## Precipitationtemperature relations

There is more to climate than temperature, and for atmospheres that contain a condensible substance - e.g. water on Earth or methane on Titan - the precipitation rate is of as much interest as the temperature. Aside from the role of rainfall in making land habitable on Earth, there are many reasons for being interested in precipitation. For example, it is the precipitation of snow that feeds the growth and flow of glaciers. Precipitation of water on Earthlike planets exerts a controlling influence on the chemical weathering processes that ultimately control atmospheric CO2, as will be discussed in Chapter 8. In the long term, the rate at which a condensible substance precipitates from an atmosphere must be balanced by the rate at which that substance evaporates or sublimates from a reservoir at the planet's surface - an ocean or glacier. Since latent heat flux can be turned into mass flux upon dividing by the appropriate latent heat, much can be learned about the evaporation or sublimation rate by a careful examination of the surface energy budget. To streamline the prose, we'll generally use the term "evaporation" to refer to evaporation or sublimation in the following, with the understanding that when the surface in question is ice the phase change is actually sublimation.

We begin by writing the surface balance in the form

FL(Tsa, Tg) = [(1 — ag)Sg — ae^] + cppsCdU(Tsa — Tg) + a(T4a — Tg4) (6.37)

In this equation we have assumed eg = 1 for simplicity. The functional form of FL given by Eq. 6.12. As usual, this equation must be solved for Tg given Tsa and the other parameters affecting the surface budget. We can distinguish two regimes: the weak evaporation regime and the strong evaporation regime. The weak evaporation limit is defined by the condition FL(Tsa, Tsa) ^ (1 — ag )Sg. This condition guarantees that FL will be negligible compare to the surface solar flux as long as the solution does not require that Tg be enormously greater than Tsa. Given the form of the Clausius-Clapeyron relation, the weak evaporation regime applies at sufficiently low temperatures, since in that case the atmosphere can carry little vapor even when it is saturated. The notion of "low" vs "high" temperature must be understood with reference to the volatility of the substance undergoing the phase change. For methane, 95K is a "high" temperature in this sense, but for water vapor even 250K is a "low" temperature.

In the weak evaporation limit, we can set the left hand side of Eq. 6.37 to zero when solving for Tg , and then use the resulting ground temperature to evaluate the evaporation by plugging it into the formula FL. Since the latent heat flux is small, leaving it out of the surface balance causes only small errors in Tg. In this limit, the evaporation is not significantly constrained by the energy supply. In the regime where Tg > Tsa - i.e. when the term in square brackets in Eq. 6.37 is positive - the exponential increase of saturation vapor pressure with temperature yielded by Clausius-Clapeyron leads to a roughly exponential increase of the latent heat flux with temperature, as Tsa is increased. Even when the surface absorbed solar radiation is so weak that an inversion forms in the surface layer, the control exerted by Clausius-Clapeyron is so strong that one still tends to get an exponential increase with temperature, unless the inversion gets very strong. The behavior is explored quantitatively in Problems ?? and ??.

At high temperatures, when the energy carried away by latent heat flux has a strong effect on the ground temperature, the behavior is very different. Intuitively, one expects that the evaporation can't increase beyond the point where the entirety of the absorbed solar radiation goes into evaporating material from the surface. It's not quite as simple as that, owing to the effect of sensible and radiative heat fluxes, but a constraint very similar to this does come into play. Let's assume that the condensible substance is like water vapor and makes e* « 0 in warm conditions. Then the surface balance becomes