Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 331 / 3 / a

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- Solution a

Solution

Substitute the stated scales and divide the momentum equation by

2Ω V

. The ratio of inertial to Coriolis acceleration is the Rossby number

Ro = \frac{V}{2Ω L},

while the dimensionless buoyancy coefficient is

\frac{αg Δ θ}{2Ω V} = \frac{B}{Ro}, B = \frac{αg Δ θ}{4 Ω ^{2} L} .

Thus

Ro (u_{t} + u \cdot \nabla u) + z \times u = - \nabla p + \frac{B}{Ro} θ z,

θ_{t} + u \cdot \nabla θ = 0, \nabla \cdot u = 0 .

The dimensionless

B

is the Burgers number for this rotating stratified flow.

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