An Analytical Model for Spatially Varying Clear-Sky CO2 Forcing

Clear-sky CO2 forcing is known to vary significantly over the globe, but the state dependence that controls this is not well understood. Here we extend the formalism of Wilson and Gea-Banacloche to obtain a quantitatively accurate analytical model for spatially varying instantaneous CO2 forcing, which depends only on surface temperature T-s, stratospheric temperature, and column relative humidity (RH). This model shows that CO2 forcing can be considered a swap of surface emission for stratospheric emission, and thus depends primarily on surface-stratosphere temperature contrast. The strong meridional gradient in CO2 forcing is thus largely due to the strong meridional gradient in T-s. In the tropics and midlatitudes, however, the presence of H2O modulates the forcing by replacing surface emission with RH-dependent atmospheric emission. This substantially reduces the forcing in the tropics, introduces forcing variations due to spatially varying RH, and sets an upper limit (with respect to T-s variations) on CO2 forcing that is reached in the present-day tropics. In addition, we extend our analytical model to the instantaneous tropopause forcing, and find that this forcing depends on T-s only, with no dependence on stratospheric temperature. We also analyze the tau = 1 approximation for the emission level and derive an exact formula for the emission level, which yields values closer to tau = 1/2 than to tau = 1.