For the axisymmetric perturbation, define
$$
\mathcal D=\partial_t+i\mathbf k\mathbin\cdot\Delta\mathbf u,
\qquad
\mathcal D_\tau=\mathcal D+\tau^{-1}.
$$
Linearization about the uniform dust density and drifting equilibrium gives
$$
\boxed{\mathcal D\sigma'
=-i\sigma_0(k_xu_x'+k_zu_z')},
$$
$$
\boxed{\mathcal D_\tau u_x'-2\Omega_Ku_y'
=\tau^{-1}v_{1x}e^{-i\omega t}},
$$
$$
\boxed{\mathcal D_\tau u_y'+\frac12\Omega_Ku_x'
=\tau^{-1}v_{1y}e^{-i\omega t}},
\qquad
\boxed{\mathcal D_\tau u_z'
=\tau^{-1}v_{1z}e^{-i\omega t}}.
$$
The factors $2\Omega_K$ and $\Omega_K/2$ are the <Coriolis acceleration> and Keplerian-shear couplings. Applying $\mathcal D_\tau(\mathcal D_\tau^2+\Omega_K^2)$ to the continuity equation and using the three momentum equations eliminates $\mathbf u'$. All remaining forcing terms are proportional to $e^{-i\omega t}$, so
$$
\boxed{\mathcal D_\tau
(\mathcal D_\tau^2+\Omega_K^2)
\mathcal D\sigma'=Fe^{-i\omega t}},
$$
where $F$ is independent of time.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-321/2/c/i/solution