Suppose a GCM model is run with 2x CO2, and WRF is nested in the GCM, but with 1x CO2. In other words, the excess CO2 is filtered out as it enters the WRF domain.
This page uses the model featured in RadiativeConvectiveEquilibrium to estimate the change, or error, in the infrared heating rate within the WRF domain.
Heating rates for radiative convective equilibrium (colors), together with the longwave heating rate (black) at the instant of CO2 reduction. All other temperatures and fluxes are held fixed at the instant of reduction, the model is not run out to a new equilibrium. (The new equilibrium would be simply the stn1x experiment). The lower troposphere cools at a rate of 0.1 K per day, the stratosphere warms at a about 1 K per day. The land surface will also cool (not shown), because downward longwave radiation to the surface is reduced from 414.7473 W m-2 to 413.7178 W m-2, a reduction of 1.03 W m-2. Even without the effect of surface cooling, the change in CAPE after one day will be of order gΔT/T * 10,000 m = 30 J/kg.