The movement of a symmetric vortex embedded in a resting environment with a constant planetary vorticity gradient (the beta drift) is investigated with a shallow-water model. The authors demonstrate that, depending on initial vortex structure, the vortex may follow a variety of tracks ranging from a quasi-steady displacement to a wobbling or a cycloidal track due to the evolution of a secondary asymmetric circulation. The principal part of the asymmetric circulation is a pair of counter-rotating gyre (referred to as beta gyres), which determine the steering flow at the vortex center. The evolution of the beta gyres is characterized by develop-ment/decay, gyration, and radial movement.
The beta gyres develop by extracting kinetic energy from the symmetric circulation of the vortex. This energy conversion is associated with momentum advection and meridional advection of planetary vorticity. The latter (referred to as "beta conversion") is a principal process for the generation of asymmetric circulation. A necessary condition for the development of the beta gyres is that the anticyclonic gyre must be located to the east of a cyclonic vortex center. The rate of asymmetric kinetic energy generation increases with increasing relative angular momentum of the symmetric circulation.
The counterclockwise rotation of inner beta gyres ( the gyres located near the radius of maximum wind) is caused by the advection of asymmetric vorticity by symmetric cyclonic flows. On the other hand, the clockwise rotation of outer beta gyres (the gyres near the periphery of the cyclonic azimuthal wind) is determined by concurrent intensification in mutual advection of the beta gyres and symmetric circulation and weakening in the meridional advection of planetary vorticity by symmetric circulation. The outward shift of the outer beta gyres is initiated by advection of symmetric vorticity by beta gyres relative to the drifting velocity of the vortex.
Tropical cyclone motion is primarily controlled by nonlinear vorticity advection. Theoretically, two distinctive mechanisms can be identified which affect adiabatic motion of a barotropic vortex: steering by environmental flows and advection by an asymmetric flow induced by interaction of the vortex circulation with the environmental absolute potential vorticity gradient. An ideal example of the environmental steering is given by Adem and Lezama (1960) who proved that a barotropic symmetric vortex embedded in a uniform flow on an f-plane moves exactly with the velocity of the environmental flow. The presence of the environmental absolute potential vorticity gradient, however, may generate asymmetric circulation through interaction with a symmetric vortex. The secondary asymmetric flow thus generated can further advect symmetric relative vorticity, causing another type of motion which was termed as propagation by Holland (1983). An ideal problem of propagation was first investigated by Rossby (1948) who showed that an isolated rigid-body-rotation vortex on a beta-plane will undergo a poleward acceleration due to the increase of the Coriolis force with latitude. The movement of a vortex embedded in such a quiescent environment on a beta-plane is now commonly referred to as beta-drift. Rossby's solution, however, did not consider the effects of secondary circulation and compensating pressure gradient forces, and hence was not in full agreement with subsequent numerical solutions (e.g., Anthes and Hoke 1975).
A fundamental theoretical problem is to explain the mechanisms causing beta-drift. Adem (1956) first worked out a series solution for a barotropic non-divergent vorticity equation which suggested a beta effect-induced asymmetric circulation that drives the vortex first westward and then poleward. Holland (1983) argued that the advection of earth vorticity by symmetric azimuthal winds could produce asymmetric vorticity which, on the one hand, drags a cyclone westward, on the other hand, creates two counter-rotating gyres, counterclockwise to the west and clockwise to the east that is often referred to as beta-gyres. The resultant poleward wind over the vortex center (referred to as "ventilation flow" by Fiorino and Elsberry (1989)) would advect the cyclone poleward. The roles of asymmetric flows on vortex propagation have received much attention in recent years (Chan and Williams 1987; Willoughby 1988; Fiorino and Elsberry 1989; Peng and Williams 1990; Shapiro and Ooyama 1990; Smith et al. 1990 and others). The propagation of a barotropic vortex is found to be intimately linked to the orientation and strength of the beta-gyres in both barotropic (e.g., Fiorino and Elsberry 1989) and baroclinic models (Wang and Li 1992). In most previous studies the beta-drift was described as a quasi-steady displacement associated with a pair of quasi-steady beta-gyres. Yet, the dynamics of the beta-gyres have not been systematically studied and thoroughly understood even within a barotropic dynamic framework.
In section 2, we describe results from extended-range time integrations for the beta-drift of vortices with different horizontal structures. We demonstrate that a cyclone may take a variety of tracks ranging from a quasi-steady displacement to a snake-shape or a cycloidal track, depending upon the symmetric circulations of the initial vortices. It is also established that different tracks of beta-drift are results of the distinctive evolution of beta-gyres involved in each individual case. The unsteady beta-gyres exhibit intensification (or decay), azimuthal movement (gyration), and radial movement (outward expansion). The primary objective of the present study is to answer the following questions: How do beta-gyres change their intensity? What are necessary and favorable conditions for beta-gyres to grow or decay? What physical processes determine the rotation and radial movement of beta-gyres? We attempt to address these questions in sections 3 and 4, respectively, via analyses of energetics and streamfunction tendency associated with the beta-gyres. In the last section, we summarize major findings regarding the barotropic dynamics of beta-gyres.
We re-examined the classical beta-drift problem with a shallow-water model. An extended-range time integration reveals that vortices with different initial symmetric structures, may take quite different tracks. The evolution of counter-rotating gyres (the beta-gyres), whose flow over the vortex center advects symmetric relative vorticity, is a good indication of vortex movement in all cases. A key to understanding the beta-drift is to explain the dynamics of the beta-gyres.
The kinetic energy for the development of beta-gyres is converted from symmetric vortex circulation. This energy conversion involves two processes: horizontal advection of relative vorticity and meridional advection of planetary vorticity. The latter, referred to as the beta-conversion, is a dominant process responsible for the generation of asymmetric kinetic energy. Further analysis of the beta-conversion reveals that (1) The development of the beta-gyres requires that the anticyclonic gyre must be located to the east of the cyclone center in Northern Hemisphere; (2) The rate of the beta-conversion depends on the covariance between the amplitude of the beta-gyres and the RAM of the symmetric vortex. Because the radial distribution of the beta-gyre intensity is also dependent on the radial distribution of the RAM, the rate of beta-conversion is determined by the radial distribution of RAM or the symmetric vortex structure. As mean RAM increases, more asymmetric kinetic energy is generated, resulting in stronger beta-gyres and a faster beta-drift. This supports the empirical relationship between the mean RAM and the beta-drift speed numerically established by Wang and Li (1992).
The azimuthal and radial movement of beta-gyres can be explained by vorticity dynamics. The total asymmetric streamfunction tendency is the sum of the advection of symmetric vorticity by the beta-gyres relative to the vortex drift (ASVA), the advection of asymmetric vorticity by symmetric flow (AAVS), the advection of planetary vorticity by symmetric flow (BETA), and the terms arising from the advection of residual vorticity by beta-gyres relative to the vortex drift and the advection of absolute vorticity of asymmetric circulation by the residual flow (ARES). For the outer beta-gyres whose centers are around the periphery of the cyclonic azimuthal wind, their clockwise rotation found in numerical experiments is caused by a decrease of BETA and a sharp increase in the sum of AAVS, ASVA and ARES. Their outward movement is mainly caused by the advection of symmetric vorticity by beta-gyres relative to the vortex drift. For the inner beta-gyres whose centers are near the radius of maximum cyclonic wind, the advection of beta-gyre vorticity by symmetric flows dominates the total tendency and is responsible for their counterclockwise rotation. The rotation rate can be estimated from the angular velocity of the azimuthal wind of the symmetric vortex at the radius of the gyre centers.
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