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and the final adjusted state can be determined without solving the time-dependent
problem.
This problem was first solved by Rossby in the 1930s and is often referred to
as the Rossby adjustment problem. As a simplified, albeit somewhat unrealistic,
example of the adjustment process, we consider an idealized shallow water system
on a rotating plane with initial conditions
u
,v
=
h
=−
0
;
h
0
sgn (x)
(7.78)
where sgn(x)
=
1 for x>0 and sgn(x)
=−
1 for x<0. This corresponds to an
initial step function in h
at x
0 , with the fluid motionless. Thus, from (7.77)
ζ
f
0
−
h
H
=
h
0
H
sgn (x)
=
(7.79)
Using (7.79) to eliminate ζ
in (7.72) yields
c
2
∂
2
h
∂x
2
∂
2
h
∂t
2
∂
2
h
∂y
2
f
0
h
=−
f
0
h
0
sgn (x)
−
+
+
(7.80)
which in the homogeneous case (h
0
=
0) yields the dispersion relation
c
2
k
2
l
2
gH
k
2
l
2
ν
2
f
0
f
0
=
+
+
=
+
+
(7.81)
This should be compared to (7.66).
Because initially h
is independent of y, it will remain so for all time. Thus, in
the final steady state (7.80) becomes
c
2
d
2
h
dx
2
f
0
h
=−
f
0
h
0
sgn (x)
−
+
(7.82)
which has the solution
h
h
0
=
−
1
+
exp (
−
x/λ
R
)
for x>0
(7.83)
+
1
−
exp (
+
x/λ
R
)
for x<0
f
−
0
√
gH is the Rossby
radius of deformation
. Hence, the radius of
deformation may be interpreted as the horizontal length scale over which the height
field adjusts during the approach to geostrophic equilibrium. For
where λ
R
≡
λ
R
the
original h
remains unchanged. Substituting from (7.83) into (7.69)-(7.71) shows
that the steady velocity field is geostrophic and nondivergent:
|
x
|
∂h
∂x
=−
g
f
0
gh
0
f
0
λ
R
u
=
v
=
0,
and
exp (
− |
x
|
/λ
R
)
(7.84)
The steady-state solution (7.84) is shown in Fig. 7.13.
