Variational principles for magnetohydrodynamics were introduced by previous authors both in Lagrangian and Eulerian form. Previously Yahalom & Lynden-Bell (2008) have introduced simpler Eulerian variational principles from which all the relevant equations of barotropic magnetohydrodynamics can be derived. The variational principle was given in terms of six independent functions for non-stationary barotropic flows with given topologies and three independent functions for stationary barotropic flows. This is less than the seven variables which appear in the standard equations of barotropic magnetohydrodynamics which are the magnetic field B the velocity field v and the density. Later Yahalom (2010) have introduced a simpler variational principle in terms of four functions for non-stationary barotropic magnetohydrodynamics. It was shown that the above variational principles are also relevant for flows of non-trivial topologies and in fact using those variational variables one arrives at additional topological conservation laws in terms of cuts of variables which have close resemblance to the Aharonov- Bohm phase (Yahalom (2013)). In previous examples (Yahalom & Lynden-Bell (2008); Yahalom (2013)) the magnetic field lines with non-trivial topology were at the intersection of two surface one which was always multivalued, here an example is introduced in which the magnetic helicity is not zero yet both surfaces are single valued.

Bibliography

Yahalom, A. & Lynden-Bell, D. 2008 Simplied variational principles for barotropic magnetohydrodynamics. Journal of Fluid Mechanics 607, 235-265.

Yahalom, A. 2010 A four function variational principle for barotropic magnetohydrodynamics. EPL 89, 34005.

Yahalom, A. 2013 Aharonov-Bohm effects in magnetohydrodynamics. Physics Letters A 377 (31-33), 1898-1904.