Geography Reference
In-Depth Information
velocity in a coordinate system rotating with the tank). Assume that the
motion is independent of depth and that V
πa
2
H . Also compute the
relative vorticity and the relative circulation.
4.14.
(a) How far must a zonal ring of air initially at rest with respect to the earth's
surface at 60˚ latitude and 100-km height be displaced latitudinally in order
to acquire an easterly (east to west) component of 10 m s
−
1
with respect
to the earth's surface? (b) To what height must it be displaced vertically in
order to acquire the same velocity? Assume a frictionless atmosphere.
4.15.
The horizontal motion within a cylindrical annulus with permeable walls of
inner radius 10 cm, outer radius 20 cm, and 10-cm depth is independent of
height and azimuth and is represented by the expressions u
=
7
−
0.2r, v
=
40
2r, where u and v are the radial and tangential velocity components
in cm s
−
1
, positive outward and counterclockwise, respectively, and r is
distance from the center of the annulus in centimeters. Assuming an incom-
pressible fluid, find
(a) the circulation about the annular ring,
(b) the average vorticity within the annular ring,
(c) the average divergence within the annular ring, and
(d) the average vertical velocity at the top of the annulus if it is zero at the
base.
+
4.16.
Prove that, as stated below Eq. (4.38), the globally averaged isentropic vor-
ticity on an isentropic surface that does not intersect the ground must be
zero. Show that the same result holds for the isobaric vorticity on an iso-
baric surface.
MATLAB EXERCISES
M4.1.
Section 4.5.2 showed that for nondivergent horizontal motion, the flow
field can be represented by a
streamfunction
ψ (x, y), and the vorticity is
then given by ζ
2
ψ . Thus, if the vorticity
is represented by a single sinusoidal wave distribution in both x and y,
the streamfunction has the same spatial distribution as the vorticity and
the opposite sign, as can be verified easily from the fact that the second
derivative of a sine is proportional to minus the same sine function. An
example is shown in the MATLAB script
vorticity 1.m
. However, when
the vorticity pattern is localized in space, the space scales of the stream-
function and vorticity are much different. This latter situation is illustrated
in the MATLAB script
vorticity demo.m
, which shows the streamfunc-
tion corresponding to a point source of vorticity at (x, y)
∂
2
ψ/∂x
2
∂
2
ψ/∂y
2
=
+
≡
∇
(0, 0).For
this problem you must modify the code in
vorticity 1.m
by specifying
=
