Geoscience Reference
In-Depth Information
From Equation (15.4), it also follows that:
(15.5)
(. )
Ba Ba
′=
′=
0
and, by analogy, that:
(15.6)
(
Ab Ab
.
′=
)
′=
0
Equations (15.5) and (15.6) can be used to demonstrate an important result. If one
takes the time-average of the cross product of two atmospheric variables, thus:
(15.7)
()(
AB A a B b AB aB Ab ab
=
+′
(
+′=
)(
+′ +
′+′′
)
and then substitutes Equations (15.5) and (15.6) it follows that:
(15.8)
()
AB
=+′ ′
A B a b
Thus, the time-average of the cross product of two atmospheric variables is equal
to the sum of two terms, the product of their mean values plus the time-average of
the instantaneous product of their fluctuating components over the period
T
. This
result is known as
Reynolds averaging
and it provides the basis for defining and
calculating measures of the strength of atmospheric turbulence and turbulent
fluxes, as described below. It is important to recognize that although Equations
(15.5) and (15.6) show the time-average of the product of a mean value with a
turbulent fluctuation is zero, in general,
the time-average of the
pro
duct
of tw
o or
more tu
rbule
nt fluctu
ations cannot be assumed to be zero
, thus
aa
′′≠
0
,
ab
′′≠
0
,
ab
′( ′) ≠
2
0
,
()()
ab
′
2
′
2
≠
0
, etc.
Variance and standard deviation
The variance of an atmospheric variable,
A
, which has been re-written in terms of
mean and fluctuating parts, is formally defined by:
(15.9)
2
() (
σ= +′ −
A
Aa A
2
) )
Multiplying out this equation and (for the purpose of illustrating their application)
applying the rules of averaging as used when deriving Equation (15.8), it follows
that:
2
(
σ
A
AaAa AAa AA
)
=
(
+′
)(
+′−
)
2
(
+′+
)
(15.10)
2
=+′ +′
AA Aa a
2
() 2
− −′ +
AA Aa AA
2
