Define the <surface density of a disk>, outward <mass flux>, internal torque, and surface magnetic torque by
$$
\Sigma=\int_{-\infty}^{\infty}\rho\,dz,
\qquad
F_M=2\pi r\int_{-\infty}^{\infty}\rho u_r\,dz,
$$
$$
\mathcal G=-2\pi r^2\int_{-\infty}^{\infty}
\left(\Pi_{r\phi}+\frac{B_rB_\phi}{4\pi}\right)dz,
\qquad
\mathcal T=2\pi r
\left[\frac{B_\phi B_z}{4\pi}\right]_{-\infty}^{\infty}.
$$
The sign convention makes $\mathcal G$ positive for outward <angular momentum transport> in an ordinary <Keplerian accretion disk>. Vertical integration of <mass conservation> gives
$$
2\pi r\,\partial_t\Sigma+\partial_rF_M=0.
$$

The <specific angular momentum> is $h=ru_\phi=r^2\Omega$. Multiply the azimuthal equation by $r$, use the continuity equation to put its left-hand side in conservative form, and integrate over $z$. The assumed decay removes the vertical mass and viscous fluxes, whereas the magnetic surface stress remains:
$$
\partial_t(2\pi r\Sigma h)+\partial_r(F_Mh)
=-\partial_r\mathcal G+r\mathcal T.
$$
Subtracting $h$ times the integrated mass equation yields
$$
F_M\frac{dh}{dr}=-\partial_r\mathcal G+r\mathcal T.
$$
Since $F_M=-(dh/dr)^{-1}(\partial_r\mathcal G-r\mathcal T)$, substitution in mass conservation gives the required one-dimensional <advection-diffusion equation>
$$
\boxed{\partial_t\Sigma
=\frac1{2\pi r}\partial_r\left[
\left(\frac{dr}{dh}\right)
(\partial_r\mathcal G-r\mathcal T)\right]}.
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-321/1/a/solution