Fully 3-D numerical simulations of thermal convection in a spherical shell have become a standard for studying the dynamics of pattern formation and its stability under perturbations to various parameter values. The question arises as to how the discretization of the governing equations affects the outcome and thus any physical interpretation. This work demonstrates the impact of numerical discretization on the observed patterns, the value at which symmetry is broken, and how stability and stationary behavior is dependent upon it. Motivated by numerical simulations of convection in the Earth's mantle, we consider isoviscous Rayleigh-Bénard convection at infinite Prandtl number, where the aspect ratio between the inner and outer shell is 0.55. We show that the subtleties involved in developing mantle convection models are considerably more delicate than has been previously appreciated, due to the rich dynamical behavior of the system. Two codes with different numerical discretization schemes-an established, community-developed, and benchmarked finite-element code (CitcomS) and a novel spectral method that combines Chebyshev polynomials with radial basis functions (RBFs)-are compared. A full numerical study is investigated for the following three cases. The first case is based on the cubic (or octahedral) initial condition (spherical harmonics of degree ℓ Combining double low line 4). How this pattern varies to perturbations in the initial condition and Rayleigh number is studied. The second case investigates the stability of the dodecahedral (or icosahedral) initial condition (spherical harmonics of degree ℓ Combining double low line 6). Although both methods first converge to the same pattern, this structure is ultimately unstable and systematically degenerates to cubic or tetrahedral symmetries, depending on the code used. Lastly, a new steady-state pattern is presented as a combination of third-and fourth-order spherical harmonics leading to a five-cell or hexahedral pattern and stable up to 70 times the critical Rayleigh number. This pattern can provide the basis for a new accuracy benchmark for 3-D spherical mantle convection codes.

The geometry of a strike-slip fault system is an important component that influences the kinematics and interactions of the various faults within the system. Discontinuities and bends in the fault geometry not only determine the types of structures and the physiography that we observe along the fault system but also have a significant influence on the propagation of earthquake ruptures. A precise knowledge of the fault geometry, especially how it is segmented and other physical parameters, is essential for seismic hazard analysis. It is known that earthquake ruptures sometimes propagate over multiple faults by jumping from one segment to the next. A fault jump is a sudden dynamic coalescence of two faults separated by a step-over. Field observations suggest that a step-over width of 5 km is an appropriate maximum jump distance. Our study shows that between 2.5 and 6.5 km of step-over width, the probability of fault jump, for both releasing and restraining step-overs, decreases significantly from 100 to <10 %.

The visualization and analysis of complex fields often requires identifying and extracting domain specific features. Through a collaboration with geophysicists we extend previous work on crease surfaces with a new and complimentary definition: extremas in principal surface curvature rather than scalar value. Using this definition, we visualize the resulting surfaces which correspond to individual wave fronts. As these wave fronts propagate through a control structure (medium), they undergo changes in intensity, shape and topology due to reflection, refraction and interference. We demonstrate our ability to effectively visualize these phenomena in complex data sets including a large-scale simulation of a hypothetical earthquake along the San Andreas fault in Southern California.