Geography Reference
In-Depth Information
2.4.1
Geostrophic Approximation and Geostrophic Wind
It is apparent from Table 2.1 that for midlatitude synoptic scale disturbances the
Coriolis force (term B) and the pressure gradient force (term F) are in approxi-
mate balance. Retaining only these two terms in (2.19) and (2.20) gives as a first
approximation the
geostrophic
relationship
1
ρ
∂p
∂x
;
1
ρ
∂p
∂y
−
fv
≈−
fu
≈−
(2.22)
where f
2sinφ is called the
Coriolis parameter
. The geostrophic balance
is a
diagnostic
expression that gives the approximate relationship between the
pressure field and horizontal velocity in large-scale extratropical systems. The
approximation (2.22) contains no reference to time and therefore cannot be used
to predict the evolution of the velocity field. It is for this reason that the geostrophic
relationship is called a diagnostic relationship.
By analogy to the geostrophic approximation (2.22) it is possible to define a
horizontal velocity field,
V
g
≡
j
v
g
, called the
geostrophic wind
, which
satisfies (2.22) identically. In vectorial form,
≡
i
u
g
+
1
ρf
∇
V
g
≡
k
×
p
(2.23)
Thus, knowledge of the pressure distribution at any time determines the geostrophic
wind. It should be kept clearly in mind that (2.23) always defines the geostrophic
wind, but only for large-scale motions away from the equator should the geostrophic
wind be used as an approximation to the actual horizontal wind field. For the scales
used in Table 2.1 the geostrophic wind approximates the true horizontal velocity
to within 10-15% in midlatitudes.
2.4.2
Approximate Prognostic Equations; the Rossby Number
To obtain prediction equations it is necessary to retain the acceleration (term A) in
(2.19) and (2.20). The resulting approximate horizontal momentum equations are
f
v
v
g
Du
Dt
=
1
ρ
∂p
∂x
=
fv
−
−
(2.24)
f
u
u
g
Dv
Dt
=−
1
ρ
∂p
∂y
=−
fu
−
−
(2.25)
where (2.23) is used to rewrite the pressure gradient force in terms of the geostrophic
wind. Because the acceleration terms in (2.24) and (2.25) are proportional to the
difference between the actual wind and the geostrophic wind, they are about an
