The lower 10-15 km of the atmosphere-the troposphere-is often in a convective steady state at constant entropy, not constant temperature. In such equilibrium pVy is independent of altitude, where γ = Cp/Cv. Use the condition of mechanical equilibrium in a uniform gravitational field to:
(a) Show that dT/dz = constant, where z is the altitude. This quantity, important in meteorology, is called the dry adiabatic lapse rate. (Do not use the barometric pressure relation that was derived in Chapter for an isothermal atmosphere.)
(b) Estimate dT/dz, in ◦C per km. Take γ = 7/5.
(c) Show that p ∞ p2, where p is the mass density.
If the actual temperature gradient is greater than the isentropic gradient, the atmosphere may be unstable with respect to convection.