Geography Reference
In-Depth Information
Substitution of (11.32) into (11.29)-(11.31) then yields a set of ordinary differential
equations in y for the meridional structure functions
v,
ˆ
u,
ˆ
ˆ
:
ik
ˆ
−
iν
u
ˆ
−
βy
v
ˆ
=−
(11.33)
∂
ˆ
−
iν
v
ˆ
+
βy
u
ˆ
=−
/∂y
(11.34)
gh
e
ik
v/∂y
=
iν
ˆ
−
+
u
ˆ
+
∂
ˆ
0
(11.35)
If (11.33) is solved for
u and the result substituted into (11.34) and (11.35), we
ˆ
obtain
β
2
y
2
ν
2
ikβy
ˆ
iν∂
ˆ
−
v
ˆ
=
+
/∂y
(11.36)
iνgh
e
∂
v
ν
2
gh
e
k
2
ˆ
v
∂y
−
k
ν
βy
ˆ
−
+
ˆ
=
0
(11.37)
ˆ
Finally, (11.37) can be substituted into (11.36) to eliminate
, yielding a second-
order differential equation in the single unknown,
v:
ˆ
ν
2
gh
e
−
ν
β
∂
2
β
2
y
2
gh
e
v
∂y
2
+
ˆ
k
k
2
−
−
v
ˆ
=
0
(11.38)
Because (11.38) is homogeneous, we expect that nontrivial solutions satisfying
the requirement of decay at large
will exist only for certain values of ν, corre-
sponding to frequencies of the normal mode disturbances.
Before discussing this equation in detail, it is worth considering the asymptotic
limits that occur when either h
e
|
y
|
0. In the former case, which is
equivalent to assuming that the motion is nondivergent, (11.38) reduces to
→∞
or β
=
ν
β
∂
2
v
∂y
2
+
ˆ
k
k
2
−
−
v
ˆ
=
0
Solutions exist of the form
v
ˆ
∼
exp (ily), provided that ν satisfies the Rossby wave
l
2
). This illustrates that for nondivergent
barotropic flow, equatorial dynamics is in no way special. The rotation of the earth
enters only in the form of the β effect; it is not dependent on f . However, if β
βk/(k
2
dispersion relationship, ν
=−
+
0,
all influence of rotation is eliminated and (11.38) reduces to the shallow water
gravity wave model, which has nontrivial solutions for
=
gh
e
k
2
l
2
1/2
ν
=±
+
Returning to (11.38), we seek solutions for the meridional distribution of
v,
ˆ
subject to the boundary condition that the disturbance fields vanish for
|
y
|→∞
.
