Equivalent potential temperature

Remarks

Equivalently, $\theta_e$ is the potential temperature of the parcel after the release of all its latent heat. A common approximation is $$\theta_e \approx \theta \exp\left( \frac{L_v r}{c_p T_L} \right).$$ Here $\theta$ is the potential temperature and $L_v$ is the latent heat of vaporisation. Also $c_p$ is the specific heat of dry air at constant pressure and $T_L$ is the temperature at the lifting condensation level. More accurate empirical formulas are used in practice.

Example: with $\theta = 300$ K, $r = 0.015$, $L_v = 2.5 \times 10^6$ J/kg, $c_p = 1005$ J/(kg K) and $T_L = 290$ K, the exponent is about $0.129$ and $\theta_e \approx 341$ K. For dry air, $r = 0$ and $\theta_e = \theta$.

Non-example: if heat is added to the parcel from outside, for example by radiation or by mixing with other air, $\theta_e$ changes. It is conserved only under dry and moist adiabatic processes, and only approximately in the second case.

A layer in which $\theta_e$ decreases with height is called potentially unstable: it becomes unstable when lifted to saturation as a whole.