Geoscience Reference
In-Depth Information
We require that the deviations be small so we can linearize about the base state.
Thus, the representation
(8.7)
is fundamentally different from the decomposition
in
Part I
of turbulence fields into ensemble-mean and fluctuating parts, where the
fluctuations need not be small.
We express the density deviation in terms of deviations of temperature and
pressure through a linear Taylor series expansion about the base state,
0
T
+
0
˜
∂ρ
∂T
∂ρ
∂p
ρ
=˜
ρ
( T,
p
˜
p)
˜
(8.8)
ρ
0
1
R
d
T
0
p
.
Pressure fluctuations in turbulence are observed to be of order
ρu
2
. If the pressure
deviation is of this order as well, then its contribution to
T
0
T
+
=−
ρ
is of order
γρ
0
u
2
/c
2
,
where
γ
=
c
p
/c
v
and
c
is the speed of sound
(γ R
d
T)
1
/
2
. Thus, by this argument
the contribution of the pressure deviation to
Eq. (8.8)
is proportional to the square
of the Mach number of the turbulence, which is very small. Thus we neglect it and
write
Eq. (8.8)
as
ρ
0
ρ
=−
T
0
T
.
˜
(8.9)
Equation (1.17)
for mass conservation can be written
∂ ρ
∂t
+
∂ ρ u
i
∂x
i
D ρ
Dt
+˜
ρ
∂ u
i
=
∂x
i
=
0
.
(8.10)
For sufficiently small density deviations this becomes
u
3
dρ
0
ρ
0
∂
u
i
˜
˜
dx
3
+
∂x
i
0
,
(8.11)
and the velocity divergence is
∂
∂x
i
−
˜
u
i
˜
u
3
ρ
0
dρ
0
u
3
˜
1
H
ρ
≡−
1
ρ
0
dρ
0
dx
3
.
dx
3
=
H
ρ
;
(8.12)
H
ρ
is a
scale height
for density. The ideal gas law then yields
(Problem 8.3)
R
d
T
0
γ
g
H
ρ
=
,
(8.13)
which is on the order of 10 km.
Presumably we can take the velocity divergence as zero if it is very small
compared to
u/
.From
Eq. (8.12)
this requires
˜
u
3
u
.
H
ρ
(8.14)
