Shearing-sheet tidal potential (${\Phi }_t$)

For a frame rotating about a Keplerian circular orbit, radial gravity and centrifugal acceleration combine to give ${\Phi }_t=-3{\Omega }^2{x}^2/2$.

For constant isothermal sound speed, time-independent shearing-sheet tidal potential ${\Phi }_t$, and inviscid flow, define $E=\Sigma u^2/2+c_s^2\Sigma \ln (\Sigma /{\Sigma }_{\mathrm{ref}})+\Sigma {\Phi }_t$. The barotropic energy density and kinetic energy balances combine to give ${\partial }_{t}E+\nabla \cdot [(E+P)\mathbf{u}]=0$. Coriolis acceleration does no work. Integral conservation requires vanishing net boundary energy flux.