlevel I and the azimuthal-mean vertical wind 
(a) Hovmöller (radius–time) diagram of I (10−3 s−1; shaded) and 
Citation: Journal of the Atmospheric Sciences 69, 9; 10.1175/JAS-D-11-0326.1
An empirical way to determine the role of efficiency in the simulated SEF process is to estimate the degree to which latent heating associated with convective and stratiform clouds is retained in the TC’s inner core as kinetic energy. Similar to Nolan et al. (2007, hereafter NMS07), we define the kinetic energy efficiency (KEE) as the ratio of the change in KE (ΔKE) over a 6-h time period to the time integral of q (Q) over a 6-h period2 (i.e., KEE = ΔKE/Q). We can compute the KEE for the entire region shown in Fig. 6a, but we instead focus on a fixed region large enough to capture the immediate surroundings of a contracting outer eyewall yet small enough so that the time-varying diabatic heating associated with the inner eyewall does not interfere with interpretation of the results. Specifically, we compute (4) and (5) with r1 = 30 km, r2 = 160 km, and zt = 13 km, as depicted by the vertical dashed lines in Fig. 6a. The area average of I at z = 5 km over this radial region is provided in Fig. 7a. There is a noteworthy jump of about 2 × 10−4 s−1 in the area-average I from 115 to 130 h corresponding in time with SEF. In this same outer eyewall region, both KE and q (Fig. 7b) evolve in a similar way as these variables over the broader domain (Fig. 6b), where a jump in q precedes the increase in KE. Some of this jump is simply the result of a temporary spike in more widespread latent heating prior to a more coherent outer eyewall structure. Not surprisingly, the area-average I correlates with KE with a Pearson correlation coefficient of 0.959, but the overall volumetric KE lags behind the midlevel I slightly such that I and q evolve more closely with one another (r = 0.895) than KE and q (r = 0.795). Figure 7c shows the 6-h change in KE and q integrated over the previous 6 h. The 6-h increase in KE jumps from negative values to over 5 × 1016 J between 111 and 121 h, which follows an uptick in the heat injection around 100 h. The KEE is plotted in Fig. 7d. Consistent with the KEE calculations in NMS07, the KEE is generally very small (<0.7%), indicating that much of the heat released never converts into localized KE and/or the KE is hampered by friction. Nonetheless, the KEE increases rapidly from 0.3% to 0.6% during SEF (115–120 h). Overall, the instantaneous inertial stability does provide a rough sense of kinetic energy efficiency, but it is more meaningful to directly compare the change in KE following a period of heating. The role of KEE in SEF will be examined more closely in the next section.
(a) The area average of I at z = 5 km (10−3 s−1), (b) KE (1017 J; solid) and q (1014 J s−1; dashed), (c) ΔKE (1016 J; solid) and Q (1018 J; dashed), and (d) KEE (%) for the radial region bounded by the vertical dashed lines in Fig. 6a.
Citation: Journal of the Atmospheric Sciences 69, 9; 10.1175/JAS-D-11-0326.1
5. Balanced aspects of secondary eyewall formation
a. Diagnosis of balanced aspects of SEF with WRF data
The simultaneous changes in I and 
To address quasi-balanced aspects of SEF, we employ the linearized, nonhydrostatic, anelastic vortex model described in Hodyss and Nolan (2007) and NMS07, now known as 3-Dimensional Vortex Perturbation Analysis and Simulation (3DVPAS). Although the model simulates unsteady nonhydrostatic dynamics, Nolan and Grasso (2003) found that when forced by fixed heat or momentum sources, the steady-state solutions produced by 3DVPAS are an excellent approximation to the response expected from a “balanced vortex” model (e.g., Eliassen 1951). This finding has been further verified by comparisons of 3DVPAS output to the analytical solutions of Schubert et al. (2007) and Rozoff et al. (2008) (not shown).
In the current study, the domain for all 3DVPAS calculations covers the radial and height ranges of 0–200 and 0–17.9 km. A staggered Arakawa C-type grid consisting of 211 radial and 69 vertical grid points is used. The radial grid spacing expands outward using the grid stretching described in Nolan and Montgomery (2002) in order to provide higher spatial resolution near the inner core while retaining computational efficiency. The grid spacing is about 0.54 km inside of r = 37 km and gently expands to 1.8 km near r = 200 km. Grid stretching is employed in the vertical direction as well in order to properly resolve the shallowness of boundary layer friction. Grid spacing is about 0.14 km below 4 km and about 0.4 km near the top of the domain. The time step of time integration is 30 s. Steady-state solutions are achieved after about 6 h of simulation time and the balanced responses are obtained from the output at 16 h.
The azimuthal-mean tangential wind from the WRF output is specified in 3DVPAS for our calculations. Figure 8a shows an example of 
Radial–height cross sections of the 97-h (a) 
Citation: Journal of the Atmospheric Sciences 69, 9; 10.1175/JAS-D-11-0326.1
The latent heating and the azimuthal component of friction from WRF are used to force 3DVPAS. Figures 9a and 9b show 
Shown are (a) 
Citation: Journal of the Atmospheric Sciences 69, 9; 10.1175/JAS-D-11-0326.1
The steady-state vortex response to stationary forcing from WRF at 105, 115, and 125 h is shown in Fig. 9c. The vertical motion in 3DVPAS mirrors the spatial distribution of 
The azimuthal-mean radial component of wind exhibits more discernible differences than 
Shown are 
Citation: Journal of the Atmospheric Sciences 69, 9; 10.1175/JAS-D-11-0326.1
Even with the significant simplifications in 3DVPAS, much can be learned about SEF in this idealized framework. As such, we will use 3DVPAS to isolate the roles of diabatic and frictional forcing throughout the SEF process. We will also use 3DVPAS to examine the role of an expanding inertial stability field on SEF.
b. Relative contributions of forcing to the secondary circulation
Diabatic and frictional forcing drive a significant portion of the secondary circulation during the SEF process. As such, it is helpful to determine the relative importance of 
Vertical wind (m s−1) from 3DVPAS due to (a) 
Citation: Journal of the Atmospheric Sciences 69, 9; 10.1175/JAS-D-11-0326.1
Examining the vortex response to the positive and negative components of 
(a) Positive and (b) negative 
Citation: Journal of the Atmospheric Sciences 69, 9; 10.1175/JAS-D-11-0326.1
c. The relative importance of forcing and radial structure on balanced aspects of SEF
To more thoroughly investigate the relative importance of forcing and vortex structure on the spinup of an outer eyewall, we use 3DVPAS for some additional idealized experiments, which are summarized in Table 1. As before, we focus our attention on the times of 105, 115, and 125 h because they represent key points of evolution during SEF. At 105 h, the rainband activity in the region of eventual SEF is just starting to increase outside of the primary eyewall and the wind field expands rapidly thereafter, yielding an increase in the kinetic energy (Figs. 6 and 7). At 115 h, SEF starts to become apparent in multiple fields (Fig. 2). By 125 h, SEF has nearly completed and a double eyewall structure is just becoming evident in horizontal plots of synthetic radar reflectivity. We only present three snapshots in time since 3DVPAS calculations for times in between show congruous results.
A summary of 3DVPAS experiments and corresponding descriptions of the times when the prescribed azimuthal-mean tangential wind 
Our primary interest here is to understand the WRF’s spinup of the outer eyewall within the idealized framework. To this end, we use 3DVPAS to diagnose symmetric tangential wind tendencies 
Before more thoroughly examining the specific vortex response to changes in vortex structure and forcing, we review changes in the vortex structure that are relevant to traditional Eliassen-type balanced vortex calculations. Important structural parameters include the inertial stability and the static stability, the latter of which will be denoted by the Brunt–Väisälä frequency 
(a) Inertial stability (10−2 s−1) at 105 h (C1) and changes in I from 105 to 115 h (C2 − C1) and from 105 to 125 h (C3 − C1). (b) Static stability (10−2 s−1) at 105 h (C1) and changes in N from 105 to 115 h (C2 − C1) and from 105 to 125 h (C3 − C1).
Citation: Journal of the Atmospheric Sciences 69, 9; 10.1175/JAS-D-11-0326.1
Figure 14a shows the diagnosis of 
(a) A diagnosis of 
Citation: Journal of the Atmospheric Sciences 69, 9; 10.1175/JAS-D-11-0326.1
Experiments E1 and E2 include the vortex structure at 115 and 125 h while using forcing from 105 h, while experiments E3 and E4 use forcing from 115 and 125 h with a vortex structure fixed at 105 h. Because the results in E1 are similar to E2, 
The difference fields provided in Fig. 14d emphasize the impact of forcing on 
To understand why the expanding wind field contributes to SEF in the balanced framework, we now consider the diagnosed transverse circulation associated with each experiment. We recall that Figs. 9 and 10 show the secondary circulations for C1, C2, and C3. The secondary circulations in experiments E1–E4 actually look qualitatively similar to C1–C3, so it is more illustrative to just consider the associated difference fields of 
(a) Difference fields of 
Citation: Journal of the Atmospheric Sciences 69, 9; 10.1175/JAS-D-11-0326.1
Figure 16 shows difference fields of 
(a) Difference fields of 
Citation: Journal of the Atmospheric Sciences 69, 9; 10.1175/JAS-D-11-0326.1
Even though the experiments that shift forcing outward produce stronger increases in low-level inflow near the nascent outer eyewall, the low-level 
(a) Difference fields of 
Citation: Journal of the Atmospheric Sciences 69, 9; 10.1175/JAS-D-11-0326.1
6. Conclusions
Axisymmetric aspects of secondary eyewall formation (SEF) are revisited using an idealized, cloud-resolving WRF simulation on the β plane. SEF rapidly commences as soon as rainband latent heating outside of the primary eyewall substantially projects onto the azimuthal-mean state. These rainbands are likely enhanced by the TC-induced β shear as described in FZ12 and convective development in these bands near the site of SEF may be further enhanced by the frontlike structure that appears in fields of equivalent potential temperature. The radial extent and intensity of the azimuthal-mean latent heating abruptly increases as soon as the simulated SEF event begins. This heating is sustained during the subsequent ERC. Absolute angular momentum, kinetic energy, and attendant inertial stability expand radially as the outer eyewall matures and contracts inward. The kinetic energy efficiency of the outer eyewall region of the TC steadily increases in turn.
An axisymmetric, linearized, nonhydrostatic vortex model is used to diagnose the secondary circulation corresponding to the azimuthal-mean tangential wind and the azimuthal-mean latent heating and friction from WRF. These numerical solutions lack some unbalanced aspects of SEF, but they provide insight into the importance of the vortex structure and forcing on SEF. The idealized solutions for the secondary circulation are qualitatively similar to the mean secondary circulation produced in WRF. In this framework, as the wind field expands in response to rainband latent heating, the low-level inflow decreases near the inner eyewall and increases in the vicinity of rainband heating. These aspects of the vortex response, which precede SEF by several hours, are consistent with a number of other recent studies (e.g., Bui et al. 2009; Xu and Wang 2010b; Fudeyasu and Wang 2011; HMW12).
Using the balanced vortex model, sensitivity experiments were carried out throughout an SEF event to isolate and quantify the relative impacts of the changes in forcing and inertial stability in the generation of the outer eyewall in the WRF. The essence of these results is that, in the vicinity of the incipient outer eyewall, an expansion of the inertial stability is more efficient in the intensification of the low-level winds than increasing diabatic heating and frictional forcing. To be clear, however, the outward shift in heating and Ekman pumping are essential to the spinup of an outer eyewall as well. In essence, there is a positive feedback suggested here, where an expanding wind field increases the efficiency of the vortex spinup associated with latent heating. Increased Ekman pumping associated with expanding friction facilitates more convection, which yields continued growth of the outer eyewall. While these results are derived from applying an idealized model to WRF output, the importance of both inertial stability and forcing in SEF are expected to remain relevant in more realistic models that account for nonlinear boundary layer dynamics in SEF (e.g., HMW12). Also, while these concepts should be further examined in analyses accounting for asymmetric dynamics, it is anticipated that these symmetric concepts will still play an essential role.
The angular momentum budget calculations in this study indicate that any forcing mechanism that produces sufficiently strong and sustained latent heating outside of the primary eyewall will promote SEF. These results support recent studies showing how higher values of relative humidity (Hill and Lackmann 2009), rainband diabatic heating (Wang 2009), surface moist enthalpy flux in the outer core (Xu and Wang 2010a), and initial vortex size (Xu and Wang 2010b) are effective means to increasing the size of a TC via the expansion of the wind field and resulting boundary layer inflow. The results ultimately confirm the relevance of the pioneering results of SW82, SH82, and HS86 as well, but they do so using more realistic evolving distributions of inertial stability and forcing—namely, in cases where a TC has a more expansive wind field, not only will the frictional inflow be enhanced, but the inertial stability will be higher as well, which will more readily support the development of an outer eyewall should latent heating activity persist in this region. It would be interesting to study large observational datasets or many model simulations to see if there is a way to anticipate the likelihood and location of SEF based on the distribution of low-level inertial stability and diabatic heating.
In a way, the role of the radial structure of inertial stability in SEF shares some similarities with the BSA mechanism proposed in Terwey and Montgomery (2008). Akin to the organizing properties of the β skirt in the BSA mechanism, the enhanced inertial stability field accompanying a more expansive β skirt increases the rainband organization and its impact on SEF. In either viewpoint, attainable convective instability and the WISHE mechanism important for the latent heat release needed for SEF should be required.
We thank Matthew Sitkowski and Daniel Stern for helpful discussions that benefitted this work. We gratefully acknowledge Chun-Chieh Wu and two anonymous reviewers for outstanding critical reviews of this manuscript that led to significant improvements. C. Rozoff was supported by ONR Grant N00014-07-1-0163 and NOAA Grants NA06NES4400002 and NA10NES4400013. D. Nolan was supported in part by NSF Grant ATM-0756308. F. Zhang was supported by ONR Grant N000140910526, NOAA HFIP, and NSF Grant 0840651. J. Fang was supported by the Nature Science Foundation of China Grants 40921160382 and 40830958, and State Key Basic Program of China 2009CB421502. The WRF simulation was carried out at the Texas Advanced Computing Center.
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.
Bui, H. H., , R. K. Smith, , M. T. Montgomery, , and J. Peng, 2009: Balanced and unbalanced aspects of tropical cyclone intensification. Quart. J. Roy. Meteor. Soc., 135, 1715–1731.
Chen, H., , D.-L. Zhang, , J. Carton, , and R. Atlas, 2011: On the rapid intensification of Hurricane Wilma (2005). Part I: Model prediction and structural changes. Wea. Forecasting, 26, 885–901.
Didlake, A. C., , and R. A. Houze, 2011: Kinematics of the secondary eyewall observed in Hurricane Rita (2005). J. Atmos. Sci., 68, 1620–1636.
Eliassen, A., 1951: Slow thermally or frictionally controlled meridional circulation in a circular vortex. Astrophys. Norv., 5, 19–60.
Fang, J., , and F. Zhang, 2012: Effect of beta shear on simulated tropical cyclones. Mon. Wea. Rev., in press.
Fudeyasu, H., , and Y. Wang, 2011: Balanced contribution to the intensification of a tropical cyclone simulated in TCM4: Outer-core spinup process. J. Atmos. Sci., 68, 430–449.
Guinn, T. A., , and W. H. Schubert, 1993: Hurricane spiral bands. J. Atmos. Sci., 50, 3380–3403.
Hack, J. J., , and W. H. Schubert, 1986: Nonlinear response of atmospheric vortices to heating by organized cumulus convection. J. Atmos. Sci., 43, 1559–1573.
Hill, K. A., , and G. M. Lackmann, 2009: Influence of environmental humidity on tropical cyclone size. Mon. Wea. Rev., 137, 3294–3315.
Hodyss, D., , and D. S. Nolan, 2007: Linear anelastic equations for atmospheric vortices. J. Atmos. Sci., 64, 2947–2959.
Hogsett, W., , and D.-L. Zhang, 2009: Numerical simulation of Hurricane Bonnie (1998). Part III: Energetics. J. Atmos. Sci., 66, 2678–2696.
Hong, S.-Y., , J. Dudhia, , and S.-H. Chen, 2004: A revised approach to ice microphysical processes for the parameterization of clouds and precipitation. Mon. Wea. Rev., 132, 103–120.
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. Soc., 69, 662–674.
Jordan, C. L., 1958: Mean soundings for the West Indies area. J. Meteor., 15, 91–97.
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.
Kossin, J. P., , and M. Sitkowski, 2009: An objective model for identifying secondary eyewall formation in hurricanes. Mon. Wea. Rev., 137, 876–892.
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.
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.
Maclay, K. S., , M. DeMaria, , and T. H. Vonder Haar, 2008: Tropical cyclone inner-core kinetic energy evolution. Mon. Wea. Rev., 136, 4882–4898.
Mallen, K. J., , M. T. Montgomery, , and B. Wang, 2005: Reexamining the near-core radial structure of the tropical cyclone primary circulation: Implications for vortex resiliency. J. Atmos. Sci., 62, 408–425.
Martinez, Y., , G. Brunet, , and M. K. Yau, 2010: On the dynamics of two-dimensional hurricane-like concentric rings vortex formation. J. Atmos. Sci., 67, 3253–3268.
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.
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.
Moon, Y., , and D. S. Nolan, 2010: The dynamic response of the hurricane wind field to spiral rainband heating. J. Atmos. Sci., 67, 1779–1805.
Moon, Y., , D. S. Nolan, , and M. Iskandarani, 2010: On the use of two-dimensional incompressible flow to study secondary eyewall formation in tropical cyclones. J. Atmos. Sci., 67, 3765–3773.
Noh, Y., , W.-G. Cheon, , S.-Y. Hong, , and S. Raasch, 2003: Improvement of the K-profile model for the planetary boundary layer based on large eddy simulation data. Bound.-Layer Meteor., 107, 401–427.
Nolan, D. S., 2007: What is the trigger for tropical cyclogenesis? Aust. Meteor. Mag., 56, 241–266.
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, 2717–2745.
Nolan, D. S., , Y. Moon, , and D. P. Stern, 2007: Tropical cyclone intensification from asymmetric convection: Energetics and efficiency. J. Atmos. Sci., 64, 3377–3405.
Nong, S., , and K. Emanuel, 2003: A numerical study of the genesis of concentric eyewalls in hurricanes. Quart. J. Roy. Meteor. Soc., 129, 3323–3338.
Qiu, X., , Z.-M. Tan, , and Q. Xiao, 2010: The roles of vortex Rossby waves in hurricane secondary eyewall formation. Mon. Wea. Rev., 138, 2092–2109.
Ritchie, E. A., , and W. M. Frank, 2007: Interactions between simulated tropical cyclones and an environment with a variable Coriolis parameter. Mon. Wea. Rev., 135, 1889–1905.
Rotunno, R., , and K. A. Emanuel, 1987: An air–sea interaction theory for tropical cyclones. Part II: Evolutionary study using a nonhydrostatic axisymmetric numerical model. J. Atmos. Sci., 44, 542–561.
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.
Schubert, W. H., , and J. J. Hack, 1982: Inertial stability and tropical cyclone development. J. Atmos. Sci., 39, 1687–1697.
Schubert, W. H., , C. M. Rozoff, , J. L. Vigh, , B. D. McNoldy, , and J. P. Kossin, 2007: On the distribution of subsidence in the hurricane eye. Quart. J. Roy. Meteor. Soc., 133, 595–605.
Shapiro, L. J., , and H. E. Willoughby, 1982: The response of balanced hurricanes to local sources of heat and momentum. J. Atmos. Sci., 39, 378–394.
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.
Sitkowski, M., , J. P. Kossin, , C. M. Rozoff, , and J. Knaff, 2012: Hurricane eyewall replacement cycle thermodynamics and the relict inner eyewall circulation. Mon. Wea. Rev., in press.
Skamarock, W. C., , J. B. Klemp, , J. Dudhia, , D. O. Gill, , D. M. Barker, , W. Wang, , and J. G. Powers, 2005: A description of the Advanced Research WRF Version 2. NCAR Tech. Note NCAR/TN-468+ST, 88 pp.
Smith, R. K., , and M. T. Montgomery, 2010: Hurricane boundary-layer theory. Quart. J. Roy. Meteor. Soc., 136, 1665–1670.
Smith, R. K., , M. T. Montgomery, , and S. Vogl, 2008: A critique of Emanuel’s hurricane model and potential intensity theory. Quart. J. Roy. Meteor. Soc., 134, 551–561.
Smith, R. K., , M. T. Montgomery, , and N. Van Sang, 2009: Tropical cyclone spin-up revisited. Quart. J. Roy. Meteor. Soc., 135, 1321–1335.
Terwey, W. D., , and M. T. Montgomery, 2008: Secondary eyewall formation in two idealized, full-physics modeled hurricanes. J. Geophys. Res., 113, D12112, doi:10.1029/2007JD008897.
Wang, Y., 2008a: Rapid filamentation zone in a numerically simulated tropical cyclone. J. Atmos. Sci., 65, 1158–1181.
Wang, Y., 2008b: Structure and formation of an annular hurricane simulated in a fully compressible, nonhydrostatic model—TCM4. J. Atmos. Sci., 65, 1505–1527.
Wang, Y., 2009: How do outer spiral rainbands affect tropical cyclone structure and intensity? J. Atmos. Sci., 66, 1250–1273.
Willoughby, H. E., , J. A. Clos, , and M. G. Shoreibah, 1982: Concentric eyewalls, secondary wind maxima, and the evolution of the hurricane vortex. J. Atmos. Sci., 39, 395–411.
Wu, C.-C., , H.-J. Cheng, , Y. Wang, , and K.-H. Chou, 2009: A numerical investigation of the eyewall evolution of a landfalling typhoon. Mon. Wea. Rev., 137, 21–40.
Wu, C.-C., , Y.-H. Huang, , and G.-Y. Lien, 2012: Concentric eyewall formation in Typhoon Sinlaku (2008). Part I: Assimilation of T-PARC data based on the ensemble Kalman filter (EnKF). Mon. Wea. Rev., 140, 506–527.
Xu, J., , and Y. Wang, 2010a: Sensitivity of tropical cyclone inner core size and intensity to the radial distribution of surface entropy flux. J. Atmos. Sci., 67, 1831–1852.
Xu, J., , and Y. Wang, 2010b: Sensitivity of the simulated tropical cyclone inner-core size to the initial vortex size. Mon. Wea. Rev., 138, 4135–4157.
Zhang, Q.-H., , Y.-H. Kuo, , and S.-J. Chen, 2005: Interaction between concentric eye-walls in super typhoon Winnie (1997). Quart. J. Roy. Meteor. Soc., 131, 3183–3204.
Zhou, X., , and B. Wang, 2011: Mechanism of concentric eyewall replacement cycles and associated intensity changes. J. Atmos. Sci., 68, 972–988.
It should be noted that (1) and (2) use ground-relative wind speeds. The storm, on average, moves toward the northwest at 1.8 m s−1 due to the β effect, but a budget for storm-relative winds produces virtually identical results and is therefore not included.
Experimentation of time intervals between 1 and 6 h for the time integration of q and for ΔKE and time lags of 1–6 h between Q and ΔKE yielded similar results.
