Past exam of the astrophysical fluid dynamics course of the University of Cambridge / 2022 / Section A / 3 / Solution

This is magnetic flux freezing in ideal magnetohydrodynamics. In addition to the supplied ideal magnetohydrodynamic induction equation, a physical magnetic field obeys Gauss's law for magnetism, $\nabla \cdot \mathbf{B}=0$.

Parameterize the material surface by two particle labels $a,b$, so $d\mathbf{S}=({\partial }_a\mathbf{x}\times {\partial }_b\mathbf{x})dadb$ and $D\mathbf{x}/Dt=\mathbf{u}$. Differentiating these tangents gives the material surface element evolution lawwhere $(\nabla \mathbf{u}{)}_{ij}={\partial }_j{u}_i$. Thus the material magnetic-flux transport theorem isExpand the curl in the ideal magnetohydrodynamic induction equation:Since $D/Dt={\partial }_t+\mathbf{u}\cdot \nabla$ is the material derivative, the first three terms cancel the corresponding terms in the surface transport formula. The remaining integrand is $\mathbf{u}\nabla \cdot \mathbf{B}=0$. ThereforeNo incompressible flow assumption is needed: the changing surface area cancels the compression term. This proof assumes smooth velocity and magnetic field and a surface carried with the fluid; the ideal magnetohydrodynamic induction equation by itself, for an arbitrary non-solenoidal vector field, would leave the final divergence term.