Thermal stability of thin disk with magnetically driven winds

The absence of thermal instability in the high/soft state of black hole X-ray binaries, in disagreement with the standard thin disk theory, is a long-standing riddle for theoretical astronomers. We have tried to resolve this question by studying the thermal stability of a thin disk with magnetically driven winds in the $\dot{M}- \Sigma$ plane. It is found that disk winds can greatly decrease the disk temperature and thus help the disk become more stable at a given accretion rate. The critical accretion rate $\dot{M}_{\rm crit}$ corresponding to the thermal instability threshold is increased significantly in the presence of disk winds. For $\alpha=0.01$ and $B_{\rm \phi}=10B_{\rm _p}$, the disk is quite stable even for a very weak initial poloidal magnetic field [$\beta_{\rm p,0}\sim 2000, \beta_{\rm p}=(P_{\rm {gas}}+P_{\rm rad})/(B_{\rm p}^2/8\pi)$]. But when $B_{\rm \phi}=B_{\rm _p}$ or $B_{\rm \phi}=0.1B_{\rm _p}$, a somewhat stronger (but still weak) field ($\beta_{\rm p,0}\sim 200$ or $\beta_{\rm p,0}\sim 20$) is required to make the disk stable. Nevertheless, despite the great increase of $\dot{M}_{\rm crit}$, the luminosity threshold corresponding to instability remains almost constant or even decreases slowly with increasing $\dot{M}_{\rm crit}$ due to the decrease of gas temperature. The advection and diffusion timescales of the large-scale magnetic field threading the disk are also investigated in this work. We find that the advection timescale can be smaller than the diffusion timescale in a disk with winds, because the disk winds take away most of the gravitational energy released in the disk, resulting in the decrease of the magnetic diffusivity $\eta$ and the increase of the diffusion timescale.