Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 321 / 3 / c / i

Table of contents
- Solution i

Solution

Let

δ = Σ^{'} / Σ_{0}

and define the derivative following the background Keplerian shear by

D_{0} = \partial_{t} - \frac{3}{2} Ω x \partial_{y} .

The linearized continuity and momentum equations are

D_{0} δ + \partial_{x} u_{x}^{'} + \partial_{y} u_{y}^{'} = 0,

D_{0} u_{x}^{'} - 2Ω u_{y}^{'} = - c_{s}^{2} \partial_{x} δ,

D_{0} u_{y}^{'} + \frac{1}{2} Ω u_{x}^{'} = - c_{s}^{2} \partial_{y} δ .

The coefficient

Ω/2

combines the background shear with the Coriolis acceleration.

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