Geography Reference
In-Depth Information
Dv
Dt
+
∂
∂y
=
fu
+
0
(9.3)
D
Dt
+
w
dθ
0
dz
=
0
(9.4)
g
θ
00
=
∂
∂z
b
≡
(9.5)
∂u
∂x
+
∂v
∂y
+
∂w
∂z
=
0
(9.6)
where b is the
buoyancy
, θ
00
is a constant reference value of potential temperature,
and
w
∂
∂z
From the discussion of the previous subsection it should be clear that the horizontal
scale of variations parallel to a front is much larger than the cross-frontal scale.
This scale separation suggests that to a first approximation we can model fronts
as two-dimensional structures. For convenience we choose a coordinate system in
which the front is stationary and take the cross-frontal direction to be parallel to
the y axis. Then L
x
D
Dt
≡
∂
∂t
+
∂
∂x
+
∂
∂y
+
u
v
L
y
, where L
x
and L
y
designate the along-front and cross-
front length scales. Similarly, U
V where U and V , respectively, designate the
along-front and cross-front velocity scales. Fig. 9.4 shows these scales relative to
the front.
Letting U
10ms
−
1
, V
1ms
−
1
, L
x
∼
100 km, we find
that it is possible to utilize the differing scales of the along-front and cross-front
motion to simplify the dynamics. Assuming that D/Dt
∼
∼
1000 km, and L
y
∼
∼
V/L
y
(the cross-front
advection time scale) and defining a Rossby number, Ro
1, the
magnitude of the ratios of the inertial and Coriolis terms in the x and y components
of the momentum equation can be expressed as
≡
V/fL
y
Ro
U
V
|
Du/Dt
|
UV/L
y
fV
∼
∼
∼
1
|
fv
|
Ro
V
U
V
2
/L
y
fU
|
Dv/Dt
|
10
−
2
∼
∼
∼
|
fu
|
Fig. 9.4
Velocity and length scales relative to a front parallel to the x axis.
