The dynamics of natural systems breaks up into a hierarchy of behaviors, with each level characterized by time scales for reactions to perturbations well-separated from adjoining levels. Scale separation is related to the combined effects of strong nonlinearity and high rates of dissipation, resulting in dynamical decoupling of behaviors at different scales. The dynamics at each level is an independent property of the system, unrelated to dynamics at any other level. The behavior of one level in the hierarchy determines the long-time-scale dynamics of the variables characterizing the next faster level by slaving these variables, as in the control a river channel exerts on a water parcel or the control a sand dune exerts on a sand grain. Conversely, slaving is related to the feedbacks inherent in the dynamics of the faster level, as in the popular view of self-organization. These two interactions, which constitute the only connections between levels and determine the behavior of fast-scale variables on long time scales, are equivalent. Because causal relationships are one way, the two-way nature of these connections implies that no cause and effect relation between the dynamics of two adjoining levels can be established. Therefore, causality in natural systems is bounded in the sense that causal relations can exist only between a system and other systems in its external environment at the same time scale.

These hypotheses appear to be consistent with behaviors from a range of environments, including arid, permafrost and nearshore landscapes. Their implications for modeling and prediction include: (i) No advantage is conferred by modeling from the bottom (using fast scales: reductionism) or the top (using slow scales: universality). (ii) The ability to use models to predict phenomena beyond time scales encompassed by the range of observations is severely restricted.

Supported by the National Science Foundation, Geology and Paleontology Program.