Geography Reference
In-Depth Information
of momentum following the total motion is then approximately equal to the rate
of change of the geostrophic momentum following the geostrophic wind:
D
V
Dt
≈
D
g
V
g
Dt
where
D
g
Dt
≡
∂
∂t
+
∂
∂t
+
∂
∂x
+
∂
∂y
V
g
·∇
=
u
g
v
g
(6.8)
Although a constant f
0
can be used in defining
V
g,
it is still necessary to retain
the dynamical effect of the variation of the Coriolis parameter with latitude in the
Coriolis force term in the momentum equation. This variation can be approximated
by expanding the latitudinal dependence of f in a Taylor series about a reference
latitude φ
0
and retaining only the first two terms to yield
f
=
f
0
+
βy
(6.9)
2 cos φ
0
a and y
where β
0atφ
0
. This approximation is
usually referred to as the
midlatitude
β
-plane
approximation. For synoptic-scale
motions, the ratio of the first two terms in the expansion of f has an order of
magnitude
≡
(df/dy)
φ
0
=
=
βL
f
0
∼
cos φ
0
sin φ
0
L
a
∼
O (Ro)
1
This justifies letting the Coriolis parameter have a constant value f
0
in the
geostrophic approximation and approximating its variation in the Coriolis force
term by (6.9).
From (6.1) the acceleration following the motion is equal to the difference
between the Coriolis force and the pressure gradient force. This difference depends
on the departure of the actual wind from the geostrophic wind. Thus, it is not
permissible to simply replace the horizontal velocity by its geostrophic value in
the Coriolis term. Rather, we use (6.6), (6.7), and (6.9) to write
×
V
g
+
V
a
−
f
k
×
V
+
∇
=
(f
0
+
βy)
k
f
0
k
×
V
g
(6.10)
≈
f
0
k
×
V
a
+
βy
k
×
V
g
where we have used the geostrophic relation (6.7) to eliminate the pressure gradient
force and neglected the ageostrophic wind compared to the geostrophic wind in
the term proportional to βy. The approximate horizontal momentum equation thus
has the form
D
g
V
g
Dt
=−
f
0
k
×
V
a
−
βy
k
×
V
g
(6.11)
Each term in (6.11) is thus O(Ro) compared to the pressure gradient force,
whereas terms neglected are O(Ro
2
) or smaller.
