Geography Reference
In-Depth Information
M6.5.
Consider a modification of the
stationary
geopotential field given in Prob-
lem 6.7:
f
0
U
0
y cos
πp
p
0
f
0
Vk
−
1
sin (kx) cos(ly)
(x, y, p)
=
0
(p)
−
+
10
−
4
s
−
1
,
where
0
is the standard atmosphere geopotential, f
0
=
1.0
×
35 m s
−
1
,V
20 m s
−
1
,k
10
6
m,l
U
0
=
=
=
(2π/L) with L
=
6.0
×
=
1.5k, and p
0
1000 hPa. Note that in this case the mean zonal wind is
negative in the lower troposphere and positive in the upper troposphere.
Again the domain of interest is
=
−
3000 km
≤
x
≤+
3000 km,
−
1000
≤
y
1000 km. (a) Neglecting the β-effect, show by evaluating the terms on
the right-hand side of the tendency equation (6.23) that the tendency van-
ishes provided that K
2
≤
σ
−
1
(f
0
π/p
0
)
2
(where the static
stability σ is treated here as a constant whose magnitude is given by
this formula). (b) Use MATLAB to plot maps of the geopotential height
(Z
k
2
l
2
≡
+
=
/g) distributions at the 750- and 250-hPa levels assuming that the
mean
geopotential height
is 2500 m at 750 hPa and 10,000 m at 250 hPa.
(c) For the conditions given above, find an expression for the horizontal
divergence and overlay maps of the divergence fields at 750 and 250 hPa
on the geopotential height maps of part (b). Explain the phase relationships
between the divergence and height fields at each level.
=
M6.6.
For the geopotential distribution of the previous problem, use the traditional
omega equation (6.34) to find an expression for ω for the conditions in
which the geopotential tendency vanishes
i.e., for K
2
σ
−
1
(f
0
π/p
0
)
2
as shown in the problem earlier. The best approach is to use a trial solution
of the form ω
=
W
0
cos (kx) cos (ly) sin
πp
p
0
where W
0
is a constant
to be determined. If you substitute this trial form into the left side of (6.34)
you can solve for W
0
. (a) Use MATLAB to plot vertical cross sections of
the divergence, the vertical motion, and the vorticity fields for this situation
at y
=
0. The script
Chapter 6 plot2.m
provides a template for contour
plotting vertical cross sections for this situation. (b) Derive an expression
for the thickness advection for this case. Use MATLAB to plot a vertical
cross section of the thickness advection. Describe the phase relationship
between thickness advection and vertical velocity and relate these to the
balance in the thermodynamic energy equation (6.21).
=
Suggested References
Wallace and Hobbs,
Atmospheric Science: An Introductory Survey,
has an excellent introductory level
description of the observed structure and evolution of midlatitude synoptic-scale disturbances.
Bluestein,
Synoptic-Dynamic Meteorology in Midlatitudes, Vol. II,
has a comprehensive treatment of
midlatitude synoptic disturbances at the graduate level.
