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of ULF noise observed prior to and during the volcano eruption. Below we choose
our own way to reproduce their result. In order to model the magma motion along
the volcano vent, we first consider an ellipsoid inclusion in the rock as a model of
magma reservoir. The origin of the coordinate system is chosen to be in the center
of the ellipsoid. The coordinate axes point from the center and coincide with the
symmetry axes, so that the ellipsoid equation is x
2
=a
2
C
y
2
=b
2
C
z
2
=c
2
D
1, where a,
b, and c are the ellipsoid semi-axes. This is only a mathematical technique because
in what follows we consider the extreme case c
!1
, which corresponds to an
infinite cylindrical channel.
As the magma can move inside the inclusion, the electric current density inside
and outside this inclusion is then
j
D
m
.
E
C
V
B
0
/ , and
j
D
r
E
;
(10.48)
where
V
denotes the local magma velocity,
B
0
is the induction of undisturbed
geomagnetic field, and
E
is the electric field. Since the magma conductivity
m
is of the order of 10
2
-10 S/m (e.g., Gaillard and Marziano
2005
), the conductivity
r
of the surrounding rock is assumed to be much smaller than
m
. Note that the
Hall/imposed current,
j
H
D
m
.
V
B
0
/ in Eq. (
10.48
) plays a role of a source of
GMPs.
For simplicity we assume that inside the ellipsoid the vector
V
is positive parallel
to
z
axis and approximately constant. Notice that this assumption can be applied to
the unclosed space such as an infinite cylinder, which will be considered below.
However the problem under consideration is similar to that for a dielectric ellipsoid
immersed in the uniform electric field. Likewise, one can find a certain similarity
of the problem to that for a magnetized ellipsoid immersed in the uniform magnetic
field. According to Landau and Lifshits (
1982
), the electromagnetic field outside of
the ellipsoid is thus determined through the effective current moment
r
m
x
O
x
r
m
y
O
y
d
D
C
r
1
n
.x/
C
C
r
1
n
.y/
;
(10.49)
m
n
.x/
m
n
.y/
where
O
x
and
O
y
are unit vectors and m
x
and m
y
are projections of the Hall current
moment
m
D
j
H
V on the x and y axes. Here V
D
4abc=3 is the ellipsoid volume
and n
.x/
and n
.y/
denote the so-called geometrical depolarization factors depending
on the parameters a, b, and c. It should be noted that the vectors
d
and
m
are not
parallel.
Following a simple model by Kopytenko and Nikitina (
2004a
,
b
), we consider
the magma motion along the infinite cylindrical vent with constant radius, that
corresponds to the case when c
!1
and a
D
b. On account that in this limit
n
.x/
D
n
.y/
D
0:5 we get
2a
2
m
r
.
V
B
0
/
m
C
r
d
c
D
d
eff
D
;
(10.50)
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