Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 314 / 3 / c / Solution

In the aligned frame, define the Alfvén numberwhere the second equality uses ${\mathbf{u}}_i\parallel {\mathbf{B}}_i$. Since $j={\rho }_i{u}_{xi}$ and $B_x$ are common,Therefore

Using alignment in the tangential momentum flux givesIts continuity then yieldsorwhenever the ratios are determinate.