Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 333 / 4 / v

Solution

Define the perturbation vorticity component

ζ^{'} = u_{z}^{'} - w_{x}^{'} .

Taking the curl of the linear momentum equation and using the buoyancy equation gives

(\partial_{t} + U \partial_{x}) ζ^{'} = - b_{x}^{'}, (\partial_{t} + U \partial_{x}) b^{'} = - N^{2} w^{'} .

Multiply the first equation by

b^{'}

, the second by

ζ^{'}

, and horizontally average. Periodicity makes the averaged

x

derivatives vanish, while

\overline{w^{'} ζ^{'}} = \overline{w^{'} u_{z}^{'} - w^{'} w_{x}^{'}} = \partial_{z} \overline{u^{'} w^{'}}

by incompressibility and integration by parts. It follows that

\partial_{t} \overline{b^{'} ζ^{'}} = - N^{2} \partial_{z} \overline{u^{'} w^{'}} .

\partial_{t} A + \partial_{z} F_{A} = 0, A = - \frac{b ^{'} ζ ^{'}}{N ^{2}}, F_{A} = - \overline{u^{'} w^{'}} .

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