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is from the southwest at 20 m s
−
1
and the relative vorticity decreases toward
the northeast at a rate of 4
10
−
6
s
−
1
per 100 km. Use the quasi-geostrophic
vorticity equation to estimate the horizontal divergence at this location on a
β-plane.
×
6.4.
Given the following expression for the geopotential field
cf
0
ct)
k
−
1
sin k (x
=
0
(p)
+
−
y [cos (πp/p
0
)
+
1]
+
−
where
0
is a function of p alone, c is a constant speed, k a zonal wave
number, and p
0
=
1000 hPa (a) Obtain expressions for the corresponding
geostrophic wind and relative vorticity fields. (b) Obtain an expression for the
relative vorticity advection. (c) Use the quasi-geostrophic vorticity equation
to obtain the horizontal divergence field consistent with this field. (Assume
that df/dy
0, obtain an expression for
ω(x, y, p, t) by integrating the continuity equation with respect to pressure.
(e) Sketch the geopotential fields at 750 and 250 hPa. Indicate regions of
maximum divergence and convergence and positive and negative vorticity
advection.
=
0.) (d) Assuming that ω(p
0
)
=
6.5.
For the geopotential distribution of Problem 6.4, obtain an expression for ω
by using the adiabatic form of the thermodynamic energy equation (6.13b).
Assume that σ is a constant. For what value of k does this expression for ω
agree with that obtained in Problem 6.4?
6.6.
As an additional check on the results of Problems 6.4 and 6.5, use the approx-
imate omega equation (6.36) to obtain an expression for ω. Note that the
three expressions for ω agree only for one value of k. Thus, for given values
of σ and f
0
, the geopotential field (x, y, p, t) of Problem 6.4 is consis-
tent with quasi-geostrophic dynamics only for one value of the zonal wave
number.
6.7.
Suppose that the geopotential distribution at a certain time has the form
f
0
ck
−
1
sin kx
(x, y, p)
=
0
(p)
−
f
0
U
0
y cos (πp/p
0
)
+
where U
0
is a constant zonal speed and all other constants are as in
Problem 6.4. Assuming that f and σ are constants, show by evaluating the
terms on the right-hand side of the tendency equation (6.23) that χ
=
0
provided that k
2
σ
−
1
(f
0
π/p
0
)
2
. Make qualitative sketches of the geopo-
tential fields at 750 and 250 hPa for this case. Indicate regions of maximum
positive and negative vorticity advection at each level. (Note: the wavelength
corresponding to this value of k is called the
radius of deformation
.)
=
