Geography Reference
In-Depth Information
We assume that the motion is two dimensional in the x, z plane. The x momentum
equation for the lower layer is then
∂u
∂t
+
u
∂u
w
∂u
gδρ
ρ
1
∂h
∂x
∂x
+
∂z
=−
(7.17)
whereas the continuity equation is
∂u
∂x
+
∂w
∂z
=
0
(7.18)
Now since the pressure gradient in (7.17) is independent of z, u will also be
independent of z provided that u
=
u(z) initially. Thus, (7.18) can be integrated
vertically from the lower boundary z
=
0 to the interface z
=
h to yield
h
∂u
∂x
w (h)
−
w (0)
=−
However, w(h) is just the rate at which the interface height is changing,
Dh
Dt
=
∂h
∂t
+
u
∂h
∂x
w (h)
=
and w(0)
0 for a flat lower boundary. Hence, the vertically integrated continuity
equation can be written
=
∂h
∂t
+
u
∂h
h
∂u
∂h
∂t
+
∂
∂x
(hu)
∂x
+
∂x
=
=
0
(7.19)
Equations (7.17) and (7.19) are a closed set in the variables u and h. We now apply
the perturbation technique by letting
u
,h
h
u
=
u
+
=
H
+
where u as before is a constant basic state zonal velocity and H is the mean depth
of the lower layer. The perturbation forms of (7.17) and (7.19) are then
∂u
∂t
+
u
∂u
∂h
∂x
=
gδρ
ρ
1
∂x
+
0
(7.20)
∂h
∂t
+
u
∂h
H
∂u
∂x
+
∂x
=
0
(7.21)
h
|
where we assume that H
|
so that products of the perturbation variables can
be neglected.
