Notes. The boundaries in the table refer only to the gas fields. ¹In the fully radiative case, the temperature at the disc surface is provided by the equilibrium between stellar heating and radiative cooling (see Section 4). This equilibrium temperature is used for the computation of both the internal and the radiative energy densities. In the MHD simulations, additional boundary conditions for the magnetic field and the emf are required. We report the details in Appendix C. All the simulations presented in this paper run on one single hemisphere. Therefore, the vertical boundaries refer to disc’s surface. On midplane, we use mirror boundary conditions.