= Isothermal shearing-sheet energy conservation
{title2=$\partial_tE+\nabla\cdot[(E+P)\mathbf u]=0$}
For constant <isothermal sound speed>, time-independent <shearing-sheet tidal potential> $\Phi_t$, and inviscid flow, define $E=\Sigma u^2/2+c_s^2\Sigma\ln(\Sigma/\Sigma_{\rm ref})+\Sigma\Phi_t$. The <barotropic energy density> and <kinetic energy> balances combine to give $\partial_tE+\nabla\cdot[(E+P)\mathbf u]=0$. <Coriolis acceleration> does no work. Integral conservation requires vanishing net boundary <energy flux>.
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