(1) Unite Mixte de
Recherche, CNRS, Toulouse, France
(2) Department of Geology and
Geophysics and Minnesota Supercomputing
Institute, University of Minnesota, Minneapolis, MN USA
(3) Department of Geological
Sciences, Bolder, CO USA
(4) Department of Geophysics,
Faculty of Mathematics and Physics, Charles University, Prague, Czech
Republic
The purpose of this work is to show how these secondary instabilities can develop as a consequence of the interaction of the shear flow developed by the large-scale circulation and the rising plume. This will be conducted first within the framework of a constant viscosity fluid. First, we will employ a two-dimensional axisymmetric spherical-shell model within the framework of a Boussinesq fluid with an aspect-ratio of around six. Second, we have used a three-dimensional Boussinesq model with an aspect-ratio of 3x3x1. Two-dimensional simulations are carried out from 3x10^7 to Ra=10^10. Secondary plume bifurcation takes place at a Ra a little bit higher than 10^7 in 2-D. As Ra moves above 10^8, the tendency to plume branching increases and is accompanied by multiple foldings. In 3-D we have gone up to 5x10^7 in a 5x5x1 box and did not find any signs of plume bending. Then at Ra=10^8 plumes are found to be bent severely by the large scale circulation produced at this high Ra and have gone to Ra=10^9. We may expect some sort of layered convection to take place in 3-D configuration for Ra between 3x10^10 and 10^11. We have demonstrated here within the framework of a constant viscosity model that the secondary instabilities can develop in a self-consistent manner in both 2-D and 3-D large aspect-ratio convection with bifurcation Rayleigh numbers of O(10^7) and O(10^8) respectively.
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