Near a finite-wavenumber threshold the growth rate of the fastest-growing modes can…

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Complex and collective systems · Complex and collective systems

Many Parts, One Behaviour, Ch Pattern formation: instabilities that choose a wavelength

Problem

Near a finite-wavenumber threshold the growth rate of the fastest-growing modes can be written as σ(q,ϵ)≈σϵ(ϵ−ϵc)−12∣σqq∣(q−qc)2,\sigma(q,\epsilon) \approx \sigma_\epsilon(\epsilon-\epsilon_{\rm c}) - \tfrac12|\sigma_{qq}|(q-q_{\rm c})^2, where ϵ\epsilon is a control parameter, ϵc\epsilon_{\rm c} its critical value, σϵ>0\sigma_\epsilon > 0, and σqq<0\sigma_{qq} < 0 at the maximum of σ(q)\sigma(q). ((a)) Derive an expression for the width Δq\Delta q of the unstable band of wavenumbers as a function of (ϵ−ϵc)(\epsilon-\epsilon_{\rm c}). ((b)) A convecting layer has σϵ=4.2 s−1\sigma_\epsilon = 4.2\ \mathrm{s^{-1}} and ∣σqq∣=15 s−1|\sigma_{qq}| = 15\ \mathrm{s^{-1}} (with qq treated as dimensionless). Find the width of the unstable band when ϵ−ϵc=0.08\epsilon-\epsilon_{\rm c}=0.08.

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