Equilibration during mantle melting: a fractal tree model.

Many basalts from oceanic islands, ridges, and arcs show strong trace element evidence for melting at great depths, where garnet is a stable phase in mantle peridotites. If partial melts ascend to the surface by porous (intergranular) flow processes, the high-pressure garnet signature will be obliterated by diffusive reequilibration at shallower depths in the mantle. Spiegelman and Kenyon [Spiegelman, M. & Kenyon, P. (1992) Earth Planet Sci. Lett. 109, 611-620] argued that partial melts must therefore be focused into a coarser transport network, for high-speed delivery to the surface. Numerous natural network systems, such as rivers and the human vascular and bronchial systems, have fractal structures that are optimal for minimizing energy expenditure during material transport. I show here that a fractal magma "tree" with these optimal properties provides a network in which magma rapidly loses diffusive chemical "contact" with its host matrix. In this fractal network, magma conduits combine by twos, with the radius and flow velocities scaling as (2)n/3, where n is the generation number. For reasonable values of volume diffusivities, viscosities, and aspect ratios, melts will experience only limited diffusive reequilibration once they have traveled some hundreds of meters from their source. Melts thus represent rather local mantle domains, and there is little problem in delivering melts with deep (<100 km) geochemical signatures to the surface.