Geography Reference
In-Depth Information
Here the total derivative in (6.1) is defined by
∂
∂t
D
Dt
≡
∂
∂p
p
+
(
V
·∇
)
p
+
ω
(6.5)
where ω
≡
Dp/Dt is the pressure change following the motion, and in (6.4)
T∂ln θ
∂p is the static stability parameter [S
p
≈
10
−
4
KPa
−
1
in the
S
p
≡−
5
×
midtroposphere].
These equations, although simplified by use of the hydrostatic approximation
and by neglect of some small terms that appear in the complete spherical coordinate
form, still contain several terms that are of secondary significance for midlatitude
synoptic-scale systems. They can be simplified further by recalling from Sec-
tion 2.4 that the horizontal flow is nearly geostrophic and that the magnitude of
the ratio of vertical velocity to horizontal velocity is of order 10
−
3
.
We first separate the horizontal velocity into geostrophic and ageostrophic parts
by letting
V
=
V
g
+
V
a
(6.6)
where the geostrophic wind is defined as
f
−
1
0
V
g
≡
k
×
∇
(6.7)
and the ageostrophic wind,
V
a
, is just the difference between the total horizontal
wind and the geostrophic wind. We have here assumed that the meridional length
scale, L, is small compared to the radius of the earth so that the geostrophic
wind (6.7) may be defined using a constant reference latitude value of the Coriolis
parameter.
2
For the systems of interest
V
g
|
V
a
|
. More precisely,
V
a
|
V
g
∼
|
O (Ro)
That is, the ratio of the magnitudes of the ageostrophic and geostrophic winds is
the same order of magnitude as the Rossby number introduced in Section 2.4.2.
The momentum can then be approximated to O(Ro) by its geostrophic value,
and the rate of change of momentum or temperature following the horizontal
motion can be approximated to the same order by the rate of change following the
geostrophic wind. Thus, in (6.5)
V
can be replaced by
V
g
and the vertical advection,
which arises only from the ageostrophic flow, can be neglected. The rate of change
2
This definition of the geostrophic wind will be referred to as
constant-f
(CF) geostrophy, whereas
the definition given in (3.4) will be called
variable-f
(VF) geostrophy. The CF geostrophic wind
is nondivergent, whereas the VF geostrophic wind has a divergent portion (see Problem 3.19). The
interpretation of the ageostrophic wind depends strongly on which type of geostrophy is used, as
explained in Blackburn (1985).
