10. In Section 3.3 we showed that the pressure of an isothermal atmosphere varies exponentially with height. Consider now an atmosphere with uniform potential temperature. Find how pressure varies with height, and show in particular that such an atmosphere has a discrete top (where p ^ 0) at altitude RTo/ (Kg), where R, k, and g have their usual meanings, and To is the temperature at 1000 mbar pressure.
11. Consider the convective circulation shown in Fig. 4.29. Air rises in the center of the system; condensation occurs at altitude zB = 1 km
(pB = 880 mbar), and the convective cell (cloud is shown by the shading) extends up to zT = 9 km (pT = 330 mbar), at which point the air diverges and descends adiabatically in the downdraft region. The temperature at the condensation level, TB, is 20°C. Assume no entrainment and that all condensate falls out immediately as rain.
(a) Determine the specific humidity at an altitude of 3 km within the cloud.
(b) The upward flux of air, per unit horizontal area, through the cloud at any level z is w(z)p(z), where p is the density of dry air and w the vertical velocity. Mass balance requires that this flux be independent of height within the cloud. Consider the net upward flux of water vapor within the cloud, and hence show that the rainfall rate below the cloud (in units of mass per unit area per unit time) is wBpB (q*B - q*r ), where the subscripts "B" and "T" denote the values at cloud base and cloud top, respectively. If wB = 5 cm s-1, and pB = 1.0 kgm-3, determine the rainfall rate in cm per day.
12. Observations show that, over the Sahara, air continuously subsides (hence the Saharan climate). Consider an air parcel subsiding in this region, where the environmental temperature Te decreases with altitude at the constant rate of 7 K km
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