Autoconvective Lapse Rate

From hydrostatic equilibrium and the ideal gas law, P = Rg rho(z) T(z) we have

rho = -(1/g) · (dP/dz) = -(Rg/g) · [ T(z) drho/dz + rho(z) dT/dz ]
dT/dz = - g/Rg - [T(z)/rho(z) · drho/dz]
But T(z) and rho(z) are always positive so that when
Gamma ident -dT/dz geq g/Rg

then drho/dz geq 0. Thus, the autoconvective criteria is that when the density increases with altitude the atmosphere will be forced to convectively adjust and this condition is met when

Gammaauto ident g/Rg