(a) Consider the level that divides the atmosphere into two equal parts by mass (i.e., one half of the atmospheric mass is above this level). What is the altitude, pressure, and density at this level?

(b) Repeat the calculation of part (a) for the level below which lies 90% of the atmospheric mass.

4. Derive an expression for the hydrostatic atmospheric pressure at height z above the surface in terms of the surface pressure ps and the surface temperature Ts for an atmosphere with constant lapse rate of temperature r = -dT/dz. Express your results in terms of the dry adiabatic lapse rate rd = g (see Section 4.3.1). Calculate the cp height at which the pressure is 0.1 of its surface value (assume a surface temperature of 290 K and a uniform lapse rate of 10 K km-1).

5. Spectroscopic measurements show that a mass of water vapor of more than 3 kgm-2 in a column of atmosphere is opaque to the "terrestrial" waveband. Given that water vapor typically has a density of 10-2 kgm-3 at sea level (see Fig. 3.3) and decays in the vertical as e-(b), where z is the height above the surface and b ~ 3 km, estimate at what height the atmosphere becomes transparent to terrestrial radiation.

By inspection of the observed vertical temperature profile shown in Fig. 3.1, deduce the temperature of the atmosphere at this height. How does it compare to the emission temperature of the Earth, Te = 255 K, discussed in Chapter 2? Comment on your answer.

6. Make use of your answer to Problem 1 of Chapter 1 to estimate the error incurred in p at 100 km through use of Eq. 3-11 if a constant value of g is assumed.

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