On trajectories of vortices in the compressible fluid on a two-dimensional manifold

TL;DR

This paper analyzes steady vortex solutions in a compressible fluid on a 2D manifold, modeling tropical cyclones, and examines their stability and shape evolution through numerical simulations.

Contribution

It introduces a class of steady vortex solutions on a rotating manifold and assesses their accuracy and stability in geophysical fluid models.

Findings

Solutions are exact or nearly exact near the vortex center.

Numerical simulations show vortex shape changes when discrepancies are neglected.

The model applies to tropical cyclones and similar geophysical phenomena.

Abstract

For the model of a compressible barotropic fluid on a two dimensional rotating Riemmanian manifold we discuss a special class of smooth solutions having a form of a steady non-singular vortex moving with a bearing field. The model can be obtained from the system of primitive equations governing the motion of air over the Earth surface after averaging over the height and therefore the solution obtained can be interpreted as a tropical cyclone which is known as a long time existing stable vortex. We consider approximations of $l$- plane and $\beta$ - plane used in geophysics for modeling of middle scale processes and equations on the whole sphere as well. We show that the solutions of the mentioned form satisfy the equations of the model either exactly or with a discrepancy which is small in a neighborhood of the trajectory of the center of vortex. We perform a numeric study of the change of the shape of the vortex affected by the neglecting the discrepancy term.