Geology Reference
In-Depth Information
the main field, while the mantle is essentially
non-magnetic. The exclusion of any mantle
contribution is also supported by the fact that
mantle convection is too slow to account for the
observed rate of variation of the geomagnetic
field. Therefore, the field must be associated
with
macroscopic
currents somewhere below
the CMB. Current models about the origin of
the Earth's main field invoke the existence of
convective motions within the outer liquid layer
of the Earth's core, which are driven by thermal
and compositional variations just above the inner
core-outer core boundary (ICB) at 2,891 km
depth (e.g., Glatzmaier and Roberts
1995
). The
Earth's core is made by an alloy of iron and
lighter elements. Iron has a good electrical
conductivity ¢
4
10
5
Sm
1
.AstheEarth
cools, iron freezes at the base of the outer core
and accretes to the inner core, determining its
progressive growth. During this process, the
latent heat of crystallization heats the residual
liquid, which is also enriched in light elements,
determining its positive buoyancy and triggering
the convective process. The upward motion of
the buoyant fluid does not occur along straight
lines, because the Coriolis' force associated with
the Earth's rotation determines the formation
of helical flows aligned as the Earth's rotation
axis. Now we are going to examine the basic
physical laws underlying the formation and the
maintenance of a magnetic field by electric
currents in the outer core. To this purpose, it
is necessary to consider again the fundamental
laws of classical electrodynamics, including the
Maxwell equations.
in motion generate magnetic fields. Charged par-
ticles also generate
electric fields
, independently
from being at rest or not, the strength of which
depends on their electric charge. An electric field,
E
, is a conservative force field that can be felt
by any other charged particle and represents the
electric force per unit charge on a test particle.
Its units are [Vm
1
]
D
[kg
m
2
electric field is said to be
electrostatic
and the
expression for
E
in the vacuum is given by
Coulomb
'
slaw
:
Z
1
4 ©
0
¡
e
.
q
/.
r
q
/dV
k
r
q
k
E .r/
D
(4.1)
3
R
where
R
is the region of distribution of the
electric charges and ©
0
D
1/(
c
2
0
)
D
8.8542
10
12
F/m (
c
being the speed of light) is the
vacuum permittivity
. The forces associated with
electric fields give rise to flow of electric charges
in the free space and in
conductors
. The constitu-
tive law that describes the resistance of materials
to be traversed by electric currents is the em-
pirical
Ohm
'
slaw
, which states that in isotropic
materials the current density
j
at a location
r
is
proportional to the electric field
E
:
j .r/
D
¢E .r/
(4.2)
where the
electric conductivity
of the material, ¢,
is measured in [S m
1
]. If the material is moving
with uniform velocity
u
,thelaw(
4.2
) is still valid
for an observator who is in motion with it, but
not
for one that is at rest in the laboratory frame of
reference. If ¡ is the density of stationary charges
in a reference frame fixed to the moving material,
then this distribution of charge is viewed, from
the perspective of the laboratory frame, as an
electric current having density ¡
u
.
The quantity:
j
c
.r/
D
¡.r/
u
(4.3)
is called
convective current density
.Let
j
0
and
E
0
be respectively the current density and the
electric field measured in the moving frame. The
invariance of (
4.2
) implies that:
j
0
D
¢E
0
(4.4)
What is the relation between the currents
j
and
j
0
? The response is simple. In general, in
the moving frame we have electric charges with
local density ¡ that move with some velocity
v
0
.
Therefore, in this frame we have a density of
current
j
0
D
¡
v
0
A
-1
]. Let
¡
D
¡(
r
) be the density function for a distribution
of electric charges (that is,theelectriccharge
per unit volume). If the charges are at rest in the
laboratory frame of reference, the corresponding
s
-3
(Fig.
4.1
). However, from the
