Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 321 / 1 / b

With $\mathcal{G}=0$ and $\mathcal{T}=2\pi v_m{h}^{\prime }\Sigma$, the evolution equation reduces to the radial continuity equationor, for the mass per unit radius $M_r=2\pi r\Sigma$,The outward mass flux is $F_M=2\pi r{v}_m\Sigma$. If $\dot{M}>0$ denotes the inward accretion rate, steady mass conservation requires $F_M=-\dot{M}$. HenceThis is positive because the magnetic drift velocity satisfies $v_m<0$.

For constant $v_m$, $M_r$ obeys the linear transport equationThe method of characteristics therefore givesThe localized profile translates inward without changing shape: material initially concentrated near $r_0$ reaches the central star after

Each annulus loses specific angular momentum as its characteristic moves to smaller $r$. The negative magnetic torque transmits that angular momentum along the large-scale field to the anchored interstellar medium. Thus the disk's angular momentum decreases even before mass crosses $r=0$; angular momentum is conserved only after the field and its anchoring medium are included.

Now the mass per unit radius satisfies the nonlinear scalar conservation lawIts characteristic curves have speedBecause $v_m<0$ and $|q^{\prime }|$ increases with $M_r$, the high central part of a localized pulse travels inward faster than its low-density edges. On the inner, rising side, faster high-density characteristics catch slower low-density ones, causing characteristic crossing and an inward-facing shock wave. On the outer, falling side, the characteristics separate and form a rarefaction wave. The initial bump therefore develops a sharp inner front and a broad outer tail while moving toward the star.