Convective available potential energy

Definition

Consider a hydrostatic atmosphere with gravitational acceleration $g$ . Lift an air parcel from near the surface, dry adiabatically up to saturation and then moist adiabatically. Let $T_{v,p}(z)$ be the virtual temperature of the parcel and $T_{v,e}(z)$ that of the environment at height $z$ . The level of free convection $z_f$ is the lowest height above which the parcel is warmer than the environment, $T_{v,p} > T_{v,e}$ . The equilibrium level $z_n > z_f$ is the height above $z_f$ at which the parcel first becomes as cold as the environment again. The convective available potential energy is $\mathrm{CAPE} = \int_{z_f}^{z_n} g \, \frac{T_{v,p}(z) - T_{v,e}(z)}{T_{v,e}(z)} \, dz.$ If there is no level of free convection, $\mathrm{CAPE} = 0$ .

Remarks

The integrand is the buoyancy acceleration of the parcel, so CAPE has units of J/kg. With the hydrostatic relation and the ideal gas law, CAPE $= R_d \int_{p_n}^{p_f} (T_{v,p} - T_{v,e}) \, d \ln p$ , where $R_d$ is the gas constant of dry air and $p_f$ , $p_n$ are the pressures at $z_f$ and $z_n$ . The negative buoyancy below $z_f$ gives the convective inhibition.

Example: if $T_{v,e} = 300$ K and $T_{v,p} - T_{v,e} = 3$ K on the whole layer from $z_f = 1$ km to $z_n = 11$ km, then CAPE $= 9.81 \cdot 0.01 \cdot 10^4 \approx 981$ J/kg. Parcel theory then bounds the updraught speed by $\sqrt{2 \, \mathrm{CAPE}} \approx 44$ m/s.

Non-example: suppose the environmental lapse rate is smaller than the moist adiabatic lapse rate at every level. Then a lifted parcel stays colder than its surroundings, there is no level of free convection, and CAPE is zero.

Reading path

To understand this page from the ground up, read these 2 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Lapse rate concept
Convective available potential energy definition

Tag	`13ZJFCMHCE`
URI	`m2://13ZJFCMHCE`
Kind	definition
Writing	W0 stub
MSC	86A10
Defines	Convective available potential energy
Uses	Lapse rate
Changed	3 November 2026

Convective available potential energy

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here