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 Incoming links: Rossby deformation radius

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 333 / 1 / b / Solution Created 2026-09-24 Updated 2026-09-25

Initially

ζ = 0

, so conservation of the linear potential vorticity gives

ζ_{f} - \frac{f}{H _{0}} η_{f} = - \frac{f}{H _{0}} η_{i} .

The final flow is in geostrophic balance, hence

ζ_{f} = \frac{g}{f} \nabla^{2} η_{f} .

Consequently the adjusted height solves the modified Helmholtz equation

(\nabla^{2} - \frac{1}{L _{R}^{2}}) η_{f} = - \frac{η _{i}}{L _{R}^{2}}, L_{R} = \frac{g H _{0}}{∣ f ∣},

where

L_{R}

is the Rossby deformation radius.

The initial condition is independent of

x

and odd in

y

. The bounded solution that is continuously differentiable at

y = 0

is

η_{f} (y) = η_{0} sgn (y) (1 - e^{- ∣ y ∣/ L_{R}}) .

It gives

u_{f} (y) = - \frac{g η _{0}}{f L _{R}} e^{- ∣ y ∣/ L_{R}}, v_{f} = 0 .

During geostrophic adjustment, the part of the initial energy incompatible with the conserved potential-vorticity distribution radiates away as inertia-gravity waves. The remaining current has width

O (L_{R})

and is geostrophically balanced.

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Rigid-lid approximation 2026-10-06

The rigid-lid approximation suppresses free surface displacement in a fluid layer while retaining a variable pressure that enforces the mass constraint. At constant depth it gives horizontally nondivergent motion. In a one-layer quasi-geostrophic approximation, removing free-surface stretching eliminates the

- ψ / R_{D}^{2}

term from shallow-water quasi-geostrophic potential vorticity. This is also the small-horizontal-scale limit compared with the Rossby deformation radius, when that term is negligible rather than exactly absent.

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