Geoscience Reference
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where
H
ρ
is the density scale height of the atmosphere, about 12 km. For boundary-
layer depths small compared to
H
ρ
the second term on the right side of
Eq. (9.24)
is
not important. To evaluate the first term on the right we turn to the vertical equation
of mean motion:
∂U
3
∂t
+
U
j
∂U
3
∂u
3
u
j
∂x
j
1
ρ
0
∂P
∂x
3
−
g
θ
0
θ
.
∂x
j
+
=−
2
3
jk
j
U
k
+
(9.25)
Let us first scale the terms in
Eq. (9.25)
. Let horizontal mean velocities scale with
U
0
, vertical mean velocity with
W
0
, turbulent velocities with
u
, horizontal gradients
with
L
x
, local time changes with
L
x
/U
, and vertical gradients with
H
. The mean
continuity equation implies
U
0
W
0
H
L
x
∼
,
(9.26)
so we have
W
0
∼
HU
0
/L
x
.
Then the order of the terms in
Eq. (9.25)
is
time change
,
mean advection:
U
0
H
L
x
,
turbulent flux divergence:
u
2
H
,
rotation:
fU
0
,
g
θ
0
θ
.
buoyancy:
(9.27)
θ
10
−
2
ms
−
2
.If
U
0
If
∼
1 K, then the buoyancy term is of order 3
×
∼
10ms
−
1
and
H
10
3
m, the rotation and turbulence terms are of order 10
−
3
ms
−
2
,
considerably smaller than the buoyancy term.
The size of the time-change and advection terms depends additionally on the char-
acteristic horizontal length scale
L
x
.If
L
x
is of the order of 10
4
m - which implies
a horizontal divergence of 10 m s
−
1
per 10 km, a fairly large value - then these
inertial terms are of order 10
−
3
ms
−
2
, again much smaller than the buoyancy term.
We conclude that if
L
x
is sufficiently large we can use the
hydrostatic
approximation
to the
U
3
-equation, writing it as
∼
1
ρ
0
∂P
∂z
=
g
θ
0
θ
.
(9.28)
Differentiating
(9.28)
with respect to
y
and using the result in
(9.24)
provides an
expression for the
z
-variation of
U
g
:
∂ θ
∂y
+
f
∂U
g
g
θ
0
fU
g
H
ρ
=−
g
T
0
∂T
∂y
+
fU
g
H
ρ
∂z
=−
.
(9.29)
