Analytic Solutions of the Rotating and Stratified Hydrodynamical Equations

In this article we investigate the two-dimensional incompressible rotating and stratified, just rotating, just stratified Euler equations, comparing with each other and with the normal Euler equations with the self-similar Ansatz. The motivation of our study is the following the presented rotating stratified fluid equations can be interpreted as a well-established starting point of various more complex and more realistic meteorologic, oceanographic or geographic models. We present analytic solutions for all four models for density, pressure and velocity fields, most of them are some kind of power-law type of functions. In general the presented solutions have a rich mathematical structure. Some solutions show nonphysical explosive properties others, however are physically acceptable and have finite numerical values with power law decays. For a better transparency we present some figs for the most complicated velocity and pressure fields. To our knowledge there are no such analytic results available in the literature till today. Our results may attract attention in various scientific fields.