Adiabatic temperature gradient (${\nabla }_{\mathrm{ad}}$)

The adiabatic temperature gradient is ${\nabla }_{\mathrm{ad}}=(\partial \log T/\partial \log P{)}_s$. For a perfect gas with constant adiabatic index, ${\nabla }_{\mathrm{ad}}=(\gamma -1)/\gamma$.

A saturated ascending air parcel releases latent heat as water vapor condenses, offsetting part of its expansion cooling. Its temperature decreases less steeply than the dry adiabatic lapse rate, with the gradient depending on temperature, pressure, moisture and whether condensate remains in the parcel. It is not a universal constant equal to the typical environmental lapse rate.

For a homogeneous dry ideal gas in a hydrostatic adiabat, $dT/dz=-g/C_p$, where $z$ increases upwards and $C_p$ is the specific heat capacity at constant pressure per unit mass. It follows from $dT/dP=1/(\rho C_p)$ and hydrostatic equilibrium. The altitude version of the Schwarzschild criterion is $dT/dz<-g/C_p$ for instability; equality is marginal. Moisture, composition gradients and nonideal equations of state modify the criterion.

A dry parcel in rapid pressure balance with an ambient ideal gas has $d{T}_g/dP=T_g\mathcal{R}/(P{C}_p)$. Ambient hydrostatic equilibrium gives $dP/dz=-Pg/(\mathcal{R}{T}_{\mathrm{env}})$, hence $d{T}_g/dz=-(g/C_p)T_g/T_{\mathrm{env}}$. At a temperature-matched launch point this equals the dry adiabatic lapse rate, supplying the local linear stability test. A substantially hotter parcel does not exactly follow the ambient density's hydrostatic adiabat.