this pressure surface slopes uniformly between 30° and 60° latitude and is flat elsewhere, use the geostrophic wind relationship (zonal component) in pressure coordinates, u = -- —

£ dz f dy to calculate the mean eastward geostrophic wind on the 200-mbar surface at 45° latitude in the winter hemisphere. Here f = 2Q sin p is the Coriolis parameter, g is the acceleration due to gravity, z is the height of a pressure surface, and dy = a x dp, where a is the radius of the Earth is a northward pointing coordinate.

From the pressure coordinate thermal wind relationship, Eq. 7-24, and approximating du du/dz dp dp/dz '

show that, in geometric height coordinates, f du dz g T

The winter polar stratosphere is dominated by the ''polar vortex,'' a strong westerly circulation at about 60° latitude around the cold pole, as depicted schematically in Fig. 7.29. (This circulation is the subject of considerable interest, because it is within the polar vortices—especially that over Antarctica in southern winter and spring—that most ozone depletion is taking place.)

Assuming that the temperature at the pole is (at all heights) 50 K colder at 80° latitude than at 40° latitude (and that it varies uniformly in between), and that the westerly wind speed at 100 mbar pressure and 60° latitude is 10 ms-1, use the thermal wind relation to estimate the wind speed at 1 mbar pressure and 60° latitude.

FIGURE 7.29. A schematic of the winter polar stratosphere dominated by the ''polar vortex,'' a strong westerly circulation at ~60° around the cold pole.
FIGURE 7.30. A schematic of surface pressure contours (solid) and mean 1000 mbar—500 mbar temperature contours (dashed), in the vicinity of a typical northern hemisphere depression (storm).

10. Starting from Eq. 7-24, show that the thermal wind equation can be written in terms of potential temperature, Eq. 4-17, thus:

dug dVg dp' dp

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