Riemann ellipsoid

Example

Take a homogeneous, incompressible, self-gravitating fluid of constant density $\rho$ that fills the ellipsoid $x_1^2/a_1^2 + x_2^2/a_2^2 + x_3^2/a_3^2 \le 1$ with semi-axes $a_1, a_2, a_3 > 0$. Let the ellipsoid rotate rigidly with angular velocity $\Omega$ about the $x_3$-axis. In the rotating frame, let the fluid move with the velocity field $$u = \lambda \left( \frac{a_1}{a_2} x_2, \; -\frac{a_2}{a_1} x_1, \; 0 \right),$$ where $\lambda$ is a real constant. This flow is tangent to the ellipses $x_1^2/a_1^2 + x_2^2/a_2^2 = \text{const}$, since $\frac{d}{dt}\big(x_1^2/a_1^2 + x_2^2/a_2^2\big) = 2\lambda\big(x_1 x_2/(a_1 a_2) - x_1 x_2/(a_1 a_2)\big) = 0$, so the surface keeps its shape. It has divergence zero and uniform vorticity $\zeta = -\lambda (a_1^2 + a_2^2)/(a_1 a_2)$ along the $x_3$-axis. A Riemann ellipsoid of type S is such a configuration with suitable $a_1, a_2, a_3$, $\Omega$ and $\zeta$, chosen as follows. The pressure given by the Euler equations with the gravity of the ellipsoid vanishes on the boundary. Then the figure is a steady equilibrium.