for a prescribed

In most MHD fluids occurring in astrophysics, the resistivity, , is extremely
small. Let us consider the perfectly conducting limit,
.
In this limit, Vainshtein and Zel'dovich, in 1978, introduced an important
distinction
between two fundamentally different classes of dynamo
solutions.^{}Suppose that we solve the eigenvalue equation (793) to obtain the
growth-rate, , of the magnetic field in the limit
.
We expect that

(794) |

It is clear, from the above discussion, that a homopolar disk generator is
an example of a slow dynamo. In fact, it is easily seen that any
dynamo which depends on the motion of a *rigid* conductor for its
operation is bound to be a slow dynamo: in the perfectly conducting
limit, the magnetic flux linking the conductor could never change, so there
would be no magnetic induction. So, why do we believe that fast dynamo
action is even a possibility for an MHD fluid? The answer is, of course, that
an MHD fluid is a *non-rigid* body, and, thus, its motion possesses
degrees of freedom not accessible to rigid conductors.

We know that in the perfectly conducting limit (
) magnetic
field-lines are frozen into an MHD fluid. If the motion is
incompressible (*i.e.*,
) then the stretching
of field-lines implies a proportionate intensification of the field-strength.
The simplest heuristic fast dynamo, first described by Vainshtein and Zel'dovich,
is based on this effect. As illustrated in Fig. 23, a magnetic
flux-tube can be *doubled* in intensity by taking it around a
stretch-twist-fold cycle. The doubling time for this
process clearly does not depend on the resistivity: in this sense, the
dynamo is a fast dynamo. However, under repeated application of this
cycle the magnetic field develops increasingly fine-scale structure.
In fact, in the limit
both the and
fields eventually become chaotic and non-differentiable.
A little resistivity is always required to smooth out the fields on
small length-scales: even in this case the fields remain *chaotic*.

At present, the physical existence of fast dynamos has not been conclusively established, since most of the literature on this subject is based on mathematical paradigms rather than actual solutions of the dynamo equation. It should be noted, however, that the need for fast dynamo solutions is fairly acute, especially in stellar dynamo theory. For instance, consider the Sun. The ohmic decay time for the Sun is about years, whereas the reversal time for the solar magnetic field is only 11 years. It is obviously a little difficult to believe that resistivity is playing any significant role in the solar dynamo.

In the following, we shall restrict our analysis to slow dynamos, which
undoubtably exist in nature, and which are characterized by *non-chaotic*
and fields.