Time-Dependent Accretion Disks with Magnetically Driven Winds: Green's Function Solutions

We present Green's function solutions for a geometrically thin, one-dimensional Keplerian accretion disk that includes angular momentum extraction and mass loss due to magnetohydrodynamic (MHD) winds. The disk viscosity is assumed to vary radially as $\nu \propto r^{n}$. We derive solutions for three types of boundary conditions applied at the inner radius $r_{\rm in}$: (i) zero torque, (ii) zero mass accretion rate, and (iii) finite torque and finite accretion rate, and investigate the time evolution of a disk with an initial surface density represented by a Dirac-delta function. The mass accretion rate at the inner radius decays with time as $t^{-3/2}$ for $n = 1$ at late times in the absence of winds under the zero-torque condition, consistent with Lynden-Bell \& Pringle (1974), while the presence of winds leads to a steeper decay. All boundary conditions yield identical asymptotic time evolution for the accretion and wind mass-loss rates, though their radial profiles differ near $r_{\rm in}$. Applying our solutions to protoplanetary disks, we find that the disk follows distinct evolutionary tracks in the accretion rate-disk mass plane depending on $\psi$, a dimensionless parameter that regulates the strength of the vertical stress driving the wind, with the disk lifetime decreasing as $\psi$ increases due to enhanced wind-driven mass loss. The inner boundary condition influences the evolution for $\psi < 1$ but becomes negligible at higher $\psi$, indicating that strong magnetically driven winds dominate and limit mass inflow near the boundary. Our Green's function solutions offer a general framework to study the long-term evolution of accretion disks with magnetically driven winds.