Fluid mechanics Magnetohydrodynamics and plasma Theorem

Cowling's antidynamo theorem

Builds on Dynamo theory Magnetohydrodynamics

Statement

Let $V \subset \R^3$ be a bounded region with smooth boundary that is symmetric under rotations about the $z$ -axis. Assume $V$ is filled with a fluid of constant magnetic diffusivity $\eta > 0$ and surrounded by an insulator. Let $\mathbf{u}$ be a smooth velocity field on $V$ , tangent to $\partial V$ and symmetric under rotations about the $z$ -axis. Let $\mathbf{B}$ be a solution of the kinematic dynamo problem for $\mathbf{u}$ that is symmetric under rotations about the $z$ -axis. Then the magnetic energy $\int_{\R^3} \abs{\mathbf{B}(x, t)}^2 \, dx$ tends to $0$ as $t \to \infty$ . In short, an axisymmetric magnetic field cannot be maintained by dynamo action.

Remarks

In cylindrical coordinates $(s, \varphi, z)$ write $\mathbf{B} = \nabla \times (A \, \mathbf{e}_\varphi) + B_\varphi \mathbf{e}_\varphi$ , a poloidal part and a toroidal part. The poloidal potential $A$ satisfies an advection–diffusion equation with no source term, so it decays. The toroidal part $B_\varphi$ has a source, the shearing of the poloidal field by the rotation $u_\varphi$ , but this source decays with $A$ , so $B_\varphi$ decays too. Thomas Cowling proved the original steady version in 1933 by looking at a point where the poloidal field vanishes: there the current cannot be sustained by $\mathbf{u} \times \mathbf{B}$ . The theorem shows that the axisymmetric part of the geomagnetic field must be maintained by non-axisymmetric motions.

Going deeper

For a student

It is the first and best-known antidynamo theorem of dynamo theory. The proof is a decoupling argument: axisymmetry splits the induction equation into a scalar equation for the poloidal flux and a forced equation for the toroidal field. The poloidal part has no way to be regenerated from the toroidal part, which is the missing link of a dynamo cycle. Mean-field dynamo theory restores this link by averaging small non-axisymmetric motions into an effective term, the alpha effect.

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Solar dynamo

Tag	`X9PZ8RJH90`
URI	`m2://X9PZ8RJH90`
Kind	statement, theorem
Writing	W0 stub
MSC	76W05, 86A25
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About	Dynamo theory
Uses	Dynamo theory, Magnetohydrodynamics
Changed	3 November 2026

Cowling's antidynamo theorem

Statement

Remarks

Going deeper

For a student

Reading path

Where this leads

Further reading

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