### Abstract

A systematic strategy for stochastic mode reduction is applied here to three prototype "toy" models with nonlinear behavior mimicking several features of low-frequency variability in the extratropical atmosphere. Two of the models involve explicit stable periodic orbits and multiple equilibria in the projected nonlinear climate dynamics. The systematic strategy has two steps: stochastic consistency and stochastic mode elimination. Both aspects of the mode reduction strategy are tested in an a priori fashion in the paper. In all three models the stochastic mode elimination procedure applies in a quantitative fashion for moderately large values of ε ≈ 0.5 or even ε ≈ 1, where the parameter ε roughly measures the ratio of correlation times of unresolved variables to resolved climate variables, even though the procedure is only justified mathematically for ε « 1. The results developed here provide some new perspectives on both the role of stable nonlinear structures in projected nonlinear climate dynamics and the regression fitting strategies for stochastic climate modeling. In one example. a deterministic system with 102 degrees of freedom has an explicit stable periodic orbit for the projected climate dynamics in two variables; however, the complete deterministic system has instead a probability density function with two large isolated peaks on the "ghost" of this periodic orbit, and correlation functions that only weakly "shadow" this periodic orbit. Furthermore, all of these features are predicted in a quantitative fashion by the reduced stochastic model in two variables derived from the systematic theory; this reduced model has multiplicative noise and augmented nonlinearity. In a second deterministic model with 101 degrees of freedom, it is established that stable multiple equilibria in the projected climate dynamics can be either relevant or completely irrelevant in the actual dynamics for the climate variable depending on the strength of nonlinearity and the coupling to the unresolved variables. Furthermore, all this behavior is predicted in a quantitative fashion by a reduced nonlinear stochastic model for a single climate variable with additive noise, which is derived from the systematic mode reduction procedure. Finally, the systematic mode reduction strategy is applied in an idealized context to the stochastic modeling of the effect of mountain torque on the angular momentum budget. Surprisingly, the strategy yields a nonlinear stochastic equation for the large-scale fluctuations, and numerical simulations confirm significantly improved predicted correlation functions from this model compared with a standard linear model with damping and white noise forcing.

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

Pages (from-to) | 1705-1722 |

Number of pages | 18 |

Journal | Journal of the Atmospheric Sciences |

Volume | 60 |

Issue number | 14 |

DOIs | |

State | Published - Jul 15 2003 |

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### ASJC Scopus subject areas

- Atmospheric Science

### Cite this

**Systematic strategies for stochastic mode reduction in climate.** / Majda, Andrew J.; Timofeyev, Ilya; Vanden-Eijden, Eric.

Research output: Contribution to journal › Article

*Journal of the Atmospheric Sciences*, vol. 60, no. 14, pp. 1705-1722. https://doi.org/10.1175/1520-0469(2003)060<1705:SSFSMR>2.0.CO;2

}

TY - JOUR

T1 - Systematic strategies for stochastic mode reduction in climate

AU - Majda, Andrew J.

AU - Timofeyev, Ilya

AU - Vanden-Eijden, Eric

PY - 2003/7/15

Y1 - 2003/7/15

N2 - A systematic strategy for stochastic mode reduction is applied here to three prototype "toy" models with nonlinear behavior mimicking several features of low-frequency variability in the extratropical atmosphere. Two of the models involve explicit stable periodic orbits and multiple equilibria in the projected nonlinear climate dynamics. The systematic strategy has two steps: stochastic consistency and stochastic mode elimination. Both aspects of the mode reduction strategy are tested in an a priori fashion in the paper. In all three models the stochastic mode elimination procedure applies in a quantitative fashion for moderately large values of ε ≈ 0.5 or even ε ≈ 1, where the parameter ε roughly measures the ratio of correlation times of unresolved variables to resolved climate variables, even though the procedure is only justified mathematically for ε « 1. The results developed here provide some new perspectives on both the role of stable nonlinear structures in projected nonlinear climate dynamics and the regression fitting strategies for stochastic climate modeling. In one example. a deterministic system with 102 degrees of freedom has an explicit stable periodic orbit for the projected climate dynamics in two variables; however, the complete deterministic system has instead a probability density function with two large isolated peaks on the "ghost" of this periodic orbit, and correlation functions that only weakly "shadow" this periodic orbit. Furthermore, all of these features are predicted in a quantitative fashion by the reduced stochastic model in two variables derived from the systematic theory; this reduced model has multiplicative noise and augmented nonlinearity. In a second deterministic model with 101 degrees of freedom, it is established that stable multiple equilibria in the projected climate dynamics can be either relevant or completely irrelevant in the actual dynamics for the climate variable depending on the strength of nonlinearity and the coupling to the unresolved variables. Furthermore, all this behavior is predicted in a quantitative fashion by a reduced nonlinear stochastic model for a single climate variable with additive noise, which is derived from the systematic mode reduction procedure. Finally, the systematic mode reduction strategy is applied in an idealized context to the stochastic modeling of the effect of mountain torque on the angular momentum budget. Surprisingly, the strategy yields a nonlinear stochastic equation for the large-scale fluctuations, and numerical simulations confirm significantly improved predicted correlation functions from this model compared with a standard linear model with damping and white noise forcing.

AB - A systematic strategy for stochastic mode reduction is applied here to three prototype "toy" models with nonlinear behavior mimicking several features of low-frequency variability in the extratropical atmosphere. Two of the models involve explicit stable periodic orbits and multiple equilibria in the projected nonlinear climate dynamics. The systematic strategy has two steps: stochastic consistency and stochastic mode elimination. Both aspects of the mode reduction strategy are tested in an a priori fashion in the paper. In all three models the stochastic mode elimination procedure applies in a quantitative fashion for moderately large values of ε ≈ 0.5 or even ε ≈ 1, where the parameter ε roughly measures the ratio of correlation times of unresolved variables to resolved climate variables, even though the procedure is only justified mathematically for ε « 1. The results developed here provide some new perspectives on both the role of stable nonlinear structures in projected nonlinear climate dynamics and the regression fitting strategies for stochastic climate modeling. In one example. a deterministic system with 102 degrees of freedom has an explicit stable periodic orbit for the projected climate dynamics in two variables; however, the complete deterministic system has instead a probability density function with two large isolated peaks on the "ghost" of this periodic orbit, and correlation functions that only weakly "shadow" this periodic orbit. Furthermore, all of these features are predicted in a quantitative fashion by the reduced stochastic model in two variables derived from the systematic theory; this reduced model has multiplicative noise and augmented nonlinearity. In a second deterministic model with 101 degrees of freedom, it is established that stable multiple equilibria in the projected climate dynamics can be either relevant or completely irrelevant in the actual dynamics for the climate variable depending on the strength of nonlinearity and the coupling to the unresolved variables. Furthermore, all this behavior is predicted in a quantitative fashion by a reduced nonlinear stochastic model for a single climate variable with additive noise, which is derived from the systematic mode reduction procedure. Finally, the systematic mode reduction strategy is applied in an idealized context to the stochastic modeling of the effect of mountain torque on the angular momentum budget. Surprisingly, the strategy yields a nonlinear stochastic equation for the large-scale fluctuations, and numerical simulations confirm significantly improved predicted correlation functions from this model compared with a standard linear model with damping and white noise forcing.

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U2 - 10.1175/1520-0469(2003)060<1705:SSFSMR>2.0.CO;2

DO - 10.1175/1520-0469(2003)060<1705:SSFSMR>2.0.CO;2

M3 - Article

AN - SCOPUS:0043127074

VL - 60

SP - 1705

EP - 1722

JO - Journals of the Atmospheric Sciences

JF - Journals of the Atmospheric Sciences

SN - 0022-4928

IS - 14

ER -