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 X 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 theologies. 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 theologies. 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.