Past exam of the advanced fluid dynamics course of the University of Oxford / 2017 / 1 / c

A linear force-free magnetic field has a spatially constant proportionality factor, $\nabla \times \mathbf{B}=\lambda \mathbf{B}$. Taking its curl and using $\nabla \cdot \mathbf{B}=0$ givesHenceThe constant $\lambda$ has dimensions of inverse length in this convention. If it varied in space, the extra term $\nabla \lambda \times \mathbf{B}$ would remain, so the same Helmholtz equation would not follow merely from a nonlinear force-free condition.