Empirical Master Equations. Part I: Numerical Properties

Abstract

In the atmospheric sciences, master equations are mainly used in a discrete time approximation to provide forecasts of the probability density function in a discretized phase space spanned by a few climate variables. The coefficients of an empirical master equation (EME) are estimated from the relative frequencies of transitions observed in time series of the variables. The quality of an EME depends on, among other things, the length and time resolution of the available time series. In this part of the paper, these dependencies are studied on the basis of data from the three-component Lorenz model with additional white noise forcing. Thus, time series of almost any length and time resolution can be generated easily, and probability density forecasts can be compared directly with the evolution of an ensemble of points. Useful results are obtained by partitioning the phase space into several hundred cells of equal grid size. The authors find that a threshold length of the time series exists beyond which improvements in the performance of the EME are hard to detect. It is even more surprising that the performance deteriorates with reduction of the time step. This is due to an increase in numerical diffusion. The choice of the dimensionality and the selection of the variables of the EME are very important. The results of this part of the paper provide useful guidelines for any application of the EME in the atmospheric sciences and elsewhere. The second part of the paper illustrates the usefulness of these guidelines through applications to stratospheric dynamics.