= Barotropic energy density
{title2=$\rho U^{\prime}-U=P$}
For a <barotropic fluid> with <pressure> $P=P(\rho)$, choose an energy density $U$ satisfying $\rho U'(\rho)-U=P$. The <continuity equation> then gives $\partial_tU+\nabla\cdot(U\mathbf u)=-P\nabla\cdot\mathbf u$. For an <isothermal equation of state> $P=c_s^2\rho$, one choice is $U=c_s^2\rho\ln(\rho/\rho_{\rm ref})$. A fixed reference density changes $U$ only by a conserved mass term. The same construction holds with <surface density> and vertically integrated <pressure>. This closure energy need not equal the microscopic internal energy of a gas maintained at fixed <temperature>.
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