Geoscience Reference
In-Depth Information
where from
Eq. (14.8)
the isotropic form for
M
is
∂u
i
∂x
j
∂u
k
M
ij km
=
∂x
m
=
αδ
ij
δ
km
+
βδ
ik
δ
jm
+
γδ
im
δ
jk
.
(14.13)
In order to evaluate these molecular destruction terms through local isotropy we
must determine the parameters
α
,
β
,and
γ
in
Eq. (14.13)
. We can identify two
constraints on
M
ij km
:
1. Summing
i
on
j
yields zero by incompressibility:
∂u
i
∂x
i
∂u
k
∂x
m
=
M
iikm
=
0
(
constraint 1
).
2. Summing
j
on
k
yields zero. This stems from
u
i
,
∂u
i
∂x
j
∂u
j
∂x
m
=
∂
∂x
j
∂u
j
∂x
m
M
ijj m
=
which is the gradient of a mean quantity. An upper limit for its magnitude is (Appendix,
Chapter 5
)
u
i
<
1
u
1
/
2
∂
∂x
j
∂u
j
∂x
m
ν
R
−
1
/
2
=
.
t
ν
Since
Eq. (14.12)
indicates that
M
ij ij
is of order
/ν
,
M
ijj m
is negligible at large
R
t
and
we can write
∂u
i
∂x
j
∂u
j
∂x
m
=
M
ijj m
=
0
(
constraint 2
).
(14.14)
These two constraints give
0
=
3
α
+
β
+
γ,
0
=
α
+
β
+
3
γ,
(14.15)
so that
α
=
γ, β
=−
4
γ,
and
γ
δ
ij
δ
km
−
δ
im
δ
jk
.
∂u
i
∂x
j
∂u
k
∂x
m
=
M
ij km
=
4
δ
ik
δ
jm
+
(14.16)
If we write from
Eq. (14.16)
∂u
1
∂x
1
2
M
1111
=
=−
2
γ,
(14.17)
then we can express the dissipation-rate tensor
M
ij km
as
∂u
1
∂x
1
2
δ
ij
δ
km
−
δ
im
δ
jk
.
∂u
i
∂x
j
∂u
k
∂x
m
=−
1
2
M
ij km
=
4
δ
ik
δ
jm
+
(14.18)
