Geography Reference
In-Depth Information
the pumping action of the Ekman layer is eliminated. The result is a baroclinic vor-
tex with a vertical shear of the azimuthal velocity that is just strong enough to bring
ζ
g
to zero at the top of the boundary layer. This vertical shear of the geostrophic
wind requires a radial temperature gradient that is in fact produced during the
spin-down phase by adiabatic cooling of the air forced out of the Ekman layer.
Thus, the secondary circulation in the baroclinic atmosphere serves two purposes:
(1) it changes the azimuthal velocity field of the vortex through the action of the
Coriolis force and (2) it changes the temperature distribution so that a thermal wind
balance is always maintained between the vertical shear of the azimuthal velocity
and the radial temperature gradient.
PROBLEMS
5.1.
Verify by direct substitution that the Ekman spiral expression (5.31) is indeed
a solution of the boundary layer equations (5.26) and (5.27) for the case
v
g
=
0.
5.2.
Derive the Ekman spiral solution for the more general case where the
geostrophic wind has both x and y components (u
g
and v
g
, respectively),
which are independent of height.
5.3.
Letting the Coriolis parameter and density be constants, show that (5.38) is
correct for the more general Ekman spiral solution obtained in Problem 5.2.
5.4.
For laminar flow in a rotating cylindrical vessel filled with water (molecular
kinematic viscosity ν
0.01 cm
2
s
−
1
), compute the depth of the Ekman
layer and the spin-down time if the depth of the fluid is 30 cm and the rotation
rate of the tank is 10 revolutions per minute. How small would the radius of
the tank have to be in order that the time scale for viscous diffusion from
the side walls be comparable to the spin-down time?
5.5.
Suppose that at 43˚ N the geostrophic wind is westerly at 15 m s
−
1
. Compute
the net cross isobaric transport in the planetary boundary layer using both the
mixed layer solution (5.22) and the Ekman layer solution (5.31). You may let
|
=
1kgm
−
3
.
0.05 m
−
1
s, and ρ
V
|=
u
g
in (5.22), h
=
De
=
1 km, κ
s
=
=
5.6.
Derive an expression for the wind-driven surface Ekman layer in the ocean.
Assume that the wind stress τ
w
is constant and directed along the x axis.
The continuity of turbulent momentum flux at the air-sea interface (z
=
0)
requires that the wind stress divided by air density must equal the oceanic
turbulent momentum flux at z
0. Thus, if the flux-gradient theory is used,
the boundary condition at the surface becomes
=
ρ
0
K
∂u
ρ
0
K
∂v
∂z
=
τ
w
,
∂z
=
0
,
at z
=
0
