 Solution

Let $\mathbf{u}=-(3/2)\Omega x{\mathbf{e}}_y+\mathbf{v}$ and $\rho ={\rho }_0(1+\delta )$, with $\delta$ and $\mathbf{v}$ small. Define the sound speed by $c^2=(dp/d\rho {)}_0$, assumed positive. The background Keplerian shear balances the radial Coriolis force and the shearing-sheet tidal potential. For disturbances preserving axisymmetry ${\partial }_y=0$, its advective derivative ${\mathbf{u}}_0\cdot \nabla$ vanishes, but $\mathbf{v}\cdot \nabla {\mathbf{u}}_0=-(3/2)\Omega v_x{\mathbf{e}}_y$ remains. The linearized Euler equations and continuity equation areThe azimuthal coefficient is $(2-3/2)\Omega =\Omega /2$, not $2\Omega$: it combines the background velocity gradient with the Coriolis force. The uniform background has no buoyancy or vertical stratification in this stated model.