B3 - stability of the atmosphere

consider at air parcel at equilibrium at a height $z_{0}$ and temperature $T_{0}$

weight = m_{p} g = m_{a} g = upthrust

raising the parcel to a new height, and calculating the acceleration in positive $z$ (upwards):

\begin{matrix} m_{p} \frac{d^{2} z}{d t^{2}} = m_{a} g - m_{p} g \\ \frac{d^{2} z}{d t^{2}} = - g (1 - \frac{ρ_{a}}{ρ_{p}}) = - g (1 - \frac{T_{p}}{T_{a}}) [∵ ρ \propto \frac{1}{T}, ideal gas law] \end{matrix}

\begin{matrix} T_{p} (z) = T_{0} - Γ Δ z \\ T_{a} (z) = T_{0} - Γ_{a} Δ z \end{matrix}

\frac{d^{2} z}{d t^{2}} = - \frac{g}{T_{a}} (Γ - Γ_{a}) Δ z

notice that this is an equation of simple harmonic motion, $\ddot{z} = - ω^{2} z$ , where $ω_{B V} = g (Γ - Γ_{a}) / T_{a}$ is the Brunt-Vaisala frequency
stable when $Γ_{a} \leq Γ$ and unstable when $Γ_{a} > Γ$