Geoscience Reference
In-Depth Information
Dividing this last equation by
and rearranging gives:
r
a
⎛
r
′
⎞
r
′
dw w
(
+′
)
1
∂
P
′
1
⎛
∂
P
⎞
1
+
a
= −
a
g
−
−
+
r
g
+
u
∇
2
(
w w
+
′
)
(17.8)
⎜
⎟
⎜
⎟
a
r
dt
r
r
∂
z
r
⎝
∂
z
⎠
⎝
⎠
a
a
a
a
The mean vertical pressure gradient in the atmosphere (around which turbulent
fluctuations occur) is in hydrostatic equilibrium, and is described by Equation (3.3).
Consequently the third term on the right hand side of Equation (17.8) is zero.
Equation (17.6) shows that fractional fluctuations in density can be estimated
from fractional fluctuations in temperature, and are of the order 10
−2
. Fractional
fluctuations in density can therefore be neglected in comparison with unity on
the left hand side of the equation, but must be retained in the first term on the
right hand side of the equation where they can be estimated from temperature
fluctuations. Hence, Equation (17.8) becomes:
dw w
(
+′
)
q
qr
′
1
∂
∂
P
′
=−
g
−
+υ∇
2
(
w w
+
′
)
(17.9)
dt
z
a
The approximation procedure just used, in which '
density fluctuations are neglected
in the inertia (storage) term but are retained in the buoyancy term
', i s t h e
Boussinesq
approximation
. In an equation of atmospheric flow, implementing the Bous
si
nesq
approximation involves simultaneously replacing each occurrence of
r
a
by
and
r
—
].
—
aa
—
] or
g
[1
a
each occurrence of
g
by
g
[(
r
+
r
a
)/
r
−
q
′
/
q
Neglecting subsidence
Observations in the ABL show that the value of
—
, the mean vertical wind speed
(which is sometimes referred to as the rate of '
subsidence
'), is usually small and
commonly less than 0.1 m s
−1
. On the other hand, the magnitude of fluctuations
around this mean value are much greater and on the order of several meters per
second. For this reason it is often assumed acceptable when writing equations
describing momentum conservation to '
neglect subsidence
, i.e., to retain terms
involving
w
in the equation while removing those involving
—
. With this assump-
tion Equation (17.9) would, for example, simplify to:
′
dw
′
q
′
1
∂
∂
P
′
(17.10)
=−
g
−
+ ∇
u
2
w
′
dt
qr
z
a
