Elementary Applications of the Basic Equations - An Introduction to Dynamic Meteorology - page 81

3.9. Show that as the pressure gradient approaches zero the gradient wind reduces
to the geostrophic wind for a normal anticyclone and to inertial flow (Sec-
tion 3.2.3) for an anomalous anticyclone.

3.10. The mean temperature in the layer between 750 and 500 hPa decreases
eastward by 3˚ C per 100 km. If the 750-hPa geostrophic wind is from the
southeast at 20 m s − 1 , what is the geostrophic wind speed and direction at
500 hPa? Let f

10 - 4 s - 1 . =

3.11. What is the mean temperature advection in the 750- to 500-hPa layer in
Problem 3.10?

3.12. Suppose that a vertical column of the atmosphere at 43˚N is initially isother-
mal from 900 to 500 hPa. The geostrophic wind is 10 m s − 1 from the south
at 900 hPa, 10 m s − 1 from the west at 700 hPa, and 20 m s − 1 from the west
at 500 hPa. Calculate the mean horizontal temperature gradients in the two
layers 900-700 hPa and 700-500 hPa. Compute the rate of advective tem-
perature change in each layer. How long would this advection pattern have
to persist in order to establish a dry adiabatic lapse rate between 600 and
800 hPa? (Assume that the lapse rate is constant between 900 and 500 hPa
and that the 800- to 600-hPa layer thickness is 2.25 km.)

3.13. An airplane pilot crossing the ocean at 45˚N latitude has both a pressure
altimeter and a radar altimeter, the latter measuring his absolute height above
the sea. Flying at an air speed of 100 m s − 1 he maintains altitude by referring
to his pressure altimeter set for a sea level pressure of 1013 hPa. He holds
an indicated 6000-m altitude. At the beginning of a 1-h period he notes that
his radar altimeter reads 5700 m, and at the end of the hour he notes that it
reads 5950 m. In what direction and approximately how far has he drifted
from his heading?

3.14. Work out a gradient wind classification scheme equivalent to Table 3.1 for
the Southern Hemisphere (f < 0) case.

3.15. In the geostrophic momentum approximation (Hoskins, 1975) the gradient
wind formula for steady circular flow (3.17) is replaced by the approximation

VV g R − 1 + fV = fV g

Compare the speeds V computed using this approximation with those
obtained in Problem 3.8 using the gradient wind formula.

3.16. How large can the ratio V g /(f R) be before the geostrophic momentum
approximation differs from the gradient wind approximation by 10% for
cyclonic flow?