Abstract
A mathematical model is considered for Rayleigh–Bénard convection of mantle where the viscosity depends strongly on both temperature and pressure defined in an Arrhenius form. The model is solved numerically for extremely large viscosity variations across a unit aspect ratio cell using a modified cut-off viscosity law, and steady solutions are obtained. The aim is to investigate the convection pattern with internal heating at a very high viscosity variation in the presence of high Rayleigh number. The study also investigates the relation between temperature dependent parameter and pressure dependent parameter in a basally heated convection cell. The numerical simulation is performed using the finite element method based PDE solver and the results are presented through figures, tables and graphs.
Similar content being viewed by others
Availability of data and materials
The data is generated through numerical simulation. No available data repository has been used for this research.
References
Ammann, M., Brodholt, J., Wookey, J., Dobson, D.: First-principles constraints on diffusion in lower-mantle minerals and a weak ${\text{ d }}^{\prime \prime }$ layer. Nature 465(7297), 462–465 (2010)
Bercovici, D., Schubert, G., Glatzmaier, G.: Influence of heating mode on three-dimensional mantle convection. Geophys. Res. Lett. 16(7), 617–620 (1989)
Bercovici, D., Ricard, Y., Richards, M.A.: The relation between mantle dynamics and plate tectonics: a primer. Geophys. Monogr. Am. Geophys. Union 121, 5–46 (2000)
Blankenbach, B., Busse, F., Christensen, U., Cserepes, L., Gunkel, D., Hansen, U., Harder, H., Jarvis, G., Koch, M., Marquart, G., et al.: A benchmark comparison for mantle convection codes. Geophys. J. Int. 98(1), 23–38 (1989)
Bunge, H.P., Richards, M.A., Baumgardner, J.R.: Effect of depth-dependent viscosity on the planform of mantle convection. Nature 379(6564), 436–438 (1996)
Bunge, H.P., Richards, M.A., Baumgardner, J.R.: A sensitivity study of three-dimensional spherical mantle convection at 108 Rayleigh number: effects of depth-dependent viscosity, heating mode, and an endothermic phase change. J. Geophys. Res. Solid Earth 102(B6), 11991–12007 (1997)
Christensen, U.: Convection with pressure-and temperature-dependent non-Newtonian rheology. Geophys. J. Int. 77(2), 343–384 (1984a)
Christensen, U.: Heat transport by variable viscosity convection and implications for the earth’s thermal evolution. Phys. Earth Planet. Inter. 35(4), 264–282 (1984b)
Cserepes, L.: Effect of depth-dependent viscosity on the pattern of mantle convection. Geophys. Res. Lett. 20(19), 2091–2094 (1993)
Deschamps, F., Lin, J.R.: Stagnant lid convection in 3D-Cartesian geometry: scaling laws and applications to icy moons and dwarf planets. Phys. Earth Planet. Inter. 229, 40–54 (2014)
Doin, M.P., Fleitout, L., Christensen, U.: Mantle convection and stability of depleted and undepleted continental lithosphere. J. Geophys. Res. Solid Earth 102(B2), 2771–2787 (1997)
Dumoulin, C., Doin, M.P., Fleitout, L.: Heat transport in stagnant lid convection with temperature-and pressure-dependent Newtonian or non-Newtonian rheology. J. Geophys. Res. Solid Earth 104(B6), 12759–12777 (1999)
Fleitout, L., Yuen, D.A.: Steady state, secondary convection beneath lithospheric plates with temperature-and pressure-dependent viscosity. J. Geophys. Res. Solid Earth 89(B11), 9227–9244 (1984)
Fowler, A.: Fast thermoviscous convection. Stud. Appl. Math. 72(3), 189–219 (1985)
Fowler, A.C.: Mathematical Geoscience, vol. 36. Springer, Berlin (2011)
Fowler, A., Howell, P., Khaleque, T.S.: Convection of a fluid with strongly temperature and pressure dependent viscosity. Geophys. Astrophys. Fluid Dyn. 110(2), 130–165 (2016)
Gurnis, M., Davies, G.F.: The effect of depth-dependent viscosity on convective mixing in the mantle and the possible survival of primitive mantle. Geophys. Res. Lett. 13(6), 541–544 (1986)
Haskell, N.A.: The viscosity of the asthenosphere. Am. J. Sci. 5(193), 22–28 (1937)
Houston, M., De Bremaecker, J.C.: Numerical models of convection in the upper mantle. J. Geophys. Res. 80(5), 742–751 (1975)
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), B05404-1 (2005)
Huang, J., Zhong, S., van Hunen, J.: Controls on sublithospheric small-scale convection. J. Geophys. Res. Solid Earth 108(B8), 2405 (2003)
Ito, E., Katsura, T.: A temperature profile of the mantle transition zone. Geophys. Res. Lett. 16(5), 425–428 (1989)
Jarvis, G.T., Peltier, W.: Mantle convection as a boundary layer phenomenon. Geophys. J. R. Astron. Soc. 68(2), 389–427 (1982)
Jimenez, J., Zufiria, J.A.: A boundary-layer analysis of Rayleigh–Bénard convection at large Rayleigh number. J. Fluid Mech. 178, 53–71 (1987)
Kameyama, M., Ogawa, M.: Transitions in thermal convection with strongly temperature-dependent viscosity in a wide box. Earth Planet. Sci. Lett. 180(3–4), 355–367 (2000)
Karato Si, W.P.: Rheology of the upper mantle: a synthesis. Science 260(5109), 771–778 (1993)
Khaleque, T.S., Fowler, A.C., Howell, P., Vynnycky, M.: Numerical studies of thermal convection with temperature-and pressure-dependent viscosity at extreme viscosity contrasts. Phys. Fluids 27(7), 076603 (2015)
King, S.D.: On topography and geoid from 2-D stagnant lid convection calculations. Geochem. Geophys. Geosyst. 10(3), Q03002 (2009)
King, S.D.: Mantle convection, the asthenosphere, and earth’s thermal history. Geol. Soc. Am. Spec. Pap. 514, SPE514-07 (2015)
Kirby, S.H.: Rheology of the lithosphere. Rev. Geophys. 21(6), 1458–1487 (1983)
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(9), Q09003 (2005)
Korenaga, J.: Pitfalls in modeling mantle convection with internal heat production. J. Geophys. Res. Solid Earth 122(5), 4064–4085 (2017)
Larsen, T.B., Malevsky, A.V., Yuen, D.A., Smedsmo, J.L.: Temperature-dependent Newtonian and non-Newtonian convection: implications for lithospheric processes. Geophys. Res. Lett. 20(23), 2595–2598 (1993)
Leitch, A., Yuen, D., Sewell, G.: Mantle convection with internal heating and pressure-dependent thermal expansivity. Earth Planet. Sci. Lett. 102(2), 213–232 (1991)
Limare, A., Fourel, L., Surducan, E., Neamtu, C., Surducan, V., Vilella, K., Farnetani, C., Kaminski, E., Jaupart, C.: Microwave-based, internally-heated convection: new perspectives for the heterogeneous case. In: AIP Conference Proceedings, vol. 1700, p. 040001. AIP Publishing LLC (2015)
McNamara, A.K., Zhong, S.: Degree-one mantle convection: dependence on internal heating and temperature-dependent rheology. Geophys. Res. Lett. 32(1), L01301-1 (2005)
Moresi, L.N., Solomatov, V.: Numerical investigation of 2D convection with extremely large viscosity variations. Phys. Fluids 7(9), 2154–2162 (1995)
Morris, S., Canright, D.: A boundary-layer analysis of Benard convection in a fluid of strongly temperature-dependent viscosity. Phys. Earth Planet. Inter. 36(3–4), 355–373 (1984)
Olson, P., Corcos, G.: A boundary layer model for mantle convection with surface plates. Geophys. J. R. Astron. Soc. 62(1), 195–219 (1980)
Parmentier, E., Turcotte, D., Torrance, K.: Studies of finite amplitude non-Newtonian thermal convection with application to convection in the earth’s mantle. J. Geophys. Res. 81(11), 1839–1846 (1976)
Reese, C., Solomatov, V., Baumgardner, J.: Scaling laws for time-dependent stagnant lid convection in a spherical shell. Phys. Earth Planet. Inter. 149(3–4), 361–370 (2005)
Roberts, G.: Fast viscous convection. Geophys. Astrophys. Fluid Dyn. 8(1), 197–233 (1977)
Schubert, G., Turcotte, D.L., Olson, P.: Mantle Convection in the Earth and Planets. Cambridge University Press, Cambridge (2001)
Shahraki, M., Schmeling, H.: Plume-induced geoid anomalies from 2D axi-symmetric temperature-and pressure-dependent mantle convection models. J. Geodyn. 59, 193–206 (2012)
Solomatov, V., Moresi, L.N.: Three regimes of mantle convection with non-Newtonian viscosity and stagnant lid convection on the terrestrial planets. Geophys. Res. Lett. 24(15), 1907–1910 (1997)
Solomatov, V.S., Moresi, L.N.: Scaling of time-dependent stagnant lid convection: application to small-scale convection on earth and other terrestrial planets. J. Geophys. Res. Solid Earth 105(B9), 21795–21817 (2000)
Stein, C., Lowman, J., Hansen, U.: The influence of mantle internal heating on lithospheric mobility: implications for super-earths. Earth Planet. Sci. Lett. 361, 448–459 (2013)
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(3–4), 223–249 (2006)
Tackley, P.J., Ammann, M., Brodholt, J.P., Dobson, D.P., Valencia, D.: Mantle dynamics in super-earths: post-perovskite rheology and self-regulation of viscosity. Icarus 225(1), 50–61 (2013)
Travis, B., Olson, P.: Convection with internal heat sources and thermal turbulence in the earth’s mantle. Geophys. J. Int. 118(1), 1–19 (1994)
Turcotte, D., Schubert, G.: Application of Continuum Physics to Geological Problems. Wiley, New York (1982)
Turcotte, D.L., Schubert, G.: Geodynamics. Cambridge University Press, Cambridge (2002)
Turcotte, D., Oxburgh, E.: Finite amplitude convective cells and continental drift. J. Fluid Mech. 28(1), 29–42 (1967)
Van Heck, H., Tackley, P.: Plate tectonics on super-earths: equally or more likely than on earth. Earth Planet. Sci. Lett. 310(3–4), 252–261 (2011)
Yamazaki, D., Si, K.: Some mineral physics constraints on the rheology and geothermal structure of earth’s lower mantle. Am. Miner. 86(4), 385–391 (2001)
Acknowledgements
Tania S. Khaleque acknowledges the valuable suggestions from Professor A. C. Fowler, University of Oxford, U.K.
Funding
The author Tania S. Khaleque received a partial funding from the University Grants Commission-Dhaka University.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflicts of interest and they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Code availability
The code and data generated by COMSOL Multiphysics can be provided if necessary.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Khaleque, T.S., Sayeed Motaleb, S.A. Effects of temperature- and pressure-dependent viscosity and internal heating on mantle convection. Int J Geomath 12, 23 (2021). https://doi.org/10.1007/s13137-021-00190-2
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13137-021-00190-2
Keywords
- Rayleigh–Bénard convection
- Mantle
- Variable viscosity
- Pressure dependence
- Internal heating
- Viscosity contrast