Canonical description of ideal magnetohydrodynamic flows and integrals of motion

Abstract

In the framework of the variational principle the canonical variables describing magnetohydrodynamic (MHD) flows of general type (i.e., with spatially varying entropy and nonzero values of all topological invariants) are introduced. It is shown that the velocity representation of the Clebsch type following from the variational principle with constraints is equivalent to that resulting from the generalization of the Weber transformation performed in the paper for the case of arbitrary MHD flows. Using such complete velocity representation enables us not only to describe the general type flows in terms of single-valued functions, but also to solve the intriguing problem of the "missing" MHD integrals of motion. The set of hitherto known MHD local invariants and integrals of motion appears to be incomplete: for the vanishing magnetic field it does not reduce to the set of the conventional hydrodynamic invariants. And if the analogs of the vorticity and helicity were discussed earlier for the particular cases, the analog of Ertel invariant has been so far unknown. It is shown that all "missing" invariants are expressed in terms of the decomposition of the velocity representation into the "hydrodynamic" and "magnetic" parts. In spite of the nonunique character of such representation it is shown that there exists a natural restriction of the gauge transformations set allowing one to make the invariants gauge independent. It is found that on the basis of the new invariants introduced a wide set of high-order invariants can be constructed. The new invariants are relevant both for the deeper insight into the problem of the topological structure of the MHD flows as a whole and for the examination of the stability problems. The additional advantage of the proposed approach is that it enables one to deal with discontinuous flows, including all types of possible breaks.

Simplified variational principles for barotropic magnetohydrodynamics

“…It can be easily shown that provided that B is in the form given in equation (17), and equation (18) is satisfied, then both equation (1) and equation (2) are satisfied. Since according to equation (16) both ∇ × B and B are parallel it follows that equation (16) can be written as:…”

“…The authors of this paper suspect that this number can be somewhat reduced. Moreover, A. V. Kats in a remarkable paper [17] (section IV,E) has shown that there is a large symmetry group (gauge freedom) associated with the choice of those functions, this implies that the number of degrees of freedom can be reduced.…”

Simplified variational principles for non-barotropic magnetohydrodynamics

“…From now on the derivation proceeds as in equations (14,15,17) resulting in equations (16,18) and will not be repeated. The difference is that now α, β and σ are not independent quantities, rather they depend through equation (32) on the derivatives of χ i , ν.…”

“…The difference is that now α, β and σ are not independent quantities, rather they depend through equation (32) on the derivatives of χ i , ν. Thus, equations (14,15,17) are not first order equations in time but are second order equations. Now let us calculate the variational derivative with respect to ν this will result in the expression:…”

“…The author of this paper suspect that this number can be somewhat reduced. Moreover, A. V. Kats in a remarkable paper [14] (section IV,E) has shown that there is a large symmetry group (gauge freedom) associated with the choice of those functions, this implies that the number of degrees of freedom can be reduced. Here we will show that only five functions will suffice to describe non barotropic magnetohydrodynamics in the case that we enforce a Sakurai [6] representation for the magnetic field.…”