Suppose the air parcel leaves height z with the environmental temperature. Assuming the displacement to be adiabatic, show that, after a time St, the parcel is warmer than its environment by an amount wAeSt, where w is the subsidence velocity and
where cp is the specific heat at constant pressure.
Suppose now that the displacement is not adiabatic, but that the parcel cools radiatively at such a rate that its temperature is always the same as its environment (so the circulation is in equilibrium). Show that the radiative rate of energy loss per unit volume must be c p pcpwAe, and hence that the net radiative loss to space per unit horizontal area must be pcpwAe dz g wAe dp, where ps is surface pressure and p is the air density.
(c) Radiative measurements show that, over the Sahara, energy is being lost to space at a net, annually-averaged rate of 20 W m-2. Estimate the vertically-averaged (and annually-averaged) subsidence velocity.
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