The climate time scale in the approach to radiative-convective equilibrium

In this paper, we discuss the importance of the surface boundary condition (fixed versus interactive surface temperature) for the long time scale of approach to Radiative-Convective Equilibrium (RCE). Using a simple linearized two-variable model for surface-atmosphere interaction, we derive an analytic expression for tau(C), a long climate relaxation time scale that remains well defined and much longer than either mixing time scale of Tompkins and Craig (1998b), even in the limit that the heat capacity of the surface vanishes. We show that the size of tau(C) is an intrinsic property of the coupling between the atmosphere and surface, and not a result of the thermal inertia of the surface alone. When the surface heat capacity is low, tau(C) can be several times longer than expected, due to the effects of moisture on the effective heat capacity of the atmosphere. We also show that the theoretical expression for tau(C) is a good predictor of best fit exponential relaxation time scales in a single-column model with full physics, across a range of surface temperatures and surface heat capacities.