The impacts of linear environmental shears on beta drift direction are assesed through numerical experiments with a single-layer, primitive equation model. It is found that cyclonic (anticyclonic) shears turn the beta drift more westward (northward) in the Northern Hemisphere. In addition, the longitudinal shear of meirdional flows ( V/ x) is much more effective than the meridional shear of zonal flows ( U/ y) in deflection of the beta drift.
A theoretical model, the beta gyre dynamic sytstem, describing evolution of the beta gyre amplitude and phase angle is advanced to interpret the numerical model results. In this model, the nonlinear energy transfer from the beta gyres to the primary vortex and higher asymmetric models was partially parameterized by linear damping. The semi-empirical theiry predicts that 1) beta drift direction is independent of the planetary vorticity gradient; 2) in a quiescent environment, the drift angle is primarily determined by the outer azimuthal flows of the vortex; and 3) in a sheared environmental flow, the deflection of beta drift induced by environmental shears depends mainly on the longitudinal shear of meridional flows. The authors show that the environmental shear changes beta drift angle by advection of beta gyre vorticity of and planetary vorticity, which affects beta gyre orientation.
In a resting atmosphere, a hurricane-like cyclonic vortex would drift approximately northwestward (southwestward) at a speed of a few meters per second in the Northern (Southern) Hemisphere due to the steering of the primary vortex by beta (planetary vorticity gradient)-induced asymmetric gyres (the beta gyres). This beta drift component accounts for systematic deviation of tropical cyclone motion from the corresponding environmental steering flow over various tropical ocean basins (Carr and Elsberry 1990). Although the meridional gradient of the Earth's vorticity and the vortex structure are essential factors determining beta drift, the presence of horizontal shears of environmental flows adds complexity to understanding the vortex motion (Elsberry 1987). Sheared environmental flows influence cyclone motion by two processes. The first (also the principal) process is their direct steering effect on the primary vortex. The second is their interaction with the primary vortex circulation which changes the strength and orientation of beta gyres, affecting beta drift.
Recent studies (Ulrich and Smith 1991; Smith 1991; Williams and Chan 1994; Wang and Li 1995, Smith et al. 1996) have found that zonal environmental flows with meridional shears may have significant impacts on beta drift. The presence of meridional shear can alter beta gyre intensity and thus the beta drift speed. In a previous paper (Li and Wang 1996, hereafter LW 96), we have analyzed beta gyre development in a general environmental flow with both meridional and longitudinal shears. It was found that a positive (negative) shear strain rate accelerates (decelerates) the beta drift by interacting with an embedded cyclone (both its axially symmetric and asymmetric components) and changing the strength of the beta gyres and thus altering beta-drift speed.
The environmental shears influence not only the beta drift speed but also the drift angle. Whereas Williams and Chan (1994) found that beta-drift direction does not change appreciably in the presence of a moderate meridional shear, we found that a longitudinal shear associated with meridional flows may significantly modify beta drift direction (LW 96). The underlying dynamics, however, remains unexplained. In fact, to our knowledge, even without the complexity of the environmental shear effects, the various factors that determine the beta drift angle are little known.
The present paper is devoted to elaborate how the horizontally sheared environmental flows influence beta drift direction. We will first assess, by means of controlled numerical experiments, impacts of environmental shears on beta drift angle (section 2). We will show that beta drift angle is primarily affected by the longitudinal shear of meridional environmental flow. To interpret the results we develop a theoretical framework the beta gyre vorticity tendency equation for study of gyration of the beta gyres and change of drift direction (section 3). Next, we advance an analytical model for analysis of beta gyre rotation (section 4). In section 5 we discuss principal processes by which environmental shears influence beta gyre rotation and interpret the results of numerical experiments. The last section presents concluding remarks and discuss the limitation and weakness of the semi-empirical theory.
Through a series of numerical experiments, we found that although both V/ x and U/ y of the environmental flow can affect beta drift angle, the longitudinal shear ( V/ x) plays a much more important role in deflecting beta drift. A cyclonic shear (either zonal or meridional or a combination of the two) turns the beta drift to the left (more westward) in the Northern Hemisphere when facing the direction of the drift. An anticyclonic shear has an opposite influence.
Numerical experiments have confirmed that in the presence of horizontally sheared environmental flows, the beta-drift direction (after removing environmental steering effect) is basically determined by the orientation of the beta gyres. The gyre orientation can be measured by the azimuthal phase angle of the anticyclonic gyre. The change of gyre phase angle is predictable in terms of the tendency of beta gyre vorticity.
With the aid of the beta gyre vorticity tendency equation and a number of simplifications, we derived a simple model for steady-state phase angle of the beta gyres. In a quiescent environment, the orientation of the beta gyres is a result of the balance between gyre vorticity advection and planetary vorticity advection, both by the symmetric vortex circulation. The environmental shears relative to the moving vortex can further advect gyre vorticity and the planetary vorticity, change the original balance in the resting environment and thus change the orientation of the beta gyres. The beta drift deviates accordingly. The deflection angle depends on a combined shear index, M, which is a weighted mean of the cyclonic shear contributed by the meridional and zonal components of the environmental flow. The present theory predicts that the longitudinal environmental shear is considerably more effective than the meridional shear in deflecting beta drift. The beta drift angle is a nonlinear function of M, but to the lowest approximation, it can be described by a linear relationship. This supports the empirical beta-drift law derived from dimensional analysis and numerical fitting by Smith et al. (1996).
Whereas previous works did not explicitly consider dissipations in beta drift problem, the dissipation term in describing beta gyre dynamics is needed, especially under the approximation R2=0 (i.e., the neglect of the radial variations in the phase angle of the beta gyres). The beta gyre grows initially by extracting energy from the primary vortex via a term designated as "beta conversion" (Wang and Li 1995). This energy source is essentially associated with the meridional transport of planetary vorticity by primary vortex circulation which continuously generates negative (positive) vorticity to the east (west) of the vortex center, sustaining the beta gyres. There are three energy sink terms that offset this beta generation in a quasi-steady state: the energy transfer from the beta gyres to the primary vortex due to inward transport of eddy momentum associated with the asymmetric gyres, the energy cascade to higher asymmetric modes, and the outward radiation of energy due to Rossby wave dispersion. The Rayleigh dissipation used in our model represents a simple parameterization of all these dissipation effects. The damping coefficient used in the model was empirically determined form the numerical experiments.
The model derivation involves critical simplifications based on intuitions gained from the numerical modeling and parameterizations which were used to surrogate neglected nonlinear processes. In this sense, the theory is semi-empirical in nature. It should be stressed that the derivation of the beta gyre phase angle equation (4.11) is based on an "after the fact" assumption (R2=0) which leads to the present simple model for beta drift angle. Other assumptions adopted in the model derivation include the specified vortex structure--- a finite size with a positive total angular momentum; the quasi-steadiness of the beta drift; the steadiness of the symmetric circulation, and the absence of feedback from vortex to environmental flow. Whereas these assumptions are valid to various degrees, they are sources for any discrepancy between the theory and numerical results. In particular, the beta gyres develop by extracting kinetic energy from the symmetric circulation and the environmental shears in the initial adjustment process (Wang and Li 1995). The energy conversion from sheared flow to gyre depends on the magnitude of the shear strain rate. To avoid instability, the strength of the shears must be constrained. The environmental flow considered in the present study has linear shears. Further study of the effects of environmental relative vorticity gradient is needed, because the latter has been shown to have significant influence on beta drift direction on certain occasions.
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