Hostname: page-component-745bb68f8f-b95js Total loading time: 0 Render date: 2025-01-08T10:54:48.609Z Has data issue: false hasContentIssue false
  • English
  • Français

Consequences of viscous anisotropy in a deforming, two-phase aggregate. Part 1. Governing equations and linearized analysis

Published online by Cambridge University Press:  11 October 2013

Yasuko Takei  and
Richard F. Katz
Yasuko Takei*
Affiliation:
Earthquake Research Institute, University of Tokyo, Tokyo 113-0032, Japan
Richard F. Katz
Affiliation:
Department of Earth Sciences, University of Oxford, Oxford OX1 3AN, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In partially molten regions of Earth, rock and magma coexist as a two-phase aggregate in which the solid grains of rock form a viscously deformable framework or matrix. Liquid magma resides within the permeable network of pores between grains. Deviatoric stress causes the distribution of contact area between solid grains to become anisotropic; in turn, this causes anisotropy of the matrix viscosity at the continuum scale. In this two-paper set, we predict the consequences of viscous anisotropy for flow of two-phase aggregates in three configurations: simple shear, Poiseuille, and torsional flow. Part 1 presents the governing equations and an analysis of their linearized form. Part 2 (Katz & Takei, J. Fluid Mech., vol. 734, 2013, pp. 456–485) presents numerical solutions of the full, nonlinear model. In our theory, the anisotropic viscosity tensor couples shear and volumetric components of the matrix stress/strain rate. This coupling, acting over a gradient in shear stress, causes segregation of liquid and solid. Liquid typically migrates toward higher shear stress, but under specific conditions, the opposite can occur. Furthermore, it is known that in a two-phase aggregate with a porosity-weakening viscosity, matrix shear causes porosity perturbations to grow into a banded or sheeted structure. We show that viscous anisotropy reduces the angle between these emergent high-porosity features and the shear plane. Laboratory experiments produce similar, high-porosity features. We hypothesize that the low angle of porosity bands in such experiments is the result of viscous anisotropy. We therefore predict that experiments incorporating a gradient in shear stress will develop sample-wide liquid–solid segregation due to viscous anisotropy.

JFM classification

Geophysical and Geological Flows: Magma and lava flow Nonlinear Dynamical Systems: Pattern formation Low-Reynolds-number flows: Porous media
Type
Papers
Information
Journal of Fluid Mechanics , Volume 734 , 10 November 2013 , pp. 424 - 455
Copyright
©2013 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bulau, J. R., Waff, H. S. & Tyburczy, J. A. 1979 Mechanical and thermodynamic constraints on fluid distribution in partial melts. J. Geophys. Res. 84, 6102–6108.CrossRefGoogle Scholar
Butler, S. L. 2012 Numerical models of shear-induced melt band formation with anisotropic matrix viscosity. Phys. Earth Planet. Inter. 200–201, 28–36.Google Scholar
Christensen, U. R. 1987 Some geodynamical effects of anisotropic viscosity. Geophys. J. R. Astron. Soc. 91 (3), 711–736.CrossRefGoogle Scholar
Cooper, R. F., Kohlstedt, D. L. & Chyung, K. 1989 Solution-precipitation enhanced creep in solid liquid aggregates which display a non-zero dihedral angle. Acta Metall. 37, 1759–1771.CrossRefGoogle Scholar
Currie, I. G. 1967 The effect of heating rate on the stability of stationary fluids. J. Fluid Mech. 29, 337–347.CrossRefGoogle Scholar
Daines, M. J. & Kohlstedt, D. L. 1997 Influence of deformation on melt topology in peridotites. J. Geophys. Res. 102, 10257–10271.CrossRefGoogle Scholar
Doumenc, F., Boeck, T., Guerrier, B. & Rossi, M. 2010 Transient Rayleigh–Benard–Marangoni convection due to evaporation: a linear non-normal stability analysis. J. Fluid Mech. 648, 521–539.Google Scholar
Drew, D. A. 1983 Mathematical modelling of two-phase flow. Annu. Rev. Fluid Mech. 15, 261–291.Google Scholar
Faul, U. H. 1997 Permeability of partially molten upper mantle rocks from experiments and percolation theory. J. Geophys. Res. 102, 10299–10311.CrossRefGoogle Scholar
Hansen, L. N., Zimmerman, M. E. & Kohlstedt, D. L. 2012 Laboratory measurements of the viscous anisotropy of olivine aggregates. Nature 492 (7429), 415–418.Google Scholar
Hewitt, I. J. & Fowler, A. C. 2008 Partial melting in an upwelling mantle column. Proc. R. Soc. A 464 (2097), 2467–2491.Google Scholar
Hirth, G. & Kohlstedt, D. 2003 Rheology of the upper mantle and the mantle wedge: a view from the experimentalists. In Inside the Subduction Factory, AGU Geophysical Monograph, vol. 138, pp. 83–105.Google Scholar
Holtzman, B. K., Groebner, N. J., Zimmerman, M. E., Ginsberg, S. B. & Kohlstedt, D. L. 2003a Stress-driven melt segregation in partially molten rocks. Geochem. Geophys. Geosyst. 4 (5), 8607.Google Scholar
Holtzman, B. K. & Kohlstedt, D. L. 2007 Stress-driven melt segregation and strain partitioning in partially molten rocks: effects of stress and strain. J. Petrol. 48, 2379–2406.CrossRefGoogle Scholar
Holtzman, B. K., Kohlstedt, D. L., Zimmerman, M. E., Heidelbach, F., Hiraga, T. & Hustoft, J. 2003b Melt segregation and strain partitioning: implications for seismic anisotropy and mantle flow. Science 301, 1227–1230.Google Scholar
Honda, S. 1986 Strong anisotropic flow in a finely layered asthenosphere. Geophys. Res. Lett. 13 (13), 1454–1457.Google Scholar
Katz, R. F., Spiegelman, M. & Holtzman, B. 2006 The dynamics of melt and shear localization in partially molten aggregates. Nature 442, 676–679.CrossRefGoogle ScholarPubMed
Katz, R. F. & Takei, Y. 2013 Consequences of viscous anisotropy in a deforming, two-phase aggregate. Part 2. Numerical solutions of the full equations. J. Fluid Mech 734, 456–485.CrossRefGoogle Scholar
Katz, R. F. & Weatherley, S. 2012 Consequences of mantle heterogeneity for melt extraction at mid-ocean ridges. Earth Planet. Sci. Lett. 335–336, 226–237.Google Scholar
Kelemen, P. B., Hirth, G., Shimizu, N., Spiegelman, M. & Dick, H. J. B. 1997 A review of melt migration processes in the adiabatically upwelling mantle beneath oceanic spreading ridges. Phil. Trans. R. Soc. Lond. A 355 (1723), 283–318.Google Scholar
King, D. S. H., Zimmerman, M. E. & Kohlstedt, D. L. 2010 Stress-driven melt segregation in partially molten olivine-rich rocks deformed in torsion. J. Petrol. 51, 21–42.CrossRefGoogle Scholar
Kohlstedt, D. L. & Holtzman, B. K. 2009 Shearing melt out of the Earth: an experimentalist’s perspective on the influence of deformation on melt extraction. Annu. Rev. Earth Planet. Sci. 37, 561–593.Google Scholar
Lee, V. W., Mackwell, S. J. & Brantley, S. L. 1991 The effect of fluid chemistry on wetting textures in novaculite. J. Geophys. Res. 96 (B6), 10023–10037.CrossRefGoogle Scholar
Lev, E. & Hager, B. H. 2008 Rayleigh–Taylor instabilities with anisotropic lithospheric viscosity. Geophys. J. Intl. 173 (3), 806–814.CrossRefGoogle Scholar
Lev, E. & Hager, B. H. 2011 Anisotropic viscosity changes subduction zone thermal structure. Geochem. Geophys. Geosyst. 12 (4), Q04009.Google Scholar
Lick, W. 1965 The instability of a fluid layer with time-dependent heating. J. Fluid Mech. 21, 565–576.Google Scholar
Malvern, L. E. 1969 Introduction to the Mechanics of a Continuous Medium. Prentice-Hall Inc.Google Scholar
McKenzie, D. 1984 The generation and compaction of partially molten rock. J. Petrol. 25, 713–765.Google Scholar
Ribe, N. M. 1985 The generation and composition of partial melts in the Earth’s mantle. Earth Planet. Sci. Lett. 73, 361–376.Google Scholar
Saito, M. & Abe, Y. 1984 Consequences of anisotropic viscosity in the Earth’s mantle. Zisin 37, 237–245.CrossRefGoogle Scholar
Simpson, G., Spiegelman, M. & Weinstein, M. I. 2010a A multiscale model of partial melts: 1. Effective equations. J. Geophys. Res. 115, B04410.Google Scholar
Simpson, G., Spiegelman, M. & Weinstein, M. I. 2010b A multiscale model of partial melts: 2. Numerical results. J. Geophys. Res. 115, B04411.Google Scholar
Spiegelman, M. 1993a Flow in deformable porous media. Part 1. Simple analysis. J. Fluid Mech. 247, 17–38.Google Scholar
Spiegelman, M. 1993b Flow in deformable porous media. Part 2. Numerical analysis—The relationship between shock waves and solitary waves. J. Fluid Mech. 247, 39–63.Google Scholar
Spiegelman, M. 2003 Linear analysis of melt band formation by simple shear. Geochem. Geophys. Geosys. 4 (9), 8615.Google Scholar
Šrámek, O., Ricard, Y. & Bercovici, D. 2007 Simultaneous melting and compaction in deformable two-phase media. Geophys. J. Intl 168 (3), 964–982.Google Scholar
Stevenson, D. J. 1989 Spontaneous small-scale melt segregation in partial melts undergoing deformation. Geophys. Res. Lett. 16, 1067–1070.Google Scholar
Takei, Y. 2010 Stress-induced anisotropy of partially molten rock analogue deformed under quasi-static loading test. J. Geophys. Res. 115, B03204.Google Scholar
Takei, Y. 2013 Elasticity, anelasticity, and viscosity of a partially molten rock. In Physics and Chemistry of the Deep Earth (ed. Karato, S.), pp. 66–93. John Wiley & Sons, Ltd.CrossRefGoogle Scholar
Takei, Y. & Holtzman, B. K. 2009a Viscous constitutive relations of solid–liquid composites in terms of grain boundary contiguity: 1. Grain boundary diffusion control model. J. Geophys. Res. 114, B06205.Google Scholar
Takei, Y. & Holtzman, B. K. 2009b Viscous constitutive relations of solid–liquid composites in terms of grain boundary contiguity: 2. Compositional model for small melt fractions. J. Geophys. Res. 114, B06206.Google Scholar
Takei, Y. & Holtzman, B. K. 2009c Viscous constitutive relations of solid–liquid composites in terms of grain boundary contiguity: 3. Causes and consequences of viscous anisotropy. J. Geophys. Res. 114, B06207.Google Scholar
von Bargen, N. & Waff, H. S. 1986 Permeabilities, interfacial areas and curvatures of partially molten systems: results of numerical computation of equilibrium microstructures. J. Geophys. Res. 91, 9261–9276.CrossRefGoogle Scholar
Waff, H. S. & Bulau, J. R. 1979 Equilibrium fluid distribution in an ultramafic partial melt under hydrostatic stress conditions. J. Geophys. Res. 84, 6109–6114.Google Scholar
Wark, D. A. & Watson, E. B. 1998 Grain-scale permeabilities of texturally equilibrated, monomineralic rocks. Earth Planet. Sci. Lett. 164, 591–605.Google Scholar
Wark, D. A., Williams, C. A., Watson, E. B. & Price, J. D. 2003 Reassessment of pore shapes in microstructurally equilibrated rocks, with implications for permeability of the upper mantle. J. Geophys. Res. 108, B1, 2050.Google Scholar
Zimmerman, M. E., Zhang, S. Q., Kohlstedt, D. L. & Karato, S. 1999 Melt distribution in mantle rocks deformed in shear. Geophys. Res. Lett. 26 (10), 1505–1508.Google Scholar