Geography Reference
In-Depth Information
where c 2
gh, with h the local deviation of
depth from the mean, and ε is an indicator function with value 0 or 1.
If the nonlinear terms on the right in (13.58)-(13.60) are omitted by letting
=
gH with H the mean depth,
=
ε
=
0, normal mode solutions can be obtained of the form
u
ˆ
ˆ
u
v
=
exp [i (kx
v
ˆ
+
ly
νt)]
(13.61)
where for this simple f -plane model it can be shown (Problem 13.10) that the
Rossby normal mode has ν R =
0, and the eastward and westward gravity modes
1/2 . Substituting (13.61) into (13.58)-(13.60)
and neglecting the nonlinear terms then yield the relationships of velocity to geopo-
tential for the three normal mode solutions:
f 2
c 2 (k 2
l 2 )
satisfy ν G ± =±[
+
+
]
ˆ
ˆ
ilf 1
ikf 1
u R =−
ˆ
v R =+
ˆ
R ,
(13.62)
R
ν G ±
f 2 1
G ± +
ilf ˆ
u G ± =
ˆ
G
±
(13.63)
ν G ±
f 2 1
G ±
ikf ˆ
v G ± =
ˆ
G
±
Here the subscripts R and G
indicate the Rossby (geostrophic) mode and the
eastward and westward gravity modes, respectively.
The normal mode solutions thus can be expressed in terms of the three indepen-
dent amplitude coefficients,
±
ˆ
ˆ
ˆ
R ,
, and
, respectively.
G
+
G
Suppose that the observed fields at t
=
0 are given by
u
v
u 0
ˆ
ˆ
=
exp [i (kx
v 0
ˆ
+
ly)]
(13.64)
0
Thus, the observations involve three independent amplitude coefficients. These
can be represented as sums over the normal modes:
u R
ˆ
u G +
u G
u 0
v R
ˆ
v G +
v G
v 0
(13.65)
ˆ
+ ˆ
+ + ˆ
= ˆ
R
G
G
0
Upon substituting from (13.62) and (13.63) into (13.65) we obtain a set of three
inhomogeneous linear equations, which can be solved for the relative geopotential
amplitudes of the three normal modes when projected onto the observed fields.
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