Axisymmetric geostrophic mode ($\omega =0,u=0,v=i{k}_x({\Phi }_1+c_s^2{\Sigma }_1/{\Sigma }_0)/(2\Omega )$)

A stationary mass density disturbance can be balanced by a changed azimuthal velocity in a shearing sheet. The continuity and azimuthal equations give zero radial velocity, while the radial equation balances Coriolis force against pressure and gravity. This mode is separate from the density-wave pair in the softened Toomre dispersion relation and is lost if one divides every equation by frequency.