A simple expression for vertical convective fluxes in planetary atmospheres

We explore the vertical convective flux F_c in a radiative-convective grey atmosphere. An expression of the form F_c=F_sτ_o/(C+Dτ_o) appears useful, where F_s is the shortwave flux absorbed at the base of an atmosphere with longwave optical depth τ_o and C and D are constants. We find excellent agreement with an idealized grey radiative-convective model with no shortwave absorption for D=1 and C=1∼2 depending on the surface-atmosphere temperature contrast and on the imposed critical lapse rate. Where shortwave absorption is correlated with longwave opacity, as in the atmospheres of Earth and Titan, C=2, D=2 provides an excellent fit, validated against the present terrestrial situation and the results of a nongrey model of Titan's strongly antigreenhouse atmosphere under a wide range of conditions. The expression may be useful for studying the energetics of planetary climates through time where there is insufficient data to constrain more elaborate models.