Geoscience Reference
In-Depth Information
If in a turbulent boundary layer we have
u
3
∼
u
, which is the case in horizontally
homogeneous conditions, then the velocity divergence is negligible if
H
ρ
.
Since
is of the order of the boundary-layer depth, in order to treat the velocity
field as divergence-free the boundary-layer depth must be small compared to the
density scale height
H
ρ
. We'll see that this is often but not always the case.
In neglecting the vertical variation of
ρ
0
we are making the
Boussinesq approx-
imation
. It allows the zero velocity divergence assumption, but we shall be on
the lookout for its side effects. As discussed by
Vallis
(
2006
), the
anelastic
approximation
allows
ρ
0
to depend on
z
and is used in deep convection.
8.2.3 Dynamics and thermodynamics
The equation of motion in an atmosphere with variable density is
˜
˜
˜
∂
u
i
u
j
∂
u
i
1
˜
p
∂x
i
+
∂
2
∂t
+˜
∂x
j
=−
g
i
−
2
ij k
j
˜
u
k
+
ν
∇
u
i
.
˜
(8.15)
ρ
The third term on the right side, which involves the cross product of the earth's
rotation vector
j
and the fluid velocity vector, is called the
Coriolis
term. It enters
because our coordinate system rotates with the earth and is therefore not an
inertial
system. In a flow of velocity scale
U
and length scale
L
, the dominant terms in
Eq. (8.15)
are of order
U
2
/L
. The Coriolis term is therefore important when
U
∼
U
2
/L
,where
=|
j
|∼
10
−
4
s
−
1
.
This is typically the case in atmospheric flows
but not in engineering flows.
In a variable-density flow the kinematic viscosity
ν
is variable as well. Variations
in
ν
influence the velocity and length scales of the dissipative eddies, but not the
dissipation rate itself
(Problem 8.4)
.
We linearize the pressure gradient in the deviations about the base state:
∂p
0
∂x
i
+
p
∂x
i
1
ρ
p
∂x
i
=−
∂
˜
1
ρ
0
+
ρ
∂
˜
−
(8.16)
p
∂x
i
+
ρ
ρ
0
1
ρ
0
∂p
0
1
ρ
0
∂
˜
˜
∂p
0
−
∂x
i
−
∂x
i
.
Using
(8.2)
and
(8.9)
then yields
p
∂x
i
−
ρ
ρ
0
δ
3
i
p
∂x
i
+
1
˜
p
∂x
i
∂
˜
1
ρ
0
∂
˜
˜
1
ρ
0
∂
˜
g
T
0
T
δ
3
i
.
−
gδ
3
i
−
g
gδ
3
i
−
(8.17)
ρ
Using
(8.17)
the equation of motion
(8.15)
becomes
p
∂x
i
−
∂
u
i
˜
u
j
∂
u
i
˜
1
ρ
0
∂
˜
g
T
0
T
δ
3
i
+
2
∂t
+˜
∂x
j
=−
2
ij k
j
˜
u
k
+
ν
∇
u
i
.
˜
(8.18)
