Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 321 / 2 / c / i

For the axisymmetric perturbation, define

D = \partial_{t} + i k \cdot Δ u, D_{τ} = D + τ^{- 1} .

Linearization about the uniform dust density and drifting equilibrium gives

D σ^{'} = - i σ_{0} (k_{x} u_{x}^{'} + k_{z} u_{z}^{'}),

D_{τ} u_{x}^{'} - 2 Ω_{K} u_{y}^{'} = τ^{- 1} v_{1 x} e^{- iω t},

D_{τ} u_{y}^{'} + \frac{1}{2} Ω_{K} u_{x}^{'} = τ^{- 1} v_{1 y} e^{- iω t}, D_{τ} u_{z}^{'} = τ^{- 1} v_{1 z} e^{- iω t} .

The factors

2 Ω_{K}

and

Ω_{K} /2

D_{τ} (D_{τ}^{2} + Ω_{K}^{2})

to the continuity equation and using the three momentum equations eliminates

u^{'}

. All remaining forcing terms are proportional to

e^{- iω t}

, so

D_{τ} (D_{τ}^{2} + Ω_{K}^{2}) D σ^{'} = F e^{- iω t},

where

F

is independent of time.

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