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Some of the peculiarities of dynamo theory are well illustrated
by the prototype example of selfexcited dynamo action, which is the
homopolar disk dynamo. As illustrated in Fig. 22, this device
consists of a conducting disk which rotates at angular
frequency about its axis under the
action of an applied torque. A wire, twisted about the axis in the
manner shown, makes sliding contact with the disc at , and with
the axis at , and carries a current . The magnetic field
associated with this current has a flux
across the disc, where is the mutual inductance between the wire and the
rim of the disc. The rotation of the disc in the
presence of this flux generates a radial electromotive
force

(783) 
since a radius of the disc cuts the magnetic flux once
every seconds. According to this simplistic
description, the equation for is written

(784) 
where is the total resistance of the circuit, and is its
selfinductance.
Figure 22:
The homopolar generator.

Suppose that the angular velocity is maintained by suitable adjustment
of the driving torque. It follows that Eq. (784) possesses an
exponential solution
, where

(785) 
Clearly, we have exponential growth of , and, hence, of the magnetic
field to which it gives rise (i.e., we have dynamo action),
provided that

(786) 
i.e., provided that the disk rotates rapidly enough. Note that
the homopolar generator depends for its success on its builtin
axial asymmetry. If the disk rotates in the opposite
direction to that shown in Fig. 22 then , and the
electromotive force generated by the rotation of the disk always acts
to reduce . In this case, dynamo action is impossible (i.e.,
is always negative). This is a troubling observation,
since most astrophysical objects, such as stars and planets, possess very
good axial symmetry. We conclude that if such bodies are to act
as dynamos then the asymmetry of their internal motions must somehow
compensate for their lack of builtin asymmetry. It is far from obvious
how this is going to happen.
Incidentally, although the above analysis of a homopolar generator
(which is the standard analysis found in most textbooks) is
very appealing in its simplicity, it cannot be entirely correct.
Consider the limiting situation of a perfectly
conducting disk and wire, in which . On the one hand,
Eq. (785) yields
, so that
we still have dynamo action. But, on the other hand, the rim of the disk
is a closed circuit embedded in a perfectly conducting medium, so the
flux freezing constraint requires that the flux, ,
through this circuit must remain a constant. There is
an obvious contradiction.
The problem is that we have neglected the currents
that flow azimuthally in the disc: i.e., the very currents
which control the diffusion of magnetic flux across the rim of
the disk. These currents become particularly important
in the limit
.
The above paradox can be resolved by supposing that the azimuthal current
is constrained to flow around the rim of the disk (e.g.,
by a suitable distribution of radial insulating strips). In this
case, the fluxes through the and circuits are
and the equations governing the current flow are
where , and refer to the circuit. Let us search
for exponential solutions,
, of the
above system of equations. It is easily
demonstrated that

(791) 
Recall the standard result in electromagnetic theory that for two
noncoincident circuits. It is clear, from the above expression, that the
condition for dynamo action (i.e., ) is

(792) 
as before. Note, however, that
as
.
In other words, if the rotating disk is a perfect conductor then dynamo
action is impossible. The above system of equations can transformed
into the wellknown Lorenz system, which exhibits chaotic behaviour
in certain parameter regimes.^{} It is noteworthy that this simplest prototype
dynamo system already contains the seeds of chaos (provided that
the formulation is selfconsistent).
It is clear from the above discussion that, whilst dynamo action requires the
resistance of the circuit, , to be low, we lose dynamo action
altogether if we go to the
perfectly conducting limit,
, because magnetic fields are unable to diffuse
into the region in which magnetic induction is operating. Thus, an efficient
dynamo requires a conductivity that is large, but not too large.
Next: Slow and Fast Dynamos
Up: Magnetohydrodynamic Fluids
Previous: MHD Dynamo Theory
Richard Fitzpatrick
20110331