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

TL;DR

This paper derives Green's function solutions for thin accretion disks with magnetically driven winds, analyzing their time evolution under various boundary conditions and wind strengths, with applications to protoplanetary disks.

Contribution

It provides a comprehensive set of solutions for accretion disk evolution including MHD winds, extending previous models to incorporate boundary conditions and wind effects.

Findings

Mass accretion rate decays as t^{-3/2} without winds

Winds cause steeper decay in accretion rate

Disk lifetime decreases with stronger winds (higher ψ)

Abstract

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.