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The cause of spontaneous generation of magnetic fields in
conducting fluid-like bodies (such as plasmas or molten metals)
is a longstanding, major problem in plasma astrophysics, solar
physics and geophysics. Magnetic fields exist in the Earth,
Sun and other stars (and perhaps in galaxies) that cannot
be explained as surviving primordial fields, and it is generally
believed that such magnetic fields are generated by plasma
flow (liquid metal for the planets).
The question of how magnetic fields are generated by unconstrained
flows of conducting fluids and plasma is referred to as the
"dynamo'' problem; theoretical research into dynamo mechanisms
has been actively pursued for several decades. However, until
quite recently our probing of the dynamo problem has been
limited to analytic calculations, numerical modeling and observational
studies;experimental validation (the critical test for any
theory) of aspects of the theory and experimental studies
of laboratory dynamos have been scarce.
Two important questions for dynamo theories and experiments
to address are:
(1) What class of flow geometries and velocity fields (presumably
turbulent) can lead to self-excitation of magnetic fields?
and (2) How does a self-generated magnetic field affect the
velocity field and bring about saturation of an otherwise
growing magnetic field?
The first question searches
for growing solutions to the magnetic induction equation:
The first question searches for growing
solutions to the magnetic induction equation:
This equation can be solved by using
a spectral technique in which both the velocity field and
magnetic field are expanded in terms of spherical vector potentials:
With this expansion, the evolution of the magentic field can
be described by a set of coupled differentical equations for
each mode:
Now, one specifies a velocity field through s(r) and t(r)
and solves for the magnetic eigenmodes and eigenfrencies of
the normalized magnetic induction equation
 ,
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Previous studies of Gubbins73 and Dudley and James,
89 have shown that several simple axisymmetric flows
are capable of generating growing magnetic fields
at sufficiently large Rm.
The topology
of the three of these flows are shown at the left,
along with the growth/damping rate o f the least damped
eigenmode as a function of Rm. The critical Rm's calculated
give evidence that an experiment
capable of producing these flows with Rm~100 should
produce a dynamo.
The t2s2 flow
in particular has a low Rmcrit and
can be produced using propellers. This flow is characterized
by two vortices counter rotating about an axis of
symmetry. Poloidal flows flow out along the poles
of the axis of symmetry and in at the equator; toroidal
flows flow clockwise in one hemisphere and counter
clockwise in the other hemisphere.
Optimization
studies have been performed which show that |
Optimization studies have found flows
with lower values of critical Rm
Velocity Streamlines of T2S2:
Magnetic Eigenmode:
![]()
The mechanism for feedback and self-generation
of the magnetic field can be understood most easily by considering
an initial seed field and tracing its evolution as it is advected
by the moving fluid. For a dynamo to exist there must be a
mechanism for amplification, or stretching, of magnetic field
lines and there must be a mechanism for positive feedback
to occur. This movie above shows the evolution of a seed magnetic
field line being distorted by the flows. Here the magnetic
field evolution is determined in the limit of infinite Rm.
In this limit, the magnetic field can be considered as frozen
into the moving fluid.
The movie shows a magnetic field line
corresponding approximately to a dipole seed field perpendicular
to the axis of symmetry of the flow.A Lagrangian integration
is used to determine the
location
of thefield line at later times. The up-welling of the vortex
in the central region stretches the field line upward, while
toroidal rotation twists the field around.
The field is stretched and reinforces the original
magnetic field. The stretching (amplification) of the magnetic
field is depicted in the figure by the increasing separation
of nearby points. Finite resistivity leads to reconnection
of field lines in the central region where field lines in
opposite directions annihilate each other.
This eigenmode has strong similarities to the
stretch-twist-fold rope dynamo of Vainshtein and Zeldovich.
By twisting and folding the flux tubes back onto themselves,
the original field is reinforced. If this process continues,
tension in the field lines would grow preventing further twisting,
and so reconnection of the field lines is an essential part
of this dynamo. As originally noted by Dudley and James, the
growth rates are found to be sensitive to the ratio of toroidal
to poloidal flow.
This observationcan now be understood geometrically
as a requirement that after one twisting, the large loop must
come back to the same azimuthal angle as the original dipole
field. If the toroidal rotation is too strong or too weak,
the stretched magnetic field generates a field at some other
toroidal angle and the feedback is insufficient to produce
self-excitation.
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