Buoyancy in tropical cyclones and other rapidly rotating atmospheric vortices

Motivated primarily by its application to understanding tropical-cyclone intensification and maintenance, we re-examine the concept of buoyancy in rapidly rotating vortices, distinguishing between the buoyancy of the symmetric balanced vortex or system buoyancy, and the local buoyancy associated with cloud dynamics. The conventional definition of buoyancy is contrasted with a generalized form applicable to a vortex, which has a radial as well as a vertical component. If, for the special case of axisymmetric motions, the balanced density and pressure distribution of a rapidly rotating vortex are used as the reference state, the buoyancy field then characterizes the unbalanced density perturbations, i.e. the local buoyancy. We show how to determine such a reference state without approximation. The generation of the toroidal circulation of a vortex, which is necessary for vortex amplification, is characterized in the vorticity equation by the baroclinicity vector. This vector depends, inter-alia, on the horizontal (or radial) gradient of buoyancy evaluated along isobaric surfaces. We show that for a tropical-cyclone-scale vortex, the buoyancy so calculated is significantly different from that calculated at constant height or on surfaces of constant sigma (sigma = (p - p*)1(p, - p*), where p is the actual pressure, p* some reference pressure and p, is the surface pressure). Since many tropical-cyclone models are formulated using or-coordinates, we examine the calculation of buoyancy on or-surfaces and derive an expression for the baroclinicity vector in a-coordinates. The baroclinic forcing term in the azimuthal vorticity equation for an axisymmetric vortex is shown to be approximately equal to the azimuthal component of the curl of the generalized buoyancy. A scale analysis indicates that the vertical gradient of the radial component of generalized buoyancy makes a comparatively small contribution to the generation of toroidal vorticity in a tropical cyclone, but may be important in tornadoes and possibly also in dust devils. We derive also a form of the Sawyer-Eliassen equation from which the toroidal (or secondary) circulation of a balanced vortex may be determined. The equation is shown to be the time derivative of the toroidal vorticity equation in which the time rate-of-change of the material derivative of potential toroidal vorticity is set to zero. In analogy with the general case, the diabatic forcing term in the Sawyer-Eliassen equation is shown to be approximately equal to the time rate-of-change of the azimuthal component of the curt of generalized buoyancy. Finally, we discuss the generation of buoyancy in tropical cyclones and contrast the definitions of buoyancy that have been used in recent studies of tropical cyclones. We emphasize the non-uniqueness of the buoyancy force, which depends on the choice of a reference density and pressure, and note that different, but equivalent interpretations of the flow dynamics may be expected to arise if different reference quantities are chosen. (c) 2005 Elsevier B.V. All rights reserved.