Self-Similar Solutions for Geometrically Thin Accretion Disks with Magnetically Driven Winds: Application to Tidal Disruption Events

We analytically derive self-similar solutions for a time-dependent, one-dimensional, magnetically driven accretion-disk-wind model based on the magnetohydrodynamic equations. The model assumes a geometrically thin, gas-pressure-dominated accretion disk and incorporates both magnetic braking and turbulent viscosity through an extended $\alpha$-viscosity prescription in the vertical and radial directions, respectively. The $\alpha$ parameter for the vertical stress is assumed to vary with the disk aspect ratio. We confirm that in the absence of a wind, our self-similar solutions agree with the classical solution of Cannizzo et al. (1990), in which the mass accretion rate follows a power-law decay with time as $t^{-19/16}$. This scaling has been widely used as a key indicator of the mass accretion rate in tidal disruption event (TDE) disks. In contrast, when a wind is present, both the mass accretion and mass loss rates decay more steeply than $t^{-19/16}$. Furthermore, we verify that the power-law indices of these rates are consistent with those obtained from the numerical simulations of Tamilan et al. (2024) at late times. In particular, our analytical solution demonstrates that magnetic braking leads to a more rapid decay of the mass accretion rate, mass loss rate, and bolometric luminosity. In the presence of a strong poloidal magnetic field, all three quantities asymptote to $t^{-5/2}$. This steep decay index can serve as a potential observational signature of magnetocentrifugally driven winds with strong poloidal magnetic fields in TDE disks.