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A maximum entropy approach to the problem of parametrization

It is shown that the principle of maximum entropy, as formulated by Jaynes, can be applied to parametrize the effect of unresolved variables on resolved variables in a dynamical system proposed by Lorenz. The starting point is the assumption that the unresolved variables are in a state of statistical equilibrium on the time‐scale of the resolved variables. The probability density function that describes the statistics of the unresolved variables is then determined by requiring that its information entropy is maximal under the constraint that the average time rate of change of the unresolved variables' energy is zero. By using this probability density function to determine the average effect of the unresolved variables on the resolved variables, a linear damping of the resolved variables is found. After incorporating the linear damping in the equations of the resolved system, the principle of maximum entropy is applied a second time to shed light on the statistics of the resolved system. The consequences are studied of using a priori probability density functions that go beyond Laplace's principle of indifference. Copyright © 2011 Royal Meteorological Society