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Effects of spatially varying roof cooling on thermal convection at high Rayleigh number in a fluid with a strongly temperature-dependent viscosity

Published online by Cambridge University Press:  15 June 2009

A. M. JELLINEK  and
A. LENARDIC
A. M. JELLINEK*
Affiliation:
Department of Earth and Ocean Sciences, The University of British Columbia, Vancouver, BC V6T1Z4, Canada
A. LENARDIC
Affiliation:
Department of Earth Science, Rice University, Houston, TX 77005, USA
*
†Email address for correspondence: mjellinek@eos.ubc.ca
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Abstract

We investigate the effects of an insulating lid of variable spatial extent on convection in the stagnant-lid regime under thermally steady-state conditions. Using a combination of laboratory experiments, numerical simulations and scaling analyses we characterize the qualitative structure and quantitative heat transfer properties of flows in terms of the fractional extent L of an insulating lid applied at the cold boundary, the thermal resistance of the lid, the magnitude of the temperature dependence of the fluid viscosity Λ and the effective Rayleigh number Rae for the composite system. A partial insulating lid has two main effects: (i) To increase the mean interior temperature and reduce the average viscosity of the system, which enhances fluid motions, and (ii) to impart a lateral asymmetry to the thermal structure of the cold boundary that leads, in turn, to lateral temperature gradients that drive an overturning flow. Consequently, whereas flow in the uninsulated stagnant-lid regime is in the form of ‘small-scale’ rising and sinking thermals, there is an additional ‘large-scale’ circulation in the presence of partial insulation. The structure, wavelength and heat transfer properties of this large-scale stirring depends on L, Λ and Rae. For given Rae – Λ conditions we find optimal values of L at which there occur well-defined maxima in the rate of overturn, the local heat flux carried into the uninsulated part of the cold boundary and in the global average heat flux Nu carried across the system. Whereas both the rate of overturning and local heat flux are associated with the largest lateral temperature gradients, the optimal basal heat flux depends also on a tradeoff with the fractional surface area of the lid. Remarkably, maximal values of the global heat flux can significantly exceed that of the uninsulated stagnant-lid case. The occurrence of such maxima is insensitive to the mechanical boundary conditions applied and is not strongly influenced by lid shape. However, the magnitude and location of optimal heat fluxes depends in a complicated way on the lid surface area and shape, as well as the structure of the hot and cold boundary layers and the wavelength of the large-scale flow.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 629 , 25 June 2009 , pp. 109 - 137
Copyright
Copyright © Cambridge University Press 2009

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References

REFERENCES

Aargard, K. & Carmack, E. C. 1989 The role of sea ice and other fresh water in Arctic circulation. J. Geophys. Res.-Climate 94, 14485–14498.CrossRefGoogle Scholar
Aargard, K. & Carmack, E. C. 1994 The Arctic ocean and climate: a perspective. In The Polar Oceans and Their Role in Shaping the Global Environment, Geophysical Monograph 85, American Geophysical Union.Google Scholar
Blackenbach, A., 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. 1989 A benchmark comparison for mantle convection codes. Geophys. J. Intl 98, 23–38.CrossRefGoogle Scholar
Booker, J. R. 1976 Thermal convection with strongly temperature-dependent viscosity. J. Fluid Mech. 76, 741–754.CrossRefGoogle Scholar
Christensen, U. R. 1984 Heat transport by variable viscosity convection. Phys. Earth Planet. Inter. 35, 264–282.CrossRefGoogle Scholar
Davaille, A. & Jaupart, C. 1993 Transient high Rayleigh number convection with large viscosity variations. J. Fluid Mech. 253, 141–166.CrossRefGoogle Scholar
Davis, E. E. & Chapman, D. S. 1989 Heat flow variations correlated with buried basement topography on the Juan de Fuca ridge flank. Nature 342, 533–37.CrossRefGoogle Scholar
Donaldson, I. G. 1982 Heat and mass circulation in geothermal systems. Annu. Rev. Earth Planet. Sci. 10, 377–395.CrossRefGoogle Scholar
Elder, J. W. 1981 Geothermal Systems. Academic Press.Google Scholar
Giannandrea, E. & Christensen, U. 1993 Variable viscosity convection experiments with a stress-free upper boundary and implications for the heat transport in the Earth's mantle. Phys. Earth. Planet. Inter. 78, 139–147.CrossRefGoogle Scholar
Gloersen, P., Campbell, W. J. Cavalieri, D. J., Comiso, J. C., Parkinson, C. L. & Zwally, H. J. 1992 Arctic and Antarctic sea ice, 1978–1987: satellite and passive microwave observations and analysis. NASA SP-511, National Air and Space Administration. NASA: Washington, DC, 290 pp.Google Scholar
Gonnermann, H. M., Jellinek, A. M., Richards, M. A. & Manga, M. 2004 Modulation of mantle plumes and heat flow at the core–mantle boundary by plate-scale flow: results from laboratory experiments. Earth Planet. Sci. Lett. 226, 53–67.CrossRefGoogle Scholar
Grigne, C. & Labrosse, S. 2001 Effects of continents on Earth cooling: thermal blanketing and depletion in radioactive elements Geophys. Res. Lett. 28, 2707–2710.CrossRefGoogle Scholar
Grigne, C., Labrosse, S. & Tackley, P. J. 2005 Convective heat transfer as a function of wavelength: implications for the cooling of the Earth. J. Geophys. Res. 110, B03409. doi: 10.1029/2004JB003376.Google Scholar
Guillou, L. & Jaupart, C. 1995 On the effect of continents on mantle convection. J. Geophys. Res. 100, 24217–24238.CrossRefGoogle Scholar
Gurnis, M. 1988 Large-scale mantle convection and the aggregation and dispersal of supercontinents. Nature 332, 695–699.CrossRefGoogle Scholar
Hartline, B. K. & Lister, C. R. B. Topographic forcing of supercritical convection in a porous medium such as the oceanic crust. Earth Planet. Sci. Lett. 55, 75–86.CrossRefGoogle Scholar
Howard, L. N. 1964 Convection at high Rayleigh number. In Proceedings of 11th International Congress of Applied Mechanics (ed. Gortler, H.). Springer.Google Scholar
Incropera, F. P. & Dewitt, D. P. 1996 Fundamentals of Heat and Mass Transfer. John Wiley & Sons.Google Scholar
Jellinek, A. M., Gonnermann, H. M. & Richards, M. A. 2003 Plume capture by divergent plate motions: implications for the distribution of hotspots, geochemistry of mid-ocean ridge basalts, and estimates of the heat flux at the core-mantle boundary. Earth Planet. Sci. Lett. 205, 361–378.CrossRefGoogle Scholar
Jellinek, A. M., Lenardic, A. & Manga, M. 2002 The influence of interior mantle temperature on the structure of plumes: Heads for Venus, Tails for the Earth. Geophys. Res. Lett. 29. doi: 10.1029/2001GL014624.CrossRefGoogle Scholar
Jellinek, A. M. & Manga, M. 2002 The influence of a chemical boundary layer on the fixity, spacing and lifetime of mantle plumes. Nature 418, 760–763.CrossRefGoogle ScholarPubMed
Jellinek, A. M. & Manga, M. 2004 Links between long-lived hot spots, mantle plumes, D” and plate tectonics. Rev. Geophys. 42, RG3002. doi: 10.1029/2003RG000144.CrossRefGoogle Scholar
King, S. D. & Ritsema, J. 2000 African hotspot volcanism: small-scale convection in the upper mantle beneath cratons. Science 290, 1137–1140.CrossRefGoogle Scholar
Korenaga, J. & Jordan, T. H. 2004 Physics of multiscale convection in Earth's mantle: Evolution of sublithospheric convection. J. Geophys. Res. 109, B01405. doi: 10.1029/2003JB002464.Google Scholar
Lathi, B. P. 1965 Linear Systems and Signals. Oxford University Press.Google Scholar
Lenardic, A. & Moresi, L.-N. 2003 Thermal convection below a conducting lid of variable extent: heat flow scalings and two-dimensional, infinite Prandtl number numerical simulations. Phys. Fluids 15, 455–466.CrossRefGoogle Scholar
Lenardic, A., Moresi, L.-N., Jellinek, A. M. & Manga, M. 2005 Continental insulation, mantle cooling, and the surface area of oceans and consitnents. Earth Planet. Sci. Lett. 234, 317–333.CrossRefGoogle Scholar
Lister, C. R. B. 1972 On the thermal balance of a mid-ocean ridge. Geophys. J. R. Astr. Soc. 26, 515–535.CrossRefGoogle Scholar
Lister, C. R. B. 1990 a An explanation for the multivalued heat-transport found experimentally for convection in a porous medium. J. Fluid Mech. 214, 287–320.CrossRefGoogle Scholar
Lister, C. R. B. 1990 b Thermal leakage from beneath sedimentary basins – an experimental test of the contribution of convective flow structures. Earth Planet. Sci. Lett. 99, 133–140.CrossRefGoogle Scholar
Lister, C. R. B. 1995 Heat transfer between magmas and hydrothermal systems, or 6 Lemmas in search of a theorem. Geophys. J. Intl 120, 45–59.Google Scholar
Lowell, R. P. & Burnell, D. K. 1991 Mathematical modeling of conductive heat transfer from a freezing, convecting magma chamber to a single pass hydrothermal system: implications for seafloor black smokers. Earth Planet. Sci. Lett. 104, 59–69.CrossRefGoogle Scholar
Manga, M. & Weeraratne, D. 1999 Experimental study of non-Boussinesq Rayleigh–Benard convection at high Rayleigh and Prandtl numbers. Phys. Fluids 11, 2969–2976.CrossRefGoogle Scholar
Manga, M., Weeraratne, D. & Morris, S. J. S. 2001 Boundary layer thickness and instabilities in Benard convection of a liquid with a temperature-dependent viscosity. Phys. Fluids 13, 802–805.CrossRefGoogle Scholar
Moresi, L.-N. & Solomatov, V. S. 1995 Numerical investigation of 2D convection with extremely large viscosity variations. Phys. Fluids 7, 2154–2162.CrossRefGoogle Scholar
Moresi, L.-N. & Solomatov, V. S. 1998 Mantle convection with a brittle lithosphere: thoughts on the global tectonic style of the Earth and Venus. Geophys. J. 133, 669–682.CrossRefGoogle Scholar
Morris, S. & Canright, D. 1984 A boundary-layer analysis of Bernard convection in a fluid of strongly temperature-dependent viscosity. Phys. Earth. Planet. Intl 36, 355–373.CrossRefGoogle Scholar
Niemela, J. J., Skrbek, L., Sreenivasan, K. R. & Donnelly, R. J. 2000 Turbulent convection at very high Rayleigh numbers. Nature 404, 837–840.CrossRefGoogle ScholarPubMed
Norton, D. 1984 A theory of hydrothermal systems. Annu. Rev. Earth Planet. Sci. 12, 155–177.CrossRefGoogle Scholar
Ogawa, M., Schubert, G. & Zebib, A. 1991 Numerical simulation of three-dimensional thermal convection in a fluid with strongly temperature-dependent viscosity. J. Fluid Mech. 233, 299–328.CrossRefGoogle Scholar
Park, J., Lindberg, C. R. & Vernon, F. L. III 1987 Multitaper spectral analysis of high-frequency seismograms. J. Geophys. Res. 92, 12675–12684.CrossRefGoogle Scholar
Parkinson, C. L. 1997 Earth From Above: Using Color-Coded Satellite Images to Examine the Global Environment. University Science Books.Google Scholar
Parkinson, C. L., Comiso, J. C., Zwally, H. Z., Cavalleri, D. J., Gloersen, P. & Campbell, W. J. 1987 Arctic sea ice, 1973–1976. NASA SP-489, National Air and Space Administration.Google Scholar
Person, M., Raffensperger, J. P., Ge, S. & Garven, G. 1996 Basin-scale hydrogeologic modeling. Revs. Geophys. 34, 61–88.CrossRefGoogle Scholar
Richter, F. M., Nataf, H. C. & Daly, S. F. 1983 Heat transfer and horizontally-averaged temperature of convection with large viscosity variations. J. Fluid Mech. 129, 173–192.CrossRefGoogle Scholar
Robin, C. M. I., Jellinek, A. M., Thayalan, V. & Lenardic, A. 2007 Transient mantle convection on Venus: the paradoxical coexistence of highlands and coronae in the BAT region. Earth Planet. Sci. Lett. 256, 100–119.CrossRefGoogle Scholar
Saar, M. O. & Manga, M. 2004 Depth dependence of permeability in the Oregon Cascades inferred from hydrogeologic, thermal, seismic, and magmatic modeling constraints. J. Geophys. Res. 109 (B4), B04204. doi: 10.1029/2003JB002855.Google Scholar
Schaeffer, N. & Manga, M. 2001 Interactions between rising and sinking mantle plumes. Geophys. Res. Lett. 28, 455–458.CrossRefGoogle Scholar
Sleep, N. H. & Jellinek, A. M. 2008 Scaling relationships for chemical-lid convection with applications to cratonal lithosphere. Geochem. Geophys. Geosyst. 9, Q12025. doi: 10.1029/2008GC002042.CrossRefGoogle Scholar
Solomatov, V. S. 1995 Scaling of temperature- and stress-dependent viscosity convection. Phys. Fluids 7, 266–274.CrossRefGoogle Scholar
Solomatov, V. S. & Moresi, L.-N. 1996 Stagnant lid convection on Venus. J. Geophys. Res. 101, 4737–4753.CrossRefGoogle Scholar
Solomatov, V. S. & Moresi, L.-N. 2000 Scaling of time-dependent stagnant lid convection: application to small-scale convection on Earth and other terrestrial planets. J. Geophys. Res. 105 21795–21817.CrossRefGoogle Scholar
Stengel, K. C., Oliver, D. S. & Booker, J. R. 1982 Onset of convection in a variable-viscosity fluid. J. Fluid Mech. 120, 411–431.CrossRefGoogle Scholar
Thayalan, V., Jellinek, A. M. & Lenardic, A. 2006 Recycling the Iid: Effects of subduction and stirring on boundary layer dynamics in bottom-heated planetary mantle convection. Geophys. Res. Lett. 33, L20318, doi:10.1029/2006GL027668.CrossRefGoogle Scholar
Thomson, D. J. 1982 Spectrum estimation and harmonic analysis. Proc. IEEE 70, 1055–1096.CrossRefGoogle Scholar
Trompert, R. A. & Hansen, U. 1998 On the Rayleigh number dependence of convection with a strongly temperature-dependent viscosity. Phys. Fluids 10, 351–360.CrossRefGoogle Scholar
Welch, P. D. 1967 The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms. IEEE Trans. Audio Electroacoust. AU-15, 70–73.CrossRefGoogle Scholar
Weeraratne, D. & Manga, M. 1998 Transitions in the style of mantle convection at high Rayleigh numbers, Earth Planet. Sci., Lett. 160, 563–568.CrossRefGoogle Scholar
Zhang, J., Childress, S. & Libchaber, A. 1997 Non-Boussinesq effect: thermal convection with broken symmetry. Phys. Fluids 9, 1034.CrossRefGoogle Scholar
Zhang, J. & Libchaber, A. 2000 Periodic boundary motion in thermal turbulence. Phys. Rev. Lett. 84, 4361–4364.CrossRefGoogle ScholarPubMed
Zhong, S. & Gurnis, M. 1993 Dynamic feedback between a continent-like raft and thermal convection. J. Geophys. Res. 98, 12219–12232.CrossRefGoogle Scholar