Equilibrium Distribution of Rectilinear Vortices in a Rotating Container

Abstract

A system of rectilinear vortices in an arbitrary multiply connected domain rotating with angular velocity $\ensuremath{\Omega}$ is studied with Lin's general formalism. In the limit of many vortices, the equilibrium distribution is shown to be a uniform vortex density $n=\frac{2\ensuremath{\Omega}}{\ensuremath{\kappa}}$ where $\ensuremath{\kappa}$ is the circulation about each vortex. If the inner boundaries are specified by a set of contours ${C}_{\ensuremath{\alpha}}$, each enclosing an area ${A}_{\ensuremath{\alpha}}$, then the equilibrium value of the circulation ${\ensuremath{\Gamma}}_{\ensuremath{\alpha}}$ about ${C}_{\ensuremath{\alpha}}$ is given by ${\ensuremath{\Gamma}}_{\ensuremath{\alpha}}=2\ensuremath{\Omega}{A}_{\ensuremath{\alpha}}$. In equilibrium, the fluid rotates as a solid body with angular momentum $L=I\ensuremath{\Omega}$ and energy $E=\frac{1}{2} I{\ensuremath{\Omega}}^{2}$, where $I$ is the moment of inertia.