Geography Reference
In-Depth Information
To demonstrate the role of nonlinearity, we assume that the troposphere has a
stable lower layer of undisturbed depth h topped by a weakly stable upper layer and
assume that the lower layer behaves as a barotropic fluid with a free surface h(x,t).
We assume that disturbances have zonal wavelengths much greater than the layer
depth. The motion of the lower layer may then be described by the shallow water
equations of Section 7.3.2, but with the lower boundary condition replaced by
Dh
M
Dt
u∂h
M
∂x
w(x, h
M
)
=
=
where h
M
again denotes the height of the topography.
We first examine the linear behavior of this model by considering steady flow
over small-amplitude topography. The linearized shallow water equations (7.20)
and (7.21) then become
u
∂u
∂h
∂x
=
gδρ
ρ
1
∂x
+
0
(9.32)
u
∂
h
−
h
M
H
∂u
+
∂x
=
0
(9.33)
∂x
Here δρ /ρ
1
is the fractional change in density across the interface between the
layers, h
=
H , where H is the mean height of the interface and h
−
h
−
h
M
is
the deviation from H of the thickness of the lower layer.
The solutions for (9.32) and (9.33) can be expressed as
h
M
u
2
c
2
1
h
M
H
u
h
=−
,u
=
u
2
c
2
u
2
c
2
(9.34)
−
1
−
where c
2
(gH δρ/ρ
1
) is the shallow water wave speed. The characteristics of
the disturbance fields h
and u
depend
o
n the magnitude of the mean-flow
Froude
number
, defined by the relation Fr
2
≡
u
2
c
2
. When Fr < 1, the flow is referred to
as
subcritical
. In subcritical flow, the shallow water gravity wave speed is greater
than the mean-flow speed, and the disturbance height and wind fields are out of
phase. The interface height disturbance is negative, and the velocity disturbance
is positive over the topographic barrier as shown in Fig. 9.9a. When Fr > 1, the
flow is referred to as
supercritical
. In supercritical flow the mean flow exceeds the
shallow water gravity wave speed. Gravity waves cannot play a role in establishing
the steady-state adjustment between height and velocity disturbances because such
waves are swept downstream from the ridge by the mean flow. In this case the fluid
thickens and slows as it ascends over the barrier (Fig. 9.9b). It is also clear from
(9.34) that for Fr
=
∼
1 the perturbations are no longer small and the linear solution
breaks down.
The nonlinear equations corresponding to (9.32) and (9.33) in the case δρ
=
ρ
1
can be expressed as
u
∂u
g
∂h
∂x
+
∂x
=
0
(9.35)
