## Abstract

The degree of gradient wind balance was investigated in a number of tropical cyclones (TCs) simulated under realistic environments. The results of global-scale numerical simulations without cumulus parameterization were used, with a horizontal mesh size of 7 km. On average, azimuthally averaged maximum tangential velocities at 850 (925) hPa in the simulated TCs were 0.72% (1.95%) faster than gradient wind–balanced tangential velocity (GWV) during quasi-steady periods. Of the simulated TCs, 75% satisfied the gradient wind balance at the radius of maximum wind speed (RMW) at 850 and at 925 hPa to within about 4.0%. These results were qualitatively similar to those obtained during the intensification phase. In contrast, averages of the maximum and minimum deviations from the GWV, in all the azimuths at the RMW, achieved up to 40% of the maximum tangential velocity. Azimuthally averaged tangential velocities exceeded the GWV (i.e., supergradient) inside the RMW in the lower troposphere, whereas the velocities were close to or slightly slower than GWV (i.e., subgradient) in the other regions. The tangential velocities at 925 hPa were faster (slower) in the right-hand (left hand) side of the TC motion. When the tangential velocities at the RMW were supergradient, the primary circulation tended to decay rapidly in the vertical direction and slowly in the radial direction, and the eyewall updraft and the RMW were at larger radii. Statistical analyses revealed that the TC with supergradient wind at the RMW at 850 hPa was characterized by stronger intensity, larger RMW, more axisymmetric structure, and an intensity stronger than potential intensity.

## 1. Introduction

Gradient wind balance (GWB) is a balanced state between the radial pressure gradient force and the sum of Coriolis and centrifugal forces. Fluid parcels that satisfy the GWB rotate around the central axis at a velocity determined by the budget of the above three forces. The GWB is widely considered to be representative of the basic state of tropical cyclones (TCs) and can be used as a basic assumption in theoretical descriptions of TC structures (e.g., Eliassen 1951; Charney and Eliassen 1964; Ooyama 1969; Shapiro and Willoughby 1982; Schubert and Hack 1982; Emanuel 1986). Recent studies have stated the importance of the unbalanced component for determining TC intensity (e.g., Smith et al. 2008; Bui et al. 2009; Bryan and Rotunno 2009). For example, Persing and Montgomery (2003) identified the superintensity problem, in which TC intensities simulated in an axisymmetric model often exceed their maximum potential intensity (MPI) as derived by Emanuel (1989) and Emanuel (1995). Smith et al. (2008) and Bryan and Rotunno (2009) observed that the existence of supergradient wind, with a tangential velocity faster than the corresponding gradient wind–balanced tangential velocity (GWV), may explain the superintensity. As can be seen from this example, it is important to understand to what degree TCs satisfy the GWB.

Based on the composited wind field obtained by flight data,^{1 }Willoughby (1990) found that the GWB is a good approximation of the azimuthally averaged tangential velocities and the root-mean-square difference from the tangential velocities is less than 1.5 m s^{−1}. That is, according to Table 1 in the report by Willoughby (1990), the TCs satisfied the GWB within about 3%. Bell and Montgomery (2008) integrated airborne data for Hurricane Isabel (2003) and showed that the hurricane approximately satisfied the GWB in the free atmosphere. Ogura (1964) carried out a scale analysis in an axisymmetric framework, and reported that the GWB-related terms are the primary order in the radial momentum equation for mature TCs. In a numerical simulation for Hurricane Andrew (1992), Zhang et al. (2001) showed that the tangential velocities in the simulated TC satisfied the GWB to within about 10%.

In contrast, previous studies that considered tangential variation indicated that tangential velocities in the TC core are often faster than the corresponding GWV. Gray and Shea (1973), using flight data, revealed that tangential velocities in the TC core region in the lower troposphere are 25% faster than the corresponding GWV (i.e., supergradient). Using in situ observation data for wind and pressure, Mitsuta et al. (1988) found that tangential velocities inside the radius of maximum winds (RMW) in Typhoons Vera (1977) and Babe (1977) were supergradient above the surface. Kepert (2006a,b) analyzed multiple flight data for Hurricanes Georges (1998) and Mitch (1998); their observations indicated that tangential velocities were remarkably supergradient in Hurricane Mitch, whereas those in Hurricane Georges satisfied the GWB well. Similarly, using observational data, Schwendike and Kepert (2008) showed that tangential velocities in Hurricane Isabel (2003) exceeded the GWV by about 15%, whereas those in Hurricane Danielle (1998) satisfied the GWB well. Thus, the degree of GWB varies from TC to TC, and differs according to location in a TC, suggesting that the degree of GWB depends on various internal and environmental parameters of the TC.

Shapiro (1983) developed a boundary layer theory under a translating TC and found that tangential velocities inside the RMW can be supergradient in a translating TC, and the maximum horizontal wind speed is located to the right-front quadrant of the TC. Kepert (2001) developed a linear TC boundary layer theory and showed that the supergradient flow forms in the upper boundary layer as a result of inward transport of large absolute angular momentum. Kepert and Wang (2001) used a nonlinear TC boundary layer model and showed that the upward advection of radial momentum also plays a key role in producing supergradient winds, and tangential velocities in the core region can be 25% faster than the GWV. Kepert (2001) and Kepert and Wang (2001) also found that tangential velocities in the TC core region tend to be supergradient when the TC is strong (as also shown by Stern and Nolan 2011) and when the RMW is large. Smith and Vogl (2008) reported that the supergradient nature is a universal characteristic in their bulk boundary layer model. These theoretical studies indicate that the tangential velocity in the TC core can often be supergradient (i.e., the GWB approximation is violated), especially when the TC is strong and large.

However, there have been only a few observational or real data simulation studies. This is a consequence of previous studies not incorporating environmental effects such as the vertical shear of background flow and feedback processes between diabatic heating in the eyewall region and the boundary layer. Because both effects play important roles in determining the structure and intensity of TCs, investigations of TCs that incorporate these factors (i.e., studies of realistic TCs) are necessary for a deeper understanding of the GWB. The purpose of this study was to reveal the statistical features of the degree of gradient wind imbalance (hereafter GIB) in realistic TCs. In particular, we focused on the GIB distribution in the whole TC, structural differences between periods when the TC core is supergradient and subgradient, and the dependence of the GIB on environmental and internal parameters.

For investigations of realistic TCs, detailed data covering an entire TC under various environments are required. Whereas observational studies have been undertaken, as mentioned above, the number of studies is limited because of the difficulty of observing the inner cores of TCs. Hence, numerical simulations of TCs under various realistic environmental conditions would be useful. However, it is not easy to conduct the numerous simulations required with a resolution fine enough to resolve the TC core. Kinter et al. (2013) carried out a series of numerical simulations using a global nonhydrostatic model, called the Nonhydrostatic Icosahedral Atmospheric Model (NICAM; Tomita and Satoh 2004; Satoh et al. 2008). The horizontal mesh size was 7 km and the total period of simulations was 24 months. These satisfy both desired conditions: high spatial resolution to resolve the inner-core structure of the TC and simulation of a number of TCs in various environments. Satoh et al. (2012) and Kinter et al. (2013) found that TCs were reasonably simulated in a series of simulations. The simulation data are useful for investigating various aspects of TCs, and we used the data in this study. The experimental settings of Kinter et al. (2013) and the methodology to extract TCs from the numerical simulations are described in section 2. The general features of the simulated TCs are shown in section 3. The key results of this study are presented in section 4: the statistics of the GIB in the simulated TCs, the structural differences in supergradient/subgradient cases, and the relationships between the GIB and related parameters. The results obtained in the present study are discussed and compared with previous studies in section 5. Section 6 presents the conclusions of this study.

## 2. Experimental settings and methodology for analyses

### a. Global nonhydrostatic model experiments

We used the results of high-resolution, long-term numerical simulations in which a number of TCs were generated under realistic conditions. The numerical simulations were performed for summer in the Northern Hemisphere, which was defined as the period from 22 May to 1 September between 2001 and 2009 except for 2003 (Project Athena; Kinter et al. 2013). The simulations were conducted using NICAM (Tomita and Satoh 2004; Satoh et al. 2008). NICAM solves five prognostic equations in five unknowns: the density, three components of the velocity vector, and internal energy in addition to the six categories of water variables (NSW6; Tomita 2008). The mesh size was 7 km, and 40 vertical layers were used, with the top of the atmosphere specified as 40 km above the sea level, with a minimum resolution of 80 m at the bottom of the atmosphere. The turbulent mixing in the boundary layer was parameterized by a modified method of Nakanishi and Niino (2006), which was based on the model of Mellor and Yamada (Mellor and Yamada 1982; Noda et al. 2010). The microphysics scheme NSW6 (Tomita 2008), which solves the tendencies of the mixing ratios for water vapor, cloud water, rainwater, cloud ice, snow, and graupel due to the diabatic processes, was used. No cumulus parameterization was included. The effects of longwave and shortwave radiation were included by incorporating the scheme of MSTRNX (Sekiguchi and Nakajima 2008). The surface fluxes for sensible and latent heat and momentum were computed by the Louis (1979) scheme. The slab ocean layer was adopted and the sea surface temperature was nudged every 5 days toward the analysis of Reynolds et al. (2007). The initial conditions were obtained from the National Centers for Environmental Prediction Global Data Assimilation System. The first 10 days, or the days from 22 to 31 May, were viewed as a spinup period and hence only the outputs of the period between 1 June and 31 August were used for the analyses. The simulated results were saved every 3 h with a horizontal mesh size of 7 km and six pressure levels in the vertical direction (925, 850, 700, 500, 250, and 200 hPa) for the horizontal velocities and five levels (850, 700, 500, 250, and 200 hPa) for the vertical velocity, temperature, and height. The relative humidity was also saved every 6 h after horizontal smoothing with 1° resolution, while the sea level pressure used for detection of low pressure disturbances was saved every 3 h with a mesh interval of 7 km. Details regarding the experimental settings of the series of numerical simulations were described by Kinter et al. (2013).

### b. Detection of tropical cyclones and definition of the quasi-steady state

To detect TCs in the simulated results, this study utilized a tracking algorithm based upon those of Oouchi et al. (2006) and Yamada et al. (2010). Because the horizontal resolution of the numerical simulations was higher than those in previous studies using global hydrostatic models (e.g., Oouchi et al. 2006), we expected the simulated TCs to have a well-defined inner-core structure by explicitly representing mesoscale circulations. Thus, we utilized conditions similar to those of Yamada et al. (2010) with some modifications. First, a low pressure disturbance is extracted at a grid point where the sea level pressure is 3 hPa smaller than that averaged in the 3° × 3° area surrounding the grid point. Then, if the maximum horizontal velocity at 850 hPa within the 3° × 3° area is faster than 21.5 m s^{−1} at any point, where *u*_{h} and *υ*_{h} are the zonal and meridional velocities, and the vertical component of relative vorticity is greater than 0 at 850 hPa, the disturbance is detected as a candidate TC. The wind speed criterion, 21.5 m s^{−1}, is determined by the horizontal velocity for TC definition, 17.2 m s^{−1}, at a height of 10 m divided by the empirical decay factor from the 850-hPa level of 0.81 (Walsh et al. 2007). The disturbance is tracked until the above conditions are not satisfied or the disturbance passes poleward of the 45° latitude. When the duration exceeds 24 h, the disturbance is regarded as a TC. The velocity field around the pressure minimum point is then transformed into the cylindrical coordinate system at each time point. Note that the simulated TCs are analyzed only when they are over ocean grid points and at least 500 km offshore from the coastal region to remove orographic effects on the GIB. In this study, the azimuthally averaged maximum tangential velocity *υ*_{m} at 850 hPa is used as a proxy for TC intensity.

This study focuses on quasi-steady TCs that are strong enough to maintain their structures. TCs are regarded as in a quasi-steady state when they satisfy all of the following conditions:

absolute rate of change in intensity at 850 hPa is less than 0.36 m s

^{−1}h^{−1};vertical average of horizontal temperature deviation, which is the difference between the center and the tangential average at the 500-km radius, between 700 and 250 hPa greater than 3 K; and

*υ*_{m}at 850 hPa minus*υ*_{m}at 250 hPa greater than 5 m s^{−1}.

The first condition is the criterion required for the quasi-steady state. The second and third conditions remove extratropical cyclones. The present method can detect more than one steady state in a TC. To avoid counting multiple pieces of information in one TC, we averaged over all the time during steady periods. Hence, one piece of time-slice data per TC was used for statistical analyses.

## 3. Simulated TCs in the numerical simulations

The overall features, such as the tracks of the simulated TCs, extracted by the present method are similar to those of Kinter et al. (2013) and Satoh et al. (2012). Note that the detection criteria in Satoh et al. (2012) are slightly different from the present study. In total, 211 TCs were extracted during the whole simulation period. Table 1 shows the number of simulated TCs for each basin, as well as the observed number [International Best Track Archive for Climate Stewardship (IBTrACS); Knapp et al. 2010] over the same period as the present analysis. It should be noted that since the conditions for both the best track TCs and simulated TCs should be consistent with each other for comparison, the thresholds for the intensity and lifetimes were applied for the calculation. The analysis period is from the beginning of June to the end of August for 8 years. As shown in Satoh et al. (2012), the overall number of simulated TCs captures the observational results, although the number in the eastern Pacific (the Atlantic) is much larger (less) than the observation.

Figure 1 shows the radius–height cross sections of the composites of the tangential velocity *υ*, radial velocity *u*, vertical velocity *w*, GIB *υ*′, and deviations of height *z* and temperature *T* from the environment. All the quantities are tangentially averaged. The environmental values for height *z*_{env} and temperature *T*_{env} are the horizontal averages in the 5° × 5° area. The GIB, which is the key parameter of the present study, is defined as

where *r*, *λ*, *p*, and *t* denote the radius, azimuth, pressure, and time, respectively; and *υ*_{g} is the GWV determined by

where *f* represents the Coriolis parameter and *ϕ* is the geopotential height. Here, the hydrostatic balance (∂*p*/∂*z* = −*ρg* where *ρ* is the density and *g* is acceleration due to gravity) is assumed to derive Eq. (2). Thus, *υ*′ > 0 (*υ*′ < 0) means that the tangential velocity is supergradient (subgradient). In Fig. 1, *υ*, *u*, and *υ*′ are normalized by *υ*_{m} at 850 hPa at each time point, whereas *w* is normalized by the value at the RMW and *p* = 500 hPa. Hereafter, the normalized quantities are denoted with an asterisk (e.g., *υ*′/*υ*_{m} = *υ*′*). The radius is normalized by the RMW *r*_{m} at 850 hPa, whereas the vertical coordinate is written in dimensional pressure. To remove the inconsistency in the number of data points used for individual TCs resulting from differences in lifetime lengths, the composites are constructed after taking a temporal average during the quasi-steady period of each simulated TC. That is, these composite figures represent the TC structure of the ensemble average of the 211 members.

As shown in Figs. 1a–c, primary and secondary circulations are present in the simulated TCs. In particular, *υ** vertically decays more significantly above than below 500 hPa (Fig. 1a). The *w** is strong through the whole layer around the RMW, and the radius of maximum of *w** at each level slants outward (Fig. 1c). The minimum of *z* − *z*_{env} or the maximum of *T* − *T*_{env} is most significant at the center of the whole layer (Figs. 1e,f). The maximum temperature deviation is located at 250 hPa. The spatial distributions of averaged quantities are consistent with observational studies (e.g., Frank 1977; Jorgensen 1984a,b; Rogers et al. 2012).

Figure 1d shows that *υ*′* is close to 0 in the whole region, indicating that the GWB is satisfied. The sign is generally negative except for immediately inside the RMW below 850 hPa. The positive peak is located at *r* ≈ 0.8*r*_{m} and *p* = 925 hPa. Thus, tangential velocities in this region are faster than the corresponding GWV (i.e., supergradient), whereas those elsewhere are slower (subgradient). Although the sign and degree of *υ*′* differ from place to place, both the positive and negative peak values are much less than 1 (−0.11 ≤ *υ*′* ≤ 0.02).

Figure 2 shows the horizontal cross sections of the composites of *υ**, *u**, *w**, *υ*′*, *z* − *z*_{env}, and *T* − *T*_{env}. The level for *υ**, *u**, and *υ*′* is 925 hPa, whereas that for the *w** and *T* − *T*_{env} composites is 500 hPa and that for *z* − *z*_{env} is 850 hPa. The horizontal velocities are earth-relative flows. The composites are constructed by the same procedure as in Fig. 1, but setting the direction of TC motion to the upward direction of the panels. The moving speed and direction of the simulated TC are determined by the temporal difference in the center position between the output interval of 3 h.

A circular distribution is shown for *υ**, which is larger in the right-hand side of the moving direction, because the moving speed is embedded in the composites (Fig. 2a). Accordingly, the distribution of *u** at 925 hPa is highly asymmetric (Fig. 2b). The minimum peak of *u** is located to the right-hand side of the TC around the RMW, and the peak rotates anticyclonically with increasing radius. In the outer region, *u** is positive (negative) at the front (rear) of TC because of TC motion. Meanwhile, *w** has a nearly circular peak around the RMW (Fig. 2c), and there are several localized peaks in the outer region. The distribution of *z* − *z*_{env} at 925 hPa is almost circular and has a minimum around the center (Fig. 2e). Similarly, *T* − *T*_{env} at 500 hPa is also distributed in a circular region (Fig. 2f).

At 925 hPa *υ*′* is basically negative (i.e., subgradient), except for the right-front quadrant (Fig. 2d). The positive *υ*′* (i.e., supergradient wind) exists from the center to *r* = 3*r*_{m} in the right-front region, and inside the RMW in the left-front region. The maximum is located immediately inside the RMW, with a value as large as 0.2. The maximum of *υ*′* is about tenfold larger than that in the radius–height composite (Fig. 1d).

## 4. Degree of gradient wind imbalance in the simulated TCs

### a. Frequency

Figure 3 shows histograms of the normalized degree of GIB *υ*′* at the RMW at 850 and 925 hPa of all the simulated TCs. In the figure, *υ*′* is averaged tangentially and also temporally during the quasi-steady period of each simulated TC. The averages *μ* of all 211 cases at 850 and 925 hPa are 0.72 × 10^{−2} and 1.95 × 10^{−2}, respectively. Thus, the tangential velocity is on average 0.72% of *υ*_{m} faster than the GWV at 850 hPa, and it is 1.95% of *υ*_{m} faster than the GWV at 925 hPa. The variances *σ*^{2} are 1.04 × 10^{−3} at 850 hPa and 1.18 × 10^{−3} at 925 hPa. The frequency distributions of *υ*′* at the two pressure levels are well represented by the Gaussian profiles determined by *μ* and *σ*. In both cases, about 75% of all the simulated TCs exist within *μ* ± *σ*. That is, in 75% of all the simulated TCs, the maximum azimuthal mean tangential velocities deviate from the corresponding GWV within a range from −2.50% (subgradient) to +3.95% (supergradient) at 850 hPa, and a range from −1.49% to +5.39% at 925 hPa.

It is of interest to examine the degree of GIB in the intensifying TCs as well as in the quasi-steady phase. Figure 4 shows the frequency distributions of *υ*′* at 850 and 925 hPa during the intensification phase defined as a period in which the intensifying rate of TC intensity is larger than 1.73 × 10^{−4} m s^{−2} (=0.62 m s^{−1} h^{−1}) and less than 15.0 × 10^{−4} m s^{−2}. The averages at 850 and 925 hPa are 0.74 × 10^{−2} and 1.42 × 10^{−2}, respectively, and the variances are 1.18 × 10^{−3} and 1.21 × 10^{−3}, respectively. The values are quantitatively similar to those in the quasi-steady phase, and the frequency distributions are similar to those in the quasi-steady state.

Figure 5 shows the frequency distributions of the maximum and minimum degrees of GIB in all the azimuths at the RMW during the quasi-steady state. It should be noted that the GIB is normalized by the azimuthally averaged maximum tangential velocity *υ** _{m}*, not the local velocity. At both pressure levels, the distributions are similar to each other and fit the Gaussian profiles. Although this feature is also seen in the case of the azimuthally averaged GIB (Fig. 3), the averages are significantly different. The averages of the maximum GIB at the 850- and 925-hPa pressure levels are 4.00 × 10

^{−1}and 3.87 × 10

^{−1}, respectively. That is, the tangential velocity at the RMW at 850 hPa locally exceeds

*υ*

_{m}by 40%. In addition, the variances are 1.03 × 10

^{−2}and 1.31 × 10

^{−2}. Hence, approximately 75% of all the TCs have a maximum GIB in the range between +30% and +50% of the maximum tangential velocities. On the other hand, the averages of the minimum GIB are −3.80 × 10

^{−1}at 850 hPa and −3.31 × 10

^{−1}at 925 hPa. The absolute values are comparable between the local maximum and minimum GIB. However, they are markedly large compared with those of the azimuthal average.

The histograms of the maximum and minimum GIB in all the azimuths at the RMW during the intensification phase are shown in Fig. 6. The distributions are qualitatively similar to those during the quasi-steady state (Fig. 5). The averages of both maximum and minimum GIB at the two pressure levels are between ±0.3 × 10^{−1} and ±0.4 × 10^{−1} and the variances are around 1.0 × 10^{−2}.

In conclusion, from an azimuthally averaged perspective, the degree of GIB is small and the simulated TCs approximately satisfy the GWB at 850 and 925 hPa. The average of the degree of GIB exceeds the maximum tangential velocity by about 1%. However, consideration of the azimuthal variation of GIB yields significantly different results. On average, the tangential velocities at the RMW deviate locally from the GWV by up to ±40% of the maximum tangential velocity.

### b. Difference in structure between the supergradient and subgradient TCs

To understand the structural differences between when *υ* at the RMW is supergradient (*υ*′ > 0) at 850 hPa and when it is subgradient (*υ*′ < 0), the differences in the six quantities in Figs. 1 and 2 are examined. Figure 7 displays the radius–height cross sections of the differences in composites of *υ**, *u**, *w**, *υ*′*, *z* − *z*_{env}, and *T* − *T*_{env}. These composites were constructed by taking the difference between the two radius–height composites for the supergradient and subgradient cases, each of which was constructed by the procedure identical to that shown in Fig. 1.

Figure 7a shows that *υ** is slower above the 850-hPa level, whereas *υ** is faster in the lower layer in the supergradient case. The zero-difference line slants outward from the RMW at the lowest layer to *r* = 5.0*r*_{m} at *p* = 400 hPa. For the supergradient case, *u** tends to be positive around the RMW compared to the subgradient case, whereas *u** tends to be negative inside the RMW above 850 hPa and outside the RMW above 300 hPa (Fig. 7b). The difference in *w** is positively large around the RMW and 850 hPa, and the positive region slants outward with height (Fig. 7c). In contrast, a negative region is present inside the positive region around the RMW and above 500 hPa, which also slants outward. That is, *w** tends to be large (small) around and outside the RMW, whereas *w** is small (large) inside the RMW in the supergradient (subgradient) case. The difference in *z* − *z*_{env} is negative in the entire region (Fig. 7e), indicating the steeper gradient of geopotential height in the supergradient case. There is also a clear difference in *T* − *T*_{env} from the center to the RMW. Because the difference is positive, the temperature deviation or the warm-core structure is more significant in the supergradient case.

Figure 7d shows that the difference in *υ*′* is generally positive inside *r* = 2.5*r*_{m}, indicating that *υ*′* is larger in the supergradient case. The difference is largest around the RMW at 925 hPa and decays in either the vertical or radial direction. Outside *r* = 3.0*r*_{m}, the difference is negative, but only to a slight degree.

Figure 8 shows horizontal sections of the differences in composites of *υ**, *u**, *w**, *υ*′*, *z* − *z*_{env}, and *T* − *T*_{env} between the supergradient and the subgradient cases. The composites for both cases are constructed in the same manner as in Figs. 2 and 7.

The difference in *υ** at 925 hPa is generally positive, and large in the front to front right of the TC (Fig. 8a). Figure 8b shows that the difference in *u** is distributed in a wavenumber-1 field in which the difference at the front (rear) of the TC is positive (negative). The difference is more significant around the RMW than in the outer region. On the other hand, the difference in *w** at 500 hPa is not clearly dominated by either positive or negative sign (Fig. 8c). The difference in *z* − *z*_{env} is distributed almost circularly with a positive sign in the entire region (Fig. 8e). The peak of the difference is located slightly to the rear of the TC center. Similarly, the distribution of the difference in *T* − *T*_{env} is nearly circular, and the sign is positive in the whole domain (Fig. 8f). Both the height and temperature anomalies indicate that the TCs are stronger in supergradient cases.

The difference in *υ*′* at 925 hPa is positive especially in the right-hand side of the TC motion within *r* = 3*r*_{m} (Fig. 8d). The maximum difference is observed at the RMW. In contrast, the difference is negative outside *r* = 3*r*_{m}.

### c. Relationships between the degree of gradient wind imbalance and related parameters

This subsection discusses the relationships between GIB at the RMW at 850 hPa and the parameters related to TC intensity or flow structure (Figs. 9, 10, and 11). Each point in the figures represents the averaged value during the quasi-steady states of a TC (i.e., one point per one TC), and hence there are 211 points in each panel. The GIB is the azimuthally averaged value as in Fig. 3. On the one hand, the line in the figures is determined by empirical orthogonal function (EOF) analysis of the 2 × 2 correlation-coefficient matrix for two selected physical quantities. Therefore, the line corresponds to the first mode of the eigenvector. To draw the line, the estimated eigenvector is dimensionalized by the standard deviation after conducting EOF analysis. In addition, the correlation coefficients for the relationships are estimated and shown on the bottom right of each panel. The line is drawn only when the correlation coefficient is statistically significant at the 95% confidence level. In the current case in which the number of samples is 211, the minimum correlation coefficient satisfying the 95% confidence level is 0.180.

Figure 9 shows the scatterplots of the relationships between *υ*′* and *υ*_{m}, the minimum sea level pressure *p*_{sm}, *r*_{m}, and axisymmetricity. The axisymmetricity describes the degree of axisymmetric structures of a TC (Miyamoto and Takemi 2013), which is defined as

Here *χ* is a physical variable and the overbar and prime stand for the axisymmetric and asymmetric components, respectively. As in the study of Miyamoto and Takemi (2013), Ertel’s potential vorticity is used here as *χ*. To represent the axisymmetric structure of a TC, *γ* is averaged in a volume inside 2 × RMW and between the 700- and 300-hPa levels in a cylindrical coordinate.

The relationship between *υ*′* and *υ*_{m} in Fig. 9a shows that *υ*′* is large in intense TCs and the correlation coefficient is 0.546. Here *υ*_{m} is distributed within the range from *υ*_{m} = 18.1 to 73.6 m s^{−1} with an average of 41.8 m s^{−1}. Similarly, the relationship between *υ*′* and *p*_{sm} in Fig. 9b indicates that *υ*′* is negatively related to *p*_{sm} with a correlation coefficient of −0.480. With a mean value of 972.9 hPa, *p*_{sm} spreads from 920.8 to 1004 hPa. It is clear from both Figs. 9a and 9b that *υ*′ tends to be larger in strong TCs. Figure 9c indicates that as *r*_{m} becomes large, *υ*′* becomes positive and large with a correlation coefficient of 0.384. Thus, large TCs (large *r*_{m}) tend to be supergradient (*υ*′* > 0). The range of *r*_{m} is between 14.0 and 99.0 km, and the average is 35.8 km. Figure 9d shows that *υ*′* is positively correlated with *γ*. The correlation coefficient is 0.420 and the average of *γ* is 0.67. The relationship shows that *υ*′* becomes larger as the TC structure becomes more axisymmetric.

Figure 10 shows the relationships of *υ*′* with the vertical shear of the horizontal background flow |Δ_{250−700}**u**_{hbg}|, the sea surface temperature averaged within the RMW *T*_{s}, the radial velocity at *r* = *r*_{m} and *p* = 850 hPa *u*_{850}, and the deviation of *υ*_{m} from Emanuel’s MPI. Note that *υ*′ at 925 hPa rather than at 850 hPa is used for the relationship with *u*_{850}. The vertical shear of background horizontal flow, , is estimated as follows. First, the horizontal velocities are averaged in the 5° × 5° area around the TC center to obtain the background velocities (i.e., *u*_{hbg} and *υ*_{hbg}) at 250 and 700 hPa. Then, after taking vertical differences for both directions, the absolute value of the shear is calculated. The MPI is estimated by Eqs. (5) and (6) of Bister and Emanuel (2002), assuming cyclostrophic balance in the TC core. As the moisture is not included in the high-resolution outputs, the relative humidity with 1° horizontal resolution is alternatively used to calculate the MPI. In addition, only when *υ*_{m} is equal to or faster than the corresponding GWV, is the data for *υ*_{m} > MPI analyzed.

Figure 10a shows that *υ*′* is not correlated with |Δ_{250−700}**u**_{hbg}|. The correlation coefficient (0.080) is not the statistically significant. As indicated in Fig. 10b, *T*_{s} is also not correlated with *υ*′*. In Fig. 10c, there is a clear positive relation between *u*_{850} and *υ*′ at 925 hPa with a coefficient of 0.243. When *υ*′* is positive (i.e., supergradient), *u*_{850} is greater than 0, meaning that the radial motion is directed outward at 850 hPa, which is almost above the boundary layer. This confirmed the ability of the model to simulate the characteristics of fluid motion when a TC is supergradient. Once a fluid parcel moving with a tangential velocity faster than the corresponding GWV in the boundary layer ejects into the free atmosphere (frictionless layer), the centrifugal force becomes too large to balance with the radial pressure gradient force. As a result, the parcel accelerates to move outward above the frictional boundary layer.

The relationship between *υ*′* and *υ*_{m} − MPI in Fig. 10d shows that the deviation of *υ*_{m} from MPI is widely distributed from −39.2 to +24.6 m s^{−1}. The present results indicating that many TCs tend to exceed their MPI are consistent with the numerical results of Persing and Montgomery (2003), who used an axisymmetric model. This large deviation may result from miscalculation for some TCs due to the coarse vertical resolution or the smoothed relative humidity data in which the horizontal resolution is 1°. The line indicates that they are positively related, and the correlation coefficient is 0.341. Thus, when TC intensity is stronger than the corresponding MPI, *υ*′* tends to be positive. The positive relation between *υ*′* and *υ*_{m} − MPI supports the results of Bryan and Rotunno (2009) indicating that the underestimation of MPI results from the neglecting of supergradient winds in the MPI theory. The positive relation appears to be due to the following reason. In strong TCs, *υ*_{m} tends to be close to MPI (i.e., *υ*_{m} − MPI is positive or small and negative). On the other hand, *υ*_{m} in weak TCs is generally far weaker than MPI. Since the stronger TCs have larger *υ*′ (Fig. 10a), *υ*′ may increase with increasing *υ*_{m} − MPI.

Figure 11 shows how *υ*′* is related to the TC translation speed *V*_{t} and the direction of TC motion. Both *V*_{t} and the direction are determined by the temporal difference in the central position of TC.

It can be seen from Fig. 11a that *V*_{t} has a mean of 6.07 m s^{−1} and the maximum and minimum values are 14.9 and 0.290 m s^{−1}, respectively. The correlation coefficient is −0.064, which is not statistically significant. Similarly, *υ*′* has almost no correlation with TC direction (Fig. 11b). In this figure, the azimuth starts from the east and rotates in a counterclockwise manner (i.e., “0” indicates eastward motion). The correlation coefficient (−0.023) is not statistically significant at the 95% confidence level. Hence, *υ*′* does not depend on the translation speed and the direction of motion.

## 5. Discussion

### a. Comparison with previous studies

Some of the results obtained are consistent with those of previous observational, theoretical, and numerical studies. The horizontal distributions of the earth-relative horizontal flow (Figs. 2a,b) in which the peaks of *υ* and *u* lie around the right-front and right-rear regions are in accordance with the idealized studies of Kepert (2001) and Kepert and Wang (2001). The positive correlations of *υ*′ with the TC intensity and RMW shown in Figs. 9a–c are also consistent with those studies. Moreover, the presence of outward motion above the height of the maximum tangential velocity (Fig. 10c) in the supergradient case has also been observed in previous studies (Gray and Shea 1973; Kepert 2001; Kepert and Wang 2001; Schwendike and Kepert 2008).

In contrast, the horizontal distribution of *w* in Fig. 2c has a nearly circular positive peak region around the RMW. This is inconsistent with the findings of Kepert (2001) and Kepert and Wang (2001), who showed that there is a wavenumber-1 structure in which the positive (negative) lies in the right (left) side. This appears to be due to the presence of the diabatic heating in the eyewall region, interaction between the eyewall and the boundary layer, and vertical wind shear.

### b. Structure differences between the supergradient and subgradient cases

The results of the present study characterize the differences in the velocity fields between the supergradient and subgradient cases. Specifically, when *υ*_{m} is supergradient, *υ* is slower in the upper layer inside the RMW, while *υ* is faster in the lower layer outside the RMW (Fig. 7a). The peak of *u* around the RMW at 850 hPa tends to be positive in the supergradient case (Fig. 7b), indicating the presence of outflow motion (Fig. 10c). The peak of *w* in the free atmosphere is located outward (Fig. 7c). In addition, *υ* and *υ*′ are stronger especially in the front-right side (Figs. 7a,d) in the supergradient case, whereas the inflow is more significant behind the TC (Fig. 7c). On the basis of the results presented in this study and also in previous studies, the differences may be interpreted as follows.

The rapid vertical decay of *υ* in the supergradient case can be understood separately in the lower layer and in the free atmosphere. In the region immediately above the height of *υ*_{m}, the outward motion (Figs. 1b, 7b, and 10c) advects a small angular momentum from inside the RMW and decreases the positive GIB (Kepert and Wang 2001). As a result, *υ* decays vertically more rapidly in the supergradient case. On the other hand, in the middle–upper troposphere, the decay may be related to the thermodynamic structure of TCs. Because *u* is negative outside the RMW at 850 hPa (Fig. 1b), the strong outward motion can enhance the horizontal convergence at the bottom of the free atmosphere compared to the subgradient case. Accordingly, the upward region, or the eyewall, is present outside relative to those in the subgradient case (Fig. 7c). In addition, the convergence may increase the diabatic heating in the eyewall region, which would result in an enhanced warm-core structure (i.e., temperature deviation) in the upper troposphere, as shown in Fig. 7f. Thus, the result that TCs approximately satisfy both the GWB (Fig. 1d) and hydrostatic balance (i.e., thermal wind balance) in the free atmosphere suggests that the tangential velocities show more rapid vertical decay in the supergradient case owing to the significant warm-core structure. Since this is a discussion to interpret the results from available dataset, more in depth analyses with higher temporal resolution, and/or additional physical quantities such as the diabatic heating rate are needed to reveal the interactions among the quantities.

The difference in the composites of horizontal structure (Fig. 8) indicates that *υ*′ is large mainly in the right-hand side of TC motion in the supergradient case. To interpret this feature, the radial profiles of *υ*′*, *u**, *υ**, and the radial gradient of *ϕ** = *z*/*z*_{env} at the four representative azimuths at 925 hPa are plotted in Fig. 12. It is clear that *υ*′* is positive (negative) in the front right (rear left) of the TC; *u** is negative (positive) in the right-hand (left hand) side, whereas *υ* exceeds (falls below) *υ*_{m} in the right-hand (left hand) side. Although the peaks of the radial gradients of *ϕ** are not markedly different, the radial position of the steepest slope is inside (outside) in the front (rear) of the TC.

These features suggest that, in the right-hand side of TCs, the strong inflow results in large *υ* through significant transport of angular momentum from the outside and producing the supergradient wind (*υ*′ > 0). In contrast, in the left-hand side, the radial motion is weak and even directed outward inside the RMW. This tends to result in smaller *υ* than in the other azimuths and subgradient wind (*υ*′ < 0). On the other hand, the TC motion may also affect the velocity profiles in which *υ*′ is large (small) in the right-hand (left hand) side. However, the relationships in the front and rear sides of TCs are not straightforward. One possible explanation for the *υ*′ distributions in the two azimuths is the tangential advection of *υ*′*. The upwind side of the front (rear) of a TC is the right-hand (left hand) side where the tangential velocities are supergradient (subgradient). Thus, the positive (negative) advection of *υ*′ seems to result in the supergradient (subgradient) motion in the front (rear) of a TC.

### c. Possible effects of model settings on the GIB features

The results presented in the present study show that the GIB in the TC core depends on the TC structure as well as TC intensity (Figs. 7, 8, and 9), both of which depend on the convective activity in the eyewall. This suggests that the GIB is sensitive to whether a cumulus parameterization is used and the type of cumulus parameterization, which may change TC structures such as eyewall convection.

The results obtained in the current experiments may also be sensitive to the horizontal mesh size. As shown in previous observational studies (Kepert 2006b; Schwendike and Kepert 2008), the GIB in some TCs can be 10%–20%. The results presented here (Fig. 3), in contrast, indicate that few TCs achieve this range. This difference appears to result from the spatial resolution, whereas this may be caused by the presence of the interaction between the boundary layer and free atmosphere (e.g., eyewall convection). Because the GIB is sensitive to the TC structure, simulations with different spatial resolutions would result in a different GIB. Higher-resolution simulations can simulate smaller-scale variability (Rotunno et al. 2009), and thus the GIB appears to temporally vary with larger amplitudes.

The series of simulations produced more (less) TCs with moderate (strong) intensity than the observation as shown in Satoh et al. (2012). Since the GIB is sensitive to the TC intensity, the GIB may be larger than that obtained in the present study, if the frequency distribution of the TC intensity becomes closer to the observational distribution.

## 6. Conclusions

We examined the degree of GWB in the TCs simulated in a set of 7-km mesh global simulations using a global nonhydrostatic model. For all of the simulated TCs during the quasi-steady state, the tangentially averaged maximum tangential velocity at 850 (925) hPa was 0.72% (1.95%) faster than the corresponding GWV. In 75% of the TCs, the average deviation from the GWB at 850 (925) hPa was less than 4.0% (5.5%). However, the averages of the maximum and minimum degrees of GIB in all the azimuths at the RMW at 850 hPa indicated that tangential velocities locally deviate from the GWV by up to 40% of the maximum tangential velocity. The degree of local maximum and minimum GIB in 75% of all the TCs ranged between 30% and 50% of the maximum tangential velocity. These results were quantitatively similar to those during the intensification phase.

The velocity fields were different between the supergradient and subgradient cases. In the supergradient case, the primary circulation decayed rapidly in the vertical direction and slowly in the radial direction, and the secondary circulation was located outward. The warm-core structure was enhanced. The peak of supergradient wind at 925 hPa appeared in the right-hand side of TC motion around *r* = *r*_{m}. In addition, the tangential velocity at 925 hPa in the supergradient case tended to be faster in the right front of TCs, whereas the inward motion was stronger to the rear of TCs. The results for the flow fields are consistent with those obtained in idealized frameworks (Shapiro 1983; Kepert 2001; Kepert and Wang 2001).

The relationships between the GIB and physical quantities related to TC intensity and structure show the following characteristics. The GIB is positively large when

the TC intensity (maximum tangential velocity and central sea level pressure) is strong,

the TC has large RMW,

the TC is axisymmetric, and

the radial velocity at 850 hPa at the RMW is positive and large.

The positive relationships of the GIB with the TC intensity and the RMW are consistent with previous studies (Kepert 2001; Kepert and Wang 2001). In addition, we found that the GIB is not sensitive to the vertical shear of horizontal background flow, the SST within the RMW, the translation speed, and the direction of motion.

Owing to the relatively coarse temporal resolution of the outputs, this study cannot provide conclusions regarding the adjustment problem and the physical interpretations of the relationships of GIB with the related parameters are not yet clear. Accordingly, the effects of interaction between the eyewall and boundary layer on the GIB cannot be examined. Further detailed studies are necessary to investigate these issues.

## Acknowledgments

The authors gratefully acknowledge Dr. S. Nishizawa for his advice on the statistical analyses. We also thank two anonymous reviewers, as well as Dr. T. Nasuno, Dr. A. Noda, and Dr. T. Yamaura for their helpful comments and discussions.

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## Footnotes

^{1}

It should be noted that Willoughby (1990) was not compositing flight level data from multiple storms together, but was compositing multiple legs from the same flight (or in some cases two or three flights) into the same storm.