Basic applications of approximate governing equations (Holton, Ch. 3)
(Homework project #4 for ATM SCI 351)
Problem #1 (Holton 3.3) A tornado rotates with constant angular velocity ω . Show that
$ −ω 2 r02 '
the surface pressure at the center of the tornado is given p = p0 exp&
)
% 2RT (
€ T is the temperature
where p0 is the surface pressure at a distance r0 from the center and
(assumed constant). If the temperature is 288 K and pressure and wind speed at 100 m
from the center are 102 kPa and 100 m s-1, respectively, what is the central pressure?
€
Problem #2 (Holton!3.6)!Show that the geostrophic balance in isothermal coordinates
may be written as fVg = k × ∇ T (RT ln p + Φ) .
Problem #3 (Holton 3.12) Suppose that a vertical column of the atmosphere at 430N is
initially isothermal from 90 to 50 kPa. The geostrophic wind is 10 m s-1 from the south at
€ m s-1 from the west at 70 kPa, and 20 m s-1 from the west at 50 kPa. Calculate
90 kPa, 10
the mean horizontal temperature gradients in the two layers 90–70 kPa and 70–50 kPa.
Compute the rate of advective temperature change in each layer. How long would this
advection pattern have to persist in order to establish a dry adiabatic lapse rate between
60 and 80 kPa? (Assume that the lapse rate is constant between 90 and 50 kPa and that
the 80–60-kPa layer thickness is 2.25 km.)
Problem #4 (Holton 3.13) An airplane pilot crossing the ocean at 450N latitude has both
a pressure altimeter and a radar altimeter, the latter measuring absolute height above the
sea. Flying at an air speed of 100 m s-1 the pilot maintains altitude by referring to the
pressure altimeter set at 101.3 kPa, holding an indicated 6000 m altitude. At the
beginning of a 1-h period the pilot notes that the radar altimeter reads 5700 m and at the
end of the hour it reads 5950 m. In what direction and approximately how far has the
pilot drifted from the heading?
Problem #5 (Holton 3.20, 3.21) The following wind data (direction and speed) were
received from 50 km to the east, north, west, and south of the station, respectively: 900,
10 m s-1; 1200, 4 m s-1; 900, 8 m s-1; 600, 4 m s-1. Calculate the approximate horizontal
divergence at the station. Suppose that the wind speeds are each in error of ±10%. What
would be the percent error in the calculated divergence in the worst case?