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

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.

Figures (2)

• Figure 1: The stationary solution (\ref{['sol-redangmom-eq']}) at $F_1=1$, $F_2=0$ is ${\bf{u}}=( \sin(\phi+\omega t), 2 \cos(\phi+\omega t)/ \sin(2 \theta) -\omega )$. It is presented at $\omega=1$, $t=0$. This solution admits singularities for all times at $\theta=0,\pi/2,\pi$. It is locally well defined solution far from the equator and poles.
• Figure 2: The stationary solution (\ref{['solCorsol-stat']}) when $\tilde{\Phi}(x)=\cos(2x)$ is ${\bf{u}}=( 1 + \cos (2(\phi + \omega t)) - \omega^2 \sin^2(\theta) ,-\omega )$. It is presented at $\omega=1$, $t=0$.