## Abstract

Based on the data of a 1-yr simulation by a global nonhydrostatic model with 7-km horizontal grid spacing, the relationships among warm-core structures, eyewall slopes, and the intensities of tropical cyclones (TCs) were investigated. The results showed that stronger TCs generally have warm-core maxima at higher levels as their intensities increase. It was also found that the height of a warm-core maximum ascends (descends) as the TC intensifies (decays). To clarify how the height and amplitude of warm-core maxima are related to TC intensity, the vortex structures of TCs were investigated. By gradually introducing simplifications of the thermal wind balance, it was established that warm-core structures can be reconstructed using only the tangential wind field within the inner-core region and the ambient temperature profile. A relationship between TC intensity and eyewall slope was investigated by introducing a parameter that characterizes the shape of eyewalls and can be evaluated from satellite measurements. The authors found that the eyewall slope becomes steeper (shallower) as the TC intensity increases (decreases). Based on a balanced model, the authors proposed a relationship between TC intensity and eyewall slope. The result of the proposed model is consistent with that of the analysis using the simulation data. Furthermore, for sufficiently strong TCs, the authors found that the height of the warm-core maximum increases as the slope becomes steeper, which is consistent with previous observational studies. These results suggest that eyewall slopes can be used to diagnose the intensities and structures of TCs.

## 1. Introduction

The warm-core structure is a unique characteristic of tropical cyclones (TCs). These structures are usually characterized by the magnitude of the temperature anomaly and the height of the warm-core maximum, and they are related to the tangential wind field via the thermal wind balance relationship. Additionally, they are closely related to the central minimum sea level pressure (SLP) through the hydrostatic balance. Therefore, improved understanding of warm-core structures will enhance our knowledge of the overall intensities and structures of TCs.

According to Durden (2013), the existence of warm-core structures had been noted at least as far back as Haurwitz (1935). The mechanisms that accomplish the warming in the eye of a TC were investigated through analyses of the potential temperature budget of both a realistic and an idealized simulation (Stern and Zhang 2013; Zhang et al. 2002). However, as recently pointed out by Stern and Nolan (2012), almost all studies examining warm-core structures have compared their results to one or more of three specific early flight-level case studies (i.e., La Seur and Hawkins 1963; Hawkins and Rubsam 1968; Hawkins and Imbembo 1976). It is recognized that the magnitude of a warm core generally increases with the intensity of a TC; however, relatively little is known about what determines the height and magnitude of a warm core. Because of limitations attributable to the differences in the vertical extent of observational data and sensitivities of observational techniques, together with differences regarding the definition of temperature anomalies, the understanding of warm-core structures and comparison with numerical simulations are not straightforward.

In situations with insufficient observational data, the use of numerically simulated data can help in the investigation of the statistical behaviors of the warm cores of TCs. Recent advances in computational resources have enabled us to conduct numerical simulations for the statistical analysis of TC activities. Manganello et al. (2014) investigated future changes in the TC activity of the western North Pacific using a 16-km-mesh general circulation model. They showed that supertyphoons in the future climate tend to have a deeper warm core accompanied by an upward shift in the outflow layer via composite analysis. Kanada et al. (2013) investigated future changes in structures of extremely intense TCs using a 2-km-mesh regional model. They found that the eyewall region of extremely intense tropical cyclones tends to be relatively smaller and taller in the future climate. Yamada et al. (2010) investigated the difference in the cloud-top height associated with TCs between present and future climate conditions using a 14-km-mesh global nonhydrostatic model. They showed that the cloud-top heights of TCs tend to be taller in future climate conditions than those in present climate conditions.

One of the advantages of using datasets output from high-resolution models is that TC structures (i.e., consistent thermodynamic and wind fields) are available. This motivated us to investigate the statistical behaviors of the warm-core structure of TCs and their relationship to TC intensities using datasets from a high-resolution global numerical model. Sanabia et al. (2014) argued that the slopes of satellite brightness temperature reflectivity surfaces can be related empirically to TC intensity, and they investigated the statistical relationship between the slopes and TC intensities using satellite brightness temperature reflectivity from the *Multifunctional Transport Satellite-2* (*MTSAT-2*). It is inferred that the slopes correspond approximately to the angular momentum surfaces because of the strong inertial stability near the center of TCs. This suggests that the intensities and structures of TCs can also be inferred to some extent from the slopes of the eyewall using the thermal wind balance relationship.

The aim of the present study was to investigate and clarify the relationship between warm-core structures and the intensities of TCs based on data from a 1-yr simulation of a 7-km-mesh global nonhydrostatic model. In section 2, the model setting and the definitions of TCs and warm cores are described. In section 3, the relationship between warm-core structures and TC intensities is investigated. The time evolutions of the warm-core structures of TCs are also examined. Furthermore, vortex structures are investigated to clarify how warm-core structures are related to TC intensities, and the extent of the thermal wind balance is investigated to determine warm-core structures. In section 4, the relationships among the height of the warm-core maximum, TC intensity, and eyewall slope, which is an indicator of the inner-core structure of a TC, are investigated. In section 5, we propose a relationship between TC intensity and eyewall slope based on the balanced model used by Emanuel and Rotunno (2011). We tested the proposed model using the simulation data. Our conclusions are presented in section 6.

## 2. Data

### a. Numerical model and experimental design

We used the output from a 1-yr simulation of the 7-km-mesh Nonhydrostatic Icosahedral Atmospheric Model (NICAM; Tomita and Satoh 2004; Satoh et al. 2008, 2014). Previous studies (e.g., Fudeyasu et al. 2010; Kinter et al. 2013; Yamada and Satoh 2013; Miyamoto et al. 2014; Kodama et al. 2015; Satoh et al. 2015) have shown that NICAM can reproduce realistic life cycles of TCs in terms of their structure and climatology. Ohno and Satoh (2015) investigated the mechanisms of upper warm cores using a modified version of NICAM to be run on an *f* plane. Furthermore, NICAM can also be used to investigate the statistical behaviors of TCs in the global domain and their changes in response to global warming by performing long-term numerical simulations with high resolution (Kinter et al. 2013). Yamada and Satoh (2013) investigated the response of the ice and liquid water paths of TCs to global warming using the same dataset as that used by Yamada et al. (2010). Miyamoto et al. (2014) examined the degree of the gradient wind balance of TCs using the 7-km-mesh NICAM. More recently, Kodama et al. (2015) have analyzed the climatology and seasonal variations of TCs based on multidecadal simulations with the 14-km-mesh NICAM. Satoh et al. (2015) proposed a constraint on the future change in global frequency of TCs due to global warming based on a convective mass flux analysis using the 14-km-mesh NICAM by Kodama et al. (2015).

The model settings of the present study were similar to those used by Noda et al. (2012). A quasi-uniform horizontal grid spacing of approximately 7 km was used. The vertical grid size increased from 80 m at the lowest grid to 3 km at the model top (38 km above ground level). Cloud microphysics were computed using the NICAM Single-Moment Water 6 (NSW6) scheme (Tomita 2008), which solves for six categories of hydrometeor: water vapor, cloud water, cloud ice, rain, snow, and graupel. Turbulent closure was calculated using level 2 of the Mellor–Yamada–Nakanishi–Niino (MYNN) model (Nakanishi and Niino 2004, 2006, 2009), which parameterizes vertical mixing and contributes to both planetary boundary and free atmosphere (Noda et al. 2010). The radiation scheme used was mstrnX (Sekiguchi and Nakajima 2008), and the bulk surface flux was calculated following Louis (1979) and Uno et al. (1995). The bottom boundary over land was computed using the Minimal Advanced Treatments of Surface Interaction and RunOff (MATSIRO) model (Takata et al. 2003), while sea surface temperatures were nudged toward the Taylor-corrected (Taylor et al. 2000) HadISST1 values using a slab-ocean model. National Centers for Environmental Prediction global analysis data at 0000 UTC 1 June 2004 were used as the initial condition and the time integration was performed over 1 yr.

As mentioned by Liu et al. (1999), numerical data with similar grid spacing (6 km) could be too coarse to resolve the deep convection of an eyewall. However, the simulated hurricane in their study agreed well with the observations of Hurricane Andrew (1992). This indicates that simulations with 6- or 7-km horizontal grid spacing can reproduce the characteristics of TCs, at least in terms of their spatial scales. Therefore, we will be able to discuss the statistical results of temperature anomaly profiles of warm-core structures, at least qualitatively, using the output of this simulation. In addition, the computational demands of a 7-km grid spacing makes it possible to perform a simulation over a year using a state-of-the-art super computer (the K computer; Hasegawa et al. 2011), even though the simulation is computationally more demanding than with a conventional global circulation model. This not only enabled a statistical analysis of TCs but also avoided the use of arbitrary or artificial initial conditions for the TCs. The 6-hourly output data were used throughout the present study. The objective method used in this study to identify TCs from the output was the same as that described in Yamada et al. (2010) and Yamada and Satoh (2013), which is based on Sugi et al. (2002) and Oouchi et al. (2006). The analysis area of this study was defined as the oceans between the latitudes of 40°S and 40°N.

### b. Definition of temperature anomaly of a warm core

The temperature anomaly of a warm core is defined as the difference between the temperature profile in the TC center and a reference profile. However, several different definitions of a reference profile have been used in previous studies. In the early flight-level studies of La Seur and Hawkins (1963), Hawkins and Rubsam (1968), and Hawkins and Imbembo (1976), the mean tropical sounding of Jordan (1958) was used. In a more recent study using the Advanced Technology Microwave Sounder (ATMS), Zhu and Weng (2013) used the average temperature within 15° latitude/longitude of a storm. In modeling studies, Zhang and Chen (2012) used a domain-averaged initial temperature profile, whereas Liu et al. (1997) used a time-varying domain-averaged temperature profile.

Durden (2013) discussed the choice of the vertical coordinate in which a temperature anomaly is evaluated. He showed that the vertical profiles of a temperature anomaly calculated in height and pressure coordinates are similar, except that the anomaly tends to be somewhat larger when evaluated in pressure coordinates.

In this study, the reference profile was defined by the time-varying mean temperature averaged over a 550–650-km annulus, which is the same definition as that used by Stern and Zhang (2013). As Jordan and Jordan (1954) highlighted, in the case of a large typhoon that passed Okinawa in 1951, significant warming occurred over a region up to 6° latitude from the typhoon center. Therefore, the above choice of radii might be somewhat smaller in some situations. However, they also showed, based on a composite analysis using radiosonde observations, that the horizontal gradient of a temperature anomaly was small around the region several hundred kilometers away from a TC center, although their data were biased toward weaker storms. However, similar results were obtained in both observational studies (e.g., Zhu and Weng 2013) and studies using idealized and realistic numerical simulations (e.g., Wang 2001; Ohno and Satoh 2015; Liu et al. 1999). This indicates that the choice of radii might have only minor effects in terms of the vertical profiles of the temperature anomalies at TC centers provided the radii are sufficiently large. Therefore, we used the above definition of the environmental profile throughout this study, although it should be noted that temperature anomalies might be underestimated for large TCs.

## 3. Results

Figure 1 shows the spatial distribution of the 85 TCs generated in the simulations as well as the observed TCs [International Best Track Archive for Climate Stewardship (IBTrACS); Knapp et al. (2010)] over the same period. The color of each point indicates the central SLP of the TCs for all the output data at 6-h intervals. The total number of samples is 4728. Each of them is defined as a sample in the following analyses. Table 1 shows the number of simulated and observed TCs for each basin. Although the overall number of simulated TCs captures the observational results, several differences between the simulation and the observation can be seen. For example, the number of TCs generated in the eastern Pacific (the western Pacific) is larger (smaller) than the observations. In addition, the tracks and intensities of TCs in the northern Atlantic are different, despite the fact that the number of simulated TCs and the number of observed ones are very close. However, as the aim of the present study is to investigate the general relationship between the inner-core structures and intensities of TCs, the simulation data can be used for the analyses.

Analysis of the relationship between the minimum sea level pressure (MSLP) and the maximum 10-m wind speed (MWS) provides a way to assess intensity and the associated structure of dynamical features of the simulated TCs, as the pressure gradient affects the wind speed. Therefore, the relationship provides information about the radial distribution of the pressure, although certain hurricane processes like eyewall replacement cycle can diversify the relationship (Kossin 2015). Here, we used 17.5 m s^{−1} as the maximum wind speed threshold (Walsh et al. 2007; Satoh et al. 2012). Figure 2 shows scatterplots of both variables for simulated TCs and observations (Atkinson and Holliday 1977). From the comparison between the simulation and observation, it can be seen that MWS is somewhat overestimated relative to MSLP in the simulation, although the relationship is generally well captured. The possible cause of this overestimation is that snapshot data is used for the evaluation of MWS in this analysis.

In this section, the statistics of the temperature anomaly maxima are shown in section 3a. It is demonstrated that the height of the warm-core maximum is relatively higher for stronger TCs. Furthermore, the warm-core maximum ascends during the development stage of a TC and descends in the decaying stage. The relationship between the vortex structures and the warm-core profiles is examined in section 3b.

### a. Statistical nature of warm core

#### 1) Warm-core maximum height and intensity

Because warm-core structures are usually characterized by the height and magnitude of the temperature anomaly maximum, we focus initially on the heights of the warm-core maxima and their relationship with TC intensities. Figure 3a shows probability density functions (PDFs) of the heights of the temperature anomaly maxima as a function of minimum SLP. The bin interval of SLP is 10 hPa. Here, we focus specifically on the levels of the largest peaks of the temperature anomalies. It can be seen that the height of the peak of the PDF increases as SLP decreases for TCs whose SLPs < 980 hPa. This indicates that stronger TCs tend to have warm-core maxima at higher levels when their SLPs < 980 hPa. We tested the values of 3, 5, and 7 hPa for the bin interval in addition to 10 hPa and verified that results are similar. Note that the number of samples for the extremely strong TCs is rather small. Then the PDFs for the stronger TCs are represented by small numbers of samples (e.g., only 15 samples fell into the bin for MSLP centered on 920 hPa).

In contrast, the warm-core maximum height does not have a clear relationship with SLPs for those TCs whose SLPs > 980 hPa. Figure 3b shows PDFs for weaker TCs with SLP > 980 hPa; here, the bin size is 3 hPa. The PDF peaks near 8-km height for weaker TCs are less significant than those for stronger TCs. However, additional PDF peaks can be seen at heights around 3.5 and 6 km. The discontinuity of PDF peaks for lower heights is unclear.

Another measure of TC intensity is the maximum tangential wind speed (MWS). Figure 4a shows a PDF of MWS at 1-km height as a function of SLP. The bin interval of SLP is 10 hPa. The close relationship that can be seen between MWS and SLP is qualitatively consistent with previous studies that have addressed the relationship between minimum SLPs and maximum surface winds (e.g., Atkinson and Holliday 1977; Knaff and Zehr 2007; Satoh et al. 2012). Figure 4b is as in Fig. 3a but shows PDFs of warm-core heights as a function of MWS at 1-km height. The bin interval is 5 m s^{−1}. As in Fig. 4a, this shows that TCs with stronger MWS tend to have warm-core maxima at higher levels.

Thus far, only the height of the temperature anomaly maximum has been considered. However, this is not a clear indicator of the height of the warm core in cases with less clear peaks. Therefore, we need to consider the vertical distribution of the temperature anomaly to the height of the warm core. We define the weighted height of the warm core by the temperature anomaly at the TC center using the following formula:

where *Z* is the weighted height of a warm core, Δ*T* is the temperature anomaly, *H* is the Heaviside step function, and *z*_{b} and *z*_{t} are the lower and upper boundaries of the integration, respectively. Here, an 18-km height and ground level are used for *z*_{t} and *z*_{b}, respectively.

Figures 5a and 5b show PDFs of weighted heights *Z* as a function of SLP and MWS, respectively. The bin intervals are 10 hPa and 5 m s^{−1}, respectively. These also clearly show that *Z* becomes higher as the TC intensities become stronger. These tendencies are statistically significant at the 99.9% confidence level with Welch’s two-sided Student’s *t* test (Welch 1947). The variance of the PDF becomes larger for weaker TCs. However, the variance of the PDF of *Z* is smaller than that of the heights of the temperature maxima shown in Fig. 3b. Figure 5a shows that, even for weaker TCs whose SLP > 980 hPa, the peak of the PDF of *Z* becomes higher as the SLP becomes smaller. The difference between the variance of *Z* and that of the warm-core maximum height suggests that the peaks of the temperature anomalies of warm cores are relatively flattened and that they are sometimes obscured by temperature perturbations that are not relevant to the vortex structures in the cases of weaker TCs.

In the current simulation, we did not detect a TC whose warm-core temperature maximum was located near the tropopause height. Such a warm core at the tropopause level was observed in Hurricane Wilma (2005) and Typhoon Megi (2010) (Zhang and Chen 2012; Wang and Wang 2014). As suggested by Ohno and Satoh (2015), warm cores near the tropopause are formed by dynamical processes near the tropopause. We speculate that the vertical resolution is insufficient for the TCs to have an upper warm core near the tropopause in this study.

#### 2) Amplitude of warm core and intensity

Next, we investigate the relationship between the magnitude of the temperature anomaly maxima and SLP. Figure 6 shows that the magnitudes of the temperature anomaly maxima clearly increase as SLP decreases. This result is consistent with that of Durden (2013). This relationship would be expected based on the hydrostatic balance and, more precisely, it could be interpreted in terms of the thermal wind balance equation because the vertical wind shear near the warm-core maximum height increases as the TC intensifies.

#### 3) Time evolution

As TCs develop, the warm-core structures also change. Because the data used in this study cover the entire life cycles of TCs, the time evolutions of warm-core structures in the life cycles of TCs can be investigated. An ascent of a warm-core maximum associated with the intensification of Hurricane Daisy was reported by Simpson et al. (1998). A similar ascent can be observed in idealized studies by Ohno and Satoh (2015). In contrast, some TCs in the data of the present study experienced warm-core descent at the decaying stage. Figure 7 exemplifies the time evolution of a warm core shown by the time–height Hovmöller plot of the temperature anomaly at the center of a TC that formed at 0600 UTC 3 June 2004 in the simulation. For this TC, a warm core emerges abruptly at around 10-km height, 120 h after genesis, when the TC intensifies rapidly. It is evident that the height of the warm-core maximum descends in the decaying stage.

Figure 8a shows the two-dimensional joint frequency distribution of the weighted height *Z* and that 24 h prior to the corresponding sampling time of the developing TC samples. The colors denote the relative frequencies normalized by the total number of samples in each horizontal bin (%). The samples are defined as in a developing stage if the SLP 24 h prior to the corresponding time decreases monotonically based on a 24-h moving average of SLP. The warm-core heights are larger than those 24 h prior to the corresponding sampling times for those TCs in the region below the black line. In contrast, the warm-core heights are smaller for TCs in the region above the black line. The behavior of TCs whose warm-core height < 6-km height is unclear. However, it can be seen that the sample distribution where *Z* > 6-km height leans to the region below the black line. This suggests that the height of the warm core for the developing TCs tends to be greater than 24 h before. Because stronger TCs tend to have larger *Z* values, it can be inferred that warm-core structures tend to ascend as TCs intensify, once the TCs have reached a sufficient intensity.

Figure 8b shows a two-dimensional joint frequency distribution for decaying TCs. Decaying TCs are similarly identified as developing TCs. In contrast to the development stage, the frequency distribution leans to the region above the black line. Therefore, it is evident that the warm-core structures tend to descend as TCs decay.

### b. Determination of warm-core structures

To clarify how the heights and amplitudes of warm cores are related to TC intensity, the vortex structures of TCs are investigated. We consider how much information is required to reconstruct the warm-core structures using simulated data. Under the assumption of the ideal gas law, the thermal wind balance equation in the radius–height coordinate is

where *r* is radius, *z* is height, *T* is temperature, *υ* is tangential wind speed, *g* is gravitational acceleration, and *f* is the Coriolis parameter. Integrating both sides of Eq. (2) with respect to *r*, we obtain the temperature anomaly Δ*T* at the TC center as

where subscripts “*c*” and “*a*” denote values at the TC center and ambient values, respectively. Here, the ambient is chosen appropriately. We specify *r*_{a} = 600 km in the following calculation.

Using Eq. (4), we calculate the warm-core structure for the case of the TC exemplified in Fig. 7. Figure 9a shows the time–height Hovmöller plot of the temperature anomaly at the TC center, which can be compared with Fig. 7. It can be seen that the time evolution of the temperature anomaly at the TC center, using Eq. (4), reproduces the structure shown in Fig. 7, particularly in the middle and upper troposphere. However, as is evident from Fig. 10, which shows the difference between actual temperature anomaly and that calculated using Eq. (4), large errors appear at about 2-km height. The reason why the temperature anomaly is overestimated in the lower levels is that the assumption of the gradient wind balance breaks down because of the associated strong inflow.

We examine the statistical behaviors of the temperature anomalies using Eq. (4) in terms of the relationship between SLP and *Z*. An 18-km height is used for *z*_{t}, as in section 3a. For the lower boundary *z*_{b}, a 4-km height is used to eliminate contributions from the abovementioned spurious near-surface temperature anomalies.

Figure 11a shows a PDF of *Z* as a function of SLP. The bin interval of SLP is 10 hPa. The contributions of the temperature anomaly in the lower levels for *Z* are relatively large for weaker TCs, and the lower region is not included in Fig. 11a. This is why the relationship between SLP and *Z* in Fig. 11a looks very different from that in Fig. 5a, especially for weaker TCs with SLP > 980 hPa. Nevertheless, the results are almost the same for stronger TCs. Thus, it can be interpreted that the warm-core structures are captured well using *z*_{b} = 4 km and *z*_{t} = 18 km for stronger TCs whose SLP < 980 hPa. Figure 11b shows a PDF of *Z* as a function of SLP, as in Fig. 11a, but with the temperature anomaly calculated using Eq. (4). The value of *Z* in Fig. 11b is underestimated by about 1 km around 1000 hPa and overestimated by up to 500 m around 920 hPa, relative to *Z* in Fig. 11a. However, the relationship between *Z* and SLP is reproduced almost as in Fig. 11a. This result suggests that the actual temperature anomaly structures are captured well by Eq. (4); that is, use of tangential winds and temperature profile with the thermal wind balance.

Next, we test the impact of the outer regions of the radii of maximum wind (RMW) on the warm-core structures. We investigate whether the central temperature anomalies can be reconstructed using information from within the inner regions of the eyewall. For this purpose, the central temperature anomalies are calculated using Eq. (4), but the outer boundaries for the integration are replaced by the radius *r*_{M}(*z*) of angular momentum surfaces that intersect the RMW at 4-km height (the lower bound for the calculation of *Z*):

The time–height Hovmöller plot of Δ*T*_{in} is similar to that calculated using Eq. (4), although the central temperature anomalies are slightly weaker (Fig. 9b). Figure 11c is a PDF of *Z* as a function of SLP, but the temperature anomalies are calculated using Eq. (5). Although the weighted heights in Fig. 11c are slightly higher (up to 500 m) than those in Fig. 11a for TCs with SLP < 970 hPa, the relationship between *Z* and SLP in Fig. 11c is similar to that in Fig. 11a. This indicates that the warm-core structures can be captured well by integrating the thermal wind equation within the inner region.

Additionally, we consider how much the temperature anomaly distribution contributes to the warm core in Eq. (5). Figure 9c shows the time–height Hovmöller plot of the temperature anomalies using Eq. (5), but the temperature on the right-hand side of Eq. (5) is replaced by the ambient temperature profiles:

The results are almost the same as those obtained using Eq. (5), as shown by Fig. 9c. The relationship between *Z* and SLP in this case, shown in Fig. 11d, is also similar to that in Fig. 11a.

We also evaluate the contributions of the terms in Eq. (2) to the central temperature anomalies. In most cases, the contribution of the first term in Eq. (2) is more than one order of magnitude larger than that of the second term. Therefore, the horizontal temperature gradient term is mostly balanced with the term of the vertical gradient of tangential wind speed.

These results suggest that warm-core structures of TCs can be reconstructed using only the inner-core tangential wind profiles with ambient temperature. Therefore, if the TC inner-core tangential wind profile were available, the warm-core structure could be inferred.

## 4. Eyewall slope

The eyewall slope characterizes the inner-core structure of a TC. This is because the shapes of the inner edges of eyewalls correspond approximately to the angular momentum surfaces. Generally, the strong inertial stability near the TC center prohibits horizontal displacement and mixing across the eye–eyewall boundary. Although the slope of the eyewall has been the focus of previous studies (e.g., Shea and Gray 1973; Corbosiero et al. 2005; Stern and Nolan 2009; Sanabia et al. 2014), the relationship between the outward slope of an eyewall and the intensity of a TC has not been explained fully. In fact, an attempt to present a theoretical explanation has been made by Stern and Nolan (2009) based on the balanced model proposed by Emanuel (1986). They concluded that the slopes of RMWs are not related to the intensities of TCs. However, their conclusion was derived by neglecting the dependency of the outflow temperature *T*_{o} on the saturated moist entropy , which is dependent on pressure. Although in the original argument by Emanuel (1986), *T*_{o} is approximated by a constant value to determine TC intensity, the dependency of *T*_{o} on is retained to determine the two-dimensional structure of a steady TC. Furthermore, Emanuel and Rotunno (2011) discussed the sensitivity of the solution of a steady TC on the outflow temperature *T*_{o} and highlighted the issue when the outflow temperature *T*_{o} is assumed constant. This implies that the assumption used in Stern and Nolan (2009) is not generally acceptable.

Here, we reconsider the relationship between the eyewall slope and TC intensity by introducing a variable that characterizes eyewall shape with the dependency of *T*_{o} on . We define a nondimensional slope parameter *S* of the angular momentum surface as follows:

where *r* and *T* (*r*_{1} and *T*_{1}) are the radius and temperature, respectively, where a given angular momentum surface intersects the level where *S* is evaluated (level 1). This variable is related to the angle of *M* surface in the following manner under the assumption of the slantwise neutrality (e.g., Emanuel 1983, 1985). Assuming slantwise neutrality and thermal wind balance, Emanuel and Rotunno (2011) presented an equation for the slope of an angular momentum surface:

where *M* is the absolute angular momentum per unit mass. The choice of level 1 is arbitrary under the assumption of slantwise neutrality. Equation (8) is integrated with respect to pressure to yield

Equation (9) represents that , which is an invariant on a specific *M* surface under the assumption of slantwise neutrality can be expressed by *r* and *T* at two different levels. Substituting Eq. (9) into Eq. (7), we write Eq. (7) as

Substituting Eq. (10) into Eq. (8), the relationship between the slope of *M* surfaces in radius–pressure coordinates and *S* is obtained by

Equation (11) shows that the slope of *M* is related to radius, *S*, temperature, and lapse rate. Figure 12 shows the relationship between the angle of *M* surface relative to the horizontal and *S* under the condition where *r* = 30, 45, and 60 km, *T* = 290 K, and the lapse rate is 5 K km^{−1}. It can be seen that as *S* decreases, the angular momentum surface becomes steeper. From Eq. (11), it can be seen that *S* is related to the slope of *M* surfaces in radius–temperature coordinates:

where *r*_{c} and *T*_{c} are constants for nondimensionalization. Equation (12) shows that *S* corresponds to slope in log *r*–log *T* coordinate. A particular *M* surface will flare outward as the height increases (Stern and Nolan 2009), and *S* is a local property. However, the constancy of makes it possible to evaluate *S* using relatively coarse spatial-resolution data. Note that the assumption of the slantwise neutrality cannot be applicable for weak TCs, as can be seen from the composited radius–height structures of TCs as shown in Fig. 13. The relationship between *S* and the angle of *M* surface represented by Eq. (11) does not hold for weak TCs.

Here, the relationship between the slope of the inner edges of the eyewall and TC intensity is investigated. As the 7-km horizontal grid spacing is somewhat coarse to resolve deep convection in the eyewalls, we define the inner edge of the eyewall as an angular momentum surface that intersects the radii located 10% of RMW inward from the RMWs at the height of *z*_{0} where *S* is evaluated. This is because the radii of the maximum convective-scale updrafts are usually located several kilometers inside the RMWs (Jorgensen 1984). We chose 4-km height for *z*_{0} in order to avoid the effect of spurious near-surface temperature anomalies as discussed in section 3b. A 12-km height is used for the upper-level *z*_{1}. The shape of the angular momentum surface is affected by baroclinicity and the warm-core height <12 km in most cases, as shown in Fig. 5. Radii of specific angular momentum surfaces tend to diverge above the height of large baroclinicity. Therefore, shape of angular momentum can be analyzed using the 7-km horizontal grid spacing data by choosing such a value as *z*_{1}.

Figure 14 shows a scatterplot for the slope parameter *S* and the height weighted by the central temperature anomaly *Z*. The color indicates the central SLP of the samples. It can be seen that the warm-core height increases as the slope decreases for sufficiently strong TCs. In other words, the angular momentum surfaces become more upright (steeper) as the warm-core heights increase. This can be observed in composited radius–height structures of TCs as shown in Fig. 15. This tendency is statistically significant at the 99.9% confidence level with the Welchs two-sided Student’s *t* test (Welch 1947). As the stronger TCs generally have higher warm-core structures, the close relationship between eyewall slopes and warm-core heights is consistent with the results obtained in section 3b.

Figure 14 shows also that stronger TCs tend to have smaller values of the slope parameter. This can be seen more clearly in Fig. 16, which shows PDFs of the slope parameter. In Fig. 16, the samples are classified according to the minimum SLP with a class interval of 10 hPa. This result is consistent with previous studies that have addressed the slope of the eyewall cloud (e.g., Sanabia et al. 2014). In contrast, for weaker TCs, the slope parameter takes a variety of values. This large variability of the slope parameter *S* is mainly due to the variability of the upper-level radius *r*_{1} (not shown). This indicates that the perturbation of the tangential wind speed from the thermal wind balance is large at the upper level where the tangential wind speed is weak, as the relationship between the slope and the warm-core height is based on the thermal wind balance. However, these results suggest that the warm-core height can be inferred from the slope of eyewall cloud, especially for stronger TCs.

The resultant relationship seems to be reasonable. From the viewpoint of the hydrostatic balance, decrease of the central sea surface pressure is due to increase of temperature and decrease of pressure above the center. The contribution of decrease of pressure above warm-core maxima tends to be small, because tangential wind speed tends to be weak owing to the large baroclinicity associated with warm cores. Thus, the deeper the barotropic region of the TC vortex becomes, the more the central pressure decreases. In addition, stronger TCs have stronger maximum tangential wind speed at lower levels. As the tangential wind speed must decay in the entire depths of TCs, these indicate the existence of the strong vertical shear layer in the upper level. This is consistent with the relationship between amplitudes of warm cores and intensities of TCs shown in section 3a.

## 5. Discussion

The previous section showed how the angular momentum surfaces become steeper as TCs intensify. We propose a relationship between TC intensity and eyewall slope based on the balanced model used by Emanuel and Rotunno (2011).

Emanuel and Rotunno (2011) presented an equation of temperature for the outflow levels of TCs in angular momentum coordinates using the assumption that the Richardson number is held at a critical value Ri_{c} at some radius *r*_{t} and the assumption of slantwise convective neutrality:

By assuming a balanced state in the boundary layer, they also derived the following relationships:

where subscript “*b*” denotes the top of the boundary layer; *r*_{b} and *T*_{b} are the radius and temperature, where the angular momentum surface of *M*_{b} intersects the top of the boundary layer; *f* is the Coriolis parameter; , where is the saturation entropy of the sea surface; and *C*_{k} and *C*_{D} are dimensionless exchange coefficients for enthalpy and momentum of the classical aerodynamic flux formulas for surface fluxes, respectively.

Assuming , , and *T*_{b} are all constants, Emanuel and Rotunno (2011) obtained the radial distributions of and *T*_{o} using Eqs. (13) and (15) and the boundary conditions

and

where *T*_{t} and *r*_{m} are the ambient tropopause temperature and the radius of the maximum winds, respectively. Additionally, they derived analytical solutions that are valid in the region where *υ* ≫ *fr*, where the terms related to *f* are negligible. In this case, the maximum wind speed *V*_{m} can be given by

where *V*_{p} is a velocity scale:

where is the environmental saturation entropy (i.e., the value of at *r* = *r*_{o}).

Assuming *r*_{1} ≫ *r*_{b} when *T*_{o} = *T*_{1}, we can write *S*_{b} as

This relationship suggests that the slope parameter *S*_{t} is related directly to TC intensity. Equation (23) can be verified easily by substituting Eq. (22) into Eq. (19). In the derivation of Eq. (23), neither the effects of dissipative heating nor the pressure dependence of are included. Therefore, it is expected that TC intensities would be underestimated using Eq. (23).

Emanuel and Rotunno (2011) discussed the steady-state structure of axisymmetric TCs by assuming *T*_{t} as the ambient tropopause temperature. The assumptions used in their model can be extended to unsteady situations (i.e., the developing and decaying stage). That is, the model is applicable to the nonsteady-state structures of TCs by replacing the boundary condition with that of TCs in nonsteady states, as long as the assumptions introduced in the above derivation are satisfied.

The proposed model was tested using the numerically simulated data by focusing on the relationship between the slope parameter *S* and TC intensity. Figure 17a shows a PDF of *S*^{−1} of the angular momentum surfaces of RMWs as a function of MWS. It can be seen that *S*^{−1} increases as MWS increases. This suggests that the angular momentum surfaces become steeper as TCs intensify. The range of *S*^{−1} is 0.16–0.28 at the most probable state for each intensity. Additionally, Fig. 17a shows the increase in the change ratio of *S*^{−1} with respect to MWS. These results are consistent with the proposed relationship expressed as Eqs. (18) and (23). The relationship between *S*^{−1} and MWS for weak TCs is not clear in Fig. 17a. This is because of the large value of *S*. The value of *S* tends to decrease from about 7 to about 6 as MWS increases from 10 to 40 m s^{−1}.

Figure 17b is as in Fig. 17a but for *S*^{−1} of the inner edges of the eyewall as a function of MWS. The change ratio of *S*^{−1} of the inner edges of the eyewall with respect to MWS is slightly moderated compared with the RMW; the upper range of *S*^{−1} is up to 0.23 at the most probable states. However, the relationship between and MWS is similar to that shown in Fig. 17a, in which *S*^{−1} increases rapidly as MWS becomes stronger. This result is consistent with the previous studies that have addressed the slope of the eyewall cloud (e.g., Sanabia et al. 2014), although the definitions of the slope are different.

## 6. Summary

Based on a 1-yr simulation using a high-resolution global nonhydrostatic atmospheric model with 7-km horizontal grid spacing, the relationship between warm-core structures and TC intensities was investigated. In this study, a warm core was defined as the temperature anomaly from the reference profile defined as the time-varying mean temperature averaged in the 550–650-km annulus.

The relationship between the intensities and heights of warm-core maxima in the simulated TCs was determined based on a statistical analysis. The relationship is significant for TCs that attain a certain level of intensity. The warm-core maxima were observed mainly at heights of around 8–10 km. In contrast, the heights of the warm-core maxima showed significant scatter for weaker TCs. For weaker TCs, the warm-core maxima tended to form at particular levels (i.e., near heights of 3.5, 6, and 8 km in this case). A similar relationship was obtained even when the height of the warm-core structure was evaluated by height weighted by the central temperature anomaly. This is because the vertical profiles of temperature are not that smooth in the warm core and the weighted maximum is more appropriate for the definition of the warm-core height. Additionally, the time evolution of the heights of TC warm cores showed that they tended to be higher (lower) than those 24 h prior to the corresponding sampling time for developing (decaying) TCs. This indicates that warm-core heights tend to ascend (descend) as TCs intensify (decay).

The relationship between the warm-core structures and TC intensities was investigated based on the thermal wind balance for the vortex structure. By gradually introducing simplifications, we found that the warm-core structures could be reconstructed well using only the tangential wind field within the inner-core region with the ambient temperature profile.

We investigated the relationships among warm-core height, TC intensity, and shape of the eyewall, which is an indicator of the TC inner-core structure. Here, we introduced a nondimensional slope parameter *S* that characterizes the eyewall shape. The parameter *S* can be evaluated from the radii and temperature of the inner edge of the eyewall. As the angular momentum surface becomes steeper (shallower), *S* decreases (increases). It was determined that the warm-core height increases as the value of *S* of the inner edge of the eyewall decreases. This is because the height of the large baroclinicity is related to the shape of the angular momentum surface in cases where the angular momentum surface distribution is sufficiently smooth. We also found that stronger TCs tend to have smaller values of *S*.

We derived a relationship between the intensity of a TC and the eyewall slope based on the balanced model introduced by Emanuel and Rotunno (2011). We examined the relationship using model output and found that the results were consistent with those of the proposed model. Conceptually, the model proposed by Emanuel and Rotunno (2011) can be applicable for unsteady situations (i.e., developing and decaying stages) if outflow temperature *T*_{o} could be obtained with a sufficient accuracy. In reality, however, that is difficult because *T*_{o} should be evaluated by its nature where the tangential wind speed vanishes. The difficulty of obtaining *T*_{o} prevents the model proposed by Emanuel and Rotunno (2011) from being applied to unsteady situations. In contrast with *T*_{o}, *S* can be evaluated where the accuracy of the gradient wind balance is relatively high. Thus, such a difficulty is avoidable by using *S* instead of *T*_{o}. Therefore, the model proposed in the present study is advantageous for the estimation and diagnosis of TC intensities.

Observationally, the variables used to evaluate *S* could be obtained from the brightness temperature reflectivity derived from geostationary satellites (e.g., *Himawari-8*; Bessho et al. 2016). For application to the estimation of TC intensity and inner-core structure, refinement of the simplifying assumptions used in the proposed model and quantification of the mixing across the eye–eyewall boundary will be needed. However, the results of the present study suggest that the shape of the inner edge of the eyewall could be useful for estimating the intensity and inner-core structure of TCs.

Our conclusions are solely based on a single numerical simulation using one specific numerical model. Thus, model uncertainties may cause a quantitative difference. However, the results are interpretable and reasonable from the viewpoint of the balanced dynamics. Therefore, we expect that qualitatively similar relationships among warm-core structures, eyewall slopes, and the intensities of TCs will be obtained, even if different models or settings are used.

## Acknowledgments

This study is supported partly by Strategic Programs for Innovative Research (SPIRE) Field 3 (Projection of Planet Earth Variations for Mitigating Natural Disasters), which is promoted by the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan. The simulations were performed using the K computer at the RIKEN Advanced Institute for Computer Science (Proposals hp120279, hp120313, hp130010, hp140219, hp150213, hp150287, and hp160230). We would like to express our gratitude to Atsushi Hamada of the Atmosphere and Ocean Research Institute for the helpful comments and discussions. Almost all of the figures were produced using the Grid Analysis and Display System (GrADS).

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