Geoscience Reference
In-Depth Information
and after applying Reynolds averaging to remove the time average of fluctuating
components, the equation reduces to:
PR R
=
r
+
r
(17.3)
T
′′
T
da
da
v
v
However, in practice,
r
a
T
′ ′
is very much less than
a
T
r
in the ABL and it can
safely be neglected, consequently:
(17.4)
PRT
=
r
vav
This equation merely says that the ideal gas law applies to average values, which is
as it should be, since it was observation of average values that originally stimulated
its discovery.
It is useful later to subtract Equation (17.3) from Equation (17.2) and then
divide by Equation (17.4) to give:
r
r
′
T
′
P
P
′
(17.5)
=+
a
v
T
a
v
In the ABL, pressure fluctuations are rarely if ever observed to be greater than
0.01 kPa and because mean pressure is on the order of 100 kPa, (
P/
—
) is on the
order of 10
−4
. On the other hand, fluctuations in mean temperature,
wh
ich is itself
on the order of 300 K, are typically on the order of 1 K, hence
(/ )
v
TT
is on the
/
—
) in comparison with
v
order of 33
×
10
−4
. Consequently it is possible to neglect (
P
′
the other terms and write:
r
′
′
′
≈≈
T
T
q
(17.6)
a
v
r
q
a
v
Using this equation, density fluctuations in the ABL (which are otherwise hard to
measure) can be estimated from the measurable fluctuations in temperature.
The Boussinesq approximation
Starting from the equation for the conservation of momentum in the vertical
direction, Equation (16.38), with the molecular flow term written in vector form
for conciseness, multiplying by
), and then expressing all the
variables as the sum of mean and fluctuating parts gives:
r
a
, recalling
m
=
(
r
a
u
dw w
(
+′
)
∂ +′
(
P P
)
2
(
rr
+′
)
=− +′
(
rr
)
g
−
+∇ +′
m
(
w w
)
(17.7)
a
a
a
a
dt
∂
z
