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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2006/iii/paper-69/2/d/solution

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Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 69 / 2 / d / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-14

Critical balance makes the nonlinear interaction time comparable to the wave period:

\frac{l _{⊥}^{2}}{v _{A} d _{i} b _{l}} \sim \frac{l _{∥} l _{⊥}}{v _{A} d _{i}}, l_{∥} \sim \frac{l _{⊥}}{b _{l}} .

An order-one interaction transfers energy in

τ_{l} \sim τ_{nl}

, so constant energy flux gives

ϵ \sim v_{A}^{3} d_{i} b_{l}^{3} / l_{⊥}^{2}

. Thus the critically balanced electron-magnetohydrodynamic cascade has

\frac{δ B _{l}}{B _{0}} \sim (\frac{ϵ}{v _{A}^{3} d _{i}})^{1/3} l_{⊥}^{2/3}, l_{∥} \sim (\frac{v _{A}^{3} d _{i}}{ϵ})^{1/3} l_{⊥}^{1/3} .

For the corresponding perpendicular magnetic energy spectrum,

δ B_{l}^{2} \sim k_{⊥} E_{B} (k_{⊥})

gives

E_{B} (k_{⊥}) \propto k_{⊥}^{- 7/3}

. The increasing ratio

l_{∥} / l_{⊥} \propto l_{⊥}^{- 2/3}

describes progressively more anisotropic fluctuations at smaller perpendicular scales, within the range in which the model and critical balance assumptions hold.

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