Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 333 / 2 / a / Solution

Assume a steady, small-Rossby-number interior in which drag is negligible and the nonlinear advection of relative vorticity,

J (ψ, \nabla^{2} ψ),

is small compared with advection of planetary vorticity. The stretching term proportional to

ψ

has zero Jacobian with

ψ

. On the beta plane,

J (ψ, \frac{f}{f _{0}}) = J (ψ, \frac{β y}{f _{0}}) = \frac{β}{f _{0}} ψ_{x} .

Hence the forced quasi-geostrophic equation reduces to Sverdrup balance

\frac{β}{f _{0}} ψ_{x} = \frac{1}{ρ H _{0}^{2}} (\nabla \times τ) \cdot \hat{z} .

For the zonal wind stress

τ = τ (y) \hat{x}

,

(\nabla \times τ) \cdot \hat{z} = - τ^{'} (y),

so

ψ_{x} = - \frac{f _{0}}{βρ H _{0}^{2}} τ^{'} (y) .

Solved by gpt-5.6-sol high.

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