Let

D / D t = \partial_{t} - S x \partial_{y}

be the material derivative along the background linear shear flow

u_{0} = - S x e_{y}

. Linearizing about constant

Σ

gives

\frac{D Σ ^{'}}{D t} = - Σ (\partial_{x} v_{x} + \partial_{y} v_{y}),

(1)

\frac{D v _{x}}{D t} - 2Ω v_{y} = - \partial_{x} Ψ - \frac{v _{s}^{2}}{Σ} \partial_{x} Σ^{'},

(2)

\frac{D v _{y}}{D t} + (2Ω - S) v_{x} = - \partial_{y} Ψ - \frac{v _{s}^{2}}{Σ} \partial_{y} Σ^{'} .

(3)

The coefficient

2Ω - S

includes the contribution

v \cdot \nabla u_{0} = - S v_{x} e_{y}

from perturbation advection of the background shear.

b

 0  0

Solution

 0  0

The first-order perturbation of the vortensity is

f^{'} = \frac{1}{Σ} (\partial_{x} v_{y} - \partial_{y} v_{x}) - \frac{2Ω - S}{Σ ^{2}} Σ^{'} .

(1)

Taking the curl of the linearized momentum equation and using the linearized continuity equation yields

\frac{D f ^{'}}{D t} = 0 .

(2)

This is linearized vortensity conservation. It is the perturbative form of material conservation of potential vorticity in an inviscid barotropic fluid; the forcing

- \nablaΨ

is a gradient and therefore creates no vorticity.

c

 0  0

Solution

 0  0

For zero forcing, the given equation for

v_{y}

(\frac{D ^{2}}{D t ^{2}} + κ_{r}^{2} - v_{s}^{2} \nabla^{2}) v_{y} = 0.

(1)

Use the shearing-wave ansatz

v_{y} = ℜ {v (t) e^{i [k_{x} (t) x + k_{y} y]}} .

(2)

The explicit

x

dependence cancels from the material derivative when

\dot{k}_{x} = S k_{y}, k_{x} (t) = k_{x} (0) + S k_{y} t .

(3)

The amplitude then obeys the time-dependent harmonic oscillator equation

\ddot{v} + g (t) v = 0, g (t) = κ_{r}^{2} + v_{s}^{2} [k_{x} (t)^{2} + k_{y}^{2}] .

(4)

This evolving wavevector is the characteristic signature of a shearing wave.

d

 0  0

Solution

 0  0

A satellite on a circular orbit is stationary in the corotating sheet. Write its tidal forcing and the response as

Ψ = ψ (x) e^{i k_{y} y}, v_{y} = v (x) e^{i k_{y} y} .

(1)

Since

D = - i S k_{y} x

, the forced wave equation becomes

- v_{s}^{2} v^{''} + [κ_{r}^{2} + v_{s}^{2} k_{y}^{2} - S^{2} k_{y}^{2} x^{2}] v = (2Ω - S) ψ^{'} - S x k_{y}^{2} ψ .

(2)

This ordinary differential equation describes a satellite-forced density wave in an astrophysical disk.

e

 0  0

Solution

 0  0

The homogeneous equation can be written

v^{''} + [\frac{S ^{2} k _{y}^{2} x ^{2}}{v _{s}^{2}} - k_{y}^{2} - \frac{κ _{r}^{2}}{v _{s}^{2}}] v = 0.

(1)

Its solutions are locally oscillatory where

S^{2} k_{y}^{2} x^{2} > κ_{r}^{2} + v_{s}^{2} k_{y}^{2},

(2)

equivalently

∣ S x ∣ > v_{s}^{2} + \frac{κ _{r}^{2}}{k _{y}^{2}} > v_{s} .

(3)

The relative background orbital speed is

∣ u_{0} ∣ = ∣ S x ∣

, so propagating wave zones occur only where the relative motion of the satellite and disk material is supersonic.

f

 0  0

Solution

 0  0

Take the Fourier transform in

x

, with convention

v (k_{x}) = \int v (x) e^{- i k_{x} x} d x

. The differentiation rules

v^{''} = - k_{x}^{2} v

and

x^{2} v = - v^{''}

turn the forced equation into

S^{2} k_{y}^{2} \frac{d ^{2} v}{d k _{x}^{2}} + [κ_{r}^{2} + v_{s}^{2} (k_{x}^{2} + k_{y}^{2})] v = i [(2Ω - S) k_{x} ψ - S k_{y}^{2} \frac{d ψ}{d k _{x}}] .

(1)

For the unforced equation,

d k_{x} / d t = S k_{y}

implies

\frac{d ^{2}}{d t ^{2}} = S^{2} k_{y}^{2} \frac{d ^{2}}{d k _{x}^{2}} .

(2)

It is therefore exactly the shearing-wave oscillator found in part c, now parametrized by radial wavenumber rather than time.

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Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 321 / 2

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