Zero-frequency balanced mode of an axisymmetric disk ($\omega =0,U_x=0,2\Omega U_y=ik{c}_s^2\sigma /{\Sigma }_0$)

Besides the two propagating inertial-acoustic wave branches, a uniform barotropic Keplerian shearing sheet has a stationary branch balancing pressure against Coriolis acceleration. For radial wavenumber $k\ne 0$, setting $\omega =0$ in continuity gives $U_x=0$, and radial momentum gives the displayed azimuthal balance. The full determinant is proportional to $\omega ({\omega }^2-{\Omega }^2-c_s^2{k}^2)$. At $k=0$ the stationary branch is a uniform column-density change; it is not a travelling density wave.