Potential Temperature and Adiabatic Processes
Potential temperature $\theta$ is the temperature a parcel would have if brought dry-adiabatically to a reference pressure $p_0 = 1000\ \text{hPa}$:
\[\theta = T\left(\frac{p_0}{p}\right)^{R/c_p}\]where $R = 287\ \text{J kg}^{-1}\text{K}^{-1}$ and $c_p = 1004\ \text{J kg}^{-1}\text{K}^{-1}$. For dry adiabatic ascent $\theta$ is conserved; it increases monotonically with height in a stable atmosphere.
Lapse Rates and Stability
| Lapse rate | Symbol | Value |
|---|---|---|
| Dry adiabatic | $\Gamma_d$ | $9.8\ \text{K km}^{-1}$ |
| Moist adiabatic | $\Gamma_s$ | $4–7\ \text{K km}^{-1}$ |
| Environmental | $\Gamma_e$ | varies |
The atmosphere is conditionally unstable when $\Gamma_s < \Gamma_e < \Gamma_d$. Saturation occurs at the lifting condensation level (LCL), above which the parcel cools at the moist adiabatic rate.
CAPE and Convective Potential
Convective Available Potential Energy measures the work done on a rising parcel between the Level of Free Convection (LFC) and the Equilibrium Level (EL):
\[\text{CAPE} = \int_{\text{LFC}}^{\text{EL}} g\,\frac{T_{v,\text{parcel}} - T_{v,\text{env}}}{T_{v,\text{env}}}\,dz\]Values above $2500\ \text{J kg}^{-1}$ indicate the potential for severe convection. The skew-T log-p diagram provides a graphical framework for reading these integrals directly from radiosonde profiles.