a. Balanced vortex dynamics

Balanced vortex dynamics assumes a quasi-balanced basic axisymmetric vortex that is in both hydrostatic and gradient wind balances. Given the spatial distributions of heat source and momentum forcing, a partial differential equation for the streamfunction of the transverse circulation, namely, the so-called Sawyer–Eliassen equation (SEQ; Eliassen 1951) can be obtained. Since the SEQ is a linear partial differential equation and its solutions to different forcings are additive and thus can be used to understand the response of the transverse circulation in a TC vortex to various heat sources or momentum forcing. Because the low-level inflow in the transverse circulation can bring absolute angular momentum inward to spin up the tangential wind, the solution of the SEQ has been used to understand the TC intensification and eyewall contraction (e.g., Shapiro and Willoughby 1982; Schubert and Hack 1982; Pendergrass and Willoughby 2009; Bui et al. 2009; Heng and Wang 2016; Heng et al. 2017), the TC outer-core size change (e.g., Fudeyasu and Wang 2011), and the secondary eyewall formation (e.g., Zhu and Zhu 2014; Wang et al. 2016).

Shapiro and Willoughby (1982) found that the low-level tangential wind tendency in response to diabatic heating in the eyewall is greater inside of the RMW than at the RMW. As a result, as the TC vortex intensifies, its RMW would move inward, namely, experiencing a contraction. The simultaneous intensification and eyewall contraction have been observed in real TCs (e.g., W82; Willoughby 1990) and in numerical simulations (e.g., Bryan 2013; Smith et al. 2014; Smith et al. 2015, Heng and Wang 2016; Deng et al. 2018). However, recent observations (e.g., Wang and Wang 2013; Stern et al. 2015; Qin et al. 2016) indicate that it is very common for the asynchrony between TC intensification and eyewall contraction. This means that although the balanced vortex dynamics has been considered as a dominant dynamical mechanism responsible for the contraction of RMW or the eyewall of a TC, it could not explain all aspects of eyewall contraction. Furthermore, it only provides a qualitative explanation but not a quantitative estimation of the RMW contraction.

b. W82

c. K12

The framework of K12 and Kieu and Zhang (2017) outlined above was challenged by S15 and Stern et al. (2017). First, S15 indicated that the total derivative of ${\overline{\mathrm{\Omega }}}^{*}$, $d{\overline{\mathrm{\Omega }}}^{*}/dt$ is not equal to its partial derivative $\partial {\overline{\mathrm{\Omega }}}^{*}/\partial t$, because ${\overline{\mathrm{\Omega }}}^{*}$ should be also a function of r, but this argument seems to be inconsistent with the assumption of K12 or in Kieu and Zhang (2017) as mentioned above. In addition, Stern et al. (2017) noted that the contraction rate in the simulations of K12 obviously increased with surface friction or C_D, which is in contradictory to the explanation of Eq. (6). The positive correlation between C_D and the eyewall contraction rate has been also reported in previous studies (e.g., Bryan 2013; Smith et al. 2014; Heng and Wang 2016). We also noticed that the dependence of RMW contraction rate on C_D should be a local behavior, that is, in addition to surface friction, the radial advection in Eq. (7) and the TC structure during the initial spinup period will also change considerably if C_D is changed (e.g., Smith et al. 2014; Heng and Wang 2016). Therefore, it is hard to tell the role of surface friction in the RMW contraction just by changing C_D from the beginning of a simulation.

The first term is related to the intensification rate of the TC, ${\left(d\overline{V}/dt\right)}_{\text{RMW}}$, and the second term is related to the local change of tangential wind at the RMW, ${\left(\partial \overline{V}/\partial t\right)}_{\text{RMW}}$. Note that Eq. (8) looks similar to Eq. (2) if one assumes that the tangential wind within and at the RMW strictly satisfies a Rankine vortex, that is, $\partial \overline{\upsilon }/\partial r{|}_{r\le \text{RMW}}=\overline{V}/\text{RMW}$, which is the key assumption of K12 and also an assumption used in estimating the RMW contraction based on the flight-level data in W82. The major difference between Eqs. (8) and (2) lies in where the local tendency of the azimuthal-mean tangential wind is defined. In Eq. (2) it is defined at the arriving point of the RMW, while in Eq. (8) it seems to be defined at the current time. Since the contraction of RMW could not be evaluated if the local change of tangential wind is defined at the current time (see Fig. 1 of Kieu and Zhang 2017), this suggests an alternative problem of K12. As a result, only when the local tendency of tangential wind or the last three terms are evaluated at the arriving point of the RMW can Eq. (7) be used to diagnose the contraction rate of the RMW. However, in this case, Eq. (7) becomes equivalent to Eq. (2), and it does not provide any extra dynamical insights into the contraction of the RMW as claimed by K12 and Kieu and Zhang (2017). Therefore, K12 will not be included in the following discussion.

d. S15

a. Model and experimental design

The Weather Research and Forecasting (WRF) Model, version 3.8.1 (Skamarock et al. 2008), was used, which is a fully compressible, nonhydrostatic model with a terrain-following vertical coordinate. The triply nested and fixed domains were used with the finest resolution of 2 km in the innermost domain. Physical parameterizations used in our simulations include the Mellor–Yamada–Nakanishi–Niino (MYNN) level 2.5 scheme (Nakanishi and Niino 2009) for the surface layer and the planetary boundary layer processes, the Dudhia shortwave radiation scheme (Dudhia 1989), the Rapid Radiative Transfer Model (RRTM; Mlawer et al. 1997) for a longwave radiation scheme, the Thompson microphysical scheme (Thompson et al. 2008) for cloud microphysics, and the Kain–Fritsch cumulus parameterization (Kain 2004) in the outermost 18-km mesh only. The average moist tropical sounding during June–October 2002 of Dunion and Marron (2008) was used as the unperturbed environment with the sea surface temperature was fixed at 29°C.

Two idealized three-dimensional simulations were conducted. In the control experiment, the experimental design followed that in Zhu and Zhu (2014) in which a strong initial TC vortex with the maximum tangential wind speed of 36 m s⁻¹ at a radius of 45 km was used. A sensitivity experiment was conducted to ensure the robustness of the results from the control experiment. The sensitivity experiment was the same as the control experiment except that a weaker initial TC vortex with the maximum tangential wind speed of 18 m s⁻¹ was used. All results discussed below were based on the model outputs from the innermost domain at 10-min intervals.

To make the finite-differencing smoother enough to facilitate the evaluation of the theoretical work summarized in section 2, similar to S15, we first filtered out the small-scale perturbations less than 40 min in time using a time mean and less than 8 km in radial direction using a spatial average, and then we interpolated all the azimuthal-mean variables from a radial grid spacing of 2 km into a 50-m grid spacing using the cubic spline interpolation. The results discussed below are all based on the filtered variables and the results are not sensitive to the filtering scale qualitatively.

b. Evaluation results

Consistent with previous studies, in the control experiment, after about 15 h of initial adjustment, a rapid contraction of the RMW occurred, which stopped well before the end of intensification (Fig. 1). The largest hourly contraction rate reached ~16 km h⁻¹ at about 17 h of simulation. The RMW remained almost constant after about 22.5 h, but the TC continued to intensify. This means that the contraction of RMW is not necessarily accompanied by TC intensification, which is consistent with observations (S15; Qin et al. 2016; Wang and Wang 2013). Figure 2a compares the time tendency of the RMW from the control experiment with those calculated using Eqs. (2) and (9), respectively, from W82 and S15 at a height of 250 m, which is the same as in S15. Note that here the finite-differencing scheme for W82 exactly followed that suggested by Kieu and Zhang (2017) using the tangential wind profile at the arriving RMW, and a centered finite-difference scheme was used for S15 in both time and space at the current time and current location of RMW. We can see that the diagnosed tendency of the RMW using Eq. (2) or (9) is almost identical to that of the model simulation. This is not surprising since both equations were derived without any mathematical simplification, although Eqs. (2) and (9) are not equivalent mathematically (Kieu and Zhang 2017). In addition, there is almost no error for W82 (Fig. 2c) because it is a fully closed kinematic model in which the arriving profile has been provided, as mentioned earlier. As we can see from Figs. 2b and 2d, the conclusion is unchanged with an initially weaker TC vortex.

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Radius–time Hovmöller plot of the azimuthal-mean tangential wind speed (shades; m s⁻¹) at a height of 250 m, overlaid by the radial position of RMW (black), and the two cyan horizontal lines mark the rapid contraction period of the RMW.

Citation: Journal of the Atmospheric Sciences 76, 10; 10.1175/JAS-D-19-0076.1

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(a) Time series of the tendency of the RMW simulated in the control experiment (red solid) and diagnosed by S15 (purple dashed) and W82 (cyan dashed). (b) As in (a), but for the sensitivity experiment with an initially weak TC vortex with a maximum wind speed of 18 m s⁻¹. (c),(d) As in (a) and (b), but showing the difference between the diagnosed and simulated results.

Citation: Journal of the Atmospheric Sciences 76, 10; 10.1175/JAS-D-19-0076.1

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To understand contributions to the RMW contraction in S15, we show in Fig. 3a the time series of both the radial gradient of the azimuthal-mean tangential wind tendency and the curvature of the azimuthal-mean tangential wind in the radial direction at the RMW and 250-m height. We can see from Fig. 3a that the curvature of tangential wind was negative definite throughout the simulation as mentioned earlier, while the radial gradient of tangential wind tendency showed large variability with both negative and positive values corresponding to the contraction and expansion of RMW in the simulation (Fig. 2). During the early stage of the rapid RMW contraction period 15–19 h (Fig. 1), the radial gradient of the azimuthal-mean tangential wind tendency at the RMW was largely negative and the curvature of the azimuthal-mean tangential wind was small, corresponding to the rapid contraction of the RMW. With the rapid contraction, the curvature of the azimuthal-mean tangential wind increased rapidly, and the tangential wind showed large sharpness near the RMW (Fig. 3b). S15 attributed the cessation of the RMW contraction to the increase in the curvature or sharpness of tangential wind. In our simulation, both the decrease in the negative radial gradient of azimuthal-mean tangential wind tendency and the increase in the sharpness of the azimuthal-mean tangential wind contributed to the cessation of the RMW contraction as we can see from Figs. 2 and 3a.

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(a) Time series of the radial gradient of time tendency of the azimuthal-mean tangential wind [$\left(\partial /\partial r\right)\left(\partial \overline{V}/\partial t\right)$; purple dashed] and the curvature of the azimuthal-mean tangential wind in radial direction at the RMW (${\partial }^2\overline{V}/\partial r^2$; green dotted) at a height of 250 m. (b) The radial profile of the azimuthal-mean tangential wind speed at a height of 250 m.

Citation: Journal of the Atmospheric Sciences 76, 10; 10.1175/JAS-D-19-0076.1

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c. The azimuthal-mean tangential wind budget

To ensure a nearly residual-free budget analysis, in addition to the instantaneous values, all the 10-min-mean $\overline{u}$, $\overline{\upsilon }$, $\overline{w}$, horizontal advection, vertical advection, turbulent vertical mixing (including surface friction), subgrid-scale horizontal diffusion, and pressure gradient force were directly calculated during the model integration. All these variables were then interpolated into the cylindrical coordinates. The TC center was defined as the circulation center, which maximizes the TC maximum tangential wind speed at 250-m height above the sea surface. Note that during the WRF integration, only the total advections $-\overline{u{\zeta }_a}$ and $-\overline{w\partial \upsilon /\partial z}$ could be explicitly given and saved. This means that both the eddy advection terms $-\overline{u^{\prime }{\zeta }_a^{\prime }}$ and $-\overline{w^{\prime }\left(\partial {\upsilon }^{\prime }/\partial z\right)}$ and the mean advections $-\overline{u{\zeta }_a}$ and $-\overline{w\partial \upsilon /\partial z}$ should be calculated offline. However, the offline calculations would induce some numerical errors in the partitioned mean and eddy advections due to the coordinate transformation, interpolation, and the use of different finite-differencing schemes. To reduce any bias in such partitioning, we first directly calculated both mean and eddy advections offline and then indirectly obtained the mean (eddy) advection terms by subtracting the offline directly calculated eddy (mean) advections from the WRF-output total advections. Finally, the mean (eddy) advections used in the final budget analysis are defined as the average of the two mean (eddy) advections obtained above. In this way, the calculation errors are distributed equally between the mean and eddy advections. Note that to minimize errors that might be introduced due to the use of different finite-differencing algorithms in calculating the advection terms in the WRF Model and in the budget, as in the WRF Model, the staggered grid finite-differencing in both vertical and radial directions was used to improve the precision. Figure 4 compares the eddy advection terms obtained from the above method and that directly calculated using the 10-min mean wind field at 19 h (Figs. 4a–d) and 21 h (Figs. 4e–h). The overall patterns are very similar except some discrepancies. Note that the 10-min mean field was used to define the storm center to avoid any inconsistencies in the calculated eddy contributions. This might introduce some extra errors. However, because the TC motion in 10 min in the simulation on an f plane in a quiescent environment is often less than one-half of the finest grid spacing (often less than 1 km), the errors would not affect our main conclusions. Therefore, the budget results can be used to understand contributions of various forcing processes to the RMW contraction.

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Radius–height cross sections of the (a),(b) eddy horizontal advection at 19 h with a 40-min average from (a) a direct offline calculation based on 10-min average wind field and (b) an average between the direct offline calculation and the difference between total horizontal advection and the offline calculated mean horizontal advection. (c),(d) As in (a) and (b), but for vertical advection. The contour interval is 2 m s⁻¹ h⁻¹. (e)–(h) As in (a)–(d), but for 21 h with a contour interval of 3 m s⁻¹ h⁻¹.

Citation: Journal of the Atmospheric Sciences 76, 10; 10.1175/JAS-D-19-0076.1

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To provide a general view on the structure of the local azimuthal-mean tangential wind tendency and their relative position to the RMW, a 40-min averaged budget during the rapid RMW contraction period (19 h) is shown in Fig. 5. Note that the azimuthal-mean tangential wind tendencies from both the model and budget are shown in Fig. 5 for a comparison. Clearly, the budget reproduced the tendency structure from the simulation very well (Figs. 5a,b), with negligible errors (~10⁻⁵ m s⁻¹ h⁻¹) from interpolations. As we can see from Fig. 5, the tangential wind tendency inside the RMW is larger than that outside, which results in the contraction of the RMW during this period (cf. Figs. 1 and 2a). Some key points can be noticed by comparing all individual terms in Figs. 5c–h. First, the contribution by mean advection varies with height. For example, the mean vertical advection (ADV_V) contributes to an expansion below ~300 m but a contraction above; the mean horizontal advection (ADV_H) contributes to a contraction of the RMW below ~400 m but an expansion above. The turbulent vertical mixing including surface friction (Ff), eddy horizontal advection (Eddy_H), and subgrid-scale horizontal diffusion (Diff) all contribute to the RMW expansion in the boundary layer, and eddy vertical advection (Eddy_V) contributes to the RMW contraction at this time.

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Radius–height cross sections of the averaged local acceleration of the azimuthal-mean tangential wind at 19 h with a 40-min average of simulation based on (a) the model simulation and (b) the budget. (c)–(h) As in (b), but for the individual contributions by mean horizontal advection, mean vertical advection, vertical mixing including surface friction, eddy horizontal advection, eddy vertical advection, and horizontal diffusion terms, respectively. The contour interval is 1 m s⁻¹ h⁻¹ in (a) and (b), 3 m s⁻¹ h⁻¹ in (c)–(g), and 0.4 m s⁻¹ h⁻¹ in (h). The red dotted line shows the RMWs at different vertical levels.

Citation: Journal of the Atmospheric Sciences 76, 10; 10.1175/JAS-D-19-0076.1

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Following Stern et al. (2017), substituting Eq. (10) into Eq. (9) and omitting the perturbation pressure gradient force, we get

${\displaystyle \frac{d\text{RMW}}{dt}}={\left\{{\displaystyle \frac{{\displaystyle \frac{\partial }{\partial r}}\left(\overline{U}{\overline{\zeta }}_a\right)}{{\partial }^2\overline{V}/\partial r^2}}+{\displaystyle \frac{{\displaystyle \frac{\partial }{\partial r}}\left(\overline{W}{\displaystyle \frac{\partial \overline{V}}{\partial z}}\right)}{{\partial }^2\overline{V}/\partial r^2}}+{\displaystyle \frac{{\displaystyle \frac{\partial }{\partial r}}\left(\overline{u^{\prime }{\zeta }_a^{\prime }}\right)}{{\partial }^2\overline{V}/\partial r^2}}+{\displaystyle \frac{{\displaystyle \frac{\partial }{\partial r}}\left[\overline{w^{\prime }\left({\displaystyle \frac{\partial {\upsilon }^{\prime }}{\partial z}}\right)}\right]}{{\partial }^2\overline{V}/\partial r^2}}-{\displaystyle \frac{{\displaystyle \frac{\partial }{\partial r}}\left({\overline{F}}_{\text{fric}}\right)}{{\partial }^2\overline{V}/\partial r^2}}-{\displaystyle \frac{{\displaystyle \frac{\partial }{\partial r}}\left({\overline{F}}_{\text{diff}}\right)}{{\partial }^2\overline{V}/\partial r^2}}\right\}|}_{\text{RMW}}.$(11)

Considering the contribution by mean advection to the RMW tendency varies with height (cf. Figs. 5c,d), and the turbulent vertical mixing is large below 500-m height (cf. Fig. 5e), two different integral averages are calculated, that is, between 0 and 500 m (the lower boundary layer) and between 500 and 1500 m (the upper boundary layer and lower troposphere), to illustrate the contribution of each tangential wind tendency forcing to the RMW tendency. Figure 6 shows each integral-averaged tangential wind tendency at the RMW [Figs. 6a and 6b; cf. Eq. (10)] and the corresponding contribution to the RMW tendency [denoted by the prefix “S15” in Figs. 6c and 6d; cf. Eq. (11)].

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(a) Time series of each tangential wind tendency at the RMW averaged below 500-m height. (b) As in (a), but averaged between 500- and 1500-m height. (c),(d) As in (a) and (b), but for the contributions by each tangential wind tendency to the RMW tendency.

Citation: Journal of the Atmospheric Sciences 76, 10; 10.1175/JAS-D-19-0076.1

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In the lower boundary layer, as expected, the mean horizontal advective forcing (S15_ADV_H) dominates the contraction of the RMW (Fig. 6c), consistent with its large magnitude (ADV_H, Fig. 6a). The subgrid-scale horizontal diffusion forcing (S15_Diff) and eddy vertical advective forcing (S15_Eddy_V) contribute marginally to the RMW contraction (Fig. 6c), consistent with their small magnitudes (Fig. 6a). In addition, although the mean vertical advection (ADV_V) at the RMW is much larger than the eddy vertical advection (Eddy_V) (Fig. 6a), its contribution to the RMW tendency is small and comparable with S15_Eddy_V (Fig. 6c). The role of mean vertical advection in the RMW tendency (S15_ADV_V) is alternately positive and negative during the simulation (Fig. 6c). Consistent with K12, the turbulent vertical mixing including surface friction (Ff) contributes to the RMW expansion (Fig. 6c). Besides, the increase of eddy horizontal advective forcing (S15_Eddy_H) also plays an important role in the cessation of RMW contraction from the later rapid contraction stage (after ~20 h) (Fig. 6c). Note that although Ff in the later rapid contraction stage is comparable with Eddy_H (Fig. 6a), its contribution (S15_Ff) to the RMW tendency is smaller than S15_Eddy_H (Fig. 6c). This difference is understandable, because ${\overline{F}}_{\mathrm{fric}}^{*}~-C_D/H\left[\overline{V}\sqrt{\left({\overline{V}}^2+{\overline{U}}^2\right)}\right]~-C_D{\overline{V}}^2/H$ (see K12), considering ${\overline{V}}^2\gg {\overline{U}}^2$, therefore, $\partial {\overline{F}}_{\mathrm{fric}}^{*}/\partial r\mathrm{~}-\left(2C_D\overline{V}/H\right)\left(\partial \overline{V}/\partial r\right)~0$, if we assume C_D and H are constant. Indeed, the contours of Ff near the RMW are almost perpendicular to the RMW (Fig. 5e), indicating a small radial gradient of Ff at the RMW, except for those levels near the surface where the approximation, ${\overline{V}}^2\gg {\overline{U}}^2$ is often invalid. However, the contours of Eddy_H near the RMW in the lower boundary layer are almost parallel to the RMW (Figs. 4a,b,e,f and 5f), which implies a large radial gradient of Eddy_H.

In the upper boundary layer and lower troposphere, consistent with the above analyses (Fig. 5), the results change a lot, especially for the mean advection terms (Figs. 6b,d), compared to that in the lower boundary layer (Figs. 6a,c). First, the sign of mean horizontal (vertical) advection term to the TC intensification changes (Fig. 6b). Second, the mean horizontal (vertical) advective forcing changes to dominate the expansion (contraction) of the RMW from the later rapid contraction stage (Fig. 6d). As a result, the increase of the mean horizontal advective forcing (S15_ADV_H) prevents further contraction of the RMW in this layer. Note that the S15_ADV_H also contributes to the RMW contraction during the early rapid contraction stage around 17 h (Fig. 6d). In addition, both the vertical mixing forcing and eddy horizontal advective forcing play a marginal role in the RMW tendency in this layer, and the eddy vertical advective forcing changes to favor the RMW contraction.

Based on the above analyses, in addition to the curvature of the azimuthal-mean tangential wind, the increase in the radial gradient of eddy horizontal advection at the RMW also plays an important role in preventing further contraction of the RMW, especially in the lower boundary layer (Fig. 6c). Since the eddy horizontal advection reflects the horizontal mixing by the resolved eddies in three dimensions, we thus can consider that the radial gradient of (both resolved and parameterized) horizontal mixing contributes to the cessation of the RMW contraction. Note that the resolved eddy mixing in three dimensions is implicitly parameterized by horizontal diffusion in axisymmetric simulations (Bryan and Fritsch 2002). Therefore, if the horizontal eddy mixing in three-dimensional simulation is really important for the cessation of RMW contraction, the subgrid-scale horizontal mixing in axisymmetric simulations should be important for the cessation of the RMW contraction. This may imply that larger horizontal diffusion (e.g., with a larger horizontal mixing length) in axisymmetric simulations may result in a larger steady-state RMW. This is indeed the case already given by Bryan (2012) and Rotunno and Bryan (2012). To further verify this implication, an axisymmetric model was used to perform several sensitivity experiments in the next section.

a. Model and experimental design

The axisymmetric model selected for our numerical experiments is the state-of-the-art cloud model CM1, version 19.4 (Bryan and Fritsch 2002), which has been used widely for understanding TC dynamics (e.g., Bryan 2012; Rotunno and Bryan 2012; Bu et al. 2017). The model resolution was 3 km within a radius of 300 km and then stretched to 13 km near the lateral boundary of the model domain at 1500 km. The model had 59 vertical levels with a stretching vertical grid spacing from 25 m at the surface to 500 m at 5.5 km and remains at 500 m above 5.5 km. The initial TC vortex was axisymmetric with the maximum tangential wind of about 15 m s⁻¹ at an 82.5-km radius, which decreases to zero with radius out to 412.5 km and the height up to 20 km. An idealized saturated and neutral sounding (Bryan and Rotunno 2009) with a fixed sea surface temperature of 28°C was used to initialize the unperturbed atmospheric environment in all simulations. The Coriolis parameter was set to 5 × 10⁻⁵ s⁻¹, corresponding to 20°N. Similar to Bryan (2012), the Morrison double-moment scheme was used for cloud microphysics (Morrison et al. 2009) and no cumulus convective parameterization was used. The Newtonian cooling capped at 2 K day⁻¹ was used to represent longwave radiation. The ratio of surface exchange coefficients for enthalpy and momentum, C_K/C_D, was fixed at 0.5 with C_D fixed at 2.4 × 10⁻³. Following Bryan (2012), the Smagorinsky scheme (Bryan and Fritsch 2002) was used to parameterize eddy mixings. Namely, the horizontal viscosity K_m,h is calculated by $K_{m,h}=l_h^2{S}_h$, and the vertical viscosity K_m,υ is calculated by $K_{m,\upsilon }=l_{\upsilon }^2{S}_{\upsilon }{\left(1-\mathrm{Ri}/\mathrm{Pr}\right)}^{1/2}$, where S_h and S_υ denote horizontal and vertical deformations; Ri and Pr denote Richardson number and Prandtl number; l_h and l_υ denote horizontal and vertical mixing lengths. The vertical mixing length at each level is determined by $l_{\upsilon }^{-2}={\left(kz\right)}^{-2}+l_{\infty }^{-2}$, where k denotes the von Kármán constant and l_∞ denotes asymptotic (z → ∞) vertical mixing length. In the control experiment, the l_h and l_∞ were set at 1000 and 50 m, respectively. In addition to the control experiment, four sensitivity experiments were conducted with the l_h and l_∞ doubled and halved, respectively, during contraction period after the initial adjustment to examine the effects of horizontal and vertical mixing on the RMW contraction. Similar to the WRF simulation, every 10-min model output of CM1 were used in the analysis.

b. Results

The temporal evolution of the maximum wind speed and RMW at the lowest model level (25 m) and 250 m in the control simulation are shown in Figs. 7a and 7b. The RMW experienced an overall contraction but with large fluctuations in the early stage of simulations. Similar to that in the three-dimensional WRF simulation, the contraction stopped at the early stage of intensification by about 60 h of simulation, but the TC intensity reached a quasi-steady state after about 140 h. Note that because the initial vortex was weaker with larger RMW than that in the WRF simulation, the TC vortex experienced a longer initial adjustment period up to 48 h during which the RMW changed more irregularly and discontinuously. Since Eq. (9) assumes a continuous change of the RMW, following the current location of the RMW to predict its radial movement, we focus on the period of a nearly continuous RMW contraction after the initial adjustment or from the later rapid contraction stage (i.e., 48 h). Similar to the analysis in section 3, the model output after the initial 48 h of simulation was also filtered and interpolated onto a 50-m radial resolution. As expected, the method of S15 and W82 can capture the RMW tendency very well in the axisymmetric simulation (Fig. 7c).

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(a) Time series of the maximum azimuthal-mean tangential wind speed (Vmax; red solid) at the lowest model level (25 m) and its corresponding radius (RMW; red dotted). (b),(c) As in Figs. 1 and 2, but for the control experiment from CM1.

Citation: Journal of the Atmospheric Sciences 76, 10; 10.1175/JAS-D-19-0076.1

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To understand the dynamics of the RMW contraction, a tangential wind budget during the contraction period (50 h) was conducted as what was done in section 3c. All the tangential wind tendency terms in Eq. (10) excepted for those eddy terms were direct output from the model simulation. Note that these tendencies were not averaged in time because the budgeted tendency at any given time (e.g., Fig. 8b) can well capture the tendency from the model (e.g., Fig. 8a). Overall, the results are consistent with those from the WRF Model simulations. The local tendency of tangential wind is larger inside the RMW than outside during the contraction period, corresponding to the RMW contraction (Fig. 7). Contributions by (both horizontal and vertical) advection terms varies with height (Figs. 8c,d). The direct contribution by vertical mixing (and surface fiction) makes the RMW expansion in the boundary layer (Fig. 8e). As expected, the horizontal mixing in the boundary layer is larger than that in the WRF simulation and contributes to the RMW expansion (Fig. 8f), similar to the eddy horizontal advection in the WRF simulations.

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As in Fig. 5, but for the control experiment from CM1 at 50 h with a 40-min average, and the eddy terms are excluded here. The contour interval is 1 m s⁻¹ h⁻¹ in (a), (b), (e), and (f) and 2 m s⁻¹ in (c) and (d).

Citation: Journal of the Atmospheric Sciences 76, 10; 10.1175/JAS-D-19-0076.1

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The individual tendencies in Eq. (10) at the RMW below and above 500-m height and their corresponding contributions to the RMW tendency in Eq. (11) are shown in Fig. 9. The overall results are consistent with those from the WRF Model simulations. First, except for the horizontal advection (ADV_H), all other terms slow down the TC intensification rate during the intensification period in the lower boundary layer (Fig. 9a), and the sign of mean horizontal (vertical) term to the TC intensification changes from the lower boundary layer (Fig. 9a) to the upper boundary layer and lower troposphere (Fig. 9b). Second, the horizontal advective forcing (S15_ADV_H) dominantly contributes to the RMW contraction in the lower boundary layer (Fig. 9c) but expansion above (Fig. 9d). In addition, the vertical advective forcing on the RMW tendency (S15_ADV_V) is mainly positive during the rapid contraction period (48–57 h) and plays a small role later in the lower boundary layer (Fig. 9c), but dominantly contributes to the RMW contraction throughout the analysis period in the upper boundary layer and lower troposphere (Fig. 9d). As expected, the horizontal mixing forcing (S15_Diff) increases during the rapid contraction period, and then makes an obvious inhibitory effect on the RMW contraction in the lower boundary layer (Fig. 9c), similar to the resolved eddy horizontal advection/mixing shown in Fig. 6c in the three-dimensional WRF simulation. Note that Diff and S15_Diff in Fig. 6 are different from those in Fig. 9, in which both the resolved and parameterized horizontal mixing are included as mentioned above. In addition, turbulent vertical mixing including surface friction (Ff) also contributes to the cessation of the RMW contraction in the lower boundary layer (Fig. 9c). Note that although the Ff in the contraction stage is about twice the value of Diff in the lower boundary layer (Fig. 9a), their contributions (S15_Ff and S15_Diff) to the RMW tendency are comparable with each other (Fig. 9c) because the contours of Diff are more parallel to the RMW than that of Ff (cf. Figs. 8e,f), consistent with the analysis in the three-dimensional WRF simulation. In addition, the roles of Ff and S15_Diff become marginal above the lower boundary layer (Fig. 9d).

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As in Fig. 6, but for the control experiment from CM1 with the eddy terms being excluded.

Citation: Journal of the Atmospheric Sciences 76, 10; 10.1175/JAS-D-19-0076.1

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The above analyses for the control axisymmetric simulation demonstrate that horizontal advective forcing predominantly contributes to the RMW contraction during the RMW contraction period in the lower boundary layer, while the horizontal mixing forcing plays an important role in preventing the RMW contraction, as the eddy horizontal advection in three-dimensional simulations. In addition, the vertical mixing (including surface friction) forcing also plays an important role in the cessation of RMW contraction. These two conclusions are further confirmed by results from four sensitivity experiments, in which the horizontal mixing length and the asymptotic vertical mixing length were doubled or halved of that used in the control experiment from 48 h of simulation in the control experiment. Figure 10 shows the evolutions of the RMW (Figs. 10a,c) and the corresponding mixing forcing (S15_Diff, S15_Ff; Figs. 10b,d). As expected, the increased horizontal and vertical mixing slowed down the RMW contraction in the rapid RMW contraction period (Figs. 10b,d) and resulted in a larger steady-state RMW (Figs. 10a,c), which are consistent with Bryan (2012) and Bu et al. (2017). Therefore, our results indicate that in addition to the sharpness of tangential wind as identified by S15, eddy and/or subgrid-scale mixing also play an important role in slowing down and finally stopping the RMW contraction in TCs.

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(a) Time series of RMW at 250 m from CM1 for control experiment (CTL; black solid) and sensitivity experiments with the horizontal mixing length doubled (2000; red dotted) and halved (500; yellow dashed). (b) As in (a), but showing S15_Diff. (c) As in (a), but with the asymptotic vertical mixing length doubled (100; red dotted) and halved (25; yellow dashed). (d) As in (c), but showing S15_Ff.

Citation: Journal of the Atmospheric Sciences 76, 10; 10.1175/JAS-D-19-0076.1

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