PX282 - D2c - convection

\begin{array}{r} material: ρ_{1}, T_{1} \to ρ_{a}, T_{a} \\ bubble: ρ_{1}, T_{1} \to ρ_{2}, T_{2} \end{array}

consider a bubble of gas rising
the bubble moves slowly, and adiabatically (no energy exchange with the environment)
as it rises, the background density and temperature decrease
this causes the bubble to expand, hence changing its density and temperature
if $ρ_{2} > ρ_{a} :$ the bubble is stable and it falls back
if $ρ_{2} < ρ_{a} :$ the bubble is unstable, and it keeps rising
the schwarzschild criterion for stability:

{| \frac{d \ln T}{d \ln P} |}_{a d} > {| \frac{d \ln T}{d \ln P} |}_{r a d}