Isobaric thermal instability ($\sigma =-c_p^{-1}(\partial \mathcal{L}/\partial T{)}_p$)

At thermal equilibrium, let $\mathcal{L}(\rho ,T)$ be net cooling per unit mass, including a specified heating law. A gas perturbation that stays at constant pressure obeys $c_{p}d\delta T/dt=-(\partial \mathcal{L}/\partial T{)}_p\delta T$. It grows when this derivative is negative. The isobaric approximation requires pressure communication faster than thermal evolution; a volumetric cooling function instead introduces a background mass density in the denominator. Thermal conduction alters the short-wavelength criterion.