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    Temporal evolution of minimum sea level pressure (dashed line), maximum total wind (solid line), maximum azimuthal mean tangential wind (thick solid line), and radius of maximum mean tangential wind (dotted line) of the simulated tropical cyclone.

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    Modeled radar reflectivity (dBZ) at surface level from hours 132 to 154 at 2-h intervals, showing the formation of secondary eyewall. The domain is 120 km × 120 km square region from the vortex center.

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    Total inner-core relative vorticity (shading, 10−3 s−1), vertical velocity (contour intervals of 1 m s−1), and asymmetric wind vector at 1.5-km altitude for some selected times during rapid intensification period.

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    Time series of azimuthal mean radial profile of angular velocity (rad s−1).

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    Calculated mean radial profile of effective beta (solid line, 10−9 m2 s−2) and filamentation time (dotted line) at 1-km altitude using temporally averaged mean quantities between (a) hours 129–131 and (b) hours 99–101.

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    Potential vorticity (PVU) distribution at 1.5-km height level from hours 132 to 154 at 2-h intervals, showing the formation of secondary eyewall. The domain is 120 km × 120 km square region from the vortex center. The hollow PV structure of eyewall is absent in the figure because of the color shading scale, as the PV value is larger than 20 PVU in the eye.

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    Time–radius Hovmöller diagram of azimuthally averaged amplitude of high wavenumber (with wavenumber >3) component of relative vorticity (10−4 s−1), superposed by the mean vertical velocity at 5-km altitude (contoured at 0.5, 1, 2, 3, and 4 m s−1).

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    Radius–height diagram of azimuthal mean tangential wind speed change at 6-h intervals: (a) hours 136–142, (b) 142–148, and (c) 148–154.

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    Time-averaged mean structure of eddy kinetic energy (m2 s−2) for (a) wavenumber 1, (b) wavenumber 2, (c) wavenumber 3, and (d) the wavenumber >3, during time period from hours 112 to 124.

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    As in Fig. 9, but averaged between hours 136 and 148.

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    As in Fig. 2, but from hours 100 to 122 at 2-h intervals, during the rapid intensification period.

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    Azimuth–height cross section of PV anomaly field (PVU) at radius of 38 km: (a) 111 h, 0 min; (b) 111 h, 10 min; and (c) 111 h, 20 min. Negative PV anomalies are shaded with blue, starting from −2 PVU. Positive PV anomalies are shaded with orange, starting from +2 PVU. The contour interval is 2 PVU. The characters A, B, and C label the three wave crests, respectively.

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    Time–radius Hovmöller diagram of amplitude of wavenumber-1–3 asymmetric vorticity (10−4 s−1) at 1-km altitude.

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    Time–radius plot of mean tangential wind speed at 1 km (shaded) and mean vertical velocity at 5-km altitude (contoured at 0.5, 1, 2, 3, and 4 m s−1) for the period from hours 100 to 124.

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    Mean tangential wind speed change (m s−1) between hours 100 and 124.

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    Time-averaged (hours 100–124) contributions to local azimuthally mean absolute angular momentum by (a) mean horizontal and vertical fluxes, contoured at ±50, ±100, ±200, and ±300 m2 s−2; and by (b) horizontal and vertical eddy momentum fluxes, contoured at ±10, ±20, ±50, and ±100 m2 s−2.

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The Roles of Vortex Rossby Waves in Hurricane Secondary Eyewall Formation

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  • 1 Key Laboratory of Mesoscale Severe Weather/MOE, and School of Atmospheric Sciences, Nanjing University, Nanjing, China
  • | 2 College of Marine Science, University of South Florida, Saint Petersburg, Florida
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Abstract

A high-resolution, full-physics model initiated with an idealized tropical cyclone–like vortex is used to simulate and investigate the secondary eyewall formation. The beta skirt axisymmetrization (BSA) hypothesis previously proposed is examined and the roles of axisymmetrizing vortex Rossby waves (VRWs) in the secondary eyewall formation are further investigated. During the formation period, convection outside the inner-core region is organized into an outer spiral rainband. The PV dipoles that are persistently generated by convective updrafts through tilting effect move along the rainband and inward toward inner-core region and are finally axisymmetrized in the preexisting beta skirt region. The formation of the secondary eyewall is preceded by a rapid intensification period, during which vortical hot towers, discrete VRWs, and sheared VRWs dominate the inner-core asymmetric structures. Sheared VRWs are repeatedly emanated from the outer edge of the eyewall and become more concentric when propagating outward, leading to the formation of a weak but nonnegligible secondary circulation near the VRWs’ stagnant radius. The mean tangential flow is accelerated by the low-level convergence associated with the secondary circulation and also by the wave–mean flow interaction mechanism, both of which are elucidated by absolute angular momentum budget calculation. The mean radial gradient of relative vorticity is enhanced across the stagnant radius, causing the extension of beta skirt to outer radii in the lower-tropospheric levels. Results from this study suggest that the stagnant radius mechanism and the BSA mechanism may work cooperatively in the sense that the former helps to establish an extensive beta skirt and the latter takes charge from then on.

Corresponding author address: Dr. Zhe-Min Tan, School of Atmospheric Sciences, Nanjing University, Nanjing 210093, China. Email: zmtan@nju.edu.cn

Abstract

A high-resolution, full-physics model initiated with an idealized tropical cyclone–like vortex is used to simulate and investigate the secondary eyewall formation. The beta skirt axisymmetrization (BSA) hypothesis previously proposed is examined and the roles of axisymmetrizing vortex Rossby waves (VRWs) in the secondary eyewall formation are further investigated. During the formation period, convection outside the inner-core region is organized into an outer spiral rainband. The PV dipoles that are persistently generated by convective updrafts through tilting effect move along the rainband and inward toward inner-core region and are finally axisymmetrized in the preexisting beta skirt region. The formation of the secondary eyewall is preceded by a rapid intensification period, during which vortical hot towers, discrete VRWs, and sheared VRWs dominate the inner-core asymmetric structures. Sheared VRWs are repeatedly emanated from the outer edge of the eyewall and become more concentric when propagating outward, leading to the formation of a weak but nonnegligible secondary circulation near the VRWs’ stagnant radius. The mean tangential flow is accelerated by the low-level convergence associated with the secondary circulation and also by the wave–mean flow interaction mechanism, both of which are elucidated by absolute angular momentum budget calculation. The mean radial gradient of relative vorticity is enhanced across the stagnant radius, causing the extension of beta skirt to outer radii in the lower-tropospheric levels. Results from this study suggest that the stagnant radius mechanism and the BSA mechanism may work cooperatively in the sense that the former helps to establish an extensive beta skirt and the latter takes charge from then on.

Corresponding author address: Dr. Zhe-Min Tan, School of Atmospheric Sciences, Nanjing University, Nanjing 210093, China. Email: zmtan@nju.edu.cn

1. Introduction

Over the past 30 years, the tropical cyclone (TC) research and operational forecast communities have witnessed vast improvements in TC track prediction. A large portion of this success lies in the combination of acquisition of better observations, advances in forecasting models and data assimilation schemes, and improved understandings of mechanisms that govern the TC motion (Wang and Wu 2004). However, there are very limited skills in predicting TC formation, rapid intensification, fluctuation, or decay (Elsberry et al. 2007). In addition to large-scale environmental factors, it is now generally recognized that TC intensity and structure changes involve life cycles of vigorous subvortex structures. The inability to represent broad scales of internal processes and their interactions with the external environments ultimately limits the skills in the present numerical forecasts of TC intensity.

Concentric eyewall replacement is one of the routes by which hurricane intensity is modulated, and potentially involves complicated dynamical interactions between spiral rainbands and the inner-core structure of hurricane vortex. Once the secondary eyewall has formed, the subsequent evolution pattern is well documented and understood (Willoughby et al. 1982, hereafter WCS82; Willoughby 1990; Black and Willoughby 1992; Houze et al. 2006; Houze et al. 2007; Rozoff et al. 2008). While the well known “cycle theory” (WCS82) has been successful in explaining the accompanying processes and intensity changes, our understanding of the formation mechanism remains limited.

Over the past years, many hypotheses have been postulated to explain the secondary eyewall formation, and the reader is referred to Terwey and Montgomery (2008, hereafter TM08) for a complete review. Although environmental forcing may play a role (Molinari and Vollaro 1990; Nong and Emanuel 2003), the frequent occurrence nature of concentric eyewall suggests that internal dynamics may be the fundamental one, since the environment may vary dramatically from case to case. Therefore, it is the internal dynamical aspect of secondary eyewall formation we focus on in this study.

The barotropic vortex interaction experiments of Kuo et al. (2004, hereafter KLCW) and Kuo et al. (2008, hereafter KSTK) suggested that axisymmetrization of peripheral cyclonic vorticity perturbations by the strong TC core is responsible for the secondary eyewall formation. Building on the axisymmetrization ideas of previous works (e.g., Melander et al. 1987; Montgomery and Kallenbach 1997, hereafter MK97; KLCW; KSTK) and borrowing the idea of β-plane turbulence from geophysical fluid dynamics, TM08 proposed the beta skirt axisymmetrization (BSA) hypothesis, in which the secondary wind maximum is hypothesized to be generated by anisotropic upscale cascading and axisymmetrization of convectively generated PV anomalies through horizontal sheared turbulence and sheared vortex Rossby waves (VRWs). The BSA theory is by far the most comprehensive one suitable for explaining the secondary eyewall formation, accounting for the continuously injection of strong, small-scale PV anomalies generated by convective cells. Moreover, the ideas of KLCW, KSTK, and TM08 share the common point of view that it is the peripheral vorticity perturbations (usually associated with convective cells embedded in the outer spiral rainbands) that are axisymmetrized and contribute to the secondary eyewall formation. Observational evidences show that one (or more) primary outer rainband with active convective cells persists for more than 12 h, and appears to wrap around the original eyewall, with moat between them (see figures in WCS82; KLCW; KSTK). By artificially adding extra heating rate outside the inner-core region in a numerical sensitivity study, Wang (2009) found that outer spiral rainbands become more active, which in turn favors the development of secondary eyewall.

In contrast, MK97 suggested the stagnant radius of outward propagating VRWs be the focal radius where active wave–mean flow interaction occurs and may contribute to the formation of secondary wind maximum. Increasing evidences show the existence of convectively coupled VRWs and their propagation characteristics are consistent with theoretical prediction (e.g., Chen and Yau 2001; Wang 2002a,b; Corbosiero et al. 2006). However, much less studies have successfully documented the wave–mean flow interaction in both observational and three-dimensional, full-physics numerical modeling contexts, or mentioned its contribution to secondary eyewall formation. This unfortunate situation is primarily due to limitations of observation and inadequate model resolutions.

Concentric eyewall cycle is a common behavior of the strong and more symmetric category of hurricanes (WCS82; Hawkins and Helveston 2004; Kossin and Sitkowski 2009). Moreover, it is often the case that concentric eyewall formation is preceded by a rapid intensification (RI) phase (see Figs. 9 and 10 in WCS82; Fig. 8 in Black and Willoughby 1992; and Fig. 2 in Willoughby and Black 1996). Therefore, a certain relationship between RI and secondary eyewall formation (SEF) can be expected. This idea originates in the fact that a distinct eyewall with annular ring of enhanced vorticity is built up rapidly during the RI period, which supports barotropic instability. Mixing events resulting from barotropic instability could effectively transport high vorticity into the eye, as well as radiate outward-propagating VRWs during the accompanying axisymmetrization process (MK97; Schubert et al. 1999, hereafter S99). Recent airborne dual-Doppler radar observation during the RI phase of Hurricane Guillermo (1997) documented periodic vorticity mixing events (Reasor et al. 2009). Using a barotropic nondivergent model with appropriately specified forcing, Rozoff et al. (2009) showed that repeated rebuilding of eyewall and episodic mixing broaden the radial profile of vorticity, leading to the formation of a vorticity skirt outside the eyewall. Furthermore, recall that nontrivial vorticity skirt with negative gradient outside TC core is a key component of priori assumptions of BSA theory, where the PV anomalies can be axisymmetrized into a cyclonic low-level jet.

The main foci of this study are to (i) examine the recently proposed BSA hypothesis for SEF and (ii) further investigate the vortex evolution during the RI period and exemplify the important role of axisymmetrizing VRWs on TC structure and intensity changes, especially its contribution to secondary eyewall formation. It is also our goal to reconcile the two distinct mechanisms, namely BSA (TM08) and wave–mean flow interaction mechanisms (MK97), proposed for secondary eyewall formation, and illustrate possible connection between them. Since high-resolution, three-dimensional mesoscale models have the capability to reproduce both RI and SEF in idealized settings (see Nguyen et al. 2008; TM08), the resulting dataset with high temporal and spatial resolution will be extremely valuable to elucidate the ideas discussed above in a coherent context and the results will add building blocks to the framework of BSA theory. Accordingly, the structure of this study is organized as follows. The model configuration and initialization are described in section 2. Section 3 contains a detailed description of intensity and structure changes of the simulated storm during the RI and SEF period. The role of VRWs in the secondary eyewall formation is further examined and discussed in section 4. The main findings are summarized in the last section along with a few concluding remarks.

2. Model configuration and experiment design

We employ the fifth-generation Pennsylvania State University–National Center for Atmospheric Research (PSU–NCAR) Mesoscale Model (MM5 version 3.7; Grell et al. 1995) in this study. Four two-way nested domains of 45-, 15-, 5-, and 1.67-km resolution, respectively, are utilized to efficiently model both the vortex scale and various subvortex structures. All domains are square, with each child domain centered at the center of mother domain. Grid points on each side are 121, 121, 121, and 241. The outermost domain is large enough so that open lateral boundary condition used here does not affect the result for the long-time model run. The innermost domain is also large enough to include most of the convective features of the modeled hurricane, such as eyewall and inner and outer spiral rainband. The model has 26 half-sigma levels, 7 of which are placed below 1.5 km above the surface.

The model domain is on an f plane centered at 20°N. The sea surface temperature is fixed throughout all domains at 28.5°C. The initial condition is similar to the one used in Nguyen et al. (2008), beginning with a weak tropical storm–strength axisymmetric vortex. Its associated tangential wind field is specified by product of two functions governing the radial and vertical profiles, respectively:
i1520-0493-138-6-2092-e1
Horizontal wind profile chosen for this study follows that of Fiorino and Elsberry (1989):
i1520-0493-138-6-2092-e2
where b equals 1, the radius of maximum wind (rmax) at 135 km, and the maximum tangential wind speed (υmax) of 15 m s−1. The strength of the swirling wind decreases sinusoidally with height, vanishing at the height of 18 km above the surface. Mean Caribbean sounding during hurricane seasons (Jordan 1958), is used to initialize the far-field temperature and humidity profiles of quiescent environment. The initial perturbation temperature field is set in gradient and hydrostatic balances with the wind field.

The high-resolution Blackadar planetary boundary layer scheme (Blackadar 1979; Zhang and Anthes 1982) and Reisner moist microphysics with Graupel scheme (Reisner et al. 1998) are applied to all model domains. Convection is explicitly simulated in the inner two domains while Betts–Miller cumulus parameterization scheme (Betts and Miller 1986) is used in the outer two domains. No radiative cooling scheme was used in this study. Sensitivity experiments carried out in a previous idealized study have shown that the evolution of the inner core is qualitatively quite similar with or without radiative cooling (Nguyen et al. 2008).

The model is integrated 192 h forward, with a 120-s time step for the coarsest domain. High temporal resolution dataset (with 10-min output interval from 100 to 160 h) is used to investigate the various processes during the period of RI and SEF, and their linkages. General vortex development is summarized in the next section using hourly output.

3. Development of simulated storm

a. Overview

Figure 1 shows the temporal evolution of the storm in terms of the minimum sea level pressure, the maximum total wind, the maximum azimuthal mean tangential wind, and the radius of maximum mean tangential wind (RMW). The maximum total wind and the maximum azimuthal mean tangential wind are calculated near 1 km above the surface, while the minimum pressure at the lowest model level is used approximately as the minimum sea level pressure.

The evolution of the vortex begins with a 6-h period of spinup for boundary layer physical processes, during which there is a discernible decrease in the maximum tangential wind and increase of the minimum sea level pressure. The inflow boundary layer air is moistened quickly, and deep convections are initiated at hour 6. These convective cells, accompanied with localized swirling flow of about 10 m s−1 in strength (estimated by comparison the maximum total wind with maximum axisymmetric tangential wind in Fig. 1), are referred to as vortical hot towers (VHTs; Hendricks et al. 2004; Montgomery et al. 2006). It is then followed by a 14-h period of spinup for moist microphysics, during which consecutive convective bursts transport moisture upward and moisten the atmospheric columns in the inner-core region of the storm, providing a favorable condition for vortex to intensify in the end.

Intensification takes place at hour 20. At hour 84, the maximum tangential wind reaches 35 m s−1 and the minimum sea level pressure drops to 973 hPa, obtaining minimal hurricane strength. During this 64-h period, the inner core is highly asymmetric and dominated by VHTs. The evolution can be generally characterized by separation of convectively generated vorticity dipoles, a vortex merger, and a subsequent axisymmetrization event, culminating in the formation of a mesoscale inner-core region rich in cyclonic vorticity (see Montgomery et al. 2006; Nguyen et al. 2008).

Deepening continues from hours 84 to about 124. During this 44-h period, the minimum sea level pressure decreases from 973 to 927 hPa, while the maximum azimuthal mean tangential wind increases from 35 to 62 m s−1, obtaining the maximum value at hour 124. The averaged mean deepening rate, in terms of the maximum azimuthal mean tangential wind, is larger than the previous stage (hours 24–80) and matches the RI definition of Kaplan and DeMaria (2003, at least 15.4 m s−1 in 24 h), derived statistically from a 12-yr sample of overwater Atlantic tropical cyclones. Thus, this stage is referred to as RI period in the following of this study. At the same time, an eyewall with distinct hollow PV structure has developed during this period and the RMW contracts steadily inward with time from 38 to 20 km.

After gaining temporal maximum intensity, there is a “V” shape in the subsequent time series of the maximum azimuthal mean tangential wind after hour 136. Furthermore, around hour 154, a sudden increase of RWM from 22 to 44 km is also observed. Both evidences indicate a concentric eyewall replacement may be occurring. Snapshots of modeled radar reflectivity at surface level from hours 132 to 154 at 2-h intervals are presented in Fig. 2. At early times (hours 132–136), inner spiral bands around the eyewall are narrow and sharp at their edges. Later, a convection active outer rainband gets closer to the inner-core region. Some of the convective features at the front end of the rainband seem to move inward, and merge with inner bands, making the appearance of the latter much smoother and stronger (hours 136–142). The inner bands are finally axisymmetrized into an intact secondary eyewall (hours 142–154).

After the formation period, the new eyewall begins contracting slowly while axisymmetrizing. Convection associated with the inner eyewall weakens appreciably and finally disappears. While the inner-core maximum wind is steady from hour 168, the minimum sea level pressure drops another 12 hPa, which may be ascribed to the vorticity mixing associated with the demise of the original eyewall (Kossin and Schubert 2001).

b. Eyewall evolution and vorticity mixing during RI period

In the early times of the RI period (hours 84–103), the eyewall evolution is dominated by VHTs, propagating azimuthally around the TC eyewall. The number of VHTs is three–seven during this period, and generally decreases with time. Figure 3 shows some selected snapshots of total relative vorticity, vertical velocity, and asymmetric wind vector field at the 1.5-km height level. Around hour 100, there are 4 VHTs existing in the eyewall. Convective updraft, as strong as 5 m s−1, stretches background vorticity, leading to the formation of localized vorticity maximum. These VHTs revolve around the vortex center, and can last for nearly 2 h. The mesovortex and straight-line segment structure of eyewall vorticity resembles the results of barotropic instability from nondivergent, barotropic modeling experiments (S99; Kossin and Schubert 2001). However, the combination of their evolutionary pictures and mean radial vorticity profile indicates that the VHTs are more likely in asymmetric mode supported by the mean vortex and contribute to the RI (Montgomery and Enagonio 1998; Möller and Montgomery 1999; Nguyen et al. 2008).

Later into the RI period (hours 103–124), a near-circular ring of enhanced vorticity has developed in the eyewall, but exhibits marked polygonal patterns. Furthermore, contrary to that during the early times of RI, the ascending potion of vertical velocity field occupies a nearly continuous region around the eyewall, with some modification by convectively coupled VRWs (Wang 2002a). These coherent wave and subvortex structures are known to be responsible for producing polygonal eyewall shape (Kuo et al. 1999; Schubert et al. 1999; Reasor et al. 2000; Wang 2002b), as well as mixing kinetic and thermodynamic feature from eyewall to eye, and vice versa (Kossin and Eastin 2001; Wang 2002b). First, note that the asymmetric wind associated with VHTs and VWRs has large radial component flowing across the eyewall, generally as strong as 10 m s−1 throughout the RI period. To further illustrate this point, the time series of the radial profile of angular velocity is shown in Fig. 4. At hour 103, a radial profile of angular velocity that has local maximum in the eyewall has already developed. Like the behavior of maximum tangential wind in response to a ring of convection (Shapiro and Willoughby 1982), angular velocity near the eyewall also increases steadily and contracts with time. However, the evolution of angular velocity in the eye is more complicated. On one hand, the transverse circulation associated with convective ring forces subsidence in the eye and divergent outflow in the low-level eye, thus deceases the tangential and angular velocity there, rendering the radial profile to be more “U” shaped. This in turn will cause the ring of vorticity to be more susceptible to barotropic instability. On the other hand, the resulting mixing event effectively transports angular momentum from the eyewall to eye, and increases tangential and angular velocity in the eye. Therefore, it is found in Fig. 4 that during the RI period, angular velocity in the eye has an increasing trend in the aggregate but with temporal variations. It is also interesting to note that the combination of simultaneous contraction of eyewall and mixing in the eye render the modeled vortex core to be characterized by a moderately thick and nearly filled one. According to the classification of the end state of hurricane-like vorticity rings (Hendricks et al. 2009), the most likely pattern of vortex during later state of RI in this experiment is polygonal, and eyewall breakdown never occurs indeed.

c. Secondary eyewall formation

The BSA hypothesis begins with the updated view of the structure of a mature tropical cyclone, which includes extensive skirt of vertical vorticity with nontrivial radial vorticity gradient in the lower troposphere outside the vortex core (Mallen et al. 2005). According to the argument of TM08, the beta skirt plays an important role in constraining the asymmetric flow within the beta skirt region to evolve approximately as quasi-linear axisymmetrization dynamics would predict, transferring perturbation vorticity and kinetic energy from sporadic deep convection into the azimuthal mean flow. With favorable convective potentials, such as large CAPE, low CIN, and long filamentation time (Rozoff et al. 2006), the persistent injection of convection induced PV anomalies will tend to form a substantial low-level jet in the beta skirt.

In our case, an extensive low-level beta skirt becomes evident at the end of the RI and the early SEF period. Azimuthally and temporally averaged mean tangential wind speed is used to calculate the mean radial profile of filamentation time and effective beta (see TM08) at 1-km altitude in order to effectively filter out small-scale perturbations. As shown in Fig. 5a, the beta skirt extends from 40 to 75 km from the TC center and the amplitude is as large as 5 × 10−8 m2 s−2. If we assume the root-mean-square of eddy velocity to be 10–20 m s−1 with an eddy scale of 20 km, the associated vortex beta Rossby number would be approximately 0.5–1. Therefore, the beta skirt is sufficient for axisymmetrizing the convection induced vorticity anomalies. Moreover, the filamentation time generally increases with radius and at >30 min where distance from the center is >45 km. According to TM08, a secondary eyewall will form between 45 and 75 km, if there is persistent PV injection.

The subsequent evolutionary pattern of the low-level PV field is shown in Fig. 6. The TC core is characterized by nearly circular region with elevated PV, but always containing substantial portion of asymmetric structures. The PV perturbations are episodically stripped off from the vortex edge, taking on long trailing spiral shape and becoming more filamented when propagating outward. As shown in Fig. 2, there is a large outer spiral rainband outside the inner-core region with embedded convective cells. The updrafts associated with these cells tilt the horizontal vorticity tube of parent vortex and generate a series of localized PV dipoles (Franklin et al. 2006; TM08), as manifested in Fig. 6. The main body of the outer rainband appears to move inward relative to TC center between hours 130 and 136. From 136 h, a large portion of the PV anomalies in the downwind side of the outer rainband is immersed in the beta skirt. With a higher moving speed, these PV anomalies propagate along the outer rainband “pouch,” and are axisymmetrized one after another in the beta skirt, with ingestion of like-sign PV anomaly and the formation of PV filament propagating outward (Montgomery and Enagonio 1998). At hour 152, another annular ring of elevated PV forms outside the primary vortex core.

To further demonstrate the connection between PV injection and the SEF, a time–radius Hovmöller diagram of the azimuthal mean amplitude of high wavenumber vorticity (with wavenumber greater than or equal to 4), superposed by the mean vertical velocity at 5 km altitude, is shown in Fig. 7. The high wavenumber vorticity is associated with convection cells in the outer spiral rainband, and therefore, is a good indicator of mean area covered by outer rainband. At hour 130, the outer rainband is located at 100–140 km. It moves 30 km inward toward the TC center in the next 5 h and positions at 70–110 km until hour 151, after which it decays. The diffusive character of high wavenumber vorticity amplitude is due to the spiral shape of the outer rainband. Following the intrusion of the PV anomalies, a 0.5 m s−1 contour of mean vertical velocity appears at hour 138, after which the incipient secondary eyewall intensifies and contracts while the primary eyewall dies out at about hour 156.

Intensification of the mean swirling flow associated with the secondary eyewall is illustrated in Fig. 8, which shows the azimuthal mean tangential wind speed change from hours 136–154 at 6-h intervals. During the 18 h, there is a persistent decay of primary eyewall and a little increase of mean flow in the TC eye region. From hours 136–142, there is a maximum value of 4 m s−1 increase of the tangential wind in the beta skirt area above the boundary layer, which extends outward with increasing height through much of the atmospheric column. During the next 6 h, the pattern shifts inward with the maximum magnitude of intensification located in the boundary layer. This feature may be ascribed to the increased boundary convergence and development of supergradient flow. The secondary eyewall continues to contract inward during the period of hours 148–154, and the intensification rate is much larger than seen in the previous time intervals.

It is clear that axisymmetrization of vorticity perturbation from the outer spiral rainband is essentially responsible for the SEF. Therefore, it can be concluded that at least one stationary outer spiral rainband is required to provide enough vorticity perturbations. Guinn and Schubert (1993) carried out an experiment of an axisymmetric vortex interacting with an isolated, spatially fixed heating source away from center. Their results showed that the vortex shifts toward the heating source and systematic wavenumber-1 asymmetry becomes evident. Since the collective effect of the outer spiral rainband can be considered as a heating source outside the primary eyewall, such an idealized scenario is pertinent to our case. During the early phase of SEF, the inner core and the outer spiral rainband moved toward each other, although it is seen from Fig. 7 that the outer spiral rainband moved toward the inner core (because of the center relative view). Accordingly, the inner-core wavenumber-1 asymmetry increased substantially (see Figs. 9a and 10a).

d. Comparison of asymmetric structure during the RI and SEF period

Mean structures of the perturbation fields during the RI period and SEF period are presented in Figs. 9 and 10, respectively, which show the height–radius plot of wavenumber-decomposed and time-averaged structures of eddy kinetic energy (EKE) for hours 112–124 and 136–148. It is noteworthy that there are a number of common features between the two figures, including (i) a significant portion of wavenumber-1 asymmetry throughout the atmospheric column near the eyewall, (ii) asymmetries of wavenumber-1 and wavenumber-2 in the lowest kilometers extend from RMW outward to 90 km, (iii) higher wavenumber (with wavenumber larger than 3) asymmetries are trapped within the radius not far from the primary eyewall, and (iv) high wavenumber asymmetries also exist outside the inner-core region (Fig. 9d). These high wavenumber asymmetries are manifestation of isolated convective cells and those embedded in the outer spiral rainband. There is also a “moat” of depressed amplitude of high wavenumber asymmetries between the eyewall and 50-km radius from the center. This region is known as rapid filamentation zone (Rozoff et al. 2006), where high wavenumber features are strongly suppressed whereas it provides a favorable environment for low wavenumber features (Wang 2008).

During the SEF period, another prominent local maximum of wavenumber-1 asymmetric structure is found around 100 km (Fig. 10a). Moreover, the amplitude of high wavenumber asymmetries during the same period has increased significantly outside the filamentation zone, and the maximum has also shifted inward to 90 km (Fig. 10d). The wavenumber-1 and high wavenumber asymmetries are associated with the band-scale and convective-scale structure of the strong outer spiral rainband. It is the strengthening and inward moving (relative to the vortex center) of this outer rainband that results in the corresponding asymmetric structure changes between Figs. 9 and 10. It is also found in Fig. 10b that wavenumber-2 asymmetry is increased around the radius of 60 km. This result is due to the axismmetrization of high wavenumber vorticity anomalies and projection onto the low wavenumber component. However, the amplitude of all wavenumbers (except for wavenumber 1) asymmetries near the eyewall are much larger during the RI period, which is due to vigorous activities of VHTs and discrete VRWs. Combined with previous analyses, it becomes clear that while the outer spiral rainband plays an important role in SEF period, the perturbation field during the RI period is dominated by VHTs and VRWs in the inner-core region.

Wang (2008) carried out EKE analysis of an idealized mature hurricane, and found that asymmetries in the eyewall are dominated by low wavenumber (see their Fig. 12). Since vigorous activities of VHTs are common features of a rapid intensifying hurricane (Hendricks et al. 2004; Montgomery et al. 2006; Nguyen et al. 2008), it is not surprising that during the RI period in our case the amplitude of high wavenumber asymmetries is of the same order with that of the low wavenumber.

4. The roles of VRWs in the secondary eyewall formation

Mixing between the eyewall and its interior by VHTs and discrete VRWs as discussed in section 3b is only one-half of the picture of axisymmetrization of convectively generated inner-core asymmetries. To conserve angular momentum during the vorticity rearrangement, some of the high eyewall vorticity is also fluxed outward, taking on the form of outward-propagating VRWs, generally coupled with convection (MK97; S99; Chen and Yau 2001). Snapshots of surface level modeled radar reflectivity during the RI period are given in Fig. 11 to show the vigorous activities of inner spiral rainbands, which are active between the outer edge of the eyewall and the radius of 60–70 km from the center. During this period, outer spiral rainbands are far away from the center (outside the radius of 120 km at large). Close examination of corresponding PV field (figure neglected) shows that inner spiral rainbands collocate with local maximum of PV filaments. These bands rotate cyclonically with the high PV core and propagate radially outward.

Figure 12 shows an azimuth–height cross section of the PV anomaly field at radius of 38 km from the storm center, during the 20 min from hour 111. This radius is within the strong negative PV gradient outside the eyewall. Three persistent local PV maxima can be identified in Figs. 12 a–c, and are labeled A, B, and C, respectively. It is clear that this distribution of PV anomaly seems more like a wavenumber-3 pattern. The anomaly B is relatively weak, while the anomalies A and C are much stronger. The azimuthal phase speed estimated from the distances traveled by anomalies A and C in 20 min at 1.5-km altitude is 41 m s−1, which is slower than the azimuthal mean tangential wind of 46 m s−1. Möller and Montgomery (2000) derived the local dispersion relation for a baroclinic wavelike disturbance that propagate on a three-dimensional stably stratified barotropic circular vortex in gradient wind and hydrostatic balance:
i1520-0493-138-6-2092-e3
where n, m, and k are the azimuthal, vertical, and radial wavenumbers, respectively; ω is the local wave frequency; R is the reference radius; η is the mean absolute vorticity; ξ is the mean inertia parameter; N2 is the mean static stability; q is the mean PV; and Ω is the mean angular velocity. The azimuthal phase speed for the wave is Cλ = ωR/n. Therefore, the wave will retrograde relative to the mean flow if the gradient of the mean background PV is negative. Taking the radial and vertical wavelength to be 30 and 10 km, the radial waveumber (k) and vertical wavenumber (m) is 2.1 × 10−4 m−1 and 6.3 × 10−4 m−1. Evaluating other terms at the reference radius (R = 38 km) and at altitude of 1.5 km using the model data, the predicted phase speed is 42.5 m s−1, which is consistent with the value estimated in Fig. 12.
The radial group velocity is obtained by differentiating the wave frequency [Eq. (3)] with respect to radial wavenumber:
i1520-0493-138-6-2092-e4
As the VRWs propagate radially outward, the radial wavenumber becomes increasingly larger, and the term associated with the radial wavenumber dominates the denominator in Eq. (4). Therefore, the radial group velocity is ∼O(k−3) and the stagnation of the wave package can be expected when k grows large enough. To show the characteristics of outward propagation of VRWs in our simulation, the time–radius Hovmöller diagram of the low wavenumber (wavenumber 1–3) asymmetric vorticity amplitude is shown in Fig. 13. It is clear from Fig. 13 that the outward-propagating VRW package is episodic and three major outburst events can be identified. The first event is relatively weak and around hour 106. The second and third events are much stronger and are around hours 116 and 124, respectively. Although modified by convective processes to some extent in our full-physics experiment, most of the low wavenumber VRWs travel outward to the radius about 60 km and never exceed 70 km, beyond which the wave activities are strongly suppressed. The average RMW during the late RI period is about 22 km, which is roughly one-third of the stagnant radius of outward propagation VRWs estimated in Fig. 13. This result is similar to earlier simulation and observation studies (e.g., MK97; Corbosiero et al. 2006).

According to MK97, this stagnant radius provides a site for absorption of wave energy and may contribute to the formation of the secondary wind maximum. Beside this wave–mean flow interaction, there is another process for convectively coupled VRWs to accelerate the local swirling flow at the stagnant radius. Being sheared by the differential rotation outside the eyewall, such convectively coupled vorticity filaments become more narrowed in the radial direction and elongated in the azimuthal direction. This effect renders the updraft associated with inner spiral rainbands to become more concentric, and have larger projection onto the axisymmetric mode. It is indeed the case as found in Fig. 14, which shows the time–radius Hovmöller plot of the azimuthal mean tangential wind speed at 1.5-km altitude and the mean vertical velocity at 5-km altitude. In the eyewall, the low-level maximum tangential wind is nearly collocated with the midlevel vertical velocity maximum, and both of them intensify and contract inward with time. A remarkable feature of Fig. 14 is that there is a local maximum of 0.5 m s−1 contour of vertical velocity appearing from time to time near the radius of 60–70 km from vortex center, and contracting from 65 km at hour 103 to 60 km at hour 124. Coincided with this is the broadening of contours of mean tangential wind speed, especially those outside the radius of 40 km from the center, although distinct secondary wind maximum is not found during this period.

To further illustrate the role of the mean secondary circulation induced by the axisymmetrizing inner spiral rainbands near the stagnant radius, the height–radius plot of the mean tangential wind change between hours 100 and 124 is shown in Fig. 15. During this 24-h period, tangential wind in the eyewall region increases notably, as large as 24 m s−1 and throughout the entire tropospheric column. Wind speed in the eye has also increased due to inward mixing. An outstanding feature in Fig. 15 is that there is a 4 m s−1 increase of mean tangential flow between the radius of 45–70 km in the lowest atmospheric levels. The coincidence of this local maximum of increase with the stagnant radius of VRWs suggests the contribution from the latter, which will be confirmed through angular momentum budget calculation later. The associated change in mean vertical component of vorticity (Δζ = Δυ/r + ∂Δυ/∂r) will cause an increase inside and at the stagnant radius and decrease outside. This greatly enhances the radial gradient of vorticity across the radius of 60 km and extends the beta skirt radially outward (cf. Figs. 5a,b). Since the existence of the beta skirt is the key necessary condition of BSA theory, it can be concluded that the stagnant radius mechanism could contribute to SEF through establishing an extensive low-level beta skirt even if itself may not be directly responsible for the SEF.

Absolute angular momentum (AAM) budget analysis is performed to ascertain the mechanisms discussed above that contribute to the acceleration of mean tangential wind speed at VRW stagnant radius during the RI phase. The AAM budget equation on an f plane is given by
i1520-0493-138-6-2092-e5
where Fr includes contributions from friction, vertical mixing, and horizontal diffusion. If we assume ρ = ρs(r, z), the continuity equation can be approximately written as
i1520-0493-138-6-2092-e6
Combining the AAM and continuity equation, decomposing each variable into mean and perturbation component, and averaging through the azimuthal direction, we get the mean AAM budget equation in the flux form:
i1520-0493-138-6-2092-e7
where the bars and primes denote azimuthal mean and eddy quantities, respectively. The contributions to local tendency of mean AAM on the right-hand side of Eq. (7) are the mean radial and vertical fluxes, the corresponding eddy fluxes, and friction. Figure 16 shows the first two contributions averaged between hours 100 and 124. In the hurricane boundary layer, the strong inflow brings larger angular momentum inward up to the radius of 10 km, and then it turns upward and radially outward, decreasing the angular momentum in the eye and outside the eyewall at about 1–1.5 km above the surface (Fig. 16a). The increase in angular momentum due to the mean inflow will be largely offset by the loss due to friction and vertical diffusion, with the latter transporting AAM upward and compensating for the AAM sink associated with outflow where the tangential wind actually increases (or at least not deceases). These features are similar to that discussed in previous studies (e.g., Wang 2002b). The upper outflow generally transports small AAM outward and causes local AAM to decrease. The largest mean transport of AAM occurs between 7 and 12 km in the slantwise eyewall. Eddies associated with VRWs transport AAM from the eyewall to the eye and increases the AAM inside the eyewall above the boundary layer up to about 5 km (Fig. 16b). A remarkable feature of Fig. 16b is that there is an AAM source at 2 km above the surface at about 60-km radius from the center, showing the absorption of wave energy at the stagnant radius. Correspondingly, a “bump” shaped AAM source is found just above the boundary layer at the same radius in Fig. 16a, which is due to the convergent flow associated with the inner-rainband-induced secondary circulation previously discussed. Comparison of the vertical location between the two momentum sources and where the tangential wind speed actually increases most (see Fig. 15) suggests that the low-level stretching effect of the weak mean secondary circulation outweighs the other contribution.

5. Summary and conclusions

A concentric eyewall cycle and the associated intensity change were stimulated in an idealized, full-physics modeling experiment. The model output was used to (i) verify the recently proposed BSA theory of SEF, and (ii) to examine the inner-core structure and intensity changes during the RI period proceeding the concentric eyewall cycle, with emphasis on elucidating the important roles of outward-propagating VRWs in the secondary eyewall formation.

During the early phase of the SEF period, a well-organized outer spiral rainband moved inward relative to the vortex center, and persisted for more than 20 h. In the convection active outer spiral rainband, PV dipoles were constantly generated through tilting effect, and moved along the band and inward toward inner-core region with a nontrivial, negative radial gradient of relative vorticity. Entering the so-called beta skirt, the PV anomalies were axisymmetrized in sequence and contributed to the formation of secondary ring of enhanced vorticity. Coupled with convection, the nascent eyewall quickly assumed the characteristics of primary eyewall. The pertinent diagnostics are consistent with the picture described by BSA hypothesis.

The formation of the secondary eyewall was preceded by an RI phase, during which a distinct eyewall quickly developed with an approximate ring of enhanced vorticity. Contrary to the formation of mesovortex and relaxation into a monopole, as simulated in the unforced nondivergent barotropic modeling experiments beginning with a thin and hollow ring (S99; Kossin and Schubert 2001; Hendricks et al. 2009), vorticity mixing between the eye and the eyewall in this experiment occurred in a more mild way. It was found that asymmetric mixing resulting from the activities of VHTs and discrete VRWs contemporaneously increases the vorticity in the eye, as the eyewall intensifies and contracts. Therefore, the vortex ring tended to be a moderately thick and nearly filled one, and became a more stable configuration.

Accompanying the vorticity mixing between the eye and the eyewall, sheared VRWs were repeatedly emanated from the eyewall, propagated outward and finally stagnated at about 60 km from the vortex center. The role of VRWs was examined through absolute angular momentum budget analysis. The result reveals that VRWs can accelerate the mean tangential flow at stagnant radius by two different contributions. One is the wave–mean flow interaction mechanism, proposed by MK97. The other is associated with the rainband-induced mean secondary circulation. As being sheared by the differential rotation outside the eyewall, the inner spiral rainbands (or the convectively coupled VRWs) became more narrowed in the radial direction and elongated in the azimuthal direction. This enabled the distribution of the updrafts associated with the rainbands to be more concentric, and have larger projection onto the axisymmetric mode, leading to the nonnegligible mean low-level convergence. The accompanied change in relative vorticity could contribute substantially to the mean vorticity radial profile around the stagnant radius, enhancing the radial gradient of vorticity there and extending the beta skirt radially outward, thereby making the interaction between vortex core and peripheral PV anomalies (i.e., BSA process) easier to realize.

It should be noted that with horizontal grid spacing of 1.67 km, our modeling experiment only marginally resolves the dynamically important stagnant radius of VRWs. Although the results show that stagnant radius mechanism does not play a direct role in the SEF, its validity could not be excluded because of the potential inability to accurately represent the stagnant radius by the current model resolution. Moreover, downdrafts from the primary outer rainband can bring low equivalent potential temperature air from the middle troposphere down to the inflow boundary layer (Barnes et al. 1983; Powell 1990a,b), thereby rendering the stagnant radius less susceptible to couple with deep convection. Such process could also frustrate one’s effort to verify the stagnant radius mechanism of the SEF. Therefore, the logical next step is to address this problem in a carefully designed numerical experiment with higher resolutions. Nevertheless, even if stagnant radius mechanism itself may not be directly responsible for the SEF, it helps to establish an extensive low-level beta skirt with which the BSA mechanism functions.

Acknowledgments

This research is sponsored by the National Key Project for Basic Research (973 Project) under Grant 2009CB421500, the National Natural Science Foundation of China with Grants 40828005 and 40325014, and the National Special Funding Project for Meteorology (GYHY200706033). The first author is supported by NCAR/ASP graduate visiting program during his stay at NCAR. The authors are grateful to Greg Holland, Rich Rotunno, Chris Davis, Chris Snyder, and other NCAR/MMM hurricane group members for the discussions and constructive comments on our initial results.

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Fig. 1.
Fig. 1.

Temporal evolution of minimum sea level pressure (dashed line), maximum total wind (solid line), maximum azimuthal mean tangential wind (thick solid line), and radius of maximum mean tangential wind (dotted line) of the simulated tropical cyclone.

Citation: Monthly Weather Review 138, 6; 10.1175/2010MWR3161.1

Fig. 2.
Fig. 2.

Modeled radar reflectivity (dBZ) at surface level from hours 132 to 154 at 2-h intervals, showing the formation of secondary eyewall. The domain is 120 km × 120 km square region from the vortex center.

Citation: Monthly Weather Review 138, 6; 10.1175/2010MWR3161.1

Fig. 3.
Fig. 3.

Total inner-core relative vorticity (shading, 10−3 s−1), vertical velocity (contour intervals of 1 m s−1), and asymmetric wind vector at 1.5-km altitude for some selected times during rapid intensification period.

Citation: Monthly Weather Review 138, 6; 10.1175/2010MWR3161.1

Fig. 4.
Fig. 4.

Time series of azimuthal mean radial profile of angular velocity (rad s−1).

Citation: Monthly Weather Review 138, 6; 10.1175/2010MWR3161.1

Fig. 5.
Fig. 5.

Calculated mean radial profile of effective beta (solid line, 10−9 m2 s−2) and filamentation time (dotted line) at 1-km altitude using temporally averaged mean quantities between (a) hours 129–131 and (b) hours 99–101.

Citation: Monthly Weather Review 138, 6; 10.1175/2010MWR3161.1

Fig. 6.
Fig. 6.

Potential vorticity (PVU) distribution at 1.5-km height level from hours 132 to 154 at 2-h intervals, showing the formation of secondary eyewall. The domain is 120 km × 120 km square region from the vortex center. The hollow PV structure of eyewall is absent in the figure because of the color shading scale, as the PV value is larger than 20 PVU in the eye.

Citation: Monthly Weather Review 138, 6; 10.1175/2010MWR3161.1

Fig. 7.
Fig. 7.

Time–radius Hovmöller diagram of azimuthally averaged amplitude of high wavenumber (with wavenumber >3) component of relative vorticity (10−4 s−1), superposed by the mean vertical velocity at 5-km altitude (contoured at 0.5, 1, 2, 3, and 4 m s−1).

Citation: Monthly Weather Review 138, 6; 10.1175/2010MWR3161.1

Fig. 8.
Fig. 8.

Radius–height diagram of azimuthal mean tangential wind speed change at 6-h intervals: (a) hours 136–142, (b) 142–148, and (c) 148–154.

Citation: Monthly Weather Review 138, 6; 10.1175/2010MWR3161.1

Fig. 9.
Fig. 9.

Time-averaged mean structure of eddy kinetic energy (m2 s−2) for (a) wavenumber 1, (b) wavenumber 2, (c) wavenumber 3, and (d) the wavenumber >3, during time period from hours 112 to 124.

Citation: Monthly Weather Review 138, 6; 10.1175/2010MWR3161.1

Fig. 10.
Fig. 10.

As in Fig. 9, but averaged between hours 136 and 148.

Citation: Monthly Weather Review 138, 6; 10.1175/2010MWR3161.1

Fig. 11.
Fig. 11.

As in Fig. 2, but from hours 100 to 122 at 2-h intervals, during the rapid intensification period.

Citation: Monthly Weather Review 138, 6; 10.1175/2010MWR3161.1

Fig. 12.
Fig. 12.

Azimuth–height cross section of PV anomaly field (PVU) at radius of 38 km: (a) 111 h, 0 min; (b) 111 h, 10 min; and (c) 111 h, 20 min. Negative PV anomalies are shaded with blue, starting from −2 PVU. Positive PV anomalies are shaded with orange, starting from +2 PVU. The contour interval is 2 PVU. The characters A, B, and C label the three wave crests, respectively.

Citation: Monthly Weather Review 138, 6; 10.1175/2010MWR3161.1

Fig. 13.
Fig. 13.

Time–radius Hovmöller diagram of amplitude of wavenumber-1–3 asymmetric vorticity (10−4 s−1) at 1-km altitude.

Citation: Monthly Weather Review 138, 6; 10.1175/2010MWR3161.1

Fig. 14.
Fig. 14.

Time–radius plot of mean tangential wind speed at 1 km (shaded) and mean vertical velocity at 5-km altitude (contoured at 0.5, 1, 2, 3, and 4 m s−1) for the period from hours 100 to 124.

Citation: Monthly Weather Review 138, 6; 10.1175/2010MWR3161.1

Fig. 15.
Fig. 15.

Mean tangential wind speed change (m s−1) between hours 100 and 124.

Citation: Monthly Weather Review 138, 6; 10.1175/2010MWR3161.1

Fig. 16.
Fig. 16.

Time-averaged (hours 100–124) contributions to local azimuthally mean absolute angular momentum by (a) mean horizontal and vertical fluxes, contoured at ±50, ±100, ±200, and ±300 m2 s−2; and by (b) horizontal and vertical eddy momentum fluxes, contoured at ±10, ±20, ±50, and ±100 m2 s−2.

Citation: Monthly Weather Review 138, 6; 10.1175/2010MWR3161.1

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