Geography Reference
In-Depth Information
We now linearize (7.25)-(7.29) by letting
ρ
u
ρ
=
ρ
0
+
u
=
u
+
p
w
p
=
p (z)
+
w
=
(7.30)
θ
θ
=
θ
(
z
)
+
where the basic state zonal flow u and the density ρ
0
are both assumed to be
constant. The basic state pressure field must satisfy the hydrostatic equation
dp
dz
=−
ρ
0
g
(7.31)
while the basic state potential temperature must satisfy (7.29) so that
γ
−
1
ln p
ln θ
=
−
ln ρ
0
+
constant
(7.32)
The linearized equations are obtained by substituting from (7.30) into (7.25)-
(7.29) and neglecting all terms that are products of the perturbation variables. Thus,
for example, the last two terms in (7.26) are approximated as
dp
dz
+
∂p
∂z
1
ρ
∂p
∂z
+
1
ρ
0
+
g
=
+
g
ρ
1
(7.33)
ρ
ρ
0
∂p
∂z
+
∂p
∂z
+
ρ
ρ
0
1
ρ
0
dp
dz
1
ρ
0
1
ρ
0
≈
−
+
g
=
g
where (7.31) has been used to eliminate p. The perturbation form of (7.29) is
obtained by noting that
ln
θ
1
γ
−
1
ln
p
1
ln
ρ
0
1
θ
θ
p
p
ρ
ρ
0
+
=
+
−
+
+
const. (7.34)
Now, recalling that ln(ab)
=
ln(a)
+
ln(b) and that ln(1
+
ε)
≈
ε for any ε
1,
we find with the aid of (7.32) that (7.34) may be approximated by
θ
θ
≈
p
p
−
ρ
ρ
0
1
γ
Solving for ρ
yields
θ
θ
+
p
c
s
ρ
≈−
ρ
0
(7.35)
where c
s
≡
p
γ
/ρ
0
is the square of the speed of sound. For buoyancy wave
motions
ρ
0
θ
θ
p
c
s
; that is, density fluctuations due to pressure changes
