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The simplest known kinematic dynamo is that
of Ponomarenko.
Consider a conducting fluid
of resistivity
which fills all space. The motion of the fluid
is confined to a cylinder of radius
. Adopting cylindrical polar coordinates
aligned with this cylinder, the flow field is written
![\begin{displaymath}
{\bf V} =\left\{
\begin{array}{lll}
(0,r\,{\Omega}, U)&\mbox...
...q a$}\\ [0.5ex]
{\bf0} && \mbox{for $r>a$}
\end{array}\right.,
\end{displaymath}](img1843.png) |
(819) |
where
and
are constants. Note that the flow is incompressible:
i.e.,
.
The dynamo equation can be written
 |
(820) |
Let us search for solutions to this equation of the form
![\begin{displaymath}
{\bf B}(r,\theta,z,t) = {\bf B}(r)\,\exp[\,{\rm i}\,(m\,\theta -k\, z)+\gamma\,t].
\end{displaymath}](img1846.png) |
(821) |
The
- and
- components of Eq. (820) are written
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
 |
 |
 |
(824) |
 |
 |
 |
(825) |
 |
 |
 |
(826) |
 |
 |
 |
(827) |
 |
 |
 |
(828) |
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
![\begin{displaymath}
y^2\,\frac{d^2 B_\pm}{dy^2} + y\,\frac{d B_\pm}{dy} -\left[(m\pm 1)^2 + q^2\,y^2\right]B_\pm = 0
\end{displaymath}](img1866.png) |
(829) |
for
, and
![\begin{displaymath}
y^2\,\frac{d^2 B_\pm}{dy^2} + y\,\frac{d B_\pm}{dy} -\left[(m\pm 1)^2 + s^2\,y^2\right]B_\pm = 0
\end{displaymath}](img1868.png) |
(830) |
for
. The above equations are immediately recognized as modified
Bessel's equations of order
.
Thus, the physical solutions of Eqs. (829) and (830), which are well behaved
as
and
, can be written
 |
(831) |
for
, and
 |
(832) |
for
. Here,
and
are arbitrary constants.
Note that the arguments of
and
are both constrained to lie in the
range
to
.
The first set of matching conditions at
are, obviously, that
are
continuous, which yields
 |
(833) |
The second set of matching conditions are obtained by integrating Eq. (823)
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
![\begin{displaymath}
a\,{\Omega}\,B_r = \frac{\eta}{\mu_0}
\left[\frac{d B_\theta}{dr}\right]_{r=a_-}^{r=a_+}.
\end{displaymath}](img1881.png) |
(834) |
Furthermore, integration of Eq. (822) tells us that
is continuous
at
. We can combine this information to give the matching
condition
![\begin{displaymath}
\left[\frac{d B_\pm}{dy}\right]_{y=1_-}^{y=1_+} =\pm{\rm i}\,{\Omega}\,\tau_R\,
\frac{B_+ +B_-}{2}.
\end{displaymath}](img1883.png) |
(835) |
Equations (831)-(835) can be combined to give the dispersion relation
 |
(836) |
where
 |
(837) |
Here,
denotes a derivative.
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
where
.
It follows that
 |
(840) |
Thus, the dispersion relation (836) reduces to
 |
(841) |
where
,
.
In the limit
, where
 |
(842) |
which corresponds to
, the simplified dispersion
relation (841) can be solved to give
 |
(843) |
Dynamo behaviour [i.e.,
] takes place
when
 |
(844) |
Note 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 Eq. (843), 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 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) |
where
 |
(846) |
is the typical time-scale for convective motion across the system. Here,
is a typical flow velocity, and
is the scale-length of the system.
Taking
, and
, we
have
 |
(847) |
for the Ponomarenko dynamo. The critical value of the Reynolds number above
which dynamo action occurs is found to be
 |
(848) |
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.
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.
Next: Magnetic Reconnection
Up: Magnetohydrodynamic Fluids
Previous: Cowling Anti-Dynamo Theorem
Richard Fitzpatrick
2011-03-31