Geography Reference
In-Depth Information
are small compared with those due to temperature changes. Therefore, to a first
approximation,
θ
θ
ρ
ρ
0
=−
(7.36)
Using (7.33) and (7.36), the linearized version of the set (7.25)-(7.28), we can
write as
∂
∂t
+
u
+
∂p
∂x
=
∂
∂x
1
ρ
0
u
0
(7.37)
∂
∂t
+
w
+
∂p
∂z
−
θ
θ
∂
∂x
1
ρ
0
u
g
=
0
(7.38)
∂u
∂x
+
∂w
∂z
=
0
(7.39)
∂
∂t
+
θ
+
∂
∂x
dθ
dz
=
w
u
0
(7.40)
Subtracting ∂(7.37)/∂z from ∂(7.38)/∂x, we can eliminate p
to obtain
∂
∂t
+
∂w
∂x
−
∂u
∂z
∂θ
∂x
=
∂
∂x
g
θ
u
−
0
(7.41)
which is just the y component of the vorticity equation.
With the aid of (7.39) and (7.40), u
and θ
can be eliminated from (7.41) to
yield a single equation for w
:
∂
∂t
+
2
∂
2
w
∂x
2
∂
2
w
∂z
2
N
2
∂
2
w
∂x
2
∂
∂x
u
+
+
=
0
(7.42)
where N
2
gdln θ/dzis the square of the buoyancy frequency, which is assumed
to be constant.
5
Equation (7.42) has harmonic wave solutions of the form
≡
Re
ˆ
w exp(iφ)
=
w
=
w
r
cos φ
−
w
i
sin φ
(7.43)
where
w
ˆ
=
w
r
+
iw
i
is a complex amplitude with real part w
r
and imaginary part
w
i
, and φ
νt is the phase, which is assumed to depend linearly on
z as well as on x and t . Here the horizontal wave number k is real because the
solution is always sinusoidal in x. The vertical wave number m
=
kx
+
mz
−
=
m
r
+
im
i
may,
5
Strictly speaking, N
2
cannot be exactly constant if ρ
0
is constant. However, for shallow disturbances
the variation of N
2
with height is unimportant.
