1. Introduction
The secondary (also often concentric) eyewall formation (SEF) mechanism of tropical cyclones (TCs) has been given more and more attention in recent years (Terwey and Montgomery 2008; Judt and Chen 2010; Qiu et al. 2010; Qiu and Tan 2013; Abarca and Corbosiero 2011; Huang et al. 2012; Bell et al. 2012; Rozoff et al. 2012; Sun et al. 2013; Abarca and Montgomery 2013, 2014; Kepert 2013; Zhu and Zhu 2014; Wang et al. 2016, Zhang et al. 2017; Dai et al. 2017; Tyner et al. 2018) because of its essential role in causing the rapid structure and intensity changes of TCs (Willoughby et al. 1982; Kossin and Sitkowski 2009; Sitkowski et al. 2011). A secondary eyewall is generally identified as the establishment of a quasi-circular convective ring at about 2–3 times of the primary radius of maximum wind, often with a secondary maximum in the azimuthal-mean tangential wind (Willoughby et al. 1982; Black and Willoughby 1992; Houze et al. 2007; Bell et al. 2012). Although the secondary eyewall phenomenon is most frequently observed in intense, highly symmetric TCs (e.g., Willoughby et al. 1982; Kuo et al. 2009), the SEF involves not only axisymmetric processes but also asymmetric processes and not only balanced dynamics but also unbalanced dynamics. SEF can be triggered not only by internal processes but also by external forcing, such as vertical wind shear (e.g., Zhu et al. 2004; Riemer 2016; Zhang et al. 2017).
Observational studies have demonstrated that most SEFs in TCs are related to the activity of spiral rainbands (Willoughby et al. 1982; Black and Willoughby 1992; Houze et al. 2007; Kuo et al. 2009; Kossin and Sitkowski 2009). Black and Willoughby (1992) studied the SEF in Hurricane Gilbert (1988) and showed that it was the outer rainbands that developed into an encircled new convective ring (the secondary eyewall). Kuo et al. (2009) found that the initiation and development of the secondary convective eyewall were favored by strong outer spiral rainbands over the western North Pacific. Kossin and Sitkowski (2009) found that the SEF is easier to form in TCs with high maximum potential intensity likely because the thermodynamic environment is more favorable for the development of strong spiral rainbands and thus the SEF.
Various mechanisms have been proposed to explain how spiral rainbands or the related asymmetric processes contribute to the formation of the secondary eyewall. One of them involves vortex Rossby waves, which are often related to the activity of inner spiral rainbands (Chen and Yau 2001; Wang 2001, 2002a,b; Corbosiero et al. 2006; Li and Wang 2012). Vortex Rossby waves propagate outward in the inner core of a TC and may stagnate at a radius where the wave energy dissipates to accelerate the local winds, forming a secondary maximum of the azimuthal-mean tangential wind and enhancing surface enthalpy flux and convection, which are favorable for the SEF (Montgomery and Kallenbach 1997). This mechanism was supported by Qiu et al. (2010), who analyzed the SEF in a simulated TC under idealized conditions. However, Judt and Chen (2010) argued that the vortex Rossby wave activity was not a contributing factor to the SEF in Hurricane Rita (2005) and that it was high potential vorticity (PV) generation and accumulation in the rainband and the subsequent axisymmetrization that led to the secondary PV maximum and the SEF in Rita. It seems to suggest that not all SEFs are related to the vortex Rossby wave activity.
Terwey and Montgomery (2008) proposed a mechanism to explain the axisymmetrization of convective disturbances. That is, convective activity in the region with negative radial gradient of the azimuthal-mean vorticity outside the eyewall can act as a source of eddy vorticity. When the resulting vorticity anomalies cascade upscale to form a secondary vorticity ring and the formation of a localized jet at low levels, surface enthalpy flux and thus convective activity could be enhanced and organized to form a secondary eyewall. This mechanism argues that the SEF is triggered by the enhanced surface moisture flux because of the secondary tangential wind maximum and the associated development of deep convection.
More recently, the unbalanced boundary layer dynamics has been proposed to be key to the SEF (Huang et al. 2012; Abarca and Montgomery 2013, 2014; Kepert 2013). Huang et al. (2012) examined the SEF process in their simulated Typhoon Sinlaku (2008) and proposed that the unbalanced boundary layer response to the outward expansion of tangential wind above the boundary layer plays a key role in the concentration and sustention of deep convection in a narrow supergradient wind zone where the secondary eyewall forms. Abarca and Montgomery (2013) confirmed that their simulated SEF was controlled by the progressive unbalanced boundary layer response to the radial broadening of tangential wind above the boundary layer. Abarca and Montgomery (2014) further indicated that the axisymmetric balanced dynamics alone could not explain the spinup of tangential winds in the boundary layer during the SEF and the unbalanced boundary layer dynamics is key. Bell et al. (2012) analyzed the concentric eyewall evolution in Hurricane Rita (2005) using multiplatform data and found that the secondary eyewall was spun up by the radial convergence of angular momentum both above and within the boundary layer, suggesting that both the balanced and unbalanced dynamics are important to the SEF. Kepert (2013) showed that the radial vorticity gradient above the boundary layer could induce frictional updrafts in the low-vorticity environment outside the inner core. He thus proposed that SEF could result from a positive feedback among the local enhancement of the radial vorticity gradient above the boundary layer, the frictionally forced updrafts, and convection.
The boundary layer mechanism above depends strongly on the radial expansion of the azimuthal-mean tangential wind, but the mechanism itself could not explain why the outward expansion occurs prior to SEF. The mechanism was introduced based on the axisymmetric argument, and thus, it could not explain the asymmetric nature of strong rainband activity prior to the SEF. Wang (2009) demonstrated that diabatic heating in outer rainbands is responsible for the size increase of a TC and a secondary eyewall could form with enhanced diabatic heating in outer rainbands. Rozoff et al. (2012) showed that when latent heating in spiral rainbands is projected onto a considerable azimuthal-mean component, the SEF rapidly occurred, and the increased vorticity in response to the outward broadening of tangential winds above the boundary layer can made this process more efficient. Based on axisymmetric balanced dynamics, Sun et al. (2013) also found that the SEF could occur if the outer-rainband convection reached a critical strength relative to the eyewall convection. Zhu and Zhu (2014) further showed that the low-level heating in rainbands is important in building up the tangential wind in the SEF region and the subsequent SEF. Although the above studies emphasized the importance of diabatic heating in outer rainbands to SEF, they have not examined how outer rainbands form and evolve into the secondary eyewall.
Some previous studies have shown that outer spiral rainbands evolving into a secondary eyewall in TCs can develop through various processes. For example, Fang and Zhang (2012) found in their idealized simulation that the beta-induced vertical wind shear could trigger the development of stratiform precipitation in the outer region. Intense convective activities can be initiated in the stratiform cloud region because of the production of the midlevel PV anomalies and then strengthened to form a quasi-symmetric convective ring outside the primary eyewall. Qiu and Tan (2013) found that the persistent asymmetric inflow in the stratiform region could descend into the boundary layer and enhance low-level convergence and convection in the SEF region and thus the SEF. Zhang et al. (2017) found that the SEF can result from the downward building and axisymmetrization of the primary rainband with the enhanced inertial stability, suggesting a top-down process (see also Wang 2006). This involves the contribution by diabatic heating in stratiform clouds to the initiation and development of the strong outer spiral rainband. Another top-down pathway to SEF was studied by Dai et al. (2017), who found that the interaction between a TC and a midlatitude westerly jet could enhance the outflow and produce asymmetric stratiform clouds outside the primary eyewall, and deep convection can be initiated and then organized into a secondary eyewall. Tyner et al. (2018) also examined a top-down pathway to SEF but with the focus on cloud microphysical processes. They showed that the SEF in their simulation began with the development of local convection triggered at the outer radii by both the fallout of hydrometeors from the primary eyewall and the cooling-induced penetrative downdrafts, and the convection-induced convergence was extended down into the boundary layer; thus, the high-entropy air was extended out to further enhance convection and the SEF. These previous studies have examined the top-down pathways mainly based on the axisymmetric viewpoint without explicitly analyzing the detailed evolution of rainbands in their simulations.
More recently, Didlake et al. (2018) studied the dynamical transition from outer rainbands to a secondary eyewall in Hurricane Earl (2010) based on airborne Doppler radar observations. They found that under the influence of environmental vertical shear, an outer rainband developed in the downshear-left quadrant of the storm. Mesoscale inflow persistently occurred in the stratiform region of the rainband, flowed inward, and descended into the boundary layer, which locally enhanced convection and tangential winds. Such steady low-level asymmetric forcing strongly projected onto the axisymmetric fields, leading to the SEF. The importance of asymmetric forcing in the boundary layer to the SEF was also noticed by Wang et al. (2016).
This study attempts to examine the detailed kinematics and dynamics of the transition from outer spiral rainbands to a secondary eyewall with the focus on addressing how spiral rainbands contributed to the spinup of tangential wind prior to and during the SEF and how spiral rainbands evolve and are axisymmetrized into the secondary eyewall based on a high-resolution simulation under idealized conditions. The rest of the paper is organized as follows. The model simulation is described in section 2. The axisymmetric and asymmetric aspects of the SEF are discussed in sections 3 and 4, respectively. The transition from the outer rainband to the secondary eyewall in the simulation is analyzed in section 5. Major results are summarized in the last section.
2. Model simulation
The model configuration and initialization followed Wang et al. (2016), and only a brief description is given here. The WRF Model, version 3.5.1, was used to conduct an idealized numerical simulation of the SEF in a TC. The model domain was quadruply nested and the horizontal domain sizes for the 45-, 15-, 5-, and 1.67-km meshes are 200 × 200, 250 × 250, 268 × 268, and 388 × 388, respectively. There were 50 vertical levels with the model top at 50 hPa. The vortex-following technique was applied to the three inner nested meshes so that they were initially located near the center of their parent domains. The Kain–Fritsch cumulus parameterization scheme (Kain and Fritsch 1993) was adopted in the two outer meshes, and convection in the inner two meshes was assumed to be explicitly resolved. Other physical parameterizations used in the four meshes were the WRF single-moment 6-class microphysics scheme (WSM6; Hong et al. 2006), the Yonsei University (YSU) planetary boundary layer scheme, the shortwave radiation scheme of Dudhia (Dudhia 1989), and the Rapid Radiative Transfer Model (RRTM) for longwave radiation calculation (Mlawer et al. 1997).
The model was initialized with an axisymmetric cyclonic vortex (Wang 2007) embedded in a quiescent tropical environment with a fixed sea surface temperature (SST) of 29°C so that the environmental asymmetric impacts were not included. The far-field unperturbed environment temperature and humidity profiles were based on the western Pacific clear-sky environment given by Gray et al. (1975). The initial vortex had the maximum tangential wind of 25 m s−1 at the radius of 80 km, and the tangential wind diminished to zero at the radius of 1000 km. The vortex was in gradient wind and hydrostatic balance and located at the center of each mesh initially. The model was configured on an f plane centered at 20°N. Further details of the model setup and simulation can be found in Wang et al. (2016), but note that the initial vortex structure was a little different from that used in Wang et al. (2016), and thus, the details of the simulated SEF was also different. The model was integrated forward for 192 h with hourly model output except that 6-min outputs were saved between 108- and 126-h integration for detailed analysis of the SEF in the simulation.
Figure 1 shows the evolution of the hourly averaged PV fields at 1.5-km height (Fig. 1a) and the hourly averaged vertically integrated cloud water mixing ratio (Fig. 1b) from 500 m to 4 km during 111–126 h at 3-h interval, showing the formation of the secondary eyewall in the simulation. PV includes both dynamical and thermodynamic properties and is widely used to understand basic dynamics, including the SEF (e.g., Judt and Chen 2010). The simulated storm showed both elevated PV and high cloud water content in both the primary eyewall and active inner and outer spiral rainbands. The secondary eyewall seemed to form from the axisymmetrization of spiral rainbands and showed a typical concentric eyewall structure with a clear moat area after 120 h of simulation. The concentric eyewall feature became much clearer in both the hourly composite PV at 1.5 km [Fig. 1a(5)] and vertically integrated cloud water content [Fig. 1b(5)] by 123 h of simulation. We will show in the next section that by common definition of the appearance of the secondary maximum in the azimuthal-mean tangential wind, the concentric eyewall completely formed by 123 h in the simulation. Therefore, in our discussion, we will refer to the 12-h period from 108 to 120 h as the stage of the SEF and the 3-h period between 120 and 123 h as the completion of the simulated SEF.
Plan view of (a) the hourly averaged PV [PV units (PVU); 1 PVU = 10−6 K kg−1 m2 s−1] at 1.5-km height and (b) the hourly averaged vertically integrated cloud water mixing ratio (g kg−1) from 500 m to 4 km during 111–126 h with a 3-h interval. The model time is noted at the top-left of each panel.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0130.1
3. The axisymmetric aspects of the simulated SEF
Figure 2 shows the time–radius cross sections of the azimuthally averaged (Fig. 2a) vertical motion at 3-km height and (Fig. 2b) tangential wind speed at 1.5-km height. Note that since our main interests are on the time evolution well before the SEF in the simulation, we show in Fig. 2 the results from 72 h of simulation. Convection, as inferred from the strong updrafts, in the primary eyewall was very intense and within a radius of 50 km prior to the SEF by about 120–123 h of simulation (Fig. 2a). The secondary eyewall formed at a radius of about 80 km and then contracted and eventually replaced the primary eyewall after 152 h of simulation as seen from the elevated vertical motion. The radius of the primary eyewall remained at about 35 km until it dissipated or was replaced by the outer eyewall. The intermittent appearance of upward motion outside the eyewall implied the active spiral rainbands. After 90 h of simulation, strong inner spiral rainbands developed outside the primary eyewall and propagated radially outward slowly (see Fig. 2a). In the meantime, strong outer rainbands developed at about 170-km radius from around 108 h and propagated radially inward (Fig. 2a). As the outer rainband propagated inward, the inner rainbands experienced a weakening and eventually merged with the outer rainband by about 120 h of simulation. This led to the formation of the secondary convective ring structure and was followed by the SEF shortly as we can see from the azimuthal-mean vertical motion in Fig. 2a.
Time–radius Hovmöller diagrams of the azimuthal-mean (a) vertical motion at 3 km (m s−1) and (b) tangential wind at 1.5-km height (m s−1) from 72 to 192 h of simulation.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0130.1
Although the azimuthal-mean tangential wind (Fig. 2b) experienced an overall outward expansion in the simulation, a rapid outward expansion occurred from 108 h of simulation prior to the SEF. This significant outward expansion seemed to be initiated from around a radius of about 150 km and shifted radially inward (Fig. 2b) following the inward-propagating outer rainband (Fig. 2a). It seems to suggest that the rapid outward expansion of the azimuthal-mean tangential wind prior to the SEF was closely related to the development and inward propagation of the outer spiral rainband. A secondary maximum in the azimuthal-mean tangential wind at 1.5-km height appearing at a radius of 75 km (Fig. 2b), together with the establishment of a quasi-axisymmetric convective ring outside the primary eyewall [Fig. 1b(5)] by about 123 h of simulation, marks the formation of the secondary eyewall. With the intensification and inward propagation of the secondary eyewall, the primary eyewall weakened gradually and eventually dissipated by around 153 h (Fig. 2a). Therefore, the double eyewall structure lasted for about 30 h in the simulated storm. As the outer eyewall contracted, the outward expansion of the azimuthal-mean tangential wind became negligible within a radius of 150 km, which is consistent with less convective activity outside the secondary eyewall afterward in the simulation (Fig. 2a).
Figure 3 shows the radius–height cross sections of 3-hourly averaged azimuthal-mean vertical motion and tangential wind speed from 108 to 125 h based on model outputs of 6-min interval. Vertical motion shows a strong updraft in the primary eyewall, which is slightly inside the radius of maximum wind during the given time period. The eyewall updraft tilts persistently outward with height, particularly in both the boundary layer and in the upper-tropospheric outflow layer. A second narrow upward-motion region appeared between radii of 50 and 80 km (Figs. 3a–d), which corresponded to the activity of inner spiral rainbands before the SEF (Figs. 1 and 2a). Another upward-motion region appeared in the upper troposphere between 7- and 13-km heights farther outside between radii of 160 and 190 km initially (Fig. 3a) and then strengthened and extended both downward and inward with time (Figs. 3b–e). This upward motion was related to the projection of vertical motion in active outer spiral rainbands (Fig. 1) onto the azimuthal mean (also see Fig. 2a). The top-hat vertical structure of vertical motion indicates that the outer rainbands was initially dominated by stratiform cloud/precipitation process and then became more convective as the rainbands intensified and propagated radially inward (Didlake and Houze 2011; Didlake et al. 2018). Note that during 117–123 h, the vertical motion in both inner and outer spiral rainbands showed temporarily weakening and then merging. This was followed by a rapid strengthening of vertical motion between 60- and 100-km radii (Fig. 3f), indicating the formation of the convective ring structure and the SEF. Note that the outward expansion of the azimuthal-mean tangential wind appeared not only in the lower troposphere (Fig. 2b) but in much of the troposphere. For example, the 40 m s−1 contour in the lower troposphere expanded from 140 km averaged between 108 and 110 h to 205 km averaged between 123 and 125 h. This outward expansion prior to the SEF has been reported in many previous studies as cited in section 1. Since the outer spiral rainband that resulted in the SEF was initiated from the upper troposphere, the SEF in the simulation can be considered as a top-down process, although the merging of inner and outer rainbands also played some roles.
The 3-hourly averaged azimuthal-mean vertical motion (shaded; m s−1) and tangential wind (contour interval of 5 m s−1) based on model outputs at 6-min intervals from 108 to 125 h of simulation. Downward and inward arrows roughly indicate downward development and inward propagation of the outer spiral rainband prior to the SEF in the simulation.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0130.1
Figure 4 shows the radius–height cross sections of 3-hourly averaged azimuthal-mean diabatic heating rate and radial wind speed from 108 to 125 h based on the model outputs of 6-min interval. The largest diabatic heating rate appeared in the outwardly sloped primary eyewall even shortly after the SEF. In the narrow radial range immediately outside the primary eyewall, the diabatic heating rate is negligible or even negative mainly because of the melting of ice-phase hydrometers and the evaporation of raindrops resulting from the lateral mixing with unsaturated air outside the eyewall. The second maximum diabatic heating rate immediately outside the narrow negative heating region centered at about 65-km radius reflects the projection of diabatic heating in inner rainbands (Fig. 1) onto the azimuthal mean. This inner rainband heating peaks at a height of about 4 km and remains strong with its appearance until the SEF by 120–122 h. Another region with positive diabatic heating rate appeared farther outside that was initially quite weak and centered between the 150- and 180-km radii (Fig. 4a) but rapidly intensified and propagated radially inward, in particular, the lower part of the heating region. This led to an outwardly sloped vertical structure of diabatic heating. This outer positive heating region reflects the projection of diabatic heating in outer spiral rainbands (Fig. 1) onto the azimuthal-mean heating rate. An apparent merging of the two positive heating regions of inner and outer spiral rainbands occurred with the initial formation stage of the secondary eyewall at around 120 h (Figs. 4d and 4e). This was followed by a sudden enhancement and concentration of diabatic heating between the radii of ~60–100 km by 123 h, namely, the SEF (Figs. 4e and 4f). Note that a broad area of positive diabatic heating rate sloped radially outward, indicating the stratiform clouds outside the newly formed secondary eyewall, and the narrow region around the radius of 40 km with negligible diabatic heating indicates the development of the moat, which became clearer by 123 h (Fig. 4f).
The 3-hourly averaged azimuthal-mean diabatic heating rate (shaded; K h−1) and radial wind (contours of −18, −15, −12, −9, −6, −3, −2, −1, 0, 1, 2, 3, 6, 9, 12, 15, and 18 m s−1) based on model outputs at 6-min interval from 108 to 125 h of simulation.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0130.1
In response to diabatic heating was the inflow in the middle to lower troposphere outside a radius of about 100 km and a broad outflow layer in the upper troposphere (Fig. 4). The strongest inflow appeared in the frictional boundary layer below about 1.5-km height. The inflow depth however shows a decreasing trend toward the radius of about 100 km, or equivalently, an increase in inflow depth radially outward from 100-km radius. This tendency seems to be consistent with the outwardly sloped diabatic heating rate. The broad outflow layer, initially in the upper troposphere, extended downward in regions of diabatic heating in both inner and outer rainbands. A shallow outflow layer immediately above the inflow layer occurred in the inner core region with two distinct centers: one in the primary eyewall region and one in the region with inner rainbands as inferred from the second maximum diabatic heating rate mentioned above. The former remained through the SEF, while the latter experienced a broadening when the inner and outer spiral rainbands merged and a sudden enhancement as the secondary eyewall formed. The radial wind distribution can be well explained by the response of the secondary circulation to diabatic heating in the primary eyewall and in spiral rainbands based on balanced dynamics (e.g., Shapiro and Willoughby 1982; Fudeyasu and Wang 2011). The shallow outflow layer immediately above the inflow boundary layer results mainly from the agradient force due to the presence of supergradient winds (not shown). Note that the shallow outflow layer outside the primary eyewall appeared in the region of inner spiral rainbands well before the SEF, suggesting that supergradient winds also exist in the inner rainbands, as well as in the region where the SEF later.
Many previous studies have already demonstrated that the heating-induced secondary circulation is responsible for the outward expansion of the azimuthal-mean tangential wind discussed above (e.g., Wang 2009; Fudeyasu and Wang 2011; Xu and Wang 2010), and the response of the boundary layer to the latter has been recognized an important process for the enhancement of vertical motion in the region of the SEF (Huang et al. 2012; Bell et al. 2012; Rozoff et al. 2012; Zhu and Zhu 2014; Wang et al. 2016). To examine this process in our simulation, we show in Fig. 5 the 3-h tendency of the azimuthal-mean tangential wind overlapped with the azimuthal-mean radial wind and vertical motion time averaged in the corresponding 3-h period from the model outputs of 6-min interval prior to and during the SEF (from 108 to 125 h). Prior to the SEF between 108 to 110 h (Fig. 5a), the azimuthal-mean tangential wind slowly increased between the radii of 50 and 150 km, showing a radial broadening of the azimuthal-mean tangential winds. The positive tendency in the azimuthal-mean tangential wind expanded radially outward initially and gradually concentrated into the region with strong upward motion between radii of 100–200 km in outer rainband region (Fig. 5b). Relatively large increases in the azimuthal-mean tangential winds appeared in the lower troposphere and mid- to upper troposphere (Figs. 5c and 5d). The positive tangential wind tendency further strengthened in the regions between radii of 60 and 120 km where the SEF occurred several hours later (Figs. 5e and 5f). The positive azimuthal-mean tangential wind tendency tilted radially outward, which was consistent with the outwardly sloped diabatic heating in outer rainbands (Fig. 4). This indicates that the positive tangential tendencies were largely contributed by the inflow and vertical motion associated with diabatic heating in spiral rainbands. Namely, the heating-induced inflow in the lower troposphere brought high absolute angular momentum inward, spinning up the tangential wind, and vertical upward motion in rainbands further transported large tangential winds upward. As a result, diabatic heating in spiral rainbands, in particular in outer rainbands, was largely responsible for the outward expansion of the azimuthal-mean tangential winds, as previously demonstrated by Fudeyasu and Wang (2011).
The 3-hourly change in the azimuthal-mean tangential wind speed (shaded; m s−1), and the corresponding 3-hourly averaged vertical motion (black contours with values of −0.25, 0, 0.25, 0.5, 1, 3, and 5 m s−1) and radial wind (m s−1; the blue contours of −15, −10, −5, −2, −1, 0, 2, 5, 10, 15, and 20 m s−1) from 108 to 125 h of simulation.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0130.1
The composite azimuthal-mean tangential wind budget over the 3-h period of 114–116 h. (a) The net tangential wind tendency [the sum of all terms on the rhs of (1)] and the individual terms on the rhs of (1): (b) mean radial advection, (c) mean vertical advection, (d) mean friction term, (e) sum of mean advection and friction terms, (f) eddy radial advection, (g) eddy vertical advection, and (h) the sum of the eddy terms. All the terms are shown in units of m s−1 h−1 except for the net tangential wind tendency, which is shown in m s−1 (3 h)−1 for a direct comparison with the corresponding model tendency shown in Fig. 5c. Contours shown are −50, −25, −10, −4, −3, −2, −1, 1, 2, 3, 4, 10, 25, and 50 m s−1 h−1 in (b)–(h) and every 1 m s−1 (3 h)−1 in (a).
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0130.1
The mean radial advection (Fig. 6b) and the mean vertical advection (Fig. 6c) are two leading terms in the azimuthal-mean tangential wind budget, but they have opposite signs within and above the boundary layer. The vertical diffusion (and surface friction) is negative in the boundary layer and becomes negligible above 2-km height (Fig. 6d). Similar to the mean advections, the eddy radial (Fig. 6f) and vertical advections (Fig. 6g) also have opposite signs with a net positive contribution to the azimuthal-mean tangential wind tangency in a layer between 2- and 8-km heights and between radii of 50 and 150 km, and also in the boundary layer between 120 and 220 km where the sum of mean advections and vertical diffusion are negative (Fig. 6e). The results suggest that although overall the mean advection process dominates the azimuthal-mean tangential wind budget and thus the outward expansion of the azimuthal-mean tangential winds, the eddy process, mainly associated with active spiral rainbands, also contributed to the azimuthal-mean tangential wind budget. In particular, the net contribution of the eddy terms was to spin up the azimuthal-mean tangential winds both in the upper troposphere and in the boundary layer outside a radius of 110 km (Fig. 6h) where the sum of the azimuthal-mean contributions was negative (Fig. 6e). This means that dynamically the eddy motion in rainbands played a role in spinning up the azimuthal-mean tangential wind in the boundary layer in the region of the SEF.
To confirm the importance of the eddy processes in spinning up the azimuthal-mean tangential winds in the boundary layer discussed above, we show in Fig. 7 the time evolutions of the total eddy contribution to the azimuthal-mean tangential wind budget and also the high-wavenumber (≥3) eddy kinetic energy (EKE) from 108 to 126 h of simulation. We can see that the eddy contribution was generally positive in the boundary layer in the outer rainband region and propagated radially inward following the inward movement of the outer rainbands. This is consistent with the time evolution of high-wavenumber EKE shown in Fig. 7b. Note that the low-level high-wavenumber EKE was considerably larger because eddies were continuously generated in the convective outer rainbands. Nevertheless, the high-wavenumber EKE showed a slightly weakening in the SEF region after the outer eyewall formed.
Radius–time plots of (a) the total eddy contribution to the azimuthal-mean tangential wind budget (m s−1h−1) and (b) the azimuthal-mean high wavenumber (≥3) EKE (m2 s−2), both averaged in the lowest 1-km layer of the model atmosphere.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0130.1
We should point out that contribution by the mean advection to the azimuthal-mean tangential wind budget also includes a large contribution by diabatic heating in spiral rainbands as indicated in Fudeyasu and Wang (2011). This is because it is the diabatic heating in spiral rainbands (projected onto the azimuthal-mean diabatic heating rate) that drove the azimuthal-mean secondary circulation as discussed in section 3 and thus the azimuthal-mean radial and vertical advections of the azimuthal-mean tangential wind. In this sense, we can conclude that active spiral rainbands are key to the outward expansion of the azimuthal-mean tangential wind and the SEF.
4. The asymmetric aspects of the SEF
To further understand how the outer spiral rainband evolved into the secondary eyewall in the simulation, we examined the detailed structure of the hourly composite rainband. As indicated in section 3, prior to the SEF, an outer spiral rainband formed and gradually propagated radially inward and downward. We first analyzed the composite in hour 116 of simulation. Figure 8 shows the hourly averaged column-maximum radar reflectivity and column-integrated cloud water mixing ratio between 500-m and 5-km heights. This is the time period about 5–6 h prior to the SEF, although the inner rainbands already showed apparent axisymmetric structure because of fast azimuthal propagation of the rainbands and the time average. As we mentioned earlier, the inner rainbands were quite active from about 80 h of simulation (Fig. 2a), much earlier than the development of a dominant outer rainband that eventually evolved into the secondary eyewall later. The outer rainband spiraled cyclonically inward from the north about 160 km away from the TC center, approached the inner core from the south, and connected to the inner rainband about 75 km south of the TC center. It is our interest to examine the along-band and cross-band structure to elucidate the dynamics of this outer spiral rainband and its contribution to the subsequent SEF several hours later. The outer rainband can be divided into four azimuthal sectors (Fig. 8a), upwind (AB), middle (BC and CD), and downwind (DE) sectors, and the corresponding four circular sectors (q1, q2, q3, and q4 in Fig. 8b, respectively.).
(a) Composite (hourly averaged) column maximum radar reflectivity (dBZ) and (b) vertically integrated cloud water mixing ratio (g kg−1) from 500-m to 5-km height over hour 116 of simulation based on model output at 6-min intervals. Segments AB, BC, CD, and DE denote the upwind, middle, and downwind sectors of the outer rainband that are considered and will be used to show the vertical cross sections in Fig. 9, and the circular sector regions q1, q2, q3, and q4 will be used to show the azimuthal average in the circular region for the upwind, middle, and downwind sectors of the rainband in Fig. 10.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0130.1
Figure 9 shows the vertical structure of the hourly mean vertical motion and diabatic heating rate together with the asymmetric tangential and radial winds along the outer rainband marked in Fig. 8a. The total fields of vertical motion and total diabatic heating rate were used to mark strong convection and forcing of diabatic heating in the outer rainband, while the asymmetric components are shown to elucidate the eddy kinematic characteristics of the rainband. The vertical motion shows both convective and stratiform natures of the outer rainband (Fig. 9a). Individual convective cells were embedded in stratiform regions with weak subsidence in the mid- to lower troposphere. Updrafts were often more intense and penetrated deeper into the upper troposphere in the upwind sector, such as along sector AB and the upwind portion of sector BC. Updrafts became weaker and shallower in the middle and downwind portions of the rainband, such as in the downwind portion of sector BC and in sectors CD and DE. This along-band convective structure of the outer rainband is consistent with observations (Barnes et al. 1983). The diabatic heating rate along the outer rainband (Fig. 9b) showed an overall distribution similar to vertical motion (Fig. 9a). Large diabatic heating rate occurred in the middle troposphere in convective updrafts, while diabatic cooling occurred in the region with subsidence or in the subcloud layer below about 500-m height in the boundary layer.
Vertical cross sections of the hourly averaged (a) vertical motion (shaded; m s−1) and asymmetric tangential wind (contours with intervals of 2 m s−1; contour 2 m s−1 is thickened) and (b) diabatic heating rate (shaded; K h−1) and asymmetric radial wind (contours at −11, −8, −5, −2, 0, 2, 5, and 8 m s−1; 0 m s−1 contour is thickened) along the four segments as shown in Fig. 8a.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0130.1
The positive asymmetric tangential winds occurred in a layer between 4- and 13-km heights in the upwind sector, and both the height and depth gradually decreased and reached the layer between 0.5- and 4-km heights in the downwind sector. The descending of the layer depth with positive asymmetric tangential winds was much more pronounced in the middle and downwind sectors of the rainband. Negative asymmetric tangential winds appeared below the positive asymmetric tangential winds in the upwind sector along AB and above the positive asymmetric tangential winds in the middle and downwind sectors along BC, CD, and DE. A deep asymmetric inflow layer1 occurred below 7-km height in the upwind sector and became shallower toward the downwind sector of the rainband (Fig. 9b). A deep asymmetric radial outflow occurred immediately above the inflow layer and also showed a descending tendency from the upwind to downwind sector along the rainband, similar to the positive asymmetric tangential wind layer. Another asymmetric inflow layer appeared above the asymmetric outflow layer in the upper troposphere and also showed a descending tendency toward the downwind sector along the rainband. The asymmetric tangential and radial winds associated with the outer rainband showed a similar inward and downward spiraling nature along the rainband and were rapidly axisymmetrized in the downwind sector DE as the rainband propagated inward toward the SEF region.
Figure 10 shows the cross-band structure of the azimuthally averaged vertical velocity and asymmetric tangential wind (Figs. 10a–d) and diabatic heating rate and asymmetric radial wind (Figs. 10e–h) in the four circular sectors marked in Fig. 8b. The azimuthal mean in each circular sector was used to show the overall cross-band structure in different sectors of the rainband. Consistent with the along-band cross section shown in Fig. 9, deep convection as inferred from strong upward motion near the radius of 150 km occurred in the upwind sector of the rainband with relatively strong positive asymmetric tangential winds in the mid- to upper troposphere. The region with strong positive asymmetric tangential winds lowered in height and shifted inward from the upwind to the downwind sectors and appeared in the boundary layer in the downwind sector between 100- and 150-km radii (Fig. 10d). This implies that the strong surface winds in the downwind sector would enhance surface enthalpy flux, favorable for convection and the subsequent SEF (Qiu and Tan 2013). The asymmetric inflow dominated the mid- to lower troposphere in the upwind sectors but became very shallow in the downwind sector of the rainband. The asymmetric outflow dominated the mid- to upper troposphere, while the asymmetric inflow dominated the lower troposphere in both the upwind and middle sectors (Figs. 10e and 10f). However, the asymmetric inflow appeared farther above the midtroposphere outflow layer in both the middle and downwind sectors (Figs. 10g and 10h). Consistent with the along-band structure, the low-level asymmetric inflow became shallower toward the downwind sector of the rainband. The results are similar to those shown by Moon and Nolan (2010) and Qiu and Tan (2013). Both studies also showed that the wind maximum was mostly evident in the downwind sector of the outer rainband in the lower troposphere. However, we found that the low-level enhanced tangential wind in the region where the SEF occurred later was a result of the overall along-band descending nature of asymmetric kinetic motion and the inward propagation of the outer rainband.
Radial–vertical cross section of the (a)–(d) hourly averaged vertical velocity (shaded; m s−1) and asymmetric tangential wind (contours with contour interval of 1 m s−1) and (e)–(h) diabatic heating rate (shaded; K h−1) and asymmetric radial wind (contours at −8, −6, −4, −2, −1, 0, 2, 4, 6, and 8 m s−1), azimuthally averaged over the four circular sectors (a),(e) q1, (b),(f) q2, (c),(g) q3, and (d),(h) q4 shown in Fig. 8b.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0130.1
5. Transition from the outer rainband to the secondary eyewall
Results discussed in sections 3 and 4 strongly suggest that the SEF in the simulation can be considered as the rapid axisymmetrization of the downwind sector of the outer rainband as the rainband spiraled cyclonically inward. To understand the transition from the outer rainband to the secondary eyewall in the simulation, we further examined the hourly averaged low-level PV fields prior to and during the SEF (as already shown in Fig. 1a). We can see clearly from Fig. 1a that, except for the high positive PV core, weak positive PV was generated mainly in the outer rainbands and spiraled cyclonically inward prior to the SEF. As the elevated PV band spiraled cyclonically inward following the outer rainband, the downwind portion of the PV band became more and more axisymmetric after 117 h of simulation and then enhanced and formed a quasi-axisymmetric, secondary elevated PV ring structure by 123 h of simulation, suggesting a rapid axisymmetrization process during the SEF. The PV evolution was very similar to that of the hourly average column-integrated cloud water content shown in Fig. 1b.
To show the overall axisymmetrization process, we calculated the axisymmetricity parameter using PV at 1.5 km following Miyamoto and Takemi [2013, see their (2)]. As we can see from Fig. 11, except for the high axisymmetricity of the primary eyewall region (between radii of 20 and 50 km) prior to the SEF, the axisymmetricity parameter in the SEF region (between radii of 50 and 120 km) showed an overall increasing trend from 108 h after the outer rainband formed and spiraled cyclonically inward and approached 0.95 after about 120 h of simulation. This high axisymmetricity of the secondary eyewall region remained until the end of the simulation. The inner eyewall region was quite axisymmetric from 72 through 126 h, but shortly after the SEF, it became less axisymmetric because the inner eyewall broke down and eventually dissipated after about 156 h of simulation (not shown). This demonstrates that the continuous axisymmetrization of the downwind end in the boundary layer of the outer rainband played an important role in the simulated SEF.
Time series of the axisymmetricity parameter using potential vorticity at 1.5 km averaged between 20- and 50-km radii (black) and between 50- and 120-km radii (red).
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0130.1
A question arises as to why the SEF occurred between radii of 60- to 100 km in the simulation. This can be understood based on the rapid filamentation process discussed in Rozoff et al. (2006) and Wang (2008). We calculated the filamentation time using the azimuthal-mean tangential wind averaged in 6 h between 116 and 121 h of simulation with the result shown in Fig. 12a. The filamentation time was less than 35 min [viz., the rapid filamentation zone (RFZ)] between the radius of maximum wind of the primary eyewall and a radius of about 75 km. Wang (2008) found that in the RFZ any high-wavenumber asymmetric eddies could be effectively dampened and thus rapidly axisymmetrized. Although previous studies all indicate the importance of axisymmetrization of asymmetric motion to the SEF (Terwey and Montgomery 2008; Qiu and Tan 2013; Sun et al. 2013; Wang et al. 2016), we found here that the continuous axisymmetrization of the asymmetric eddies in the downwind sector of the rainband contributed to the spinup of the azimuthal-mean tangential wind mainly in the boundary layer and triggered the SEF in our simulation. This is consistent with the results of our azimuthal-mean tangential wind budget analysis discussed in section 3. As we show in Fig. 7a, the region with large positive azimuthal-mean tangential wind tendencies induced by the net eddy process in the boundary layer was farther outside the SEF region, shifted inward as the outer rainband spiraled cyclonically inward, and reached the SEF region later with the SEF.
Radius–height cross section of (a) filamentation time (min) and (b) relative vorticity (1.0 × 10−4 s−1) calculated using the azimuthal-mean tangential wind averaged in the 6 h between 116 and 121 h of simulation.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0130.1
We should point out here that the axisymmetrization of high-wavenumber asymmetric eddies does not mean the eddies were actually dampened. Figure 13 shows the azimuthal-mean EKE of wavenumbers 3 and above. We can see that the high-wavenumber asymmetric eddies were always active in the rainband region (as marked with vertical motion in contours) and under the outwardly tilted rainband. The EKE is particularly high below 2-km height where eddies contributed considerably to the spinup of the azimuthal-mean tangential wind in the boundary layer as discussed in section 3 (Fig. 7). Note that eddies did not show any significant weakening with the axisymmetrization process because eddies were continuously generated in the convective rainband and then in the newly formed secondary eyewall. There was a relatively small high-wavenumber EKE region between the primary eyewall and the newly secondary eyewall, which was coincident with the almost precipitation-free area (Fig. 1), namely, the moat area in the RFZ as previously documented by Rozoff et al. (2006).
Radius–height cross sections of the azimuthal-mean high-wavenumber (≥3) EKE (shaded; m2 s−2) overlain by the azimuthal-mean vertical motion (contours; m s−1) based on model outputs at 6-min interval.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0130.1
6. Conclusions
The SEF numerically simulated in a TC under idealized conditions, namely, in a quiescent environment on an f plane, with the advanced WRF Model was analyzed in this study. Both the axisymmetric and asymmetric aspects of the SEF were examined to understand the SEF processes. Consistent with previous findings (Huang et al. 2012; Rozoff et al. 2012; Wang et al. 2016; Zhang et al. 2017), our results also showed that prior to the SEF, the tangential wind of the TC vortex experienced an outward expansion both above and within the boundary layer near and outside the region where the SEF occurred later. This outward expansion was shown to result primarily from the top-down development and inward propagation of a strong outer spiral rainband, which was characterized by deeper and more intense convection upwind and shallower and weaker convection downwind. In response to diabatic heating in the rainband was inflow in the mid- to lower troposphere, which brought the absolute angular momentum inward and spun up tangential wind in the region of inflow, and also in the convective updrafts due to vertical advection. As a result, as the outer rainband intensified and spiraled cyclonically inward, perturbation (asymmetric) tangential and radial winds also spiraled cyclonically inward and downward along the rainband, and correspondingly, the outward expansion of the azimuthal-mean tangential wind showed an inward penetration toward the region where the SEF occurred later. Since the TC vortex maintained an RFZ between the primary eyewall and the radius where the SEF occurred later, as it approached the inner core, the downwind sector in the boundary layer of the outer rainband was rapidly axisymmetrized near the outer edge of the RFZ. Continuous inward propagation of the outer rainband and axisymmetrization of its downwind sector led to the spinup of tangential wind in the boundary layer, which enhanced surface enthalpy flux and convection and, eventually, the SEF. Figure 14 shows a schematic diagram demonstrating the evolution and contribution of the outer spiral rainband to the SEF in our simulation. We noticed that although the overall process of the simulated SEF was dominated by the outer spiral rainband, the merging of the inward-propagating outer rainband with the outward-propagating inner rainbands also played some important roles.
Schematic diagram showing the evolution of an outer spiral rainband that led to the SEF in the simulation. The inner dashed circle indicated the inner edge of the newly formed eyewall and the outer edge of the moat area after the formation of the secondary eyewall. Inner rainbands are shown in gray in the RFZ, while the gray near the outer edge of the RFZ shows the outer spiral rainband with embedded convective cells, in particular, in the upwind sections. Both updrafts and downdrafts associated with the outer rainband are shown in red and purple arrows. The dashed red arrow indicates the cyclonically spiraling inward and downward motions of the outer rainband, leading to the SEF in the simulation.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0130.1
The outer rainband that led to the SEF was initiated from the upper-troposphere upwind, developed both downward and inward, and spiraled cyclonically inward with a coherent tilted along-band structure. This strongly suggests that the SEF in our simulation is a top-down process, which is similar to that discussed in some previous studies (Wang 2006; Fang and Zhang 2012; Sun et al. 2013; Zhang et al. 2017; Dai et al. 2017; Tyner et al. 2018; Didlake et al. 2018) but is different not only in details but also in fundamental processes. We have revealed that the top-down process was closely related to the sloped along-band structure and evolution of an outer spiral rainband (Fig. 14). This provides a three-dimensional picture of the SEF process. We showed that active convective cells often formed in the upwind sector of the outer rainband with deep inflow under and immediately outside the rainband, which led to the spinup and outward expansion of tangential wind mainly in the mid- to upper troposphere. In the middle sector of the outer rainband, convection is less penetrative with midtropospheric heating maximum, which induced inflow in the mid- to lower troposphere and thus the spinup of tangential wind in the inflow region as well as in the region with updrafts due to vertical advection. In the downwind sector of the rainband, convection was generally suppressed in its vertical penetration, and thus, inflow and tangential wind spinup often occurred in the lower troposphere and in the boundary layer. As the outer rainband spiraled cyclonically inward and the inner downwind sector of the rainband approached the RFZ, asymmetric eddies in the rainband were effectively axisymmetrized to spin up the azimuthal-mean tangential wind primarily in the boundary layer. This was also demonstrated by results from the azimuthal-mean tangential wind budget. We showed that although the azimuthal-mean advection predominantly contributed to the spinup of the azimuthal-mean tangential wind in the SEF region, eddy processes also played an important role in the spinup of tangential wind in the boundary layer, in particular in the rainband region. Note that part of the azimuthal-mean advection associated with the secondary circulation also resulted from the azimuthally averaged diabatic heating in spiral rainbands.
Note that although this study has further emphasized the importance of the asymmetric processes in the simulated SEF, the asymmetric processes are closely tied with the axisymmetric dynamics as well. We showed that the azimuthal-mean diabatic heating projected from diabatic heating in outer rainbands contributed to the outward expansion of tangential wind by driving inflow below the level of maximum heating and thus enhanced the inertial stability and relative vorticity outside the eyewall (Rozoff et al. 2012). In addition to diabatic heating, perturbation tangential and radial winds in the outer spiral rainbands also contributed to the spinup of the azimuthal-mean tangential wind by eddy angular momentum flux, in particular in the downwind sector in the boundary layer where they were axisymmetrized by the rapid filamentation process. In this sense, both axisymmetric and asymmetric processes are not independent but cooperatively led to the SEF in the simulation. We also noticed that the axisymmetrization in the SEF region does not mean significant weakening of high-wavenumber EKE because eddies were continuously generated in the outer rainband and in the newly formed secondary eyewall.
Our results strongly support previous findings that most SEFs in TCs are related to the activity of strong spiral rainbands. Therefore, any internal and external process that can trigger strong spiral rainbands could be considered as a forcing mechanism of SEF, such as the beta effect (Wang 2006; Fang and Zhang 2012), environmental vertical wind shear (Zhu et al. 2004; Riemer 2016; Zhang et al. 2017), outflow–jet interaction (Dai et al. 2017), and stratiform precipitation originating from TC outflow layer (Tyner et al. 2018). However, it is still an issue why some spiral rainbands propagate radially outward while some others propagate radially inward. This indeed is the key for the SEF, but we still have no answer to this issue at this moment and will address this issue in a future study by conducting sensitivity experiments. Another issue is what determines the radial location of the SEF. Both enhanced inertial stability outside the eyewall and the width of the RFZ are candidates and are often considered to be important in previous studies. However, an increase in inertial stability often results in the increase of the filamentation time, or vice versa. This seems to suggest that a secondary eyewall would form near the outer edge of the RFZ if the region where the inertial stability is enhanced with active convection, such as the downwind sector of the outer spiral rainband as examined in this study. To understand these details will be a topic of our future study as well.
Acknowledgments
This study has been supported in part by the National Key R&D Program of China under Grants 2017YFC1501602 and 2015CB452805; National Natural Science Foundation of China under Grants 41730960, 41675044, and 41875080; and the Basic Research Fund of CAMS 2016Z003.
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Here, we refer to negative (positive) asymmetric radial wind as the asymmetric inflow (outflow), which is relative to the azimuthal-mean radial wind since we are focusing on the eddy motion in the rainband region relative to the azimuthal mean at the same radius.
