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    (a) Time series of MAXW associated with the primary eyewall (thick black), the secondary MAXW (thick gray), the radius of maximum tangential wind (thin black), and the radius of secondary maximum tangential wind (thin gray) for the time period 48–192 h. (b) Low-level (z = 1.1 km) radial profiles of the azimuthal-mean tangential wind (black) and relative vorticity (gray) at 132 h. (c) As in (b), but for 140 h.

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    Modeled surface rain rate (mm h−1) from 127 to 160 h at 3-h intervals, showing the formation of a secondary eyewall and the subsequent eyewall replacement cycle. The plot domain is a 140 km × 140 km square following the TC center. The direction moving toward (white arrow) and speed (text in the bottom-right corner) of storm motion are shown in each plot. The circles are at every 30-km radius from the TC center.

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    Time–azimuth Hovmöller diagram of the radially averaged (between radii of 60 and 100 km) wavenumber-1 (a) radial wind and (b) tangential wind at z = 3 km. Hourly averaged quantities are used. Units are m s−1.

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    (a) Time–radius Hovmöller diagram of the azimuthal-mean vertical velocity (light gray, dark gray, and black lines are the 0.2, 0.5, and 1 m s−1 contours, respectively) at z = 5.4 km. (b) Azimuthal-mean vertical velocity (m s−1) at z = 5.4 km, radially averaged between 55 and 70 km. (c) Integrated kinetic energy (×1016 J) within the annulus of 55–120 km over the depth of 1–6 km. (d) Azimuthal-mean agradient wind (m s−1) at z = 470 m, radially averaged between 30 and 55 km. (e) As in (d), but for convergence (10−4 s−1). Hourly averaged quantities are used.

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    Three-hourly change of the azimuthally averaged tangential wind speed: (a) 130–133, (b) 133–136, (c) 136–139, and (d) 139–142 h. Positive values are contoured at 1, 2, 3, and 4 m s−1 (red lines). The blue line outlines the area with values <−1 m s−1.

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    As in Fig. 5, but for radial wind speed: (a) 130–133 h; positive values are contoured at 0.5, 1, and 2 m s−1 (red lines) and negative values are contoured at −0.5, −1, and −2 m s−1 (blue lines). The gray (black) line is 1 (−1) m s−1 isotach of radial velocity at 130 h. (b)–(d) As in (a) but for 133–136, 136–139, and 139–142 h, respectively.

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    (a) Time–azimuth Hovmöller diagram of the radially averaged (55–70 km) vertical velocity (m s−1) at z = 5.4 km. The black dashed line approximately marks the border where the convection is enhanced, and the gray dashed line tracks the azimuthal propagation of the enhanced convection. (b) The temporally averaged (over 1-h period) and spatially filtered (wavenumber 0–4) midlevel vertical velocity at 136 h [shaded at 0.25 (dark) and 0.5 m s−1 (light)]. The circles are at every 20-km radius from the TC center. The three sectors, labeled A (230°–255°), B (255°–280°), and C (280°–305°) are for use in Figs. 8 and 9. Azimuth rotates counterclockwise from due east at 0° (the same below).

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    Reflectivity (dBZ) at 136 h averaged over the three sectors (a) A, (b) B, and (c) C shown in Fig. 7b and vertical velocity contours with the thin (thick) gray line equal to 0.5 (1) m s−1 and the black line, −0.05 m s−1. (d)–(f) As in (a)–(c), but for diabatic heating (K h−1).

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    As in Figs. 8a–c, but for the asymmetric components of (a)–(c) radial wind (m s−1), (d)–(f) equivalent potential temperature (K), (g)–(i) tangential wind (m s−1), and (j)–(l) pressure (×10 Pa), respectively.

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    Time–azimuth Hovmöller diagrams of radial velocity at z = 470 m, radially averaged (a) between 30 and 50 km and (b) between 50 and 70 km. (c) Difference between (a) and (b). Contour interval is 2 m s−1. Hourly averaged and azimuthally filtered (wavenumber 0–4) quantities are used.

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    (a) Time–azimuth Hovmöller diagram of divergence (shading represents with increasing darkness −4, −5, and −6 × 10−4 s−1) at z = 470 m, radially averaged between 30 and 55 km. (b) As in (a), but for the local agradient wind (m s−1). Hourly averaged and azimuthally filtered (wavenumber 0–4) quantities are used.

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    Horizontal structures of (a) radial velocity (shading interval is 2 m s−1, given at bottom left), (b) divergence (shaded is −5 × 10−4 s−1), and (c) local agradient wind (shading breaks are at 4, 6, 8, and 10 m s−1 given at bottom right) at z = 470 m for 131 h. (d)–(f) As in (a)–(c), but for z = 780 m. Hourly averaged and azimuthally filtered (wavenumber 0–4) quantities are used. The circles are at every 20-km radius from the TC center. The sector bounded by two dashed lines in (b) is for use in Fig. 15a.

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    As in Fig. 12, but for 136 h. The sector bounded by two dashed lines in (b) is for use in Fig. 15b.

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    (top) Tendency of tangential wind due to (a) radial transportation of absolute vorticity, (b) tangential advection, and (c) tangential component of pressure gradient force at z = 470 m for 136 h. (bottom) Tendency of radial wind due to (d) radial advection, (e) tangential advection, and (f) agradient force. Hourly averaged and azimuthally filtered (wavenumber 0–4) quantities are used to calculate each term. Shading intervals are 2 × 10−3 m s−2. The circles are at every 20-km radius from the TC center.

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    (a) Radial velocity (red and blue contours; m s−1), divergence (shaded with intervals of −2, −4, and −6 × 10−4 s−1 from light gray to dark gray), and vertical velocity (thin black is 0.5 m s−1, thick black is 1 m s−1) averaged over the sector shown in Fig. 12b. (b) As in (a), but averaged over the sector shown in Fig. 13b.

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The Roles of Asymmetric Inflow Forcing Induced by Outer Rainbands in Tropical Cyclone Secondary Eyewall Formation

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  • 1 Key Laboratory of Mesoscale Severe Weather/MOE, and School of Atmospheric Sciences, Nanjing University, Nanjing, China
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Abstract

This study analyzes the secondary eyewall formation (SEF) process in an idealized cloud-resolving simulation of a tropical cyclone. In particular, the unbalanced boundary layer response to asymmetric inflow forcing induced by outer rainbands (ORBs) is examined in order to understand the mechanisms driving the sustained convection outside the primary eyewall during the early phase of SEF.

The enhancement of convection in the SEF region follows the formation and inward contraction of an ORB. The azimuthal distribution of the enhanced convection is highly asymmetric but regular, generally along a half circle starting from the downwind portion of the ORB. It turns out that the descending radial inflow in the middle and downwind portions of the ORB initiates/maintains a strong inflow in the boundary layer. The latter is able to penetrate into the inner-core region, sharpens the gradient of radial velocity, and reinforces convergence. Consequently, warm and moist air is continuously lifted up at the leading edge of the strong inflow to support deep convection. Moreover, the inflow from the ORB creates strong supergradient winds that are ejected outward downwind, thereby enhancing convergence and convection on the other side of the storm. The results provide new insight into the key processes responsible for convection enhancement during the early phase of SEF in three dimensions and suggest the limitations of axisymmetric studies. There are also implications regarding the impact of the asymmetric boundary layer flow under a translating storm on SEF.

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

Abstract

This study analyzes the secondary eyewall formation (SEF) process in an idealized cloud-resolving simulation of a tropical cyclone. In particular, the unbalanced boundary layer response to asymmetric inflow forcing induced by outer rainbands (ORBs) is examined in order to understand the mechanisms driving the sustained convection outside the primary eyewall during the early phase of SEF.

The enhancement of convection in the SEF region follows the formation and inward contraction of an ORB. The azimuthal distribution of the enhanced convection is highly asymmetric but regular, generally along a half circle starting from the downwind portion of the ORB. It turns out that the descending radial inflow in the middle and downwind portions of the ORB initiates/maintains a strong inflow in the boundary layer. The latter is able to penetrate into the inner-core region, sharpens the gradient of radial velocity, and reinforces convergence. Consequently, warm and moist air is continuously lifted up at the leading edge of the strong inflow to support deep convection. Moreover, the inflow from the ORB creates strong supergradient winds that are ejected outward downwind, thereby enhancing convergence and convection on the other side of the storm. The results provide new insight into the key processes responsible for convection enhancement during the early phase of SEF in three dimensions and suggest the limitations of axisymmetric studies. There are also implications regarding the impact of the asymmetric boundary layer flow under a translating storm on SEF.

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

1. Introduction

Accumulated aircraft and satellite observations revealed that intense tropical cyclones (TCs) frequently develop a secondary ring of deep convection outside the primary eyewall (Willoughby et al. 1982; Hawkins and Helveston 2004; Hawkins et al. 2006; Kossin and Sitkowski 2009; Kuo et al. 2009). The common route for the subsequent evolution of these concentric ring patterns is known as eyewall replacement cycle (ERC), during which the primary inner eyewall weakens continuously and finally replaced by a contracting and intensifying secondary outer eyewall. The overall changes of TC intensity and structure during ERCs have been well documented by numerous observational and theoretical studies (e.g., Shapiro and Willoughby 1982; Willoughby 1990; Black and Willoughby 1992; Houze et al. 2006; Rozoff et al. 2008; Sitkowski et al. 2011, hereafter SKR11). To get deeper understanding of how and when ERCs happen is of high priority, since the intensity fluctuations occurring in conjunction with ERCs pose a big challenge for the current numerical prediction of TCs (Elsberry et al. 2007; Houze et al. 2007). Moreover, ERCs often result in rapid broadening of destructive force winds and significant increase of integrated kinetic energy (Maclay et al. 2008; SKR11), thereby influencing a larger coastal area and reducing evacuation time. However, the governing mechanisms of secondary eyewall formation (SEF)—the first of a sequence of structure changes during an ERC—are still not well understood.

Although specific synoptic conditions such as upper-level eddy momentum fluxes (Molinari and Vollaro 1990; Nong and Emanuel 2003) and relaxation of vertical wind shear (Hogsett and Zhang 2009) have been suggested as triggers for SEF, the internal dynamics of TCs may play fundamental roles given the frequency of SEF occurrence. Secondary eyewalls usually develop at about 2–3 times the radius of maximum wind (SKR11; Rogers et al. 2012). This location happens to match the stagnation radius of convectively coupled vortex Rossby waves (VRWs) propagating azimuthally and radially outward from the primary eyewall (Montgomery and Kallenbach 1997, hereafter MK97; Chen and Yau 2001; Wang 2002a,b; Corbosiero et al. 2006). MK97 hypothesized that VRWs can accelerate the mean tangential flow at their stagnation radius through VRW-mean flow interactions and may contribute to the formation of the secondary wind maximum (see also Martinez et al. 2011; Menelaou et al. 2012). Borrowing the idea of two-dimensional β-plane turbulence from geophysical fluid dynamics and incorporating three-dimensional convective processes, Terwey and Montgomery (2008) proposed the β-skirt axisymmetrization (BSA) mechanism of SEF. In the BSA mechanism, the secondary wind maximum is hypothesized to be generated through the anisotropic upscale cascading of convectively generated vorticity anomalies in the β-skirt region. Qiu et al. (2010, hereafter QTX10) found that sustained activity of VRWs led to the outward expansion of the β skirt, thus providing enough radial extent for the axisymmetrization of the convection moving in from outer rainbands.

More recent cloud-resolving model simulations as well as observational studies emphasized that a well-established circular secondary wind maximum (typical characteristics of an eyewall) is a late manifestation of SEF, preceded by a period of enhanced azimuthal-mean diabatic heating associated with convective rainbands outside the primary eyewall (e.g., Abarca and Corbosiero 2011; SKR11; Rozoff et al. 2012). The axisymmetric analysis along with the calculations using an balanced vortex model presented in Rozoff et al. (2012) further suggested the existence of a positive feedback, where an expanding wind field in response to diabatic heating increases inertial stability and hence the efficiency of vortex spinup. However, the mechanisms forcing the sustained convection in the SEF region during the early phase of SEF are not clearly recognized.

Observations indicated that the presence of strong outer rainbands might favor the initiation of a secondary ring of convection (e.g., Kuo et al. 2008, their Fig. 1; Kuo et al. 2009, their Fig. 9). Numerical sensitivity studies demonstrated that either artificially increasing the diabatic heating rate in outer rainbands (Wang 2009) or moistening the environmental humidity (Hill and Lackmann 2009) could enhance the activity of outer rainbands and lead to larger storms and SEF events. Huang et al. (2012, hereafter HMW12) found that the precursor to the simulated SEF of Typhoon Sinlaku (2008) was a broadening of the tangential winds, followed by progressive strengthening of boundary layer inflow and enhancement of convergence. Based on the axisymmetric diagnostics, HMW12 concluded that the unbalanced boundary layer response to an expanding wind field in a narrow zone of supergradient winds is important for concentrating and sustaining deep convection. Moreover, the fundamental asymmetric processes associated with outer rainbands may play important roles. Moon and Nolan (2010, hereafter MN10) examined the dynamic response of TC wind field to realistic distributions of diabatic heating from stratiform and convective precipitation in outer rainbands. Common kinematic features of outer rainbands such as horizontal secondary wind maximum could be recovered. In addition, MN10 found that the descending midlevel radial inflow is strongest in the downwind portion owing to pronounced stratiform diabatic heating. This inflow is likely to enhance the boundary layer convergence and trigger convection in the inner-core region of TCs. Fang and Zhang (2012) suggested that the development of new convection preceding SEF might be facilitated with a frontlike feature in the low-level equivalent potential temperature field. The sharpening of equivalent potential temperature gradient was due to the evaporative cooling of rainwater under a wavenumber-1 stratiform cloud deck induced by β shear. Furthermore, statistical analysis of satellite radar reflectivity data suggested that secondary eyewalls probably form from rainbands undergoing axisymmetrization and bottom-up development (Hence and Houze 2012a).

Given the intimate connections between outer rainbands and SEF discussed above, the objective of this study is to get a detailed understanding of how outer rainbands enhance convection activity in the SEF region during the early phase of SEF. It is accomplished by analyzing the unbalanced boundary layer processes in response to asymmetric inflow forcing induced by outer rainbands in an idealized cloud-resolving simulation of SEF. The remainder of this paper is organized as follows. The model configuration and initialization is described in section 2. An overview of the simulated SEF event is provided in section 3. Simple diagnostics are utilized in section 4 to identify the early phase of the simulated SEF. The evolution of the azimuthal-mean structure during the early phase of SEF is presented in section 5. Section 6 details the key dynamical processes responsible for initiating and maintaining the enhanced convection near the site of SEF. Finally, the main findings are summarized and discussed in section 7.

2. Model setup

The model configuration and initialization follows QTX10, and a brief description is provided here. The fifth-generation Pennsylvania State University–National Center for Atmospheric Research (PSU–NCAR) Mesoscale Model (MM5, version 3.7; Grell et al. 1995) is employed and configured with four two-way nested computational domains with 45-, 15-, 5-, and 1.67-km horizontal grid spacing, respectively. All domains are square, and grid points on each side are 121, 121, 121, and 241. The model top is placed at 50 hPa (about 20 km from the surface). There are 26 half-sigma levels in the vertical direction, 7 of which lie below 1.5-km altitude. The Betts–Miller cumulus parameterization scheme (Betts and Miller 1986) is used in the outer two domains, whereas convection in the inner-two domains is explicitly simulated. Other physical parameterizations applied to all model domains include the Blackadar planetary boundary layer scheme (Blackadar 1979; Zhang and Anthes 1982) and the Reisner with graupel scheme (Reisner et al. 1998) for moist microphysical processes.

As described in QTX10, the model begins with an axisymmetric cyclonic vortex embedded in a quiescent tropical environment. The initial vortex is of weak tropical storm strength with the maximum tangential wind speed of 15 m s−1 at a radius of 135 km. The magnitude of tangential wind speed decreases sinusoidally with height, vanishing at 18 km from the surface. The far-field temperature and humidity profiles are based on the mean Caribbean sounding during hurricane seasons (Jordan 1958). The Coriolis parameter is spatially invariant with its value at 20°N. The sea surface temperature is set to a constant 28.5°C.

The model is integrated 192 h forward with a 10-min output interval. The intensity measure, in terms of the maximum azimuthal-mean tangential wind, is calculated using wind filed at 1.1-km altitude.

3. Overview of the SEF process

The simulated TC undergoes a rapid intensification from 84 to 124 h (Fig. 1a). During this 40-h period, the maximum azimuthal-mean tangential wind (MAXW) increases from 35 to 62 m s−1, and the radius of maximum tangential wind decreases steadily from 40 to 22 km. After the rapid intensification phase concludes, there are only moderate oscillations in the MAXW. Meanwhile, the radius of maximum tangential wind stays approximately at the same radius. Consistent with the observations of the near-core radial structure of mature TCs (Mallen et al. 2005), the simulated TC contains a low-level vorticity skirt extending out from the eyewall with broadly negative radial gradient (Fig. 1b). The secondary MAXW along with a localized secondary vorticity maximum emerges at 140 h (Fig. 1c). Thereafter, the MAXW of the primary eyewall exhibits a steady weakening trend (−0.68 m s−1 h−1 on average) and disappears at 156 h. In the meantime, the secondary MAXW intensifies rapidly and contracts inward. The MAXW of the secondary eyewall surpasses that of the inner eyewall at 152 h. Between 140 and 155 h, the mean intensification rate of the secondary MAXW is about 1 m s−1 h−1, and the mean inward contraction rate is about 1 km h−1.

Fig. 1.
Fig. 1.

(a) Time series of MAXW associated with the primary eyewall (thick black), the secondary MAXW (thick gray), the radius of maximum tangential wind (thin black), and the radius of secondary maximum tangential wind (thin gray) for the time period 48–192 h. (b) Low-level (z = 1.1 km) radial profiles of the azimuthal-mean tangential wind (black) and relative vorticity (gray) at 132 h. (c) As in (b), but for 140 h.

Citation: Journal of the Atmospheric Sciences 70, 3; 10.1175/JAS-D-12-084.1

Snapshots of surface rain rate for the periods of SEF and the following ERC are shown in Fig. 2. At the mature stage of the simulated TC, the prominent features of inner-core precipitation include a closed quasi-circular eyewall and the azimuthally elongated inner rainbands that populate the region between radii of 30 and 70 km (Fig. 2a). Isolated convective elements and rainbands are active in the outer region, but the azimuthal preference is not very obvious at this time. From 127 to 142 h, the TC vortex undergoes an apparent trochoidal motion. It is clearly seen in Figs. 2a–f that the motion vector rotates cyclonically with time, at a speed of about 7.27 × 10−5 rad s−1 (or 15° h−1). The motion is closely related to the wavenumber-1 asymmetry in the wind field, which becomes well pronounced at 130 h. For example, the time–azimuth evolution of the wavenumber-1 radial wind and tangential wind at 3-km altitude is shown in Fig. 3. The magnitude of the wavenumber-1 radial wind increases from about 1 m s−1 at 127 h to 3 m s−1 at 130 h, and remains 2–3 m s−1 during the next 12 h. It then decays rapidly when the trochoidal motion concludes. In addition, the phase of the wavenumber-1 radial wind propagates cyclonically at a speed very close to that of the motion vector. The magnitude of the wavenumber-1 tangential wind is similar to that of radial wind, but the phase lags by about 90°. The strengthening of the wavenumber-1 asymmetry and the accompanying trochoidal motion might be due to the wavenumber-1 instability of the TC vortex (e.g., Nolan et al. 2001). It will be shown later in section 6 that a pronounced wavenumber-1 asymmetry of radial wind also exists in the boundary layer, which facilitates the enhancement of inner-core convection during the early phase of SEF through its interaction with the inflow induced by an outer rainband.

Fig. 2.
Fig. 2.

Modeled surface rain rate (mm h−1) from 127 to 160 h at 3-h intervals, showing the formation of a secondary eyewall and the subsequent eyewall replacement cycle. The plot domain is a 140 km × 140 km square following the TC center. The direction moving toward (white arrow) and speed (text in the bottom-right corner) of storm motion are shown in each plot. The circles are at every 30-km radius from the TC center.

Citation: Journal of the Atmospheric Sciences 70, 3; 10.1175/JAS-D-12-084.1

Fig. 3.
Fig. 3.

Time–azimuth Hovmöller diagram of the radially averaged (between radii of 60 and 100 km) wavenumber-1 (a) radial wind and (b) tangential wind at z = 3 km. Hourly averaged quantities are used. Units are m s−1.

Citation: Journal of the Atmospheric Sciences 70, 3; 10.1175/JAS-D-12-084.1

Returning to Fig. 2, the convection outside the radius of 90 km increases significantly in the left quadrants relative to the motion vector between 127 and 130 h as motion speed doubles. The convective cells become clustered together and organize into a continuous rainband. The developing outer rainband moves inward relative to the TC center during the next 3 h. It is located between 120 and 140 km at 130 h and between 90 and 120 km at 133 h. As the downwind part of the outer rainband gets closer to the inner-core region, the precipitation associated with the inner rainbands near the radius of 60 km is strengthened (Fig. 2d). The inner rainbands develop a pronounced wavenumber-2 pattern at 142 h, with one rainband (the other) spiraling from 45-km radius to the east (west) of the TC center to 60-km radius to the west (east). Thereafter, the rainbands evolve into a quasi-closed, elliptical outer eyewall (Figs. 2g–i). A clear moat between the inner and outer eyewalls is evident at 151 h when the MAXWs of the two eyewalls are of equal strength. After 151 h, the secondary outer eyewall continues to intensify and contract inward while the primary inner eyewall decays rapidly. The simulated TC returns back to a single eyewall state at 160 h.

4. Identification of the early phase of SEF

Using flight-level aircraft observations, SKR11 defined the start of SEF as when the first of a coherent cluster of outer wind maxima in space and time is detected. These outer wind maxima are highly correlated with actively convective rainbands (Samsury and Zipser 1995). The diabatic heating from the latter is strong enough to project substantially onto the azimuthal-mean state and promote SEF (e.g., Judt and Chen 2010; Rozoff et al. 2012). From perspectives of axisymmetric balanced dynamics, the azimuthal-mean secondary circulation induced by the enhanced diabatic heating will bring inward large angular momentum, thereby increasing the tangential wind and relative vorticity near and outside the heating source (i.e., an expansion of the tangential wind field and inertial stability field). With elevated level of inertial stability, the increase in tangential wind is larger and more confined near the heat source, leading to the rapid spinup of the secondary wind maximum (e.g., Shapiro and Willoughby 1982; Schubert and Hack 1982; Hack and Schubert 1986; Rozoff et al. 2012). Moreover, the unbalanced processes occurring during the intensification of primary eyewalls (e.g., Smith et al. 2009) are also relevant to the development of secondary eyewalls. Observations showed that the mature secondary eyewalls contain strong supergradient winds within and just above the boundary layer (e.g., Didlake and Houze 2011). HMW12 and Bell et al. (2012) found that following the initial broadening of the tangential wind field, the boundary layer inflow toward the SEF region strengthened steadily. In the meantime, the low-level tangential winds within the SEF region became increasingly more supergradient, which resulted in decelerated inflow and enhanced convergence.

Based on results from the above-mentioned studies, this study divides SEF process into two phases: an early phase followed by a developing phase. The early phase of SEF starts when the diabatic heating from the convective rainbands outside the primary eyewall projects substantially onto the azimuthal-mean state. Alternatively, it can be determined as when notable changes in the azimuthal-mean secondary circulation and primary circulation take place (e.g., enhancement of upward motion outside the primary eyewall and an expansion of the tangential wind field). The latter criterion is adopted in the current study, since it is easier to understand and makes the comparison with other studies more convenient. The start time of the developing phase, or the end time of the early phase, is defined as when a coherent increasing trend of supergradient winds and convergence in the boundary layer near the site of SEF is initiated. A secondary quasi-circular ring of convection, collocated with a well-defined wind maximum, culminates the developing phase. It is worthy to note that this definition of SEF process is essentially consistent with Wu et al. (2012) and HMW12. The dynamical characteristics in the two phases of SEF process are quite similar to the features identified prior to SEF time in Wu et al. (2012) and HMW12. The division of SEF process proposed above is unique in that it provides an effective means to identify the early phase of SEF with a clear dynamical meaning. The rationality of this division is further supported by the evolution of the simulated TC shown below and in section 5.

A time–radius Hovmöller diagram of the midlevel (z = 5.4 km) vertical velocity is shown in Fig. 4a. Before 133 h, there is a weak secondary maximum of upward motion outside the primary eyewall, which results from activity of convectively coupled VRWs (see QTX10). Following the inward contraction of the outer rainband, the upward motion from the outer rainband projects itself onto the azimuthal mean. A 0.2 m s−1 contour of vertical velocity appears at 134 h in the region between radii of 80 and 100 km. In the meantime, the upward motion associated with the inner rainbands around the radius of 65 km is reinforced before the rainbands consolidate to an incipient secondary eyewall (cf. Fig. 2). It is worthy to note that similar results are found in Rozoff et al. (2012), where the SEF process rapidly commences following the sudden expansion of the 0.2 m s−1 contour of the midlevel vertical velocity. Marked change in the azimuthal-mean secondary circulation occurs after 133 h. Take the region bounded by 55 and 70 km radius for example: the mean vertical velocity within the annulus is 0.21 m s−1 on average during the 6-h period of 127–133 h (Fig. 4b). It increases abruptly to 0.38 m s−1 at 135 h and remains this value over the next 3 h. Following the abrupt change in the secondary circulation, rapid expansion of the tangential wind field takes place. The time evolution of integrated kinetic energy due to the azimuthal-mean tangential wind is shown in Fig. 4c. It is clearly seen that the time rate of change of integrated kinetic energy is much larger after 134 h (2.4 × 1015 J h−1) than that over the previous 7-h period (0.78 × 1015 J h−1). Concurrent with the reinforcement of the midlevel upward motion is a sudden jump in the azimuthal-mean agradient wind and convergence in the boundary layer (Figs. 4d,e). After the jump, the magnitude of the azimuthal-mean agradient wind and convergence keeps the elevated state up to 138 h when a persistent increasing trend is initiated.

Fig. 4.
Fig. 4.

(a) Time–radius Hovmöller diagram of the azimuthal-mean vertical velocity (light gray, dark gray, and black lines are the 0.2, 0.5, and 1 m s−1 contours, respectively) at z = 5.4 km. (b) Azimuthal-mean vertical velocity (m s−1) at z = 5.4 km, radially averaged between 55 and 70 km. (c) Integrated kinetic energy (×1016 J) within the annulus of 55–120 km over the depth of 1–6 km. (d) Azimuthal-mean agradient wind (m s−1) at z = 470 m, radially averaged between 30 and 55 km. (e) As in (d), but for convergence (10−4 s−1). Hourly averaged quantities are used.

Citation: Journal of the Atmospheric Sciences 70, 3; 10.1175/JAS-D-12-084.1

In view of the results presented above, the early phase of the simulated SEF in this study is determined roughly as the period between 133 and 138 h.

5. Evolution of axisymmetric structure

Three-hourly changes of the azimuthal-mean tangential and radial wind during the 12-h period from 130 to 142 h are shown in Figs. 5 and 6, respectively. Between 130 and 133 h, there is little change in the tangential and radial winds between radii of 40 and 90 km inside and above the boundary layer (Figs. 5a and 6a). The maximum acceleration of tangential wind is about 1 m s−1 near the radius of 110 km from the TC center. In the boundary layer, there is a 1 m s−1 increase of radial inflow between radii of 100 and 130 km. As shown previously in Fig. 2, the convective elements and rainbands at the same radii are evolving into an outer rainband during this time interval. Therefore, the increases in the tangential and radial winds seen in Figs. 5a and 6a are probably due to the projection of the kinematic features associated with the developing outer rainband into the azimuthal-mean state.

Fig. 5.
Fig. 5.

Three-hourly change of the azimuthally averaged tangential wind speed: (a) 130–133, (b) 133–136, (c) 136–139, and (d) 139–142 h. Positive values are contoured at 1, 2, 3, and 4 m s−1 (red lines). The blue line outlines the area with values <−1 m s−1.

Citation: Journal of the Atmospheric Sciences 70, 3; 10.1175/JAS-D-12-084.1

Fig. 6.
Fig. 6.

As in Fig. 5, but for radial wind speed: (a) 130–133 h; positive values are contoured at 0.5, 1, and 2 m s−1 (red lines) and negative values are contoured at −0.5, −1, and −2 m s−1 (blue lines). The gray (black) line is 1 (−1) m s−1 isotach of radial velocity at 130 h. (b)–(d) As in (a) but for 133–136, 136–139, and 139–142 h, respectively.

Citation: Journal of the Atmospheric Sciences 70, 3; 10.1175/JAS-D-12-084.1

Following the inward contraction of the outer rainband during the next 3 h (133–136 h), the area with increasing tangential wind above the boundary layer also shifts inward (Fig. 5b). The magnitude is larger than 1 m s−1 between radii of 90 and 110 km throughout the lower- and midtropospheric levels. Moreover, there is a 1–2 m s−1 acceleration of tangential wind inside and just above the boundary layer over a substantial radial span (45 < r < 100 km). The spinup of the low-level tangential wind is associated with an abrupt strengthening of boundary layer inflow over the radial interval of 50 < r < 110 km (Fig. 6b). Concurrent with the increase of boundary layer inflow outside the radius of 50 km is a decrease of inflow toward the primary eyewall. The sharpening of radial inflow gradient between radii of 30 and 55 km results in a narrow zone of concentrated boundary layer convergence (cf. Fig. 4e). Within this radial interval, the boundary layer inflow is turned upward to initiate deep convection, as can be inferred from the upward extension of the outflow above the boundary layer (Figs. 6b and 6c). Note that there is a 1 m s−1 acceleration of tangential wind near the radius of 65 km in the middle levels of the troposphere. This is most likely related to the reinforced secondary circulation outside the primary eyewall, which could transport excessive angular momentum in the boundary layer upward into the free atmosphere (e.g., Rozoff et al. 2012, their Fig. 4).

The pattern of change in the low-level radial wind between 136 and 139 h (Fig. 6c) is similar to that over the previous 3-h period, although the magnitude is smaller. As shown previously in Fig. 4e, the boundary layer convergence between radii of 30 and 55 km is generally maintained during this time interval. In addition, the increase of tangential wind within the boundary layer is less (Fig. 5c). Above the boundary layer, the area with strengthening tangential wind extends inward to the radius of 60 km. The extra area of increase (between radii of 60 and 80 km) is collocated with the updraft associated with the reinforced secondary circulation outside the primary eyewall. It is worthy to note that the average value of inertial stability in the region given by 50 < r < 90 km and 1 < z < 6 km increases by 17% (1.1 × 10−4 s−1) during the 6-h period between 133 and 139 h.

The wind fields evolve in a substantially different way over the next 3-h period (139–142 h). The location of the maximum increase in the boundary layer inflow has shifted inward to the radius of 50 km. As a result, the boundary layer convergence near the radius of 40 km is greatly enhanced (Fig. 6d). Above the boundary layer, airflow is getting more confluent in a limited area as well. These changes in the low-level radial wind field suggest that the enhanced convection is evolving into an incipient eyewall. In the meantime, the largest increase in the low-level tangential wind field is now concentrated near the radius of 60 km (Fig. 5d). Recall that the secondary wind maximum appears during this time interval. Moreover, the ongoing acceleration of the low-level tangential wind outside the radius of 80 km has largely ceased.

The evolution of the azimuthal-mean structure of the simulated TC during SEF presented above is similar to the recent findings of HMW12 in several aspects, such as the expansion of the tangential wind field concurred with the reinforcement of radial inflow and convergence in the boundary layer. Moreover, it is found that the acceleration of the tangential winds occurs in the region occupied by the outer rainband and the reinforced inner rainbands. This is consistent with the results from Rozoff et al. (2012), which demonstrated that the outward expansion of the tangential wind field was predominantly a symmetric response to the azimuthal-mean and wavenumber-1 components of the transverse circulation induced by rainband activity. It will be shown in the next section that the initiation and maintenance of radial inflow in the boundary layer during the early phase of SEF is closely related to the three-dimensional kinematic features of the outer rainband.

6. Enhancement of inner-core convection

To understand how the outer rainband enhances convection activity in the SEF region during the early phase of SEF, it is worthwhile to examine the azimuthal distribution of the enhanced convection as well as the mesoscale structures of the outer rainband first.

a. Azimuthal distribution

The composites of airborne Doppler radar analyses in Rogers et al. (2012) showed that a secondary maximum of the midlevel upward motion with a magnitude of about 0.5 m s−1 exists outside the primary eyewall. Since over one-third of the storms in the sample contain mature secondary eyewalls, this value is a reasonable reference for the definition of enhanced rainband convection. In addition, Rozoff et al. (2012) demonstrated that similar magnitude of vertical motion could be reproduced with a balanced vortex model using the diabatic forcing from active rainbands preceding SEF. Therefore, it is suggested here that rainband convection be considered as enhanced when the midlevel upward motion is larger than 0.5 m s−1 covering a substantial azimuthal span in order to contribute positively to the azimuthal-mean and promote SEF.

A time–azimuth Hovmöller diagram of the radially averaged (between radii of 55 and 70 km) midlevel vertical velocity is shown in Fig. 7a. Consistent with the axisymmetric view in Fig. 4b, the convection activity within the annulus is strengthened at 134 h. Over the next 6-h period, the azimuthal distribution of enhanced convection is strongly modulated by a wavenumber-1 signal, indicating that there are critical quadrants providing favorable conditions for initiating and sustaining deep convection. Once initiated, the convection propagates downstream at a speed of 4.85 × 10−4 rad s−1 (or 100° h−1), which is about 80% of the advection speed by the mass-weighted mean tangential wind (averaged within the annulus and over the depth between 1.5 and 5 km). The preferred azimuth for the initiation of deep convection (indicated by the black dashed line in Fig. 7a) also rotates cyclonically with time, but at a much slower speed similar to that of the outer rainband (cf. Fig. 2). It is interesting to note that a notable wavenumber-2 asymmetry of upward motion emerges at 142 h (during the developing phase of SEF), which is accompanied by the wavenumber-2 pattern in the precipitation field (Fig. 2f).

Fig. 7.
Fig. 7.

(a) Time–azimuth Hovmöller diagram of the radially averaged (55–70 km) vertical velocity (m s−1) at z = 5.4 km. The black dashed line approximately marks the border where the convection is enhanced, and the gray dashed line tracks the azimuthal propagation of the enhanced convection. (b) The temporally averaged (over 1-h period) and spatially filtered (wavenumber 0–4) midlevel vertical velocity at 136 h [shaded at 0.25 (dark) and 0.5 m s−1 (light)]. The circles are at every 20-km radius from the TC center. The three sectors, labeled A (230°–255°), B (255°–280°), and C (280°–305°) are for use in Figs. 8 and 9. Azimuth rotates counterclockwise from due east at 0° (the same below).

Citation: Journal of the Atmospheric Sciences 70, 3; 10.1175/JAS-D-12-084.1

A plane view of the midlevel vertical velocity at 136 h is given in Fig. 7b. Hourly averaged and azimuthally filtered (wavenumber 0–4) quantities are used in order to remove small-scale features in time and space. This method of data manipulation is also adopted in Figs. 1013. It is clearly seen in Fig. 7b that the convection within the annulus of 55 < r < 70 km is enhanced in the southeastern quadrant, northeastern quadrant, and half of the northwestern quadrant. The upwind end of the enhanced convection (60–80 km south to the TC center) is on the radially inward side of the outer rainband. The concentrated upward motion in the outer rainband is located between radii of 80 and 100 km south to the TC center over an azimuthal span of 230° < α < 290°. As shown previously in Fig. 2d, convection within this region is clustered together showing a continuous band structure, while convection on the upwind side is broken into discrete cells. Note that the intense updrafts associated with the isolated convective cells on the upwind side are not well depicted in Fig. 7b because of azimuthal filtering.

b. Outer rainband structure

Figure 8 shows modeled reflectivity, diabatic heating, as well as vertical velocity azimuthally averaged over the three consecutive sectors encompassing the region of concentrated upward motion in the outer rainband. The azimuthal span of each sector is large enough so that the averaged quantities are representative of the mesoscale features of the rainband, but small enough so that the change of characteristics along the rainband could be identified. The asymmetric components of radial wind, equivalent potential temperature, tangential wind, and pressure are given in Fig. 9. With the azimuthal-mean state removed, the asymmetric component of a quantity effectively distinguishes the rainband-associated feature from the background.

Fig. 8.
Fig. 8.

Reflectivity (dBZ) at 136 h averaged over the three sectors (a) A, (b) B, and (c) C shown in Fig. 7b and vertical velocity contours with the thin (thick) gray line equal to 0.5 (1) m s−1 and the black line, −0.05 m s−1. (d)–(f) As in (a)–(c), but for diabatic heating (K h−1).

Citation: Journal of the Atmospheric Sciences 70, 3; 10.1175/JAS-D-12-084.1

Fig. 9.
Fig. 9.

As in Figs. 8a–c, but for the asymmetric components of (a)–(c) radial wind (m s−1), (d)–(f) equivalent potential temperature (K), (g)–(i) tangential wind (m s−1), and (j)–(l) pressure (×10 Pa), respectively.

Citation: Journal of the Atmospheric Sciences 70, 3; 10.1175/JAS-D-12-084.1

The outer rainband region in sector A contains intense convective cells embedded in stratiform precipitation (Fig. 8a). A strong overturning circulation is evident in this sector, composed of a low-level radial inflow, an outward-leaning updraft core, and an upper-level radial outflow (Fig. 9a). Warm and moist air is transported upward by the updraft, leading to a 2-K anomaly in equivalent potential temperature field near the radius of 90 km throughout the lower and middle tropospheric levels (Fig. 9d). On the radially outward side of the updraft core, there is a positive anomaly in the tangential wind field over the depth between 1 and 4 km (Fig. 9g). Note that care should be taken when interpreting the actual magnitude of this tangential jet. As shown previously in Fig. 3b, the tangential winds in the southern quadrants are about 2–3 m s−1 less than the azimuthal-mean at 136 h, owing to the wavenumber-1 asymmetry in the wind field. It is also seen in Figs. 9g–i that negative anomalies fill the entire domain except for those regions strongly modified by the rainband circulation.

Also in sector A, there is a radial inflow, which originates from about 3-km altitude at the radius of 160 km and descends slantwise to the surface. As demonstrated by MN10, this descending radial inflow occurs largely as a response to the particular distribution of diabatic heating from stratiform precipitation. The outflow above 3-km altitude transports hydrometers radially outward, leading to the outward expansion of the stratiform cloud deck (Figs. 8a and 9a). Between radii of 130 and 160 km, diabatic heating is positive above 3-km altitude, and generally negative below (Fig. 8d). The descending radial inflow is collocated with the area of strong diabatic cooling, and both follow the outer periphery of the stratiform precipitation. Large vertical gradient of reflectivity within the area indicates that the diabatic cooling is due to evaporation of rainwater. The strong diabatic cooling, on one hand, decreases air temperature and initiates downward motion; on the other, it induces a positive anomaly of air pressure below 3-km altitude outside the radius of 100 km (Fig. 9j). The outer periphery of the descending region approximately follows the central axis of the outward-leaning pressure anomaly. Within the descending region, the radial pressure gradient force associated with the pressure anomaly is inward pointing, thus accelerating the inward motion of air parcels. The largest positive anomaly of pressure is located below 1.5-km altitude at the radius of 130 km, which results in the rapid acceleration of radial inflow in the boundary layer between radii of 110 and 130 km. In addition, the updraft warms the tropospheric column and induces a negative anomaly of pressure at the radius of 90 km. This low-pressure anomaly, along with the high-pressure anomaly resulting from diabatic cooling, contributes to the inflow maximum near the radius of 95 km.

Taken together, the rainband region in sector A contains an overturning circulation, a low-level jet on the outward side of the updraft, as well as a descending radial inflow initiated under the stratiform cloud deck. These circulation features are reminiscent of the convective-scale motions commonly observed in the middle portion of the principal rainband (Barnes et al. 1983; Powell 1990; Hence and Houze 2008). If viewed as the aggregate reflection of convective structures upon the mesoscale, the rainband identified in sector A is consistent with the results from the above-mentioned observational studies. Following the classification of different portions (i.e., upwind, middle, and downwind) in the principal rainband, the region in sector A is referred to as the middle portion of rainband.

Proceeding downwind, the convective cells gradually collapse into heavy stratiform precipitation (Figs. 8b,c). The overturning circulation, as well as the convectively generated low-level jet, fades accordingly (Figs. 9b,c,h,i). Radial inflow even appears in the middle levels of sector C, which is most likely due to the enhanced diabatic cooling over the depth of 3–5 km outside the radius of 130 km in sector B (Fig. 8e). As shown in MN10, such a distribution of diabatic heating from stratiform precipitation will generate a positive PV anomaly and hence a cyclonic circulation in the middle levels, leading to the enhancement of radial inflow on the downwind side (sector C) and reinforcement of the outflow on the upwind side (sector A). This midlevel cyclonic circulation could further modify the distribution of hydrometers and hence diabatic heating. It is seen in Figs. 8a–c that the stratiform cloud deck is more (less) expansive in sector A (sector C) than that in sector B.

In contrast to the results of MN10, which suggested that the descending radial inflow is strongest in the downwind portion of rainbands, it is found here that the descending radial inflow, diabatic cooling rate, and the pressure anomaly in sector B are of comparable magnitude to those in sector A. The discrepancy might result from the idealized setup of the distribution of diabatic heating in MN10. In their experiments, the rainband-induced circulations (e.g., the convective overturning circulation in the middle portion and the midlevel cyclonic circulation in the stratiform portion) do not feed back to the distribution of diabatic heating. Moreover, without including environmental flow in the current simulation, the thermodynamic properties of the air parcels moving into sector B have already been heavily modified by the stratiform precipitation in sector A. For the same reason, the descending radial inflow, diabatic cooling, and the pressure anomaly are much less evident in sector C. However, the horizontal wind maximum, which follows the path of the descending radial inflow, becomes increasingly more pronounced from sector A to sector C (Figs. 9g–i). Different from the convective nature of the low-level jet in the middle portion, this horizontal wind maximum is due to the stratiform characteristics of the rainband. The descending radial inflow brings inward larger momentum and increases the tangential winds along its path. Meanwhile, the positive anomaly of tangential wind is azimuthally advected downwind. As a result, the horizontal wind maximum is most evident in sector C.

As shown above, the stratiform characteristics in terms of diabatic cooling and the descending radial inflow are qualitatively similar among the three sectors. This is quite different from the situation in a typical principal rainband, where the stratiform characteristics are much more prominent in the downwind portion. However, as in the principal rainband, the fading of convective structures along the rainband is apparent. Taken as a whole, the transition from convective overturning circulation in the middle portion to stratiform circulation features downwind (i.e., the midlevel cyclonic circulation, and the horizontal wind maximum following the path of the descending radial inflow) is still evident. In this aspect, the region encompassed by sectors B and C shows similarity to the downwind portion of the principal rainband and therefore is referred to as the downwind portion in the following. In the boundary layer, the inflow maximum moves progressively inward to the radii of 60–80 km from the TC center (Figs. 9a–c), which is primarily due to the contributions from the tangential and radial advection (i.e., the three dimensionality of the rainband). In addition, once the enhanced inner-core convection is initiated, it induces a low pressure anomaly between the radii of 60 and 80 km (Figs. 9k,l). The resulting increased radial pressure gradient force also leads to the reinforcement of the boundary layer inflow. Furthermore, the cooler air carried by the descending radial inflow does not seem to be detrimental to the inner-core convection (cf. Barnes et al. 1983; Powell 1990), since its magnitude is relatively small (3–4 K) and becomes less when entering the boundary layer (Figs. 9d–f).

c. Boundary layer response

A time-azimuth Hovmöller diagram of radial velocity at the midlevel of the boundary layer is shown in Fig. 10. At 127 h, a wavenumber-3 pattern of inflow is evident, especially between radii of 50 and 70 km. The two branches of radial inflow in the eastern quadrants of the TC are much stronger than that to the west of the TC center, indicating that the wavenumber-3 pattern of boundary layer inflow is strongly modulated by a wavenumber-1 asymmetry. Following the strengthening of the wavenumber-1 asymmetry in the free atmosphere and the initiation of the trochoidal motion (Figs. 2 and 3), the wavenumber-1 asymmetry of radial inflow in the boundary layer becomes increasingly more apparent from 127 h and well pronounced at 131 h. From 132 to 135 h, the boundary layer inflow increases substantially between radii of 50 and 70 km in the southwestern quadrant, where the negative phase of the preexisting wavenumber-1 asymmetry of inflow originally resides (Fig. 10b). In the meanwhile, the location of the strengthening inflow rotates cyclonically, approximately following the azimuthal propagation of the outer rainband. As shown previously in Fig. 9, the outer rainband induces radial inflow that descends to the surface, leading to the initiation and maintenance of the strong boundary layer inflow toward the inner-core region. Consequently, the gradient of radial velocity is greatly sharpened between radii of 40 and 60 km south of the TC center (Fig. 10c), which results in a narrow zone of concentrated convergence in the boundary layer (Fig. 11a). Warm and moist air is lifted up at the leading edge of the strong inflow to reinforce the updrafts feeding the deep convection (Figs. 9b,c,e,f). In addition to the convergence zone to in the southern quadrants, there is another convergence zone in the boundary layer, located in the northern quadrants. The latter convergence zone is transient between 130 and 134 h, while it becomes well persistent and rotates cyclonically with time from 135 h (Fig. 11a). It is found that the regularization of this convergence zone is also due to the inflow forcing induced by the outer rainband, but the corresponding dynamical processes are more involved. The governing equations for tangential and radial wind in cylindrical coordinates centered over the TC center are used to aid the explanation.

Fig. 10.
Fig. 10.

Time–azimuth Hovmöller diagrams of radial velocity at z = 470 m, radially averaged (a) between 30 and 50 km and (b) between 50 and 70 km. (c) Difference between (a) and (b). Contour interval is 2 m s−1. Hourly averaged and azimuthally filtered (wavenumber 0–4) quantities are used.

Citation: Journal of the Atmospheric Sciences 70, 3; 10.1175/JAS-D-12-084.1

Fig. 11.
Fig. 11.

(a) Time–azimuth Hovmöller diagram of divergence (shading represents with increasing darkness −4, −5, and −6 × 10−4 s−1) at z = 470 m, radially averaged between 30 and 55 km. (b) As in (a), but for the local agradient wind (m s−1). Hourly averaged and azimuthally filtered (wavenumber 0–4) quantities are used.

Citation: Journal of the Atmospheric Sciences 70, 3; 10.1175/JAS-D-12-084.1

The original form of tangential and radial wind equations is given as follows:
e1
e2
where , , and are the radial, tangential, and vertical components of the wind vector; , , , and are the time, radial, tangential, and vertical (constant height) coordinates; , , and are the Coriolis parameter, air density, and pressure; and and are the radial and tangential components of diffusion, respectively. After some regrouping of terms, one can obtain a transformed set of equations:
e3
e4
e5
e6
where is the vertical component of absolute vorticity, and AF stands for the agradient force defined as the sum of the Coriolis force, the centrifugal force, and the radial component of pressure gradient force. An air parcel experiences a radially outward acceleration when the value of the agradient force is positive and the dissipation effects (i.e., ) are negligible. Another way of measuring the degree of local unbalance among the three forces is to introduce the local gradient wind balance relationship, the local gradient wind and the local agradient wind following the definitions of their axisymmetric counterparts:
e7
e8
For convenience, local agradient wind is referred to as agradient wind in the following. It is clear from the definitions that a positive agradient wind, or supergradient wind, corresponds to an outward-pointing agradient force.

The strengthening of boundary layer inflow in the downwind portion of the outer rainband leads up to a sequence of structure changes in the downstream boundary layer. A substantial increase in tangential winds occurs as a result of the inward transportation of absolute vorticity. Since supergradient winds are common in the inner-core boundary layer of a mature TC (e.g., Smith et al. 2009), a sudden increase in tangential winds results in a corresponding increase in agradient winds. It is clearly seen in Fig. 11b that the agradient winds build up rapidly from 134 h in the eastern quadrants, following the strengthening of the southern branch of inflow (cf. Fig. 10). That the localized maximum of agradient winds is situated on the downwind side of the inflow maximum is primarily due to the contribution from the azimuthal advection term of tangential winds, which will be demonstrated later via detailed calculations. It is also found in Figs. 10a and 11b that, as the supergradient winds strengthen, the radial inflow within and on the downwind side of the supergradient-wind zone decreases continuously (owing to the continued outward acceleration within the zone). For example, the −6 m s−1 isotach of radial inflow at 134 h to the east of the TC center decreases to less than −4 m s−1 in the next 2 h. In addition, the decreased radial velocity is advected downwind, as indicated by the packing of contours in the northeast quadrant around 136 h. Meanwhile, the radial inflow between radii of 50 and 70 km over the corresponding azimuthal span largely maintains its strength (Fig. 10b). As a result, the difference of radial velocity between the radii of 40 and 60 km is increased (Fig. 10c), leading to a coherent convergence zone in time and space to the north of the TC center (Fig. 11a).

While Hovmöller diagrams are useful for highlighting the important kinematic and dynamical changes in the boundary layer, the magnitudes and locations of the inflow maximum, the agradient-wind zone, and the convergence zone induced by the outer rainband are not precisely delineated, owing to their spiral structures and the radial averaging applied to Figs. 10 and 11. Therefore, planar views of radial velocity, divergence, and agradient wind for 131 and 136 h, which serve as examples representative of other neighboring times, are displayed in Figs. 12 and 13, respectively. At 131 h, there is a notable wavenumber-1 pattern of radial velocity in the boundary layer (Figs. 12a,d), which is related to the preexisting wavenumber-1 asymmetry (Fig. 3a). With a slow translating speed, the inflow maxima in the boundary layer are generally located in the right-front quadrant relative to the storm motion (cf. Fig. 2), which is consistent with the results from early theoretical studies (e.g., Shapiro 1983). Supergradient-wind zones are on the downwind side of the inflow maxima (Figs. 12c,f). The convergence zone to the east of the TC center is only apparent at the midlevel of the boundary layer (Figs. 12b,e). At 136 h, there are two branches of strong inflow in the boundary layer (Fig. 13a). The one to the north is associated with the preexisting wavenumber-1 asymmetry. The southern branch, which exhibits a typical spiral pattern, is initiated and maintained by the outer rainband. The inflow from the outer rainband penetrates into the inner-core region, which leads to the sharpening of the gradient of radial velocity and hence the formation of the convergence zone in the southeastern quadrant between radii of 40 and 60 km (Fig. 13b).

Fig. 12.
Fig. 12.

Horizontal structures of (a) radial velocity (shading interval is 2 m s−1, given at bottom left), (b) divergence (shaded is −5 × 10−4 s−1), and (c) local agradient wind (shading breaks are at 4, 6, 8, and 10 m s−1 given at bottom right) at z = 470 m for 131 h. (d)–(f) As in (a)–(c), but for z = 780 m. Hourly averaged and azimuthally filtered (wavenumber 0–4) quantities are used. The circles are at every 20-km radius from the TC center. The sector bounded by two dashed lines in (b) is for use in Fig. 15a.

Citation: Journal of the Atmospheric Sciences 70, 3; 10.1175/JAS-D-12-084.1

Fig. 13.
Fig. 13.

As in Fig. 12, but for 136 h. The sector bounded by two dashed lines in (b) is for use in Fig. 15b.

Citation: Journal of the Atmospheric Sciences 70, 3; 10.1175/JAS-D-12-084.1

To demonstrate the responses of downstream boundary layer to asymmetric inflow forcing from outer rainband, several terms in Eqs. (3) and (4) are calculated and shown in Fig. 14. The southern branch of inflow strengthens the tangential winds through inward transportation of absolute vorticity (Fig. 14a). The maximum increase in the tangential wind field due to inward transportation of absolute vorticity (near the radius of 50 km to the southeast of the TC center) is not collocated with the inflow maximum associated with the outer rainband, since the absolute vorticity between radii of 40 and 60 km is much larger than that near the inflow maximum (cf. Fig. 1b). In the inner-core region, where the tangential winds are already supergradient, an increase of tangential winds corresponds to an increase of agradient winds. By such means, the inflow from the outer rainband creates strong supergradient winds in the southeastern quadrant. Note that the maximum agradient wind is slightly on the downwind side of the area where the largest increase of tangential wind due to inward transportation of absolute vorticity occurs (cf. Figs. 13c and 14a). This is because of the azimuthal advection term that tends to shift the agradient wind zone downstream (Fig. 14b). The tangential component of pressure gradient force is minor in the balance of forces (Fig. 14c). Within the zone of strong supergradient winds, the radial inflow loses its strength steadily because of the continuous outward acceleration by the positive agradient force (Fig. 14f). The decreased radial velocity is advected downstream to the radially inward side of the northern strong inflow (Figs. 13a and 14e), which leads to the enhancement of the convergence zone to the north of the TC center (Figs. 13b,e). The radial advection term shifts large radial velocities inward, which is beneficial for the maintenance of convergence (Fig. 14d).

Fig. 14.
Fig. 14.

(top) Tendency of tangential wind due to (a) radial transportation of absolute vorticity, (b) tangential advection, and (c) tangential component of pressure gradient force at z = 470 m for 136 h. (bottom) Tendency of radial wind due to (d) radial advection, (e) tangential advection, and (f) agradient force. Hourly averaged and azimuthally filtered (wavenumber 0–4) quantities are used to calculate each term. Shading intervals are 2 × 10−3 m s−2. The circles are at every 20-km radius from the TC center.

Citation: Journal of the Atmospheric Sciences 70, 3; 10.1175/JAS-D-12-084.1

At 0.78-km altitude, the strength of both inflow maxima is about 6 m s−1 weaker. Strong outflow even appears in the region where the minimum of inflow resides at the midlevel of the boundary layer (Figs. 13a,d). With less dissipation effects, the tangential winds and hence agradient winds are larger, especially in the southeast quadrant (Fig. 13f). Moreover, convergence in the northeastern quadrant is enhanced at this altitude and is similar in magnitude to that in the southeastern quadrant (Fig. 13e).

To further explore how the supergradient winds created by the inflow from the outer rainband affect the convergence and upward motion within this region, the radius–height structures of radial velocity, divergence, and vertical velocity averaged over the sectors in Figs. 12b and 13b are displayed in Fig. 15. At 131 h, convergence within the boundary layer is relatively disorganized and not well coupled with the upward motion above. The outflow above the boundary layer is very weak also (Fig. 15a). It is clearly seen that, after the strong supergradient winds are initiated, the boundary layer inflow between the radii of 40 and 50 km is decreased appreciably (see the pronounced concave shaping of contours of radial velocity in Fig. 15b). As a result, convergence is significantly enhanced on the radially inward side of the inflow maximum throughout the depth of the boundary layer, which leads to the massive eruption of air from the boundary layer (see the 0.5 m s−1 isotach of vertical velocity at the radius of 55 km just above the boundary layer). Above the boundary layer, strong outflow, which also results from the supergradient winds, decelerates rapidly when it impinges upon the deep radial inflow associated with the preexisting wavenumber-1 asymmetry. Consequently, convergence over the depth between 1 and 2 km is greatly enhanced, and the upward motion is much stronger.

Fig. 15.
Fig. 15.

(a) Radial velocity (red and blue contours; m s−1), divergence (shaded with intervals of −2, −4, and −6 × 10−4 s−1 from light gray to dark gray), and vertical velocity (thin black is 0.5 m s−1, thick black is 1 m s−1) averaged over the sector shown in Fig. 12b. (b) As in (a), but averaged over the sector shown in Fig. 13b.

Citation: Journal of the Atmospheric Sciences 70, 3; 10.1175/JAS-D-12-084.1

It is worthy to note that the enhanced convergence and convection will further modify airflow in the boundary layer. The stretching effect amplifies relative vorticity within the convergence zones. The radial inflow toward each convergence zone is also strengthened because of increased inward-pointing pressure gradient force and vertical advection of radial velocity. With larger vorticity and stronger inflow, the increase in tangential winds and hence agradient winds is faster. It follows that the radial inflow on the inward side of each convergence zone is decelerated to a greater extent, which in turn reinforces convergence. This hypothesized feedback process between two convergence zones may play an important role in the transition from the early phase to the developing phase of SEF and the intensification of the incipient secondary eyewall.

7. Conclusions and discussion

The key dynamical processes responsible for enhancing the inner-core convection outside the primary eyewall during the early phase of SEF are examined in an idealized three-dimensional simulation of a TC using a cloud-resolving model. It is found that the onset of the enhanced convection in the SEF region immediately follows the inward contraction of an organized outer rainband. The outer rainband induces a radial inflow descending to the surface, which reinforces boundary layer inflow and convergence in the inner-core region of the simulated TC. As a result, warm and moist air is continuously lifted up at the leading edge of the strong boundary layer inflow to support deep convection. Moreover, the strong inflow from the outer rainband has some quite fundamental consequences for the downstream boundary layer flow, including the buildup of supergradient winds, the deceleration of radial inflow due to the outward-pointing agradient force, the blocking of the strong inflow associated with a preexisting wavenumber-1 inflow asymmetry, and the enhancement of convergence and convection on the other side of the TC. Both convergence zones are maintained during the early phase of SEF by the persistent inflow forcing induced by the outer rainband.

It should be noted that the rapid expansion of the tangential wind field starts only when the mesoscale circulation features of the outer rainband are becoming evident. This result is consistent with some recent studies, which showed that the spinup of the outer-core circulation occurs largely as a symmetric response to the diabatic heating in outer rainbands (e.g., Fudeyasu and Wang 2011; Rozoff et al. 2012). Moreover, it is found that the strengthening of boundary layer inflow during the early phase of SEF involves the fundamental three-dimensional processes of outer rainbands (i.e., the descending radial inflow in the middle and downwind portions), which naturally poses questions with respect to the applicability of axisymmetric models (e.g., HMW12). Using an idealized model of vortex, MN10 demonstrated that the response of TC wind field to the diabatic heating associated with an outer rainband could be essentially viewed as the sum of the azimuthal-mean secondary transverse circulation and a nonnegligible contribution from the asymmetric component of potential vorticity (at least in the linear approximation). Therefore, only considering the symmetric boundary layer response to an expanding wind field as in HMW12 is incomplete for describing the strengthening of the boundary layer inflow. Nevertheless, the sequence of structural changes in the boundary layer of a mature TC (i.e., the development of supergradient winds and the enhancement of convergence) following the strengthening of the boundary layer inflow, as found in the axisymmetric study of HMW12 and the present study, plays a critical role in initiating and sustaining deep convection in the SEF region.

Finally, although the wavenumber-1 flow asymmetry emerges spontaneously in the current simulation without environmental forcing, it is common in the boundary layer of a moving TC vortex (e.g., Shapiro 1983; Wu and Braun 2004). As can be inferred from the results presented in this study, the optimal configuration for convection enhancement during the early phase of SEF requires the outer rainband should be collocated with the negative phase of the wavenumber-1 asymmetry of radial inflow. This inference can also be generalized to the case in which two outer rainbands are in opposite directions with each other. In addition, moderate-to-strong vertical wind shear can have a determining effect on the azimuthal orientation of outer rainbands (e.g., Corbosiero and Molinari 2002, 2003; Chen et al. 2006; Hence and Houze 2012b). It would be interesting to examine SEF processes in a set of idealized full-physics model simulations initiated with a TC-like vortex embedded in various environmental flows and to evaluate the environmental favorability for SEF. The corresponding results will be presented in due course.

Acknowledgments

This research is sponsored by the National Key Project for Basic Research (973 Project) under Grant 2009CB421500; National Natural Science Foundation of China through Grants 41130964, 41105034, and 41275059; the National Special Funding Project for Meteorology (GYHY-201006004); the Fundamental Research Funds for the Central Universities; and the Priority Academic Program Development of Jiangsu Higher Education Institutions. Valuable feedback from Drs. Juan Fang and Yi Zhang during the early stages of this work is appreciated. The authors would also like to thank Dr. Chun-Chieh Wu and two anonymous reviewers for their careful reading, critical comments, and helpful suggestions.

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