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.
Bibliographical noteFunding Information:
We thank Bobby Bolshoi and Minye Liu for encouragementa nd discussions and Bill Newman for an enlightening review. E. Pachepsky was supported by a summer internship program of the Minnesota Supercomputer Institute. We are grateful for the technical assistancep rovided by Y. Itoh and D.M. Reuteler. This researchw as supportedb y Cray Research Inc., Ocean Sciences Program of N.S.F., the Danish ResearchC ouncil, the Geoscienceso f D.O.E. and the Universitair Stimulerings Fonds of the Vrije Universiteit, Amsterdam, Netherlands. Much of the computerr esourcesa nd visualization assistancec ame from the Laboratory for Computational Science and Engineering (LCSE) at the University of Minnesota.