Past exam of the mathematics course of the University of Cambridge / 2008 / ib / Paper 3 / 18B / iii / Solution

The Euler equations for an inviscid fluid are ${\partial }_t\mathbf{v}+(\mathbf{v}\cdot \nabla )\mathbf{v}=-\nabla p/\rho +\mathbf{f}$. Here the vorticity is $2\Omega {\mathbf{e}}_z$, and direct differentiation gives $(\mathbf{v}\cdot \nabla )\mathbf{v}=-{\Omega }^2(x,y,0)$, whose curl vanishes. Taking the curl therefore givessoFor constant force coefficients and the initial rest condition, $\Omega (t)=(\gamma -\beta )t/2$. Only the antisymmetric part of the horizontal force gradient generates rotation; its symmetric part is balanced by pressure.