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
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
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.