Notes and correspondence on the momentum equation for the quasi-geostrophic model

The momentum equation for the quasi-geostrophic (QG) model derived based on the conventional Rossbynumber expansions does not uniquely determine the QG motion up to first order in the Rossby number. There are infinitely many ways of closing the equations. The momentum equation for QG derived by Holm and Zeitlin in 1998 based on a variational formulation for QG is compared with that for the conventional Rossby-number expansions. The underlying assumption in the construction of the variational formulation is geostrophic velocity for the particles. It is shown that the variational momentum equation corresponds to a particular way of closing the conventional momentum equation for QG. The numerical results for potential vorticity (PV) inversion on a circular vortex indicate a smaller range of applicability and loss of accuracy for the variational momentum equation for QG when compared with the QG one that sets the first-order linearized potential vorticity to zero.