Communications in Mathematical Sciences

Volume 14 (2016)

Number 6

Predicting fat-tailed intermittent probability distributions in passive scalar turbulence with imperfect models through empirical information theory

Pages: 1687 – 1722



Di Qi (Department of Mathematics and Center for Atmosphere and Ocean Science, Courant Institute of Mathematical Sciences, New York University, New York, N.Y., U.S.A.)

Andrew J. Majda (Department of Mathematics and Center for Atmosphere and Ocean Science, Courant Institute of Mathematical Sciences, New York University, New York, N.Y., U.S.A.)


Modelling extreme events is a central issue in climate science and engineering. The capacity of imperfect models to capture intermittent behavior with fat-tailed probability distributions of a passive scalar field advected by turbulent flow systems is investigated here. We consider the effects with complicated flow systems including strong nonlinear and non-Gaussian interactions, and construct much simpler and cheaper imperfect models with model error to capture the crucial statistical features in the stationary tracer field. The Lorenz ’96 (L-96) system is utilized as a test model to generate the turbulent advection flow field. Tracer statistics under this L-96 flow field are analyzed both theoretically and numerically, and strong intermittent fat tails can be observed in different dynamical regimes of the flow system with distinct statistical features. The complexity and large computational expense in resolving the true advection flow require the introduction of simpler and more tractable imperfect models which still maintain the ability to capture the key intermittent features in the tracer field. The simplest linear stochastic models containing no positive Lyapunov exponents are proposed here to approximate the tracer advected by the original L-96 system with large degrees of internal instabilities. It is demonstrated that the prediction skill of this imperfect linear model can be greatly improved through fitting the autocorrelation functions with empirical information theory. A systematic framework of measuring the autocorrelation function under spectral representation with the help of empirical information theory is developed, and the optimal model parameters under this unbiased information measure can be achieved easily in a training phase before running the predictions. This imperfect model using optimal parameters achieved through the information-theoretic framework is tested in a variety of dynamical regimes of the L-96 system. Uniformly high skill of the optimal model is displayed in accurately capturing the crucial tracer statistical features in a stationary statistical steady state, especially in getting accurate intermittent fat tails in tracer density distributions. This information framework for tuning autocorrelation functions can be further generalized to more complicated turbulent models and should have many applications.


turbulent diffusion, passive scalar field, intermittency, Gaussian velocity model, information metric

2010 Mathematics Subject Classification

62B10, 76F25, 86A32

Published 12 August 2016