Geography Reference
In-Depth Information
11.2
SCALE ANALYSIS OF LARGE-SCALE TROPICAL MOTIONS
Despite the uncertainties involved in the interaction between convective and syn-
optic scales, some information on the character of synoptic-scale motions in the
tropics can be obtained through the methods of scale analysis. The scaling argu-
ments can be carried out most conveniently if the governing equations are written
in the log-pressure coordinate system introduced in Chapter 8:
∂
∂t
+
V
∂
∂z
∗
w
∗
·∇
+
+
×
=−
∇
V
f
k
V
(11.1)
∂/∂z
∗
=
RT /H
(11.2)
∂w
∗
/∂z
∗
−
w
∗
/H
∂u/∂x
+
∂v/∂y
+
=
0
(11.3)
∂
∂t
+
T
w
∗
N
2
H/R
V
·∇
+
=
J/c
p
(11.4)
We wish to compare the magnitudes of the various terms in (11.1)-(11.4) for
synoptic-scale motions in the tropics. We first note that an upper limit on the vertical
velocity scale W is imposed by the continuity equation (11.3). Thus, following the
discussion of Section 4.5,
∂u/∂x
+
∂v/∂y
≤
U/L
However, for motions with vertical scales comparable to the density scale
height H ,
∂w
∗
/∂z
∗
−
w
∗
/H
∼
W/H
so that the vertical velocity scale must satisfy the constraint W
HU/L if the
horizontal divergence and vertical stretching terms are to balance in the continuity
equation. We next define characteristic scales for the various field variables as
follows:
≤
10
4
m
H
∼
vertical length scale
10
6
m
L
∼
horizontal length scale
10 m s
−
1
U
∼
horizontal velocity scale
W
≤
HU/L
vertical velocity scale
δ
geopotential fluctuation scale
10
5
s
L/U
∼
timescale for advection
The magnitudes chosen for the horizontal length and velocity scales are typical
for observed values in synoptic-scale systems both in the tropics and at midlati-
tudes. We now wish to show how the corresponding characteristic scales for ver-
tical velocity and geopotential fluctuations are limited by the dynamic constraints
imposed by conservation of mass, momentum, and thermodynamic energy.