Statistical-Dynamical Predictions

By analogy with Gibbs' statistical mechanics, a statistical-dynamical theory is described which can be applied to the problem of atmospheric predictions from synoptic charts. The m variables and constants whose measurement is needed to characterize an initial state are regarded as coordinates of an m dimensional phase space. In this space, the probability density ψ of the ensemble of possible initial measurements is used to define the "true" values of these quantities and to furnish probabilities of initial measurements lying within prescribed limits of true values. Dynamical equations provide standard deterministic predictions here, while a general continuity equation for ψ transforms initial probability distributions into final ones which, in turn, yield probability forecasts. This continuity equation is resolvable into component equations of probability diffusion for all coordinates of the phase space. It is found that predictability increases (decreases) with time if the atmosphere is divergent (convergent), is undergoing a net loss (gain) of energy, or there is an energy balance but the loss (gain) is occurring predominantly at the lower temperatures. In general, divergence (convergence) in real space implies convergence (divergence) in phase space. In the special case of an incompressible, conservative atmosphere, the continuity equation for ψ reduces to Liouville's theorem in statistical mechanics. In application to the real atmosphere, a linear, m variate, normal distribution is assumed which satisfies the general continuity equation and component diffusion equations obtained previously by the author. These diffusion equations resemble the Fokker-Planck equation for one-dimensional stochastic processes. Quantitative results include limiting values of ordinary, conditional and compound probabilities for physical variables, as well as theoretical variances, covariances and correlations of these variables, for initial and forecast times. A simple numerical example is presented to illustrate the theory and its use.