Dead zone midplane turbulent viscosity parameter α_d. If the disc is not self-graviting (i.e., Q>Q_crit), then α_d = 0. Otherwise, its value is depicted by Eq. (7). We note that we manually smooth the viscosity variation (hence α_d) over five grid points to avoid code divergence. For Ṁ_infall = 10^-6 M_⊙ yr^-1 the smoothing is between 2.2 and 2.5 AU and for Ṁ_infall = 2 × 10^-5 M_⊙ yr^-1 between 5.2 and 5.5 AU.