Geology Reference
In-Depth Information
outer core? The response to this question comes
from one of the fundamental laws of classical
electrodynamics. In Sect.
3.3
we introduced only
two of the four Maxwell equations, in the special
case of stationary fields in the empty space. Now
we must consider a third equation in the more
general context of time-varying fields. This equa-
tion, which is known as
Faraday
'
slaw
,predicts
the existence of a
non
-
conservative
(i.e., not com-
ing from a potential) electric field
E
associated
with temporal variations of
B
:
u
is zero, the first term at the right-hand side of
(
4.14
) disappears. In this
diffusion limit
we have:
@
@t
D
ǜ
r
2
B
(4.15)
This equation is well known in mathematics.
It is called the
diffusion equation
and is found in
many applications. For example, in Chap.
12
we
shall see that it describes the non-stationary con-
duction of heat. To understand the significance
of the diffusion term in the magnetic induction
equation, let us consider a field depending only
from
x
and assume that at time
t
D
0wehave:
@
@t
r
E
D
(4.11)
C
B
0
I
x>0
B
0
I
x<0
This equation describes the
electromagnetic
induction
determined by a time-varying magnetic
field. Soon we shall further investigate the conse-
quences and the significance of this fundamental
law. For the moment, it will be used just to
eliminate
E
from (
4.10
):
B.x;0/
D
(4.16)
Let us also assume that the field is held fixed
at points
˙
L
,sothat:
B.L;t/
D
B.
L;t/
D
B
0
(4.17)
1
0
¢
r
B
u
B
@
@t
Dr
In this instance, it is possible to show that
the solution to the diffusion equation is (e.g.,
Wilmot-Smith et al.
2005
):
1
0
¢
rr
B
Cr
.
u
B/
(4.12)
D
X
1
n
exp
n
2
2
ǜt=L
2
sin
n x
L
x
L
C
2B
0
B.x;t/
D
B
0
The first term at the right-hand side of (
4.12
)
can be simplified using a simple rule of vector
n
D
1
(4.18)
2
2
rr
B
Dr
.
r
B/
r
B
Dr
B
(4.13)
A graphical representation of this solution is
illustrated in Fig.
4.2
. It suggests that the field
diffuses gradually into the fluid, thus removing
local inhomogeneities. If
L
is interpreted as the
length scale of magnetic inhomogeneities, then
the
diffusion time
is defined as follows:
Substituting in (
4.12
) gives the fundamental
equation of magnetohydrodynamics (MHD):
@
@t
Dr
.
u
B/
C
ǜ
r
2
B
(4.14)
L
2
ǜ
£
D
(4.19)
where ǜ
1/(
0
¢) is called
magnetic diffusivity
.
Equation
4.14
is known as the
magnetic induction
equation
and plays a key role in the study of the
Earth's magnetic field and in plasma physics. It
allows to determine the magnetic field associated
with currents that are originated by electromag-
netic induction as a consequence of convective
motions within the Earth's core. If the velocity
This is the time interval required to smooth
away any local anomaly of the field. For exam-
ple, Fig.
4.2
shows that after a time
t
D
£
D
the
magnetic field distribution is within a factor 10
4
from the stationary solution
B
(
x
)
D
B
0
x
/
L
.In
the general case of an infinite range (
L
!1
)
