Essential Dynamics of the Vertical Wind Shear Affecting the Secondary Eyewall Formation in Tropical Cyclones

Yi-Fan Wang aSchool of Atmospheric Sciences, Nanjing University, Nanjing, China
bNanjing Joint Institute for Atmospheric Sciences, Nanjing, China

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Zhe-Min Tan aSchool of Atmospheric Sciences, Nanjing University, Nanjing, China

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Abstract

This study investigated the effects of vertical wind shear (VWS) with varying magnitudes on secondary eyewall formation (SEF). It turns out that weak-to-moderate VWS advances the timing of SEF. Strong VWS, however, is unfavorable for SEF in our idealized simulations. VWS affecting SEF mainly lies on its influence on the outer rainbands (ORBs). Under weak-to-moderate VWS, ORBs develop more quickly in the downshear side and have distinct stratiform features in the upshear-left quadrant. The asymmetric inflow associated with the stratiform cooling descends into the boundary layer, reinforcing radial convergence at the radially inward side of ORBs. The radial convergence enhances the low-level convection, resulting in strengthened boundary layer inflow and accelerated low-level tangential wind jet. A budget analysis reveals that tangential advection extends a tangential wind jet farther downwind, forming supergradient winds above the boundary layer in the upshear-right quadrant. As the ORBs propagate into the upshear-right quadrant, the pre-existing supergradient winds enhances the low-level convection, facilitating the closing of the secondary convective ring. The evolution in the upshear side exhibit quadrant-dependent interactions between ORBs and boundary layer. Following that, azimuthal-mean tangential wind acceleration becomes visible, forming the secondary tangential wind maximum. Under strong VWS, the storm is weakened and the boundary layer in the upshear-left quadrant is invaded by low-entropy air, resulting in decreased conditional instability and low-level thermal buoyancy. The decreased stratiform precipitation due to weakened convective activity in the upshear-left quadrant prevents the upshear propagation of ORBs and thus is unfavorable for SEF.

© 2022 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: Zhe-Min Tan, zmtan@nju.edu.cn.

Abstract

This study investigated the effects of vertical wind shear (VWS) with varying magnitudes on secondary eyewall formation (SEF). It turns out that weak-to-moderate VWS advances the timing of SEF. Strong VWS, however, is unfavorable for SEF in our idealized simulations. VWS affecting SEF mainly lies on its influence on the outer rainbands (ORBs). Under weak-to-moderate VWS, ORBs develop more quickly in the downshear side and have distinct stratiform features in the upshear-left quadrant. The asymmetric inflow associated with the stratiform cooling descends into the boundary layer, reinforcing radial convergence at the radially inward side of ORBs. The radial convergence enhances the low-level convection, resulting in strengthened boundary layer inflow and accelerated low-level tangential wind jet. A budget analysis reveals that tangential advection extends a tangential wind jet farther downwind, forming supergradient winds above the boundary layer in the upshear-right quadrant. As the ORBs propagate into the upshear-right quadrant, the pre-existing supergradient winds enhances the low-level convection, facilitating the closing of the secondary convective ring. The evolution in the upshear side exhibit quadrant-dependent interactions between ORBs and boundary layer. Following that, azimuthal-mean tangential wind acceleration becomes visible, forming the secondary tangential wind maximum. Under strong VWS, the storm is weakened and the boundary layer in the upshear-left quadrant is invaded by low-entropy air, resulting in decreased conditional instability and low-level thermal buoyancy. The decreased stratiform precipitation due to weakened convective activity in the upshear-left quadrant prevents the upshear propagation of ORBs and thus is unfavorable for SEF.

© 2022 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: Zhe-Min Tan, zmtan@nju.edu.cn.

1. Introduction

Secondary eyewall formation (SEF) has received a lot of attention as an important structural change in intense tropical cyclones (TCs). Although the majority of these studies focus on internal dynamics of SEF in real cases (e.g., Willoughby et al. 1982; Houze et al. 2006; Rozoff et al. 2008; Wu et al. 2012; Huang et al. 2012; Didlake et al. 2018) and idealized simulations without environmental flow (e.g., Nong and Emanuel 2003; Terwey and Montgomery 2008; Qiu et al. 2010; Rozoff et al. 2012; Kepert 2013; Qiu and Tan 2013; Abarca and Montgomery 2014; Wang et al. 2016, 2019; Wang and Tan 2020, WT20 hereafter), environmental conditions also have nonnegligible influences on SEF [e.g., low-level relative humidity (Hill and Lackmann 2009), beta effect (Fang and Zhang 2012), westerly jet (Dai et al. 2017), shortwave radiation (Tang et al. 2017), vertical wind shear (VWS) (Zhang et al. 2017)].

VWS is one of the most important environmental factors influencing TC intensity (e.g., DeMaria 1996; Frank and Ritchie 2001; Riemer et al. 2010; Tang and Emanuel 2010; Gu et al. 2015) and structure (e.g., Black et al. 2002; Corbosiero and Molinari 2002, 2003; Zhang et al. 2013; Gu et al. 2016). However, the impacts of VWS on SEF are still unclear. Despite the SEF climatology indicating that SEF corresponds with weaker VWS (Kossin and Sitkowski 2009), cases of SEF under strong VWS are also documented in observational studies (Zhang and Perrie 2018; Dougherty et al. 2018). In idealized simulation studies, Zhang et al. (2017) found that 38 out of 40 members undergo complete or partial SEF under VWS of 6 m s−1, indicating that sheared environmental conditions do not prevent SEF. Menelaou et al. (2014), on the other hand, suggested that VWS has a negative effect on SEF because it disrupts the development of a closed secondary potential vorticity ring. As a result, the influence of VWS on SEF is still debatable.

Recent studies have widely acknowledged the initiating role of the asymmetric forcing associated outer rainbands (ORBs) in SEF (Qiu and Tan 2013; Didlake et al. 2018; Wang et al. 2019; WT20). The division of ORBs and inner rainbands is 3 times the radius of maximum wind (RMW) (Wang 2009). Compared to inner rainbands, ORBs exhibit asymmetric structures that consist of convective precipitation at the upwind and stratiform precipitation at the downwind end (Moon and Nolan 2010; Didlake and Houze 2013b). Qiu and Tan (2013) first identified that the asymmetric inflow at the downwind portion of ORBs descends into boundary layer (BL) and helps initiate convection during the early stage of SEF. The existence and role of the descending inflow in triggering convection were corroborated by observations of Hurricane Earl (2010) (Didlake et al. 2018). Wang et al. (2019) emphasized that the axisymmetrization of asymmetric winds associated with ORBs contributes to accelerating the secondary tangential wind maximum of SEF. WT20 pointed out that the asymmetric radial inflow reinforces BL convergence and stretches up the BL relative vorticity at the radially inward side of ORBs, which was a crucial point in the development of the secondary convective ring and the BL tangential wind maximum. In comparison, the vortex dominated by inner rainbands fails to drive the secondary tangential wind maximum pathways, demonstrating the ORBs are the internal triggering mechanism of SEF.

A canonical SEF, based on previous observations and simulations, should have two basic characteristics: a secondary convective ring and an associated secondary low-level tangential wind maximum. WT20 pointed out that the two characteristics of secondary eyewall are governed by distinct dynamical pathways. In the case of active outer rainbands of TC, the two features are codeveloped in SEF. In the case of inner rainband dominance, the secondary convective ring occurs independently without a secondary wind maximum, which is referred to as “fake SEF.” Therefore, the influence on both two features should be taken into consideration when assessing the impact of VWS on SEF. Moreover, because the asymmetric forcing of ORBs is vital to SEF, the effect of VWS on ORBs can be viewed as a starting point for investigating the influences on SEF.

VWS facilitates the formation of ORBs by inducing vortex tilt and asymmetric thermodynamic conditions (Willoughby et al. 1984; Jones 1995; Frank and Ritchie 2001; Riemer et al. 2010; Molinari et al. 2012; Reasor et al. 2013; Li et al. 2017; Rios-Berrios and Torn 2017; Schecter 2022). Riemer et al. (2010) suggest that VWS induces wavenumber-1 vorticity asymmetry atop the BL at the downshear-right (DR) quadrant through outer-vortex tilting, which forces the development of ORBs through BL frictional convergence (Chen and Yau 2001). Li et al. (2017) also demonstrate that VWS facilitates the formation of ORBs from original inner rainbands at the downshear side. Moreover, VWS induces asymmetric thermodynamical conditions, including higher CAPE and conditional instability at the downshear side (particularly at DR quadrant) and lower CAPE and conditional instability at the upshear side (Chen et al. 2006; Molinari et al. 2012; DeHart et al. 2014; Li and Dai 2020). As a consequence, convection exhibits different features and constitutes different rainbands structure among shear-relative quadrants. Typically, the convective cells of ORBs develop at DR quadrant, mature at downshear-left (DL) quadrant, and collapse into stratiform at upshear-left (UL) quadrant, which respectively corresponds to the upwind, middle, and downwind portions of the ORBs (Hence and Houze 2008). In this regard, VWS facilitates the development of ORBs and fixes the structures of ORBs in shear-relative quadrants, providing the asymmetric forcing necessary for SEF.

However, on the negative side, thermodynamic conditions at the upshear side are unfavorable for convective activities (Molinari and Vollaro 2010; Schecter 2022), and thus is unfavorable for the closing of the secondary convective ring associated with SEF. Although Riemer et al. (2010) noted that VWS promotes the development of rainbands at downshear side, the resulted convective downdrafts bring low entropy air into BL and suppress convection downwind. Li and Dai (2020) also suggest that in strong sheared TCs, downdrafts of rainbands reduce the equivalent potential temperature (θe) of BL at DL, resulting in lower conditional instability at DL. Moreover, UR is characterized by weaker BL inflow and radial divergence (Zhang et al. 2013), both of which are detrimental to the development of secondary eyewall convection and spinup of low-level tangential wind. In other words, while the VWS may facilitate the asymmetric forcing of ORBs for SEF at the downshear side, the projection of asymmetric forcing onto the axisymmetric state is more difficult under VWS. Therefore, the effects of VWS on SEF should be bifurcated and dependent on the magnitudes of VWS.

Through a set of ensemble experiments, Zhang et al. (2017) reveal that VWS-induced outer-core convection causes the expansion of wind field, which facilitates the top-down building and enhancement of ORBs from an axisymmetric view. However, the asymmetric dynamical process of ORBs’ forcing and BL response during the early stage of SEF under VWS needs further investigation. Moreover, the effects of VWS with varying magnitudes on SEF also remain answering. Specifically, the initiation/enhancement of convection and acceleration of tangential wind at different VWS-relative quadrants, especially at upshear side, warrants detailed investigation.

Based on the above questions, the impacts of VWS on SEF and associated dynamics are being investigated in this study, aiming to answer the following questions: (i) How does VWS with varying magnitudes affect SEF? (ii) How do the structures of ORBs and BL responses related to SEF evolve in different VWS-relative quadrants? (iii) How do the secondary convective ring and tangential wind maximum form at upshear side under VWS?

The remainder of this paper is organized as follows. Model setup and experiment design are introduced in section 2. Results and an overview of the simulations are presented in section 3. Section 4 compares rainband structures and BL response at left of shear side. The initiation of convection at UR is examined in section 5. Section 6 gives a preliminary discussion on the influence of VWS on SEF with varying storm outer-core sizes. Finally, main findings are summarized in section 7.

2. Model setup and experiment design

In WT20, a set of triplet experiment without environmental flow was introduced, which utilize different initial outer-core sizes controlling rainbands activities (Fig. 1). Among the triplets, the vortex with medium initial outer-core size (named as B05 in WT20) develops into a fake SEF during the first 90 h, which exhibits a concentric convective ring but without a secondary tangential wind maximum. In this study, B05 is chosen as the control run (CTRL) to assess the influence of VWS on SEF. VWSs of varying magnitudes are added on B05 to determine whether VWSs are beneficial to SEF by inducing the formation of the secondary tangential wind maximum, or detrimental to SEF by breaking the secondary convective ring.

Fig. 1.
Fig. 1.

Radial profiles of tangential wind (m s−1, solid line) and relative vorticity (10−4 s−1, dashed line) of the initial vortex in WT20. B05 (red lines) is taken as CTRL of this study. B02 and B08 are used only in section 6.

Citation: Journal of the Atmospheric Sciences 79, 11; 10.1175/JAS-D-21-0340.1

The initial vortex is integrated with Weather Research and Forecasting (WRF) Model (version 3.8.1) on an f plane at 20°N over a quiescent ocean with a constant SST of 28°C. There are three nested domains (361 × 361, 181 × 181, 361 × 361) with horizontal grid spacing of 18, 6, and 2 km, respectively. The parameterization schemes are identical to those used in WT20, including the Thompson (Thompson et al. 2004, 2008) and the Mellor–Yamada–Janjić (MYJ) (Janjić 1996, 2002) for the parameterization of the microphysical and planetary boundary layer processes, respectively; the RRTM longwave (Mlawer et al. 1997); and Goddard shortwave radiative scheme (Chou and Suarez 1999). The Kain–Fritch cumulus scheme (Kain 2004) is applied in the outer two domains. Open (periodic) lateral boundary conditions are applied at the south–north (east–west) direction onto the outermost domain.

VWS is introduced at 48 h when the simulated storm is formed in CTRL (Fig. 2a). The introduction of VWS here follows that in Gu et al. (2015). First, spin up the vortex without environmental flow till 48 h. Then, create a background state with temperature and pressure fields that balance the desired zonal wind field using the point-downscaling method proposed by Nolan (2011). Finally, merge the simulated vortex with the baroclinic environment and adjust the pressure to ensure gradient wind balance before restarting. Shear magnitudes of 4, 8, and 12 m s−1 are adopted to represent the weak, moderate, and strong VWS, respectively (Rios-Berrios and Torn 2017). Although the profiles of environmental wind are more complex in the real atmosphere and have different influences on the TC evolution (e.g., Onderlinde and Nolan 2014, 2016; Gu et al. 2018, 2019; Chen et al. 2018, 2019), here we adopt the vertical profiles of the zonal wind, which changes linearly between the heights of 2 and 12 km with zero flow in the lower levels, since we mainly focus on the influence of shear magnitudes in this study (Fig. 2b).

Fig. 2.
Fig. 2.

(a) Time–radius evolution of the azimuthal-mean tangential wind at z = 1 km (m s−1, shading) overlaid with the vertical velocity z = 5 km (m s−1, contours at 0.25, 0.5 and 1.0 m s−1) of the CTRL; (b) the vertical wind shear profiles with the magnitudes of 0, 4, 8, and 12 m s−1.

Citation: Journal of the Atmospheric Sciences 79, 11; 10.1175/JAS-D-21-0340.1

3. Overview and results

a. Axisymmetric evolution

Figure 3 shows the axisymmetric structural evolutions of CTRL, SH04, SH08, and SH12 during 48–120 h. In CTRL, vertical motions propagate radially outward from 80 to 120 km radius during 48–84 h, indicating that inner rainbands dominate convective activities outside the primary eyewall (Fig. 3a). The simulated reflectivity shows a secondary convective ring between 80 and 120 km radii (Figs. 4c,d). During this period, however, there is no obvious enhanced BL inflow and accelerated tangential wind associated with the inner rainbands (Fig. 3e), exhibiting a “fake SEF” as shown in WT20. After that, ORBs (indicated by w ≥ 0.25 m s−1 beyond 120 km radius) contract radially inward from 160 km radius and merge with inner rainbands at 96 h. The BL inflow increases from 84 h along with the inward contraction of ORBs, and a secondary tangential wind maximum develops at around 74 km radius at 100 h.

Fig. 3.
Fig. 3.

(a)–(d) Time–radius evolution of the azimuthal-mean tangential wind at z = 1 km (m s−1, shading) overlaid with the vertical velocity z = 5 km (m s−1, contours at 0.25, 0.5 and 1.0 m s−1) during 48–120 h. (e)–(h) As in (a)–(d), but showing the BL inflow (m s−1, averaged between z = 0.2–1 km, shading) overlaid with the divergence (contours at −1, −3, −5, −10, and −20 × 10−4 s−1, averaged between z = 0.2 and 1 km). The secondary RMW is marked by white dashed lines.

Citation: Journal of the Atmospheric Sciences 79, 11; 10.1175/JAS-D-21-0340.1

Fig. 4.
Fig. 4.

Plan view of the modeled reflectivity (dBZ) at z = 3 km during 63–84 h. (a)–(f) CTRL, (g)–(l) SH08, and (m)–(r) SH12. Circles are shown at every 40 km from the storm center and the black circle highlights the 120 km radius. The red arrow indicates the shear direction. The orange arrows indicate the secondary convective ring in CTRL.

Citation: Journal of the Atmospheric Sciences 79, 11; 10.1175/JAS-D-21-0340.1

By comparison, SH04 and SH08 develop secondary tangential wind maxima at 92 and 84 h, respectively (Figs. 3b,c), suggesting that weak-to-moderate VWS promotes earlier SEF in these simulations. In particular, the increasing VWS favors the earlier development and organization of ORBs within 120–160 km radii. Especially in SH08, ORBs are well organized beyond 120 km radius at 63 h and enhanced at 69 h (w ≥ 0.5 m s−1). The locally enhanced BL inflow between 80 and 120 km radii becomes evident from 69 h, reinforcing BL convergence at the radially inward side of ORBs (Fig. 3g). Moreover, the tangential wind shows acceleration along with the inward contraction of ORBs. As a result of the earlier development and contraction of ORBs, weak-to-moderate VWS advances the timing of SEF. It should be noted that the secondary RMW is located at a radius of 70 km in SH04 and 66 km radius in SH08 averaged over 96–108 h, implying weak-to-moderate VWS reduces the radial location of the secondary eyewall compared to CTRL.

In SH12 where strong VWS is placed, there is no SEF since the vortex intensity and the BL inflow are significantly weakened (Figs. 3d,h). Although there are convective activities beyond 120-km radius after 72 h, they are loosely organized and move radially outward, which is different from the ORBs associated with SEF. Therefore, strong VWS is unfavorable for the organization and inward contraction of ORBs, which is detrimental to SEF.

b. Rainbands evolution

As shown above, VWSs of varying magnitudes influence the organization of ORBs, which is associated with variations of BL inflow and tangential wind. Specifically, wake-to-moderate VWS advances the timing of SEF by advancing inward-contracting ORBs, whereas strong VWS disrupts the secondary maximum ascending motion outside the primary eyewall. To further investigate rainband characteristics leading to advanced SEF in SH08, the simulated reflectivity of CTRL, SH08, and SH12 during 63–84 h is compared in Fig. 4. Since the vortex evolution of SH04 is in between of CTRL and SH08, its results are omitted.

During this period, convection in CTRL is dominated by inner rainbands (within the radius of 120 km), especially before 72 h (Figs. 4a–d). Convection outside of 120 km radius has not been organized into ORBs. Although the inner rainbands form a secondary convective ring between 80 and 120 km radii in CTRL at 69 and 72 h (Figs. 4c,d), there is no local acceleration of the tangential wind (Fig. 3a), exhibiting the fake-SEF stage before the development of ORBs.

In SH08, asymmetric ORBs form at the downshear side between 120 and 200 km radii during 63–66 h (Figs. 4g,h). In DR with favorable thermodynamic conditions (shown below in Figs. 6 and 7b,e), the inner rainbands become more convective (indicated by higher reflectivity convective cells) as they propagate outward from 60 to 120 km radius, corresponding to the upwind portion of ORBs (Li et al. 2017). Farther downwind, the scattered convective cells gathered in the DL, forming the middle portion of ORBs. Moreover, the simulated reflectivity indicates scattered convective cells embedded in stratiform precipitation in the UL (shown below in Fig. 5). By comparison, the UR quadrant has the weakest convection outside the primary eyewall. During 69–72 h (Figs. 4i,j), the radial coverage of stratiform precipitation in UL increases, corresponding to contraction of ORBs and enhanced vortex-scale BL inflow around 120 km radius (Fig. 3). Afterward, ORBs propagate to UR quadrant with enhanced convection at 78 h, suggesting the ORBs are symmetrized into the secondary convective ring at 80 km radius (Fig. 4k). By comparing with Fig. 3c, it can be noticed that the acceleration of tangential wind outside the primary eyewall becomes noticeable after the closing of the secondary convective ring.

Fig. 5.
Fig. 5.

(a)–(c) The convective–stratiform partitioning (shading) for (left) CTRL, (center) SH08, and (right) SH12 at 69 h following the algorithm of Rogers (2010). White region denotes “no rain.” Contours represent radial velocity averaged (−12, −14, −16, and −18 m s−1 with line colors from light to dark) at z = 600 m. (d)–(f) As in (a)–(c), but for 72 h.

Citation: Journal of the Atmospheric Sciences 79, 11; 10.1175/JAS-D-21-0340.1

In SH12, the wavenumber-1 convective asymmetry outside 120 km radius is more visible under strong VWS (Figs. 4k–o). In DR and DL quadrants, high reflectivity convective cells are dispersed over 120–200 km radii. In comparison to the developing stratiform precipitation of UL in SH08, the evolution of reflectivity of SH12 indicates that strong VWS may weaken ORBs at upshear side, especially before 72 h (Fig. 4p). Although there are developing rainbands signals beyond 160 km after 78 h (Fig. 4q), the rainbands propagate radially outward from storm center (Fig. 3d), which is beyond the scope of this study. Therefore, strong VWS is unfavorable to SEF in part because it prevents the downwind propagating of ORBs in UL, destroying the axisymmetric convective ring outside the primary eyewall.

To show the evolution of rainband structure more precisely, precipitation partition is conducted in Fig. 5. Following Rogers (2010), the grid is classified as convective if the reflectivity at z = 3 km exceeds 40 dBZ and z = 1–3 km averaged vertical motion exceeds 0.5 m s−1. The criteria for stratiform type are 3 km height reflectivity ≥ 20 dBZ and no convective flagging. A grid with a reflectivity between 0 and 20 dBZ is flagged as “other” [indicating anvil-type precipitation according to Rogers (2010)], while a grid with reflectivity ≤ 0 is flagged as “no rain.” As shown in Fig. 5, the separation of stratiform and convective precipitation in CTRL is less visible than that in SH08 (Figs. 5a,d). The majority of the BL inflow exceeding 12 m s−1 is concentrated within a 120-km radius. By comparison, SH08 exhibits obvious stratiform precipitation at the left of shear side and relative weaker precipitation in UR at 69 h (Fig. 5b). There is locally enhanced BL inflow around 120 km radius in the DL, and a weaker BL inflow in the UR. As a result, BL convergence is stronger at the leading edge of BL inflow in the DL and weaker in the UR. At 72 h, the stratiform precipitation in UL is further enhanced and extended to UR (Fig. 5e). The asymmetric BL inflow extended downwind to UL at 72 h accompanied by increased stratiform precipitation. Compared with Fig. 3e, the expansion of stratiform precipitation is concurrent with an increased azimuthal mean inflow. In SH12, the stratiform precipitation at the left of shear weakens into anvil-type precipitation and fails to extend to UR. Moreover, BL inflow is restricted at the downshear side, especially in the DL, indicating exacerbated radial divergence of UR.

The convective–stratiform variation is related to thermodynamic conditions (Didlake and Houze 2013b). To investigate the asymmetric thermodynamic conditions under different shear magnitudes, the azimuthal distribution of CAPE and Δθe (difference between z = 4 and 1 km, representing the low-level stability) averaged over 80–160 km radii (the radial range of concentrated convection outside the primary eyewall) are shown in Fig. 6. Because this study focuses on the asymmetric structure of ORBs prior to the closing of the secondary convective ring, 66–72 h is chosen as the period for time average. Under larger VWS, there are larger (smaller) CAPE and low-level convective instability in the DR (UL), which is consistent with observations (Molinari et al. 2012). SH12 shows the most evident decreased low-level convective instability in the UL quadrant, which is consistent with the weakened rainband convection at UL (Figs. 4n–p). According to Riemer et al. (2010), the low-θe air is brought downward into BL by the convective downdrafts in the DL, which is advected downwind and stabilize the lower troposphere at upshear side. To examine this mechanism, θe at z = 800 m during 66–72 h is shown in Figs. 7a–c. The low-level thermal buoyancy is also exhibited.

Fig. 6.
Fig. 6.

The azimuthal distribution of the (a) CAPE (J kg−1) and (b) Δθe (K, difference between z = 4 and 1 km, representing the low-level instability) average over 80–160 km radii during 66–72 h.

Citation: Journal of the Atmospheric Sciences 79, 11; 10.1175/JAS-D-21-0340.1

Fig. 7.
Fig. 7.

Plan view of (a)–(c) θe (K) at z = 800 m and thermal buoyancy (cm s−2) averaged between z = 0.8 and 1.2 km averaged over 66–72 h for (left) CTRL, (center) SH08, and (right) SH12.

Citation: Journal of the Atmospheric Sciences 79, 11; 10.1175/JAS-D-21-0340.1

The thermal buoyancy is calculated following Eastin et al. (2012) as
BT=gθeθe¯
where g is taken as 9.81 m s−2 and θe is equivalent potential temperature. Overbar and prime denote the reference state and perturbation. Following Ahern et al. (2021), the azimuth-mean θe averaged over a 20 km radial width was chosen as the reference state. As shown in Fig. 7, the mean θe between 80 and 120 km radii of UL is approximately 348 K in SH12, which exhibits a 3 K deficit compared to the same area of SH08 (∼351 K), suggesting that the decreased instability is due to cooler BL air. Meanwhile, in SH12, negative thermal buoyancy spirals from a radius of 160 km of DL to the inner-core region of UR (Fig. 7f). The spiral pattern of negative buoyancy is striking in the UL outside a radius of 80 km, indicating that negative θe perturbations are associated with ORBs. In comparison to SH12, SH08 has much smaller negative buoyancy in the UL (Fig. 7e). Moreover, the UR quadrant of SH08 is dominated by positive thermal buoyancy (with little negative buoyancy) and favorable convective instability conditions (Fig. 6), implying the weak convection at UR is caused by BL divergence associated with weak inflow before 72 h (Figs. 5b,e).

In summary, weak-to-moderate VWS facilitates stratiform precipitation and increases BL inflow at the left of shear side. However, the stratiform precipitation in the UL quadrant is weakened due to the lower BL θe under strong VWS. Earlier studies suggest that the low-θe air descending into the BL is not necessarily detrimental to convection since it may trigger the lifting of high-θe air along the edge of cold pool (Qiu and Tan 2013; Didlake et al. 2018; Chen et al. 2021). However, in this case the negative buoyancy and weaker BL inflow of the UL are unfavorable for the lifting of high-θe air from BL. Therefore, strong VWS is unfavorable for the development/maintenance of stratiform precipitation in the UL.

4. Evolution and BL response of rainbands at left of shear

a. Rainbands structure and radial inflow

As stated above, moderate VWS induces the development of stratiform precipitation of ORBs at left of shear. The asymmetric inflow associated with the stratiform sector of ORBs is initiative to SEF (Qiu and Tan 2013; Didlake et al. 2018; WT20). To further demonstrate that moderate VWS facilitates SEF by inducing the stratiform sector of ORBs, the radius–height structures of diabatic heating and BL response at 69 h are compared in Fig. 8. In this study, the top of BL is defined as the altitude of radial inflow diminishing to 0 m s−1.

Fig. 8.
Fig. 8.

The radius–height structure of (a)–(c) the diabatic heating rates (K h−1) overlaid by vertical velocities (solid line at 0.25 and 0.5 m s−1; dashed line at −0.025 and −0.05 m s−1) and (d)–(f) radial divergence (10−5 s−1) overlaid by radial velocities (blue line at −2, −4, −8, −12, −16, and −20 m s−1 and red line at 2, 4, and 6 m s−1) averaged over the DL (southwest quadrant for CTRL) quadrant at 69 h. (g)–(i) As in (a)–(c), but for UL and (j)–(l) as in (d)–(f), but for UL (southeast quadrant for CTRL). Columns show (left) CTRL, (center) SH08, and (right) SH12.

Citation: Journal of the Atmospheric Sciences 79, 11; 10.1175/JAS-D-21-0340.1

In CTRL without VWS, the azimuthal average is conducted over the southwest (Figs. 8a,d) and southeast (Figs. 8g,j) quadrants. Comparing to SH08 and SH12, the difference between DL and UL in CTRL is minimal. Diabatic heating and BL convergence are associated with inner rainbands within 80 km radius. As shown in Fig. 4c, ORBs of CTRL are relatively weaker, with a more uniform azimuthal distribution. Therefore, outside 120 km radius, the local acceleration of BL inflow and convergence are less evident in CTRL.

By comparison, SH08 exhibits the strongest diabatic heating between 120 and 160 km, with cooling along the outer edge of the ORBs in the DL (Fig. 8b). Convective cells in the DL are indicated by scattered updrafts (w ≥ 0.5 m s−1). Farther downwind in the UL, diabatic heating is weaker than of DL around 120 km radius, and diabatic cooling is stronger outside 140 km radius (Fig. 8h), indicating stratiform features of UL. That is, from DL to UL in SH08, ORBs exhibit decaying convective features and strengthening stratiform features. The BL inflow and convergence are also distinct between DL and UL. In response to the intense diabatic heating of DL, a secondary BL inflow maximum is well-developed around 160 km radius (Fig. 8e), reinforcing BL convergence between 120 and 160 km radii (Fig. 8e). In the UL, the stratiform cooling induces downward motion below 6 km height and penetrates into the BL between 120 and 160 km radii. As proposed by Qiu and Tan (2013), the asymmetric inflow accelerated by the stratiform cooling reinforces BL convergence between 80 and 120 km radii. Moreover, there is a distinct outflow above the BL over 80–120 km radii, which collides with the midlevel inflow outside 160 km radius and reinforces radial convergence above the BL between 120 and 160 km radii (Fig. 8k). The outflow above the BL is accelerated by the supergradient force associated with the low-level tangential wind jet. The formation of the low-level tangential wind jet is related to the descending inflow of the stratiform precipitation, which accelerates the low-level tangential wind through triggering convective updraft at its at its radially inward edge (Didlake et al. 2018; Yu et al. 2021).

The quadrant variation of rainband structures is stronger in SH12. The scattered convective cells in the DL are indicated by the fragmented updrafts, which are interspersed with low-level downdrafts (Fig. 8c). The diabatic heating and BL convergence between 80 and 120 km of DL in SH12 is weaker than that in SH08 (Fig. 8f). In the UL quadrant, the region outside the primary eyewall is dominated by diabatic cooling, which corresponds to weakened rainband convection there (Fig. 8i). As a result, the enhancement of BL inflow and convergence outside the primary eyewall are subtle in the UL quadrant in SH12 (Fig. 8l).

b. Enhancement convection in the UL

From the above analysis, it can be concluded that the stratiform precipitation at UL of SH08 induces radial convergence both within and above BL. As suggested in WT20, the connection of two patches of convergence facilitates high-θe air out of BL, which enhances the convection associated with SEF. Convection between 80 and 120 km radii is strengthened at UL after the formation of two patches of convergence at 72 h (Figs. 4j and 5e). To examine the enhancement of convection in UL of SH08, the evolution of BL convergence during 69–78 h is exhibited in Fig. 9.

Fig. 9.
Fig. 9.

Radius–height structures averaged over the UL of SH08 during 69–78 h with 3 h interval. (a)–(d) Diabatic heating rates (K h−1) overlaid by tangential velocities (at every 5 m s−1 from 45 to 65 m s−1). (e)–(h) Shading represents radial divergence (10−5 s−1). Radial velocities are shown in blue (−2, −4, −8, −12, −16, −18, and −22 m s−1) and red contours (2, 4, 6, and 8 m s−1). Vertical velocities are shown in black contours (solid line at 0.25 and 0.5 m s−1; dashed line at −0.025 and −0.05 m s−1).

Citation: Journal of the Atmospheric Sciences 79, 11; 10.1175/JAS-D-21-0340.1

At 69 h, there are two ascending motion regions related to convergence within and above BL, respectively (Fig. 9e). At 72 h, the two patches of convergence become more pronounced during the inward contraction of ORBs (Fig. 9f). Diabatic heating associated with the two patches of convergence is also enhanced compared to 69 h (Figs. 9a,b). Meanwhile, although the secondary tangential wind maximum has not developed at this time, the acceleration of low-level tangential wind between 80 and 120 km radii can be observed, and locally enhanced relative vorticity (υ/r+υ/r) can be expected. As suggested in Kepert (2013), the increased relative vorticity atop BL promotes BL convergence via the Ekman-like pumping effect. At 75 h, the two patches of convergence are concentrated into enhanced convergence below 3 km height between 60 and 80 km, with diabatic heating connected below 6 km height (Figs. 9c,d). Following that, the connected convergence and diabatic heating are further amplified, accompanied by an accelerated low-level tangential wind jet (Figs. 9d,h). During this stage, persistent outflow above BL around 80 km radius indicates the maintenance/enhancement of supergradient force at radially inward side of ORBs, which contributes to the enhancement of rainband convection by reinforcing convergence above BL.

Overall, the interaction of ORBs and BL and the evolution of convection in the UL of SH08 is consistent with WT20, which can be summarized as follows: stratiform heating structure at the downwind end of ORBs induces convergence within and above BL, which enhances convection during the inward contraction of ORBs. The enhanced convection facilitates strengthening of BL inflow and acceleration of tangential wind, which in turn reinforces radial BL convergence.

5. Convection initiation at UR

As mentioned above, UR is at the negative phase of azimuthal convective asymmetry caused by VWS. Convection of UR determines whether the secondary convective ring of SEF can form. As shown in Fig. 10, the initiation of convection in the SEF region of UR is investigated. At 69 and 72 h, diabatic heating associated with ORBs in the UR is unnoticeable (Figs. 10a,b). However, radial outflow greater than 6 m s−1 occurs above BL around 80 km radius, reinforcing convergence at the radially outward side at 72 h (Fig. 10f). The elevated tangential wind jet between 80 and 120 km indicates that the outflow is accelerated by supergradient force (Smith et al. 2009). At 75 h, outflow and convergence above BL are further enhanced. Meanwhile, diabatic heating within 80–120 km radii and diabatic cooling outside 120 km radius become evident, with the occurrence of the descending inflow. In combination with Figs. 4i–k, it can be concluded that the low-level convection of ORBs is enhanced by the low-level convergence associated with the outflow above the BL as the downwind end of ORBs propagates to the UR quadrant. The enhanced diabatic heating and low-level convergence at 78 h in the UR quadrant indicate the secondary convective ring associated with SEF is closed.

Fig. 10.
Fig. 10.

As in Fig. 9, but averaged over UR.

Citation: Journal of the Atmospheric Sciences 79, 11; 10.1175/JAS-D-21-0340.1

By comparing Figs. 9 and 10, it can be concluded that the deep convection develops at the radially side of stratiform precipitation during 69–78 h in the UL. Meanwhile, rainbands with stratiform features are developed in the UR, indicating the downwind end of rainbands extending from UL to UR. Different from UL where outflow occurs simultaneously with rainbands, outflow above BL occurs prior to rainbands in the UR and maintains/enhances rainbands convection with favorable low-level thermal buoyancy and conditional instability (Figs. 6 and 7e).

To further demonstrate the relationship between the occurrence of supergradient winds and enhancement of convection in SEF region of UR, the quadrant-averaged evolution of agradient winds and vertical velocity atop BL are shown in Fig. 11. The agradient force (AGF), gradient wind balance, and agradient wind are given as follows:
AGF=fυ+υ2r1ρPr,
fυg+υg2r1ρPr=0,
υa=υυg,
where υ stands for total storm-relative tangential velocities, υg indicates gradient wind, and υa indicates agradient wind; f, ρ, and p are Coriolis parameter, air density, and pressure, respectively.
Fig. 11.
Fig. 11.

Quadrant-averaged time–radius evolution of the agradient wind average between z = 0.6 and 1.0 km (m s−1, shading) overlaid with the vertical velocity at z = 1 km (m s−1, contours at 0.2, 0.5, 1.0, and 2.0 m s−1) in SH08. The red arrow between the rows indicates the shear direction.

Citation: Journal of the Atmospheric Sciences 79, 11; 10.1175/JAS-D-21-0340.1

In the DR and DL quadrants, the development of ORBs causes supergradient winds at the upper BL (Figs. 11a,b), which is due to the convective cells at the upwind and middle portion increase the tangential wind more rapidly than the pressure field can adjust (Didlake and Houze 2013a). In the UL quadrant (Fig. 11d), there is little ascending motion or supergradient wind before 66 h. After 66 h, there are developing supergradient winds in accompany with weak ascending motion, indicating the presence of stratiform precipitation of ORBs. The supergradient winds are associated with a low-level tangential wind jet accelerated by the descending inflow of the stratiform sector of ORBs (Didlake et al. 2018; Yu et al. 2021). Along with the inward contraction of ORBs, the supergradient winds reinforce the radial convergence and enhance the advection of ORBs in return (Fig. 9). By contrast, in the UR, the development of the supergradient wind is well before the vertical motion atop the BL (Fig. 11c). Combined with Fig. 10, it can be concluded the pre-existing supergradient force accelerates radial outflow and thus reinforces radial convergence above the BL. As the downwind end of ORBs extended to UR, radial convergence above BL enhances the low-level convection of ORBs. The BL inflow associated with ORBs reinforces the low-level radial convergence and accelerates tangential wind in return, which ultimately leads to enhanced convection in the UR. Therefore, the interactions of ORBs and BL are distinct between UL and UR quadrants, demonstrating a quadrant-dependent interaction between ORBs and BL process.

As seen from Eq. (2), the acceleration of tangential wind causes supergradient force. To determine the physical process responsible for the spinup of the tangential wind at UR of SH08, a budget analysis of tangential wind tendency is conducted. The governing equation of tangential wind tendency in cylindrical coordinates is given as follows:
υt=u(f+ζ)υυrλwυz1ρprλ+Fλ,
where u, υ, and w are the radial, tangential, and vertical components of the storm-relative velocities, respectively; r, λ, and z indicate radial, tangential, and vertical direction of cylindrical coordinates, respectively; ζ=υ/r+υ/r is the vertical component of relative verticity; f, ρ, and p are Coriolis parameter, air density, and pressure, respectively. The terms on the right-hand side are radial advection, tangential advection and vertical advection, tangential pressure gradient force, and diffusion term, respectively. The diffusion term includes subscale wind tendencies, which are directly output from model before interpolated to cylindrical coordinates.

Figure 12 shows the plan view of the tangential wind tendency averaged during 69–72 h. Figure 12a shows the tendency output from the WRF model, and Fig. 12b represents the sum of the rhs of Eq. (5). The budget diagnostic detects the positive tendency of tangential wind between 80 and 120 km radii at the upshear side. The distribution of radial advection and radial velocities reveals the radial inflow transports absolute vorticity inward between 80 and 120 km radii in DR, DL, and UL quadrants (Fig. 12c). The radial advection is much weaker in the UR due to weak inflow before 75 h (Fig. 10c). Between 80 and 120 km radii, the tangential advection term has a positive tendency in the UR, and a negative tendency in the UL (Fig. 12d). The maximum tangential wind between 80 and 120 km radii is located at the border between UL and UR, indicating the tangential wind jet at downwind end of ORBs. At z = 1 km, the vertical advection term is negative at DR, DL, and UL, indicating that ascending motion transports momentum upward (Fig. 12e). Due to the lack of ascending motion in UR, the BL wind speed loss due to vertical advection is reduced. The diffusion term is relatively weaker in the outer-core region at z = 1 km height (Fig. 12f). The tangential pressure gradient force is negligible (Fig. 12g).

Fig. 12.
Fig. 12.

Plan view of tangential wind budget at z = 1 km during 69–72 h in SH08: (a) modeled tangential wind tendency (m s−1 h−1), (b) calculated tangential wind speed tendency (m s−1 h−1), (c) radial advection term (m s−1 h−1, shading) and mean radial velocities (contours from 2 to 8 at 2 m s−1 interval) during 69–72 h, (d) tendency of tangential advection (m s−1 h−1, shading) and mean tangential wind speed (contours from 39 to 60 at 3 m s−1 interval), (e) vertical advection term (m s−1 h−1, shading) and mean vertical velocities (contours from 0.25 to 1 at 0.25 m s−1 interval), (f) diffusion term (m s−1 h−1), and (g) tangential pressure gradient force (m s−1 h−1). The red arrow in the bottom left denotes the shear direction. Values within 40 km radius are omitted.

Citation: Journal of the Atmospheric Sciences 79, 11; 10.1175/JAS-D-21-0340.1

Based on this budget analysis, it can be concluded that the inflow accelerates tangential wind through radial advection term in the UL. The elevated momentum is transported downwind by tangential advection term, which accelerates the tangential wind and forms supergradient winds in the UR. The tangential advection is important for the quadrant-dependent interaction between ORBs and BL at upshear side.

6. Discussion

In this study, B05 in WT20 is taken as CTRL (Fig. 1). Compared to B05, B02 and B08 in WT20 represent initial vortex with larger outer-core size and smaller outer-core size, respectively. Two additional groups of experiments with B02 and B08 initial vortex wind profiles are conducted to assess the responses of a vortex with varying initial outer-core sizes to VWS in terms of SEF. The experiments named as B02_SH00, B02_SH04, B02_SH08, and B02_SH12 denote VWS with the magnitudes of 0, 4, 8, and 12 m s−1 imposed on B02 after a 48 h integration in static environmental flow. The naming convention of B08 group is same to that of B02.

The secondary tangential wind maximum forms at 84 h in B02_SH00 (Fig. 13a). In comparison, the secondary tangential wind maxima in B02_SH04 and B02_SH08 form at 74 and 70 h, respectively (Figs. 13b,c), which is consistent with the conclusion that weak-to-moderate VWS promotes earlier SEF. In B02_SH12 with strong VWS, the secondary tangential wind maximum is largely attenuated and merged with the primary eyewall (Fig. 13d), demonstrating the detrimental effects of strong VWS on SEF. However, unlike B05_SH12, where SEF is completely destroyed, B02_SH12 still has inward contractions of ORBs and secondary convective ring. The secondary convective ring diminishes when 16 m s−1 VWS is added to B02 (not shown). In terms of SEF, vortex with a larger outer-core size has higher resistance to VWS. Additionally, Dougherty et al. (2018) document the SEF under strong VWS in Hurricane Bonnie (1998). They proposed that the broadening wind field supports the secondary eyewall that extends to the upshear side, implying the importance of a larger outer-core wind for SEF under strong VWS.

Fig. 13.
Fig. 13.

(a)–(d) Time–radius evolution of the azimuthal-mean tangential wind at z = 1 km (m s−1, shading) overlaid with the vertical velocity z = 5 km (m s−1, contours at 0.25, 0.5, and 1.0 m s−1) during 48–120 h for B02 group. (e)–(h) As in (a)–(d), but for B08 group. The secondary RMW is marked by white dashed lines.

Citation: Journal of the Atmospheric Sciences 79, 11; 10.1175/JAS-D-21-0340.1

There is no SEF in B08_SH00 (Fig. 13e). As is shown in Figs. 13f–h, although adding VWS activates convective activities outside the primary eyewall, they are loosely organized and fail to form the secondary eyewall. It is indicated that, while the environmental conditions influence SEF, the initial outer-core size largely determines the organization of ORBs and whether SEF can form. The effect of VWS on vortex with varying initial outer-core size merits further investigation.

7. Conclusions

In this study, the effects of VWS on SEF and associated dynamics are investigated using idealized full-physics simulations. In the control experiment without VWS, the simulated vortex is dominated by inner rainbands outside the primary eyewall before 96 h, forming a “fake SEF” with a secondary convective ring but no secondary tangential wind maximum. To investigate the effect of VWS on SEF, VWS with weak, moderate, and strong magnitudes are added to the control experiment at 48 h. The results show that the effects of VWS on SEF is bifurcated depending on shear magnitudes. Weak-to-moderate VWS advances the timing of SEF while decreasing the radius of secondary tangential wind maximum. Strong VWS, however, causes significant TC weakening and breaks the secondary convective ring and therefore is unfavorable for SEF.

The dynamics of SEF under weak-to-moderate VWS are investigated. Weak-to-moderate VWS promotes earlier SEF by advancing the organization of ORBs and inducing asymmetric structures in different VWS-relative quadrants. Specifically, at DR with the most favorable thermodynamic conditions, active convective cells form the upwind portion of ORBs. In the DL, convective cells gathered and matured, forming the middle portion of ORBs. Compared to the inner rainbands in the experiment without VWS, the stronger diabatic heating of ORBs in the DL quadrant under moderate VWS induces stronger BL inflow and radial convergence in SEF region.

At UL where thermodynamic conditions for convection are reduced, the downwind end of ORBs exhibits obvious stratiform features. The asymmetric inflow induced by diabatic cooling descends from 6 km height into BL along the outer edge of the stratiform deck, reinforcing radial BL convergence at the radially inward side of ORBs. Radial BL convergence enhances convection of ORBs in return, resulting in increased BL inflow and accelerated low-level tangential wind jet. The positive feedback between ORBs and BL convergence promotes the enhancement of convection outside the primary eyewall and spinup of low-level tangential wind in the UL.

By comparison, in UR quadrant which is originally dominated by the weakest BL inflow and BL convergence, the enhancement of outer convection is preceded by the supergradient wind above BL. A budget analysis is conducted to investigate the occurrence of supergradient wind prior to the enhanced convection. It turns out the BL inflow in UL accelerates tangential wind through radial advection of absolute vorticity. The tangential wind jet transports increased tangential momentum downwind, accelerating the tangential wind and forming supergradient winds above the BL in the UR. As the downwind end of ORBs extends to UR, the pre-existing supergradient winds strengthen the low-level convection of ORBs, resulting in enhanced diabatic heating at UR. The initiation/enhancement of convection in UR promotes the closing of the secondary convective ring. Following that, enhanced BL inflow and acceleration of tangential wind are projected onto the azimuthal-mean state, forming the secondary tangential wind maximum.

Through the analysis of SEF under moderate VWS, it can be concluded that the interaction between ORBs and BL is still essential for the development of the secondary convective ring and secondary tangential wind maximum in sheared conditions, which is consistent with WT20. But compared with WT20, the interaction between ORBs and BL shows quadrant-dependent features at the upshear side. The stratiform convection of ORBs triggers unbalanced BL responses in the UL quadrant and the supergradient winds above the BL maintain/enhance rainbands convection in the UR quadrant. Hereinto, UL can be viewed as the crucial quadrant for the initiation of SEF under VWS.

This study also looks into the negative effects of strong VWS on SEF. It turns out that convective downdrafts in DL bring low θe into the BL. The low-θe air is advected farther downwind, causing decreased instability and negative thermal buoyancy in UL. As a result, the stratiform precipitation in UL is weakened due to suppressed convective activities, thus cutting off the ORBs extending to upshear side. The breaking off of SEF under strong VWS also supports that the stratiform sector of ORBs in the UL is vital for SEF under VWS.

Finally, this study also reveals that in addition to the magnitudes of VWS, the effects of VWS on SEF also depend on vortex outer-core size. It is found that vortex with larger outer-core size has a higher resistance to VWS in terms of SEF. The synergistic effects of VWS and vortex structure on SEF merit further investigation in future work.

Acknowledgments.

This work is jointly supported by the National Natural Science Foundation of China under Grant 42192555 and the National Key R&D Program of China under Grant 2017YFC1501601. Wang Y.-F. is also supported by the Basic Research Fund of CAMS under Grant 2022Y022 and the Jiangsu Innovative and Entrepreneurial Talent Programme JSSCBS20221644. We thank Dr. Gu Jian-Feng for providing the initialization program of adding vertical wind shear. The authors also thank Dr. Rozoff and three anonymous reviewers for their careful reading and helpful suggestions that brought significant improvements to the manuscript.

Data availability statement.

The data used in this study are output by the WRF Model (version 3.8.1) and visualized by NCL (https://www.ncl.ucar.edu). Main model setup and method are described in section 2. Namelist is shared via https://box.nju.edu.cn/d/77a4fe523e734094b5f9/.

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