Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-05T09:45:24.591Z Has data issue: false hasContentIssue false
  • English
  • Français

Consequences of viscous anisotropy in a deforming, two-phase aggregate. Part 2. Numerical solutions of the full equations

Published online by Cambridge University Press:  11 October 2013

Richard F. Katz  and
Yasuko Takei
Richard F. Katz*
Affiliation:
Department of Earth Sciences, University of Oxford, Oxford OX1 3AN, UK
Yasuko Takei
Affiliation:
Earthquake Research Institute, University of Tokyo, Tokyo 113-0032, Japan
*
†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; this, in turn, causes anisotropy of the matrix viscosity at the continuum scale. In the second of a two-paper set, we use numerical methods to solve the full, nonlinear, time-dependent equations governing this system. We consider porosity evolution in simple shear, Poiseuille and torsional flow. Under viscous anisotropy, there are two modes of porosity evolution: base-state segregation, which modifies the domain-scale porosity distribution, and growth of porosity perturbations into melt-rich bands. Simulation results with fixed anisotropy confirm and extend the linearized analysis of Part 1 (Takei & Katz, J. Fluid Mech., vol. 734, 2013, pp. 424–455). Most importantly, numerical solutions capture the interaction of the two modes: under Poiseuille flow, base-state segregation enhances band formation; under torsional flow, bands are suppressed. Simulations also show that low band angle is maintained by nonlinear processes such as reconnection of high-porosity segments and by back-rotation of the compacted regions between bands. Simulations with dynamic anisotropy modify these results, further lowering the average band angle. The effective viscosity of each flow is controlled by base-state segregation; it does not evolve under simple shear, decreases in Poiseuille flow and increases in torsion. We propose a reinterpretation of experimental results in terms of the consequences of viscous anisotropy.

JFM classification

Complex Fluids: Complex Fluids Geophysical and Geological Flows: Geophysical and Geological Flows Low-Reynolds-number flows: Stokesian dynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 734 , 10 November 2013 , pp. 456 - 485
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

Amestoy, P. R., Duff, I. S., L’Excellent, J. Y. & Koster, J. 2001 A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM J. Matrix Anal. Appl. 23 (1), 15–41.Google Scholar
Balay, S., Buschelman, K., Gropp, W. D., Kaushik, D., Knepley, M., McInnes, L. C., Smith, B. F. & Zhang, H. 2001 Portable, Extensible Toolkit for Scientific Computation, http://www.mcs.anl.gov/petsc.Google Scholar
Balay, S., Buschelman, K., Gropp, W. D., Kaushik, D., Knepley, M., McInnes, L. C., Smith, B. F. & Zhang, H. 2004 PETSc Users Manual. Tech. Rep. Argonne National Lab.Google Scholar
Butler, S. L. 2009 The effects of buoyancy on shear-induced melt bands in a compacting porous medium. Phys. Earth Planet. Inter. 173 (1–2), 51–59.Google 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
Cai, X.-C., Keyes, D. E. & Venkatakrishnan, V. 1997 Newton–Krylov–Schwarz: an implicit solver for CFD. In Proceedings of the Eighth International Conference on Domain Decomposition Methods, pp. 387–400.Google Scholar
Holtzman, B. K., Groebner, N. J., Zimmerman, M. E., Ginsberg, S. B. & Kohlstedt, D. L. 2003 Stress-driven melt segregation in partially molten rocks. Geochem. Geophys. Geosyst. 4, 1227–1230.Google Scholar
Holtzman, B. K., Kohlstedt, D. L. & Morgan, J. Phipps 2005 Viscous energy dissipation and strain partitioning in partially molten rocks. J. Petrol. 46 (12), 2569–2592.CrossRefGoogle Scholar
Katz, R. F., Knepley, M. G., Smith, B., Spiegelman, M. & Coon, E. T. 2007 Numerical simulation of geodynamic processes with the Portable Extensible Toolkit for Scientific Computation. Phys. Earth Planet. Inter. 163, 52–68.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
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.Google Scholar
King, D. S. H, Holtzman, B. K. & Kohlstedt, D. L. 2011 An experimental investigation of the interactions between reaction-driven and stress-driven melt segregation: 1. Application to mantle melt extraction. Geochem. Geophys. Geosyst. 12.Google 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 37, 561–593.Google Scholar
McKenzie, D. 1984 The generation and compaction of partially molten rock. J. Petrol. 25, 713–765.Google Scholar
Spiegelman, M. 2003 Linear analysis of melt band formation by simple shear. Geochem. Geophys. Geosyst. 4 (9), 8615.CrossRefGoogle 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. & 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
Takei, Y. & Katz, R. F 2013 Consequences of viscous anisotropy in a deforming, two-phase aggregate. Part 1. Governing equations and linearized analysis. J. Fluid Mech. 734, 424–455.Google Scholar
Trompert, R. A. & Hansen, U. 1996 The application of a finite volume multigrid method to three-dimensional flow problems in a highly viscous fluid with variable viscosity. Geophys. Astrophys. Fluid Dyn. 83, 261–291.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