Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 331 / 4 / c / Solution

The full perturbation equations arewhere the streamfunction advection bracket is $J(f,g)=f_z{g}_x-f_x{g}_z$. Put $R=R_c+{ϵ}^2$ and introduce the slow time $T={ϵ}^{2}t$, so ${\partial }_t={ϵ}^2{\partial }_T$ on the amplitude-dependent fields. ThenThe weakly nonlinear expansion balances the small linear growth against the cubic saturation on this slow time scale.