 Solution

For steady depth and no sources of fluid mass, the continuity equation gives ${\nabla }_h\cdot (H\mathbf{u})=0$. In a simply connected region this defines an ocean transport streamfunction:A multiply connected basin additionally needs its circulation or boundary-constant data. Taking the vertical curl of $\mathbf{u}=(-{\psi }_y,{\psi }_x)/H$ gives the exact relative vorticity from a variable-depth transport streamfunction relationNo small-depth-variation expansion has been made. No normal flow at a basin wall means that $\psi$ is constant along that connected wall.