Ponomarenko Dynamo

The simplest known kinematic dynamo is that of Ponomarenko (Ponomarenko 1973). Consider a conducting fluid of resistivity $\eta$ that fills all space. The motion of the fluid is confined to a cylinder of radius $a$ . Adopting standard cylindrical coordinates $(r,\,\theta,\,z)$ aligned with this cylinder, the flow field is written

$${\bf V} =\left\{ \begin{array}{lll} (0,\,r\,{\mit\Omega}, \,U)&\m... ... \mbox{for $r\leq a$}\\ [0.5ex] {\bf0} && \mbox{for $r>a$} \end{array} \right.,$$ (7.139)

where ${\mit\Omega}$ and $U$ are constants. Note that the flow is incompressible. In other words, $\nabla\cdot{\bf V} = 0$ .

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

$$\frac{\partial{\bf B}}{\partial t} = ({\bf B}\cdot\nabla)\,{\bf V} -({\bf V}\cdot\nabla)\,{\bf B} + \frac{\eta}{\mu_0}\,\nabla^2{\bf B},$$ (7.140)

where use has been made of $\nabla\cdot{\bf B}=\nabla\cdot{\bf V}=0$ . Let us search for solutions to this equation of the form

$${\bf B}(r,\theta,z,t) = {\bf B}(r)\,\exp[\,{\rm i}\,(m\,\theta -k\, z)+\gamma\,t].$$ (7.141)

The $r$ - and $\theta$ -components of Equation (7.140) are written (Huba 2000a)

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

$$\phantom{=}+ \frac{\eta}{\mu_0} \left[\frac{d^2 B_r}{dr^{\,2}} + ... ...k^2 r^{\,2} +1)\,B_r}{r^{\,2}}- \frac{{\rm i}\,2\,m\,B_\theta}{r^{\,2}}\right],$$ (7.142)

and

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

$$\phantom{=}+ \frac{\eta}{\mu_0} \left[\frac{d^2 B_\theta}{dr^{\,2... ...k^2 r^{\,2} +1)\,B_\theta}{r^{\,2}}+ \frac{{\rm i}\,2\,m\,B_r}{r^{\,2}}\right],$$ (7.143)

respectively. In general, the term involving $d{\mit\Omega}/dr$ is zero. In fact, this term is only included in the analysis to enable us to evaluate the correct matching conditions at $r=a$ . We do not need to write the $z$ -component of Equation (7.140), because $B_z$ can be obtained more directly from $B_r$ and $B_\theta$ via the constraint $\nabla\cdot{\bf B}=0$ .

Let

$$B_\pm = B_r \pm {\rm i}\,B_\theta,$$ (7.144)

$$y = \frac{r}{a},$$ (7.145)

$$\tau_R =\frac{\mu_0\,a^2}{\eta},$$ (7.146)

$$q^2 = k^2\, a^2 + \gamma\,\tau_R + {\rm i}\,(m\,{\mit\Omega} - k\,U)\,\tau_R,$$ (7.147)

$$s^2 = k^2 \,a^2 + \gamma\,\tau_R.$$ (7.148)

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

$$y^2\,\frac{d^2 B_\pm}{dy^2} + y\,\frac{d B_\pm}{dy} -\left[(m\pm 1)^2 + q^2\,y^2\right]B_\pm = 0$$ (7.149)

for $y\leq 1$ , and

$$y^2\,\frac{d^2 B_\pm}{dy^2} + y\,\frac{d B_\pm}{dy} -\left[(m\pm 1)^2 + s^2\,y^2\right]B_\pm = 0$$ (7.150)

for $y>1$ . The previous equations are modified Bessel's equations of order $m\pm 1$ (Abramowitz and Stegun 1965b). Thus, the physical solutions of Equations (7.149) and (7.150) that are well behaved as $y\rightarrow 0$ and $y\rightarrow \infty$ can be written

$$B_\pm(y) = C_\pm\,\frac{I_{m\pm 1}(q\,y)}{I_{m\pm 1}(q)}$$ (7.151)

for $y\leq 1$ , and

$$B_\pm(y) = D_\pm \, \frac{K_{m\pm 1}(s\,y)}{K_{m\pm 1}(s)}$$ (7.152)

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 matching condition at $y=1$ is the continuity of $B_\pm$ , which yields

$$C_\pm = D_\pm.$$ (7.153)

The second matching condition is obtained by integrating Equation (7.143) from $r=a-\delta$ to $r=a-\delta$ , where $\delta$ is an infinitesimal quantity, and making use of the fact that the angular velocity ${\mit\Omega}$ jumps discontinuously to zero at $r=a$ . It follows that

$$a\,{\mit\Omega}\,B_r = \frac{\eta}{\mu_0} \left[\frac{d B_\theta}{dr}\right]_{r=a_-}^{r=a_+}.$$ (7.154)

Furthermore, integration of Equation (7.142) tells us that $dB_r/dr$ is continuous at $r=a$ . We can combine this information to give the matching condition

$$\left[\frac{d B_\pm}{dy}\right]_{y=1_-}^{y=1_+} =\pm{\rm i}\,{\mit\Omega}\,\tau_R\left( \frac{B_+ +B_-}{2}\right).$$ (7.155)

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

$$G_+\,G_- = \frac{{\rm i}}{2}\,{\mit\Omega}\,\tau_R\,(G_+-G_-),$$ (7.156)

where

$$G_\pm = q\,\frac{I_{m\pm 1}'(q)}{I_{m\pm 1}(q)} - s\,\frac{K_{m\pm 1}'(s)} {K_{m\pm 1}(s)}.$$ (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 $\vert q\vert, \vert s\vert \gg 1$ , which corresponds to that of small wavelength (i.e., $k\,a\gg 1$ ). The large argument asymptotic behavior of the Bessel functions is specified by (Abramowitz and Stegun 1965b)

$$\sqrt{\frac{2\,z}{\pi}}\,K_m(z) = {\rm e}^{\,-z}\left[1+\frac{4\,m^2-1}{8\,z} +{\cal O}\left(\frac{1}{z^2}\right)\right],$$ (7.158)

$$\sqrt{2\pi\,z}\,I_m(z) = {\rm e}^{\,+z}\left[1-\frac{4\,m^2-1}{8\,z} +{\cal O}\left(\frac{1}{z^2}\right)\right],$$ (7.159)

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

$$G_\pm = q + s + \left(\frac{m^2}{2}\pm m + \frac{3}{8}\right)\,\l... ... \frac{1}{s}\right) + {\cal O}\left(\frac{1}{q^{\,2}}+\frac{1}{s^{\,2}}\right).$$ (7.160)

Thus, the dispersion relation (7.156) reduces to

$$(q+s)\,q\,s = {\rm i}\,m\,{\mit\Omega}\,\tau_R,$$ (7.161)

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

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

$$\mu= (m\,{\mit\Omega}-k\,U)\,\tau_R,$$ (7.162)

which corresponds to $({\bf V}\cdot\nabla)\,{\bf B} \rightarrow 0$ , the simplified dispersion relation (7.161) can be solved to give

$$\gamma\,\tau_R \simeq {\rm e}^{\,{\rm i}\,\pi/3}\left(\frac{m\,{\mit\Omega}\,\tau_R} {2}\right)^{2/3} - k^2\,a^2 - {\rm i}\,\frac{\mu}{2}.$$ (7.163)

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

$${\mit\Omega}\,\tau_R > \frac{2^{\,5/2}\,(k\,a)^{\,3}}{m}.$$ (7.164)

Observe 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 Equation (7.163), 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 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 $m$ and $k$ , in order to find the dynamo mode that grows exponentially in time for the smallest values of ${\mit\Omega}$ and $U$ . It is conventional to parameterize the flow in terms of the magnetic Reynolds number,

$$S = \frac{\tau_R}{\tau_H},$$ (7.165)

where

$$\tau_H = \frac{L}{V}$$ (7.166)

is the typical timescale for convective motion across the system. Here, $V$ is a typical flow velocity, and $L$ is the characteristic lengthscale of the system. Taking $V=\vert{\bf V}(a)\vert= ({\mit\Omega}^{\,2}\,a^2+U^{\,2})^{\,1/2}$ , and $L=a$ , we have

$$S = \frac{\tau_R\,({\mit\Omega}^{\,2}\,a^2+U^{\,2})^{\,1/2}}{a}$$ (7.167)

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

$$S_c = 17.7.$$ (7.168)

The most unstable dynamo mode is characterized by $m=1$ , $U/({\mit\Omega}\,a)=1.3$ , $k\,a=0.39$ , 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.

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.