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
kν
G
±
+
ilf
ˆ
u
G
±
=
ˆ
G
±
(13.63)
ν
G
±
−
f
2
−
1
lν
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
ˆ
u
G
+
+ˆ
u
G
−
=ˆ
u
0
v
R
+ˆ
ˆ
v
G
+
+ˆ
v
G
−
=ˆ
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.