Horizontal spreading of planetary debris accreted by white dwarfs

White dwarfs with metal-polluted atmospheres have been studied widely in the context of accretion rocky debris from evolved planetary systems. One open question is geometry and how material arrives mixes white dwarf surface layers. Using 3D radiation-hydrodynamics code CO$^5$BOLD, we present first transport coefficients degenerate star which describe advection-diffusion a passive scalar across surface-plane. We couple newly derived horizontal diffusion previously published vertical to provide theoretical constraints on spreading metals dwarfs. Our grid simulations probes vast majority parameter space convective dwarfs, pure-hydrogen effective temperature range 6000-18000 K pure-helium 12000-34000 K. results suggest that warm hydrogen-rich (DA; $\gtrsim$13000 K) helium-rich (DB, DBA; $\gtrsim$30000 are unable efficiently spread accreted their surface, regardless time dependence accretion. This result may be at odds current non-detection abundance variations discs. For cooler hydrogen- atmospheres, predict largely homogeneous distribution within timescale. typically less than 0.1 per cent disc lifetime estimates, quantity revisited this paper using overshoot results. These relevance for studies bulk composition systems models physics.