### Abstract

A shell model is introduced to study a turbulence driven by the thermal instability (Rayleigh-Bénard convection). This model equation describes cascade and chaos in the strong turbulence with high Rayleigh number. The chaos is numerically studied based on this model. The characteristics of the turbulence are analyzed and compared with those of the Gledzer-Ohkitani-Yamada (GOY) model. Quantities such as a mean value of total fluctuation energy, it's standard deviation, time averaged wave spectrum, probability distribution function, frequency spectrum, the maximum instantaneous Lyapunov exponent, distribution of instantaneous Lyapunov exponents, are evaluated. The dependences of these quantities on the error of numerical integration are also examined. There is not a clear correlation between the numerical accuracy and the accuracy of these quantities, since the interaction between a truncation error and an intrinsic nonlinearity of the system exists. A finding is that the maximum Lyapunov exponent is insensitive to a truncation error.

Original language | English |
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Pages (from-to) | 393-402 |

Number of pages | 10 |

Journal | Chaos |

Volume | 9 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jun 1999 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics
- Physics and Astronomy(all)
- Applied Mathematics

### Cite this

*Chaos*,

*9*(2), 393-402. https://doi.org/10.1063/1.166416

**Numerical study of chaos based on a shell model.** / Yagi, M.; Itoh, S. I.; Itoh, K.; Fukuyama, A.

Research output: Contribution to journal › Article

*Chaos*, vol. 9, no. 2, pp. 393-402. https://doi.org/10.1063/1.166416

}

TY - JOUR

T1 - Numerical study of chaos based on a shell model

AU - Yagi, M.

AU - Itoh, S. I.

AU - Itoh, K.

AU - Fukuyama, A.

PY - 1999/6

Y1 - 1999/6

N2 - A shell model is introduced to study a turbulence driven by the thermal instability (Rayleigh-Bénard convection). This model equation describes cascade and chaos in the strong turbulence with high Rayleigh number. The chaos is numerically studied based on this model. The characteristics of the turbulence are analyzed and compared with those of the Gledzer-Ohkitani-Yamada (GOY) model. Quantities such as a mean value of total fluctuation energy, it's standard deviation, time averaged wave spectrum, probability distribution function, frequency spectrum, the maximum instantaneous Lyapunov exponent, distribution of instantaneous Lyapunov exponents, are evaluated. The dependences of these quantities on the error of numerical integration are also examined. There is not a clear correlation between the numerical accuracy and the accuracy of these quantities, since the interaction between a truncation error and an intrinsic nonlinearity of the system exists. A finding is that the maximum Lyapunov exponent is insensitive to a truncation error.

AB - A shell model is introduced to study a turbulence driven by the thermal instability (Rayleigh-Bénard convection). This model equation describes cascade and chaos in the strong turbulence with high Rayleigh number. The chaos is numerically studied based on this model. The characteristics of the turbulence are analyzed and compared with those of the Gledzer-Ohkitani-Yamada (GOY) model. Quantities such as a mean value of total fluctuation energy, it's standard deviation, time averaged wave spectrum, probability distribution function, frequency spectrum, the maximum instantaneous Lyapunov exponent, distribution of instantaneous Lyapunov exponents, are evaluated. The dependences of these quantities on the error of numerical integration are also examined. There is not a clear correlation between the numerical accuracy and the accuracy of these quantities, since the interaction between a truncation error and an intrinsic nonlinearity of the system exists. A finding is that the maximum Lyapunov exponent is insensitive to a truncation error.

UR - http://www.scopus.com/inward/record.url?scp=0005510660&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0005510660&partnerID=8YFLogxK

U2 - 10.1063/1.166416

DO - 10.1063/1.166416

M3 - Article

AN - SCOPUS:0005510660

VL - 9

SP - 393

EP - 402

JO - Chaos

JF - Chaos

SN - 1054-1500

IS - 2

ER -