A balanced approximation of the one-layer shallow-water equations on a sphere

A global version of the equivalent barotropic vorticity equation is derived for the one-layer shallow-water equations on a sphere. The equation has the same form as the corresponding beta plane version, but with one important difference: the stretching (Cressman) term in the expression of the potential vorticity retains its full dependence on the Coriolis parameter. As a check of the resulting system we consider the dynamics of linear Rossby waves. It is shown that these waves are rather accurate approximations of the westward propagating waves of the second class of the original shallow-water equations. It is also concluded that for Rossby waves with short meridional wavelengths the squared Coriolis parameter in the stretching term can be replaced by a constant, corresponding to the Coriolis parameter at 45 degrees latitude.