Geoscience Reference
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Fig. 1.5
A scenario of the
generation of a toroidal
magnetic field from a
poloidal one for the case of
nonuniform rotation of a
conducting medium. Taken
from the site
http://www.
1.1.7
Inhomogeneous Rotation
An astronomical observation has shown that stars, interstellar material, and galaxies
rotate nonuniformly (differentially) that means that the angular velocity of the
material rotation depends on the coordinates. The same phenomenon is believed to
be a necessary accompaniment of the Earth rotation. Irregularities of the Earth spin
are due to such factors as the presence of liquid core (Fig.
1.1
) and the precession
of the Earth rotation axis. This latter factor can play a role of driving force. The
angular velocity of the precession depends on dynamical compression of a solid
and thus the angular velocity in the mantle differs from that in the outer core. This
difference may give rise to a motion of the molten material filling the space between
the mantle and inner solid core. As a result the core will rotate slowly compared to
the mantle.
The nonuniform rotation of a conducting medium may greatly affect the gener-
ation of magnetic field. To illustrate the mechanism of the phenomenon, consider
the magnetic field frozen in the conducting media. This implies that the energy
dissipation can be neglected. We now show that the nonuniform rotation generates a
toroidal (azimuthal) magnetic field
B
t
D
B
'
'
from the poloidal (meridional) field
B
p
D
B
r
O
r
C
B
O
O
stand for unit vectors. Magnetic field lines of the
poloidal field are in meridional planes which contain the spin axis, while the toroidal
field lines are in the orthogonal planes. Since the different parts of the “frozen-in”
field line rotate at different angular velocities !
D
!.r;/, the magnetic field line
stretches in the azimuthal direction. As shown in Fig.
1.5
, the heterogenous rotation
of the conducting medium results in generation of the toroidal magnetic field
B
t
in
a way that the field lines of the toroidal field are perpendicular to the meridional
planes. It should be noted that the generation of toroidal field from poloidal one due
to the heterogeneous rotation is described, in a mathematical sense, via the term
r
.
h
V
ih
B
i
/ on the right-hand side of Eq. (
1.25
).
, where
O
r
,
, and
'
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