Ponomarenko Dynamos

The simplest known kinematic dynamo is that of Ponomarenko. Consider a conducting fluid of resistivity $\eta$ which fills all space. The motion of the fluid is confined to a cylinder of radius $a$. Adopting cylindrical polar coordinates $(r,\theta,z)$ aligned with this cylinder, the flow field is written

$${\bf V} =\left\{ \begin{array}{lll} (0,r\,{\Omega}, U)&\mbox... ...q a$}\\ [0.5ex] {\bf0} && \mbox{for $r>a$} \end{array}\right.,$$ (819)

where ${\Omega}$ and $U$ are constants. Note that the flow is incompressible: i.e., $\nabla\cdot{\bf V} =0$.

The dynamo equation can be written

$$\frac{\partial{\bf B}}{\partial t} = ({\bf B}\cdot\nabla){\b... ...f V}\cdot\nabla){\bf B} + \frac{\eta}{\mu_0}\,\nabla^2{\bf B}.$$ (820)

Let us search for solutions to this equation of the form

$${\bf B}(r,\theta,z,t) = {\bf B}(r)\,\exp[\,{\rm i}\,(m\,\theta -k\, z)+\gamma\,t].$$ (821)

The $r$- and $\theta$- components of Eq. (820) are written

$$\gamma\,B_r = -{\rm i}\,(m\,{\Omega}-k \,U)\,B_r$$ (822)

$$+ \frac{\eta}{\mu_0} \left[\frac{d^2 B_r}{dr^2} + \frac{1}{r}\fra... ... \frac{(m^2+k^2 r^2 +1)\,B_r}{r^2}- \frac{{\rm i}\,2\,m\,B_\theta}{r^2}\right],$$

and

$$\gamma\,B_\theta = r\,\frac{d{\Omega}}{dr}\,B_r -{\rm i}\,(m\,{\Omega}-k\, U)\,B_\theta$$ (823)

$$+ \frac{\eta}{\mu_0} \left[\frac{d^2 B_\theta}{dr^2} + \frac{1}{r... ... \frac{(m^2+k^2 r^2 +1)\,B_\theta}{r^2}+ \frac{{\rm i}\,2\,m\,B_r}{r^2}\right],$$

respectively. In general, the term involving $d{\Omega}/dr$ is zero. In fact, this term is only included in the analysis to enable us to evaluate the correct matching conditions at $r=a$. Note that we do not need to write the $z$-component of Eq. (820), since $B_z$ can be obtained more directly from $B_r$ and $B_\theta$ via the constraint $\nabla\!\cdot\!{\bf B} =0$.

Let

$$B_\pm = B_r \pm {\rm i}\,B_\theta,$$ (824)

$$y = \frac{r}{a},$$ (825)

$$\tau_R = \frac{\mu_0\,a^2}{\eta},$$ (826)

$$q^2 = k^2 a^2 + \gamma\tau_R + {\rm i}\,(m\,{\Omega} - k\,U)\,\tau_R,$$ (827)

$$s^2 = k^2 a^2 + \gamma\tau_R.$$ (828)

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

$$y^2\,\frac{d^2 B_\pm}{dy^2} + y\,\frac{d B_\pm}{dy} -\left[(m\pm 1)^2 + q^2\,y^2\right]B_\pm = 0$$ (829)

for $y\leq 1$, and

$$y^2\,\frac{d^2 B_\pm}{dy^2} + y\,\frac{d B_\pm}{dy} -\left[(m\pm 1)^2 + s^2\,y^2\right]B_\pm = 0$$ (830)

for $y>1$. The above equations are immediately recognized as modified Bessel's equations of order $m\pm 1$. Thus, the physical solutions of Eqs. (829) and (830), which are well behaved as $y\rightarrow 0$ and $y\rightarrow \infty$, can be written

$$B_\pm = C_\pm\,\frac{I_{m\pm 1}(q\,y)}{I_{m\pm 1}(q)}$$ (831)

for $y\leq 1$, and

$$B_\pm = D_\pm \, \frac{K_{m\pm 1}(s\,y)}{K_{m\pm 1}(s)}$$ (832)

for $y>1$. Here, $C_\pm$ and $D_\pm$ are arbitrary constants. Note that the arguments of $q$ and $s$ are both constrained to lie in the range $-\pi/2$ to $+\pi/2$.

The first set of matching conditions at $y=1$ are, obviously, that $B_\pm$ are continuous, which yields

$$C_\pm = D_\pm.$$ (833)

The second set of matching conditions are obtained by integrating Eq. (823) from $r=a-\delta$ to $r=a-\delta$, where $\delta$ is an infinitesimal quantity, and making use of the fact that the angular velocity ${\Omega}$ jumps discontinuously to zero at $r=a$. It follows that

$$a\,{\Omega}\,B_r = \frac{\eta}{\mu_0} \left[\frac{d B_\theta}{dr}\right]_{r=a_-}^{r=a_+}.$$ (834)

Furthermore, integration of Eq. (822) tells us that $dB_r/dr$ is continuous at $r=a$. We can combine this information to give the matching condition

$$\left[\frac{d B_\pm}{dy}\right]_{y=1_-}^{y=1_+} =\pm{\rm i}\,{\Omega}\,\tau_R\, \frac{B_+ +B_-}{2}.$$ (835)

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

$$G_+\,G_- = \frac{{\rm i}}{2}\,{\Omega}\,\tau_R\,(G_+-G_-),$$ (836)

where

$$G_\pm = q\,\frac{I_{m\pm 1}'(q)}{I_{m\pm 1}(q)} - s\,\frac{K_{m\pm 1}'(s)} {K_{m\pm 1}(s)}.$$ (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 $\vert q\vert, \vert s\vert \gg 1$, which corresponds either to that of small wave-length (i.e., $k\,a\gg 1$), or small resistivity (i.e., ${\Omega}\,\tau_R\gg 1$). The large argument asymptotic behaviour of the Bessel functions is specified by

$$\sqrt{\frac{2\,z}{\pi}}\,K_m(z) = {\rm e}^{-z}\left(1+\frac{4\,m^2-1}{8\,z} +\cdots\right),$$ (838)

$$\sqrt{2\,z\,\pi}\,I_m(z) = {\rm e}^{+z}\left(1-\frac{4\,m^2-1}{8\,z} +\cdots\right),$$ (839)

where $\vert\arg(z)\vert<\pi/2$. It follows that

$$G_\pm = q + s + (m^2/2\pm m + 3/8)(q^{-1} + s^{-1}) + O(q^{-2}+s^{-2}).$$ (840)

Thus, the dispersion relation (836) reduces to

$$(q+s)\,q\,s = {\rm i}\,m\,{\Omega}\,\tau_R,$$ (841)

where $\vert\arg(q)\vert$, $\vert\arg(s)\vert<\pi/2$.

In the limit $\mu\rightarrow 0$, where

$$\mu= (m\,{\Omega}-k\,U)\,\tau_R,$$ (842)

which corresponds to $({\bf V}\cdot\nabla){\bf B} \rightarrow 0$, the simplified dispersion relation (841) can be solved to give

$$\gamma\,\tau_R \simeq {\rm e}^{\,{\rm i}\,\pi/3}\left(\frac{... ...,\tau_R} {2}\right)^{2/3} - k^2\,a^2 - {\rm i}\,\frac{\mu}{2}.$$ (843)

Dynamo behaviour [i.e., ${\rm Re}(\gamma)>0$] takes place when

$${\Omega}\,\tau_R > \frac{2^{5/2}\,(ka)^3}{m}.$$ (844)

Note that ${\rm Im}(\gamma)\neq 0$, 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 $m=0$] 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 $m$ and $k$ in order to find the dynamo mode which grows exponentially in time for the smallest values of ${\Omega}$ and $U$. It is conventional to parameterize the flow in terms of the magnetic Reynolds number

$$S = \frac{\tau_R}{\tau_H},$$ (845)

where

$$\tau_H = \frac{L}{V}$$ (846)

is the typical time-scale for convective motion across the system. Here, $V$ is a typical flow velocity, and $L$ is the scale-length of the system. Taking $V=\vert{\bf V}(a)\vert= \sqrt{{\Omega}^2\,a^2+U^2}$, and $L=a$, we have

$$S = \frac{\tau_R\,\sqrt{{\Omega}^2\,a^2+U^2}}{a}$$ (847)

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

$$S_c = 17.7.$$ (848)

The most unstable dynamo mode is characterized by $m=1$, $U/{\Omega}\,a=1.3$, $k\,a=0.39$, and ${\rm Im}(\gamma)\,\tau_R = 0.41$. As the magnetic Reynolds number, $S$, is increased above the critical value, $S_c$, 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 $80\%$ 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.