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Variational formulation for weakly nonlinear perturbations of ideal magnetohydrodynamics

理想磁気流体力学の弱非線形摂動に対する変分原理の定式化

廣田 真

Hirota, Makoto

理想磁気流体力学(MHD)の弱非線形現象を支配する新たな運動方程式を、よく知られた線形の運動方程式の自然な拡張として導いた。この導出はMHDに対するラグランジアンを、プラズマの変位に関して三次オーダーまで摂動展開することによって可能となるが、それにはLie級数展開の応用が必要となる。得られた運動方程式は二次の非線形項を含み、これがモード間の結合を引き起こす。その際、三次オーダーのポテンシャルエネルギーはモード間の結合の強さ(結合係数)を評価するための指標となる。従来の理論と比べると、結合係数は線形理論で馴染みの深い変位ベクトル場を用いて表現され、固定境界と自由境界の場合を両方扱うことができる。

A new equation of motion that governs weakly nonlinear phenomena inideal magnetohydrodynamics (MHD) is derived as a natural extension of the well-known linearized equation of motion for the displacement field. This derivation is made possible by expanding the MHD Lagrangianexplicitly up to third order with respect to the displacement of plasma, which necessitates an efficient use of the Lie series expansion. The resultant equation of motion (i.e., the Euler-Lagrange equation) includes a new quadratic force term which is responsible for various mode-mode coupling due to the MHD nonlinearity. The third-order potential energy serves to quantify the coupling coefficient among resonant three modes and its cubic symmetry proves the Manley-Rower elations. In contrast to earlier works, the coupling coefficient is expressed only by the displacement vector field, which is already familiar in the linear MHD theory, and both the fixed and free boundary cases are treated systematically.

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パーセンタイル:23.46

分野:Physics, Fluids & Plasmas

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