TY - JOUR

T1 - On general transformations and variational principles for the magnetohydrodynamics of ideal fluids. Part 4. Generalized isovorticity principle for three-dimensional flows

AU - Vladimirov, V. A.

AU - Moffatt, H. K.

AU - Ilin, K. I.

PY - 1999/7/10

Y1 - 1999/7/10

N2 - 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 δ2E is constructed. It is shown that δ2E is a quadratic functional of the first-order variations δ1u, δ1h of u and h (from a steady state U(x), H(X)), and that δ2E is an invariant of the linearized MHD equations. Linear stability is then assured provided δ2E 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 1 of this 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.

AB - 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 δ2E is constructed. It is shown that δ2E is a quadratic functional of the first-order variations δ1u, δ1h of u and h (from a steady state U(x), H(X)), and that δ2E is an invariant of the linearized MHD equations. Linear stability is then assured provided δ2E 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 1 of this 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.

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U2 - 10.1017/S0022112099004991

DO - 10.1017/S0022112099004991

M3 - Article

AN - SCOPUS:0032662208

SN - 0022-1120

VL - 390

SP - 127

EP - 150

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

ER -