Communications in Mathematical Sciences

Volume 11 (2013)

Number 2

Regression models with memory for the linear response of turbulent dynamical systems

Pages: 481 – 498



John Harlim (Department of Mathematics, North Carolina State University, Raleigh, N.C., U.S.A.)

Emily L. Kang (Department of Mathematical Sciences, University of Cincinnati, Ohio, U.S.A.)

Andrew J. Majda (Courant Institute of Mathematical Sciences, New York University, New York, N.Y.)


Calculating the statistical linear response of turbulent dynamical systems to the change in external forcing is a problem of wide contemporary interest. Here the authors apply linear regression models with memory, AR(p) models, to approximate this statistical linear response by directly fitting the autocorrelations of the underlying turbulent dynamical system without further computational experiments. For highly nontrivial energy conserving turbulent dynamical systems like the Kruskal-Zabusky (KZ) or Truncated Burgers-Hopf (TBH) models, these AR(p) models exactly recover the mean linear statistical response to the change in external forcing at all response times with negligible errors. For a forced turbulent dynamical system like the Lorenz-96 (L-96) model, these approximations have improved skill comparable to the mean response with the quasi-Gaussian approximation for weakly chaotic turbulent dynamical systems. These AR(p) models also give new insight into the memory depth of the mean linear response operator for turbulent dynamical systems.


fluctuation-dissipation theory, autoregressive models, linear response, climate change

2010 Mathematics Subject Classification

37N10, 76F20

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