Geography Reference
In-Depth Information
4.6.
Show that the anomalous gradient wind cases discussed in Section 3.2.5
have negative absolute circulation in the Northern Hemisphere and hence
have negative average absolute vorticity.
4.7.
Compute the rate of change of circulation about a square in the (x, y) plane
with corners at (0, 0), (0, L),(L, L), and (L, 0) if temperature increases
eastward at a rate of 1˚C per 200 km and pressure increases northward at
a rate of 1 hPa per 200 km. Let L
=
1000 km and the pressure at the point
(0, 0) be 1000 hPa.
4.8.
Verify the identity (4.18) by expanding the vectors in Cartesian components.
4.9.
Derive a formula for the dependence of depth on radius for an incompressible
fluid in solid-body rotation in a cylindrical tank with a flat bottom and free
surface at the upper boundary. Let H be the depth at the center of the tank,
the angular velocity of rotation of the tank, and a the radius of the tank.
4.10.
By how much does the relative vorticity change for a column of fluid
in a rotating cylinder if the column is moved from the center of the tank
to a distance 50 cm from the center? The tank is rotating at the rate of
20 revolutions per minute, the depth of the fluid at the center is 10 cm, and
the fluid is initially in solid-body rotation.
4.11.
A cyclonic vortex is in cyclostrophic balance with a tangential velocity
profile given by the expression V
V
0(
r/r
0
)
n
where V
0
is the tangential
velocity component at the distance r
0
from the vortex center. Compute the
circulation about a streamline at radius r, the vorticity at radius r, and the
pressure at radius r. (Let p
0
be the pressure at r
0
and assume that density is
a constant.)
4.12.
A westerly zonal flow at 45˚ is forced to rise adiabatically over a north-
south-oriented mountain barrier. Before striking the mountain, the westerly
wind increases linearly toward the south at a rate of 10 m s
−
1
per 1000 km.
The crest of the mountain range is at 800 hPa and the tropopause, located
at 300 hPa, remains undisturbed. What is the initial relative vorticity of the
air? What is its relative vorticity when it reaches the crest if it is deflected
5˚ latitude toward the south during the forced ascent? If the current assumes
a uniform speed of 20 m s
−
1
during its ascent to the crest, what is the radius
of curvature of the streamlines at the crest?
4.13.
A cylindrical vessel of radius a and constant depth H rotating at an angular
velocity about its vertical axis of symmetry is filled with a homoge-
neous, incompressible fluid that is initially at rest with respect to the vessel.
A volume of fluid V is then withdrawn through a point sink at the center of
the cylinder, thus creating a vortex. Neglecting friction, derive an expression
for the resulting relative azimuthal velocity as a function of radius (i.e., the
=
