Geoscience Reference
In-Depth Information
We denote 2
3
2
sin
θ,
where
θ
is latitude, as the
Coriolis parameter f
.In
steady, laminar
geostrophic
flow these equations reduce to
=
1
ρ
0
∂P
∂x
+
1
ρ
0
∂P
∂y
−
0
=−
fV,
0
=−
fU,
(9.18)
so that
1
fρ
0
∂P
∂y
,V
1
fρ
0
∂P
∂x
.
U
=
U
g
=−
=
V
g
=
(9.19)
This
geostrophic balance
between Coriolis and pressure-gradient forces makes the
wind vector perpendicular to the mean pressure gradient:
U
i
∂P
1
fρ
0
∂P
∂y
∂P
∂x
+
∂P
∂x
∂P
∂y
∂x
i
=
−
=
0
.
(9.20)
In the northern hemisphere (where
f
is positive) geostrophic flow is clockwise
around a high-pressure center and counterclockwise around a low; the rotation is
reversed in the southern hemisphere.
We often use the solution
(9.19)
for geostrophic flow to express the components
of the horizontal pressure gradient as
1
fρ
0
∂P
∂y
,
1
fρ
0
∂P
∂x
.
U
g
=−
g
=
(9.21)
and to write the mean momentum
equations (9.17)
as
∂U
∂t
+
∂uw
∂z
=
f(V
−
V
g
),
(9.22)
∂V
∂t
+
∂vw
∂z
=
f(U
g
−
U).
U
g
and
V
g
are not to be interpreted as components of a mean wind in the ABL;
rather,
fU
g
and
fV
g
are mean kinematic horizontal pressure gradients, as indicated
in
Eq. (9.19)
.
The stress-divergence and geostrophic-departure terms in the mean momentum
equations (9.22)
typically range from 10
−
4
to 10
−
3
ms
−
2
in magnitude. The time-
change and advection terms must be much smaller (10
−
5
ms
−
2
, say) for the mean
momentum balance to be considered steady and horizontally homogeneous. This
would mean the local wind acceleration cannot exceed about 4 cm s
−
1
per hour in
order to neglect time changes. If the mean wind speed is 5 m s
−
1
, its horizontal
variation cannot exceed about 2 cm s
−
1
per ten kilometers in order to neglect
mean advection. These severe requirements suggest that the steady, horizontally
homogeneous model of the mean momentum balance does not often apply to real
ABL flows.