### Abstract

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.

Original language | English (US) |
---|---|

Pages (from-to) | 481-498 |

Number of pages | 18 |

Journal | Communications in Mathematical Sciences |

Volume | 11 |

Issue number | 2 |

State | Published - 2013 |

### Fingerprint

### Keywords

- Autoregressive models
- Climate change
- Fluctuation-dissipation theory
- Linear response

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications in Mathematical Sciences*,

*11*(2), 481-498.

**Regression models with memory for the linear response of turbulent dynamical systems.** / Kang, Emily L.; Harlim, John; Majda, Andrew J.

Research output: Contribution to journal › Article

*Communications in Mathematical Sciences*, vol. 11, no. 2, pp. 481-498.

}

TY - JOUR

T1 - Regression models with memory for the linear response of turbulent dynamical systems

AU - Kang, Emily L.

AU - Harlim, John

AU - Majda, Andrew J.

PY - 2013

Y1 - 2013

N2 - 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.

AB - 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.

KW - Autoregressive models

KW - Climate change

KW - Fluctuation-dissipation theory

KW - Linear response

UR - http://www.scopus.com/inward/record.url?scp=84870408387&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84870408387&partnerID=8YFLogxK

M3 - Article

VL - 11

SP - 481

EP - 498

JO - Communications in Mathematical Sciences

JF - Communications in Mathematical Sciences

SN - 1539-6746

IS - 2

ER -