On the Ionisation Fraction in Protoplanetary Disks II: The Effect of Turbulent Mixing on Gas–phase Chemistry

We calculate the ionisation fraction in protostellar disk models using two different gas–phase chemical networks, and examine the effect of turbulent mixing by modelling the diffusion of chemical species vertically through the disk. The aim is to determine in which regions of the disk gas can couple to a magnetic field and sustain MHD turbulence. The disk models are conventional α–disks, and the primary source of ionisation is X–ray irradiation from the central star. We assume that the vertical mixing arises because of turbulent diffusion, and accordingly equate the vertical diffusion coefficient, D, with the kinematic viscosity, ν. We find that the effect of diffusion depends crucially on the elemental abundance of heavy metals (magnesium) included in the chemical model. In the absence of heavy metals, diffusion has essentially no effect on the ionisation structure of the disks, as the recombination time scale is much shorter than the turbulent diffusion time scale. When metals are included with an elemental abundance above a threshold value, the diffusion can dramatically reduce the size of the magnetically decoupled region (“dead zone”), or even remove it altogther. This arises when recombination is dominated by metal ions, and the recombination time exceeds the vertical diffusion time scale. For a complex chemistry the elemental abundance of magnesium required to remove the dead zone is xMg = 10 – 10. We also find that diffusion can modify the reaction pathways, giving rise to dominant species when diffusion is switched on that are minor species when diffusion is absent. This suggests that there may be chemical signatures of diffusive mixing that could be used to indirectly detect turbulent activity in protoplanetary disks. We find examples of models in which the dead zone in the outer disk region is rendered deeper when diffusion is switched on. This is caused by turbulent mixing diluting the electron fraction in regions where the ionisation degree is marginally above that required for good coupling. Overall these results suggest that global MHD turbulence in protoplanetary disks may be self–sustaining under favourable circumstances, as turbulent mixing can help maintain the ionisation fraction above that necessary to ensure good coupling between the gas and magnetic field.