Deductions from a simple climate model: Factors governing surface temperature and atmospheric thermal structure

Radiative equilibrium solutions are the starting point in our attempt to understand how the atmospheric composition governs the surface and atmospheric temperatures, and the greenhouse effect. The Schwarzschild analytical grey gas model (SGM) was the workhorse of such attempts. However, the solutions suffered from serious deficiencies when applied to Earth's atmosphere and were abandoned about 3 decades ago in favor of more sophisticated computer models. Here we show that simple heuristic modifications to the SGM resolve these deficiencies, at least the catastrophic ones. The modifications include the addition of a spectral window, as well as allowing the scale height of optical depth to be different from that of the atmospheric pressure.

The modified SGM reveals the fundamental factors that govern the radiative equilibrium thermal structure. (1) The presence of a spectral window allows the temperature jump between the surface and the immediately overlying atmosphere to become small, irrespective of the magnitude of the surface temperature. (2) In an optically thick atmosphere (such as Earth and Venus), the surface temperature and the runaway greenhouse effect depend inversely (as the one-fourth power) on the Planck function-weighted width of the window and are independent of the optical depth and other atmospheric parameters. The vertical variation of temperature within the atmosphere, however, is determined by the vertical variation of optical depth. (3) The degree of convective instability of the radiative equilibrium thermal structure γ is proportional to the ratio H/H_τ, where H is the atmospheric scale and H_τ is the scale height of the vertical variation of optical depth. Here, γ is the ratio of the radiative equilibrium lapse rate and the neutral lapse rate for dry convection; γ>1 indicates that the profile is unstable to free convection. Thus H_τ<H emerges as one of the fundamental criteria for convective instability. These factors lead to several corollaries regarding the specific details of atmospheric thermal structure on Earth and Venus.

For example, according to the modified SGM, the radiative equilibrium temperature profile is strongly superadiabatic in Earth's lower atmosphere because H_τ is dominated by the water vapor's scale height being much less (2 compared with 8 km) than the atmospheric scale height. In the case of Venus, the pressure broadening of the CO₂ rotational lines makes H_τ a factor of 2 smaller than H. Some of these results have been obtained using more detailed, multispectral radiative equilibrium models.