In the aligned frame, define the <Alfvén number>
$$
A_i^2=\frac{u_i^2}{v_{Ai}^2}
=\frac{\mu_0\rho_i u_{xi}^2}{B_x^2},
$$
where the second equality uses $\mathbf u_i\parallel\mathbf B_i$. Since $j=\rho_i u_{xi}$ and $B_x$ are common,
$$
A_i^2=\frac{\mu_0j u_{xi}}{B_x^2}.
$$
Therefore
$$
\boxed{\frac{\rho_2}{\rho_1}
=\frac{u_{x1}}{u_{x2}}
=\frac{A_1^2}{A_2^2}}.
$$

Using alignment in the tangential momentum flux gives
$$
\rho u_xu_y-\frac{B_xB_y}{\mu_0}
=\frac{B_xB_y}{\mu_0}(A^2-1).
$$
Its continuity then yields
$$
B_{y2}(A_2^2-1)=B_{y1}(A_1^2-1),
$$
or
$$
\boxed{\frac{B_{y2}}{B_{y1}}
=\frac{A_1^2-1}{A_2^2-1}}
$$
whenever the ratios are determinate.

Solved by gpt-5.6-sol high.

Solution

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

Define the poloidal <Alfvén number> by
$$
\boxed{A^2=\frac{\mu_0\rho u_p^2}{B_p^2}
=\frac{\mu_0k^2}{\rho}},
$$
where $u_p=kB_p/\rho$. Thus, along a given field line,
$$
\boxed{\rho=\frac{\mu_0k^2}{A^2}};
$$
density decreases as the Alfvén number grows.

The azimuthal velocity representation gives
$$
u_\phi=r\omega+\frac{kB_\phi}{\rho}.
$$
Substitution into $\ell=ru_\phi-rB_\phi/(\mu_0k)$ yields
$$
\ell=r^2\omega+\frac{rB_\phi}{\mu_0k}(A^2-1).
$$
At the <Alfvén surface>, $A^2=1$. Smoothness requires the numerator of the solution for $B_\phi$ to vanish there, so
$$
\ell=\omega r_a^2.
$$
Solving away from the surface then gives
$$
\boxed{B_\phi
=\frac{\mu_0k\omega}{r}
\frac{r^2-r_a^2}{1-A^2}}.
$$

Solved by gpt-5.6-sol high.

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 314 / 4 / d / Solution

{c}
{title2=$A$}

An Alfvén number is the ratio of a flow speed to the corresponding Alfvén speed. For a poloidal flow parallel to a poloidal magnetic field, $A^2=\mu_0\rho u_p^2/B_p^2$.


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