Geography Reference
In-Depth Information
standard pressure p
0
(z), which is the horizontally averaged pressure at each height,
and a corresponding standard density ρ
0
(z), defined so that p
0
(z) and ρ
0
(z) are in
exact
hydrostatic balance:
1
ρ
0
dp
0
dz
≡−
g
(2.26)
We may then write the total pressure and density fields as
p
(x,y,z,t)
p(x, y, z, t)
=
p
0
(z)
+
(2.27)
ρ
(x,y,z,t)
ρ(x, y, z, t)
=
ρ
0
(z)
+
where p
and ρ
are deviations from the standard values of pressure and density.
For an atmosphere at rest, p
and ρ
would thus be zero. Using the definitions of
(2.26) and (2.27) and assuming that ρ
/ρ
0
is much less than unity in magnitude so
that (ρ
0
+
=
ρ
−
1
0
ρ
)
−
1
(1 -ρ
/ρ
0
), we find that
∂z
p
0
+
p
−
1
ρ
∂p
∂z
−
1
(ρ
0
+
∂
−
g
=−
g
ρ
)
ρ
ρ
0
ρ
g
(2.28)
∂p
∂z
∂p
∂z
1
ρ
0
dp
0
dz
−
1
ρ
0
≈
=−
+
For synoptic scale motions, the terms in (2.28) have the magnitudes
δP
ρ
0
H
∂p
∂z
∼
ρ
g
ρ
0
∼
1
ρ
0
10
−
1
ms
-2
,
10
−
1
ms
-2
∼
Comparing these with the magnitudes of other terms in the vertical momentum
equation (Table 2.2), we see that to a very good approximation the perturbation
pressure field is in hydrostatic equilibrium with the perturbation density field so that
∂p
∂z
+
ρ
g
=
0
(2.29)
Therefore, for synoptic scale motions, vertical accelerations are negligible and
the vertical velocity cannot be determined from the vertical momentum equation.
However, we show in Chapter 3 that it is, nevertheless, possible to deduce the
vertical motion field indirectly.
2.5
THE CONTINUITY EQUATION
We turn now to the second of the three fundamental conservation principles, con-
servation of mass. The mathematical relationship that expresses conservation of
mass for a fluid is called the
continuity equation
. This section develops the con-
tinuity equation using two alternative methods. The first method is based on an
Eulerian control volume, whereas the second is based on a Lagrangian control
volume.
