Equilibrium Distribution of Rectilinear Vortices in a Rotating Container

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