(819) |

The dynamo equation can be written

(821) |

and

respectively. In general, the term involving is zero. In fact, this term is only included in the analysis to enable us to evaluate the correct matching conditions at . Note that we do not need to write the -component of Eq. (820), since can be obtained more directly from and via the constraint .

Let

Here, is the typical time for magnetic flux to diffuse a distance under the action of resistivity. Equations (822)-(828) can be combined to give

for , and

for . The above equations are immediately recognized as modified Bessel's equations of order .

for , and

(832) |

The first set of matching conditions at are, obviously, that are
continuous, which yields

(833) |

(834) |

Equations (831)-(835) can be combined to give the dispersion relation

(837) |

Unfortunately, despite the fact that we are investigating the simplest known dynamo,
the dispersion relation (836) is sufficiently complicated that it can only
be solved numerically. We can simplify matters considerably
taking the limit
, which corresponds either
to that of small wave-length (*i.e.*, ), or small
resistivity (*i.e.*,
).
The large argument asymptotic behaviour of the Bessel functions is
specified by^{}

(838) | |||

(839) |

where . It follows that

(840) |

where , .

In the limit
, where

(842) |

Dynamo behaviour [

Note that , implying that the dynamo mode

It is clear from Eq. (844) that dynamo action occurs whenever the flow
is made sufficiently rapid. But, what is the minimum amount of flow
which gives rise to dynamo action?
In order to answer this question we
have to solve the full dispersion relation, (836), for various values
of and in order to find the dynamo mode which grows exponentially in time
for the smallest values of and . It is conventional
to parameterize the flow in terms of the *magnetic Reynolds number*

(845) |

(846) |

(847) |

(848) |

Interestingly enough, an attempt was made in the late 1980's to construct a
Ponomarenko dynamo by rapidly pumping liquid sodium through a cylindrical
pipe equipped with a set of twisted vanes at one end to induce helical
flow. Unfortunately, the experiment failed due to mechanical vibrations,
after achieving a Reynolds number which was of the critical value required
for self-excitation of the magnetic field, and was not repaired due to budgetary
problems.^{} More recently, there has been renewed
interest worldwide in the idea of
constructing a liquid metal dynamo, and two such experiments (one in Riga, and one in Karlsruhe) have demonstrated
self-excited dynamo action in a controlled
laboratory setting.