### Abstract

The recently developed time-periodic fluctuationdissipation theorem (FDT) provides a very convenient way of addressing the climate change of atmospheric systems with seasonal cycle by utilizing statistics of the present climate. A triad nonlinear stochastic model with exactly solvable first and second order statistics is introduced here as an unambiguous test model for FDT in a time-periodic setting. This model mimics the nonlinear interaction of two Rossby waves forced by baroclinic processes with a zonal jet forced by a polar temperature gradient. Periodic forcing naturally introduces the seasonal cycle into the model. The exactly solvable first and second order statistics are utilized to compute both the ideal mean and variance response to the perturbations in forcing or dissipation and the quasi-Gaussian approximation of FDT (qG-FDT) that uses the mean and the covariance in the equilibrium state. The time-averaged mean and variance qG-FDT response to perturbations of forcing or dissipation is compared with the corresponding ideal response utilizing the triad test model in a number of regimes with various dynamical and statistical properties such as weak or strong non-Gaussianity and resonant or non-resonant forcing. It is shown that even in a strongly non-Gaussian regime, qG-FDT has surprisingly high skill for the mean response to the changes in forcing. On the other hand, the performance of qG-FDT for the variance response to the perturbations of dissipation is good in the near-Gaussian regime and deteriorates in the strongly non-Gaussian regime. The results here on the test model should provide useful guidelines for applying the time-periodic FDT to more complex realistic systems such as atmospheric general circulation models.

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

Pages (from-to) | 1741-1757 |

Number of pages | 17 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 239 |

Issue number | 17 |

DOIs | |

State | Published - Sep 1 2010 |

### Fingerprint

### Keywords

- Exactly solvable model
- Fluctuationdissipation theorem
- Linear response
- Time-periodic statistics

### ASJC Scopus subject areas

- Condensed Matter Physics
- Statistical and Nonlinear Physics

### Cite this

*Physica D: Nonlinear Phenomena*,

*239*(17), 1741-1757. https://doi.org/10.1016/j.physd.2010.05.009

**A test model for fluctuationdissipation theorems with time-periodic statistics.** / Gershgorin, Boris; Majda, Andrew J.

Research output: Contribution to journal › Article

*Physica D: Nonlinear Phenomena*, vol. 239, no. 17, pp. 1741-1757. https://doi.org/10.1016/j.physd.2010.05.009

}

TY - JOUR

T1 - A test model for fluctuationdissipation theorems with time-periodic statistics

AU - Gershgorin, Boris

AU - Majda, Andrew J.

PY - 2010/9/1

Y1 - 2010/9/1

N2 - The recently developed time-periodic fluctuationdissipation theorem (FDT) provides a very convenient way of addressing the climate change of atmospheric systems with seasonal cycle by utilizing statistics of the present climate. A triad nonlinear stochastic model with exactly solvable first and second order statistics is introduced here as an unambiguous test model for FDT in a time-periodic setting. This model mimics the nonlinear interaction of two Rossby waves forced by baroclinic processes with a zonal jet forced by a polar temperature gradient. Periodic forcing naturally introduces the seasonal cycle into the model. The exactly solvable first and second order statistics are utilized to compute both the ideal mean and variance response to the perturbations in forcing or dissipation and the quasi-Gaussian approximation of FDT (qG-FDT) that uses the mean and the covariance in the equilibrium state. The time-averaged mean and variance qG-FDT response to perturbations of forcing or dissipation is compared with the corresponding ideal response utilizing the triad test model in a number of regimes with various dynamical and statistical properties such as weak or strong non-Gaussianity and resonant or non-resonant forcing. It is shown that even in a strongly non-Gaussian regime, qG-FDT has surprisingly high skill for the mean response to the changes in forcing. On the other hand, the performance of qG-FDT for the variance response to the perturbations of dissipation is good in the near-Gaussian regime and deteriorates in the strongly non-Gaussian regime. The results here on the test model should provide useful guidelines for applying the time-periodic FDT to more complex realistic systems such as atmospheric general circulation models.

AB - The recently developed time-periodic fluctuationdissipation theorem (FDT) provides a very convenient way of addressing the climate change of atmospheric systems with seasonal cycle by utilizing statistics of the present climate. A triad nonlinear stochastic model with exactly solvable first and second order statistics is introduced here as an unambiguous test model for FDT in a time-periodic setting. This model mimics the nonlinear interaction of two Rossby waves forced by baroclinic processes with a zonal jet forced by a polar temperature gradient. Periodic forcing naturally introduces the seasonal cycle into the model. The exactly solvable first and second order statistics are utilized to compute both the ideal mean and variance response to the perturbations in forcing or dissipation and the quasi-Gaussian approximation of FDT (qG-FDT) that uses the mean and the covariance in the equilibrium state. The time-averaged mean and variance qG-FDT response to perturbations of forcing or dissipation is compared with the corresponding ideal response utilizing the triad test model in a number of regimes with various dynamical and statistical properties such as weak or strong non-Gaussianity and resonant or non-resonant forcing. It is shown that even in a strongly non-Gaussian regime, qG-FDT has surprisingly high skill for the mean response to the changes in forcing. On the other hand, the performance of qG-FDT for the variance response to the perturbations of dissipation is good in the near-Gaussian regime and deteriorates in the strongly non-Gaussian regime. The results here on the test model should provide useful guidelines for applying the time-periodic FDT to more complex realistic systems such as atmospheric general circulation models.

KW - Exactly solvable model

KW - Fluctuationdissipation theorem

KW - Linear response

KW - Time-periodic statistics

UR - http://www.scopus.com/inward/record.url?scp=77955426833&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77955426833&partnerID=8YFLogxK

U2 - 10.1016/j.physd.2010.05.009

DO - 10.1016/j.physd.2010.05.009

M3 - Article

VL - 239

SP - 1741

EP - 1757

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 17

ER -