On general transformations and variational principles for the magnetohydrodynamics of ideal fluids. IV: Generalized isovorticity principle for three-dimensional flows. (English) Zbl 1003.76096
[For part I see the first two authors, ibid. 283, 125-139 (1995; Zbl 0834.76095); for part II see the authors, ibid. 329, 187-205 (1996; Zbl 0891.76093); for part III see the authors, J. Plasma Phys. 57, No. 3, 89-120 (1997).]
Summary: The equations of magnetohydrodynamics (MHD) of an ideal fluid have two families of topological invariants: the magnetic helicity invariants and the cross-helicity invariants. It is first shown that these invariants define a natural foliation (described as isomagnetovortical, or imv for short) in the function space in which solutions \(\{u(x,t), h(x,t)\}\) of the MHD equations reside. A relaxation process is constructed whereby total energy (magnetic plus kinetic) decreases on an imv folium (all magnetic and cross-helicity invariants being thus conserved). The energy has a positive lower bound determined by the global cross-helicity, and it is thus shown that a steady state exists having the (arbitrarily) prescribed families of magnetic and cross-helicity invariants.
The stability of such steady states is considered by an appropriate generalization of (Arnold) energy techniques. The first variation of energy on the imv folium is shown to vanish, and the second variation \(\delta^2E\) is constructed. It is shown that \(\delta^2E\) is a quadratic functional of the first-order variations \(\delta^1u\), \(\delta^1h\) of \(u\) and \(h\) (from a steady state \(U(x)\), \(H(x))\), and that \(\delta^2E\) is an invariant of the linearized MHD equations. Linear stability is then assured provided \(\delta^2E\) is either positive-definite or negative-definite for all imv perturbations. It is shown that the results may be equivalently obtained through consideration of the frozen-in ‘modified’ vorticity field introduced in part I of his series.
Finally, the general stability criterion is applied to a variety of classes of steady states \(\{U(x)\), \(H(x)\}\), and new sufficient conditions for stability to three-dimensional imv perturbations are obtained.
Summary: The equations of magnetohydrodynamics (MHD) of an ideal fluid have two families of topological invariants: the magnetic helicity invariants and the cross-helicity invariants. It is first shown that these invariants define a natural foliation (described as isomagnetovortical, or imv for short) in the function space in which solutions \(\{u(x,t), h(x,t)\}\) of the MHD equations reside. A relaxation process is constructed whereby total energy (magnetic plus kinetic) decreases on an imv folium (all magnetic and cross-helicity invariants being thus conserved). The energy has a positive lower bound determined by the global cross-helicity, and it is thus shown that a steady state exists having the (arbitrarily) prescribed families of magnetic and cross-helicity invariants.
The stability of such steady states is considered by an appropriate generalization of (Arnold) energy techniques. The first variation of energy on the imv folium is shown to vanish, and the second variation \(\delta^2E\) is constructed. It is shown that \(\delta^2E\) is a quadratic functional of the first-order variations \(\delta^1u\), \(\delta^1h\) of \(u\) and \(h\) (from a steady state \(U(x)\), \(H(x))\), and that \(\delta^2E\) is an invariant of the linearized MHD equations. Linear stability is then assured provided \(\delta^2E\) is either positive-definite or negative-definite for all imv perturbations. It is shown that the results may be equivalently obtained through consideration of the frozen-in ‘modified’ vorticity field introduced in part I of his series.
Finally, the general stability criterion is applied to a variety of classes of steady states \(\{U(x)\), \(H(x)\}\), and new sufficient conditions for stability to three-dimensional imv perturbations are obtained.
