Cowling Anti-Dynamo Theorem

One of the most important results in slow, kinematic dynamo theory is credited to Cowling (Cowling 1934; Cowling 1957b). The so-called Cowling anti-dynamo theorem states that:

An axisymmetric magnetic field cannot be maintained via dynamo action.

Let us attempt to prove this proposition.

We adopt standard cylindrical coordinates: $(\varpi,\,\theta,\,z)$. The system is assumed to possess axial symmetry, so that $\partial/\partial\theta \equiv 0$. For the sake of simplicity, the plasma flow is assumed to be incompressible, which implies that $\nabla\cdot{\bf V} = 0$.

It is convenient to split the magnetic and velocity fields into poloidal and toroidal components:

$${\bf B} = {\bf B}_p + {\bf B}_t,$$ (8.115)

$${\bf V} = {\bf V}_p + {\bf V}_t.$$ (8.116)

Here, a “poloidal” vector only possesses non-zero $\varpi$- and $z$-components, whereas a “toroidal” vector only possesses a non-zero $\theta$-component.

The poloidal components of the magnetic and velocity fields are written (Richardson 2019),

$${\bf B}_p = \nabla\times\left(\frac{\psi}{\varpi}\,{\bf e}_\theta\right) \equiv \frac{\nabla\psi\times{\bf e}_\theta}{\varpi},$$ (8.117)

$${\bf V}_p = \nabla\times\left(\frac{\phi}{\varpi}\,{\bf e}_\theta\right) \equiv \frac{\nabla\phi\times{\bf e}_\theta}{\varpi},$$ (8.118)

where $\psi=\psi(\varpi,z,t)$ and $\phi=\phi(\varpi,z,t)$. The toroidal components are given by

$${\bf B}_t = B_t(\varpi,z,t)\,{\bf e}_\theta,$$ (8.119)

$${\bf V}_t = V_t(\varpi,z,t)\,{\bf e}_\theta.$$ (8.120)

Note that by writing the ${\bf B}$ and ${\bf V}$ fields in the previous form we ensure that the constraints $\nabla\cdot{\bf B}=0$ and $\nabla\cdot{\bf V} = 0$ are automatically satisfied. Note, further, that because ${\bf B}\cdot\nabla\psi=0$ and ${\bf V}\cdot\nabla\phi=0$, we can regard $\psi$ and $\phi$ as stream-functions for the magnetic and velocity fields, respectively.

The condition for the magnetic field to be maintained by dynamo currents, rather than by currents at infinity, is

$$\psi\rightarrow \frac{1}{r} \mbox{\hspace{1cm}as $r\rightarrow\infty$} \,$$ (8.121)

where $r=(\varpi^{2}+z^{2})^{1/2}$. We also require the flow stream-function, $\phi$, to remain bounded as $r\rightarrow\infty$.

Consider the MHD Ohm's law for a resistive plasma:

$${\bf E} + {\bf V} \times{\bf B} = \eta\,{\bf j}.$$ (8.122)

Taking the toroidal component of this equation, we obtain

$$E_t + ({\bf V}_p\times{\bf B}_p)\cdot {\bf e}_\theta=\eta\, j_t.$$ (8.123)

It is easily demonstrated from the Faraday-Maxwell equation that

$$E_t =-\frac{1}{\varpi}\frac{\partial\psi}{\partial t}.$$ (8.124)

Furthermore,

$$({\bf V}_p\times{\bf B}_p)\cdot{\bf e}_\theta =\frac{(\nabla\phi\... ... z} -\frac{\partial\phi}{\partial\varpi}\frac{\partial\psi}{\partial z}\right),$$ (8.125)

and (Richardson 2019)

$$\mu_0\,j_t = \nabla\times {\bf B}_p \cdot {\bf e}_\theta= -\left[... ...\frac{\partial\psi}{\partial\varpi}+\frac{\partial^2\psi}{\partial z^2}\right).$$ (8.126)

Thus, Equation (8.123) reduces to

$$\frac{\partial\psi}{\partial t} - \frac{1}{\varpi}\left(\frac{\pa... ...frac{\partial\psi}{\partial\varpi} +\frac{\partial^2\psi}{\partial z^2}\right).$$ (8.127)

Multiplying the previous equation by $\psi$ and integrating over all space, we obtain

$$\frac{1}{2}\frac{d}{dt}\!\int\psi^2\,dV -\int\!\!\int 2\pi\,\psi\left(\frac{\partial\psi}{\partial\varpi}... ...partial\phi}{\partial\varpi}\frac{\partial\psi}{\partial z}\right) d\varpi \,dz$$

$$=\frac{\eta}{\mu_0}\int\!\!\int2\pi\,\varpi\,\psi\left(\frac{\par... ...al\psi}{\partial\varpi} +\frac{\partial^2\psi}{\partial z^2}\right)d\varpi\,dz.$$ (8.128)

The second term on the left-hand side of the previous expression can be integrated by parts to give

$$-\int\!\!\int 2\pi\left[-\phi\,\frac{\partial}{\partial z} \left(... ...varpi}\left(\psi\,\frac{\partial\psi}{\partial z} \right)\right] d\varpi\,dz=0,$$ (8.129)

where surface terms have been neglected, in accordance with Equation (8.121). Likewise, the term on the right-hand side of Equation (8.128) can be integrated by parts to give

$$\frac{\eta}{\mu_0}\int\!\!\int 2\pi\left[-\frac{\partial(\varpi\,... ...}\right)^2 +\left(\frac{\partial\psi}{\partial z}\right)^2\right]\,d\varpi\,dz.$$ (8.130)

Thus, Equation (8.128) reduces to

$$\frac{d}{dt}\!\int\psi^2\,dV = -2\,\frac{\eta}{\mu_0}\int \vert\nabla\psi\vert^{2}\,dV\leq 0.$$ (8.131)

It is clear, from the previous expression, that the poloidal stream-function, $\psi$—and, hence, the poloidal magnetic field, ${\bf B}_p$—decays to zero under the influence of resistivity. We conclude that the poloidal magnetic field cannot be maintained via dynamo action.

Of course, we have not ruled out the possibility that the toroidal magnetic field can be maintained via dynamo action. In the absence of a poloidal field, the curl of the poloidal component of Equation (8.122) yields

$$-\frac{\partial {\bf B}_t}{\partial t} +\nabla\times({\bf V}_p\times {\bf B}_t) = \eta\,\nabla\times {\bf j}_p,$$ (8.132)

which reduces to

$$-\frac{\partial B_t}{\partial t} + \nabla\times({\bf V}_p\times {... ..._\theta=- \frac{\eta}{\mu_0}\,\nabla^2(B_t\,{\bf e}_\theta)\cdot{\bf e}_\theta.$$ (8.133)

Now (Richardson 2019),

$$\nabla^2(B_t\,{\bf e}_\theta)\cdot{\bf e}_\theta =\frac{\partial^... ...t}{\partial\varpi} + \frac{\partial^2 B_t}{\partial z^2} -\frac{B_t}{\varpi^2},$$ (8.134)

and (Richardson 2019)

$$\nabla\times({\bf V}_p\times {\bf B}_t) \cdot{\bf e}_\theta= \fra... ...rtial z}\! \left(\frac{B_t}{\varpi}\right) \frac{\partial\phi}{\partial\varpi}.$$ (8.135)

Thus, Equation (8.133) yields

$$\frac{\partial\chi}{\partial t} -\frac{1}{\varpi}\left(\frac{\par... ...rac{\partial\chi} {\partial\varpi} +\frac{\partial^2\chi}{\partial z^2}\right),$$ (8.136)

where

$$B_t = \varpi\,\chi.$$ (8.137)

Multiply Equation (8.136) by $\chi$, integrating over all space, and then integrating by parts, we obtain

$$\frac{d}{dt}\!\int\chi^2\,dV = -2\,\frac{\eta}{\mu_0}\int\vert\nabla\chi\vert^{2}\,dV\leq 0.$$ (8.138)

It is clear, from this equation, that $\chi$—and, hence, the toroidal magnetic field, ${\bf B}_t$—decays to zero under the influence of resistivity. We conclude that no axisymmetric magnetic field—either poloidal or toroidal—can be maintained by dynamo action, which proves Cowling's theorem.

Cowling's theorem is the earliest, and most significant, of a number of anti-dynamo theorems that severely restrict the types of magnetic fields that can be maintained via dynamo action. For instance, it is possible to prove that a two-dimensional magnetic field cannot be maintained by dynamo action (Moffatt 1978). Here, “two-dimensional” implies that in some Cartesian coordinate system, $(x,\,y,\,z)$, the magnetic field is independent of $z$. The suite of anti-dynamo theorems can be summed up by saying that successful dynamos possess a rather low degree of symmetry.