Geoscience Reference
In-Depth Information
17
Equations of
Turbulent Flow
in the ABL
Introduction
The set of equations which describe the movement and evolution of the atmosphere
in the ABL, at any point in time, were introduced in Chapter 16. In this chapter
these equations are re-written to describe the mean flow for time-average values of
atmospheric variables, including the influence on these mean flow equations of
turbulent fluctuations on the high frequency side of the spectral gap, see Fig. 15.3
and associated text. Doing this involves expressing the value of each variable
described by an equation introduced in Chapter 16 in terms of a mean and a
fluctuating part, then applying the Reynolds averaging rules (see Table 15.2) to
derive the equivalent equation for mean flow, and finally introducing any
simplifications and approximations that are appropriate in the ABL.
Fluctuations in the ideal gas law
Consider first the effect of re-expressing the ideal gas law in terms of atmospheric
variables which recognize mean and fluctuating components. It is convenient to
rearrange the equation
P
=
r
a
R
d
T
v
into the form
P/R
d
=
r
a
T
v
before substituting
—
P
=
+
P
',
rrr
=+′
and
TTT
to give:
=+′
a
a
a
v
v
v
PP
+
(17.1)
RR
rr
= +′
(
)( )
a a v v
TT TT T T
+′ =
r
+
r r
′ +
′ +′
r
′
av va av a v
d
d
Averaging this equation it becomes:
PP
+= + ′ + ′ +′ ′
a
(17.2)
RR
r
r
r
r
T
TT T
a
a
a
v
v
v
v
d
d
