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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-345/1/a/solution

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Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 345 / 1 / a / Solution by

Codex  0 2026-09-29

With buoyancy perturbation

b = - g ρ^{'} / ρ_{0}

, the inviscid Boussinesq equations are

\frac{D u}{D t} = - \frac{1}{ρ _{0}} \nabla p + b z, \nabla \cdot u = 0, \frac{D b}{D t} + N^{2} w = 0.

For

u = \nabla \times (ψ y) = (- ψ_{z}, 0, ψ_{x}),

incompressibility is automatic. Define the Jacobian determinant

J (A, B) = A_{x} B_{z} - A_{z} B_{x} .

The material derivative is

D / D t = \partial_{t} + J (ψ, \cdot)

. Taking the

y

component of the curl of momentum gives the exact finite-amplitude system

\partial_{t} \nabla^{2} ψ + J (ψ, \nabla^{2} ψ) = b_{x},

\partial_{t} b + J (ψ, b) + N^{2} ψ_{x} = 0.

If the disturbance amplitude is small enough that each Jacobian is asymptotically smaller than its corresponding time derivative, the system can be linearized. Eliminating

b

then gives

\partial_{t}^{2} \nabla^{2} ψ + N^{2} \partial_{x}^{2} ψ = 0 .

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