Fractal features in mixing of non-Newtonian and Newtonian mantle convection

Mixing processes in mantle convection depend on the rheology. We have investigated the dynamical differences for both non-Newtonian and Newtonian rheologies on convective mixing for similar values of the effective Rayleigh number. A high-resolution grid, consisting of up to 1500 × 3000 bi-cubic splines, was employed for integrating the advection partial differential equation, which governs the passive scalar field carried by the convecting velocity. We show that, for similar magnitudes of the averaged velocities and surface heat flux, the local patterns of mixing are quite different for the two rheologies. There is a greater richness in the scales of the spatial heterogeneities of the passive scalar field exhibited by the non-Newtonian flow. We have employed the box-counting technique for determining the temporal evolution of the fractal dimension, D, passive scalar field of the two rheologies. We have explained theoretically the development of different regimes in the plot of N, the number of boxes, covered by a range of colors in the passive scalar field, and S, the grid size used in the box-counting. Mixing takes place in several stages. There is a transition from a fractal type of mixing, characterized by islands and clusters to the complete homogenization stage. The manifestation of this transition depends on the scales of the observation, and the initial heterogeneity and on the rheology. Newtonian mixing is homogenized earlier for long-wavelength observational scales, while a very long time would transpire before this transition would take place for non-Newtonian rheology. These results show that mixing dynamics in the mantle have properties germane to fluid turbulence and self-similar scaling.