## 1. Introduction

Eyewall contraction is closely related to intensification, often rapid intensification, of a tropical cyclone (TC). Therefore, understanding the dynamics of eyewall contraction is of fundamental importance for understanding the dynamics of TC intensification. However, although eyewall contraction is a common feature during the TC intensification, its dynamics has not been well understood so far. Because eyewall contraction is closely tied with the contraction of the radius of maximum wind (RMW), the dynamics of eyewall contraction is thus often studied by examining the change of the RMW in previous studies (Willoughby et al. 1982, hereafter W82; Kieu 2012, hereafter K12; Stern et al. 2015, hereafter S15). Note that although the inner core of a TC could be quite asymmetric and the RMW may vary azimuthally, the RMW is often defined using the azimuthal-mean tangential wind, which may change with height (S15).

Although no theory exists for the size of the RMW or its change, previous studies have attempted to examine key processes to the size of the RMW based on idealized high-resolution numerical simulations (e.g., Xu and Wang 2010a,b). Wang and Xu (2010) and Xu and Wang (2010a) showed that the RMW would contract more if the surface enthalpy flux outside 2–2.5 times the RMW was removed. This is because the removal of surface enthalpy flux in the outer region suppressed outer rainband activities and thus diabatic heating outside the eyewall, which otherwise would reduce low-level inflow into the eyewall and reduce eyewall contraction (cf. Xu and Wang 2010a). Xu and Wang (2010b) found that both the initial size of the RMW and the initial moisture in the lower troposphere could affect the rainband activities and thus the quasi-steady RMW of a simulated TC. They also proposed that the effects of the initial vortex size and initial moisture are coupled with each other, although the quasi-steady RMW is more sensitive to the initial vortex size. Rotunno and Bryan (2012) found that the steady-state RMW is insensitive to the vertical mixing length, but sensitive to the horizontal mixing length in their simulations using an axisymmetric TC model, while Bu et al. (2017) showed that the RMW tends to be larger with stronger vertical mixing in the boundary layer or higher sea surface temperature. Bu et al. (2017) also found that cloud-radiative forcing can increase the RMW because the cloud-radiative forcing can promote rainband activities.

Note that the mechanisms for the steady-state RMW in the mature phase and RMW contraction during the intensification stage could be different. For example, a larger surface drag coefficient *C*_{D} usually results in a faster contraction rate (e.g., Bryan 2013; Smith et al. 2014), but there was no obvious relationship between *C*_{D} and the steady-state RMW (e.g., Bryan 2012, 2013) unless *C*_{D} was rather small or zero (Kilroy et al. 2017). Similar to the steady-state RMW, the contraction rate of RMW has been shown to be also sensitive to many parameters in previous numerical studies. For instance, a faster contraction rate can occur in simulations with a smaller Coriolis parameter (Smith et al. 2015; Deng et al. 2018). These are only some qualitative results from numerical simulations since the contraction rate of the RMW has not been the main focus of these studies.

Shapiro and Willoughby (1982) proposed a mechanism to explain the contraction of the RMW in a TC based on balanced vortex dynamics. They showed that the tangential wind tendency in response to diabatic heating in the eyewall is greater inside of the RMW than at the RMW, leading not only to the intensification of a TC vortex but also to the contraction of the RMW. This has become the major dynamical mechanism used for the explanation of the contraction of the RMW or the eyewall of a TC. However, this conceptual explanation could not be used to quantitatively estimate the contraction of the RMW. W82 proposed a kinematic RMW contraction equation following a moving frame of reference of the RMW, which showed good agreement with observations for the hurricane cases diagnosed.

Some recent efforts have been devoted to quantitatively estimate the contraction rate of RMW (K12; S15). K12 derived an equation for the contraction rate of RMW based on the tangential wind equation and a kinematic equation of the RMW under some assumptions/approximations (see section 2c). K12 proposed a dependence of the contraction rate of RMW on both the radial inflow and surface friction, with the former favoring the inward penetration of angular momentum and thus the RMW contraction and the latter being responsible for the slowdown and termination of the contraction. S15 proposed a method to diagnose the contraction rate of RMW based on the geometrical definition of the RMW, which was attributed to the radial gradient of tangential wind tendency and the curvature (or sharpness) of radial profile of tangential wind at the RMW. Kieu and Zhang (2017) presented concerns with the work of S15 in terms of the lack of dynamics to the contraction rate of the RMW because the method was based purely on kinematics. S15 and Stern et al. (2017) also argued that the equation of K12 could not explain the contraction rate of RMW because of some contradictory mathematical assumptions used in the derivation of the equation (see section 2c). All of the concerns in S15 and Stern et al. (2017) for K12 were disputed by Kieu and Zhang (2017).

In this study, the dynamics of the RMW contraction is revisited based on both theoretical consideration and diagnostics of numerical simulations. We first review the existing theories and compare their contraction rates of RMW with those from idealized numerical simulations. Both axisymmetric and three-dimensional simulations are conducted to understand the dynamics of eyewall contraction. The rest of paper is organized as follows. Section 2 briefly reviews the main existing theories on eyewall contraction, including the balanced dynamics and those discussed in W82, K12, and S15. An evaluation of W82 and S15 using results from idealized simulations and an azimuthal-mean tangential wind budget is discussed in section 3, in which a three-dimensional, cloud-permitting high-resolution model is used. The axisymmetric dynamics of the RMW contraction and the roles of horizontal and vertical mixings are discussed in section 4 using a series of axisymmetric simulations. Concluding remarks are given in section 5.

## 2. A brief review of existing theories

### 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

*r*is the radial distance from the TC center and the overbar denotes azimuthal mean or the variable in an axisymmetric framework as in W82. Note that the lowercase subscript “rmw” denotes the arriving radial location of the RMW. Equation (1) indicates that the change in tangential wind

### c. K12

*z*and

*f*denote height and Coriolis parameter; the uppercases and the superscript asterisk denote those variables at the RMW exactly;

*U*and

*W*denote radial and vertical velocity at the RMW; and

_{RMW}denotes an operator, in which the derivation follows the RMW rather than the air parcel. By definition, the

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 *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,

### d. S15

## 3. Evaluation of W82 and S15

In this section, the performances of W82 and S15 in capturing the contraction rate of the RMW are evaluated using the outputs from three-dimensional high-resolution idealized numerical simulations and a tangential wind budget analysis.

### 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.

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.

### c. The azimuthal-mean tangential wind budget

*ζ*

_{a}= ∂

*υ*/∂

*r*+

*υ*/

*r*+

*f*is the absolute vertical vorticity,

*p*is air pressure,

*λ*is the azimuth, and

*α*is the specific volume of dry air. The first two terms on the rhs of Eq. (10) denote the mean horizontal (radial) advection (ADV_H) and mean vertical advection (ADV_V); followed by eddy horizontal advection (Eddy_H), eddy vertical advection (Eddy_V), perturbation pressure gradient force, turbulent vertical mixing including surface friction (Ff), and subgrid-scale horizontal diffusion (Diff), respectively. Note that our results show that the perturbation pressure gradient force is rather small, at least four orders smaller in magnitude than the leading terms in Eq. (10), and thus is omitted in the following discussions. Note also that the RMW tilts mostly outward with height after the initial adjustment. This implies the importance of boundary layer dynamics for the contraction of RMW. Therefore, the budget analysis is confined below 3 km.

To ensure a nearly residual-free budget analysis, in addition to the instantaneous values, all the 10-min-mean *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.

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.

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 *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,

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.

## 4. Axisymmetric dynamics of the RMW contraction

This section gives insights into the axisymmetric dynamics of the RMW contraction with the focus on examining the role of horizontal mixing processes in preventing the RMW contraction as implied from the three-dimensional simulations discussed in section 3.

### 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,υ} is calculated by *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 *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).

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.

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).

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.

## 5. Concluding remarks

In this study, we have revisited the dynamics of the RMW contraction in TCs based on both theoretical consideration and diagnostics of high-resolution axisymmetric and three-dimensional numerical simulations. The existing theories are first reviewed and evaluated using idealized numerical simulation results. Dynamically, the RMW contraction results from larger tangential wind tendency inside the RMW than that outside it. The balanced response of a TC vortex to eyewall heating, which shows a larger tangential wind tendency inside the RMW, is considered to be the major reason for TC eyewall contraction (Shapiro and Willoughby 1982). This concept was quantified based on the definition of the directional derivative in a moving frame of reference following the RMW of a TC (W82).

More recently, K12 and S15 developed different frameworks for RMV contraction. Although the equation of K12 could be simplified to the RMW tendency equation of W82 under some assumptions, the framework of S15 can provide a tendency equation of the RMW without any assumption/simplification. We have shown that both W82 and S15 can reproduce precisely the changing rate of RMW in idealized high-resolution numerical simulations in both three-dimensional and axisymmetric models. However, compared with that of W82, an extension of equation of S15 can be further used to provide insights into the dynamics of the RMW contraction. Based on S15, the rate of the RMW change is directly proportional to the radial gradient of local tangential wind and inversely proportional to the curvature or sharpness of tangential wind at the RMW. In addition to the increase of the sharpness, as suggested by S15, this study indicates that the decrease in the negative radial gradient of azimuthal mean tangential wind tendency also contributes to the cessation of the RMW contraction.

The azimuthal-mean tangential wind budget, based on the three-dimensional and axisymmetric idealized simulations, indicates that the mean horizontal advective forcing contributes predominantly to the RMW contraction (expansion) in the lower boundary layer (the upper boundary layer and lower troposphere) in the later rapid contraction stage. The mean vertical advective forcing is secondary but also plays a role in the lower boundary layer, particularly during the rapid contraction stage, but contributes predominantly to the RMW contraction above. Overall, the vertical mixing and surface friction often lead to the RMW expansion and the cessation of RMW mainly in the lower boundary layer. An interesting finding is that in addition to the increase of sharpness of tangential wind, with the TC intensification, the increase of radial gradient of (both resolved and parameterized) horizontal mixing also slows down the RMW contraction, mainly in the lower boundary layer, and subsequently contributes to the cessation of the RMW contraction. Note that although the conclusions here are similar to those in K12, Kieu and Zhang (2017), and Qin et al. (2018), in which the mean advection effects make a net positive while the mixing effects make a net negative impact on the RMW contraction, we refer to the radial gradient of these processes, while K12 and KZ17 referred to those processes themselves.

Finally, note that the framework of S15 assumes continuous changes of the RMW in both time and space. As a result, the method cannot be applied to understand a discontinuity or jump of the RMW, such as that prior to the formation of an eyewall structure and the concentric eyewall replacement. Note that in the diagnostic viewpoint W82 is still valid even though the RMW change is discontinuous because the directional derivative is involved. In addition, although the resolved eddy mixing prevents the RMW contraction, it may contribute positively to the initial organization of the eyewall through the eddy–mean flow interaction. Therefore, eddy processes could play different roles in different stages of a TC. This remains an issue for a future study.

## Acknowledgments

The authors thank three anonymous reviewers for their thoughtful review comments. This study has been supported in part by National Natural Science Foundation of China under Grant 41730960 and in part by NSF Grants AGS-1326524 and AGS-1834300. Y. Li and Y. Lin are supported by the National Key Research Project of China (Grant 2018YFC1507001). Y. Li is funded by China Scholarship Council (File 201806210324).

## REFERENCES

Bryan, G. H., 2012: Effects of surface exchange coefficients and turbulence length scales on the intensity and structure of numerically simulated hurricanes.

,*Mon. Wea. Rev.***140**, 1125–1143, https://doi.org/10.1175/MWR-D-11-00231.1.Bryan, G. H., 2013: Comments on ‘Sensitivity of tropical-cyclone models to the surface drag coefficient.’

,*Quart. J. Roy. Meteor. Soc.***139**, 1957–1960, https://doi.org/10.1002/qj.2066.Bryan, G. H., and J. M. Fritsch, 2002: A benchmark simulation for moist nonhydrostatic numerical model.

,*Mon. Wea. Rev.***130**, 2917–2928, https://doi.org/10.1175/1520-0493(2002)130<2917:ABSFMN>2.0.CO;2.Bryan, G. H., and R. Rotunno, 2009: Evaluation of an analytical model for the maximum intensity of tropical cyclones.

,*J. Atmos. Sci.***66**, 3042–3060, https://doi.org/10.1175/2009JAS3038.1.Bu, Y. P., R. G. Fovell, and K. L. Corbosiero, 2017: The influences of boundary layer mixing and cloud-radiative forcing on tropical cyclone size.

,*J. Atmos. Sci.***74**, 1273–1292, https://doi.org/10.1175/JAS-D-16-0231.1.Bui, H. H., R. K. Smith, M. T. Montgomery, and J. Peng, 2009: Balanced and unbalanced aspects of tropical cyclone intensification.

,*Quart. J. Roy. Meteor. Soc.***135**, 1715–1731, https://doi.org/10.1002/qj.502.Deng, L., T. Li, M. Bi, J. Liu, and M. Peng, 2018: Dependence of tropical cyclone development on Coriolis parameter: A theoretical model.

,*Dyn. Atmos. Oceans***81**, 51–62, https://doi.org/10.1016/j.dynatmoce.2017.12.001.Dudhia, J., 1989: Numerical study of convection observed during the winter monsoon experiment using a mesoscale two-dimensional model.

,*J. Atmos. Sci.***46**, 3077–3107, https://doi.org/10.1175/1520-0469(1989)046<3077:NSOCOD>2.0.CO;2.Dunion, J. P., and C. S. Marron, 2008: A reexamination of the Jordan mean tropical sounding based on awareness of the Saharan air layer: Results from 2002.

,*J. Climate***21**, 5242–5253, https://doi.org/10.1175/2008JCLI1868.1.Eliassen, A., 1951: Slow thermally of frictionally controlled meridional circulation in a circular vortex.

,*Astrophys. Nor.***5**, 19–60.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**, 430–449, https://doi.org/10.1175/2010JAS3523.1.Heng, J., and Y. Wang, 2016: Nonlinear response of a tropical cyclone vortex to prescribed eyewall heating with and without surface friction in TCM4: Implications for tropical cyclone intensification.

,*J. Atmos. Sci.***73**, 1315–1333, https://doi.org/10.1175/JAS-D-15-0164.1.Heng, J., Y. Wang, and W. Zhou, 2017: Revisiting the balanced and unbalanced aspects of tropical intensification.

,*J. Atmos. Sci.***74**, 2575–2591, https://doi.org/10.1175/JAS-D-17-0046.1.Kain, J. S., 2004: The Kain–Fritsch convective parameterization: An update.

,*J. Appl. Meteor.***43**, 170–181, https://doi.org/10.1175/1520-0450(2004)043<0170:TKCPAU>2.0.CO;2.Kieu, C. Q., 2012: An investigation into the contraction of the hurricane radius of maximum wind.

,*Meteor. Atmos. Phys.***115**, 47–56, https://doi.org/10.1007/s00703-011-0171-7.Kieu, C. Q., and D.-L. Zhang, 2017: Comments on “Revisiting the relationship between eyewall contraction and intensification.”

,*J. Atmos. Sci.***74**, 4265–4274, https://doi.org/10.1175/JAS-D-17-0011.1.Kilroy, G., M. T. Montgomery, and R. K. Smith, 2017: The role of boundary-layer friction on tropical cyclogenesis and subsequent intensification.

,*Quart. J. Roy. Meteor. Soc.***143**, 2524–2536, https://doi.org/10.1002/qj.3104.Mlawer, E. J., S. J. Taubman, P. D. Brown, M. J. Iacono, and S. A. Clough, 1997: Radiative transfer for inhomogeneous atmospheres: RRTM, a validated correlated-k model for the longwave.

,*J. Geophys. Res.***102**, 16 663–16 682, https://doi.org/10.1029/97JD00237.Morrison, H., G. Thompson, and V. Tatarskii, 2009: Impact of cloud microphysics on the development of trailing stratiform precipitation in a simulated squall line: Comparison of one- and two-moment scheme.

,*Mon. Wea. Rev.***137**, 991–1007, https://doi.org/10.1175/2008MWR2556.1.Nakanishi, M., and H. Niino, 2009: Development of an improved turbulence closure model for the atmospheric boundary layer.

,*J. Meteor. Soc. Japan***87**, 895–912, https://doi.org/10.2151/jmsj.87.895.Pendergrass, A. G., and H. E. Willoughby, 2009: Diabatically induced secondary flows in tropical cyclones. Part I: Quasi-steady forcing.

,*Mon. Wea. Rev.***137**, 805–821, https://doi.org/10.1175/2008MWR2657.1.Qin, N., D.-L. Zhang, and Y. Li, 2016: A statistical analysis of steady eyewall sizes associated with rapidly intensifying hurricanes.

,*Wea. Forecasting***31**, 737–742, https://doi.org/10.1175/WAF-D-16-0016.1.Qin, N., D.-L. Zhang, W. Miller, and C. Q. Kieu, 2018: On the rapid intensification of Hurricane Wilma (2005). Part IV: Inner-core dynamics during the steady radius of maximum wind stage.

,*Quart. J. Roy. Meteor. Soc.***144**, 2508–2523, https://doi.org/10.1002/qj.3339.Rotunno, R., and G. H. Bryan, 2012: Effects of parameterized diffusion on simulated hurricanes.

,*J. Atmos. Sci.***69**, 2284–2299, https://doi.org/10.1175/JAS-D-11-0204.1.Schubert, W. H., and J. J. Hack, 1982: Inertial stability and tropical cyclone development.

,*J. Atmos. Sci.***39**, 1687–1697, https://doi.org/10.1175/1520-0469(1982)039<1687:ISATCD>2.0.CO;2.Shapiro, L. J., and H. E. Willoughby, 1982: The response of balanced hurricanes to local sources of heat and momentum.

,*J. Atmos. Sci.***39**, 378–394, https://doi.org/10.1175/1520-0469(1982)039<0378:TROBHT>2.0.CO;2.Skamarock, W. C., and Coauthors, 2008: A description of the Advanced Research WRF version 3. NCAR Tech. Note NCAR/TN-475+STR, 113 pp., https://doi.org/10.5065/D68S4MVH.

Smith, R. K., M. T. Montgomery, and G. L. Thomsen, 2014: Sensitivity of tropical-cyclone models to the surface drag coefficient in different boundary-layer schemes.

,*Quart. J. Roy. Meteor. Soc.***140**, 792–804, https://doi.org/10.1002/qj.2057.Smith, R. K., G. Kilroy, and M. T. Montgomery, 2015: Why do model tropical cyclones intensify more rapidly at low latitudes?

,*J. Atmos. Sci.***72**, 1783–1804, https://doi.org/10.1175/JAS-D-14-0044.1.Stern, D. P., J. L. Vigh, D. S. Nolan, and F. Zhang, 2015: Revisiting the relationship between eyewall contraction and intensification.

,*J. Atmos. Sci.***72**, 1283–1306, https://doi.org/10.1175/JAS-D-14-0261.1.Stern, D. P., J. L. Vigh, D. S. Nolan, and F. Zhang, 2017: Reply to “Comments on ‘Revisiting the relationship between eyewall contraction and intensification.’”

,*J. Atmos. Sci.***74**, 4275–4286, https://doi.org/10.1175/JAS-D-17-0120.1.Tang, X., Z. Tan, J. Fang, E. B. Munsell, and F. Zhang, 2019: Impact of the diurnal radiation contrast on the contraction of radius of maximum wind during intensification of Hurricane Edouard (2014).

,*J. Atmos. Sci.***76**, 421–432, https://doi.org/10.1175/JAS-D-18-0131.1.Thompson, G., P. R. Field, R. M. Rasmussen, and W. D. Hall, 2008: Explicit forecasts of winter precipitation using an improved bulk microphysics scheme. Part II: Implementation of a new snow parameterization.

,*Mon. Wea. Rev.***136**, 5095–5115, https://doi.org/10.1175/2008MWR2387.1.Wang, H., C. Wu, and Y. Wang, 2016: Secondary eyewall formation in an idealized tropical cyclone simulation: Balanced and unbalanced dynamics.

,*J. Atmos. Sci.***73**, 3911–3930, https://doi.org/10.1175/JAS-D-15-0146.1.Wang, Y., and J. Xu, 2010: Energy production, frictional dissipation, and maximum intensity of a numerically simulated tropical cyclone.

,*J. Atmos. Sci.***67**, 97–116, https://doi.org/10.1175/2009JAS3143.1.Wang, Y., and H. Wang, 2013: The inner-core size increase of Typhoon Megi (2010) during its rapid intensification phase.

,*Trop. Cyclone Res. Rev.***2**, 65–80, https://doi.org/10.6057/2013TCRR02.01.Willoughby, H. E., 1990: Temporal changes of the primary circulation in tropical cyclones.

,*J. Atmos. Sci.***47**, 242–264, https://doi.org/10.1175/1520-0469(1990)047<0242:TCOTPC>2.0.CO;2.Willoughby, H. E., J. A. Clos, and M. G. Shoreibah, 1982: Concentric eye walls, secondary wind maxima, and the evolution of the hurricane vortex.

,*J. Atmos. Sci.***39**, 395–411, https://doi.org/10.1175/1520-0469(1982)039<0395:CEWSWM>2.0.CO;2.Xu, J., and Y. Wang, 2010a: Sensitivity of tropical cyclone inner-core size and intensity to the radial distribution of surface entropy flux.

,*J. Atmos. Sci.***67**, 1831–1852, https://doi.org/10.1175/2010JAS3387.1.Xu, J., and Y. Wang, 2010b: Sensitivity of the simulated tropical cyclone inner-core size to the initial vortex size.

,*Mon. Wea. Rev.***138**, 4135–4157, https://doi.org/10.1175/2010MWR3335.1.Zhu, Z., and P. Zhu, 2014: The role of outer rainband convection in governing the eyewall replacement cycle in numerical simulations of tropical cyclones.

,*J. Geophys. Res.***119**, 8049–8072, https://doi.org/10.1002/2014JD021899.