Mixing in the transition to strong turbulence in infinite Prandtl number thermal convection as applied to the Earth’s mantle
Authors: Yuen David A. , Langenberger Sherri E. , Hansen Ulrich 
Published: July 2002

Recently there has developed a great deal of interest in the transition to hard turbulence in thermal convection. This transition and the corresponding hard turbulence regime has been the subject of laboratory and numerical experiments. Our study of convection at high Rayleigh numbers (Ra) has been motivated by the phenomenon of subsolidus mantle convection, whose Ra may range between 5×105 and O(108), depending on the still uncertain estimates of the lower-mantle viscosity. We have conducted a series of calculations at Ra spanning between 5×105 and 108 because we are interested in the transition from weak to strong turbulence in mantle convection and the effects this transition would have on mixing. We have employed a two-dimensional finite-element method, combined with an implicit, predictor–corrector, time-stepping scheme to advance the evolutionary temperature equation. The biharmonic equation for the streamfunction is solved exactly by a variational equation at each time step. We present direct numerical simulations of two-dimensional, high Ra thermal convection for constant property fluids with both base-heated and partially internally heated configurations in large-aspect-ratio boxes (up to 10).
We will emphasize the importance of visualization of these strongly time-dependent results, as they present a formidable challenge in the management and analysis of the data. First, an adequate resolution of the boundary layer and mixing layer requires high resolution. In our large (10) aspect-ratio box runs we have employed around 27 000 unevenly spaced elements, resulting in about 105 unknowns per time step. Each run goes up to between 105 and 106 time steps in order to get many overturns of the dominant cells. We will present videos displaying the evolution of the physical fields, in particular the temperature and vorticity fields, which give a vivid portrayal of the mixing dynamics. Visualization is a handy medium for illustrating and also for discovering the richness of the mixing process, its multiple spatial and time scales in the transition from weak to strong turbulence. At Ra around 106, we have found that convection does not take place in a strictly cellular manner. Thermals emanate from the hot and cold boundary layers, which are superimposed on a large-scale type of circulation. These boundary layer instabilities enhance mixing of the interior.
The fate of these instabilities is determined ultimately by the large-scale flow. With greater amounts of internal heating, the large-scale flow becomes smaller and mixing between distant regions is inhibited, but that between neighboring cells is enhanced by the changes of the flow pattern induced by internal heating. We have found a ‘‘mixing layer’’ above the thermal boundary layer in the hard-turbulent regime. At high Ra, between 107 and 108, we find breakdown of globally connected thermal plumes for base-heated convection. In this hard-turbulent regime the plumes become disconnected, ‘‘podlike’’ structures, thus inhibiting efficient vertical mixing. This disconnection of plumes in base-heated convection is to be regarded as a manifestation of the soft to hard turbulence transition. In the presence of internal heating and high Ra the descending cold boundary layers become dominant and serve to promote the interaction between the top and bottom, while the ascending plumes diminish in strength and disappear altogether. Mixing in the mantle thus influenced by the vigor in convection, the amount of internal heating, and the aspect ratio of the global configuration.