 Solution

ID: past-exam-of-the-astrophysical-fluid-dynamics-course-of-the-university-of-cambridge/2021/section-b/6/c/solution

Expand the axisymmetric Euler equations about the rotating equilibrium, retaining only terms linear in

u_{r}, u_{ϕ}, u_{z}

π (r, z, t) = \frac{p ( r , z , t ) - p _{0} ( r )}{ρ}, [π] = m^{2} s^{- 2} .

The linearized centripetal term in the radial equation is

- 2Ω u_{ϕ}

, after subtracting the equilibrium term

- r Ω^{2}

. Radial advection of the background azimuthal velocity, together with the cylindrical-coordinate term

u_{r} v_{ϕ} / r

, gives

[\frac{d}{d r} (r Ω) + Ω] u_{r} = (2Ω + r Ω^{'}) u_{r} = \frac{1}{r} \frac{d}{d r} (r^{2} Ω) u_{r} .

This is the azimuthal coupling in the second equation; it also expresses conservation of specific angular momentum

j = r^{2} Ω

for a radially displaced element. The axial equation is acceleration balanced by

- \partial_{z} π

. Finally, the axisymmetric incompressible flow condition in cylindrical coordinates is

\frac{1}{r} \partial_{r} (r u_{r}) + \partial_{z} u_{z} = 0,

which expands to the last supplied equation. Background azimuthal advection contributes no

Ω \partial_{ϕ}

term because the perturbations are axisymmetric.

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