Geomorphology is the study of the small-scale surface evolution of the earth under the forces resulting from such agents as wind, water, gravity, and ice. Understanding and prediction in geomorphology are critically dependent upon the ability to model the processes that shape the landscape. Because these processes in general are too complicated on large scales to describe in detail, it is necessary to adopt a system of hierarchical models in which the behavior of small systems is summarized by a set of rules governing the next larger system; in essence, these rules constitute a simplified algorithm for the physical processes in the smaller system that cannot be treated fully at larger scales. A significant fraction of the processes in geomorphology involve entrainment, transport and deposition of particulate matter. Where the intergrain forces become comparable to or greater than the forces arising from the transporting agents, consideration of the properties of a granular material, a system of grains which collide with the slide against neighboring grains, is warranted. A micromechanical description of granular materials has proved difficult, except in energetic flow regimes [Haff:83a], [Jenkins:83a]. Thus, researchers have turned to dynamical and computer simulations at the level of individual grains in order to elucidate some of the basic mechanical properties of granular materials ([Cundall:79a], [Walton:83a] pioneered this simulation technique). In this section, we discuss the role that hypercube concurrent processing has played and is expected to play both in grain-level dynamical simulations and in relating these simulations to modelling the formation and evolution of landforms.

As an example of this approach to geomorphology, we shall consider efforts to model transport of sand by the wind based upon the grain-to-grain dynamics. Sand is transported by the wind primarily in saltation and in reptation [Bagnold:41a]. Saltating grains are propelled along the surface in short hops by the wind. Each collision between a saltating sand grain and the surface results in a loss of energy which is compensated, on the average, by energy acquired from the wind. Reptating grains are ejected from the sand surface by saltating grain-sand bed impacts; they generally come to rest shortly after returning to the sand surface.

Computer simulations of saltating grain impacts upon a loose grain bed were performed on an early version of the hypercube [Werner:88a;88b]. Collisions between a single impacting grain and a box of 384 circular grains were simulated. The grains interact through stiff, inelastic compressional contact forces plus a Coulomb friction force. The equations of motion for the particles are integrated forward in time using a predictor-corrector technique. At each step in time, the program checks for contacts between particles and, where contacts exist, computes the contact forces. Dynamical simulations of granular materials are computationally intensive, because the time scale of the interaction between grains (tens of microseconds) is much smaller than the time scale of the simulation (order one second).

The simulation was decomposed on a Caltech hypercube by assigning the processors to regions of space lying on a rectangular grid. The computation time is a combination of calculation time in each processor due to contact searches and to force computations, and of communication time in sending information concerning grain positions and velocities to neighboring processors for interparticle force calculations on processor boundaries. Because the force computation is complicated, the communication time was found to be a negligible fraction of the total computation time for granular materials in which enduring intergrain contacts are dominant. The boundaries between processors are changed incrementally throughout the calculation in order to balance the computational load among the processors. The optimal decomposition has enough particles per processor to diminish the relative importance of statistical fluctuations in the load, and a system of boundaries which conforms as much as possible to the geometry of the problem. For grain-bed impacts, efficiencies between 0.89 and 0.97 were achieved [Werner:88a].

Irregularities of the geometry are important in determining which sand grains interact with each other. Thus, it is not possible to find an efficient synchronous algorithm for this and many other particle interaction problems. The very irregular inhomogeneous astrophysical calculations described in Section 12.4 illustrate this point clearly. One also finds the same issue in molecular dynamics codes, such as CHARMM which are extensively used in chemistry. This problem is, however, loosely synchronous as we can naturally macroscopically synchronize the calculation after each time step-thus a MIMD implementation where each processor processes its own irregular collection of grains is very natural and efficient. The sand grain problem, unlike that of Section 12.4, has purely local forces as the grains must be in physical contact to affect each other. Thus, only very localized communication is necessary. Note that Section 4.5 describes a synchronous formulation of this problem.

The results of the grain-bed impact simulations have facilitated treatment of two larger scale problems. A simulation of steady-state saltation in which calculation of saltating grain trajectories and modifications to the wind velocity profile, due to acceleration of saltating grains, were combined with a grain-bed impact distribution function derived from experiments and simulations. This simulation yielded such characteristics of saltation as flux and erosive potential [Werner:90a]. A simulation of the rearrangement of surface grains in reptation led to the formation of self-organized small-scale bedforms, which resemble wind ripples in both size and shape [Landry:93a], [Werner:91a;93a]. Larger, more complicated ripple formation simulations and a simulation of sand dune formation, using a similar approach which is under development, are problems that will require a combination of processing power and memory not available on present supercomputers. Ripple and dune simulations are expected to run efficiently with a spatial decomposition on a hypercube.

Water is an important agent for the transport of sediment. Unlike wind-blown sand transport, underwater sand transport requires simultaneous simulation of the grains and the fluid because water and sand are similar in density. We are developing a grain/fluid mixture simulation code for a hypercube in which the fluid is modelled by a gas composed of elastic hard circles (spheres in three dimensions). The simulation steps the gas forward at discrete time intervals, allowing the gas particles to collide (with another gas particle or a macroscopic grain) only once per step. The fluid velocity and the fluid force on each grain are computed by averaging. Since a typical void between macroscopic grains will be occupied by up to 1000 gas particles, the requisite computational speed and memory capacity can be found only in the hypercube architecture. Communication is expected to be minimal and load balancing can be accomplished for a sufficiently large system. It is expected that larger scale simulations of erosion and deposition by water [Ahnert:87a] will benefit from the findings of the fluid/grain mixture simulations. Also, these large-scale landscape evolution simulations are suitable themselves for a MIMD parallel machine.

Computer simulation is assuming an increasing role in geomorphology. We suggest that the development and availability of high-performance MIMD concurrent processors will have considerable influence upon the future of computing in geomorphology.

Wed Mar 1 10:19:35 EST 1995