Scale heights from vertical hydrostatic equilibrium, h_p (blue), from the first (h₁, orange) and second (h₂, green) moment of mass, plotted against the radius. Top two panels: R_ℓ14 and R_40ky_ℓ18 at 103 kyr. Bottom two panels: R_ℓ14 and R_80ky_ℓ18 at 138 kyr. The horizontal and the left vertical gray lines mark the numerical resolution (dx, outside the figure extent, thus not shown for the high-resolution runs). The values are averaged within rings of thickness dx and plotted against the mean radius of the rings. The second vertical gray line marks the radius of the numerical sink accretion scheme (4dx). The two vertical dotted lines mark the characteristic radii of the disk (R_kep and R_mag), as we defined in Sect. 5.1.2. Within R_kep the disk is close to vertical thermal equilibrium and the three definitions of scale height coincide very well. Beyond this radius, magnetic support becomes strong and deviation occurs. The thermal scale height, h_p (r), is fitted to apower law, which is shown in the legend (in units of AU).