The behavior of magnetic Prandtl number on the Rossby wave instability in the protoplanetary discs

In recent years, the Rossby wave instability (RWI) has become the target of intense theoretical and simulation investigations in dealing to some ambiguous problems such as the planet formation and angular momentum transport in the protoplanetary discs. The role of hydrodynamic turbulence on the RWI theory has been well understood by many theoretical and simulation works. However, less attention has been paid to the magnetohydrodynamic (MHD) turbulence in theoretical works related to the RWI. However, the turbulent magnetic Prandtl numbers (Prm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathrm{Pr}_{m}$\end{document}), i.e., the ratio of the turbulent viscosity to the turbulent magnetic diffusivity, is one of the significant parameters in (MHD) turbulence. On the other hand, the gradient and strength of the toroidal magnetic field can affect some variables and parameters in the stationary and perturbation states in some radii which may be lead to important results. In this paper, the whole range of Prm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathrm{Pr}_{m}$\end{document} is considered in details with considering the gradient and strength of the toroidal magnetic field. Although complicating the problem, it gives us a comprehensive view on the RWI occurrence in the cold and hot discs. The results show that the Prm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathrm{Pr}_{m}$\end{document} can significantly control the RWI occurrence as well as the growth rate of unstable modes. While the magneto-rotational instability cannot be responsible for the angular momentum transport in the protoplanetary discs, our results indicate that the RWI is responsible about this subject in these discs.