### Abstract

We consider an example of strongly non-local interaction in incompressible magnetohydrodynamic (MHD) turbulence which corresponds to the case where the Alfvén waves travelling in the opposite directions have essentially different characteristic wavelengths. We use two approaches to the dynamics of turbulent Alfvénic wavepackets: the first is a geometrical WKB theory [Phys. Lett. A 165 (1992) 330] and the second one is a three-wave kinetic equation derived for weakly turbulent waves [J. Plasma Phys., in press]. We show that these theories have a common limit of weak turbulence with scale separation in which they both predict the same Fokker-Planck equation for the wave power spectrum. In both cases the packet wavenumbers (and therefore the Lagrangian field-line separations) are allowed to experience order 1 changes. The WKB theory developed here formalises an intuitive geometrical argument of Goldreich and Sridhar [ApJ 485 (1997) 680] and allows one to see where such an intuition leads to a wrong conclusion about the inapplicability of the three-wave kinetic equation for order 1 wavepacket distortions. We show that the exponent of the constant flux non-local spectrum matches the value previously found for local turbulence at the boundary of the locality interval. The relationship between the WKB theory and the weak turbulence theory found in this paper for an ensemble of Alfvén waves seems to be general for three-wave systems.

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

Pages (from-to) | 646-652 |

Number of pages | 7 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 152-153 |

DOIs | |

State | Published - May 15 2001 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Applied Mathematics
- Statistical and Nonlinear Physics

### Cite this

*Physica D: Nonlinear Phenomena*,

*152-153*, 646-652. https://doi.org/10.1016/S0167-2789(01)00197-X

**Non-local MHD turbulence.** / Nazarenko, S. V.; Newell, Alan C; Galtier, S.

Research output: Contribution to journal › Article

*Physica D: Nonlinear Phenomena*, vol. 152-153, pp. 646-652. https://doi.org/10.1016/S0167-2789(01)00197-X

}

TY - JOUR

T1 - Non-local MHD turbulence

AU - Nazarenko, S. V.

AU - Newell, Alan C

AU - Galtier, S.

PY - 2001/5/15

Y1 - 2001/5/15

N2 - We consider an example of strongly non-local interaction in incompressible magnetohydrodynamic (MHD) turbulence which corresponds to the case where the Alfvén waves travelling in the opposite directions have essentially different characteristic wavelengths. We use two approaches to the dynamics of turbulent Alfvénic wavepackets: the first is a geometrical WKB theory [Phys. Lett. A 165 (1992) 330] and the second one is a three-wave kinetic equation derived for weakly turbulent waves [J. Plasma Phys., in press]. We show that these theories have a common limit of weak turbulence with scale separation in which they both predict the same Fokker-Planck equation for the wave power spectrum. In both cases the packet wavenumbers (and therefore the Lagrangian field-line separations) are allowed to experience order 1 changes. The WKB theory developed here formalises an intuitive geometrical argument of Goldreich and Sridhar [ApJ 485 (1997) 680] and allows one to see where such an intuition leads to a wrong conclusion about the inapplicability of the three-wave kinetic equation for order 1 wavepacket distortions. We show that the exponent of the constant flux non-local spectrum matches the value previously found for local turbulence at the boundary of the locality interval. The relationship between the WKB theory and the weak turbulence theory found in this paper for an ensemble of Alfvén waves seems to be general for three-wave systems.

AB - We consider an example of strongly non-local interaction in incompressible magnetohydrodynamic (MHD) turbulence which corresponds to the case where the Alfvén waves travelling in the opposite directions have essentially different characteristic wavelengths. We use two approaches to the dynamics of turbulent Alfvénic wavepackets: the first is a geometrical WKB theory [Phys. Lett. A 165 (1992) 330] and the second one is a three-wave kinetic equation derived for weakly turbulent waves [J. Plasma Phys., in press]. We show that these theories have a common limit of weak turbulence with scale separation in which they both predict the same Fokker-Planck equation for the wave power spectrum. In both cases the packet wavenumbers (and therefore the Lagrangian field-line separations) are allowed to experience order 1 changes. The WKB theory developed here formalises an intuitive geometrical argument of Goldreich and Sridhar [ApJ 485 (1997) 680] and allows one to see where such an intuition leads to a wrong conclusion about the inapplicability of the three-wave kinetic equation for order 1 wavepacket distortions. We show that the exponent of the constant flux non-local spectrum matches the value previously found for local turbulence at the boundary of the locality interval. The relationship between the WKB theory and the weak turbulence theory found in this paper for an ensemble of Alfvén waves seems to be general for three-wave systems.

UR - http://www.scopus.com/inward/record.url?scp=18244420829&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=18244420829&partnerID=8YFLogxK

U2 - 10.1016/S0167-2789(01)00197-X

DO - 10.1016/S0167-2789(01)00197-X

M3 - Article

AN - SCOPUS:18244420829

VL - 152-153

SP - 646

EP - 652

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

ER -