Ocean transport streamfunction ($Hu=-{\psi }_y,Hv={\psi }_x$)

An ocean transport streamfunction defines depth-integrated horizontal transport. For constant depth $H$, it is $H$ times the velocity streamfunction. Its difference between two boundaries gives signed transport across a connecting line.

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.