Effects of Diabatic Heating and Cooling in the Rapid Filamentation Zone on Structure and Intensity of a Simulated Tropical Cyclone

Qingqing Li State Key Laboratory of Severe Weather, Chinese Academy of Meteorological Sciences, Beijing, and Shanghai Typhoon Institute and Laboratory of Typhoon Forecast Technique, China Meteorological Administration, Shanghai, China

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

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

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Abstract

The effects of diabatic heating and cooling in the rapid filamentation zone (RFZ), within which inner rainbands are often active, on tropical cyclone (TC) structure and intensity are investigated based on idealized numerical experiments using a cloud-resolving TC model (TCM4). The results show that removal of heating (cooling) in the RFZ would reduce (increase) the TC intensity. Diabatic heating in the RFZ plays an important role in increasing the inner-core size whereas diabatic cooling tends to limit the inner-core size increase or even reduce the inner-core size of a TC. Removal of both diabatic heating and cooling in the RFZ greatly suppresses the activity of inner rainbands but leads to the quasi-periodic development of a convective ring immediately outside of the inner core. A similar convective ring also develops in an experiment with the removal of diabatic heating only in the RFZ. With diabatic cooling removed only in the RFZ, an annular-hurricane-like structure arises with the outer rainbands largely suppressed.

Corresponding author address: Dr. Qingqing Li, Shanghai Typhoon Institute, 166 Puxi Road, Shanghai 200030, China. E-mail: liqq@mail.typhoon.gov.cn

Abstract

The effects of diabatic heating and cooling in the rapid filamentation zone (RFZ), within which inner rainbands are often active, on tropical cyclone (TC) structure and intensity are investigated based on idealized numerical experiments using a cloud-resolving TC model (TCM4). The results show that removal of heating (cooling) in the RFZ would reduce (increase) the TC intensity. Diabatic heating in the RFZ plays an important role in increasing the inner-core size whereas diabatic cooling tends to limit the inner-core size increase or even reduce the inner-core size of a TC. Removal of both diabatic heating and cooling in the RFZ greatly suppresses the activity of inner rainbands but leads to the quasi-periodic development of a convective ring immediately outside of the inner core. A similar convective ring also develops in an experiment with the removal of diabatic heating only in the RFZ. With diabatic cooling removed only in the RFZ, an annular-hurricane-like structure arises with the outer rainbands largely suppressed.

Corresponding author address: Dr. Qingqing Li, Shanghai Typhoon Institute, 166 Puxi Road, Shanghai 200030, China. E-mail: liqq@mail.typhoon.gov.cn

1. Introduction

It has been long known that one of the necessary conditions for tropical cyclone (TC) development and maintenance is the diabatic heating due to condensation in moist convection (Malkus and Riehl 1960; Riehl and Malkus 1961; Yanai 1961; Möller and Shapiro 2002; Bui et al. 2009; and others). Many numerical studies have documented the effects of diabatic heating and cooling associated with the overall TC circulation on TC structure and intensity and their changes. Wang (2002c) conducted a series of numerical sensitivity experiments of idealized TCs and found that downdrafts are significantly reduced owing to removal of cooling as evaporation of rain or melting of snow and graupel are excluded in the simulation, leading to a greatly increased TC intensification rate and ultimate intensity. Similar experiments were performed by Zhu and Zhang (2006) to investigate the influences of melting and evaporative cooling on determining the intensity of Hurricane Bonnie (1998). They also indicated the important roles of melting and evaporative cooling in slowing the amplification and determining the final intensity of the simulated hurricane. The results of Frisius and Hasselbeck (2009) are akin to the above; that is, latent cooling causes a delay of development and a decrease in the final intensity of a simulated storm. Furthermore, they also found that the mature TC undergoes intensity fluctuations when evaporation is switched off, seemingly resulting from nonaxisymmetric processes. More recently, Sawada and Iwasaki (2010a,b) investigated the impacts of evaporative cooling from raindrops on the structure of a simulated TC by ignoring evaporative cooling but allowing conversion from rainwater into water vapor in a high-resolution numerical model. They showed that evaporative cooling slows down the early development of a TC because of the stabilization in the boundary layer around the eyewall, while at the mature stage it steadily intensifies the TC for a longer period and widens the TC size as a result of the presence of outer rainbands.

It is well known that diabatic heating and cooling in different parts of a TC may play different roles in TC structure and intensity and their changes. From an axisymmetric perspective, Vigh and Schubert (2009) analytically confirmed that azimuthally averaged diabatic heating in the low-inertial-stability region outside the radius of maximum wind (RMW) is inefficient to generate a warm core, while the heating in the high-inertial-stability region inside the RMW is highly efficient to produce a localized warming tendency. This result suggests that rapid intensification favorably occurs as at least some of the eyewall convection (diabatic heating) exists inside the RMW, which is consistent with the results from a composite study of airborne Doppler observations in Rogers et al. (2013) and the finding from idealized simulations in Nguyen et al. (2011). Wang (2008b) explored the formation and maintenance of an annular hurricane and pointed out that the tilt and radial location of the eyewall heating can change the strength and location of the strongest mean tangential wind. Pendergrass and Willoughby (2009) diagnosed the balanced response of the secondary circulation to diabatic heating at different radial locations. They found that diabatic heating in the eyewall could enhance the primary circulation primarily in the inner core and thus the TC intensity, whereas heating outside the eyewall could contribute to the change in the outer-core primary circulation and thus the size of a TC. Fudeyasu and Wang (2011) further examined the balanced contribution to the spinup of the outer core based on a full-physics model simulation and found that the midtropospheric inflow in response to diabatic heating in mid- to upper-tropospheric stratiform clouds outside the eyewall transports absolute angular momentum inward to accelerate the outer-core circulation. Wang (2009) showed that cooling in outer spiral rainbands maintains both the intensity of a TC and the compactness of its inner core, whereas heating in outer rainbands limits the intensity but widens the size of a TC.

Inner spiral rainbands are generally explained as convectively coupled sheared vortex Rossby wave (VRW) activity (Guinn and Schubert 1993; Montgomery and Kallenbach 1997; Chen and Yau 2001; Wang 2002a,b; Li and Wang 2012b; Otto and Soderholm 2012). The aforementioned work has not paid much attention to the effects of diabatic heating and cooling in inner spiral rainbands on TC structure and intensity changes. In a related study, Moon and Nolan (2010) examined the response of winds to specified asymmetric rotating spiral heat source using a three-dimensional, nonhydrostatic, linear model. They found the resultant overturning secondary circulation, descending midlevel radial inflow, and accelerated cyclonic tangential wind radially outside of inner rainbands. The overturning secondary circulation results mostly from the convective part of the rainband and is stronger in the upwind region, while the midlevel radial inflow descending to the surface is due to the stratiform characteristics of the rainband and is stronger in the downwind region. The results in Moon and Nolan (2010) indicate that the primary effects of inner rainbands on the wind field are caused by the direct response to asymmetric diabatic heating of embedded convection and that the structure of the heating is primarily responsible for the unique kinematic structures.

A unique feature of inner rainbands is their prominent occurrence in the rapid filamentation zone (RFZ) where the filamentation time is less than the typical overturning time scale of individual convective clouds (~30–45 min; Rozoff et al. 2006; Wang 2008a; Li and Wang 2012a,b). In particular, the straining deformation in the RFZ plays an important role in the formation of curved spiral rainbands. Adverse shear in the RFZ is contributive critically to damping high azimuthal wavenumber asymmetries in the inner-core region by filamentation and axisymmetrization (Smith and Montgomery 1995; Montgomery and Kallenbach 1997; Wang 2008a). As a result, TC inner rainbands are generally characterized by low azimuthal wavenumber (usually wavenumber ≤ 4) structure. The objective of this study is to understand how overall TC structure and intensity may respond to diabatic heating and cooling in the RFZ within which inner spiral rainbands are often active by artificially modifying the heating and/or cooling rate in idealized numerical experiments with a nonhydrostatic, cloud-resolving TC model. We first introduce the model used and the design of experiments in section 2. Section 3 discusses the results from sensitivity numerical experiments. Major conclusions are drawn in the last section.

2. Model and numerical experiment design

The model employed here is the fully compressible, nonhydrostatic atmospheric model TCM4 as used in Li and Wang (2012a,b). A description of TCM4 was detailed in Wang (2007). The model has been used for studying many aspects of TCs, including the inner-core asymmetric structure (Wang 2007), RFZ (Wang 2008a), annular hurricane structure (Wang 2008b), size change (Wang and Xu 2010; Xu and Wang 2010a,b), and inner and outer spiral rainbands (Li and Wang 2012a,b).

The model equations are formulated in Cartesian coordinates in the horizontal and in mass coordinates in the vertical. The model assumes a flat lower boundary at the ocean surface with a uniform unperturbed surface pressure of 1010 hPa. The model top is set at about 40 km MSL. A sponge upper boundary condition similar to that used in Durran and Klemp (1983) is applied to absorb the upward-propagating sound and gravity waves. The physical parameterizations in the model include an Eε turbulence closure scheme for subgrid-scale vertical turbulent mixing (Langland and Liou 1996), a modified Monin–Obukhov scheme for surface flux calculation (Fairall et al. 2003), an explicit treatment of mixed-phase cloud microphysics, a nonlinear fourth-order horizontal diffusion for all prognostic variables except for that related to the mass conservation equation, a simple Newtonian cooling term added to the perturbation potential temperature equation to mimic the longwave radiative cooling (Rotunno and Emanuel 1987), and the dissipative heating related to the turbulent kinetic energy dissipation rate ε from the Eε turbulence closure scheme.

The model domain is quadruply nested with two-way interactive nesting and with the all inner meshes automatically moving to follow the model storm. The model has 26 levels in the vertical and has mesh sizes of 201 × 181, 109 × 109, 127 × 127, and 163 × 163 grid points with their horizontal grid spacings of 67.5, 22.5, 7.5, and 2.5 km for the four meshes, respectively. No large-scale environmental flow is considered in this study, and no cumulus parameterization is employed even in the two outermost meshes since convection occurs mainly within 200 km from the storm center and is covered by the innermost mesh. The model is initialized with an axisymmetric cyclonic vortex on an f plane at 18°N over the ocean with a uniform sea surface temperature of 29°C. The initial thermodynamic profile of the unperturbed model atmosphere is defined as the western Pacific clear-sky environment given by Gray et al. (1975). Given the tangential wind field for the initial cyclonic vortex, which has a maximum wind speed of 25 m s−1 at a radius of 80 km at the surface and decreases sinusoidally with height, the corresponding mass and thermodynamic fields are obtained by solving the nonlinear balance equation (Wang 2001).

The experimental design follows Wang (2009). In a control experiment (CTL), the default model settings were used and the model was integrated for 96 h. Three sensitivity experiments were conducted with an artificially modified diabatic heating rate Q from moist physics (cloud microphysics) in the model, similar to that in Wang (2009). The total Q calculated from the model cloud microphysics at a given time step at a grid point in TCM4 can be decomposed into the heating rate and the cooling rate (Wang 2009):
eq1
In the first sensitivity experiment (HC00), the total diabatic heating rate beneath 10-km altitude is set to be zero after 42 h spinning up as in the CTL experiment between radii of 40 and 60 km from the TC center with a transition zone between 60- and 70-km radii given below:
eq2
where the coefficient b is a function of the radius r from the TC center (Fig. 1). Note that by 42 h of spinning up, the model storm had an RMW at about 20 km with no active outer spiral rainbands (not shown). The inner rainbands predominantly occurred immediately outside of the eyewall and inside a radius of 60 km from the storm center where the azimuthal-mean filamentation time is less than 45 min in the troposphere (Fig. 1)—namely in the RFZ. The artificial removal of Q between radii of 40 and 60 km in HC00 means the ignorance of diabatic heating and cooling associated with inner rainbands in the RFZ.
Fig. 1.
Fig. 1.

The radius–height cross section of the azimuthal-mean filamentation time (shading; min) after 42 h of simulation in CTL, and the value of coefficient b (solid line) as a function of the radius.

Citation: Journal of the Atmospheric Sciences 71, 9; 10.1175/JAS-D-13-0312.1

In the second sensitivity experiment (H00), only is reset to be zero between radii of 40 and 60 km from the TC center after 42 h of spinning up:
eq3
Similarly, in the third sensitivity experiment (C00), is reset to be zero between radii of 40 and 60 km with the heating effect maintained:
eq4

Although filamentation is used to kinematically represent the deformation effects of fluid rather than the thermodynamic forcing discussed in the present study, inner rainbands are active and predominant in the RFZ as noted in Wang (2008a) and Li and Wang (2012b). As shown in Fig. 1, modification to diabatic heating rate by b(r) appears mainly in the RFZ where diabatic heating and/or cooling is primarily associated with the activity of inner rainbands. Therefore, the use of the RFZ is more physically based in reflecting the region with active inner spiral rainbands. Note that in all three sensitivity experiments, only was the heating and/or cooling rate below 10-km altitude artificially modified so that diabatic heating in the upper troposphere associated with the eyewall anvil clouds was not affected by the experimental design. This is different from what was done in Wang (2009) for diabatic heating in outer rainbands region where the heating rate was modified throughout the troposphere. Diabatic heating and cooling from the cloud microphysics scheme in the model result from phase changes. The heating rate is due to condensation and fusion of water vapor and freezing of cloud and rain droplets. The cooling is due to evaporation of cloud droplets or rain drops, melting of snow or graupel, or sublimation of snow and cloud ice. Both heating and cooling are associated with the activity of inner spiral rainbands and are determined by moisture conditions in the interested region, in particular the relative humidity.

3. Results

a. Intensity

Figures 2a and 2b depict the evolution of the minimum sea level pressure (MSLP) and maximum wind speed at the lowest model level, respectively, of the simulated storms in the four experiments. After a 24-h spinup, the TC intensified steadily through 42 h. In the CTL experiment, the storm continued intensifying through about 60 h of simulation (Figs. 2a and 2b) with an averaged deepening rate of 1.0 hPa h−1. Thereafter, the storm became mature with quasi-periodic intensity oscillations resulting from the quasi-periodic behavior of outer rainbands in the simulated storm as revealed in Li and Wang (2012a).

Fig. 2.
Fig. 2.

Time series of simulated (a) minimum sea level pressure (hPa) and (b) maximum wind speed (m s−1) at the lowest model level.

Citation: Journal of the Atmospheric Sciences 71, 9; 10.1175/JAS-D-13-0312.1

In HC00, the storm slightly intensified for about 6 h after both the heating and cooling between 40- and 60-km radii were removed. This is followed by a slow weakening with the MSLP rising from 937 hPa at 54 h to 943 hPa at 72 h of simulation. The storm slightly intensified again during 72–90 h with the MSLP of 933 hPa and the maximum wind speed at the lowest model level of 58.3 m s−1 by 96 h (Figs. 2a and 2b). Note that the storm in HC00 was slightly weaker than that in CTL, suggesting that diabatic heating in the RFZ or the activity of inner spiral rainbands contribute to the storm intensity in the CTL experiment.

With the heating switched off between 40- and 60-km radii only in H00, the storm intensified very slightly for the first 2 h (Fig. 2) while decayed steadily through 75 h of simulation with the MSLP increased by about 22 hPa (Fig. 2a). A relatively weak intensity oscillation of the TC appeared thereafter as well (Figs. 2a,b). As a result, the storm in H00 is the weakest through the 96-h simulation among the four experiments. This indicates that heating outside the eyewall in the RFZ is also important for TC intensity. This is in sharp contrast to the role of heating in the outer spiral rainband region shown in Wang (2009), where heating is generally negative to TC intensity.

With the cooling ignored between 40- and 60-km radii only in C00, the storm intensified rapidly up to about 60 h of simulation with the MSLP decreased by 40 hPa (Fig. 2a). The storm reached its steady state after 80 h of simulation with the maximum wind speed of about 74 m s−1 at 96 h (Fig. 2b)—namely, the most intense storm among the four experiments. The results thus suggest that cooling outside the eyewall in the RFZ may substantially limit the TC intensity. Cooling would result in significant downdrafts, which often bring cool and dry air into the inflow boundary layer. The low–equivalent potential temperature (θe) air can be easily transported and mixed into the eyewall, weakening the eyewall convection and thus reducing the storm intensity.

The above argument for the different intensity changes in the different experiments can be demonstrated by examining the time evolution of low-level θe fields (Fig. 3). As we can see, θe is always higher within the RMW and relatively low outside the eyewall in all experiments. The alternative high–low–high θe outside the eyewall in CTL (Fig. 3a) reflects quasi-periodic behavior of outer spiral rainbands as studied in Li and Wang (2012a). Removal of both heating and cooling in the RFZ in HC00, the θe value outside the RMW is considerably reduced (Fig. 3b), consistent with the slightly weaker storm than in CTL. In addition, the air with relatively low θe, resulting most likely from the convective ring discussed below, is advected radially inward to the outer edge of the eyewall between 54 and 72 h of simulation in HC00 (Fig. 3b), responsible for the slight weakening of the storm during that period (Fig. 2). With the removal of heating only in H00, low-level θe becomes much lower outside the RMW with the minimum value less than 339 K (Fig. 3c). The low-θe air is advected radially inward, diluting the local θe in the eyewall region, suppressing the eyewall convection, and thus reducing the storm intensity (Fig. 2). With the removal of cooling only in the RFZ in C00, the boundary layer θe became much higher in both eyewall and outside the eyewall region (Fig. 3d). This is consistent with the most intense storm in this experiment. This result is in sharp contrast to the effect of heating and/or cooling associated with the outer spiral rainbands outside the RFZ (Wang 2009), where enhanced heating (cooling) can reduce (increase) the storm intensity. Regardless of the effect of intrusion of low-enthalpy air, Wang (2009) documented that heating in outer rainbands decreases the surface pressure on the inward side of the rainbands, lessening the horizontal pressure gradient across the radius of maximum wind and thereby reducing the TC intensity. However, the key to understand how diabatic heating and/or cooling in the RFZ affects the storm intensity depends on how θe in the boundary layer is changed in the eyewall region.

Fig. 3.
Fig. 3.

Radius–time Hovmöller diagram of θe (K) at z = 0.3 km in (a) CTL, (b) HC00, (c) H00, and (d) C00.

Citation: Journal of the Atmospheric Sciences 71, 9; 10.1175/JAS-D-13-0312.1

b. Structural evolution

Diabatic heating in the RFZ is directly related to the activity of inner spiral rainbands (Wang 2008a). As indicated by Wang (2008a), inner rainbands are often sheared, elongated, and well organized and can be viewed as quasi-balanced convectively coupled VRWs with low-azimuthal-wavenumber structure in potential vorticity (PV) field (Smith and Montgomery 1995; Wang 2001, 2002a,b; Li and Wang 2012b). Therefore, we first examine the low-wavenumber PV structure in the simulated storms. The relatively small-amplitude low-wavenumber asymmetric PV outside the RMW in CTL (Fig. 4a) reflects the existence of active inner rainbands, as can be seen from the azimuthal-mean rain rate in Fig. 5a.

Fig. 4.
Fig. 4.

Radius–time Hovmöller diagram of the amplitude of the azimuthal wavenumber ≤ 3 asymmetric PV at z = 3 km in (a) CTL, (b) HC00, (c) H00, and (d) C00.

Citation: Journal of the Atmospheric Sciences 71, 9; 10.1175/JAS-D-13-0312.1

Fig. 5.
Fig. 5.

Radius–time Hovmöller diagram of the azimuthal-mean surface rain rate (mm h−1) in (a) CTL, (b) HC00, (c) H00, and (d) C00.

Citation: Journal of the Atmospheric Sciences 71, 9; 10.1175/JAS-D-13-0312.1

With the removal of both heating and cooling in the RFZ in HC00, the amplitude of wavenumber ≤ 3 asymmetric PV at 3-km height is highly depressed immediately outside of the bimodal asymmetric PV amplitude structure in the eyewall region (Fig. 4b). This asserts the significant suppressed activity of inner rainbands between 40 and 60 km in this experiment (Fig. 5b). With the removal of heating only between 40 and 60 km in H00, the characteristics of low-wavenumber asymmetric PV are quite different from those in HC00. The initial bimodal asymmetric PV structure in the eyewall became a unimodal structure after 52 h of simulation in H00, and the decrease in the amplitude of wavenumber ≤ 3 asymmetric PV between 40 and 60 km (Fig. 4c) is not as dramatic as in HC00 (Fig. 4b). Nevertheless, the activity of inner rainbands was largely suppressed between 40 and 60 km in H00 (Fig. 5c), indicative of other asymmetric PV sources in the simulation. With the removal of cooling only between 40 and 60 km, a very clear bimodal asymmetric PV structure is present in the eyewall region. In this case the amplitude of low-wavenumber asymmetric PV decreased rapidly outside of the 60-km radius (Fig. 4d). Note that in this case, even though cooling was switched off between 40 and 60 km, the rain structure both in the inner core and outside the 60-km radius is largely different from that in CTL (Fig. 5d). Namely, precipitation becomes very high within a radius of 60 km while it almost disappears outside of about 90-km radius (Fig. 5d), indicating the suppression of outer spiral rainbands. In CTL, very heavy precipitation occurs near the radius of 20 km in the eyewall region and quasi-periodically outward-propagating precipitation outside a radius of 60 km is related to the activity of outer rainbands (Fig. 5a). There are two active episodes of outer spiral rainbands as shown in Fig. 5a, with a period of approximately 22 h between the active episodes. Li and Wang (2012a) pointed out that the quasi-periodic nature of outer spiral rainbands is associated with the boundary layer recovery from the effect of convective downdrafts and the consumption of convective available potential energy (CAPE) by convection in previous outer spiral rainbands (Fig. 6a).

Fig. 6.
Fig. 6.

Radius–time Hovmöller diagram of the CAPE (J kg−1) in (a) CTL, (b) HC00, (c) H00, and (d) C00. Contours indicate the smoothed azimuthal-mean surface rain rate less than 10 mm h−1 with an interval of 2 mm h−1.

Citation: Journal of the Atmospheric Sciences 71, 9; 10.1175/JAS-D-13-0312.1

Many previous studies demonstrated that rapid filamentation tends to damp high-wavenumber kinematic asymmetries (Smith and Montgomery 1995; Montgomery and Kallenbach 1997; Wang 2008a), favoring the formation of low-azimuthal-wavenumber inner spiral rainbands as shown in CTL. Of interest is how diabatic heating and cooling changes in the RFZ in the sensitivity experiments. We found that the amplitude of low-wavenumber (wavenumber ≤ 4) diabatic heating in the RFZ is comparable to that of the high-wavenumber (e.g., wavenumber 5–10) components in CTL and H00, whereas the amplitude of low-wavenumber diabatic heating is much larger than that of the high-wavenumber components in C00 (not shown). Different from the kinematic variables, changes in the amplitudes of low- and high-wavenumber diabatic heating are generally in phase in CTL, H00, and C00 (not shown). The results suggest that although dynamical fields are dominated by low-azimuthal-wavenumber structure, heating and/or cooling associated with convective activity still occurs at small scales owing to the nature of moist convection even though they are largely suppressed by the rapid filamentation.

The symmetric rain rate in HC00 displays three adjacent regions with distinct precipitating patterns (Fig. 5b). Elevated rainfall occurs in the eyewall region inside a radius of 40 km and a precipitation-suppressed band presents between the radii of 40 and 70 km. Quasi-periodic rainfall signals are found outside the 70-km radius, similar to those in CTL. The azimuthal-mean rain rate between 70- and 90-km radii is much stronger than in CTL. Examination of the horizontal distribution of radar reflectivity indicates that different from the activity of outer spiral rainbands in CTL (Li and Wang 2012a), the quasi-periodic behavior in the outward-propagating azimuthal-mean rain rate in Fig. 5b is mainly related to the periodically outward-propagating convective rings in HC00 (Fig. 7). For example, at 81 h of simulation, a narrow convective ring develops between 70- and 90-km radii, accompanying the outward propagation of a preexisting convective ring farther outside. Later, the previous convective ring gradually collapses as a result of the reduced CAPE (Fig. 6b), convection in the newly generated convective ring becomes more vigorous, and another convective-ring cycle seems to begin.

Fig. 7.
Fig. 7.

Plan view of the simulated reflectivity (dBZ) at z = 3 km from 66 to 89 h 45 min of simulation in HC00. Inner (outer) circles show the location of the 60 (120)-km radius from the storm center.

Citation: Journal of the Atmospheric Sciences 71, 9; 10.1175/JAS-D-13-0312.1

Consistent with the continuous weakening of the storm in H00, the azimuthal-mean surface rain rate in the eyewall tends to decrease with time (Fig. 5c) and is much smaller than that in both CTL and HC00. The minimum in precipitation in the annular region between 40- to 60-km radii (Fig. 5c) is due to the prevailing subsidence induced by evaporative cooling and melting of snow and graupel. Similar to that in HC00, precipitation outside the 70-km radius also shows quasi-periodic behavior with enhanced rain rates between 70 and 90 km (Fig. 5c). As we can see from Fig. 8 for a complete cycle, similar to that in HC00, a convective ring developed between 70 and 90 km in the beginning of the cycle (e.g., at 62.5 h in Fig. 8) with embedded cellular convection. A distinct feature is the existence of a moat between 40- and 60-km radii, which can also be seen from the azimuthal-mean rainfall shown in Fig. 5c. The moat region is dominated by evaporative cooling and melting of snow and graupel as we can see from the azimuthal-mean diabatic heating and vertical motion in Fig. 9c. Subsequently, the convective ring becomes radially broader and less organized as it propagates radially outward (e.g., at 70 h in Fig. 8). The physical processes responsible for the outward propagation and quasi-periodic appearance follow those discussed in Li and Wang (2012a)—namely, the consumption of CAPE by convection (Fig. 6c) and recovery of the boundary layer due to the surface entropy flux.

Fig. 8.
Fig. 8.

As in Fig. 7, but from 60 to 83 h 45 min of simulation in H00.

Citation: Journal of the Atmospheric Sciences 71, 9; 10.1175/JAS-D-13-0312.1

Fig. 9.
Fig. 9.

Radius–height cross sections of the azimuthal-mean AAM (green contours; 105 m2 s−1), diabatic heating rate (shading; 10−3 K s−1), and secondary circulation (arrows; m s−1) in (a) CTL averaged between 70 and 71 h, (b) HC00 averaged between 70 and 71 h, (c) H00 averaged between 64 and 65 h, and (d) C00 averaged between 70 and 71 h.

Citation: Journal of the Atmospheric Sciences 71, 9; 10.1175/JAS-D-13-0312.1

With cooling removed in C00, the azimuthal-mean surface rainfall becomes much broader and stronger in the eyewall region and also within a radius of 80 km owing to active inner spiral rainbands, but become much weaker outside of the 90-km radius owing to the lack of any active outer spiral rainbands (Fig. 5d). Note that although the local CAPE is high (Fig. 6d), the negative effect of strong downdrafts beneath 9-km height outside the inner core (Fig. 9d) plays an important role in suppressing the development of outer rainbands. Descending motion above 5-km altitude is likely forced by sublimation cooling and that below 5-km altitude results mainly from cooling due to melting and evaporation. Nevertheless, the spiral structure in vertical motion outside the eyewall (Fig. 10) still reflects active inner spiral rainbands and the associated convectively coupled VRWs (Wang 2002a,b). The distinctive feature in C00 from HC00 and H00 is its lack of convective-ring structure outside the inner-core region and the inactive outer spiral rainbands, resulting in a storm very similar to the annular hurricane (Knaff et al. 2003; Wang 2008b). We found that the main physical processes for the formation of the annular hurricane structure follows those discussed in Wang (2008b) as well and are not given here.

Fig. 10.
Fig. 10.

Plan view of the simulated vertical velocity (m s−1) at z = 5 km from 63 to 82 h of simulation in C00. Inner (outer) circles show the location of the 60 (120)-km radius from the storm center.

Citation: Journal of the Atmospheric Sciences 71, 9; 10.1175/JAS-D-13-0312.1

To further investigate the dynamical response to the diabatic heating and/or cooling change, we examine the structure of absolute angular momentum (AAM; , where r is radius, f is the Coriolis parameter, and υ is the tangential wind) and the AAM budget. The azimuthal-mean AAM budget equation in cylindrical coordinates centered at the storm center can be written as
e1
where u and w are radial and vertical components of wind, z is the height, and t, F, and D are time, surface friction of tangential winds, and horizontal diffusion of tangential winds, respectively. The overbars and primes denote azimuthal-mean and eddy quantities, respectively. Note that is the azimuthal-mean vertical component of absolute vorticity, and the eddy pressure gradient term is omitted in (1) owing to its very negligible value compared with other terms. The contributions to the local tendency of the azimuthal-mean AAM on the right-hand side of (1) are the mean radial flux of absolute vorticity, the mean vertical flux of M, the corresponding eddy fluxes, and friction and diffusion terms.

As we can see in Fig. 9, the M surfaces slope largely inward with height near the surface, suggesting that the tangential wind increases with height and that M is being lost to the underlying ocean in all three experiments (Figs. 11g, 12g, and 13g). The radial wind between 40 and 60 km is much weaker in HC00 (Fig. 9b), so that the contributions by the mean-flow advection to the AAM budget are relatively smaller therein (Figs. 11a–c). A significant fraction of the azimuthal-mean AAM increase in the boundary layer results from the inward flux of by the azimuthal-mean inflow (Fig. 11a), which is largely offset by the negative contributions by the azimuthal-mean vertical AAM advection (Fig. 11b) and surface friction and vertical mixing (Fig. 11g). Vertical advection of AAM enhances AAM throughout the troposphere in the eyewall and convective ring (Fig. 11b) even where diabatic heating is clustered (Fig. 9b). The total eddy flux associated with convection in the convective ring in HC00 only shows slightly positive AAM tendency between 4- and 8-km altitudes (Fig. 11f). The convective ring looks like a classic secondary eyewall (Willoughby et al. 1982), because a secondary tangential wind maximum associated with the convective ring in HC00 is found in the lower troposphere on the late stage of simulation (not shown). It is suggested that the positive contribution by the azimuthal-mean secondary circulation in the convective ring and the negative contribution in the RFZ (Fig. 11c) to the AAM tendency are responsible for the presence of the secondary tangential wind maximum.

Fig. 11.
Fig. 11.

(a) The azimuthal-mean radial vorticity flux, (b) azimuthal-mean vertical AAM advection, (c) mean secondary circulation, (d) radial eddy vorticity flux, (e) vertical eddy AAM advection, (f) total eddy flux, and (g) frictional and horizontal diffusion contributions to (h) the AAM budget averaged from 70 to 71 h in HC00. Tendency in AAM is expressed in units of meters squared per second squared.

Citation: Journal of the Atmospheric Sciences 71, 9; 10.1175/JAS-D-13-0312.1

Fig. 12.
Fig. 12.

As in Fig. 11, but averaged from 64 to 65 h in H00.

Citation: Journal of the Atmospheric Sciences 71, 9; 10.1175/JAS-D-13-0312.1

Fig. 13.
Fig. 13.

Radius–time Hovmöller diagram of the azimuthal-mean divergence (shading; 10−4 s−1) at z = 500 m in (a) HC00 and (b) H00, overlaid by the azimuthal-mean radial wind contoured at −6, −5, −4, −3, −2, −1, and 0 m s−1.

Citation: Journal of the Atmospheric Sciences 71, 9; 10.1175/JAS-D-13-0312.1

Figure 9c illustrates apparent cooling-induced descending outflow above the inflow boundary layer and below 4-km height in the RFZ in H00. The outflow leads to a negative contribution of the azimuthal-mean radial AAM flux (Fig. 12a), resulting in the inward tilting of the AAM surfaces with height in the RFZ (Fig. 9c). Additionally, downdrafts advect high AAM downward into the boundary layer in the RFZ (Fig. 12b). Accompanying with the downdrafts is the weak inflow in the middle troposphere as a result of mass conservation (Fig. 9c). The inflow together with the relatively strong midtroposphere inflow related to diabatic heating in the convective ring brings high AAM inward (Figs. 12a,c,h). As the convection in the eyewall fades in H00 resulting from the intrusion of low-θe air in the boundary layer, the negative frictional effect overrides the positive contribution of the boundary layer inflow to the AAM budget. As a result, the net AAM tendency becomes predominantly negative in the boundary layer in H00 (Fig. 12h), causing the weakening of the storm and the shrink of the inner-core size, which will be discussed later.

It is interesting to examine the convective-ring structure appeared in both HC00 and H00. We found that the occurrence of the convective rings results from the boundary layer convergence (Fig. 13). With the removal of heating and cooling between 40 and 60 km in HC00, the boundary layer inflow shows an obvious minimum near the radius of 60 km, leading to a divergence region between 40 and 60 km and enhanced convergence immediately outside of the 60-km radius (Fig. 13a). As a result, the regional convergence in the boundary layer together with the high CAPE triggers robust convective cells between 70 and 90 km, forming a convective ring (Fig. 7). Similar divergence and convergence occurred in H00, where noticeable divergence between 40 and 60 km and convergence outside of 60 km result from the reduction in boundary layer inflow near the 60-km radius (Fig. 13b). Note that the magnitude of divergence between 40 and 60 km in H00 is much greater than that in HC00 (Fig. 13) because of the prevailing downdrafts in the lower troposphere and boundary layer in the former (Fig. 9c).

Previous studies have demonstrated that diabatic heating in outer spiral rainbands can cause an increase in the inner-core size of a TC (Wang 2009; Hill and Lackmann 2009). It is worth examining how diabatic heating and cooling associated with inner rainbands in the RFZ affect the inner-core size of the simulated TCs in our various experiments. Following Xu and Wang (2010a,b), the inner-core size is simply referred to as the radius of azimuthal-mean near-surface damaging-force wind (25.7 m s−1) outside the eyewall. Figure 14 presents the temporal evolution of the inner-core size of the storms simulated in the four experiments. In CTL, the storm inner-core size increased by about 10 km from 42 to 96 h. With the cooling in the RFZ removed in C00, the inner-core size of the storm persistently increased with time from about 65 km at the beginning of the removal of cooling to about 100 km at 96 h. On the contrary, the storm inner-core size tends to decrease when either both the heating and cooling rates are removed (in HC00) or only the heating rate is removed (in H00). Compared with the inner-core size in CTL, that in HC00 is reduced by about 30 km while that in H00 is reduced drastically by about 60 km by 96 h of simulation.

Fig. 14.
Fig. 14.

Time evolution of the radius of the azimuthal-mean damaging-force wind (25.7 m s−1).

Citation: Journal of the Atmospheric Sciences 71, 9; 10.1175/JAS-D-13-0312.1

Diabatic heating in spiral rainbands can drive strong inflow both in the middle troposphere and in the boundary layer outside the eyewall, bringing high AAM inward and increasing tangential wind outside the eyewall (Fudeyasu and Wang 2011). The wind field therefore expands outward so that the inner-core size increases. Results from both CTL and C00 clearly demonstrate that diabatic heating (cooling) in inner rainbands in the RFZ is important for the increase (decrease) in the storm inner-core size. For example, there are three typical inflow jets outside of the 60-km radius in the TC simulated in C00 (Fig. 9d), located in the boundary layer, the midtroposphere, and about 10-km altitude. They are driven by diabatic heating in the eyewall, in the inner rainbands in the RFZ, and in the upper-tropospheric anvil clouds, respectively (Fudeyasu and Wang 2011). The inflow in the midtroposphere and under the upper-tropospheric anvil clouds brings high AAM inward to accelerate the tangential wind (Fig. 15a). Note that the strengthened convection in the RFZ (between 40- and 60-km radii) also contributes positively to the AAM budget above 2-km altitude through the azimuthal-mean vertical advection of AAM (Fig. 15b). On the other hand, the region with high inertial stability extends outward in C00 (not shown). Therefore, diabatic heating in that region facilitates the enhancement of the inflow (Fig. 9d) and positive AAM tendency in the boundary layer (Fig. 15a). As a result, the positive net AAM tendency prevails both outside of the 20-km radius below 6-km height and in the upper-tropospheric outflow layer (Fig. 15h). This indicates the intensification and the inner-core size increase of the storm simulated in C00 (Fig. 14).

Fig. 15.
Fig. 15.

As in Fig. 11, but averaged from 70 to 71 h in C00.

Citation: Journal of the Atmospheric Sciences 71, 9; 10.1175/JAS-D-13-0312.1

The results here further confirm that diabatic heating outside the eyewall is necessary for the size growth of a TC while diabatic cooling in spiral rainbands generally reduces the inner-core size of a TC. We also can see from Fig. 14 that the effect of diabatic heating on the inner-core size increase is larger than the effect of diabatic cooling on the inner-core size decrease, because the difference in inner-core size at 96 h between CTL and H00 is much larger than that between CTL and C00.

4. Conclusions

Inner spiral rainbands are generally active in the RFZ in a TC. Diabatic heating due to condensation and sublimation and diabatic cooling due to evaporation and melting in inner spiral rainbands can considerably affect the structure and intensity (and their changes) of a TC. In this study, idealized numerical experiments are conducted with the nonhydrostatic TC model TCM4 to investigate how diabatic heating and cooling in the RFZ affect the structure and intensity of a TC in a quiescent environment. In addition to a control experiment in which all default model settings were used, several sensitivity experiments were conducted by artificially removing either diabatic heating (H00) or diabatic cooling (C00) or both (HC00) below 10-km altitude in the RFZ (between 40- and 60-km radii) of the simulated storm after a 42-h spinup.

The results show that diabatic heating in the RFZ contributes positively to the storm intensity whereas diabatic cooling in the RFZ substantially limits the TC intensity. The latter is mainly due to the boundary layer low-θe air associated with downdrafts being transported and mixed into the eyewall, which suppresses eyewall convection, reducing the secondary circulation associated with the eyewall heating and the inward transport of AAM, and thus leading to the weakening of the TC. These effects are opposite to the effects of diabatic heating in outer rainbands where heating (cooling) is generally negative (positive) to TC intensity (Wang 2009).

Removal of diabatic heating in the RFZ highly suppresses the activity of inner rainbands. Removal of diabatic cooling in the RFZ results in an annular-hurricane-like storm without active outer rainbands owing to the cooling-related subsidence beneath 9-km height outside the inner core. Therefore, diabatic cooling in the RFZ appears unfavorable for the development of an annular hurricane structure. Since downdrafts associated with inner rainbands are closely related to relative humidity outside the eyewall in the middle troposphere, higher relative humidity in the middle troposphere may thus be favorable for the formation of annular hurricane structure as suggested in previous studies (Knaff et al. 2003; Wang 2009; Xu and Wang 2010a).

Different from that in Li and Wang (2012a), which showed that outer rainbands are often triggered by convective remnants of inner rainbands immediately outside of the RFZ, with inner rainbands restrained in both HC00 and H00, quasi-axisymmetric convective rings rather than typical outer rainbands form quasi periodically. The convective rings are initialized immediately outside the inner core and the quasi-periodic nature results from the consumption of CAPE by convection and the recovery of the boundary layer from the effect of downdrafts as documented in Li and Wang (2012a). The occurrence of the convective rings is subject to the local boundary layer convergence outside of the RFZ. Although the convective ring looks like a secondary eyewall in radar reflectivity, the secondary tangential wind maximum associated with the ring appears only in the late simulation in the lower troposphere but not obvious in the boundary layer. The appearance of the secondary tangential wind maximum is mainly due to the reduction of the tangential wind in the RFZ resulting from the reduced boundary layer inflow because of the suppression of inner rainbands. Moreover, the convective-ring-related low-θe air in the inflow boundary layer and the reduced mass and moisture convergence into the eyewall are negative to eyewall convection and thus the TC intensity.

Previous studies have revealed that diabatic heating in spiral rainbands outside the eyewall is the key to size change of a TC. Results from the present study have demonstrated that diabatic heating in inner spiral rainbands in the RFZ is essential for the increase in the inner-core size of a TC, whereas diabatic cooling in the region limits the increase in inner-core size.

Note that there are still some caveats in this study. One of the problems is that the heating region artificially modified in the sensitivity experiments is fixed between 40 and 60 km from the storm center and is referred to as the RFZ. However, the storm structure and intensity both evolve with time in the simulation and thus the actual region of the RFZ may change with time in the simulations. This has not been taken into account when we interpreted our results. The modification to diabatic heating and cooling has been limited up to 10 km. In reality, inner rainbands are subject to outward vertical tilt and convection in inner rainbands could also penetrate into the upper troposphere. This possibility has been ignored as well because of the difficulties in separating diabatic heating in inner rainbands and in the outflow layer from the outwardly tilted eyewall. Inner spiral rainbands are featured by low-azimuthal-wavenumber structure in the RFZ. The particular roles of individual low-wavenumber moving heating and cooling asymmetries related to inner rainbands in TC intensity and structure changes are not investigated in the present study either. Furthermore, all simulations were performed on an f plane in an environment at rest in this study. It is unclear how the results could be altered if any complex environmental flow is considered. Nevertheless, the results from this study still provide some new insights into the physical processes that lead to TC structure and intensity changes and also provide a basis for future studies that include complex interactions with various environmental settings, such as the beta effect, uniform environmental flow, and vertical wind shear.

Acknowledgments

This work was supported by the National (Key) Basic Research and Development (973) Program of China under Grants 2009CB421505 and 2013CB430300; the National Natural Science Foundation of China under Grants 40775060, 41005033, 40975035, 40921160381, 41130964, and 41375068; and the China Meteorological Administration Special Public Welfare Research Fund under Grants GYHY201006008 and GYHY200906002. YW has been also partially supported by NSF Grant AGS-1326524.

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  • Bui, H. H., R. K. Smith, M. T. Montgomery, and J. Y. Peng, 2009: Balanced and unbalanced aspects of tropical cyclone intensification. Quart. J. Roy. Meteor. Soc., 135, 17151731, doi:10.1002/qj.502.

    • Search Google Scholar
    • Export Citation
  • Chen, Y., and M. K. Yau, 2001: Spiral bands in a simulated hurricane. Part I: Vortex Rossby wave verification. J. Atmos. Sci., 58, 21282145, doi:10.1175/1520-0469(2001)058<2128:SBIASH>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Durran, D. R., and J. B. Klemp, 1983: A compressible model for the simulation of moist mountain waves. Mon. Wea. Rev., 111, 23412361, doi:10.1175/1520-0493(1983)111<2341:ACMFTS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Fairall, C. W., E. F. Bradley, J. E. Hare, A. A. Grachev, and J. B. Edson, 2003: Bulk parameterization of air–sea fluxes: Updates and verification for the COARE algorithm. J. Climate, 16, 571591, doi:10.1175/1520-0442(2003)016<0571:BPOASF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Frisius, T., and T. Hasselbeck, 2009: The effect of latent cooling processes in tropical cyclone simulations. Quart. J. Roy. Meteor. Soc., 135, 17321749, doi:10.1002/qj.495.

    • Search Google Scholar
    • Export Citation
  • Fudeyasu, H., and Y. Wang, 2011: Balanced contribution to the intensification of a tropical cyclone simulated in TCM4: Outer-core spinup process. J. Atmos. Sci., 68, 430449, doi:10.1175/2010JAS3523.1.

    • Search Google Scholar
    • Export Citation
  • Gray, W. M., E. Ruprecht, and R. Phelps, 1975: Relative humidity in tropical weather systems. Mon. Wea. Rev., 103, 685690, doi:10.1175/1520-0493(1975)103<0685:RHITWS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Guinn, T. A., and W. H. Schubert, 1993: Hurricane spiral bands. J. Atmos. Sci., 50, 33803403, doi:10.1175/1520-0469(1993)050<3380:HSB>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Hill, K. A., and G. M. Lackmann, 2009: Influence of environmental humidity on tropical cyclone size. Mon. Wea. Rev., 137, 32943315, doi:10.1175/2009MWR2679.1.

    • Search Google Scholar
    • Export Citation
  • Knaff, J. A., J. P. Kossin, and M. DeMaria, 2003: Annular hurricanes. Wea. Forecasting, 18, 204223, doi:10.1175/1520-0434(2003)018<0204:AH>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Langland, R. H., and C. S. Liou, 1996: Implementation of an Eε parameterization of vertical subgrid-scale mixing in a regional model. Mon. Wea. Rev., 124, 905918, doi:10.1175/1520-0493(1996)124<0905:IOAPOV>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Li, Q., and Y. Wang, 2012a: Formation and quasi-periodic behavior of outer spiral rainbands in a numerically simulated tropical cyclone. J. Atmos. Sci., 69, 9971020, doi:10.1175/2011JAS3690.1.

    • Search Google Scholar
    • Export Citation
  • Li, Q., and Y. Wang, 2012b: A comparison of inner and outer spiral rainbands in a numerically simulated tropical cyclone. Mon. Wea. Rev., 140, 27822805, doi:10.1175/MWR-D-11-00237.1.

    • Search Google Scholar
    • Export Citation
  • Malkus, J. S., and H. Riehl, 1960: On the dynamics and energy transformations in steady-state hurricanes. Tellus, 12, 120, doi:10.1111/j.2153-3490.1960.tb01279.x.

    • Search Google Scholar
    • Export Citation
  • Möller, J. D., and L. J. Shapiro, 2002: Balanced contributions to the intensification of Hurricane Opal as diagnosed from a GFDL model forecast. Mon. Wea. Rev., 130, 18661881, doi:10.1175/1520-0493(2002)130<1866:BCTTIO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Montgomery, M. T., and R. J. Kallenbach, 1997: A theory for vortex Rossby-waves and its application to spiral bands and intensity changes in hurricanes. Quart. J. Roy. Meteor. Soc., 123, 435465, doi:10.1002/qj.49712353810.

    • Search Google Scholar
    • Export Citation
  • Moon, Y., and D. S. Nolan, 2010: The dynamic response of the hurricane wind field to spiral rainband heating. J. Atmos. Sci., 67, 17791805, doi:10.1175/2010JAS3171.1.

    • Search Google Scholar
    • Export Citation
  • Nguyen, M. C., M. J. Reeder, N. E. Davidson, R. K. Smith, and M. T. Montgomery, 2011: Inner-core vacillation cycles during the intensification of Hurricane Katrina. Quart. J. Roy. Meteor. Soc., 137, 829844, doi:10.1002/qj.823.

    • Search Google Scholar
    • Export Citation
  • Otto, P., and J. Soderholm, 2012: The convective features within and surrounding severe tropical cyclone Larry (2006). Trop. Cyclone Res. Rev., 1, 143162, doi:10.6057/2012TCRR02.07.

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  • Fig. 1.

    The radius–height cross section of the azimuthal-mean filamentation time (shading; min) after 42 h of simulation in CTL, and the value of coefficient b (solid line) as a function of the radius.

  • Fig. 2.

    Time series of simulated (a) minimum sea level pressure (hPa) and (b) maximum wind speed (m s−1) at the lowest model level.

  • Fig. 3.

    Radius–time Hovmöller diagram of θe (K) at z = 0.3 km in (a) CTL, (b) HC00, (c) H00, and (d) C00.

  • Fig. 4.

    Radius–time Hovmöller diagram of the amplitude of the azimuthal wavenumber ≤ 3 asymmetric PV at z = 3 km in (a) CTL, (b) HC00, (c) H00, and (d) C00.

  • Fig. 5.

    Radius–time Hovmöller diagram of the azimuthal-mean surface rain rate (mm h−1) in (a) CTL, (b) HC00, (c) H00, and (d) C00.

  • Fig. 6.

    Radius–time Hovmöller diagram of the CAPE (J kg−1) in (a) CTL, (b) HC00, (c) H00, and (d) C00. Contours indicate the smoothed azimuthal-mean surface rain rate less than 10 mm h−1 with an interval of 2 mm h−1.

  • Fig. 7.

    Plan view of the simulated reflectivity (dBZ) at z = 3 km from 66 to 89 h 45 min of simulation in HC00. Inner (outer) circles show the location of the 60 (120)-km radius from the storm center.

  • Fig. 8.

    As in Fig. 7, but from 60 to 83 h 45 min of simulation in H00.

  • Fig. 9.

    Radius–height cross sections of the azimuthal-mean AAM (green contours; 105 m2 s−1), diabatic heating rate (shading; 10−3 K s−1), and secondary circulation (arrows; m s−1) in (a) CTL averaged between 70 and 71 h, (b) HC00 averaged between 70 and 71 h, (c) H00 averaged between 64 and 65 h, and (d) C00 averaged between 70 and 71 h.

  • Fig. 10.

    Plan view of the simulated vertical velocity (m s−1) at z = 5 km from 63 to 82 h of simulation in C00. Inner (outer) circles show the location of the 60 (120)-km radius from the storm center.

  • Fig. 11.

    (a) The azimuthal-mean radial vorticity flux, (b) azimuthal-mean vertical AAM advection, (c) mean secondary circulation, (d) radial eddy vorticity flux, (e) vertical eddy AAM advection, (f) total eddy flux, and (g) frictional and horizontal diffusion contributions to (h) the AAM budget averaged from 70 to 71 h in HC00. Tendency in AAM is expressed in units of meters squared per second squared.

  • Fig. 12.

    As in Fig. 11, but averaged from 64 to 65 h in H00.

  • Fig. 13.

    Radius–time Hovmöller diagram of the azimuthal-mean divergence (shading; 10−4 s−1) at z = 500 m in (a) HC00 and (b) H00, overlaid by the azimuthal-mean radial wind contoured at −6, −5, −4, −3, −2, −1, and 0 m s−1.

  • Fig. 14.

    Time evolution of the radius of the azimuthal-mean damaging-force wind (25.7 m s−1).

  • Fig. 15.

    As in Fig. 11, but averaged from 70 to 71 h in C00.

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