Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 331 / 1 / b

Table of contents
- Solution b

Solution

Write

u = u^{'}

and

θ = - z + θ^{'}

. Dropping quadratic perturbation terms gives the Linearized Boussinesq equations

\partial_{t} u^{'} + λ z \times u^{'} + \nabla p^{'} = σ R a θ^{'} z + σ \nabla^{2} u^{'}, \nabla \cdot u^{'} = 0,

and

\partial_{t} θ^{'} - W = \nabla^{2} θ^{'}, W = u^{'} \cdot z .

The fixed temperatures give

θ^{'} = 0

at

z = 0, 1

. Impermeable stress-free boundary conditions give

W = D^{2} W = 0, D ω = 0, ω = z \cdot \nabla \times u^{'} .

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