Geostrophic Adjustment in an Axisymmetric Vortex

Abstract

A linearized system of equations for the atmosphere's first internal mode in the vertical is derived. The system governs small-amplitude, forced, axisymmetric perturbations on a basic-state tangential flow which is independent of height. When the basic flow is at rest, solutions for the transient and final adjusted state are found by the method of Hankel transforms. Two examples are considered, one with an initial top hat potential vorticity and one with an initial Gaussian-type potential vorticity. These two examples, which extend the work of Fischer (1963) and Obukhov (1949), indicate that the energetical efficiency of cloud-cluster-scale heating in producing balanced vortex flow is very low, on the order of a few percent. The vast majority of the energy is simply partitioned to gravity-inertia waves. In contrast the efficiency of cloud-cluster-scale vorticity transport is very high. When the basic state possesses positive relative vorticity in an inner region, the energy partition can be substantially modified, and cloud-cluster-scale heating can become considerably more efficient. The energy partition results have important implications for the lateral boundary condition used in tropical cyclone models. Faced with the fact that a perfect non-reflecting condition is possible but impractical to implement, one is forced to use an approximate condition which causes some reflection of gravity-inertia waves and hence some distortion of the geostrophic adjustment process. The distortion can be kept small by the use of a suitable radiation condition.