Clausius-Clapyron implies roughly a 6% increase in saturation vapor pressure per degree of warming. The low level moisture field scales with Clausius-Clapyron, but the strength of the global hydrological cycle does not, typically increasing much more slowly with increasing temperature. The implication is that the exchange of mass between the boundary layer and the free troposphere is reduced in strength. In the tropics, this conclusion is often reached by invoking the balance between subsidence heating and radiative cooling, and noting that the lapse rate decreases following a warmer moist adiabat. These arguments are essentially equivalent. Additional assumptions are required to infer that the mean tropical (Hadley/Walker) overturning will decrease in strength, as is often observed in climate models.
Less familiarly, the moist entropy budget potentially constrains the extent to which the generation of kinetic energy can change. The advantage of this constraint is that it applies to the kinetic energy flowing through all scales of motion, including the convective scales unresolved in GCMs. Unfortuntely, the entropy budget is made more complex by the possibility of changes in the partitioning of the irreversible entropy generation due to mixing of momentum and of moisture, which is difficult to estimate. I discuss whether it is possible to rule out climates with both stronger and more frequent "storms" using this entropy constraint.
Supplementary URL: