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By deriving analytical solutions to radiative–convective equilibrium (RCE), it is shown mathematically that convective available potential energy (CAPE) exhibits Clausius–Clapeyron (CC) scaling over a wide range of surface temperatures up to 310 K. Above 310 K, CAPE deviates from CC scaling and even decreases with warming at very high surface temperatures. At the surface temperature of the current tropics, the analytical solutions predict that CAPE increases at a rate of about 6%–7% per kelvin of surface warming. The analytical solutions also provide insight on how the tropopause height and stratospheric humidity change with warming. Changes in the tropopause height exhibit CC scaling, with the tropopause rising by about 400 m per kelvin of surface warming at current tropical temperatures and by about 1–2 km K−1 at surface temperatures in the range of 320–340 K. The specific humidity of the stratosphere exhibits super-CC scaling at temperatures moderately warmer than the current tropics. With a su...

Clausius–Clapeyron Scaling of CAPE from Analytical Solutions to RCE