A simple computer-simulation algorithm for the transport of sand by wind produces forms resembling barchan, crescentic ridge, linear, and star natural dune classes. Sand is moved as slabs composed of many grains that are picked up at random, transported in a specified direction, and deposited (1) with a probability that depends on the local presence or absence of sand or (2) in shadow zones in the lee of dunes. The simulated dune fields are interpreted as complex systems, with sand-dune classes being dynamical attractors of these systems. The evolution of dunes once formed becomes decoupled from the details of eolian sand transport.

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Striking circular, labyrinthine, polygonal, and striped patterns of stones and soil self-organize in many polar and high alpine environments. These forms emerge because freeze-thaw cycles drive an interplay between two feedback mechanisms. First, formation of ice lenses in freezing soil sorts stones and soil by displacing soil toward soil-rich domains and stones toward stone-rich domains. Second, stones are transported along the axis of elongate stone domains, which are squeezed and confined as freezing soil domains expand. In a numerical model implementing these feedbacks, circles, labyrinths, and islands form when sorting dominates; polygonal networks form when stone domain squeezing and confinement dominate; and stripes form as hillslope gradient is increased.

Patterns in nature, such as meandering rivers and sand dunes, display complex behavior seemingly at odds with their simplicity of form. Existing approaches to modeling natural landform patterns, reductionism and universality, are incompatible with the nonlinear, open nature of natural systems. An alternative modeling methodology based on the tendency of natural systems to self-organize in temporal hierarchies is described.A river channel meanders in wide, sweeping loops through its floodplain. Sand dunes mantle vast expanses of arid regions with crescentic, linear, and star-shaped forms. Shorelines are molded into smooth arcuate bays interrupted by cuspate horns. Frozen soils throughout the Arctic are broken by a latticework of thermal contraction fractures filled with wedges of ice. A photographer's dream, these and other natural landform patterns exhibit complex behavior that belies their apparent simplicity of form.Complicated behavior arising from a simple form is a classic symptom of a complex system. Complexity in natural landform patterns is a manifestation of two key characteristics (1). Natural patterns form from processes that are nonlinear, those that modify the properties of the environment in which they operate or that are strongly coupled; and natural patterns form in systems that are open, driven from equilibrium by the exchange of energy, momentum, material, or information across their boundaries. A surf zone, where waves break near the shoreline, exemplifies these characteristics. Wave propagation, generation of currents, turbulent fluid flow, and transport of sediment all are nonlinear processes. The energy and momentum for driving fluid and sediment motion is derived from waves, generated in deeper water, entering the open system. The surf zone and other open natural systems are subject to external forcing on a broad range of temporal scales, for example, 10-s waves to changes in sea level over thousands of years.These properties can lead to complexity in natural patterns in two stages. First, development of collective behavior by selforganization (2) reduces the very large number of degrees of freedom (for example, those characterizing sand grains on a beach) to a much smaller number of independent dynamical variables (beach profile or shoreline position). Second, these variables evolve and interact nonlinearly to produce rich, potentially emergent behavior that is only weakly related to the original numerous degrees of freedom or the processes operating on them.Most natural patterns exhibit some form of complex behavior. Bedforms, patterns in a sediment bed such as ripples and dunes in rivers, oceans, and deserts, generally start small and disorganized; they grow in spacing and become better organized through interactions and mergers between bedforms (3-5). Crescent-shaped wind-blown (barchan) dunes spawn new dunes from their downwind pointing horns, an emergent behavior (1). Sand bars on beaches undergo transitions between different shapes, such as linear and crescentic, based on a complica...