Influence of dihedral angle on the seismic velocities in partially molten rocks
Introduction
Earth's deep interior is characterized by a number of low seismic velocity zones. As they pass through these regions, both S and P wave velocities decline greatly, with S waves slowing down more than P waves. A number of thin patches of such ultralow velocity zones (ULVZ) have been observed at the Earth's core–mantle boundary (Hutko et al., 2009, Rost & Revenaugh, 2003, Rost et al., 2005, Williams & Garnero, 1996). Typically, the magnitude of the velocity reductions in the ULVZ is ascribed to the melt volume fraction or the degree of melting. A body of recent theoretical and experimental works demonstrates that the fractional area of intergranular contact, contiguity, of a partially molten aggregate exerts the primary control on their effective elastic properties (Hier-Majumder, 2008, Takei, 1998, Takei, 2000, Takei, 2002). While the melt volume fraction or the degree of melting controls the contiguity of a partially molten aggregate, other controls on the contiguity are also capable of influencing the seismic velocities.
Besides contiguity, another important textural quantity in partially molten rocks is the dihedral angle at grain–melt interfaces. The dihedral angles in a given mineral matrix are sensitive to the chemical composition of the melt. For example, under upper mantle conditions, basaltic melts subtend a dihedral angle of approximately 34° (Cooper and Kohlstedt, 1982) while an aqueous fluid subtends an angle of 76° (Hier-Majumder and Kohlstedt, 2006), in an olivine-rich matrix. A recent compilation of laboratory experiments on the steady-state microstructures indicate that the contiguity in partially molten rocks display a systematic variation with dihedral angles (Yoshino et al., 2005). Such a variation in contiguity and dihedral angle also influences the seismic velocities of these melt and fluid bearing rocks. Evidence from direct measurement of seismic velocity in partially molten analogue materials with controlled dihedral angles, also supports this inference (Takei, 2000). In aggregates with low dihedral angles, both shear and P waves travel slower than through aggregates containing the same melt fraction but a higher dihedral angle (Takei, 2000). These experimental results indicate that besides the degree of melting, variation in melt composition (through dihedral angle) can also produce a distinct seismic signature.
Following a recent work (Hier-Majumder, 2008), this article explores the correlation between the dihedral angle and contiguity in a partially molten aggregate. The model incorporates dynamic interaction among a number of contiguous grains surrounding a melt pocket in two dimensions. Interfacial tension along grain–grain and grain–melt interfaces excites a viscous flow in the interior of the grains and the melt pocket until a steady-state microstructure is reached (Hopper, 1990, Hopper, 1993a, Hopper, 1993b, Kang, 2005, Kuiken, 1993).The viscous flow within the grains, part of a process named viscous sintering, is controlled by the conservation of mass and momentum within each grain and the melt pocket, supplemented by suitable boundary conditions. We employ a boundary integral formulation to solve the governing nonlinear equations in a hexagonal grain geometry. This numerical solution is also supplemented by an analytical solution for pressure, velocity, and steady-state grain shape in an aggregate with a four-fold packing symmetry. Contiguity and dihedral angles measured from the numerical experiments are compared with experimental measurements of contiguity by Yoshino et al. (2005). Finally, we calculate the elastic moduli and seismic velocities of the aggregates from the numerical models, and discuss the implications for Earth's deep interior. The results from our numerical models are also compared to the direct measurement of the influence of dihedral angle on seismic velocity by Takei (2000).
Section snippets
Methods
A detailed derivation of the formulation is presented in a previous article (Hier-Majumder, 2008). Only essential equations are outlined here. In this article, we present two sets of solutions. In one set of analysis, we present analytical solutions for the velocity, pressure, and steady-state shape of a grain immersed in the melt and acted upon by surface tension, representing a four-fold symmetry. In the second set, we present a two-dimensional numerical solution for the steady-state shape of
Numerical solution
Results from the numerical solutions display that the melt geometry varies monotonically with the prescribed value of surface tension γ in Eq. (12). Within the range of γ explored in this article, dihedral angles decrease steadily with an increase in γ from − 1 to 1. The plot in Fig. 1 displays an aggregate from the numerical solution, with the triangle indicating the unit cell. In the inset, two end members of the numerical results for γ = 1 and − 1 are displayed. At a lower dihedral angle,
Comparison with experimental results
Experimental measurements of contiguity from a variety of different solid–fluid aggregates is compiled by Yoshino et al. (2005). We compare their data with the results from our numerical experiments. Two important distinctions between the numerical and laboratory data, however, must be recognized. First, the laboratory experiments were performed over a range of temperatures, leading to a variation of the volume fraction of melt in the samples. All of our numerical experiments were carried out
Conclusions
We carried out a series of numerical experiments on a three grain aggregate at a constant melt volume fraction of 0.09 by varying the surface tension and generating melt geometries with different dihedral angles. Our results indicate that the contiguity of a partially molten aggregate generally increases with an increase in the dihedral angle. This result is in broad agreement with laboratory experiments on partially molten rocks. When effective seismic properties are calculated, our results
Acknowledgment
This research was supported by an NSF grant EAR 0809689. Insightful reviews by Lars Stixrude and two anonymous reviewers are gratefully acknowledged.
References (31)
Numerical evaluation of singular boundary integrals — theory and fortran code
J. Comput. Appl. Math.
(2006)- et al.
Role of grain boundaries in magma migration and storage
Earth Planet. Sci. Lett.
(2006) - et al.
Effect of pressure on Fe–Mg interdiffusion in (FexMg1 − x)O, ferropericlase
Phys. Earth Planet. Inter.
(2003) - et al.
Localized double-array stacking analysis of pcp: D ′ ′ and ulvz structure beneath the cocos plate, mexico, central pacific and north pacific
Phys. Earth Planet. Int.
(2009) - et al.
Viscosity of peridotite liquid up to 13 GPa: implications for magma ocean viscosities
Earth Planet. Sci. Lett.
(2005) Interfacial dynamics for stokes flow
J. Comput. Phys.
(2001)- et al.
A generalized formulation of interfacial tension driven fluid migration with dissolution/precipitation
Earth Planet. Sci. Lett.
(2009) - et al.
High Pressure Research in Geophysics, Advances in Earth and Planetary Sciences. Cent. Acad. Pub., Tokyo. Chapter Interfacial Energies in Olivine-Basalt System
(1982) - et al.
Low Reynolds Number Hydrodynamics
(1983) Influence of contiguity on seismic velocities of partially molten aggregates
J. Geophys. Res.
(2008)