 Depth independence of hydrostatic shallow-water flow

In a homogeneous layer under the hydrostatic approximation, the horizontal pressure gradient is independent of depth. The depth derivatives of velocity obey $(u_z{)}_t-f{v}_z=0$, $(v_z{)}_t+f{u}_z=0$. If the motion starts without shear, these derivatives remain zero. Hydrostatic pressure alone does not eliminate a pre-existing freely oscillating shear.