Geography Reference
In-Depth Information
Dρ
Dt
+
ρ
∂u
∂x
=
0
(7.7)
D ln θ
Dt
=
0
(7.8)
where for this case D/Dt
u∂/∂x. Recalling from (2.44) and the ideal
gas law that potential temperature may be expressed as
=
∂/∂t
+
=
p
ρR
p
s
p
R
c
p
θ
where p
s
=
1000 hPa, we may eliminate θ in (7.8) to give
1
γ
D ln p
Dt
D ln ρ
Dt
−
=
0
(7.9)
where γ
=
c
p
/c
v
. Eliminating ρ between (7.7) and (7.9) gives
1
γ
D ln p
Dt
∂u
∂x
=
+
0
(7.10)
The dependent variables are now divided into constant basic state portions
(denoted by overbars) and perturbation portions (denoted by primes):
u
(x, t)
u(x, t)
=
u
+
p
(x, t)
p(x, t)
=
p
+
(7.11)
ρ
(x, t)
ρ(x,t)
=
ρ
+
Substituting (7.11) into (7.6) and (7.10) we obtain
∂t
u
u
+
u
u
∂x
u
u
+
∂x
p
p
∂
∂
1
∂
+
+
+
+
=
0
ρ
)
(ρ
+
∂t
p
p
+
u
u
∂x
p
p
+
γ
p
p
∂x
u
u
∂
∂
∂
+
+
+
+
+
=
0
We next observe that provided
ρ
/ρ
1 we can use the binomial expansion to
approximate the density term as
1
−
1
1
ρ
ρ
ρ
ρ
1
1
ρ
1
ρ
ρ
)
=
+
≈
−
(ρ
+
