Shallow-water quasi-geostrophic potential vorticity ($Q={\nabla }^2\psi -\psi /L_D^2+\beta y$)

For small Rossby number and small free-surface displacement, shallow-water potential vorticity reduces, up to a constant factor and additive constant, to relative vorticity plus background PV gradients and the stretching term $-\psi /L_D^2$.

The barotropic deformation radius is the horizontal scale at which free-surface gravity and rotation balance. It contributes $L_D^{-2}$ to the denominator of the one-layer Rossby-wave dispersion relation.