Geography Reference
In-Depth Information
(a) Show that in the natural coordinate system the force balances parallel
and perpendicular to the velocity vector in the mixed layer model (see
Fig. 5.3) can be expressed respectively as
fκ
s
V
2
=
fu
g
cos β,
and
fV
=
fu
g
sin β
where it is here assumed that the pressure gradient force is directed north-
ward so that fu
g
p
ρ
−
1
0
, and β designates the angle between the
pressure gradient force vector and the mixed layer velocity,
V
. (Other
notation as defined in Section 5.3.1.)
(b) Use MATLAB to solve for V and β given u
g
in the range of 1 to
50ms
-
1
and plot V and β versus u
g
.
Hint
: Solve for V in the above
two equations and for each value of u
g
vary β until the two solutions for
V agree.
M5.2.
Suppose that the geopotential distribution at the top of the mixed layer can
be expressed in the form
(
x, y
)
=
∇
=
0
−
f
0
U
0
y
+
A sin kx sin ly, where
9800 m
2
s
−
2
, f
0
10
−
4
s
−
1
, U
0
5ms
−
1
, A
1500 m
2
s
−
2
,
0
=
=
=
=
πL
−
1
, and, l
πL
−
1
, where L
k
6000 km. (a) Use the technique
given in the demonstration script
mixed layer wind1.m
to determine the
wind distribution in the mixed layer for this situation for the case where
κ
s
=
=
=
0.05. (b) Using the formula obtained in Problem 5.10, compute
the vertical velocity distribution at the top of the mixed layer for this
distribution of geopotential when the depth of the mixed layer is 1 km.
(The MATLAB script
mixed layer wind 2.m
has a template that you can
use to contour the vertical velocity field and vorticity fields.)
=
M5.3.
For the geopotential distribution of Problem M5.2, use the Ekman layer
theory to derive the pattern of vertical velocity at the top of the boundary
layer. Assume that K
m
=
10 m
2
s
−
1
and again use MATLAB to contour the
fields of vorticity and vertical velocity. Explain why the vertical velocity
patterns derived from the mixed layer and Ekman theories differ for this
situation.
Suggested References
Arya,
Introduction to Micrometeorology,
gives an excellent introduction to boundary layer dynamics
and turbulence at the beginning undergraduate level.
Garratt,
The Atmospheric Boundary Layer
is an excellent graduate-level introduction to the physics of
the atmospheric boundary layer.
Stull,
An Introduction to Boundary Layer Meteorology
, provides a comprehensive and very nicely
illustrated treatment of all aspects of the subject at the beginning graduate level.
