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

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- Solution c

Solution

The full perturbation equations are

\nabla^{2} ψ^{'} + R θ_{x}^{'} = 0, \nabla^{2} θ^{'} - ψ_{x}^{'} = θ_{t}^{'} + J (ψ^{'}, θ^{'}),

where 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. Then

\nabla^{2} ψ^{'} + R_{c} θ_{x}^{'} = - ϵ^{2} θ_{x}^{'}, \nabla^{2} θ^{'} - ψ_{x}^{'} = ϵ^{2} θ_{T}^{'} + J (ψ^{'}, θ^{'}) .

The weakly nonlinear expansion balances the small linear growth against the cubic saturation on this slow time scale.

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