= Solution
For <slow cooling of an astrophysical disk>, let $t_{\rm cool}\gg\Omega^{-1}$ and assume a vertically symmetric laminar column with no significant radial mass exchange. Its vertical velocity is of order $H/t_{\rm cool}$, so the vertical <material derivative> of that velocity is of order $H/t_{\rm cool}^2$. Relative to gravity or <pressure> acceleration $\sim\Omega^2H$, the inertial correction is $O[(\Omega t_{\rm cool})^{-2}]$.
The <hydrostatic approximation> therefore holds to leading order while the column slowly cools and contracts. Non-turbulent evolution removes the prescribed turbulent heating, but not the compressional work in the <ideal gas> energy equation:
$$
\boxed{\partial_zP=-\Omega^2z\rho,\qquad
\frac1{\gamma-1}\left(D_tP-\frac{\gamma P}{\rho}D_t\rho\right)=-AT^{1+\beta},\qquad
P=\mathcal R\rho T.}
$$
The <continuity equation> must still determine the slow vertical flow; $D_t$ is not generally $\partial_t$ at fixed height. This is a quasi-static approximation for a well-prepared, mechanically stable column, with fast free oscillations or growing <convection> excluded from the assumed slow solution.
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