Dynamo theory

Definition

Let $V \subset \R^3$ be a bounded region with smooth boundary, filled with a fluid of constant magnetic diffusivity $\eta > 0$. Let the exterior $\R^3 \setminus \overline{V}$ be an electrical insulator. Let $\mathbf{u}$ be a given smooth velocity field on $V$, tangent to $\partial V$. The kinematic dynamo problem is the problem for a magnetic field $\mathbf{B}(x, t)$ on $\R^3$ such that:

1. in $V$, $\partial_t \mathbf{B} = \nabla \times (\mathbf{u} \times \mathbf{B}) + \eta \Delta \mathbf{B}$;
2. outside $V$, $\nabla \times \mathbf{B} = 0$, and $\mathbf{B}(x) = O(\abs{x}^{-3})$ as $\abs{x} \to \infty$;
3. $\nabla \cdot \mathbf{B} = 0$ everywhere, and $\mathbf{B}$ is continuous across $\partial V$.

The flow $\mathbf{u}$ is a kinematic dynamo if some solution has magnetic energy $\int_{\R^3} \abs{\mathbf{B}}^2 \, dx$ that does not tend to $0$ as $t \to \infty$. Dynamo theory is the study of this problem and of its nonlinear version. In the nonlinear version, $\mathbf{u}$ solves the momentum equation of magnetohydrodynamics with the Lorentz force.