Geoscience Reference
In-Depth Information
Conservation of mass of air
The rate of change with time in
r
a
, the density of the parcel of air with volume
V
shown in Fig. 16.7 is given by the difference between the incoming and outgoing
fluxes of air mass along all three coordinates. The contribution from along the
x axis is given by:
⎛
∂
r
⎞
(16.39)
V
a
=
r
A u
(
−
u
)
⎜
⎟
a
x
x
+
d
x
⎝
∂
t
⎠
x
where
A
is the cross-sectional area of the parcel of air in the plane perpendicular
to the X axis. By taking the first two terms in a Taylor expansion, this can be re-
written as:
⎛
∂
r
⎞
⎛
∂
u
⎞
∂
u
⎡
⎤
a
V
=
r
A u
−
u
+
d
x
= −
r
V
(16.40)
⎜
⎟
⎜
⎢
⎥
⎟
a
x
x
a
⎝
⎠
∂
t
⎝
∂
x
⎠
∂
x
⎣
⎦
x
In three dimensions, the total change in density is therefore:
∂
r
⎛
⎞
∂∂∂
uvw
(16.41)
a
=−
r
+
+
⎜
⎟
a
∂
t
⎝
∂
x
∂
y
∂
z
⎠
This is the
Continuity Equation
for the mass of air and applies everywhere in the
atmosphere.
It can be shown that in atmospheric domains where
f
is the maximum frequency
of pressure waves and
c
s
is the speed of sound, and where typical air velocity is less
than 100 m s
−1
and length scale is less than 12 km, (
c
s
2
/
g
), and (
c
s
2
/
f
), pressure forces
are able to equilibrate density fluctuations in the atmosphere sufficiently quickly
Changing Internal Density,
r
w
in the Volume
V
=
A
d
z
Cross Sectional
Area,
,
Perpendicular
to the x axis
A
W
z
+
d
z
V
y
+
d
y
W
z
V
y
Figure 16.7
Axial
contributions to the time rate
of change of mass in a parcel
of air.
u
x
u
x
+
d
x
