Geography Reference
In-Depth Information
Fig. 2.6
Change in Lagrangian control volume (shown by shading) due to fluid motion parallel to the
x axis.
so that in the limit δV
0, (2.32) reduces to the continuity equation (2.31); the
divergence of the three-dimensional velocity field is equal to the fractional rate of
change of volume of a fluid parcel in the limit δV
→
0. It is left as a problem for
the student to show that the divergence of the
horizontal
velocity field is equal to
the fractional rate of change of the horizontal area δA of a fluid parcel in the limit
δA
→
→
0.
2.5.3
Scale Analysis of the Continuity Equation
Following the technique developed in Section 2.4.3, and again assuming that
ρ
ρ
0
1, we can approximate the continuity equation (2.31) as
1
ρ
0
∂ρ
∂t
+
w
ρ
0
dρ
0
dz
+
∇·
ρ
U
·∇
+
U
≈
0
(2.33)
A
B
C
where ρ
designates the local deviation of density from its horizontally averaged
value, ρ
0
(z). For synoptic scale motions ρ
/ρ
0
∼
10
−
2
so that using the charac-
teristic scales given in Section 2.4 we find that term A has magnitude
∂ρ
∂t
+
ρ
ρ
ρ
0
1
ρ
0
U
L
≈
10
−
7
s
−
1
U
·∇
∼
For motions in which the depth scale H is comparable to the density scale height,
d ln ρ
0
dz
H
−
1
, so that term B scales as
∼
w
ρ
0
dρ
0
dz
W
H
≈
10
−
6
s
−
1
Expanding term C in Cartesian coordinates, we have
∂u
∂x
+
∂v
∂y
+
∂w
∂z
∇·
U
=
