1. Introduction
Tropical cyclones (TCs) often have several distinct concentric eyewalls (CEs). Some TCs with CEs undergo an eyewall replacement cycle (ERC), a phenomenon whereby the outer eyewall contracts after secondary eyewall formation (SEF) and the inner eyewall gradually dissipates. TCs undergoing an ERC exhibit rapid intensity changes. On the basis of aircraft observations, Black and Willoughby (1992), reported that Hurricane Gilbert (1988) had two clear CEs with peaks in the tangential wind field in both eyewalls, and then the maximum wind peak shifted from the inner eyewall to the outer eyewall, and the radius of maximum wind increased during the ERC. During an ERC, the rapid changes that occur in a TC intensity and wind field influence the accuracy of TC prediction. However, an ERC does not always occur after SEF. Yang et al. (2013, 2014), who used satellite data to study the characteristics of typhoons with CEs in the western North Pacific from 1997 to 2011, indicated that 23% of typhoons with CEs did not undergo an ERC; moreover, the intensity of these typhoons was higher than that of typhoons that experienced an ERC. Therefore, it is important for more accurate prediction of TC intensity changes to understand not only the mechanisms of SEF and ERCs but also the mechanism by which long-lived CEs are maintained by some TCs.
Proposed mechanisms of SEF include purely barotropic vorticity dynamics (Kuo et al. 2004, 2008), β-skirt axisymmetrization (Terwey and Montgomery 2008), a supergradient wind effect in the planetary boundary layer (PBL) (Huang et al. 2012; Abarca and Montgomery 2013), persistent rainband convection (e.g., Wang 2009; Judt and Chen 2010; Moon and Nolan 2010; Rozoff et al. 2012), and an unbalanced eddy process as a spinup mechanism of the outer eyewall within the PBL in particular (Wang et al. 2016). Studies of the ERC mechanisms have focused mainly on the dissipation of the inner eyewall. Rozoff et al. (2008) studied the analytical profiles of diabatic heating and tangential winds using a balanced vortex model and indicated that diabatic heating in the outer eyewall region statically stabilizes the inner eyewall region, allowing updrafts associated with the inner eyewall to gradually weaken. This result suggests that the inner eyewall does not decay as a direct result of subsidence associated with the outer eyewall; instead, the mechanism is indirect warming due to diabatic heating in the outer eyewall. On the basis of radar observations of Hurricane Rita (2005), which had clear CEs, Houze et al. (2007) reported that, after the outer eyewall forms, it chokes the inner eyewall by blocking the entropy supply to the inner eyewall. Zhou and Wang (2011) used a three-dimensional, nonhydrostatic, full-physics model and showed that the high entropy supply to the inner eyewall is cut off in the PBL below the outer eyewall. They also reported that more time is required for the ERC when the radius at which the outer eyewall forms is larger. This finding suggests that the time required for the ERC is determined by the radius of the outer eyewall and that the contraction of the outer eyewall controls the dissipation of the inner eyewall. Kepert (2013) showed by using boundary layer models and an imposed gradient wind field that a small perturbation in vorticity outside of the primary radius of maximum wind results in a relatively strong updraft, which, through feedbacks between the updraft, convection, and vorticity, can lead to formation and intensification of a secondary eyewall. Kepert and Nolan (2014) found by using boundary layer models and the gradient wind field based on a numerical simulation that the vertical velocity field can largely be reproduced as the frictional response to the gradient wind field.
The maintenance mechanism of long-lived CEs, however, is not well studied. Yang et al. (2013) classified typhoons with CEs into three types: 1) an ERC type, in which the inner eyewall dissipates less than 20 h after SEF; 2) a no replacement cycle (NRC) type, in which the outer eyewall dissipates less than 20 h after SEF; and 3) a CE maintenance (CEM) type, in which CEs are maintained for more than 20 h. They reported that the ERC, NRC, and CEM types comprise 53%, 24%, and 23%, respectively, of all typhoons with CEs. Although the CEM type is least common among typhoons with CEs, the intensity of CEM typhoons is higher than that of ERC and NRC typhoons (see Yang et al. 2013, their Fig. 4). Thus, the type should be taken into account for more accurate predictions of TC intensity changes. Yang et al. (2013) proposed that barotropic stability of core and ring vortices (Kossin et al. 2000) is a possible mechanism of CE maintenance. Kossin et al. (2000) performed a stability analysis with a nondivergent barotropic model and proposed that outer eyewall contraction and inner eyewall dissipation occur as a result of barotropic instability caused by the interaction of a core vortex (i.e., inner eyewall) with a ring vortex (i.e., outer eyewall). Indeed, Zhou and Wang (2011) indicated that barotropic instability between the inner and outer eyewalls acted during the ERC. Yang et al. (2013) proposed that CEs of CEM typhoons can be maintained for a long time if barotropic instability does not arise between the CEs (i.e., the core and ring vortices) after SEF. However, the model proposed by Kossin et al. (2000) does not include any diabatic heating or variations with height of the dynamic and thermodynamic fields. In real TCs, temporal variations of the storm are dominated by processes, such as acceleration of tangential winds due to diabatic heating (e.g., Shapiro and Willoughby 1982) and strong inflow in the PBL (e.g., Huang et al. 2012). It is difficult, however, to obtain information about such processes from satellite and radar observations. To investigate the maintenance mechanism of long-lived CEs, therefore, it is necessary to use a three-dimensional full-physics atmospheric model to perform numerical simulations of real TCs with long-lived CE.
Typhoon Bolaven (2012) is one example of a typhoon with long-lived CEs. Because the CEs were maintained for more than one day, Bolaven was a CEM-type typhoon (Yang et al. 2013). The operational radars of the Japan Meteorological Agency (JMA 2002) observed Bolaven and its CEs as the typhoon passed over the Okinawa islands. The purpose of the present study is to propose a possible maintenance mechanism of long-lived CEs through diagnostic analyses of a numerical simulation of Typhoon Bolaven (2012) with a three-dimensional nonhydrostatic full-physics model. To examine the maintenance mechanism, we focus on 1) maintenance of the inner eyewall and 2) contraction of the outer eyewall. We investigate the first by an entropy budget and backward trajectory analysis of the inner eyewall and the second by diagnosis of the potential vorticity (PV) budget. In section 2, the configuration of our numerical experiment is documented. Section 3 is an overview of Bolaven based on JMA’s operational observations. Section 4 is an overview of the simulated Bolaven and its CEs. We discuss the maintenance of the inner eyewall in section 5 and contraction of the outer eyewall in section 6. Finally, we present our conclusions regarding the maintenance mechanism of the long-lived CEs associated with the simulated Bolaven in section 7.
2. Numerical model, experimental design, and analysis method
The Typhoon Bolaven numerical experiment was conducted with the Cloud Resolving Storm Simulator (CReSS 3.4.2), which is a three-dimensional, regional, compressible nonhydrostatic model (Tsuboki and Sakakibara 2002). This numerical model uses a Cartesian horizontal coordinate system with a Lambert conical projection and a terrain-following coordinate with stretching. It solves 15 prognostic variables: three-dimensional wind velocities, pressure perturbation, potential temperature perturbation, turbulent kinetic energy (TKE), mixing ratios of water vapor, cloud water, rain, cloud ice, snow, and graupel, and number densities of cloud ice, snow, and graupel. The CReSS model has been used to study many aspects of TCs (e.g., Akter and Tsuboki 2012; Wang et al. 2012; Tsuboki et al. 2015).
The cloud physics scheme was an explicit bulk cold rain scheme, which is a double moment for ice, snow, and graupel, based on Murakami (1990), Ikawa and Saito (1991), and Murakami et al. (1994). Cumulus parameterization was not utilized in this model. Subgrid-scale turbulent mixing was parameterized using 1.5-order closure with TKE prediction (Deardorff 1980). Momentum and energy fluxes at the surface were formulated according to Kondo (1975) for ocean and Louis et al. (1981) for land. No atmospheric radiation scheme was included because the sensitivity of the atmospheric and cloud radiation processes on the intensity and track was not significant when a simulation with radiation for only domain 2 was performed as a sensitivity test.
To simulate a realistic CE structure, the horizontal grid spacing of a numerical model must be less than 2 km (e.g., Wang et al. 2013). In general, to save numerical resources, two-way nesting methods are used in a typhoon simulation with high resolution. Use of the two-way nesting can change an environmental field in which the storm is embedded because of interactive communication of the two-way nesting between the fine-resolution and coarse-resolution domains. Two-way nesting is not supported in our model, and a one-way nesting method was used in our simulation with three computational domains (Fig. 1). Note that the influence of the storm-induced circulation on the environmental field cannot be estimated in our numerical simulation. The area of the outermost domain was 2140 km × 2390 km with a horizontal grid spacing of 5 km (domain 1), the middle domain was 1700 km × 1800 km with a 2.5-km resolution (domain 2), and the finest domain was 1200 km × 1500 km with a 1-km resolution (domain 3). The detailed configuration of the numerical model in each domain is summarized in Table 1. There were 45 vertical levels, the lowest grid interval was 200 m, and the upper boundary condition was a rigid lid. To prevent reflection of vertically propagating waves, a sponge layer was applied from a height of 17 km to the top of each calculation domain. The model tops of domains 1, 2, and 3 were at 25, 22.5, and 20 km, respectively. The initialization times of domains 1, 2, and 3 were 0000 UTC 22 August, 0600 UTC 23 August, and 1200 UTC 24 August 2012, respectively. The ending time of integration of all experiments was 0600 UTC 27 August 2012. The sea surface temperature (SST) was calculated by using a one-dimensional, vertical heat diffusion equation (Segami et al. 1989). The SST calculation did not take account of cooling due to vertical mixing or Ekman upwelling processes (e.g., Yablonsky and Ginis 2009), although in strong typhoons, SST cooling due to these processes can be large (Lin et al. 2008). Typhoon Bolaven was a category 3 typhoon, and the magnitude of SST cooling during its passage was about 1 K, based on the JMA’s Merged Satellite and In Situ Data Global Daily SST (MGDSST) dataset (JMA 1996). Thus, we considered SST cooling due to mixing and upwelling to be small in this case. For the initial SSTs, the MGDSST dataset, which has a 0.25° resolution, was used. The lateral boundary conditions were wave radiating with constant phase velocity (e.g., Klemp and Wilhelmson 1978). In the experiment, 6-hourly Global Spectral Model (GSM; JMA 2007) output data (horizontal resolution, 0.5° × 0.5°) provided by JMA (2002) were used for the initial atmospheric and boundary conditions of domain 1 (Davies 1983). Shuttle Radar Topography Mission data with a horizontal resolution of 30 s from the U.S. National Aeronautics and Space Administration were used for the terrain (U.S. Geological Survey 2005). No data assimilation was performed in our experiment. The best-track data were used for comparing the simulated and observed tracks of Bolaven.
Model domains and Typhoon Bolaven’s track. Domains 1, 2, and 3 are the areas enclosed by the solid line, the short-dashed line, and the long-dashed line, respectively. The simulated track in domain 3 is shown by dots, and the observed track is shown by cross marks. Open circles and squares show the location of Bolaven along each track at the initial time of 1200 UTC 24 Aug and the end time of 0600 UTC 27 Aug 2012 in domain 3.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0236.1
Specifications of the numerical experiment in each calculation domain. The variable 
3. Overview of Typhoon Bolaven
Typhoon Bolaven formed in the western North Pacific (14.1°N, 142.1°E) on 19 August 2012. It slowly moved northwestward and passed over Okinawa Island on 26 August 2012 (Fig. 1). Bolaven made landfall on the coast of North Korea on 28 August 2012 and changed to an extratropical cyclone on 29 August 2012. The lifetime maximum wind speed and minimum central pressure of Bolaven of 100 kt (1 kt = 0.51 m s−1) and 910 hPa, respectively (i.e., category 3), were estimated at 1200 UTC 25 August 2012, before the TC passed over the Okinawa Islands. The estimation is based on the JMA best-track data (i.e., the maximum wind speed is the 10-min average value). On the other hand, estimation of the maximum wind speed and minimum central pressure based on the JTWC best-track data is 125 kt (1-min average) and 929 hPa at 1800 UTC 24 August 2012, respectively.
To check the eyewall structure observed in Bolaven, microwave imagery provided by the Naval Research Laboratory (NRL; https://www.nrlmry.navy.mil/TC.html) Marine Meteorology Division in Monterey, California (Hawkins et al. 2001), is shown in Fig. 2. As shown in Fig. 2b, Typhoon Bolaven qualitatively had triple eyewalls (McNoldy 2004; Zhao et al. 2016), although the innermost eyewall was very small and enclosed a pinhole eye (e.g., McNoldy 2004). Yang et al. (2013) developed an objective method to detect CEs of typhoons using microwave imagery, and Abarca et al. (2014) modified the method so that it could be used for the detection of triple eyewalls. These methods were used to examine the maintenance period of Bolaven’s triple eyewalls. In a typhoon with triple eyewalls, formation of the secondary eyewall is defined by Abarca et al. (2014) as the secondary minimum (≤230 K) of the brightness temperature, but its dissipation is not defined, unlike in the case of a typhoon with double eyewall. Thus, in the present study, the dissipation of the secondary eyewall was defined as the dissipation of the secondary minimum of the brightness temperature, similar to Yang et al. (2013). As shown in the right panels of Fig. 2, the triple-eyewall structure can also be detected quantitatively. In particular, the triple eyewalls were clearly formed at 1100 UTC 25 August (Figs. 2a,b). They continued to be maintained at 0800 UTC 26 August (Fig. 2c), but by 2000 UTC 26 August they had collapsed (Fig. 2d). Thus, the triple eyewalls were maintained for at least 20 h. Thus, in the classification of Yang et al. (2013), Bolaven was a CEM typhoon. The secondary eyewall had significant asymmetry, however, and the asymmetry had a linkage with the tertiary eyewall (left panel of Fig. 2c).
(left) Color-enhanced microwave (89-GHz band) CE imageries of Typhoon Bolaven provided by the NRL website. (right) The averaged brightness temperature 
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0236.1
Radar observations made as Bolaven passed through the Okinawa region revealed a multiple-ring shaped precipitation pattern (Fig. 3), which in this study is interpreted as CEs. The horizontal resolution of the JMA radar observation network is 1 km, and its temporal resolution is 10 min. Bolaven’s CEs had already formed before the typhoon came into the JMA radar range, so the SEF was not observed by the radar (Fig. 3a). Consistent with the microwave imagery in Fig. 2, the JMA radar observations showed that Bolaven had three eyewalls (Fig. 3b). The eyewall distribution of precipitation associated with Bolaven persisted until 1800 UTC 26 August (Fig. 3c), except for asymmetry of the primary and secondary eyewalls. Although the primary and secondary precipitation corresponding to the eyewalls have the asymmetry, the triple eyewall structure does not collapse. It is consistent with the microwave imagery (Fig. 2d). In addition, the outermost eyewall and the inner or middle eyewall still remain at 0700 UTC 27 August based on satellite observations (not shown). Thus, the JMA radar observations showed that the CEs of Bolaven were maintained for at least one day.
Distribution of precipitation intensity (color scale; mm h−1) estimated from radar reflectivity, observed by the JMA radar network in the Okinawa region at (a) 1800 UTC 25 Aug, (b) 0600 UTC 26 Aug, and (c) 1800 UTC 26 Aug 2012. The Okinawa Main Island is located in the center of each panel.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0236.1
Precipitation intensity was azimuthally averaged at each time from 
Time–radius cross section of the azimuthal mean of precipitation intensity (color scale; mm h−1) observed by the JMA radars from 
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0236.1
The three-dimensional structure of Bolaven’s observed CEs was revealed by Constant Altitude Plan Position Indicator radar reflectivity data obtained by the JMA radar on Okinawa Main Island. At 
Latitude (from south to north) and vertical cross section of radar reflectivity (color scale; dBZ), observed by the JMA radar on Okinawa Main Island, through the center of Bolaven at 
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0236.1
4. Results
a. Track and intensity of simulated Typhoon Bolaven
In this paper, only the results of the domain 3 simulation are presented. The time length of integration is 66 h. The simulated track (Fig. 1) is estimated by the method of Braun (2002), which calculates the azimuthal variance of sea level pressure for 50 radii between 1 and 50 km at each grid point within a radius of 65 km from the minimum sea level pressure grid and searches for the point of minimum variance. This method is often used to determine TC centers in simulations performed with a high-resolution nonhydrostatic model (e.g., Mashiko 2005). The root-mean-square error of the simulated track to the observed track was 168 km. In Fig. 1, the western bias of simulated Bolaven (i.e., the error in the initial location on domain 3) was mainly influenced by the error of the track arising from the simulations of domain 1 and domain 2 (not shown). In the domain 1 simulation, another typhoon, Tembin, was present over the sea to the east of Taiwan, and the track of simulated Bolaven in domain 1 was influenced by the wind field of typhoon Tembin through vortex–vortex interaction (e.g., Fujiwhara effect). However, the simulated track of Bolaven was nearly parallel to the observed track (Fig. 1). The time series of central pressure of the JMA best track and simulated Bolaven (Fig. 6a) showed that the simulated central pressure was 926 hPa when the minimum central pressure of 910 hPa was estimated on the basis of satellite observations. For the time series of wind speed, the simulated wind speed was about 40 m s−1 during the maximum wind speed of 47.5 m s−1 estimated by satellites. The intensification rates of the simulated central pressure and wind speed are different from the estimated pressure and wind speed in the best track during 
Time series of (a) central pressure and (b) maximum wind speed of Bolaven in the best-track data of the JMA (red dots) and in the simulation by the CReSS model (black lines). The simulation time 
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0236.1
b. Distribution and lifetime of precipitation
At 
Distribution of precipitation intensity (color scale; mm h−1) and sea level pressure (contour; hPa) simulated by the CReSS model at Ts = (a) 15, (b) 24, and (c) 48 h. Black stars denote the center of Bolaven.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0236.1
The axisymmetric distribution of the simulated CEs and its time variation can be shown by a radius–time cross section of the azimuthally averaged precipitation intensity (Fig. 8). Two peaks of intense precipitation were simulated: one between radii of 40 and 60 km and the other between 100 and 160 km. The width of the simulated moat was about 40 km. The simulated inner eyewall was located at mostly the same position as the observed middle eyewall, and the simulated outer eyewall was at the same position as the observed outermost eyewall. In the numerical experiment, the precipitation peak associated with the innermost eyewall of observed Bolaven was not simulated. However, the two simulated precipitation peaks were maintained from 
Time series of the azimuthal average of precipitation intensity (color scale; mm h−1) simulated by the CReSS model from 
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0236.1
The radius of the simulated outer eyewall gradually decreased from 
c. Vertical structure of the CEs
To compare the structure of radar reflectivity between observed and simulated Bolaven, we estimate the simulated radar reflectivity by the method of Murakami (1990) from the simulated mixing ratios of liquid and ice hydrometeors. At 
Vertical cross section of relative humidity (contours; %) and simulated radar reflectivity (color scale; dBZ) from south to north through the center of simulated Bolaven at 
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0236.1
To determine whether the dynamical fields of the simulated CEs were consistent with the findings of previous numerical studies, we examined the azimuthally averaged fields of the three-dimensional wind in the simulated Bolaven at 
Azimuthal-mean wind field structure at 
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0236.1
5. Maintenance of the simulated inner eyewall
Some studies have proposed a possible mechanism by which the moist entropy supply to the inner eyewall is intercepted by the outer eyewall leading to the dissipation of the inner eyewall after the SEF (e.g., Houze et al. 2007; Rozoff et al. 2008). In a numerical experiment performed with the Weather Research and Forecasting Model, Zhou and Wang (2011) found that dissipation of the inner eyewall during an ERC was caused by a low-entropy flow in the inner eyewall. At least, these studies suggest that a decrease of moist entropy in the inner eyewall is important for the dissipation of the inner eyewall. If such a decrease is essential for the dissipation of the inner eyewall, in the case of the long-lived CEs the entropy supply to the inner eyewall will be sufficient after SEF. Although different dissipation mechanisms have been proposed by other studies (e.g., Kossin et al. 2000; Rozoff et al. 2008; Kepert 2013), we examine only the mechanism of decreasing moist entropy in the inner eyewall. To investigate the maintenance mechanism of the inner eyewall in simulated Bolaven after the SEF, we performed an equivalent potential temperature budget analysis of the inner eyewall. Then, to investigate the path of the moist air mass carrying sufficient entropy to maintain the inner eyewall of the long-lived CEs, we carried out a backward trajectory analysis.
a. Equivalent potential temperature budget analysis
The accuracy of the diagnosed budget is shown by comparing the diagnosed total budget (solid black line in Fig. 11) with the actual tendency (dashed black line in Fig. 11). Before the SEF (from 
Time series of each term on the right-hand side of Eq. (3) and their sum (TOTAL): ADV; LHSH; and dPTdt, the net tendency of 
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0236.1
The 
We caution that the budget analysis is diagnosed for only the long-lived CEs in the simulation, and the result cannot be applied to short-lived CEs (e.g., Zhou and Wang 2011) directly. In other words, we cannot conclude that, for the short-lived CEs, 
b. Trajectory analysis
The 
Result of the backward trajectory analysis. Bold vectors denote four paths in total parcels (350) at each time (
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0236.1
Parcels arriving at a radius of 60 km at 
Results of backward trajectory analysis from Ts = (a)–(d) 36 and (e)–(h) 46 h. The black dots denote air parcels passing through the radius of 120 km (black bold circles) below the height of 1 km: that is, within the PBL in the outer eyewall. The color scale shows precipitation intensity (mm h−1). Times are Ts = (a) 36 h, (b) 34 h 20 min, (c) 30 h 40 min, (d) 24 h, (e) 46 h, (f) 44 h, (g) 42 h 30 min, and (h) 34 h.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0236.1
In contrast, parcels arriving at a radius of 60 km at 
Horizontal distribution of precipitation intensity (mm h−1). (a),(c),(e) The simulation, and (b),(d),(f) the radar observations. Stars denote the center of the simulated typhoon.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0236.1
In the first region (i.e., the weak precipitation region), the net buoyancy2 in the PBL was weaker than that in the other regions of the outer eyewall (not shown), and the air mass (parcel group) experienced weak buoyancy when passing through this region. Therefore, it was difficult for the outer eyewall to intercept the moist air mass in the first region. In the second region (i.e., the strong precipitation region), the air mass experienced stronger net buoyancy by comparison, but inflow in the PBL below that region was also much stronger than in the first region (not shown). If the inflow is sufficiently strong in the PBL below the outer eyewall, the upward forcing that the parcels experience during passing in the PBL is insufficient to lift the parcels from the PBL to the free atmosphere. As a result, the parcels can subsequently pass through the second region and arrive at the inner eyewall. During PS2, the entropy supply to the inner eyewall was therefore controlled mainly by the nonaxisymmetric dynamic fields of the outer eyewall.
6. Contraction of the simulated outer eyewall
Figure 15 (
Radius–height cross sections of the major terms of the PV budget [Eq. (6)] at 
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0236.1
As in Fig. 15, but at 
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0236.1
In radius–height cross sections at 
Radius–height cross sections of the sum of axisymmetric terms in Eq. (6) at 
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0236.1
Abarca and Montgomery (2015), however, indicated that strong inflow associated with the eyewall in the PBL plays an important role in the contraction of the eyewall. The strong inflow is not estimated in a diagnosis obtained with the Sawyer–Eliassen equation, which is based on a gradient wind balance vortex. According to Abarca and Montgomery (2015), the maximum speed of the strong inflow associated with the eyewall within the PBL exceeds about 28 m s−1 in their numerical simulation. Based on the Sawyer–Eliassen equation, the maximum speed of the PBL inflow diagnosed from their model output is only 16 m s−1. In their simulation, the unbalanced inflow associated with the eyewall is very strong. However, in our simulation the unbalanced inflow is very small. Figure 18 shows transverse circulation diagnosed by the model output of tangential wind, diabatic heating, and frictional forcing at 
Radius–height cross section of (a) radial wind (m s−1) and (b) vertical wind (m s−1) diagnosed by the Sawyer–Eliassen equation at 
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0236.1
7. Summary and conclusions
To investigate a possible maintenance mechanism of long-lived CEs associated with a typhoon, a high-resolution numerical experiment for Typhoon Bolaven (2012) was performed with the CReSS model, which is a three-dimensional, nonhydrostatic, full-physics model. The validation of the numerical experiment using observed data showed that the typhoon track, intensity, and maintenance of CEs were reasonably simulated, except for the observed innermost eyewall. The maintenance period could be split into two phases: one of gradual contraction (PS1; about 16 h after SEF) and one of mostly steady maintenance (PS2; about 8 h after PS1).
The CE maintenance mechanism is explained by two features: 1) the lack of dissipation of the inner eyewall and 2) the constancy of the large radius of the outer eyewall. We have summarized our results in a schematic diagram (Fig. 19). With regard to feature 1, to examine the entropy supply to the inner eyewall, we carried out a 
A conceptual model of the maintenance mechanism of the long-lived CEs of simulated Bolaven. (a) Height z and radius r cross section, and (b) horizontal (plan) view. In (a) solid vectors show the axisymmetric flows. The shaded regions show the axisymmetric terms of Eq. (6) during PS1 and PS2, and the region enclosed by the red line shows the contribution to contraction of the outer PV peak associated with ASYMM during PS2. Solid and dashed lines, which bound these regions, denote positive and negative contributions to contraction of the outer PV peak, respectively. The solid straight line shows the axis of the outer PV peak. In (b), regions of precipitation are shaded. Solid and dashed arrows indicate the movement of moist air masses associated with the nonaxisymmetric and axisymmetric boundary layer inflows, respectively. In regions bounded by dashed lines, interception of the nonaxisymmetric flows was insufficient.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0236.1
With regard to feature 2, we investigated the axisymmetric PV budget around the outer eyewall. Positive contributions to the contraction of the outer eyewall are made mainly by the generation of PV as a result of the radial gradient of axisymmetric diabatic heating and PV advection due to axisymmetric updrafts in the outer eyewall. These terms lead to a negative radial gradient of the PV tendency on the outer PV peak and the following contraction of the outer eyewall. In contrast, the generation of PV as a result of the vertical gradient of diabatic heating and PV advection due to the axisymmetric midlevel outflow inside of the outer eyewall prevented the outer eyewall from contracting. Thus, the radial gradient 
The results of the present study reveal that the maintenance of long-lived CEs is controlled by both the axisymmetric and nonaxisymmetric structure of the outer eyewall. This finding suggests that the dissipation of the inner eyewall associated with an ERC depends not only on the radius of the SEF (Zhou and Wang 2011) but also on the structure of the outer eyewall. Moreover, long-lived CEs are maintained by the distribution of diabatic heating (i.e., PV source) in the outer eyewall.
Here, we focused on only one TC with long-lived CEs, and the proposed maintenance mechanism is a prototype for long-lived CEs. Without more numerical simulations and observations of CEM-type typhoons, it is impossible to say whether the proposed mechanism of long-lived CEs is applicable to all TCs with long-lived CEs. For example, Yang et al. (2013) noted that the characteristic structure of the moat between CEs is important for their maintenance, on the basis of the barotropic framework. Moreover, the lack of the unbalanced dynamics proposed by Abarca and Montgomery (2015) may be important for the maintenance of the outer eyewall. To understand the essential mechanism of the maintenance of long-lived CEs, idealized numerical experiments and advanced observations of long-lived CEs are needed in addition to more numerical simulations of real TCs having CEs.
We caution that in our experiment the observed innermost eyewall was not simulated, but the lack of this eyewall has small influence in our hypothesis (see appendix B). Thus, our mechanism is limited on the explanation of the maintenance between the secondary and tertiary eyewalls in the observed Bolaven. The maintenance mechanism of the observed innermost eyewall should be addressed by a future study of Typhoon Bolaven (2012). In addition, our numerical simulation was performed using the one-way nesting method. Note that the influence of the storm-induced circulation on the environmental field in which the storm is embedded is not represented in our numerical simulation. Therefore, our hypothesis cannot explain the influences of the environmental condition on long-lived CEs as reported by Yang et al. (2013). It should be investigated by another future study.
Acknowledgments
The authors thank Dr. S. Kanada and Mr. M. Kato of Nagoya University, Dr. J. D. Keppert of the Centre for Australian Weather and Climate Research, and Dr. K. Ito of the University of the Ryukyus, for their excellent suggestions and comments. The authors thank the editor, Dr. Chun-Chieh Wu, and three anonymous reviewers for their helpful and thoughtful comments and suggestions in the manuscript. Some of the results were obtained by using the K computer at the RIKEN Advanced Institute for Computational Science (Proposal hp120282). The satellite microwave images were made available by the Naval Research Laboratory Marine Meteorology Division in Monterey, California. This study was supported by the “formation of a virtual laboratory for diagnosing the Earth’s climate system (VL)” project, supported by the Ministry of Education, Culture, Sports, Science and Technology of Japan (MEXT). We used the drawing library of Dennou Common Library (DCL) developed and maintained by GFD Dennou Club (http://www.gfd-dennou.org/library/dcl/) for this study.
APPENDIX A
Derivation of the PV Budget Equation
APPENDIX B
Limitations of the Simulation
(a),(b) Distributions of analytical diabatic heating (color; K h−1) and tangential wind (contour; m s−1), and (c),(d) radius–height cross sections of radial (contour; m s−1) and vertical (color; m s−1) winds diagnosed from their distributions. (left) The case of a double eyewall, and (right) the case of a triple eyewall.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0236.1
Figures B1c and B1d show the diagnosed transverse circulation for each vortex. Note that no frictional forcings are imposed on the analytical vortices (Fig. B1) in the diagnosis of the transverse circulation, unlike in the case of the simulated typhoon (Fig. 18). Around the outer eyewall, the differences between the two vortices are within about 2 m s−1 (radial wind; Fig. B2) and 0.5 m s−1 (vertical wind) in the lower and middle troposphere. The relative differences in the maximum values of the radial and vertical flows around the outermost eyewall between the double- and triple-eyewall vortices are less than 10% (Fig. B2).
Difference of radial velocity between double and triple eyewalls. Color and contour denote the relative (%) and absolute differences (m s−1). The relative difference is normalized by the mean radial wind speed (20 m s−1) below the height of 1 km in the outermost eyewall.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0236.1
For the PV budget analysis, the low-level inflow is related to MADVR, MADVZ, MDIAR, and MDIAZ, which are major terms in the PV budget, in the outer eyewall of the simulated Bolaven. Thus, the difference in low-level inflows between the double and triple eyewalls causes fluctuation of the major terms only within 10%. In the moist entropy budget analysis, the low-level inflow directly influences the lateral moisture supply. To check the influence of the innermost eyewall on the budget, at a radius of 60 km of each vortex, the lateral mass flux (
On the basis of the qualitative discussion above, we conclude that the innermost eyewall has only a small effect on our hypothesized CE maintenance mechanism. Thus, we consider that the proposed mechanism is applicable to the outer eyewall, even though the innermost eyewall was not simulated.
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In the CReSS model, the lowest level of the vertical grid is at 100-m height. Thus, we cannot trace back to where the parcels were before they appear at the lowest level.
In this context, net buoyancy is the sum of thermal buoyancy, water loading, and dynamic pressure gradient force.
Originally the CReSS model used the Cartesian coordinate, but we applied a cylindrical coordinate system for our diagnosis. Thus, the PV budget equation is derived in cylindrical coordinates.
The maxima, widths, and slopes of the diabatic heating are estimated from the simulation result.
