A useful way to relate vorticity & pressure especially when momentum advection (e.g. \(u\frac{\partial u}{\partial x}=0\)) is important. Recall 1-D shallow water equation:
$$
\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=-g\frac{\partial \eta}{\partial x} - Ru.
$$
For steady, frictionless flow,