1. vv Abstract In order to study the flow of highly deformable yet heterogeneous material, we investigate the application of the Smoothed Particle Hydrodynamics modeling method to geophysical fluid dynamics. SPH is a grid-free, Lagrangian method that solves the continuum equations (continuity, momentum, energy) on a set of interpolation points (particles). These interpolation points correspond to representative volume elements which track the material points of the fluid and it's state variables (density, stress, temperature). The evolution of the state variables for a given volume element is calculated by considering the influence of each neighboring element and summing over all neighbors. Volume elements interact with boundaries through mirror particles which enforce a no- slip boundary condition by mirroring the element's material properties, stress and velocity. The method includes several advantages over irregular grid-based approaches including easy parallelization of the calculations and efficient tracking of material properties in regions of high deformation rate.We test two-dimensional, mechanical models of unconfined, viscous fluid flow for both quasi-static flows, applicable to tectonics, and dynamic flows such as landslides. Our SPH implementation is tested by comparing model results to analytic solutions for several confined and unconfined fluid flow problems. In particular, we compare our model with three test cases for homogeneous viscous flow; Poiseuille channel flow, Stoke's flow past a cylinder, and unconfined flow down an inclined plane. Model testing 1-D Motivation Tectonic deformation is often modeled numerically, especially using grid-based methods, as a continuous process when, in fact, deformation can be highly localized along distinct fault planes. Particle methods are useful in studying this localized deformation due their ability to localize displacement, track interfaces and model large deformation. Analog sandbox models (Figure 1) provide examples of particle-based, frictional-plastic deformation. Sandbox models provide high-resolution simulation of fault-dominated deformation, but are limited in their representation of alternative rheologies. SPH Methodology Smoothed Particle Hydrodynamics (SPH) is a particle-based method that solves the full continuum equations. The SPH method of modeling was formulated in 1977 for astrophysical fluid flows (Gingold and Monaghan, 1977). The method is based on Monte-Carlo interpolation theory where the state variables (pressure, density, temperature) are calculated at interpolation points. Distinct element modeling is the numerical method that has seen the most application to geologic problems. In this method, particles interact with each other through contact forces as show schematically in Figure 2. This method has been used to simulate Mohr-Coulomb behavior in tectonics (Figure 3) and predicts localization (faulting). However, it is difficult to relate bulk rheological properties to particle contact parameters or to simulate viscous constitutive behavior. The fundamental assumption behind SPH is that the values of a function at a point x can be approximated by the integral convolved with a kernel W of width h. This integral can be represented simply as a summation over a set of interpolation points (particles) with the differential volume element being approximated by the ratio of the particle’s mass, m, to its density, . The value of a function can then be found by summing the contribution of all points weighted according to the kernel function (Figure 4). The kernal function W(r,h) has the characteristic that it integrates to unity and has a finite characteristic width, h. In this study, we use a cubic spline. The SPH formulation of the derivative of a function,Ψ, is shown on the right. An important point to note with this formulation is that the gradient operator has been transferred from the variable to the kernel function. This is achieved by applying the divergence theorem to the integral/kernel approximation of the function. This results in two integrals, one over the volume of the domain and one over it's boundary. For kernels that decay sufficiently fast, the boundary integral approaches zero. Transferring the gradient operator to the kernel allows the approximation of gradients of the variables by calculating the gradient of the known kernel function. To calculate the evolutions of a system using SPH, the equations of motion need to be recast in the discretized, form. The SPH formulation of the field equations are shown below In these equations,  is density, v is velocity,  is the stress tensor, b is the body force vector, D is the rate of deformation tensor, r is the heat source, q is the heat flux vector, T is the temperature, k is the thermal conductivity, and where the latin and greek indicies correspond to particles and dimensions respectively. To test the SPH formulation of viscous fluid flow, 1-D Matlab scripts were written to calculate velocity profiles for the two time-dependent cases of Couette flow and Poiseulle flow. The results are shown in Figures 5 and 6 respectively. In these test cases, the particles are held fixed while only the transverse velocity is solved. Figure 7 shows the results for time-dependent thermal diffusion problem where there is a temperature contrast between the right and left side of 1000° C. 2-D 3-D A classic problem for inviscid free-surface fluid flow is the bam burst problem. In this problem, a volume of fluid is held back by a dam. At time t=0, the dam is removed and the column of fluid collapses. The problem was first addressed using SPH by Monaghan (1994) who used a slightly different form of the momentum equation which considered only pressure forces (Figure 9). The solution using the full momentum equation is shown in Figure 10. In this calculation, the shear and bulk viscosities are set to zero. In Figure 11, the viscous dam break problem is solved for different values of viscosity. Time Viscosity 10 2 Pa s 0.50 s1.0 s1.5 s 10 3 Pa s 10 4 Pa s 0.25 s Time 0.50 s1.5 s2.5 s Figure 5: Couette Flow (after Morris, et al, 1997)Figure 6: Poiseulle Flow (after Morris, et al, 1997)Figure 7: Thermal Diffusion Figure 8: Flow down an incline plane Figure 9: Dam burst (Monaghan, 1994) Figure 10: Dam burst Figure 4: SPH Kernel Figure 1: Sandbox Modeling (Persson and Sokoutis, 2002) Figure 3: DEM modeling of faulting (Saltzer, et al, 1992) Figure 11: Viscous Dam burst Figure 13: Viscous Accretionary Wedges Viscosity 10 2 Pa s 10 3 Pa s 10 4 Pa s 10 5 Pa s Figure 12: 3-D Viscous Droplet A simple test case for 2-D unconfined flow is given by the viscosity dependent velocity profile of a sheet of fluid flowing down an incline plane. While this is actually a 1-D solution, but provides a test of the 2-D code and boundary conditions. Our 2-D model (Figure 8) does predict the correct shape of the profile, however particles are slipping along the boundary resulting in a shift in the velocity profile suggesting the no-slip boundary is not well-implemented. Development in three-dimensions is straightforward. This preliminary solution shows the collapse of a viscous cube with time. A classic tectonics problem considers the growth of a convergent wedge in response to motion of a rigid backstop. Below are models showing development of viscous wedges for three values of viscosity. With increasing viscosity, the deformation is more localized near the moving backstop. Accretionary wedges Figure 2: DEM Interparticle interaction (Ferrez, 2001) V = -0.05 m/s SummaryReferences Our preliminary results show that the method of Smoothed Particle Hydrodynamics can successfully model simple heat and Newtonian fluid flow. We are currently working to implement a non-linear viscosity in order to model Mohr- Coulomb rheology. This method shows promise in studying systems in tectonics such as a brittle upper crust overlying a viscous lower crust, where deformation is localized along distinct faults in the upper layer while deformation is distributed in the underlying viscous layer. Ferrez, J., Dynamic triangulations for efficient 3D simulation of granular material, PhD dissertation, Ecole Polytechnique Federale de Lausanne, 2001. Ginggold and Monaghan; Smoothed particle hydrodynamics: theory and application to non-spherical stars; Monthly Notices of the Royal Astronomical Society; v. 181; p.375-89; 1977. Monaghan; Simulating Free Surface Flows with SPH; Journal of Computational Physics; v. 110; p.399-406; 1994. Morris, et. al.; Modeling Low Reynolds Number Incompressible Flows Using SPH; Journal of Computational Physics; v. 136; p.214-26; 1997. Saltzer, S., et. al., Distinct element modeling of structures formed by extensional reactivation of basement normal faults, Tectonics, v.11, p.165-174, 1992. Persson and Sokoutis; Analogue models of orogenic wedges controlled by erosion; Tectonophysics; v. 356; p.323-36; 2002.