Problems

1. Consider a zonally symmetric circulation (i.e., one with no longitudinal variations) in the atmosphere. In the inviscid upper troposphere, one expects such a flow to conserve absolute angular momentum, so that DA/Dt = 0, where A = Qa2 cos 2p + ua cos p is the absolute angular momentum per unit mass (see Eq. 8-1) where Q is the Earth rotation rate, u the eastward wind component, a the Earth's radius, and p latitude.

(a) Show for inviscid zonally symmetric flow that the relation DA/Dt = 0 is consistent with the zonal component of the equation of motion (using our standard notation, with Fx the x-component of the friction force per unit mass)

Du Dt

1 dp

p dx in (x, y, z) coordinates, where y = ap (see Fig. 6.19).

(b) Use angular momentum conservation to describe how the existence of the Hadley circulation explains the existence of both the subtropical jet stream in the upper troposphere and the near-surface trade winds.

(c) If the Hadley circulation is symmetric about the equator, and its edge is at 20° latitude, determine the strength of the subtropical jet stream.

2. Consider the tropical Hadley circulation in northern winter, as shown in Fig. 8.16. The circulation rises at 10° S, moves northward across the equator in the upper troposphere, and sinks at 20° N. Assuming that the circulation outside the near-surface

FIGURE 8.16. A schematic diagram of the Hadley circulation rising at 10° S, crossing the equator, and sinking at 20° N.

boundary layer is zonally symmetric (independent of x) and inviscid (and thus conserves absolute angular momentum about the Earth's rotation axis), and that it leaves the boundary layer at 10° S with zonal velocity u = 0, calculate the zonal wind in the upper troposphere at (a) the equator, (b) 10° N, and (c) 20° N.

3. Consider what would happen if a force toward the pole was applied to the ring of air considered in Problem 1 and it conserved its absolute angular momentum, A. Calculate the implied relationship between a small displacement 5p and the change in the speed of the ring Su. How many kilometers northwards does the ring have to be displaced to change its relative velocity 10 m s-1? How does your answer depend on the equilibrium latitude? Comment on your result.

4. An open dish of water is rotating about a vertical axis at 1 revolution per minute. Given that the water is 1°C warmer at the edges than at the center at all depths, estimate, under stated assumptions and using the following data, typical azimuthal flow speeds at the free surface relative to the dish. Comment on, and give a physical explanation for, the sign of the flow.

How much does the free surface deviate from its solid body rotation form?

Briefly discuss ways in which this rotating dish experiment is a useful analogue of the general circulation of the Earth's atmosphere.

Assume the equation of state given by Eq. 4-4 with pref = 1000 kgm-3, a = 2 x 10-4 K-1, and Tref = 15° C, the mean temperature of the water in the dish. The dish has a radius of 10 cm and is filled to a depth of 5 cm.

5. Consider the incompressible, baroclinic fluid [p = p(T)] sketched in Fig. 8.17 in which temperature surfaces slope upward toward the pole at an angle si. Describe the attendant zonal wind field assuming it is in thermal wind balance.

By computing the potential energy before and after interchange of two rings of fluid (coincident with latitude circles y at height z) along a surface of slope s, show that the change in potential energy APE = PEfiml- PEMml is given by

APE = PrefN2 y - y^2 s (s - s1), where N2 = -g/pref dp/dz is the buoyancy frequency (see Section 4.4),

FIGURE 8.17. Schematic for energetic analysis of the thermal wind considered in Problem 5. Temperature surfaces have slope s1; parcels of fluid are exchanged along surfaces which have a slope s.

pref is the reference density of the fluid, and y1, y2 are the latitudes of the interchanged rings. You will find it useful to review Section 4.2.3.

Hence show that for a given meridional exchange distance (y2 - y1):

(b) the energy released is a maximum when the exchange occurs along surfaces inclined at half the slope of the temperature surfaces.

This is the wedge of instability discussed in Section 8.3.3 and illustrated in Fig. 8.10.

6. Discuss qualitatively, but from basic principles, why most of the Earth's desert regions are found at latitudes of ± (20° - 30°).

7. Given that the heat content of an elementary mass dm of air at temperature T is cpT dm (where cp is the specific heat of air at constant pressure), and that its northward velocity is v:

(a) show that the northward flux of heat crossing unit area (in the x - z plane) per unit time is pcpvT;

(b) hence, using the hydrostatic relationship, show that the net northward heat flux H in the atmosphere, at any given latitude, can be written

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