Geoscience Reference
In-Depth Information
Taking into account that the spatially averaged mean values of fluctuated
variables are vanish, that is
=0
,
and
∂δϕ
∂x
k
=0
δσ
ik
then the equation mean current (10.19) becomes
δσ
ik
∂δϕ
∂x
k
.
j
i
=
σ
ik
E
k
−
(10.20)
The physical meaning of two terms is the following. The first term is a part
of the total current defined by the mean electric field and mean conductivity.
The second term is the contribution into the total current due to the in-
homogeneities. Therefore, the problem reduces to finding
δϕ
(
r
) with known
coordinate dependency
σ
ik
(
r
) or, equivalently,
δσ
ik
(
r
)
.
The unknown vari-
able is fluctuations of the electric potential
δϕ
which can be found from the
condition that
j
i
is divergence free current. From
∇
·
j
(
r
) = 0 follows that
σ
ik
∂ϕ
∂x
k
=
∂
2
ϕ
∂x
i
∂x
k
∂
∂x
i
∂
∂x
i
∂ϕ
∂x
k
(
δσ
ik
)
+
σ
ik
= 0
(10.21)
and with (10.17) the last equation becomes
∂
2
δϕ
∂x
i
∂x
k
∂
2
δϕ
∂x
i
∂x
k
−
∂
(
δσ
ik
)
∂x
i
∂δϕ
∂x
k
∂
(
δσ
ik
)
∂x
i
·
+
σ
ik
+
δσ
ik
E
k
=0
.
(10.22)
Therefore, the problem reduces to finding
δϕ
(
r
) with known coordinate
dependency
σ
ik
(
r
) or, equivalently,
δσ
ik
(
r
)
.
In the Appendix it is shown a formal solution procedure reducing to an
expansion of the fluctuating variables into a Fourier series. Substitution of
the series into (10.22) yields a much more tractable equation for an individual
spatial Fourier harmonic and after considerable manipulation of Ohm's law
for the average current (10. A.5):
+
q
B
l
(
q
)
j
i
=
σ
ik
E
k
δσ
il
(
−
q
)
q
i
q
m
E
l
q
l
(10.23)
σ
im
which is a basis for the calculation of effective conductivity
σ
eff
ik
. Here
B
i
(
q
)
is defined by (10. A.3). By definition,
σ
e
xx
=
σ
e
P
is the transverse effective
conductivity for
E
x
and
B
0
z
. Equation (10. A.5), can be rewritten as
=
i
=
x,y
+
q
δσ
kl
(
q
)
q
l
B
i
(
q
)
j
k
σ
ki
E
i
;
(10.24)
σ
mn
q
m
q
n
Here,
k
=
x, y
for the corresponding current components.
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