Gradient adjustment in an axisymmetric vortex

Abstract

We consider gradient adjustment in an axisymmetric vortex. From an initially unbalanced state, the final adjusted state is obtained. Initially, the geopotential is flat and the velocity field is given by the Rankine vortex. (Note: To view the equation, open the pdf of the document.) We derive a system of equations governing the final adjusted state. This system of equations is solved numerically using the shooting method. We also present solutions for the geostrophic adjustment problem and compare these to the solutions for the gradient adjustment problem. In the gradient adjustment case, the results indicate that (i) the geopotential decrease at the centre is proportional to E/a. (ii) the decrease of the maximum tangential wind is dependent on E for smaller a (a is less than 1). For a greater or equal to 1, the decrease is independent of E. (iii) the stretching is proportional to E/a. (iv) the mass removed from within a given region tends to E for large r. Comparison between the solutions for the geostrophic and gradient adjustment problems shows significant differences for small a.