## Problems

1. Figure 5.5 shows the net incoming solar and outgoing long-wave irra-diance at the top of the atmosphere. Note that there is a net gain of radiation in low latitudes and a net loss in high latitudes. By inspection of the figure, estimate the magnitude of the poleward energy flux that must be carried by the atmosphere-ocean system across the 30° latitude circle, to achieve a steady state.

2. Suppose that the Earth's rotation axis were normal to the Earth-Sun line. The solar flux, measured per unit area in a plane normal to the Earth-Sun line, is So. By considering the solar flux incident on a latitude belt bounded by latitudes (<, < + d<), show that F, the 24-hr average of solar flux per unit area of the Earth's surface, varies with latitude as

F S0

(a) Using this result, suppose that the atmosphere is completely transparent to solar radiation, but opaque to infrared radiation such that, separately at each latitude, the radiation budget can be represented by the ''single slab'' model shown in Fig. 2.7. Determine how surface temperature varies with latitude.

(b) Calculate the surface temperature at the equator, 30°, and 60° latitude if Earth's albedo is 30% and S0 = 1367 Wm-2. Compare your result with observations shown in Fig. 5.7.

3. Use the hydrostatic relation and the equation of state of an ideal gas to show that the 1000-500 mbar ''thickness,'' Az = z(500mbar) -z(1000 mbar) is related to the mean temperature (T) of the 1000-500 mbar layer by

Az = RT. In 2, g where where the integrals are from 500 mbar to 1000 mbar. (Note that

(a) Compute the thickness of the surface to 500-mbar layer at

30° and 60° latitude, assuming that the surface temperatures computed in Problem 2b extend uniformly up to 500 mbar.

(b) Figures 7.4 and 7.25 (of Chapter 7) show 500 mbar and surface pressure analyses for 12 GMT on June 21, 2003. Calculate (T) for the 1000-mbar to 500-mbar layer at the center of the 500-mbar trough at 50°N, 120°W, and at the center of the ridge at 40°N, 90°W. [N.B. You will need to convert from surface pressure, ps, to height of the

1000 hPa surface, zi000; to do so use the (approximate) formula

Z1000 = 10 {ps - 1000), where z1000 is in meters and ps is in hPa.]

Is (T) greater in the ridge or the trough? Comment on and physically interpret your result.

4. Use the expression for saturated specific humidity, Eq. 4-24, and the empirical relation for saturated vapor pressure es (T), Eq. 1-4 (where A = 6.11 mbar, p = 0.067 °C-1, and T is in °C), to compute from the graph of T(p) in the tropical belt shown in Fig. 4.9, vertical profiles of saturated specific humidity, q*(p). You will need to look up values of R and Rv from Chapter 1.

Compare your q* profiles with observed profiles of q in the tropics shown in Fig. 5.15. Comment?

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