### Abstract

Multiple-scales perturbation methods are used to study wave interactions in magnetohydrodynamics (MHD), in one Cartesian space dimension, with application to cosmic-ray-modified shocks. In particular, the problem of the propagation and interaction of short wavelength MHD waves, in a large-scale background flow, modified by cosmic rays is studied. The wave interaction equations consist of seven coupled evolution equations for the backward and forward Alfven waves, the backward and forward fast and slow magnetoacoustic waves and the entropy wave. In the linear wave regime, the waves are coupled by wave mixing due to gradients in the background flow, cosmic-ray squeezing instability effects, and damping due to the diffusing cosmic rays. In the most general case, the evolution equations also contain nonlinear wave interaction terms due to Burgers self wave steepening for the magnetoacoustic modes, resonant three wave interactions, and mean wave field interaction terms. The form of the wave interaction equations in the ideal MHD case is also discussed. Numerical simulations of the fully nonlinear cosmic ray MHD model equations are compared with spectral code solutions of the linear wave interaction equations for the case of perpendicular, cosmic-ray-modified shocks. The solutions are used to illustrate how the different wave modes can be generated by wave mixing, and the modification of the cosmic ray squeezing instability due to wave interactions. It is shown that the Alfven waves are coupled to the magnetoacoustic and entropy waves due to linear wave mixing, only in background flows with non-zero field aligned electric current and/or vorticity (i.e. if B · Δ × B ≠ 0 and/or B · Δ × u ≠ 0, where B and u are the magnetic field induction and fluid velocity respectively).

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

Pages (from-to) | 295-346 |

Number of pages | 52 |

Journal | Journal of Plasma Physics |

Volume | 61 |

Issue number | 2 |

DOIs | |

State | Published - Feb 1999 |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Condensed Matter Physics

### Cite this

*Journal of Plasma Physics*,

*61*(2), 295-346. https://doi.org/10.1017/S0022377898007399

**Wave interactions in magnetohydrodynamics, and cosmic-ray-modified shocks.** / Webb, G. M.; Zakharian, A.; Brio, Moysey; Zank, G. P.

Research output: Contribution to journal › Article

*Journal of Plasma Physics*, vol. 61, no. 2, pp. 295-346. https://doi.org/10.1017/S0022377898007399

}

TY - JOUR

T1 - Wave interactions in magnetohydrodynamics, and cosmic-ray-modified shocks

AU - Webb, G. M.

AU - Zakharian, A.

AU - Brio, Moysey

AU - Zank, G. P.

PY - 1999/2

Y1 - 1999/2

N2 - Multiple-scales perturbation methods are used to study wave interactions in magnetohydrodynamics (MHD), in one Cartesian space dimension, with application to cosmic-ray-modified shocks. In particular, the problem of the propagation and interaction of short wavelength MHD waves, in a large-scale background flow, modified by cosmic rays is studied. The wave interaction equations consist of seven coupled evolution equations for the backward and forward Alfven waves, the backward and forward fast and slow magnetoacoustic waves and the entropy wave. In the linear wave regime, the waves are coupled by wave mixing due to gradients in the background flow, cosmic-ray squeezing instability effects, and damping due to the diffusing cosmic rays. In the most general case, the evolution equations also contain nonlinear wave interaction terms due to Burgers self wave steepening for the magnetoacoustic modes, resonant three wave interactions, and mean wave field interaction terms. The form of the wave interaction equations in the ideal MHD case is also discussed. Numerical simulations of the fully nonlinear cosmic ray MHD model equations are compared with spectral code solutions of the linear wave interaction equations for the case of perpendicular, cosmic-ray-modified shocks. The solutions are used to illustrate how the different wave modes can be generated by wave mixing, and the modification of the cosmic ray squeezing instability due to wave interactions. It is shown that the Alfven waves are coupled to the magnetoacoustic and entropy waves due to linear wave mixing, only in background flows with non-zero field aligned electric current and/or vorticity (i.e. if B · Δ × B ≠ 0 and/or B · Δ × u ≠ 0, where B and u are the magnetic field induction and fluid velocity respectively).

AB - Multiple-scales perturbation methods are used to study wave interactions in magnetohydrodynamics (MHD), in one Cartesian space dimension, with application to cosmic-ray-modified shocks. In particular, the problem of the propagation and interaction of short wavelength MHD waves, in a large-scale background flow, modified by cosmic rays is studied. The wave interaction equations consist of seven coupled evolution equations for the backward and forward Alfven waves, the backward and forward fast and slow magnetoacoustic waves and the entropy wave. In the linear wave regime, the waves are coupled by wave mixing due to gradients in the background flow, cosmic-ray squeezing instability effects, and damping due to the diffusing cosmic rays. In the most general case, the evolution equations also contain nonlinear wave interaction terms due to Burgers self wave steepening for the magnetoacoustic modes, resonant three wave interactions, and mean wave field interaction terms. The form of the wave interaction equations in the ideal MHD case is also discussed. Numerical simulations of the fully nonlinear cosmic ray MHD model equations are compared with spectral code solutions of the linear wave interaction equations for the case of perpendicular, cosmic-ray-modified shocks. The solutions are used to illustrate how the different wave modes can be generated by wave mixing, and the modification of the cosmic ray squeezing instability due to wave interactions. It is shown that the Alfven waves are coupled to the magnetoacoustic and entropy waves due to linear wave mixing, only in background flows with non-zero field aligned electric current and/or vorticity (i.e. if B · Δ × B ≠ 0 and/or B · Δ × u ≠ 0, where B and u are the magnetic field induction and fluid velocity respectively).

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U2 - 10.1017/S0022377898007399

DO - 10.1017/S0022377898007399

M3 - Article

AN - SCOPUS:0032672380

VL - 61

SP - 295

EP - 346

JO - Journal of Plasma Physics

JF - Journal of Plasma Physics

SN - 0022-3778

IS - 2

ER -