Variational Principles and Applications of Local Topological Constants of Motion for Non-Barotropic Magnetohydrodynamics (original) (raw)
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Variational principles for magnetohydrodynamics were introduced by previous authors both in Lagrangian and Eulerian form. In a previous work Yahalom & Lynden-Bell introduced a simpler Eulerian variational principles from which all the relevant equations of magnetohydrodynamics can be derived. The variational principle was given in terms of six independent functions for non-stationary flows and three independent functions for stationary flows. This is less then the seven variables which appear in the standard equations of magnetohydrodynamics which are the magnetic field B the velocity field v and the density ρ. In this work I will improve on the previous results showing that non-stationary magnetohydrodynamics should be described by four functions .
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Arxiv preprint arXiv:1005.3977, 2010
Variational principles for magnetohydrodynamics were introduced by previous authors both in Lagrangian and Eulerian form. In previous works [1] Yahalom & Lynden-Bell and later Yahalom [2] introduced a simpler Eulerian variational principle from which all the relevant equations of Magnetohydrodynamics can be derived. The variational principles were given in terms of four independent functions for non-stationary flows and three independent functions for stationary flows. This is less than the seven variables which appear in the standard equations of magnetohydrodynamics which are the magnetic field B r , the velocity field v r and the density ρ . In the case that the magnetohydrodynamic flow has a non trivial topology such as when the magnetic lines are knotted or magnetic and stream lines are knotted, some of the functions appearing in the Lagrangian are non-single valued. Those functions play the same rule as the phase in the Aharonov-Bohm celebrated effect .
Barotropic magnetohydrodynamics as a four-function field theory
EPL, 2010
Variational principles for magnetohydrodynamics were introduced by previous authors both in Lagrangian and Eulerian form. In a previous work Yahalom & Lynden-Bell introduced a simpler Eulerian variational principles from which all the relevant equations of magnetohydrodynamics can be derived. The variational principle was given in terms of six independent functions for non-stationary flows and three independent functions for stationary flows. This is less then the seven variables which appear in the standard equations of magnetohydrodynamics which are the magnetic field B the velocity field v and the density ρ. In this work I will improve on the previous results showing that non-stationary magnetohydrodynamics should be described by four functions .
Journal of Fluid Mechanics, 1996
The techniques developed in Part 1 of the present series are here applied to two-dimensional solutions of the equations governing the magnetohydrodynamics of ideal incompressible fluids. We first demonstrate an isomorphism between such flows and the flow of a stratified fluid subjected to a field of force that we describe as ‘pseudo-gravitational’. We then construct a general Casimir as an integral of an arbitrary function of two conserved fields, namely the vector potential of the magnetic field, and the analogous potential of the ‘modified vorticity field’, the additional frozen field introduced in Part 1. Using this Casimir, a linear stability criterion is obtained by standard techniques. In §4, the (Arnold) techniques of nonlinear stability are developed, and bounds are placed on the second variation of the sum of the energy and the Casimir of the problem. This leads to criteria for nonlinear (Lyapunov) stability of the MHD flows considered. The appropriate norm is a sum of the ...
Canonical description of ideal magnetohydrodynamic flows and integrals of motion
Physical Review E, 2004
In the framework of the variational principle there are introduced canonical variables describing magnetohydrodynamic (MHD) flows of general type without any restrictions for invariants of the motion. It is shown that the velocity representation of the Clebsch type introduced by means of the variational principle with constraints is equivalent to the representation following from the generalization of the Weber transformation for the case of arbitrary MHD flows. The integrals of motion and local invariants for MHD are under examination. It is proved that there exists generalization of the Ertel invariant. It is expressed in terms of generalized vorticity field (discussed earlier by Vladimirov and Moffatt (V. A. Vladimirov, H. K. Moffatt, J. Fl. Mech., 283, pp. 125-139, 1995) for the incompressible case). The generalized vorticity presents the frozen-in field for the barotropic and isentropic flows and therefore for these flows there exists generalized helicity invariant. This result generalizes one obtained by Vladimirov and Moffatt in the cited work for the incompressible fluid. It is shown that to each invariant of the conventional hydrodynamics corresponds MHD invariant and therefore our approach allows correct limit transition to the conventional hydrodynamic case. The additional advantage of the approach proposed enables one to deal with discontinuous flows, including all types of possible breaks.
Canonical description of ideal magnetohydrodynamics and integrals of motion
2002
In the framework of the variational principle there are introduced canonical variables describing magnetohydrodynamic (MHD) flows of general type without any restrictions for invariants of the motion. It is shown that the velocity representation of the Clebsch type introduced by means of the variational principle with constraints is equivalent to the representation following from the generalization of the Weber transformation for the case of arbitrary MHD flows. The integrals of motion and local invariants for MHD are under examination. It is proved that there exists generalization of the Ertel invariant. It is expressed in terms of generalized vorticity field (discussed earlier by Vladimirov and Moffatt (V. A. Vladimirov, H. K. Moffatt, J. Fl. Mech., 283, pp. 125-139, 1995) for the incompressible case). The generalized vorticity presents the frozen-in field for the barotropic and isentropic flows and therefore for these flows there exists generalized helicity invariant. This result generalizes one obtained by Vladimirov and Moffatt in the cited work for the incompressible fluid. It is shown that to each invariant of the conventional hydrodynamics corresponds MHD invariant and therefore our approach allows correct limit transition to the conventional hydrodynamic case. The additional advantage of the approach proposed enables one to deal with discontinuous flows, including all types of possible breaks.