Calibration of the mixing-length theory for structures of helium-dominated atmosphere white dwarfs

We perform a calibration of the mixing-length parameter at the bottom boundary of the convection zone for helium-dominated atmospheres of white dwarfs. This calibration is based on a grid of 3D DB (pure-helium) and DBA (helium-dominated with traces of hydrogen) model atmospheres computed with the CO5BOLD radiation-hydrodynamics code, and a grid of 1D DB and DBA envelope structures. The 3D models span a parameter space of hydrogen-to-helium abundances in the range -10.0 ≤ log (H/He) ≤-2.0, surface gravities in the range 7.5 ≤ log g ≤ 9.0, and effective temperatures in the range 12 000 K ≲ T_eff ≲ 34 000 K. The 1D envelopes cover a similar atmospheric parameter range, but are also calculated with different values of the mixing-length parameter, namely 0.4 ≤ ML2/α ≤ 1.4. The calibration is performed based on two definitions of the bottom boundary of the convection zone: the Schwarzschild and the zero convective flux boundaries. Thus, our calibration is relevant for applications involving the bulk properties of the convection zone including its total mass, which excludes the spectroscopic technique. Overall, the calibrated ML2/α is smaller than what is commonly used in evolutionary models and theoretical determinations of the blue edge of the instability strip for pulsating DB and DBA stars. With calibrated ML2/α we are able to deduce more accurate convection zone sizes needed for studies of planetary debris mixing and dredge-up of carbon from the core. We highlight this by calculating examples of metal-rich 3D DBAZ models and finding their convection zone masses. Mixing-length calibration represents the first step of in-depth investigations of convective overshoot in white dwarfs with helium-dominated atmospheres.