A Numerical Study of Outer Rainband Formation in a Sheared Tropical Cyclone

Qingqing Li Pacific Typhoon Research Center, Key Laboratory of Meteorological Disaster, Ministry of Education, Nanjing University of Information Science and Technology, Nanjing, and State Key Laboratory of Severe Weather, Chinese Academy of Meteorological Sciences, Beijing, China

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Yuqing Wang International Pacific Research Center, and Department of Meteorology, School of Ocean and Earth Science and Technology, University of Hawai‘i at Mānoa, Honolulu, Hawaii

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Yihong Duan State Key Laboratory of Severe Weather, Chinese Academy of Meteorological Sciences, Beijing, China

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Abstract

The dynamical process of outer rainband formation in a sheared tropical cyclone (TC) is examined in this study using the fully compressible, nonhydrostatic TC model. After the easterly vertical wind shear of 10 m s−1 was imposed upon an intensifying strong TC, an outer rainband characterized by a wavenumber-1 structure formed as a typical principal rainband downshear. Further analysis indicates that the outer rainband formation was closely connected to the activity of the inner rainband previously formed downshear. Moving radially outward, the inner rainband tended to be filamented owing to the strong radial gradient of angular velocity. As the inner rainband approached the outer boundary of the inner core, convection in its middle and upwind segments reinvigorated and nascent convective cells formed upwind of the rainband, caused mainly by the decreased filamentation and stabilization. Subsequently, the rainband reorganized into a typical outer rainband. Three different scenarios are found to be responsible for the outer rainband formation from downshear inner rainbands. The first is the outer rainband forming from an inner rainband downshear as a sheared vortex Rossby wave. The second is the outer rainband forming directly from a single deformation-induced inner rainband. The third is the outer rainband developing from an inner rainband downshear organized from a blend–merger of inner rainbands that were initiated from locally deformed convection upshear right.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author e-mail: Qingqing Li, liqq@nuist.edu.cn

Abstract

The dynamical process of outer rainband formation in a sheared tropical cyclone (TC) is examined in this study using the fully compressible, nonhydrostatic TC model. After the easterly vertical wind shear of 10 m s−1 was imposed upon an intensifying strong TC, an outer rainband characterized by a wavenumber-1 structure formed as a typical principal rainband downshear. Further analysis indicates that the outer rainband formation was closely connected to the activity of the inner rainband previously formed downshear. Moving radially outward, the inner rainband tended to be filamented owing to the strong radial gradient of angular velocity. As the inner rainband approached the outer boundary of the inner core, convection in its middle and upwind segments reinvigorated and nascent convective cells formed upwind of the rainband, caused mainly by the decreased filamentation and stabilization. Subsequently, the rainband reorganized into a typical outer rainband. Three different scenarios are found to be responsible for the outer rainband formation from downshear inner rainbands. The first is the outer rainband forming from an inner rainband downshear as a sheared vortex Rossby wave. The second is the outer rainband forming directly from a single deformation-induced inner rainband. The third is the outer rainband developing from an inner rainband downshear organized from a blend–merger of inner rainbands that were initiated from locally deformed convection upshear right.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author e-mail: Qingqing Li, liqq@nuist.edu.cn

1. Introduction

Spiral rainbands are distinct features in tropical cyclones (TCs). Two kinds of rainbands in mature TCs were sketched in Willoughby et al. (1984) and Willoughby (1988): principal and secondary rainbands. The former is stationary relative to the core of the storm and exhibits an evident azimuthal wavenumber-1 characteristics, and the latter is regularly located radially inward of the principal rainband with a smaller scale and shorter life cycle. Houze (2010) introduced another type of TC rainbands called distant rainbands. Distant rainbands lie far beyond the principal rainband and have more active convective structures than principal and secondary rainbands. On the other hand, spiral rainbands can also simply classified into inner and outer rainbands (Guinn and Schubert 1993; Wang 2009; Li and Wang 2012a,b; Moon and Nolan 2015). Inner rainbands are those spirally banded structures in the inner-core region within about 3 times the radius of maximum wind (RMW) from the TC center, as defined by Wang (2009), where the flow tends to be rapidly filamented. Some of the secondary rainbands are outward-propagating vortex Rossby waves (VRWs). Outer rainbands, as the term implies, lie outside the inner core. Thus, both principal and distant rainbands may be termed outer rainbands.

Formation and subsequent behavior of inner and outer spiral rainbands within TCs are fruitful areas of research and have been long addressed observationally and theoretically. After Montgomery and Kallenbach (1997) coined the term “vortex Rossby waves” to describe waves whose restoring force is the radial potential vorticity (PV) gradient, many observational and modeling studies attributed the inner rainbands to the activity of convectively coupled VRWs (Reasor et al. 2000; Wang 2001, 2002a,b; Chen and Yau 2001; Corbosiero et al. 2006; Li and Wang 2012b; Hall et al. 2013) because inner rainbands exhibited many characteristics of VRWs. More recently, Cotto et al. (2015) and Nikitina and Campbell (2015) pointed out that VRWs propagate in narrow annular waveguides bounded by inner turning and outer critical radii. At inner turning radii the waves are partially reflected and their frequencies are Doppler shifted to the local propagation frequency of a one-dimensional VRW, and at outer critical radii they are ultimately absorbed. The initially outward-propagating waves transport energy toward the critical radius and eddy angular momentum toward the eyewall. They also related the activity of VRWs to some observed inner rainbands. However, Moon and Nolan (2015) showed in a cloud-resolving simulation of Hurricane Bill (2009) that the behavior of the inner rainbands was not consistent with VRWs. The simulated inner rainbands had horizontal dipoles of PV anomalies rather than positive anomalies as inferred from VRWs, and their propagation did not agree with the VRW theory. Moon and Nolan (2015) proposed that the inner rainbands were convective clouds that were advected by the rotating TC flow and filamented into spirals by the strong flow deformation.

Earlier studies predominantly regarded outer bands in TCs as inertia–gravity waves (Kurihara and Tuleya 1974; Diercks and Anthes 1976; Kurihara 1976; Willoughby 1978; Chow et al. 2002), while recent observational and modeling results (Sawada and Iwasaki 2010a,b; Yu and Tsai 2010; Li and Wang 2012a) indicate that outer rainbands move radially outward at a much slower speed than inertia–gravity waves. Sawada and Iwasaki (2010a,b) suggested that the formation of outer rainbands in a quiescent environment in their idealized simulation was closely related to cold-pool dynamics in the boundary layer. Reexamination of the impacts of rainwater evaporation on TC structure and intensity by Li et al. (2015), however, demonstrated that cold-pool dynamics contributed more to the growth and maintenance of outer rainbands rather than to their initiation. Li and Wang (2012a) investigated the formation of outer rainbands in a TC simulated with the fully compressible, nonhydrostatic, cloud-resolving Tropical Cyclone Model version 4 (TCM4) and emphasized the close link between the formation of outer rainbands and the activity of inner rainbands. In their simulation, there existed a rapid filamentation zone (RFZ; Rozoff et al. 2006; Wang 2008) between the outer edge of the eyewall and approximately the radius 3 times the RMW where the filamentation time scale was less than 45 min. Filamentation may weaken convection because of the reduction of the radial inflow by the centrifugal and Coriolis forces as a result of the continuity constraint and, in the presence of horizontal shear, of a reduction of cloud buoyancy caused by entrainment (Rozoff et al. 2006; Wissmeier and Smith 2011; Kuo et al. 2012). As the inner rainband remnants moved outside the rapid filamentation zone and the boundary layer recovered from the drying and cooling effect of convective downdrafts in previous outer rainbands (Molinari et al. 2013), they could convectively reinvigorate to form new outer rainbands.

Ambient environments would complicate the formation of outer rainbands. For example, Willoughby et al. (1984) proposed that a principal rainband might develop because of the convergence between the storm flow and the environmental flow. Akter and Tsuboki (2012) addressed the formation of an outer rainband in Cyclone Sidr (2007) in the north Indian Ocean, indicating that the rainband originated in the low troposphere between the boundary of the weakly sheared, gradient-balanced wind on the inner side and the highly sheared, agradient wind on the outer side.

It has been well known that vertical shear (of the large-scale horizontal wind) is a predominant environmental factor contributing to the intensity and structure changes of TCs. In general, vertical shear has a detrimental effect on TC intensity (DeMaria and Kaplan 1994; DeMaria 1996; Elsberry and Jeffries 1996; DeMaria and Kaplan 1999; Wong and Chan 2004; Wu and Braun 2004; DeMaria et al. 2005; Paterson et al. 2005; Zeng et al. 2010; Riemer et al. 2010; Gu et al. 2015; Wang et al. 2015). Most significant impact of vertical shear is the forcing of convective asymmetries (Jones 1995; Wang and Holland 1996; Bender 1997; Frank and Ritchie 2001; Black et al. 2002; Rogers et al. 2003; Wu and Braun 2004; Heymsfield et al. 2006; Ueno 2007, 2008; Li et al. 2008; Xu and Wang 2013; Reasor et al. 2013; Barnes and Barnes 2014; DeHart et al. 2014). A wavenumber-1 pattern prevails with the convective maximum downshear left of the shear vector. Several mechanisms have been proposed to explain the occurrence of such an asymmetry. One is a balanced response of the tilting vortex, which leads to updrafts downtilt (Jones 1995; Wang and Holland 1996; Frank and Ritchie 1999). A second mechanism is the adiabatic motion of the vortex cyclonic flow along the tilted isentropic surface in response to temperature anomalies required for the thermal wind balance of the tilted vortex. After the first mechanism takes effect, the upward motion shifts 90° to the right of the tilt direction (Jones 1995). The third mechanism derives from storm-relative flow (Willoughby et al. 1984; Wang and Holland 1996; Bender 1997; Frank and Ritchie 2001). Specifically, vertical shear imposes opposite flow patterns onto the storm at the lower and upper levels: on the downshear (upshear) side, relative flow is radially inward at the lower (upper) levels and radially outward in the upper (lower) levels. Because of the strong radial vorticity gradient, this vorticity advection by the relative flow is approximately balanced by vortex stretching, leading to upward motion downshear and downward motion upshear. More recently, Xu and Wang (2013) addressed the initial development of convective asymmetry in the inner core for a TC embedded in vertical shear. They pointed out that the TC eyewall acts as a vessel in vertical shear, and convergence and downdrafts arise along the outer edge of the eyewall upshear while divergence and updrafts develop downshear shortly after the vertical shear was imposed.

Since the formation of outer rainbands seems to coincide with the activity of inner rainbands (Li and Wang 2012a) and the environmental vertical shear plays an important role in structure change in the inner core where inner rainbands are active, an issue worth addressing is how outer rainbands form in a TC embedded in an environment with vertical shear, which is the subject of this study. This paper is organized as follows. Section 2 describes the model configuration and experimental design. Section 3 presents the evolution of the simulated storm structure. The outer rainband formation and the associated physical processes are discussed in section 4. The main conclusions are given in the last section.

2. Model setup and experimental design

The model used in this study is the fully compressible, nonhydrostatic model TCM4. The model assumes a flat lower boundary at the ocean surface with a uniform unperturbed surface pressure of 1010 hPa. The model physics include an Eε turbulence closure scheme for subgrid-scale vertical turbulent mixing (Langland and Liou 1996), a modified Monin–Obukhov scheme for surface flux calculations (Fairall et al. 2003), an explicit treatment of mixed-phase cloud microphysics (Wang 2001), a nonlinear fourth-order horizontal diffusion for all prognostic variables except for those related to the mass conservation equation, a simple Newtonian cooling term added to the perturbation potential temperature equation to roughly account for the longwave radiative cooling (Rotunno and Emanuel 1987), and the dissipative heating related to turbulent kinetic energy dissipation rate (ε), derived directly from the prognostic turbulent closure scheme. A detailed description of TCM4 can be found in Wang (2007).

The model domain is quadruply nested with two-way interactive nesting. All three inner meshes automatically follow the center of the simulated storm. The grid has 32 levels in the vertical and the four horizontal meshes have grid points of 241 × 201, 127 × 127, 163 × 163, and 313 × 313, respectively, with their horizontal grid intervals of 54, 18, 6, and 2 km. No cumulus parameterization is employed, even in the two outermost meshes since convection occurs mainly within 250 km of the storm center. The model is initialized with an axisymmetric cyclonic vortex on an f plane at 18°N over ocean with a fixed sea surface temperature (SST) of 29°C. The initial vortex has a maximum tangential wind speed of 20 m s−1 at the 90-km radius near the surface and decreases sinusoidally with pressure to vanish at 100 hPa. The initial thermodynamic profile of the unperturbed model atmosphere is defined as the moist-tropical sounding of Dunion (2011).

Because the focus of this study is on the formation of outer rainbands in a sheared TC, an environment with an easterly shear of 10 m s−1 and the zonal wind speed increasing from 0 m s−1 at about z = 1.5 km to 10 m s−1 at about z = 13.5 km with a bellramp function with height (see the inset in Fig. 1) was introduced after 60 h when the vortex was well spun up. As we can see from Fig. 1, after the shear was imposed, the storm intensified for about 2 h, followed by a rise in the minimum sea level pressure from 963 hPa at 62 h to 967 hPa at 66 h. Later on, the minimum sea level pressure sharply dropped by approximately 35 hPa within 22 h (Fig. 1), indicative of rapid intensification of the simulated TC during this period. Although ambient shear is generally detrimental to TC intensification, the 10 m s−1 shear is not strong enough to prevent the storm from intensifying, as found in previous studies (Wang et al. 2004; Shelton and Molinari 2009; Molinari and Vollaro 2010b; Nguyen and Molinari 2012; Stern and Zhang 2013). Since the effect of vertical shear on storm intensity change is not the focus of this study, the following analyses/discussions will focus on mechanisms related to the formation of outer rainbands.

Fig. 1.
Fig. 1.

Time series of the simulated TC minimum sea level pressure (hPa), which is plotted in black (blue) before (after) the vertical shear was imposed (after 60 h of simulation as indicated with the dashed vertical line). The inset shows the vertical profile of zonal wind (m s−1) with an easterly vertical shear of 10 m s−1.

Citation: Journal of the Atmospheric Sciences 74, 1; 10.1175/JAS-D-16-0123.1

3. Evolution of the simulated storm structure

Here, the structural changes of the simulated TC after the introduction of the environmental shear are discussed briefly. Figure 2a shows the time–azimuth Hovmöller diagram of vertical velocity radially averaged between 20- and 40-km radii and vertically averaged between 1- and 3-km heights, roughly representing the low-level convective characteristics in the eyewall region during the 48 h after the vertical shear was introduced. Note that the RMW slightly decreased from about 28 to 22 km during this period (not shown). Consistent with previous studies, a wavenumber-1 convective asymmetry immediately developed in the eyewall as the easterly shear was imposed, with enhanced ascent in the downshear-left quadrant and descent in the upshear-right quadrant of the eyewall (Fig. 2a). The eyewall convection was initially enhanced downshear-right (Fig. 2a) along with near-surface inflow and convergence (Fig. 3a) and decayed upshear left (Fig. 2a) along with near-surface outflow and divergence (Fig. 3a). Late on in this period, the convective maximum shifted to the south, left of the shear vector. In the region between 40- and 80-km radii, where the RFZ was located (discussed later), the downshear–upshear difference in convection was also significant (Fig. 2b). The most active convection in this region tended to occur downshear, azimuthally upwind of the strongest eyewall convection, which was consistent with observations (Hence and Houze 2012; Didlake and Houze 2013). In contrast, the low-level downdrafts in the RFZ predominantly occurred upshear (Fig. 2b), slightly upwind of the weakest eyewall convection. The descent in the RFZ was coincident with divergence near the surface (Fig. 3b). Of interest is that the convection initiation in the RFZ between 40- and 80-km radii tended to be upshear right (Fig. 2b), azimuthally upwind of the convection initiation in the eyewall (Fig. 2a). This upshear-right convection initiation was correlated with the formation and development of deformation-induced inner rainbands as will be discussed below.

Fig. 2.
Fig. 2.

Time–azimuth distributions of vertical velocity (m s−1) radially averaged between (a) 20- and 40-km radii and (b) 40- and 80-km radii, both vertically averaged between 1- and 3-km heights. The plots have been extended to two revolutions around the simulated TC center. Shear-relative quadrants defined as upshear-left (UL), upshear-right (UR), downshear-right (DR), and downshear-left (DL) quadrants are indicated as well.

Citation: Journal of the Atmospheric Sciences 74, 1; 10.1175/JAS-D-16-0123.1

Fig. 3.
Fig. 3.

Time–azimuth distributions of radial winds (color shading; m s−1) radially averaged between (a) 20- and 40-km radii and (b) 40- and 80-km radii, both vertically averaged from the surface to 1-km height. Contours denote the average divergence of 0 s−1, with convergence outside the contours stippled.

Citation: Journal of the Atmospheric Sciences 74, 1; 10.1175/JAS-D-16-0123.1

Figure 4 shows the horizontal distribution of the modeled reflectivity at z = 3 km from 67.4 to 70.7 h every 18 min. Not surprisingly, an evident wavenumer-1 convective asymmetry dominated the inner-core region (within a radius of 80 km from the storm center), with the highest inner-core reflectivity downshear-left and little rainfall upshear right. A fairly broad stratiform precipitation region prevailed outside the eyewall in the downshear-left quadrant of the storm, with quite uniform high reflectivity at low levels. It is supposed that lots of ice particles from inner-core convection in the downshear-right quadrant traveled downwind to fall out as primarily stratiform rain in the downshear-left quadrant, together with ice particles extruded from the eyewall (Hence and Houze 2012). This stratiform region was the downwind sector of outer rainbands (Hence and Houze 2008; Li and Wang 2012b), and the simulated precipitation pattern was consistent with observations (Hence and Houze 2012).

Fig. 4.
Fig. 4.

Plan view of the simulated reflectivity (dBZ) at z = 3 km from (top left) 67.4 to (bottom center) 70.4 h of simulation at every 18-min interval. Black concentric circles are every 40 km from the storm center. The black and dark blue dashed curves mark the inner rainbands and outer rainband of interest, respectively. See the text for detail. Shear direction is indicated by the black arrow and the shear-relative quadrants in the bottom-right corner of the figure, with “UL,” “UR,” “DR,” and “DL” indicating upshear left, upshear right, downshear right, and downshear left, respectively.

Citation: Journal of the Atmospheric Sciences 74, 1; 10.1175/JAS-D-16-0123.1

It has been long recognized that there exists a strain-dominated region outside the eyewall of a strong TC (Guinn and Schubert 1993; Shapiro and Montgomery 1993; Kossin et al. 2000; Rozoff et al. 2006; Wang 2008), within which convection is filamented. The filamentation time () is proposed to quantitatively measure the strength of strain, which is defined as
eq1
where (1/2)S1 and (1/2)S2 are the stretching and shearing deformations, respectively, and ζ is the relative vorticity. A detailed description of the basic concepts of the filamentation time can be found in Rozoff et al. (2006). Figure 5 shows the filamentation time temporally averaged 68–70, 79–81, and 94–96 h at the 3-km height. The filamentation time tended to be axisymmetric inside of a radius of 80 km, with its value increasing radially outward as found in previous studies (Rozoff et al. 2006; Wang 2008; Li and Wang 2012a,b). An area prevailed by the filamentation time of less than 45 min was generally located outside the vorticity-dominated region (the whitened-out area in Fig. 5) and inside a radius of 80 km. This area is referred to as the RFZ, which is conducive to the organization of inner rainbands and the damping of high azimuthal–wavenumber asymmetries (Wang 2008). Therefore, inner rainbands tended to develop inside the 80-km radius in the current simulation.
Fig. 5.
Fig. 5.

Filamentation time (min) at z = 3 km averaged between (a) 68 and 70, (b) 79 and 81, and (c) 94 and 96 h of simulation. Dashed concentric circles are every 40 km from the storm center. Shear direction is indicated by the black arrow and the shear-relative quadrants at the top of the figure.

Citation: Journal of the Atmospheric Sciences 74, 1; 10.1175/JAS-D-16-0123.1

Li and Wang (2012b) systematically compared the inner rainbands and outer rainbands in a simulated TC in a quiescent environment and found that the major difference between these two types of rainbands was in the convective behavior. Inner rainbands tend to appear in the RFZ and are organized in an azimuthally elongated banded structure with smooth boundaries in convection, whereas outer rainbands are generally characterized by extensive stratiform precipitation regions with embedded cellular convective features having heterogeneous organization in their upwind, middle, and downwind sectors. Figure 4 depicts the evolution of a typical outer rainband (marked by the dark blue dashed curve) after 70 h of simulation. This rainband spiraled cyclonically from the north outside a radius of 80 km to the south near the outer edge of the eyewall. As the rainband moved radially outward, nascent convective cells developed upwind frequently, matured in the middle sector of the band and finally collapsed in the stratiform region downwind (not shown). This outer rainband exhibited the apparent principal rainband feature with a secondary rainband radially inside of it after 70.4 h of simulation (Fig. 4). Similar characteristics and behavior of outer rainbands were seen at other times as well after the vertical wind shear was imposed.

4. Formation scenarios of outer rainbands

Li and Wang (2012a) shed light on the formation of outer rainbands in a simulated TC without environmental flow. They showed that their simulated outer rainbands were bound up with the reinvigoration of convective remnants of outward-propagating inner rainbands immediately outside of the RFZ. Prior studies showed that there exist two types of inner rainbands which are forced in distinct ways, as mentioned in section 1. One is related to sheared VRWs stemming from eyewall convection (herein called “VRW-related inner rainbands”; Reasor et al. 2000; Wang 2001, 2002a,b; Chen and Yau 2001; Corbosiero et al. 2006; Li and Wang 2012b) and the other originates from convection outside the eyewall in the RFZ, which is quickly filamented by deformation and cyclonically advected by the TC rotating flow (herein called “deformation-induced inner rainbands”; Moon and Nolan 2015). In the present study, we find that the formation of outer rainbands in the sheared TC was subject to the activity of both types of inner rainbands. Moreover, three distinct scenarios contributed to the initiation of outer rainbands, as elaborated in the following subsections.

a. Scenario I: Outer rainband formation from a VRW-related inner rainband

Many previous studies (Montgomery and Kallenbach 1997; Kuo et al. 1999; Wang 2001, 2002a,b; Chen and Yau 2001; Nolan and Montgomery 2002; Schecter and Montgomery 2006; Li and Wang 2012b) have suggested that convectively coupled sheared VRWs may contribute to the formation of inner spiral rainbands, namely VRW-related inner rainbands. Once excited on the eyewall flank by convective processes, VRWs are characterized by PV filaments owing to the effects of strain and adverse shear surrounding the PV core (Wang 2001, 2002a,b; Chen and Yau 2001; Chen et al. 2003; Menelaou et al. 2013; Naylor and Schecter 2014).

To illustrate the development of the VRW-related inner rainbands and the subsequent initiation of an outer rainband, we show in Fig. 6 the upward vertical motion field vertically averaged between 1- and 3-km heights and the divergence field averaged between the surface and 1-km height from 94.2 to 96.4 h. At 94.2 h, an updraft band (band A; marked by the dark green dashed curve) skirted the downshear eyewall of the storm in the vicinity of a radius of 30 km (Figs. 6 and 7). A meticulous examination indicates that this banded structure formed from convection in the downshear portion of the eyewall and slowly propagated radially outward (Figs. 6 and 7). Figure 8 depicts the wavenumber-1 and wavenumber-2 asymmetric components of vertical relative vorticity at z = 3 km and upward motion averaged between 1- and 3-km heights from 94.2 to 94.5 h of simulation. The wavenumber-1 structure in the eyewall was nearly stationary with positive asymmetric vorticity and enhanced upward motion downshear left (Figs. 8a,c,e,g), which corresponded to the easterly vertical shear rather than a wavenumber-1 VRW. At 94.2 h, positive asymmetric vorticity in wavenumber 2 was seen both downshear and upshear (Fig. 8b), spiraling cyclonically toward the eyewall. It is visible that the downshear positive vorticity in wavenumber 2 was associated with the development of band A, and no spiral rainband appeared in its upshear counterpart because of the convective suppression by vertical shear-induced wavenumber-1 subsidence. About 6 min later, the wavenumber-2 feature rotated cyclonically and moved radially outward (Fig. 8d). The estimated azimuthal phase speed of the wavenumber-2 wave was about 39.0 m s−1, approximately ⅔ of the mean tangential wind speed, indicative of the typical retrograde propagation of VRWs found in previous studies (Wang 2001, 2002a; Chen and Yau 2001; Corbosiero et al. 2006; Li and Wang 2012b; Hall et al. 2013). The calculated azimuthal phase speed of the wavenumber-2 VRW based on the local dispersion relation for VRWs formulated in Möller and Montgomery (2000) was 36.3 m s−1, also very close to the phase speed seen in Fig. 8. At the same time, band A was also accompanied by the wavenumber-2 VRW, propagating radially outward at a radial speed of about 9.5 m s−1. This feature agreed with the previous finding that wavenumber-2 VRWs tend to contribute to the formation and activity of inner rainbands (Chen and Yau 2001; Wang 2002a; Li and Wang 2012a; Hall et al. 2013; Cotto et al. 2015). Figures 8c and 8d further illustrate that the evolution of the upwind and middle portions of band A tended to connect closely to the wavenumber-2 vorticity asymmetry, whereas its downwind portion tended to connect to the shear-forced wavenumber-1 asymmetry. These characteristics appeared to indicate that the wavenumber-2 VRW could be initially forced by the wavenumber-1 asymmetry in the eyewall—a feature that will be investigated in separate study. A similar feature was also found in a high-resolution simulation of Typhoon Morakot (2009) in Hall et al. (2013). In that case, the vertical shear between 200 and 850 hPa was around 12–14 m s−1. In addition, the evolution of wavenumber-2 vorticity seemed to share some features with that discussed in Cotto et al. (2015). For instance, the perturbation vorticity tended to become more filamented and tightly wound as the waves propagated radially toward the critical radius and vorticity decreased in amplitude (Fig. 8) because most of the wave energy propagated away.

Fig. 6.
Fig. 6.

Divergence (shading; 10−4 s−1) averaged between the surface and 1-km height and upward vertical motion averaged between 1- and 3-km heights contoured at 0.4, 0.8, 1.5, and 3.0 m s−1 from (top left) 94.2 to (bottom center) 96.2 h of simulation. Dashed purple concentric circles are every 40 km from the storm center. The deep green and red dashed curves mark the inner rainbands and outer rainband of interest, respectively. See the text for detail. Shear direction is indicated by the black arrow and the shear-relative quadrants in the bottom-right corner of the figure.

Citation: Journal of the Atmospheric Sciences 74, 1; 10.1175/JAS-D-16-0123.1

Fig. 7.
Fig. 7.

As in Fig. 4, but for the period from 94.2 to 96.4 h of simulation.

Citation: Journal of the Atmospheric Sciences 74, 1; 10.1175/JAS-D-16-0123.1

Fig. 8.
Fig. 8.

(left) Wavenumber-1 and (right) wavenumber-2 vertical relative vorticity (color shading; 10−4 s−1) at z = 3 km at (a),(b) 94.2; (c),(d) 94.3; (e),(f) 94.4; and (g),(h) 94.5 h of simulation. Full vertical motion averaged from 1- to 3-km heights is contoured in black at 0.4, 0.8, 1.2, 1.6, 2.5, and 4.0 m s−1. Dashed white concentric circles are every 20 km from the storm center. The deep green curve marks the VRW-related inner rainband. Shear direction is indicated by the black arrow and shear-relative quadrants at the top of the figure, with “UL,” “UR,” “DR,” and “DL” indicating upshear left, upshear right, downshear right, and downshear left, respectively.

Citation: Journal of the Atmospheric Sciences 74, 1; 10.1175/JAS-D-16-0123.1

Later, the wavenumber-2 VRW together with band A continued moving radially outward to a radius of 40 km (Figs. 8f and 8h), and convection in band A became active (Figs. 7 and 8h) resulting from the locally enhanced confluence related to the asymmetric inflow at low levels (Fig. 9). Figures 9a, 9c, and 9e show the radial wind and asymmetric winds temporally averaged between 94 and 96 h of simulation at z = 11.6, 5.8, and 0.96 km, respectively. The radial wind fields were suggestive of dominant wavenumber-1 asymmetries. At z = 0.96 km inflow and outflow occurred in and outside the downshear left and upshear right in the eyewall (Fig. 9e), respectively, whereas at z = 5.8 km inflow and outflow were seen in and outside the upshear right and downshear left in the eyewall (Fig. 9c). At the upper levels, inflow and outflow were in the upshear and downshear portions in the inner-core region (Fig. 9a), respectively. The vertical distribution of radial winds clearly indicated an in–up–out secondary circulation downshear and an in–down–out flow upshear. These characteristics of the vertical shear-forced secondary circulation were consistent with previous results from numerical simulations and observational studies (Wong and Chan 2004; Zhang and Kieu 2006; Xu and Wang 2013; Barnes and Dolling 2013; DeHart et al. 2014; Moon and Nolan 2015).

Fig. 9.
Fig. 9.

(left) Radial wind (color shading; m s−1) and (right) divergence (color shading; 10−4 s−1) averaged between 94 and 96 h of simulation at (a),(b) z = 11.6-; (c),(d) z = 5.8-; and (e),(f) z = 0.96-km height. Contours are the time-mean vertical velocity contoured at (top),(middle) −2.0, −1.5, −1.0, 1.0, 2.0, 3.0, and 5.0 m s−1 and at (bottom) −0.5, −0.2, 0.2, 0.5, 1.0, 2.0, 3.0, and 5.0 m s−1. Vectors are time-mean storm-relative asymmetric winds (m s−1). Dashed concentric purple circles are every 40 km from the storm center. Shear direction is indicated by the black arrow and shear-relative quadrants at the top of the figure, with “UL,” “UR,” “DR,” and “DL” indicating upshear left, upshear right, downshear right, and downshear left, respectively.

Citation: Journal of the Atmospheric Sciences 74, 1; 10.1175/JAS-D-16-0123.1

As band A moved radially outward from 40-km radius, associated convection weakened (Figs. 6 and 7) owing to the persistent effect of rapid filamentation (Fig. 5c). In the presence of boundary layer convergence induced by outflow to the north outside the eyewall region (Fig. 9f), deformation-induced inner rainbands with weak convection were generated there as well (e.g., at 95.2 h in Fig. 6). These bands occasionally mixed with the upwind segment of the VRW-related inner rainbands and became elongated as a result of shearing deformation. The upwind and middle portions of band A continued moving radially outward and gradually became more elongated because of the blend of deformation-induced inner rainbands (95.4–95.8 h in Figs. 6 and 7).

Figure 10e shows the distribution of convective available potential energy (CAPE) averaged from 94 to 96 h. High CAPE appeared in the downshear side particularly outside the 80-km radius and low CAPE appeared on the upshear side. This upshear–downshear difference in CAPE agreed with recent observational results (Molinari and Vollaro 2010a; Molinari et al. 2012), providing potential support for convective cell growth in the upwind portion of the outer rainband. With the upwind and middle portions of band A approaching the northwestern boundary of the inner core (around the 80-km radius), convection in the upwind portion was reinvigorated, and nascent convective cells formed upstream where moist enthalpy was much higher (Figs. 10e and 10f) and the filamentation effect was largely restrained (Fig. 5c). Meanwhile, the downwind portion of the band was located in the stratiform region downshear left in the inner core. It has been noted that the difference in the convective behavior between inner rainbands and outer rainbands is that prevailing cellular convection was embedded in stratiform clouds in outer rainbands but smooth banded convective structure prevails in inner rainbands (Wang 2009; Li and Wang 2012b). Finally, a rainband (marked by red and dark blue dashed curves in Figs. 6 and 7, respectively) bearing outer rainband characteristics formed around 96 h of simulation, and the formation process of this outer rainband was somehow akin to the mechanism proposed in Li and Wang (2012a).

Fig. 10.
Fig. 10.

(left) CAPE (J kg−1) and (right) equivalent potential temperature (K) at z = 100-m height, both averaged (a),(b) 68–70; (c),(d) 79–81; and (e),(f) 94–96 h of simulation. Dashed concentric black circles are every 40 km from the storm center. Shear direction is indicated by the black arrow and shear-relative quadrants at the top of the figure, with “UL,” “UR,” “DR,” and “DL” indicating upshear left, upshear right, downshear right, and downshear left, respectively.

Citation: Journal of the Atmospheric Sciences 74, 1; 10.1175/JAS-D-16-0123.1

It has been recognized that VRWs may lead to angular momentum divergence near the critical radius by inward-transporting eddy angular momentum (Montgomery and Kallenbach 1997; Cotto et al. 2015), thus spinning down the tangential wind inside the critical radius and making its radial gradient less negative there. An alternative way of VRW processes to lead to outer rainband formation might be that additional convergence in the boundary layer, associated with the decrease in frictional inflow due to the less negative radial gradient of tangential wind, locally reinforces convection near the critical radius. However, the simulated tangential wind in the vicinity of the upwind and middle sectors of band A did not exhibit significant dilution (not shown) as the band approached the boundary of the inner core. A possible reason is that the convection (diabatic heating) associated with the band accelerated the local tangential wind (Hence and Houze 2008; Moon and Nolan 2010), offsetting the effect of wave angular momentum divergence. In addition, Moon and Nolan (2015) argued that it was likely questionable to consider inner bands to be VRWs in those simulations at relatively coarser resolutions, because they could not resolve the finescale structures of inner rainbands well. However, a simulation of Hurricane Wilma (2005) at a 1-km resolution in Menelaou et al. (2013) and the empirical normal mode analysis employed in that study also evidenced the modeled inner bands in association with convectively coupled VRWs.

b. Scenario II: Outer rainband formation directly from a single deformation-induced inner rainband

Figures 11 and 12 show another scenario of the outer rainband formation. A convective cell forced by low-level convergence outside the eyewall (Figs. 3b and 11) appeared upshear right inside a radius of 80 km at 78. 8 h and moved cyclonically (Fig. 11), subsequently evolving into a relatively short, banded cloud structure with weak low-level upward motion (marked by the thick green dashed curve) in the inner core. This structure was further deformed, showing a more banded shape owing to rapid filamentation (Fig. 5b), and later interacted with convection within the RFZ to form a well-organized band B at 79.6 h to the north in the inner core (Fig. 11). Band B showed common features of the inner rainbands detailed in Moon and Nolan (2015), thereby perceived as a deformation-induced inner rainband. The downwind sector of band B tended to move radially inward to interconnect with the eyewall downshear (at 79.8 h in Figs. 11 and 12). Later, the middle and upwind sectors of band B moved cyclonically and a new inner band C formed, apparently by detaching from the inward side of band B at 79.9 h (not shown). As most of band B moved downshear, its convection became more active from 79.9 to 80.1 h because of the enhanced convergence in the boundary layer and divergence above (Figs. 13d and 13f). This tendency was followed by subtle convective dilution as a result of rapid filamentation in the inner core (Fig. 5b). By 80.4 h, band C became less active (Figs. 11 and 12). At 80.6 h, the upwind sector of band B was connected with weak-precipitation clouds in the upshear-right quadrant outside a radius of 80 km (Fig. 12). Several isolated convective cells developed on the outer edge in the upwind and middle sectors of the rainband (Figs. 11 and 12) because of the subsiding filamentation effect (Fig. 5b), and the downwind part of band B also moved into the inner-core region downshear left. It was thus indicated that an outer rainband formed from inner band B around this time. Thereafter, nascent convective cells continuously formed in the upwind sector of this rainband on account of enhanced local convergence at low levels (Fig. 13f) and high CAPE in the downshear-right quadrant outside the 80-km radius (Fig. 10b). They tended to mature and move downwind along the outer edge, with broad stratiform precipitation areas in its downwind sector (Figs. 11 and 12), which was featured by a typical outward-moving outer rainband.

Fig. 11.
Fig. 11.

As in Fig. 6, but for the period from 78.8 to 80.8 h of simulation. Color shading is divergence (10−4 s−1) averaged between the surface and 1-km height and black contours are upward vertical motion averaged between 1- and 3-km heights at 0.4, 0.8, 1.5, and 3.0 m s−1.

Citation: Journal of the Atmospheric Sciences 74, 1; 10.1175/JAS-D-16-0123.1

Fig. 12.
Fig. 12.

As in Fig. 4, but for the period from 78.8 to 80.8 h of simulation.

Citation: Journal of the Atmospheric Sciences 74, 1; 10.1175/JAS-D-16-0123.1

Fig. 13.
Fig. 13.

As in Fig. 9, but temporally averaged between 79 and 81 h of simulation. Color shading indicates (left) radial wind (m s−1) and (right) divergence (10−4 s−1). Contours are time-mean vertical velocity contoured at (top),(middle) −2.0, −1.5, −1.0, 1.0, 2.0, 3.0, and 5.0 m s−1 and (bottom) −0.5, −0.2, 0.2, 0.5, 1.0, 2.0, 3.0, and 5.0 m s−1. Vectors are time-mean storm-relative asymmetric winds.

Citation: Journal of the Atmospheric Sciences 74, 1; 10.1175/JAS-D-16-0123.1

c. Scenario III: Outer rainband formation from a downshear inner rainband organized from a blend of deformation-induced inner rainbands

A typical example for this mechanism for outer rainband formation is that associated with the outer rainband mentioned in section 3. At 67.4 h, two relatively short banded cloud structures with weak low-level upward motion were located in the northeastern quadrant in the inner core inside a radius of 80 km. These two cloud systems arose from convection forced by low-level convergence outside the eyewall (Figs. 3b and 14) and became banded quickly (not shown). These inner rainbands were thus characterized by deformation-induced inner rainbands. In the following 18 min, the two rainbands merged to form a single band to the north of the eyewall.

Fig. 14.
Fig. 14.

As in Fig. 6, but for the period from 67.4 to 70.4 h of simulation. Color shading is divergence (10−4 s−1) averaged between the surface and 1-km height and contours are upward vertical motion averaged between 1- and 3-km heights at 0.4, 0.8, 1.5, and 3.0 m s−1.

Citation: Journal of the Atmospheric Sciences 74, 1; 10.1175/JAS-D-16-0123.1

As they moved cyclonically, sporadic updraft cores formed in the upshear-right quadrant within the 80-km radius. At 68.0 h, the preexisting bands and nascent updraft cores merged to form a single well-organized spiral band D north in the inner core (Fig. 14). The banded structure also appeared clearly in the reflectivity field as shown Fig. 4. The formation of this inner rainband is similar to the generation of banded feature in the tracer simulations in Moon and Nolan (2015). That is, neighboring convective cells were deformed into spiral shapes by high horizontal deformation in the RFZ outside the eyewall (Rozoff et al. 2006; Wang 2008), eventually forming a single banded structure. Figure 5a suggests that the region between 40- and 80-km radii was the RFZ within which the filamentation time was less than 45 min. As a result, convection between 40- and 80-km radii tended to be deformed into an elongated banded pattern.

Of interest is band D which subsequently became more spiraled with its upwind section apt to shift radially outward and its downwind section inward, leading it to attach to the eyewall downshear. This evolution was associated with the TC flow in the lower troposphere. When band D was well organized and elongated in the north (e.g., at 68.0 h in Fig. 14), its upwind and downwind sectors were located right in the low-level outflow and inflow regions (Fig. 15e), respectively. Consequently, the upwind and downwind sectors of the inner rainband were steered radially outward and inward, respectively, making the band to attach to the eyewall downshear and become stationary.

Fig. 15.
Fig. 15.

As in Fig. 9, but temporally averaged between 68 and 70 h of simulation. Shading indicates (left) radial wind (m s−1) and (right) divergence (10−4 s−1). Contours are time-mean vertical velocity contoured at (top),(middle) −2.0, −1.5, −1.0, 1.0, 2.0, 3.0, and 5.0 m s−1 and at (bottom) −0.5, −0.2, 0.2, 0.5, 1.0, 2.0, 3.0, and 5.0 m s−1. Vectors are time-mean storm-relative asymmetric winds.

Citation: Journal of the Atmospheric Sciences 74, 1; 10.1175/JAS-D-16-0123.1

During the early period (e.g., from 68.0 to 68.3 h) when the downwind portion of band D moved downshear toward the eyewall, updrafts in the downwind sector of the inner rainband were transiently enhanced (Fig. 14) likely because of the strengthened convergence in the lower-tropospheric asymmetric inflow (Figs. 15e and 15f). Subsequently, the rapid filamentation effect (Fig. 5a) and relatively weak local low-level convergence (Fig. 14) resulted in the weakening of convection in the inner rainband (e.g., from 68.4 to 68.8 h). Note that new convective bands steadily formed in the upshear-right quadrant in the inner core, developed and coalesced into new band E to the north (at 68.9 h in Fig. 14). This new inner rainband moved cyclonically and became more spiraled as well. Later, it caught up with the middle part of the preexisting and weakening band D, evolving into a more elongated inner band F that spiraled from the downshear eyewall to the north with its upwind portion extending outside the RFZ (at 69.5 h in Figs. 4 and 14). The low-level outflow (Fig. 15e) tended to steer the upwind sector of band F radially outward (e.g., from 69.5 to 69.8 h in Fig. 4). In contrast, convection in the middle sector was healthier (Figs. 4 and 14) because of the relatively weaker filamentation there. The middle sector of the rainband moved cyclonically and radially outward during that period mainly because of the outflow in the midtroposphere (Fig. 15e). When most of the inner rainband (particularly its middle and upwind sectors) was located near and outside the outer boundary of the RFZ (Fig. 5a), cellular convection appeared upwind and moved downwind after 69.8 h (Figs. 4 and 14). As the bulk of the rainband was located outside the inner-core region and its downwind sector approached the downshear-left stratiform region of the inner core, an outer rainband was thus identified to form from band F and moved radially outward after 70.0 h of simulation. Of interest is a typical secondary rainband appearing downshear at 70.4 h (Fig. 4). The life cycle of this secondary rainband was consistent with the deformation-induced inner rainband (not shown), suggesting that secondary rainbands are one type of inner rainbands.

The outer rainband discussed in this subsection formed from an inner rainband organized from a blend of preexisting deformation-induced inner rainbands rather than directly from a deformation-induced inner rainband as noted in the last subsection. Furthermore, it appears that whether an outer rainband forms from a blend of preexisting deformation-induced inner rainbands or directly from a single deformation-induced inner rainband is controlled by convective instability on the downshear side. The downshear CAPE associated with the outer rainband formation examined in the last subsection (Fig. 10c) was generally over 1100 J kg−1 inside 120 km, a radius much larger than that related to the outer rainband formation discussed in this subsection (Fig. 10a). The low-downshear CAPE inside a radius of 160 km in Fig. 10a was likely due to the incomplete boundary layer recovery (Fig. 10b) from evaporatively cooled downdrafts in the outer rainbands (not shown here; Li and Wang 2012a; Molinari et al. 2013), while surface heat and moisture fluxes produced substantial increases in boundary layer entropy downshear during 79–81 h (Fig. 10d). Thus, the convective continuation and/or development of band B became more likely in the presence of the favorable downshear thermodynamic conditions during 79–81 h (Figs. 10c and 10d), and the deformation-induced inner band B eventually evolved into an outer rainband without a significant weakening as it moved downshear (Figs. 11 and 12).

5. Conclusions

Vertical wind shear has a significant effect on TC structure and intensity changes. Observational and modeling studies indicate that the vortex tilt and asymmetric inner-core convection regularly occur in highly sheared TCs, and strong shear tends to weaken TCs or reduce their intensification rate. Outer rainbands are prominent features of TCs, except in annular hurricanes (Knaff et al. 2003), and their activity may also influence the storm structure and intensity (Wang 2009). Although several previous studies have been devoted to understanding of outer rainband formation in complex environments, an unanswered question is how they form in sheared storms. In this study, a high-resolution numerical experiment was conducted to examine the physical processes of outer rainband formation for a TC embedded in a unidirectional 10 m s−1 easterly vertical shear.

Shortly after the shear was imposed, active inner-core convection arose downshear left (Fig. 16), with the shear-forced in–up–out secondary circulation downshear. Apparent subsidence in the eyewall occurred upshear right, accompanied by upshear outflow and corresponding divergence outside the eyewall near the surface. It is shown that outer rainbands, characterized by principal rainbands, tended to be present downshear (Fig. 16). As principal rainbands continue moving radially outward, they became distant rainbands (Fig. 16).

Fig. 16.
Fig. 16.

Schematic diagram of spiral rainbands for a TC embedded in an easterly vertical wind shear, adapted from Fig. 30 in Houze (2010).

Citation: Journal of the Atmospheric Sciences 74, 1; 10.1175/JAS-D-16-0123.1

The simulation indicates a close connection between the formation of outer bands and the activity of inner rainbands, as previously shown in Li and Wang (2012a). The outer rainbands tended to develop from the well-organized inner rainbands downshear, as illustrated schematically in Fig. 17. These inner rainbands tended to be elongated because of the shearing deformation of the TC flow and to move radially outward. As the upwind and middle sectors of inner rainbands approached the outer boundary of the inner core, convection reinvigorated with enhanced convective cells growing upwind because of the reduced filamentation and high boundary layer entropy. The rainbands were thus characterized by nascent cellular convection upwind, mature convective cells in the middle, and predominant stratiform clouds with collapsing convection downwind. This configuration is a typical outer rainband structure—namely, the formation of an outer rainband downshear.

Fig. 17.
Fig. 17.

Schematic diagrams showing (left to right) three scenarios of inner rainband behavior contributing to outer rainband formation in an easterly shear environment. The region within the dashed circle indicates the inner core of a TC. the letters IR and OR indicate the inner and outer rainband, respectively. The first scenario is the outer rainband forming from an inner rainband downshear as a sheared vortex Rossby wave. The second is the outer rainband forming directly from a single deformation-induced inner rainband. The third is the outer rainband developing from an inner rainband downshear organized from a blend/merger of inner rainbands that were initiated from locally deformed convection upshear right.

Citation: Journal of the Atmospheric Sciences 74, 1; 10.1175/JAS-D-16-0123.1

Apart from the fact that outer rainbands preferably formed on the downshear side and were related to the activity of inner rainbands, three scenarios were found to contribute to the initiation of outer rainbands for a TC embedded in a unidirectional shear (Fig. 17). In the first scenario, the inner rainband, which later developed into an outer rainband, was initially excited downshear near the outer edge of the eyewall (Fig. 17) and appeared to be a typical convectively coupled VRW. This type of outer rainband formation was previously illustrated in a quiescent environment in Li and Wang (2012a). In the second scenario, the downshear inner rainband, which eventually led to the formation of an outer rainband, was generated as a result of the storm’s deformational flow. Low-level convergence associated with the shear-induced outflow forced several weak convective cells upshear right in the RFZ outside the eyewall. These cells were rapidly filamented into a band. Low-level inflow on the downshear side steered the downwind portion of the rainband radially inward to connect with the eyewall downshear when the rainband propagated cyclonically and became more tightly wound. The formation of the rainband was a result of convection deformed by local TC flow as demonstrated in Moon and Nolan (2015), referred to as a deformation-induced inner rainband in this study. As the deformation-induced inner rainband became well organized on the downshear side and elongated, convection in its middle and upwind sectors became active and eventually evolved into an outer rainband (Fig. 17). In the third scenario, a deformation-induced inner rainband similar to that in the second scenario first formed downshear. Convection associated with the inner rainband occasionally disorganized because of shear-induced asymmetric flow and rapid filamentation. Other deformation-induced inner rainbands that formed later could catch up with the preexisting downshear inner rainband (Fig. 17). The blend of the inner rainbands often leads to a well-organized and more elongated inner-rainband downshear. Moving radially outward, this newly engendered inner rainband was eventually transformed into an outer rainband. Whether an outer rainband forms from a blend of preexisting deformation-induced inner rainbands or directly from a single deformation-induced inner rainband depends on convective instability on the downshear side. The CAPE downshear in the inner core in the second scenario was much higher than that in the third scenario, facilitating the convective growth in the inner rainband.

This study has examined specifically the formation of outer rainbands in a sheared TC simulated with TCM4. There are still many issues that have not been addressed, such as the proportion of the three scenarios contributing to outer rainband formation. The inner rainband evolution and outer rainband formation process proposed in the present study also need to be verified from observations and in simulations of real TC cases. More recently, the impact of shear on TC intensity change in observations has been noticed to depend on the vertical shear profile (Wang et al. 2015). How outer rainbands form in vertical shear with different vertical shear profiles and the strength of the shear requires further investigation.

Acknowledgments

The authors thank three anonymous reviewers for their constructive comments and suggestions. This work was jointly supported by the National (Key) Basic Research and Development (973) Program of China under Grant 2015CB452803, the National Natural Science Foundation of China under Grant 41475058, the Open Project of the State Key Laboratory of Severe Weather under 2016LASW-B08, the Top-notch Academic Programs Project of Jiangsu Higher Education Institutions (TAPP), and the Special Program for Applied Research on Super Computation of the NSFC-Guangdong Joint Fund (the second phase). Y. Wang has been supported in part by the National Natural Science Foundation of China under Grant 41130964 and in part by the NSF Grant AGS-1326524. Most of the data visualizations are postprocessed by the NCAR Command Language Version 6.3.0 (http://dx.doi.org/10.5065/D6WD3XH5).

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