Analytic solutions of the rotating and stratified hydrodynamical equations

TL;DR

This paper derives analytic solutions for rotating and stratified hydrodynamical equations, revealing complex mathematical structures and physical properties, including both physically acceptable and unphysical solutions.

Contribution

It provides explicit analytic solutions for four models of rotating and stratified Euler equations using a self-similar Ansatz, highlighting their mathematical richness.

Findings

Solutions include compound power-law functions.

Some solutions exhibit unphysical explosive behavior.

Physically acceptable solutions have finite values and decay with power laws.

Abstract

In this article we investigate the two-dimensional incompressible rotating and stratified, just rotating, just stratified Euler equations with each other and with the normal Euler equations with the self-similar Ansatz. There are analytic solutions available for all four models, for density, pressure and velocity fields, some of them are compound power-law dependent functions. In general the solutions have a rich mathematical structure. Some solutions show unphysical explosive properties others, however are physically acceptable and have finite numerical values with power law decays. For a better transparency we present some figures for the most complicated velocity and pressure fields.