### Abstract

We derive the closed system of averaged magnetohydrodynamic (MHD) equations for general oscillating flows. The used small parameter of our asymptotic theory is the dimensionless inverse frequency, and the leading term for a velocity field is chosen to be purely oscillating. The employed mathematical approach combines the two-timing method and the notion of a distinguished limit. The properties of commutators are used to simplify calculations. The derived averaged equations are similar to the original MHD equations, but surprisingly (instead of the commonly expected Reynolds stresses) a drift velocity plays a part of an additional advection velocity. In the special case of a vanishing magnetic field h=0, the averaged equations produce the Craik-Leibovich equations for Langmuir circulations (which can be called 'vortex dynamo'). We suggest that, since the mathematical structure of the full averaged equations for h ≠ 0 is similar to those for h = 0, these full equations could lead to a possible mechanism of MHD dynamo, such as the generation of the magnetic field of the Earth.

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

Pages (from-to) | 51-61 |

Number of pages | 11 |

Journal | Journal of Fluid Mechanics |

Volume | 698 |

DOIs | |

Publication status | Published - May 10 2012 |

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### Keywords

- dynamo theory
- general fluid mechanics
- magnetohydrodynamics

### ASJC Scopus subject areas

- Condensed Matter Physics
- Mechanics of Materials
- Mechanical Engineering

### Cite this

**Magnetohydrodynamic drift equations : From Langmuir circulations to magnetohydrodynamic dynamo?** / Vladimirov, V. A.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Magnetohydrodynamic drift equations

T2 - From Langmuir circulations to magnetohydrodynamic dynamo?

AU - Vladimirov, V. A.

PY - 2012/5/10

Y1 - 2012/5/10

N2 - We derive the closed system of averaged magnetohydrodynamic (MHD) equations for general oscillating flows. The used small parameter of our asymptotic theory is the dimensionless inverse frequency, and the leading term for a velocity field is chosen to be purely oscillating. The employed mathematical approach combines the two-timing method and the notion of a distinguished limit. The properties of commutators are used to simplify calculations. The derived averaged equations are similar to the original MHD equations, but surprisingly (instead of the commonly expected Reynolds stresses) a drift velocity plays a part of an additional advection velocity. In the special case of a vanishing magnetic field h=0, the averaged equations produce the Craik-Leibovich equations for Langmuir circulations (which can be called 'vortex dynamo'). We suggest that, since the mathematical structure of the full averaged equations for h ≠ 0 is similar to those for h = 0, these full equations could lead to a possible mechanism of MHD dynamo, such as the generation of the magnetic field of the Earth.

AB - We derive the closed system of averaged magnetohydrodynamic (MHD) equations for general oscillating flows. The used small parameter of our asymptotic theory is the dimensionless inverse frequency, and the leading term for a velocity field is chosen to be purely oscillating. The employed mathematical approach combines the two-timing method and the notion of a distinguished limit. The properties of commutators are used to simplify calculations. The derived averaged equations are similar to the original MHD equations, but surprisingly (instead of the commonly expected Reynolds stresses) a drift velocity plays a part of an additional advection velocity. In the special case of a vanishing magnetic field h=0, the averaged equations produce the Craik-Leibovich equations for Langmuir circulations (which can be called 'vortex dynamo'). We suggest that, since the mathematical structure of the full averaged equations for h ≠ 0 is similar to those for h = 0, these full equations could lead to a possible mechanism of MHD dynamo, such as the generation of the magnetic field of the Earth.

KW - dynamo theory

KW - general fluid mechanics

KW - magnetohydrodynamics

UR - http://www.scopus.com/inward/record.url?scp=84859977826&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84859977826&partnerID=8YFLogxK

U2 - 10.1017/jfm.2012.40

DO - 10.1017/jfm.2012.40

M3 - Article

VL - 698

SP - 51

EP - 61

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

SN - 0022-1120

ER -