Chaos and predictability in ocean water levels

The classical problem of characterizing and classifying ocean water levels (all fluctuations that are greater than a few minutes duration) is examined using methods derived from studies of nonlinear dynamical systems. The motivation for this study is the difficulty of characterizing coastal water level dynamics and tide zones with existing methods. There is also long-standing evidence that coastal water levels are not a simple linear superposition of astronomical tides and other influences. Thus it can be appropriate to view water levels as a single, nonlinear, dynamical system. We show that it is appropriate to treat water levels as chaotic by virtue of the existence of a positive Lyapunov exponent for the seven data sets studied. The integer embedding space (the number of state space coordinates) needed to reconstruct an attractor for data collected from sensors exposed to the open ocean is five. Four dynamical degrees of freedom appear to be required to describe the observed dynamics in a state space reconstructed solely from the observations themselves. Water levels in a complex estuary (Chesapeake Bay) have a global dimension of six and have five dynamical degrees of freedom. The largest global Lyapunov exponents, a measure of predictability, vary from 0.57 h(-1) for a station relatively well exposed to the ocean (Charleston, South Carolina) to 4.6 h(-1) for a station well inside a complex estuary (Baltimore, Maryland). The larger values are generally associated with stations that are less predictable, which is consistent with the errors of the astronomical estimator currently used by the U.S. government to generate tide predictions. Lower values are associated with water levels where the estimator errors are smaller. These results are consistent with the interpretation of the Lyapunov exponents as a measure of dynamical predictability. The dynamical characteristics, notably the Lyapunov exponents, are shown to be good candidates for characterizing water level variability and classifying tide zones.