On solutions of the Euler equation for incoherent fluid on a rotating sphere

TL;DR

This paper investigates solutions to the Euler equation for an inviscid, compressible fluid on a rotating sphere, presenting hodograph equations, explicit solutions, blow-up curves, and effects of rotation speed.

Contribution

It introduces a class of solutions parameterized by arbitrary functions, derives hodograph equations, and analyzes blow-up behavior and rotation effects in the fluid dynamics context.

Findings

Derived hodograph equations for the Euler system.

Provided explicit solutions and characterized blow-up curves.

Analyzed the impact of rotation speed on fluid deformation.

Abstract

The motion of compressible, inviscid fluid under the constant pressure on a rotating sphere is studied. The hodograph equations for the corresponding Euler equation are presented. They provide us with the class of solutions of the Euler equation parameterized by two arbitrary functions of two variables. Several particular explicit solutions are given. The blow-up curves, on which the derivatives of velocitiy blows up, are described. The limiting cases of slowly and rapidly rotating sphere are considered. The equation describing the deformations of elliptic functions modulus is presented.