Testing the tracer ratio method for modeling active compositional fields in mantle convection simulations

[1] Tracer methods are attractive for modeling compositional fields because they offer the potential of zero numerical diffusion. Composition is typically taken to be proportional to the absolute local concentration of tracers, but an increasingly popular method is to have dense'' and regular'' tracers with composition being equal to the local fraction of dense'' tracers. This paper tests this ratio'' method using established benchmarks and by comparing the performance of the two tracer methods and grid-based methods for simulating the long-term evolution of a convecting mantle with a thick, dense, stable layer. For this scenario the ratio method is found to have several advantages, giving sharp, stable long-term layering with no tracer settling, minimal statistical noise'' and low entrainment, even with only -5 tracers per cell. The method is equally applicable to finite volume and finite element treatments of the underlying flow. Entrainment in grid-based advection methods is heavily dependent on resolution and numerical details, and is reduced -1 order of magnitude by the filter proposed by A. Lenardic. Numerical determination of physically correct entrainment rates remains a challenging problem. Comparing tracer and grid based methods, the spatial pattern of the thermal and chemical fields appear to be converging on the finest grids; however the estimated entrainment differs significantly.