Geoscience Reference
In-Depth Information
Similarly, we have
f
∂V
g
fV
g
H
ρ
g
T
0
∂T
∂x
+
∂z
=
.
(9.30)
In general the first term in these expressions is by far the more important.
According to
Eqs. (9.29)
and
(9.30)
, a horizontal gradient of mean virtual tem-
perature of 3 K per 100 km in the lower atmosphere, which is not uncommon,
generates a geostrophic wind shear, or baroclinity, of 10 m s
−
1
per 1000 m. Thus,
the mean horizontal pressure gradient can change radically across the depth of the
ABL. Horizontal density gradients near fronts can cause comparable effects in the
upper ocean.
9.6.3 Ekman pumping
In the absence of significant advection of mean momentum and horizontal variabil-
ity of the geostrophic wind, the mean vertical vorticity equation in the quasi-steady
ABL reduces to a balance between turbulent friction and Coriolis effects:
∂x ∂z
=
f
∂U
=−
f
∂W
∂z
.
∂
2
uw
∂y ∂z
−
∂
2
vw
∂V
∂y
∂x
+
(9.31)
Integrating
(9.31)
from the surface to the top of the ABL at height
h
,wherethe
turbulent stress vanishes, gives
∂τ
23
(
0
)
∂x
∂τ
13
(
0
)
∂y
∂vw(
0
)
∂x
∂uw(
0
)
∂y
1
ρ
0
−
=
−
=−
f W (h).
(9.32)
(τ
13
(
0
), τ
23
(
0
))
is the vector shearing stress
on the fluid
in the surface plane. Its
negative, the shearing stress
on the surface
, is conventionally taken as proportional
to the product of mean wind speed and the mean wind vector
(U
ref
,V
ref
)
at some
reference height (e.g., at 10 m):
(τ
13
(
0
), τ
23
(
0
))
(τ
13
(
0
), τ
23
(
0
))
=−
=
C
d
ρ
0
S
ref
(U
ref
,V
ref
),
(9.33)
with
C
d
the
drag coefficient
. For circularly symmetric flow around a pressure min-
imum or maximum (a
low
or a
high
) with
S
ref
and
ρ
0
taken as constant we can then
use
Eq. (9.33)
to write
(9.32)
as
∂V
ref
∂x
∂U
ref
∂y
C
d
S
ref
ω
3
=
f W (h),
ω
3
=
−
,
(9.34)
