The effects of high Rayleigh number convection for temperature-dependent viscosity at infinite Prandtl number are studied with three-dimensional direct numerical simulations (using a fully spectral method) in a wide box with dimensions 5 × 5 × 1. The form of the temperature-dependent viscosity decreases exponentially with the temperature. Solutions for an volumetrically averaged Rayleigh numbers up to 6.25 × 106 have been obtained for a viscosity contrast of 25. Both models with and without viscous dissipation and adiabatic heatings have been considered. There is a distinct difference in the plume dynamics between models with a dissipation number D = 0 and D = 0.3. At zero dissipation number the hot plumes extend up to the top, while plumes at D = 0.3 are cooled due to adiabatic expansion and do not extend throughout the entire layer. The cold descending flows occur in sheets and they form longer wavelength networks and can reach the bottom, regardless of the value of D. A time-varying depth-dependent mean horizontal flow is produced from the correlation between the laterally varying viscosity field and the velocity gradients. At high Rayleigh number there is also a change in the surface toroidal velocity field to a coherent network of river-like structures and compact vortices. Viscous dissipation is found to increase with the Rayleigh number and is particularly strong in regions of downwelling flows. In the context of mantle convection, these localized heat sources are observed to be strong with local magnitudes in some places exceeding what Earth-like radioactive heating would be by more than an order of magnitude.
Bibliographical noteFunding Information:
This research has been supported by Geochemistry program and Di- vision of Mathematical Sciences of National Science Foundtaton (NSF DMS 93-01042), National Aeronaturics and space Administration. We thank the anoymous referees for their valuable comments.
- Extended Boussinesq approximation
- Mantle convection
- Temperature-dependent viscosity
- Three-dimensional numerical simulation
- Toroidal motion
- Viscous dissipation