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