Top left: sinking timescales of ⁴⁰Ca on a hydrogen-atmosphere white dwarf (0.6 M_⊙, ﹩{\mathrm{log}}\,g=8﹩, abbreviated as DA) calculated by MESA. Two cases are simulated—full convection with ML2/α = 0.8 and no convection. The latter case represents the limit where the magnetic field significantly affects the base of the convection zone. The α = 0.8 curve corresponds to the fiducial mixing length value used in the MESA prescription of E. B. Bauer & L. Bildsten (2019). Bottom left: the pollution gradient parameter f_cov (Equation (28)) or the timescale ratio τ_sink/τ_spread. The horizontal diffusion coefficient D_surf is taken from the 3D radiation hydrodynamics code CO⁵BOLD (T. Cunningham et al. 2021), which is then used to calculate τ_spread with Equation (27). When f_cov is of order unity, the pollution spot can spread over the entire white dwarf surface, but there is still a strong pollution gradient. Homogeneous pollution can be possible when f_cov ≪ 1. The two shaded regions with their corresponding labels show these two global and homogeneous pollution limits. Partial convection may be possible (Section 4.3), resulting in covering fractions between the convective and nonconvective limits shown in this figure. Right: same as the left panels except now calculated for a helium-atmosphere white dwarf (﹩{\mathrm{log}}\,g=8﹩, abbreviated as DB). Three cases are simulated—convection with ML2/α = 1.25 (D. Koester et al. 2020), convection with ML2/α = 0.8 (E. Cukanovaite et al. 2019), and no convection. Note that the vertical scales differ for all four panels.