Geoscience Reference
In-Depth Information
Table 15.2
Averaging rules for time-dependent variables
A
and
B
re-written in terms of their mean and fluctuating components and a time
independent constant
C
.
Number
Averaging rule
Number
Averaging rule
_
A1
C
= C
A6
dA
dA
⎛⎞
=
⎜
⎝⎠
dt
dt
A2
A7
(
AB AB
+=+
)
⎛⎞
=
∂
AA
t
∂
⎜
⎝⎠
∂
∂
t
A3
A8
(
CA
)
=
CA
⎛
∂′
a
⎞
=
∂′
a
⎜
⎟
⎝
⎠
∂
t
∂
t
A4
(AA
=
and
(BB
=
A9
⎛
∂′
a
⎞
∂′
a
A
=
A
⎜
⎟
⎝
⎠
∂
t
∂
t
⎛
2
⎞
2
∂′
()
a
∂′
()
a
A5
A10
(
AB
)
=
AB
=
⎜
⎟
⎝
∂
t
⎠
∂
t
that follow when two time dependent variables,
A
and
B,
are decomposed into
mean and fluctuating components and the mean value of their fluctuating
components is zero. In Table 15.2,
A
and
B
are as follows:
AAa
=+′
(15.1)
and
BBb
=+′
(15.2)
while
C
is a fixed constant that does not vary over the averaging period used to
define the mean and fluctuating components of
A
and
B
. In this table the presence
of an overbar over a variable implies that an average of that variable is taken over
the time period
T
.
One important (though obvious) result that immediately follows from these
rules is that because:
(15.3)
AAa AaAa
=+′ = +′ =+′
(
)
(
)
(
)
it follows that:
a
′=
0
(15.4)
