Geodynamo

Definition

Let $V$ be the Earth's liquid outer core, a spherical shell filled with an electrically conducting fluid of magnetic diffusivity $\eta > 0$. Let $\mathbf{u}$ be the velocity of the fluid and $\mathbf{B}$ the magnetic field. In magnetohydrodynamics, $\mathbf{B}$ obeys the induction equation $$\frac{\partial \mathbf{B}}{\partial t} = \nabla \times (\mathbf{u} \times \mathbf{B}) + \eta \nabla^2 \mathbf{B}, \qquad \nabla \cdot \mathbf{B} = 0.$$ Couple this equation with the equations of buoyancy-driven convection of the fluid in a frame rotating with the Earth. Let $L$ be the depth of the shell, so that $L^2/\eta$ is the free decay time of a field. The geodynamo is the regime of this coupled system in the core in which the field is self-sustained. This means that $\mathbf{B}$ does not decay to zero over times much longer than $L^2/\eta$.