Protoplanetary migration in non-isothermal discs with turbulence driven by stochastic forcing

Low-mass objects embedded in isothermal protoplanetary discs are known to suffer rapid inwards Type I migration. In non-isothermal discs, recent work has shown that a decreasing radial profile of the disc entropy can lead to a strong positive corotation torque which can significantly slow down or reverse Type I migration in laminar viscous disc models, depending on the amount of viscous and thermal diffusion operating in the planet's horseshoe region. Since the latter is a fraction of the pressure scale height of the disc, it is not clear, however, how this picture changes in turbulent disc models. The aim of this study is to examine the impact of turbulence on the torque experienced by a low-mass planet embedded in a non-isothermal protoplanetary disc. We particularly focus on the role of turbulence on the corotation torque whose amplitude depends on the efficiency of diffusion processes in the planet's horseshoe region. The main issues we want to address are whether the part of the corotation torque scaling with the entropy gradient can remain unsaturated in the presence of turbulence and whether the saturation process in non-isothermal discs can be satisfactorily modelled using laminar disc models. We performed 2D numerical simulations using a grid-based hydrodynamical code in which turbulence is modelled as stochastic forcing. In order to provide estimations for the viscous and thermal diffusion coefficients as a function of the amplitude of turbulence, we first set up non-isothermal disc models for different values of the amplitude of the turbulent forcing. We then include a low-mass planet and determine the evolution of its running time-averaged torque. We show that in non-isothermal discs, the entropy-related corotation torque can indeed remain unsaturated in the presence of turbulence. For turbulence amplitudes that do not strongly affect the disc temperature profile, we find that the running time-averaged torque experienced by an embedded protoplanet is in fairly good agreement with laminar disc models with appropriate values for the thermal and viscous diffusion coefficients and with the formulae of Paardekooper, Baruteau & Kley for the total torque in non-isothermal discs. In discs with turbulence driven by stochastic forcing, the corotation torque therefore behaves similarly as in laminar viscous discs and can be responsible for significantly slowing down or reversing Type I migration.