Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 321 / 3 / b

For $u_x=0$ and $u_y=u_y(x)$, the material acceleration vanishes. The radial component of force balance isThereforeThe first term is the background Keplerian shear; the second is the geostrophic balance correction produced by the surface-density gradient.

Differentiating the equilibrium velocity givesOn the other hand, the definition of vortensity and $\Sigma ={\Sigma }_0{e}^{\sigma }$ giveMultiplying by $2/\Omega$, and introducing the dimensionless variablesyields

Linearizing $e^{\sigma }=1+\sigma +O({\sigma }^2)$ givesThe prescribed $q$ and the decay conditions are even in $\xi$, so the localized solution is even. Continuity of $\sigma$ and ${\sigma }^{\prime }$ at $\xi =\pm d$ determines all constants:Thus $\Sigma ={\Sigma }_0{e}^{\sigma }\simeq {\Sigma }_0(1+\sigma )$ is a symmetric density bump: it has a concave parabolic core, reaches its maximum at $\xi =0$, and has exponential tails approaching ${\Sigma }_0$. The linearization is self-consistent when $d+d^2/2\ll 1$.