The beta-effect on translation of cyclonic and anticyclonic vortices with height-dependent circulation (the beta-drift problem) is investigated via numerical experiments using a dry version of a multi-level primitive equation model (Florida State University model).
The vertical structure of vortex circulation influences steady translation in a manner similar to that of the horizontal structure. Both spatially changes the mean relative angular momentum (MRAM) of the vortex. The translation speed and its meridional component are both approximately proportional to the square root of the magnitude of MRAM of the initial (or quasi steady-state) symmetric circulation. The latitude is another important factor controlling the speed of the beta-drift. The meridional component decreases by about 45% when the central latitude of the vortex increases from 10oN to 30oN.
The beta-drift speed is intimately related to the axially asymmetric pressure field. During quasi-steady vortex translation the asymmetric pressure field maintains a stationary wavenumber one pattern in azimuthal direction with a high in the northeast and a low in the southwest quadrant of a northern hemisphere cyclone. The beta-drift velocity is approximately equal to the geostrophic flow implied by the asymmetric pressure gradient at the vortex center. If the Rossby number associated with the asymmetric flow is small, to the lowest order, the asymmetric pressure gradient force at the vortex center is balanced by the Coriolis force associated with the beta-drift of the vortex.
INTRODUCTION
Beta-drift is basic component of tropical cyclone motion. It arises from the interaction between the gradient of Earth's vorticity (or in more general terms, the absolute vorticity gradient) and the vortex circulation. Beta-drift may create a deviation from environmental steering and may play a dominant role if the ambient steering current is weak or indefinite, particularly in the deep tropics. Study of beta-drift also provides fundamental understanding of nonlinear interaction between the vortex dynamics and the environment.
The investigation of motion of an isolated barotropic vortex in a quiescent environment dates back to Rossby (1948). Using a solid-body rotation vortex and assuming a balance between the integrated pressure force by the surrounding fluid and the Coriolis force associated with the vortex translation, Rossby argued that an axially symmetric cyclonic vortex is driven poleward due to the latitudinal variation of the Coriolis parameter. Adem (1956) reexamined the translation of a geostrophic vortex on beta-plane in terms of a Taylor expansion of streamfunction in time. In the Northern Hemisphere a cyclonic vortex was shown to first move westward then turn northward. Adem first noticed a relation between vortex translation and its horizontal structure: the initial westward and subsequent northward components are proportional to the vortex radius and the maximum wind speed, respectively.
Numerical experiments using barotropic models all indicate that an initially symmetric vortex on a beta-plane moves consistently northwestward in the Northern Hemisphere (e.g. Anthes and Hoke, 1975, Madala and Piacsek, 1975) rather than northward or first westward then northward. Recent numerical studies of barotropic beta-drift have further focussed on the effects of the horizontal vortex structure on the beta-drift. DeMaria (1985) showed that the vortex track is much more sensitive to changes in the outer region (size change) than to changes in the inner regions (intensity change). Holland (1983, 1984) postulated that the vortex motion depends on cyclone-environment interaction at an effective radius of interaction which is an envelope defined by the region of rapid increase in inertial instability. He intuitively predicted that the maximum wind variation should not affect the motion, while the changes in the size and strength of the outer circulation affect the motion by changing the effective radius. On the other hand, Chan and Williams (1987) showed that for a constant-shape vortex the northward movement increases with both the maximum wind speed and the radius of maximum wind. Fiorino and Elsberry (1989a) examined contributions of small, medium, and large scale components of some typical tangential wind profiles to the vortex motion. They concluded that the speed of the motion is primarily determined by the strength of large scale components (the flow in the outer region), whereas the medium (and small) scales have a significant effect on the direction of the motion by influencing the orientation of the asymmetric gyres that are induced mainly by the large scales.
Rossby's theoretical explanation of beta-drift was based on the analysis of resultant force and energetics of a solid-body rotation vortex in which asymmetric vortex circulation was not permitted. However, vortex translation is closely related to this asymmetric circulation. The asymmetric gyres associated with translation were termed "beta gyres" since they arise from the beta-effect (or the effect of the absolute vorticity gradient) (Fiorino and Elsberry, 1989b). Fiorino and Elsberry found a close association of the speed and direction of the vortex motion with the asymmetric flow at the vortex center (the ventilation flow).
In his linear analysis Willoughby (1988) derived a proportionality of the northward speed to the initial total relative angular momentum (TRAM) in the presence of asymmetric circulation. However, Shapiro and Ooyama (1990) pointed out that for an initial symmetric vortex with nonzero TRAM the TRAM within a large circle centered on the vortex would decrease with time due to Rossby wave dispersion (McWilliams and Flierl, 1979) and thus the motion of the vortex could not be related to the initial TRAM in any simple way.
Identification of controlling factors responsible for beta-drift remains a necessary step toward understanding the beta-drift dynamics. In this regard, previous analyses using different barotropic models and approaches have not reached a generally accepted conclusion. In addition, the study of the beta-drift has been primarily confined to the framework of barotropic models. However, the structure of tropical cyclones is height-dependent. It is not clear how the vertical circulation structure influences vortex motion.
The present study uses numerical experiments to study beta-drift of vortices with height-dependent structure. Section 2 briefly introduces the numerical model and results of sensitivity tests. Our first focus is to identify possible dynamic factors and processes that are responsible for the beta-drift, in particular, the influence of the vertical structure of vortex on its motion (Section 3). The second focus is to examine the relation between the axially asymmetric pressure field and the beta-drift as well as the initial symmetric vortex circulation in an effort to explain the mechanism for beta-drift (Section 4). The last section summarizes our primary findings.
SUMMARY
The movement of an initially isolated symmetric vortex embedded in a quiescent environment due to the interaction between the vortex circulation and the gradient of the Earth's vorticity (the beta-drift) was investigated via numerical experiments. The multi-level primitive equation model (a dry version of the FSU model) used in this study allows examination of the influences on vortex translation of both horizontal and vertical structures.
The cyclonic vortices initially (about 12 hours) accelerate then translate quasi-steadly (Table 2a). Although the total relative angular momentum is not conserved due to Rossby wave dispersion, the mean relative angular momentum (MRAM) of the cyclonic circulation alone exhibits little change during the quasi-steady translation after an initial drop accompanying the acceleration (Table 2b). The ratio of MRAM at quasi-steady state to that of initial state is approximately a constant for different vortices used in our experiments. The weak anticyclonic circulation in the far-field outside the circle with a radius of 2 Ro (Ro being the initial vortex radius), has insignificant effect on vortex translation in a 48-hour integration. The existence of a quasi-steady phase of beta-drift and the proportionality between initial and quasi-steady state MRAM makes it meaningful to examine the relationship between the characteristics of the initial symmetric circulation and the quasi-steady vortex translation.
Differences in the meridional and zonal beta-drift are noticeable. First of all, the sense of the meridional component of beta-drift depends on the signs of both relative angular momentum and the Coriolis parameter (the vertical component of the earth's vorticity): it is directed poleward for cyclones and equatorward for anticyclones. The zonal component of the beta-drift, on the other hand, is always directed westward regardless of the signs of the relative angular momentum and the Coriolis parameter. Secondly, the poleward component of beta-drift decreases significantly with increasing latitude whereas the westward component does not have a simple relationship with the beta-parameter or Coriolis parameter (Table 4). Lastly, the westward drift seems to be weakly linearly correlated with MRAM for selected samples (Fig. 3b), while the magnitude of the meridional drift speed is found to be roughly proportional to the square root of the MRAM of the initial vortex (Fig. 3a). These differences suggest that the meridional and zonal vortex translations are controlled by different physical processes. The westward drift which is sensitive to vortex size and maximum winds within limited range is related to beta-induced wave dispersion which is directed westward. The meridional drift that depends on the sense of the circulation and the Coriolis parameter may be due to the Coriolis force and beta-induced pressure gradient force applied to the entire vortex. Both processes result from the meridional variation of planetary vorticity, an inherent property of the ambient medium. Since the poleward drift speed is generally larger than the westward drift speed for cyclonic vortices, the total beta-drift speed of a cyclone is also approximately proportional to the square root of the MRAM of the initial vortex (Eq. (7)) and decreases with increasing latitude (Table 4).
The dependence of meridional beta-drift on MRAM is clearly demonstrated for vortices whose tangential winds vary with height but do not reverse their sense. For these vortices, the area-averaged relative angular momentum at each level varies with height, but the beta-drift speed is nearly height-independent (Fig. 4) and is determined by the vertical mean of the area-averaged RAM at each level, i. e., the MRAM (Fig. 3a). This results from the vertical coupling of the vortex circulation by a secondary circulation developed in the deep cyclonic (or anticyclonic) vortex. It is also demonstrated that in the absence of the vertical coupling as in the case of a compound dry baroclinic vortex, the upper anticyclonic part translates southwestward while the lower cyclonic part drifts northwestward for the Northern Hemisphere case.
The relationship between MRAM and the beta-drift has practical implications. For vortices having similar strength measured by horizontal area-averaged RAM, the depth of the vortex is a crucial factor in considering its translation due to the beta-effect: a deep vortex can be markedly faster than a shallow one. Further study of the beta-drift in the presence of latent heat is necessary. In that case, the vertical coupling provided by latent heat-induced divergent circulation may effectively maintain the vortex as an integral system with a coherent upper-level anticyclonic and lower-level cyclonic circulation. The opposite circulations aloft and below tend to have opposite meridional drifts that may compensate each other. The resultant beta-drift is expected to be mainly westward or to substantially differ, in general, from that of a dry baroclinic vortex.
The beta-effect interacting with an initial axisymmetric vortex circulation induces an axially asymmetric pressure field. There exists an intimate relationship between the asymmetric pressure field and beta-drift. Our numerical experiments show that the asymmetric pressure field maintains a stationary pattern in azimuthal direction with a high pressure in the northeast and a low in the southwest quadrant (Fig. 9). Our experiments also reveal that for the vortices with height dependent tangential wind (and angular momentum) the mean asymmetric pressure gradient at the vortex center does not vary with height, and the beta-drift speed is approximately proportional to the asymmetric pressure gradient at the vortex center.
Simple scale analysis shows that if the Rossby number associated with asymmetric motion is small, to the lowest order, the asymmetric pressure gradient force at the vortex center is balanced by the Coriolis force associated with the vortex translation. We speculate that if we approximate the vortex circulation as an idealized solid-body, the asymmetric pressure field around the vortex would exert a resultant pressure gradient force to the vortex which is directed southwestward for a northern-hemisphere cyclone. Rossby (1948) estimated the resultant environmental pressure gradient force on a solid-body rotation vortex which is directed westward and exactly balances the Coriolis force, so that the vortex moves northward. The translation of the cyclone is also subject to a Coriolis force. To the lowest order, the vortex translation will reach geostrophic balance as long as the Rossby number associated with asymmetric flow is small. In this sense, the beta-drift may be viewed as a steady geostrophic motion implied by the beta-induced asymmetric pressure gradient force.
We have found that the asymmetric pressure gradient at the vortex center in the quasi-steady translation stage is roughly in proportion to the square root of the MRAM (Fig. 10b) and increases with the latitude where the vortex is located (Table 4). Further study is needed, however, to understand the nonlinear dependence of the beta-induced asymmetric pressure gradient at the vortex center upon the MRAM of the initial vortex and the Coriolis parameter. We intend to continue our study of these issues.