Problem 1.40. In Problem 1.16 you calculated the pressure of earth's atmosphere as a function of altitude, assuming constant temperature. Ordinarily, however, the temperature of the bottommost 10-15 km of the atmosphere (called the tropo- sphere) decreases with increasing altitude, due to heating from the ground (which is warmed by sunlight). If the temperature gradient (df /de| exceeds a certain critical value, convection will occur: Warm, low-density air will rise, while cool, high-density air sinks. The decrease of pressure with altitude causes a rising air mass to expand adiabatically and thus to cool. The condition for convection to occur is that the rising air mass must remain warmer than the surrounding air despite this adiabatic cooling. (a) Show that when an ideal gas expands adiabatically, the temperature and pressure are related by the differential equation dT 2 T dP f+2 P (b) Assume that d / de is just at the critical value for convection to begin, so that the vertical forces on a convectiong air mass are always approximately in balance. Use the result of Problem 1.16(b) to find a formula for d7 / dz in this case. The result should be a constant, independent of temperature and pressure, which evaluates to approximately -10"C/km. This fundamental meteorological quantity is known as the dry adiabatic lapse rate.with this additional part: c] What is T[z] in this case? At what altitude must this model be wrong? Problem 1.16. The exponential atmosphere. (a) Consider a horizontal slab of air whose thickness (height) is dz. If this slab is at rest, the pressure holding it up from below must balance both the pressure from above and the weight of the slab. Use this fact to find an expression for dP/de, the variation of pressure with altitude, in terms of the density of air. (b) Use the ideal gas law to write the density of air in terms of pressure, tem- perature, and the average mass m of the air molecules. (The information needed to calculate m is given in Problem 1.14.) Show, then, that the pressure obeys the differential equation dP P. dz KI called the barometric equation. (c) Assuming that the temperature of the atmosphere is independent of height ( not a great assumption but not terrible either), solve the barometric equa- tion to obtain the pressure as a function of height: P(2) = P(0je-my=/KT Show also that the density obeys a similar equation.(d) Estimate the pressure, in atmospheres, at the following locations: Ogden, Utah (4700 ft or 1430 m above sea level); Leadville, Colorado (10.150 ft, 3090 m); Mt. Whitney, California (14,500 ft, 4420 m); Mt. Everest, Nepal/ Tibet (29,000 ft, 8850 m). (Assume that the pressure at sea level is 1 atm.)