A Three-Dimensional Balance Theory for Rapidly Rotating Vortices

A three-dimensional balance formulation for rapidly rotating vortices, such as hurricanes, is presented. The asymmetric balance (AB) theory represents a new mathematical framework for studying the slow evolution of rapidly rotating fluid systems. The AB theory is valid for large Rossby number; it makes no formal restriction on the magnitude of the divergence or vertical advection, which need not be small. The AB is an ordered expansion in the square of the ratio of orbital to inertial frequencies, the square of a local Rossby number. The approximation filters gravity and inertial waves from the system. Advantage is taken of the weak asymmetries near the vortex care as well as the tendency for low azimuthal wavenumber asymmetries to dominate. Linearization about a symmetric balanced vortex allows the three-dimensional asymmetric dynamics to be deduced properly. The AB formulation has a geopotential tendency equation with a three-dimensional elliptic operator. The AB system has a uniformly valid continuation to nonlinear quasigeostrophic theory in the environment. It includes the full inertial dynamics of the vortex core, and reduces to Eliassen's formulation for purely axi-symmetric flow. It has a full set of conservation laws on fluid parcels analogous to those for primitive equations, including conservation of potential temperature, potential vorticity, three-dimensional vorticity, and energy. A weakly nonlinear extension of the formulation in the near-vortex region is presented. Appropriate physical applications for the AB system, as well as its limitations, are discussed.