Relative vorticity from a variable-depth transport streamfunction ($\zeta =\nabla \cdot (H^{-1}\nabla \psi )$)

The ocean transport streamfunction convention $Hu=-{\psi }_y$, $Hv={\psi }_x$ gives relative vorticity $\zeta =H^{-1}{\nabla }^2\psi -H^{-2}\nabla H\cdot \nabla \psi$. This identity is exact for variable positive depth and does not assume that depth variations are small.