Geography Reference
In-Depth Information
Thus, the potential energy released per unit length in y during adjustment is
+∞
+∞
ρgh
2
2
ρgh
0
2
dx
−
dx
=
−∞
−∞
2
+∞
0
1
e
−
x
/
λ
R
2
dx
1
ρgh
0
2
3
2
ρgh
0
λ
R
−
−
=
(7.85)
) all potential energy available initially is
released (converted to kinetic energy) so that there is an infinite energy release.
(Energy is radiated away in the form of gravity waves, leaving a flat free surface
extending to
In the nonrotating case (λ
R
→∞
.)
In the rotating case only the finite amount given in (7.85) is converted to kinetic
energy, and only a portion of this kinetic energy is radiated away. The rest remains
in the steady geostrophic circulation. The kinetic energy in the steady-state per
unit length is
|
x
|→∞
as t
→∞
ρH
gh
0
fλ
R
2
+∞
2
+∞
0
ρH
v
2
2
1
2
ρgh
0
λ
R
e
−
2x/λ
R
dx
dx
=
=
(7.86)
0
Thus, in the rotating case a finite amount of potential energy is released, but only
one-third of the potential energy released goes into the steady geostrophic mode.
The remaining two-thirds is radiated away in the form of inertia-gravity waves.
This simple analysis illustrates the following points: (a) It is difficult to extract
the potential energy of a rotating fluid. Although there is an infinite reservoir of
potential energy in this example (because h
), only a finite
amount is converted before geostrophic balance is achieved. (b) Conservation of
potential vorticity allows one to determine the steady-state geostrophically adjusted
velocity and height fields without carrying out a time integration. (c) The length
scale for the steady solution is the Rossby radius λ
R
.
The dynamics of the adjustment process plays an essential role in initialization
and data assimilation in numerical prediction (see Section 13.7). For example,
under some conditions the adjustment process may effectively damp out new height
data inserted at a gridpoint, as the new data will generally be unbalanced and hence
will tend to adjust toward geostrophic balance with the existing wind field.
is finite as
|
x
|→∞
7.7
ROSSBY WAVES
The wave type that is of most importance for large-scale meteorological processes
is the
Rossby wave
,or
planetary wave
. In an inviscid barotropic fluid of con-
stant depth (where the divergence of the horizontal velocity must vanish), the
Rossby wave is an absolute vorticity-conserving motion that owes its existence to
