multi phase flows and computational aspects of plate tectonics

MULTI PHASE FLOWS AND COMPUTATIONAL ASPECTS OF PLATE TECTONICS Louis Moresi and Hans-Bernd MŸhlhaus Australian Geodynamics Cooperative Research Centre CSIRO division of Exploration and Mining PO Box 437, Nedlands WA 6009, Australia Abstract A Lagrangian Particle Finite Element scheme is presented which is suited to problems in which material composition and history must be tracked through very large deformations associated with creeping fluid flow. The method is applied to large-scale geodynamic modeling in which some parts of the system are actively convecting which others remain nearly stagnant. Introduction Geology records the slow movements of the continents with respect to one another; the theory of plate tectonics describes the manner in which this happens. Plate tectonics is regarded as the surface manifestation of solid-state convection in underlying rocky mantle which is responsible for releasing the EarthÕs inner heat. In the past it has proved to be very difficult to reproduce plate-like behaviour selfconsistently at the surface of a convection experiment. The intimate connection between plate boundaries and earthquakes suggests that this is because such experiments lack a description of the brittle nature of the cool lithosphere. History-dependent, visco-elastic-plastic rheologies have been used in modelling crustal deformation for some time, and there has been considerable work recently to incorporate some of these ideas into mantle convection / plate tectonics models. Two serious difficulties arise, one is that it is necessary to track history parameters despite the existence of enormous strains within the convecting fluid. The second is the difference in scale between the brittle plate boundary zones (km's) and the plates themselves and their associated convection cells (thousands of km's). Recent modeling has mainly ignored the first of these difficulties, concentrating on the influence of highly non-linear rheological laws on the surface motions of convecting systems [e.g. References 1,2,3]

Buoyant Crust

Stiff Lithosphere

Convecting mantle

Figure 1: Lagrangian particle FEM simulation of 2D mantle convection with history dependent rheology and tracking of materials of differing composition

Figure 1 is a numerical simulation which illustrates the components of the system which must be modeled accurately if the surface motions and stresses are to be predicted accurately. Temperature dependent viscosity results in a very low strain rate within the cold boundary layer (lithosphere) relative to the strain rate in the convecting interior. Chemically buoyant continental crust stabilizes some parts of the cold

lithosphere against thermally driven recycling. Brittle effects near plate boundaries are considered to be responsible for allowing plate-like surface velocities to develop by they localizing deformation at narrow shear bands. The location of plate boundaries is stabilized by the history dependence of the brittle rheology (strain weakening, for example) and by the presence of the chemical heterogeneity in the lithosphere. Mathematical Model We solve the equations of incompressible StokesÕ flow and thermal convection which are relevant to the EarthÕs mantle. In our model, the viscosity, η , is given by η(T ) = η0 exp( E T ) (1) where T is the absolute temperature, η0 was chosen to be 1020 Pa s, and the activation energy, E, was taken to be 3600 kJ mol-1. This relationship gives a viscosity of 1021 PaÊs at an average mantle temperature of 1300°C. At a surface temperature of 0°C, the viscosity is 5x1025 PaÊs. This temperature dependence is smaller than predicted by small-scale laboratory measurements but is sufficiently large to capture the stiffness of the upper boundary layer without introducing any difficulties for the numerical solver. The yield criterion is τ y = c(ε p , ε˙ p ) + µ (ε p , ε˙ p ) P′ (2) where τ y is the maximum value of the second invariant of the stress tensor, c is the equivalent value at zero pressure (a cohesion term), and µ is a Ôfriction coefficientÕ. The pressure, PÕ, is assumed to be hydrostatic for the sake of numerical stability at low resolution. Both c and µ can be functions of the accumulated plastic strain, ε p , and the strain rate. The accumulated plastic strain for a particle is obtained by evaluating t

ε p (t ) = ∫ ε˙ p dt ′

(3)

0

along a particle path, where ε˙ p is the second invariant of the strain rate tensor obtained during plastic yielding Ñ it is assumed to be zero during pure viscous deformation below the yield stress. What is the best choice for the functional form of c(ε p , ε˙ p ) and µ (ε p , ε˙ p ) ? It is, of course, possible to obtain suitable relationships for individual samples at moderate strains, but it is not clear whether these measurements can be extrapolate to the scale of the entire lithosphere, to great faults which have accumulated slip for millions of years, and to ductile shear zones which may exist in the mantle lithosphere. For this reason, we assume a relationship which is simple but illustrative:

µ (ε ) = µ0 + ( µε − µ0 )(ε ε 0 ) µ (ε ) = µ ε

n

(ε ≤ ε 0 ) (ε > ε 0 )

(4)

This form also ignores strain rate dependence which does not require tracking of the deformation history. We used values of ε 0 = 1 , and n=3 for all the simulations presented here. We ignored the cohesion term but this necessitated truncating the smallest value of the viscosity so that the total contrast did not exceed five orders of magnitude. For simulations lasting for the order of a convective overturn time, it is necessary to consider a healing term which complements Eqn 3. In this case we simply assumed that the accumulated strain would be reset when a parcel of material was heated above 300°C. Numerical Method We start from a standard Eulerian, multigrid finite element code, CITCOM [Reference 4] which has been thoroughly benchmarked for convection in highly viscous fluids where viscosity is a very strong function of temperature. It is extremely troublesome to deal with the convective terms in Eulerian formulations. Although there has been a lot of work in this area for two-component chemical advection (see Reference

5), where numerical diffusion and dispersion can be partially compensated, when tracking smoothly varying quantities such as history variables, some form of Lagrangian reference frame is almost unavoidable. Therefore, to permit the treatment of strain- and composition-dependent viscosities, we have made some straightforward but unusual modifications to the Eulerian code. The modifications are based upon the Material Point Method (MPM) of Sulsky et al. [Reference 6] which was developed for momentum dominated flow problems. MPM is a form of particle-in-cell method which is derived using the formalism of the finite element method. In MPM, numerical integration over element volumes is achieved by summation over a set of particle locations. The particles are advected with the fluid flow and carry composition and accumulated strain information. The material nonlinearity introduced by yielding is treated exactly as in the original CITCOM code [Reference 3]. Stresses are computed on the finite element mesh and evaluated at the individual particle locations using the nodal shape functions. These stresses are then tested against the yield criterion for each particle in turn Ñ after yield, the material is modeled using a nonlinear viscosity which is iterated to ensure the yield stress is never exceeded

(a)

(b)

(c)

(d)

Figure 2: Temperature field for mature convection runs with various values of brittle parameters in the lithosphere. Darker shades indicate cooler temperatures and higher viscosities except where high plastic strain rates occur in the lithosphere — these regions have been lightened. Graphs indicate horizontal surface velocity.

Results The simulations shown in this section were run in a 7x1 Cartesian layer using periodic vertical boundary conditions, free-slip upper surface and no-slip lower boundary. The lower boundary temperature was fixed at 1500°C and a depth of 700km. Internal heating was set at 5x10-12Wkg-1. This combination, in conjunction with the viscosity law in (1), gave a surface velocity of 1-5 cm/yr, and an internal temperature near 1300°C. The initial thermal state came from a lower resolution calculation which had achieved statistical steady state with a single downwelling located in the left side of the box, and a spreading region in the right half. Fig. 2 shows four simulations which compare strain-rate localization in cases with small and large values of µ0 , and both with and without pronounced strain weakening. The graph above each simulation is the horizontal surface velocity with the tick marks indicating 1cm/yr. Regions with very high plastic strain rates are lightened in the image. Fig 2a is a reference case which has µ0 = µε = 0.04 . The initial pattern of upwellings and downwellings was not stable and soon broke down. In general, the downwellings in this simulation were more inclined to wander than those in simulations with strain weakening. The runs shown in Figs. 2b,c did have strain weakening with µ0 = 0.4, µε = 0.04 , and µ0 = 0.8, µε = 0.04 respectively. Distinct shear bands established themselves rapidly without altering the initial pattern of convection which remained unchanged for the first 2x108 years of the simulation (~109 years total). The surface velocity was very small in the ÔvÕ between the two shear bands where stagnant regions developed. Fig 2 d shows another end-member example in which µ0 = µε = 0.8 . The region in which the yield stress was exceeded was confined to the uppermost part of the layer Ñ not sufficient to mobilize the material in the cold downwelling. The focused downwelling disappeared to be replaced by a broadly thickened region of the boundary layer. Fig. 3a plots the surface heat flux for the initial evolution of a suite of runs using with different µ0 \ µε as indicated. The runs with lower µ0 had higher initial heat flux, corresponding to faster surface motion. However, the negative buoyancy in the lithosphere was consumed faster than cooling could renew the boundary layer and so the runs with initially high surface heat flow later slumped to nearly stagnant surface

Figure 3: Surface heat flow evolution with time for runs with different brittle parameters.

conditions. Ultimately, however, runs with a common µε settled down to a comparable surface heat flow (~50mWm-2) once a steady state had been reached (Fig. 3b), whereas the final simulation with high friction coefficient and no strain-weakeninghad a much lower average surface heat flux (~25 mWm-2). Note, for very high µ0 , the low strain-rate solution seen in Fig 2d is also stableÑ it is possible for the system to cross into this state during the adjustment transient. Discussion The shear bands which develop in the lithsphere are highly symmetric, and vertical downwellings are very stable. The rheology contains no mechanism for favouring one of the shear bands over the other. The only way for a downwelling to become asymmetric (as is the case for subduction) is geometrical: when buckling occurs in the downgoing slab that redistributes the near-surface stresses. Even then material is consumed from both plates. In further work, the role of buoyant continental crust should be examined to determine if the presence of continents is significant in determining the nature of oceanic plate motions. Conclusions The Lagrangian Particle Method which we have described is very well suited to modeling of problems where history and composition dependent materials undergo very large strains. This method has allowed us to model the convecting mantle in which persistent shear zones develop to mobilize the very viscous oceanic lithosphere, and to incorporate material property differences associated with the presence of buoyant, compositionally distinct continental crust REFERENCES 1. Tackley, P. J., Self-consistent generation of tectonic plates in three-dimensional mantle convection, Earth. Planet. Sci. Lett., 157, 9-22.,1998. 2. Trompert, R., and Hansen, U., Mantle convection simulations with rheologies that generate plate-like behaviour, Nature, 395, p.686-689, 1998. 3. Moresi, L. & Solomatov, V. , Mantle convection with a brittle lithosphere: thoughts on the global tectonic styles of the Earth and Venus, Geophys. J. Int., 133, 669-682, 1998. 4. Moresi, L. & Solomatov, V. S., Numerical investigation of 2D convection with extremely large viscosity variations, Phys. Fluids, 7, 2154-2162, 1995 5. Lenardic, A. & Kaula, W. M., A numerical treatment of geodynamic viscous flow problems involving the advection of material interfaces, J. Geophys. Res. 98, 8243-8269, 1993. 6.. Sulsky, D., Zhou, S.-J., & Schreyer, H.L., Application of a particle in cell method to solid mechanics, Comp. Phys. Comm., 87, 236-252. 1995