Linear and nonlinear signatures in the planetary wave dynamics of an AGCM: Probability density functions

To identify and quantify indications of linear and nonlinear planetary wave behavior and their impact on the distribution of atmospheric states, characteristics of a very long integration of an atmospheric general circulation model (GCM) in a four-dimensional phase space are examined. The phase space is defined by the leading four empirical orthogonal functions of 500-hPa geopotential heights. First it is established that nonlinear tendencies similar to those reported in an earlier study of the phase space behavior in this GCM have the potential to lead to non-Gaussian features in the probability density function (PDF) of planetary waves. Then using objective measures it is demonstrated that the model's distribution of states has distinctive non-Gaussian features. These features are characterized in various subspaces of dimension as high as four. A key feature is the presence of three radial ridges of enhanced probability emanating from the mode, which is shifted away from the climatological mean. There is no evidence of multiple maxima in the full PDF, but the radial ridges lead to three distinct modes in the distribution of circulation patterns. It is demonstrated that these key aspects of non-Gaussianity are captured by a two-Gaussian mixture model fitted in four dimensions. The two circulation states at the centroids of the component Gaussians are very similar to those associated with two nonlinear features identified by Branstator and Berner in their analysis of the trajectories of the GCM. These two dynamical features are locally linear, so it is concluded that the behavior of planetary waves can be conceptualized as being approximately piecewise-linear, leading to a two-Gaussian mixture with three preferred patterns.