Chapter 17, Problem 66P

A theoretical atmospheric lapse rate. Section 16.7 described experimental data on the decrease in temperature with altitude in the Earth’s atmosphere. Model the troposphere as an ideal gas, everywhere with equivalent molar mass M and ratio of specific heats y. Absorption of sunlight at the Earth’s surface warms the troposphere from below, so vertical convection currents are continually mixing the air. As a parcel of air rises, its pressure drops and it expands. The parcel does work on its surroundings, so its internal energy decreases and it drops in temperature. Assume that the vertical mixing is so rapid as to be adiabatic. (a) Show that the quantity TP ^{(1 − γ)/γ} has a uniform value through the layers of the troposphere. (b) By differentiating with respect to altitude y, show that the lapse rate is given by

$\frac{dT}{dy}=\frac{T}{P}\left(1-\frac{1}{\lambda }\right)\frac{dP}{dy}$

(c) A lower layer of air must support the weight of the layers above. From Equation 15.4, observe that mechanical equilibrium of the atmosphere requires that the pressure decrease with altitude according to dP/dy = −ρg.

The depth of the troposphere is small compared with the radius of the Earth, so you may assume that the free-fall acceleration is uniform. Proceed to prove that the lapse rate is

$\frac{dT}{dy}=-\left(1-\frac{1}{\gamma }\right)\frac{Mg}{R}$

Problem 16.50 in Chapter 16 calls for evaluation of this theoretical lapse rate on the Earth and on Mars and for comparison with experimental results.