Geography Reference
In-Depth Information
order of magnitude smaller than the Coriolis force and the pressure gradient force
in agreement with our scale analysis. The fact that the horizontal flow is in approx-
imate geostrophic balance is helpful for diagnostic analysis. However, it makes
actual applications of these equations in weather prognosis difficult because accel-
eration (which must be measured accurately) is given by the small difference
between two large terms. Thus, a small error in measurement of either velocity or
pressure gradient will lead to very large errors in estimating the acceleration. This
problem is discussed in some detail in Chapter 13.
A convenient measure of the magnitude of the acceleration compared to the
Coriolis force may be obtained by forming the ratio of the characteristic scales
for the acceleration and the Coriolis force terms: (U
2
/L)/(f
0
U). This ratio is a
nondimensional number called the
Rossby number
after the Swedish meteorologist
C. G. Rossby (1898-1957) and is designated by
R
0
≡
U/(f
0
L)
Thus, the smallness of the Rossby number is a measure of the validity of the
geostrophic approximation.
2.4.3 The Hydrostatic Approximation
A similar scale analysis can be applied to the vertical component of the momentum
equation (2.21). Because pressure decreases by about an order of magnitude from
the ground to the tropopause, the vertical pressure gradient may be scaled by P
0
/H ,
where P
0
is the surface pressure and H is the depth of the troposphere. The terms in
(2.21) may then be estimated for synoptic scale motions and are shown in Table 2.2.
As with the horizontal component equations, we consider motions centered at 45˚
latitude and neglect friction. The scaling indicates that to a high degree of accuracy
the pressure field is in
hydrostatic equilibrium
; that is, the pressure at any point is
simply equal to the weight of a unit cross-section column of air above that point.
The above analysis of the vertical momentum equation is, however, somewhat
misleading. It is not sufficient to show merely that the vertical acceleration is small
compared to g. Because only that part of the pressure field that varies horizontally
is directly coupled to the horizontal velocity field, it is actually necessary to show
that the horizontally varying pressure component is itself in hydrostatic equilibrium
with the horizontally varying density field. To do this it is convenient to first define a
Table 2.2
Scale Analysis of the Vertical Momentum Equation
(u
2
v
2
)/a
ρ
−
1
∂p/∂z
z -Eq.
Dw
/
Dt
−
2u cos φ
−
+
=−
−
g
+F
rz
U
2
/a
gν
WH
−
2
Scales
UW
/L
f
0
U
P
0
/(ρ
H
)
ms
−
2
10
−
7
10
−
3
10
−
5
10
−
15
10
10
