## Introduction

Tropical cyclones are known to move by the largescale environmental flow and the ventilation flow of beta gyre or the beta drift (Holland, 1983;Holland et al., 1992;Chan and Williams, 1987;Fiorino and Elsberry, 1989;Shapiro and Ooyama, 1990;Carr and Elsberry, 1990;Elsberry, 1990;Wang and Li, 1992;Yasunaga et al., 2016;Emanuel, 2018). In the lower latitudes, they tend to move westward by the easterly and the ventilation flow while in the middle latitudes they make recurving and move northeastward by the westerlies (Knapp et al., 2010). The beta-drift, the northwest movement of tropical cyclone induced by the asymmetric flow of tropical cyclone itself, is called a self-steering, which is caused by the effect of the latitudinally varying Coriolis parameter, i.e., the beta effect (Holland, 1983;Shapiro and Ooyama, 1990;Smith and Ulrich, 1990;Li and Wang, 1994;Sutyrin and Flierl, 1994;Wang and Li, 1995;Willoughby and Jones, 2001;Zhao et al., 2009;Fang and Zhang, 2012).

The beta gyre is an asymmetric, azimuthal wavenumber one component of the disturbance surrounding a tropical cyclone, which is generated by the beta effect of the rotating spherical Earth (Reeder et al., 1991). Since the largest beta effect is found at the Equator and decreases with the latitude, the beta gyre is expected to depend on the latitudinal location of tropical cyclone. Numerical studies showed that the beta gyre, which is affected significantly by the presence of the large-scale environmental flows, appears not only in the barotropic models but also in threedimensional models (Wang and Li, 1992;Li and Wang, 1994;Wu and Wang, 2000). Therefore, the translation speed of tropical cyclones depends to a large extent on the structure of large-scale flows. This means that the large-scale environmental flow directly influences the translation of a tropical cyclone, and at the same time, it controls indirectly the movement through the beta gyre.

The beta gyre is known to be a part of the Rossby waves that are induced by tropical cyclone (Holland, 1983;Shapiro and Ooyama, 1990). Generation of Rossby waves by tropical cyclone is considered as a dispersion process of the Rossby waves that constitute the tropical cyclone. In the framework of non-divergent barotropic motion, a strong vortex-like tropical cyclone can be represented as a sum of infinite number of Rossby-Haurwitz waves which are defined as Rossby waves with a global structure on the sphere (Haurwitz, 1940;Thuburn and Li, 2000). The Rossby waves tend to disperse with time because the phase speed of westward translation is different from one another: Rossby waves with a smaller wavenumber propagates faster than those with a larger wavenumber. Since the Rossby waves are related to the beta-effect, a betaplane model is usually used when studying the dynamics of them (Chan and Williams, 1987;Li and Wang, 1994;Gonzalez et al., 2015). Although being capable of providing a simple and effective way to explore the basic dynamics associated with the Rossby waves, the beta-plane approximation is not suitable for studying the Rossby waves that disperse over a large area on the sphere (Hoskins et al., 1977). Since the beta effect decreases with latitude, the beta-plane model may give more westward translation of tropical cyclone-scale vortices compared to the models employing the spherical domain.

In this study, the tropical cyclone-scale vortices are simulated using global-domain barotropic models with a focus on the movement of the vortex, the beta gyre, and the Rossby waves associated with it. The models used for simulations are divergent- and nondivergentbarotropic models which are spatially discretized using the spherical harmonic (or associated Legendre) functions as basis functions. The paper is organized as follows. Next section will be served to explain the governing equation, discretization method, and initial condition. Section 3 presents the results of numerical simulations of tropical cyclone scale vortices focusing on the Rossby wave and the beta gyre. Summary and conclusions are presented in the final section.

## Barotropic Models on the Sphere and Initial Conditions

Divergent barotropic model (or shallow water model) on the sphere, which is scaled by the radius of the Earth (*α*) and the rotation rate (*Ω*), is written as (Bourke, 1972;Haltiner and Williams, 1980;Williamson et al., 1992;Cheong and Park, 2007):

where *λ* (*θ*) means longitude (latitude), *u*(*υ*) means the velocity component in *λ* (*θ*) direction, *ζ* (*D*) is the vorticity (divergence), *f* (≡2sin*θ*) implies the Coriolis parameter, (*U*, *V*) is defined as (*u*cos*θ*, *υ*cos*θ*), and *Φ* is the geopotential with the bar and prime denoting the global mean and deviation from it, respectively. The model only with the vorticity equation is referred to as the non-divergent model. Vorticity and divergence are represented with the velocities as follows

Velocities are expressed in terms of the streamfunction (*ψ*) and velocity potential (*χ*) as

where *ψ* and *χ* are related with the vorticity and the divergence as following

Shallow water equations (1a-c) are discretized with the spherical harmonics spectral method, where dependent variables are expanded with a finite number of normalized spherical harmonic functions (Haltiner and Williams, 1980;Williamson et al., 1992), for instance,

where $i=\sqrt{-1}$, *m* represents the zonal wavenumber, and ${P}_{n}^{\left|m\right|}\left(\text{sin}\hspace{0.17em}\theta \right)$ means associated Legendre function with the order *m* and degree *n* (total wavenumber or spherical wavenumber). Re[•] denotes the real part of •, and *ζ _{m,n}* means the spectral coefficient which is complex-valued. All dependent variables are realvalued, hence, the identity ${\zeta}_{n,m}=\zeta {*}_{n,-m}$ holds for with the superscript of asterisk meaning the complex conjugate. Spectral coefficients are obtained from the orthogonality (Williamson et al., 1992)

with *δ _{n,l}* being the Kronecker delta. Unlike the models based on the finite-element Galerkin method (e.g., Cheong et al., 2015), the Gaussian grid is used in the spherical harmonics model, for which the spectral transform via the orthogonality in equation (6) should be carried out with Gaussian quadrature (Haltiner and Williams, 1980;Williamson et al., 1992).

The initial condition for the vortex (in Fig. 1) is given in terms of geopotential (*Φ*) and tangential velocity (*υ _{t}*) as follows (Holland, 1983;Kwon and Cheong, 2010)

where *Φ _{c}* means the amplitude of the vortex, given as 2,500 m

^{2}s

^{−2},

*r*means the distance from the center of vortex, and the parameters for the radius of maximum wind and width of the vortex are specified as

*r*=100 km and

_{m}*b*=1.5, respectively. Four kinds of basic zonal states, defined in terms of zonal-mean zonal flow, vorticity, and vorticity gradient are presented in Fig. 2. Horizontal resolution of the model is approximately 0.12 degrees in both longitude and latitude, which corresponds to the grid-size of about 13.3 km at the Equator. Triangular truncation was employed in the spectral model with the maximum total-wavenumber (

*n*

_{max}) being set as 1,000. With this resolution, the vortices of tropical cyclone-scale were resolved sufficiently well, therefore, the dynamics associated with them is expected to be simulated properly (Shapiro and Ooyama, 1990;Kwon and Cheong, 2010). Time integrations of the model were carried out with the leapfrog-Asselin method for 7 days, where the timestep size is given as 2.5 minutes.

## Results of Simulations

### Track of the vortices

Before showing the numerical results, sensitivity of the tracks was tested for the case of no zonal-mean zonal flow by changing the resolution from *n*_{max}=1,000 to 800 and 1,200 which correspond to the number of grid points of 2,400 (1,200) and 3,600 (1,800) in longitude (latitude), respectively. Results with the reduced resolution showed a faster translation speed than the control case by about 2.6 and 2.5% in the longitudinal direction and latitudinal direction, respectively. However, tracks obtained with an increased resolution provided a slower translation speed than the control case by about only 1.7 and 1.4% in the longitude and latitude, respectively. Therefore, it is here assumed that the numerical solution for the initial conditions and the zonal basic flows as stated above has converged.

Figure 3 shows the tracks of eight simulations, of which four cases used non-divergent model while other four cases used divergent models. First of all, it is noted that the tracks for the divergent models do not exhibit a significant difference from those for the nondivergent models (Shapiro and Ooyama, 1990;Montgomery et al., 1999). Distance of northward translation is largest for the case of II with 22.5 degrees for 7 days, while it is smallest for the case of IV with 9.7 degrees for 7 days. For the case I (control case with no zonal-mean zonal flow), the vortex starts to propagate relatively slowly: The moving velocity, V_{c}[≡(*u _{c}*,

*υ*], was (−0.5, 1.5) m s

_{c}^{−1}between

*t*=0 and

*t*=0.25 day, but after that shows an almost steady northwestward translation. At later stages of simulations, e.g., day 5-7, the translation speed decreased to a certain extent. The largest track-difference for case II seems to be a consequence of propagation of the vortex into the middle latitude of strong zonal flow. It should be emphasized that although the track appears to have a northwest translation as can be imagined from nearly straight line, it is basically different from those for the beta-plane models (Shapiro and Ooyama, 1990;Li and Wang, 1994) because of the sphericity of the model domain. This means that tracks for the spherical surface show more eastward deflection as the vortex moves northward, compared to the betaplane model. For case II, the translation speed appears to vary significantly with time due to the combined effect of zonal-mean steering flow and the selfadvection (that is, beta drift). In the first 3 days and a half, the vortex remains in the north-westward translation with the speed increasing with time, which is followed by northeastward translation after day 3. It is interesting to note that the meridional translation speed is kept nearly unchanged after making a recurving by day 3.5. On the other hand, the zonal translation speed exhibits a steady increase with time after day 1 under the influence of the zonal-mean zonal flow. The zonal translation speed looks quite well matched with the zonal-mean zonal flow, marking the maximum value of 14.5 m s

^{−1}by day 7, as is consistent with Wang and Li (1995) and Knapp et al. (2010). For the case III, the tracks can be divided into two different stages: In the first stage, which corresponds to the initial one and a half day when the velocity vector increases with time, whereas in the second stage (i.e., from day 1.5) the translation speed in both westward and northward direction increases with time but a reduced rate compared to the first stage. Case IV presents a northeastward translation during the first one day but a steady decrease of both zonal and meridional speed after day 1. From the tracks for cases III and IV, it is likely that the northward translation is suppressed by the easterly in the subtropics, which is consistent with previous study (Wang and Li, 1995).

### Rossby wave field and beta gyre

Figure 4 shows the streamfunction by day 4 for the cases I, II, III, and IV simulated with the nondivergent (indicated by NVE) and divergent barotropic model (indicated by SWE). Rossby-wave train with alternating signs of anomalies is well established in the southeastward or northeastward direction. As a whole, the wave pattern resembles closely those produced by the beta-plane models (e.g., Shapiro and Ooyama, 1990), which is a kind of Rossby wave generated by the variable Coriolis effect (Longuet-Higgins, 1964;Montgomery et al., 1999). Compared to the waves simulated with a linear model on the sphere, such as Hoskins et al. (1977), the wave train is less dispersed. As is consistent with the results for tracks, the wave patterns do not show a significant difference between those produced by the divergent model and nondivergent model. The spatial distribution and magnitude of anomalies simulated by two models look quite similar to each other. It is worthy to mention that the northeastward oriented wave train is only observed in case II. Another notable feature that distinguishes case II from other cases is the increased magnitude of the anomalies. The magnitude of the positive streamfunction anomaly for case II reaches about 1.5 times of other cases. As is consistent with Hoskins et al. (1977), the anomalies associated with the Rossby wave seems to have spread over quite a large area, which suggests the necessity of sufficiently large domain even in the case of simulating the tropical cyclones as well as tropical cyclone-scale vortices on a beta-plane.

Figure 5 presents the beta gyre by day 4, which is obtained by taking the wavenumber-one component by azimuthal Fourier decomposition referenced to the center of vortex. It is found that the orientation of the beta gyre around the center of vortex is northwest for all cases except for case II which shows an almost northerly orientation. This kind of orientation is necessary for the beta gyre to extract kinetic energy from the symmetric part of the vortex, as was analyzed and termed it as the beta conversion by Wang and Li (1995). As the angle that the beta gyre axis (a straight line that the streamline of zero streamfunction is optimally fitted) makes from due north decreases, the kinetic energy conversion from the symmetric part to the beta gyre increases. In this regard, the kinetic energy conversion for case II is likely to be more than is estimated by considering only the amplitude of the beta gyre. In addition to the orientation, case II presents relatively larger amplitude of beta gyre compared to other cases by about one and a half times. Interestingly, however, the vortex for case II has a smaller intensity than others. This is because the beta gyre is determined not by the vortex alone but by the pair of anomalies constituting the Rossby wave. This means that the strength of the beta gyre is proportional to the strength of the positive streamfunction anomaly existing to the right of the vortex. Indeed, the strongest positive streamfunction anomaly is found for case II, as can be seen in Fig. 4. The beta gyre takes the horizontal pattern in spiral shape with the amplitude decreasing as the distance from the center of vortex increases, as is evident in Fig. 4. Note that the direction of the spiral is counterclockwise for case II while it is clockwise for other cases.

Shown in Fig. 6 is the time variation of the ventilation vector, defined as the wind vector associated with the beta gyre at the center of vortex, for four cases. Common to three cases I, III, and IV, the orientation of ventilation vector V_{p}[≡(*u _{p}*,

*υ*] gradually turns from northeast to northwest. It may be stated that the ventilation vector at day 4 well corresponds to the structure of beta gyre as illustrated in Fig. 5. It is interesting to note that the ventilation vector for case II is significantly different from other cases, as it displays a gradual clockwise turning with time from day 1 to 7. Mean ventilation speed in zonal and meridional direction between day 1 to 7 are approximately

_{p}*u*=0 and

_{p}*υ*=5 m s

_{p}^{−1}. The amplitude of clockwise turning appears with a larger amplitude in the case of divergent barotropic model than the nondivergent model. Furthermore, in the case of divergent model, the amplitude of ventilation vector in zonal direction is greater than that in meridional direction. On the other hand, the amplitudes of oscillation in zonal- and meridional- direction are almost the same in the case of nondivergent model. The oscillation of ventilation vector of beta gyre may be related to the structural change of the Rossby wave with time: To see this, the Rossby waves, represented with the perturbation streamfunction without zonalmean part, are plotted in Fig. 7 with one day interval for the case of II. It is clearly observed in this figure that the wave train is oriented northeasterly within initial 5 days but beyond that, it does not provide a well-defined wave train. As it translates northeast, the amplitude of the Rossby wave increases with time for initial 5 days. After day 5, Rossby waves are dispersed away from the center of the vortex, and as a result, the wave train seems to have almost disappeared.

As is well known, the tropical cyclones (and also the tropical cyclone-scale vortices) are steered not only by the large-scale environmental flow and the beta gyre but also by other minor factors (Carr and Elsberry, 1990;Zhao et al., 2009;Yasunaga et al., 2016). Minor factors are largely canceled out if an average is taken over many cases, and hence the translation (movement) of tropical cyclone can be accounted for exclusively by the two main factors. In fact, however, even in the case of minor factors are not present at all as in the simulations shown here, the roles of two main factors are not likely to be defined with clarity because of difficulty of determining the environmental flow and the beta gyre. Figure 8 presents the difference of two vectors [called as the propagation vector (Carr and Elsberry, 1990)], that is, the translation vector subtracted by the zonal-mean zonal flow. If this vector matches well with the ventilation vector shown in Fig. 6, the instantaneous translation of the vortices can be explained by the two factors, environmental steering flow and the beta gyre ventilation vector calculated at the center of the vortex. As a whole, the patterns in Fig. 8 are similar to those in Fig. 6, but there is appreciable discrepancy in the aspect of the magnitude and direction. The most noticeable discrepancy is observed in the meridional speed, which indicates that the ventilation vector was not properly estimated: Since there is no basic meridional flow, the discrepancy should be attributed to the beta gyre. Defining the beta gyre ventilation vector and the steering flow in different ways would result in a better match of the propagation vectors. This subject is one of the central issues with regards to the dynamics of the tropical cyclones (e.g., Yasunaga et al., 2016), but unfortunately, there is no uniquely accepted definition for this. The complexity of the concept of steering effect comes from the fact that the beta gyre is a part of the complicated Rossby wave field produced by the tropical cyclone itself.

## Summary and Conclusions

In this study, tropical cyclone-scale vortices were investigated using numerical simulations with a focus on the movement, Rossby wave train, and the asymmetric wavenumber one structure. The model equations used are the divergent and nondivergent barotropic equations, and they are discretized with the spherical harmonics spectral method. A vortex in the gradient wind was used as an initial condition. The tracks on the sphere were found to have less westward translation speed than those produced with the simulations with a beta-plane approximation. It was revealed that the difference between the divergent model and the nondivergent model was almost negligible. It was shown, however, that tracks were significantly affected by the presence of the basic zonal state. Zonal-mean shear flow, which resembles the climatological zonal flow, was shown to enhance the northward translation noticeably, compared to the case of no zonal-mean zonal flow. On the other hand, for the shear flow that has the reversed sign of the climatological mean-like structure, the northward propagation was suppressed to a large extent. It was found that the northward translation speed was associated with the orientation and strength of the beta gyre which is asymmetric, azimuthal wavenumber-one component with respect to the vortex centered coordinates. Moreover, the beta gyre was found to be directly related to the Rossby wave train generated by the vortex itself. That is, the strongest beta gyre in the case of climatological meanlike basic zonal state was found to be associated with the strongest positive streamfunction anomaly of the Rossby wave train. The beta gyre turned out to give rise to an enhanced northward translation of the vortices by maintaining its orientation closer to the due north. Further analysis revealed that in the case of climatological-mean zonal flow the beta gyre, as well as the Rossby wave train, appeared to rotate clockwise with time. Unexpectedly, the beta gyre presented a northwest orientation even in the case of reversed climatological-mean zonal flow but with an appreciably reduced strength. The results shown in the present study indicates that the spherical domain is essential in investigating the behavior of Rossby wave and beta gyre associated with tropical cyclones.