Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 321 / 3

Use a local Cartesian frame rotating at $\Omega =\sqrt{GM/r_0^3}$ about a circular orbit at $r_0$, with $x=r-r_0\ll r_0$, and neglect curvature beyond leading order. The radial gravitational-plus-centrifugal potential isIts first derivative vanishes at $r_0$ and its second derivative there is $-3{\Omega }^2$. Dropping a constant gives the shearing-sheet tidal potential

For ${\mathbf{u}}_0=-\frac{3}{2}\Omega x{\mathbf{e}}_y$, advection and viscosity vanish, while its Coriolis acceleration cancels $-\nabla {\Phi }_t$; constant pressure and $\theta =0$ complete the equilibrium. For axisymmetric perturbations,

For a mode $e^{i\mathbf{k}\cdot \mathbf{x}+st}$, put $D=s+\nu k^2$. The linear equations becomewith $s\theta =u_x$ and $k_x{u}_x+k_z{u}_z=0$. Eliminating $u_y,u_z,P,\theta$ gives the dispersion relationExpanding,

When $N=0$, the factorized relation is $s(D^2+{\varpi }^2)=0$, soThe oscillatory roots are inertial waves restored by rotation; viscosity leaves their leading oscillation frequency unchanged and damps their amplitude at rate $\nu k^2$. The zero root is the decoupled buoyancy-variable mode.

For $\nu =0$,An exponentially growing root exists exactly when $1+n^2<0$, orThus an adverse radial entropy gradient $N^2<0$ must overcome the epicyclic restoring effect of rotation. Rotation stabilizes gradients with $-{\Omega }^2<N^2<0$.

Let $ϵ=\Omega /(\nu k^2)\ll 1$ and $s=ϵ\Omega s_1+\cdots$. Since $D=\Omega /ϵ+O(ϵ\Omega )$, the leading terms ofgiveThe mode is unstable for every $\boxed{N^2<0}$, a weaker criterion than the inviscid condition. Strong viscosity damps the velocity response and removes the rapid epicyclic restoration, permitting a slow viscous-convective instability; its growth rate nevertheless tends to zero on very small scales.