### Abstract

The decay form of the time correlation function Un (t) of a state variable un (t) with a small wave number kn has been shown to take the algebraic decay 1/ { 1+ (γna t) 2 } in the initial regime t< τn (γ) and the exponential decay αne exp (- γne t) in the final regime t> τn (γ), where τn (γ) denotes the decay time of the memory function Γn (t). This dual structure of Un (t) is generated by the deterministic short orbits in the initial regime and the stochastic long orbits in the final regime, thus giving the outstanding features of chaos and turbulence. The kn dependence of γna, αne, and γne is obtained for the chaotic Kuramoto-Sivashinsky equation, and it is shown that if kn is sufficiently small, then the dual structure of Un (t) obeys a hydrodynamic scaling law in the final regime t> τn (γ) with scaling exponent z=2 and a dynamic scaling law in the initial regime t< τn (γ) with scaling exponent z=1. If kn is increased so that the decay time τn (u) of Un (t) becomes equal to the decay time τn (γ), then the decay form of Un (t) becomes the power-law decay t-3/2 in the final regime.

Original language | English |
---|---|

Article number | 051124 |

Journal | Physical Review E - Statistical, Nonlinear, and Soft Matter Physics |

Volume | 80 |

Issue number | 5 |

DOIs | |

Publication status | Published - Nov 24 2009 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Statistics and Probability
- Condensed Matter Physics

### Cite this

**Dual structures of chaos and turbulence, and their dynamic scaling laws.** / Mori, Hazime; Okamura, Makoto.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Dual structures of chaos and turbulence, and their dynamic scaling laws

AU - Mori, Hazime

AU - Okamura, Makoto

PY - 2009/11/24

Y1 - 2009/11/24

N2 - The decay form of the time correlation function Un (t) of a state variable un (t) with a small wave number kn has been shown to take the algebraic decay 1/ { 1+ (γna t) 2 } in the initial regime t< τn (γ) and the exponential decay αne exp (- γne t) in the final regime t> τn (γ), where τn (γ) denotes the decay time of the memory function Γn (t). This dual structure of Un (t) is generated by the deterministic short orbits in the initial regime and the stochastic long orbits in the final regime, thus giving the outstanding features of chaos and turbulence. The kn dependence of γna, αne, and γne is obtained for the chaotic Kuramoto-Sivashinsky equation, and it is shown that if kn is sufficiently small, then the dual structure of Un (t) obeys a hydrodynamic scaling law in the final regime t> τn (γ) with scaling exponent z=2 and a dynamic scaling law in the initial regime t< τn (γ) with scaling exponent z=1. If kn is increased so that the decay time τn (u) of Un (t) becomes equal to the decay time τn (γ), then the decay form of Un (t) becomes the power-law decay t-3/2 in the final regime.

AB - The decay form of the time correlation function Un (t) of a state variable un (t) with a small wave number kn has been shown to take the algebraic decay 1/ { 1+ (γna t) 2 } in the initial regime t< τn (γ) and the exponential decay αne exp (- γne t) in the final regime t> τn (γ), where τn (γ) denotes the decay time of the memory function Γn (t). This dual structure of Un (t) is generated by the deterministic short orbits in the initial regime and the stochastic long orbits in the final regime, thus giving the outstanding features of chaos and turbulence. The kn dependence of γna, αne, and γne is obtained for the chaotic Kuramoto-Sivashinsky equation, and it is shown that if kn is sufficiently small, then the dual structure of Un (t) obeys a hydrodynamic scaling law in the final regime t> τn (γ) with scaling exponent z=2 and a dynamic scaling law in the initial regime t< τn (γ) with scaling exponent z=1. If kn is increased so that the decay time τn (u) of Un (t) becomes equal to the decay time τn (γ), then the decay form of Un (t) becomes the power-law decay t-3/2 in the final regime.

UR - http://www.scopus.com/inward/record.url?scp=71849095351&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=71849095351&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.80.051124

DO - 10.1103/PhysRevE.80.051124

M3 - Article

C2 - 20364964

AN - SCOPUS:71849095351

VL - 80

JO - Physical Review E

JF - Physical Review E

SN - 2470-0045

IS - 5

M1 - 051124

ER -