where and are constants. Note that the flow is incompressible. In other words, .

The MHD kinematic dynamo equation, (7.113), can be written

where use has been made of . Let us search for solutions to this equation of the form

The - and -components of Equation (7.140) are written (Huba 2000a)

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 . We do not need to write the -component of Equation (7.140), because can be obtained more directly from and via the constraint .

Let

(7.144) | ||

(7.145) | ||

(7.146) | ||

(7.147) | ||

(7.148) |

Here, is the typical time required for magnetic flux to diffuse a distance under the action of resistivity. Equations (7.142)-(7.148) can be combined to give

for , and

for . The previous equations are modified Bessel's equations of order (Abramowitz and Stegun 1965b). Thus, the physical solutions of Equations (7.149) and (7.150) that are well behaved as and can be written

for , and

(7.152) |

for . Here, and are arbitrary constants. Note that the arguments of and are both constrained to lie in the range to .

The first matching condition at is the continuity of , which yields

(7.153) |

The second matching condition is obtained by integrating Equation (7.143) from to , where is an infinitesimal quantity, and making use of the fact that the angular velocity jumps discontinuously to zero at . It follows that

(7.154) |

Furthermore, integration of Equation (7.142) tells us that is continuous at . We can combine this information to give the matching condition

Equations (7.151)-(7.155) yield the dispersion relation

where

(7.157) |

Here, denotes a derivative with respect to argument.

Unfortunately, despite the fact that we are investigating the simplest known kinematic dynamo, the dispersion relation (7.156) is sufficiently complicated that it can only be solved numerically. We can simplify matters considerably taking the limit , which corresponds to that of small wavelength (i.e., ). The large argument asymptotic behavior of the Bessel functions is specified by (Abramowitz and Stegun 1965b)

(7.158) | ||

(7.159) |

where . It follows that

(7.160) |

Thus, the dispersion relation (7.156) reduces to

where , .

In the limit , where

(7.162) |

which corresponds to , the simplified dispersion relation (7.161) can be solved to give

Dynamo behavior [i.e., ] takes place when

Observe that , implying that the dynamo mode oscillates, or rotates, as well as growing exponentially in time. The dynamo generated magnetic field is both non-axisymmetric [note that dynamo activity is impossible, according to Equation (7.163), if ] and three-dimensional, and is, thus, not subject to either of the anti-dynamo theorems mentioned in the preceding section.

It is clear, from Equation (7.164), that dynamo action occurs whenever the flow
is made sufficiently rapid. But, what is the minimum amount of flow
needed to give rise to dynamo action?
In order to answer this question, we
have to solve the full dispersion relation, (7.156), for various values
of
and
, in order to find the dynamo mode that grows exponentially in time
for the smallest values of
and
. It is conventional
to parameterize the flow in terms of the *magnetic Reynolds number*,

(7.165) |

where

(7.166) |

is the typical timescale for convective motion across the system. Here, is a typical flow velocity, and is the characteristic lengthscale of the system. Taking , and , we have

(7.167) |

for the Ponomarenko dynamo. The critical value of the Reynolds number above which dynamo action occurs is found to be (Ponomarenko 1973)

(7.168) |

The most unstable dynamo mode is characterized by , , , and . As the magnetic Reynolds number, , is increased above the critical value, , other dynamo modes are eventually destabilized.

In 2000, the Ponomarenko dynamo was realized experimentally by means of a tall cylinder filled with liquid sodium in which helical flow was excited by a propeller (Gailitis et al. 2000). More information on laboratory dynamo experiments can be found in Verhille et alia 2009.