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Numerical studies of thermal convection with temperature- and pressure-dependent viscosity at extreme viscosity contrasts. (English) Zbl 1326.76094

Summary: Motivated by convection of planetary mantles, we consider a mathematical model for Rayleigh-Bénard convection in a basally heated layer of a fluid whose viscosity depends strongly on temperature and pressure, defined in an Arrhenius form. The model is solved numerically for extremely large viscosity variations across a unit aspect ratio cell, and steady solutions for temperature, isotherms, and streamlines are obtained. To improve the efficiency of numerical computation, we introduce a modified viscosity law with a low temperature cutoff. We demonstrate that this simplification results in markedly improved numerical convergence without compromising accuracy. Continued numerical experiments suggest that narrow cells are preferred at extreme viscosity contrasts, and this conclusion is supported by a linear stability analysis.{
©2015 American Institute of Physics}

MSC:

76R10 Free convection
76E20 Stability and instability of geophysical and astrophysical flows
85A30 Hydrodynamic and hydromagnetic problems in astronomy and astrophysics
76M25 Other numerical methods (fluid mechanics) (MSC2010)
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[1] Haskell, N. A., The viscosity of the asthenosphere, Am. J. Sci., 33, 22-28 (1937) · doi:10.2475/ajs.s5-33.193.22
[2] Cathles, L. M., The Viscosity of the Earth’s Mantle (1975)
[3] Paulson, A.; Zhong, S.; Wahr, J., Inference of mantle viscosity from grace and relative sea level data, Geophys. J. Int., 171, 2, 497-508 (2007) · doi:10.1111/j.1365-246X.2007.03556.x
[4] Turcotte, D. L.; Oxburgh, E. R., Finite amplitude convective cells and continental drift, J. Fluid Mech., 28, 29-42 (1967) · Zbl 0147.45902 · doi:10.1017/S0022112067001880
[5] Roberts, G. O., Fast viscous Bénard convection, Geophys. Astrophys. Fluid Dyn., 12, 235-272 (1979) · Zbl 0419.76066 · doi:10.1080/03091927908242692
[6] Jimenez, J.; Zufiria, J. A., A boundary-layer analysis of Rayleigh-Bénard convection at large Rayleigh number, J. Fluid Mech., 178, 53-71 (1987) · Zbl 0633.76092 · doi:10.1017/S0022112087001113
[7] Moore, D. R.; Weiss, N. O., Two-dimensional Rayleigh-Bénard convection, J. Fluid Mech., 58, 2, 289-312 (1973) · Zbl 0261.76028 · doi:10.1017/S0022112073002600
[8] Jarvis, G. T.; Peltier, W. R., Mantle convection as a boundary layer phenomenon, Geophys. J. R. Astron. Soc., 68, 389-427 (1982) · Zbl 0512.76053 · doi:10.1111/j.1365-246X.1982.tb04907.x
[9] Kirby, S. H., Rheology of the lithosphere, Rev. Geophys., 21, 6, 1458-1487 (1983) · doi:10.1029/RG021i006p01458
[10] Karato, S.; Wu, P., Rheology of the upper mantle-A synthesis, Science, 260, 5109, 771-778 (1993) · doi:10.1126/science.260.5109.771
[11] Armann, M.; Tackley, P. J., Simulating the thermochemical magmatic and tectonic evolution of venus’s mantle and lithosphere: Two-dimensional models, J. Geophys. Res.: Planets, 117, E12003 (2012) · doi:10.1029/2012JE004231
[12] Huang, J.; Yang, A.; Zhong, S., Constraints of the topography, gravity and volcanism on Venusian mantle dynamics and generation of plate tectonics, Earth Planet. Sci. Lett., 362, 207-214 (2013) · doi:10.1016/j.epsl.2012.11.051
[13] Schubert, G.; Turcotte, D. P.; Olson, P., Mantle Convection in the Earth and Planets (2001)
[14] Bercovici, D.; Ricard, Y.; Richards, M. A.; Richards, M.; Gordon, R.; van der Hilst, R., The relation between mantle dynamics and plate tectonics: A primer, The History and Dynamics of Global Plate Motions, 121, 5-46 (2000)
[15] Bercovici, D., A simple model of plate generation from mantle flow, Geophys. J. Int., 114, 3, 635-650 (1993) · doi:10.1111/j.1365-246X.1993.tb06993.x
[16] Bercovici, D., The generation of plate tectonics from mantle convection, Earth Planet. Sci. Lett., 205, 3, 107-121 (2003) · doi:10.1016/S0012-821X(02)01009-9
[17] Tackley, P. J., Self-consistent generation of tectonic plates in three-dimensional mantle convection, Earth Planet. Sci. Lett., 157, 1, 9-22 (1998) · doi:10.1016/S0012-821X(98)00029-6
[18] Tackley, P. J., Self-consistent generation of tectonic plates in time-dependent, three-dimensional mantle convection simulations. 1. Pseudoplastic yielding, Geochem., Geophys., Geosyst., 1, 8, 1021 (2000) · doi:10.1029/2000gc000036
[19] Tackley, P. J., Self-consistent generation of tectonic plates in time-dependent, three-dimensional mantle convection simulations 2. Strain weakening and asthenosphere, Geochem., Geophys., Geosyst., 1, 8, 1026 (2000) · doi:10.1029/2000gc000043
[20] Fowler, A. C., Boundary layer theory and subduction, J. Geophys. Res., 98, 21997-22005 (1993) · doi:10.1029/93JB02040
[21] Fowler, A. C.; O’Brien, S. B. G., Lithospheric failure on venus, Proc. R. Soc. A, 459, 2039, 2663-2704 (2003) · Zbl 1044.86504 · doi:10.1098/rspa.2002.1083
[22] Christensen, U. R., Heat transport by variable viscosity convection and implications for the earth’s thermal evolution, Phys. Earth Planet. Inter., 35, 264-282 (1984) · doi:10.1016/0031-9201(84)90021-9
[23] Christensen, U. R., Convection with pressure- and temperature-dependent non-Newtonian rheology, Geophys. J. R. Astron. Soc., 77, 343-384 (1984) · doi:10.1111/j.1365-246X.1984.tb01939.x
[24] Christensen, U. R.; Yuen, D. A., Layered convection induced by phase transitions, J. Geophys. Res., 90, B12, 10291-10300 (1985) · doi:10.1029/JB090iB12p10291
[25] Christensen, U.; Harder, H., 3-d convection with variable viscosity, Geophys. J. Int., 104, 1, 213-226 (1991) · doi:10.1111/j.1365-246X.1991.tb02505.x
[26] Moresi, L.-N.; Solomatov, V. S., Numerical investigation of 2D convection with extremely large viscosity variations, Phys. Fluids, 7, 9, 2154-2162 (1995) · Zbl 1126.76364 · doi:10.1063/1.868465
[27] Moresi, L.; Solomatov, V., Mantle convection with a brittle lithosphere: Thoughts on the global tectonic styles of the earth and venus, Geophys. J. Int., 133, 3, 669-682 (1998) · doi:10.1046/j.1365-246X.1998.00521.x
[28] Reese, C. C.; Solomatov, V. S.; Moresi, L.-N., Non-newtonian stagnant lid convection and magmatic resurfacing on venus, Icarus, 139, 1, 67-80 (1999) · doi:10.1006/icar.1999.6088
[29] Reese, C. C.; Solomatov, V. S., Mean field heat transfer scaling for non-newtonian stagnant lid convection, J. Non-Newtonian Fluid Mech., 107, 1, 39-49 (2002) · Zbl 1038.76043 · doi:10.1016/S0377-0257(02)00140-4
[30] Fleitout, L.; Yuen, D. A., Steady state, secondary convection beneath lithospheric plates with temperature-and pressure-dependent viscosity, J. Geophys. Res., 89, B11, 9227-9244 (1984) · doi:10.1029/JB089iB11p09227
[31] Doin, M. P.; Fleitout, L.; Christensen, U., Mantle convection and stability of depleted and undepleted continental lithosphere, J. Geophys. Res., 102, B2, 2771-2787 (1997) · doi:10.1029/96JB03271
[32] Stemmer, K.; Harder, H.; Hansen, U., A new method to simulate convection with strongly temperature-and pressure-dependent viscosity in a spherical shell: Applications to the earth’s mantle, Phys. Earth Planet. Inter., 157, 3, 223-249 (2006) · doi:10.1016/j.pepi.2006.04.007
[33] Fowler, A. C., Mathematical Geoscience (2011) · Zbl 1219.86001
[34] Blankenbach, B.; Busse, F.; Christensen, U.; Cserepes, L.; Gunkel, D.; Hansen, U.; Harder, H.; Jarvis, G.; Koch, M.; Marquart, G.; Moore, D.; Olson, P.; Schmeling, H.; Schnaubelt, T., A benchmark comparison for mantle convection codes, Geophys. J. Int., 98, 1, 23-38 (1989) · doi:10.1111/j.1365-246X.1989.tb05511.x
[35] Koglin, D. E. Jr.; Ghias, S. R.; King, S. D.; Jarvis, G. T.; Lowman, J. P., Mantle convection with reversing mobile plates: A benchmark study, Geochem., Geophys., Geosyst., 6, Q09003 (2005) · doi:10.1029/2005GC000924
[36] King, S. D., On topography and geoid from 2-D stagnant lid convection calculations, Geochem., Geophys., Geosyst., 10, Q03002 (2009) · doi:10.1029/2008GC002250
[37] Solomatov, V. S.; Moresi, L.-N., Stagnant lid convection on venus, J. Geophys. Res., 101, E2, 4737-4753 (1996) · doi:10.1029/95JE03361
[38] Huang, J.; Zhong, S.; van Hunen, J., Controls on sublithospheric small-scale convection, J. Geophys. Res.: Solid Earth, 108, B8, 2405 (2003) · doi:10.1029/2003JB002456
[39] Huang, J.; Zhong, S., Sublithospheric small-scale convection and its implications for the residual topography at old ocean basins and the plate model, J. Geophys. Res.: Solid Earth, 110, B5, B05404 (2005) · doi:10.1029/2004JD005101
[40] Stengel, K. C.; Oliver, D. S.; Booker, J. R., Onset of convection in a variable-viscosity fluid, J. Fluid Mech., 120, 411-431 (1982) · Zbl 0534.76093 · doi:10.1017/S0022112082002821
[41] Christensen, U., Effects of phase transitions on mantle convection, Annu. Rev. Earth Planet. Sci., 23, 65-87 (1995) · doi:10.1146/annurev.ea.23.050195.000433
[42] van Keken, P. E.; Ballentine, C. J., Whole-mantle versus layered mantle convection and the role of a high-viscosity lower mantle in terrestrial volatile evolution, Earth Planet. Sci. Lett., 156, 19-32 (1998) · doi:10.1016/S0012-821X(98)00023-5
[43] Ogawa, M., Mantle convection: A review, Fluid Dyn. Res., 40, 379-398 (2008) · Zbl 1184.86009 · doi:10.1016/j.fluiddyn.2007.09.001
[44] Tackley, P. J.; Bercovici, D., Mantle geochemical dynamics, Mantle Dynamics, 7, 437-505 (2009)
[45] Parsons, B.; McKenzie, D., Mantle convection and the thermal structure of the plates, J. Geophys. Res., 83, B9, 4485-4496 (1978) · doi:10.1029/JB083iB09p04485
[46] Fowler, A. C., Towards a description of convection with temperature-and pressure-dependent viscosity, Stud. Appl. Math., 88, 113-139 (1993) · Zbl 0768.76064
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