### Abstract

The projection operator formalism yields a time evolution equation for the time correlation function Un (t) of the chaotic modes of interest in terms of the memory function Γn (t). On the assumption of similarity between Un (t) and Γn (t), this equation leads to a closed equation for Un (t), which yields the asymptotic behavior of the time correlation function Un (t) and the corresponding power spectrum In (ω) analytically. Thus it turns out that the time correlation function takes the algebraic form 1/ (1+ t2) for t→0 as predicted previously, and can be classified into three decay forms for t→ according to the wave number kn: the exponential decay e-t, the oscillatory exponential decay e-t cost, and the oscillatory power-law decay t-3/2 cost. All the corresponding power spectra form a dual structure which is Lorentzian as ω→0 and decays exponentially as ω→. In the entire domain 0≤t<, solutions to the closed equation are quite consistent with the numerical results for small kn, while they are consistent with those for large kn, except for the phase. In the case that the integral time scale of Un (t) is equal to that of Γn (t), the closed equation is identical to the direct interaction approximation equation for fluid turbulence in the limit kn →.

Original language | English |
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Article number | 056312 |

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

Volume | 79 |

Issue number | 5 |

DOIs | |

Publication status | Published - May 26 2009 |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

**Time correlation functions in a similarity approximation for one-dimensional turbulence.** / Okamura, Makoto; Mori, Hazime.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Time correlation functions in a similarity approximation for one-dimensional turbulence

AU - Okamura, Makoto

AU - Mori, Hazime

PY - 2009/5/26

Y1 - 2009/5/26

N2 - The projection operator formalism yields a time evolution equation for the time correlation function Un (t) of the chaotic modes of interest in terms of the memory function Γn (t). On the assumption of similarity between Un (t) and Γn (t), this equation leads to a closed equation for Un (t), which yields the asymptotic behavior of the time correlation function Un (t) and the corresponding power spectrum In (ω) analytically. Thus it turns out that the time correlation function takes the algebraic form 1/ (1+ t2) for t→0 as predicted previously, and can be classified into three decay forms for t→ according to the wave number kn: the exponential decay e-t, the oscillatory exponential decay e-t cost, and the oscillatory power-law decay t-3/2 cost. All the corresponding power spectra form a dual structure which is Lorentzian as ω→0 and decays exponentially as ω→. In the entire domain 0≤t<, solutions to the closed equation are quite consistent with the numerical results for small kn, while they are consistent with those for large kn, except for the phase. In the case that the integral time scale of Un (t) is equal to that of Γn (t), the closed equation is identical to the direct interaction approximation equation for fluid turbulence in the limit kn →.

AB - The projection operator formalism yields a time evolution equation for the time correlation function Un (t) of the chaotic modes of interest in terms of the memory function Γn (t). On the assumption of similarity between Un (t) and Γn (t), this equation leads to a closed equation for Un (t), which yields the asymptotic behavior of the time correlation function Un (t) and the corresponding power spectrum In (ω) analytically. Thus it turns out that the time correlation function takes the algebraic form 1/ (1+ t2) for t→0 as predicted previously, and can be classified into three decay forms for t→ according to the wave number kn: the exponential decay e-t, the oscillatory exponential decay e-t cost, and the oscillatory power-law decay t-3/2 cost. All the corresponding power spectra form a dual structure which is Lorentzian as ω→0 and decays exponentially as ω→. In the entire domain 0≤t<, solutions to the closed equation are quite consistent with the numerical results for small kn, while they are consistent with those for large kn, except for the phase. In the case that the integral time scale of Un (t) is equal to that of Γn (t), the closed equation is identical to the direct interaction approximation equation for fluid turbulence in the limit kn →.

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U2 - 10.1103/PhysRevE.79.056312

DO - 10.1103/PhysRevE.79.056312

M3 - Article

VL - 79

JO - Physical Review E

JF - Physical Review E

SN - 2470-0045

IS - 5

M1 - 056312

ER -