 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/iii/paper-75/2/a/solution

 Top articles  Latest articles New article in topic

Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 75 / 2 / a / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-20

At fixed number density

n

, the electron current gives

u_{e} = - \frac{J}{e n} = - α \nabla \times B, α = \frac{c}{4 π e n}, α B_{0} = v_{A} d_{i} .

The second equality uses Ampère's law without displacement current and stationary ions. A perturbation with characteristic perpendicular wave number

k_{⊥}

therefore produces

δ u_{e} \sim α k_{⊥} δ B

. Its nonlinear magnetic-advection rate is

τ_{nl}^{- 1} \sim k_{⊥} δ u_{e} \sim v_{A} d_{i} k_{⊥}^{2} \frac{δ B}{B _{0}} .

For the anisotropic electron magnetohydrodynamics wave,

k ≃ k_{⊥}

and

∣ ω ∣ \sim v_{A} d_{i} ∣ k_{∥} ∣ k_{⊥}

. Equating this to the nonlinear rate by critical balance gives

\frac{δ B}{B _{0}} \sim \frac{∣ k _{∥} ∣}{k _{⊥}} \sim ϵ ≪ 1.

This is small-amplitude but strong turbulence: a small perturbation can have order-one nonlinear distortion over one wave period because its perpendicular gradients are much larger than its parallel gradients.

 Read the full article

 New to topics? Read the docs here!