Quasi-geostrophic approximation (QG approximation)

The quasi-geostrophic approximation describes slowly evolving, nearly geostrophic flow with small Rossby number. Its dynamics are governed by advection and forcing of potential vorticity.

The quasi-geostrophic streamfunction defines horizontal velocity by $u=-{\psi }_y$ and $v={\psi }_x$. For a one-layer free-surface model, geostrophic balance gives $\psi =g\eta /f_0$.

For constant buoyancy frequency $N$ on a beta plane, three-dimensional quasi-geostrophic potential vorticity can be written

A quasi-geostrophic vertical mode separates horizontal and vertical dependence of the streamfunction. With rigid horizontal boundaries and constant stratification, the vertical eigenfunctions are $\cos (n\pi z/H)$.

An isopycnal displacement is the vertical displacement $\xi$ of a constant-density surface. In linear stratified quasi-geostrophic flow,