Abstract
The current paper is devoted to the introduction of simpler Eulerian variational principles from which all the relevant equations of nonbarotropic stationary magnetohydrodynamics can be derived for magnetic fields that lie on surfaces. A variational principle is given in terms of three independent variables for stationary nonbarotropic magnetohydrodynamic flows. This is a smaller number of variables than the eight variables that appear in the standard equations of nonbarotropic magnetohydrodynamics, which are the magnetic field, the velocity field, the specific entropy, and the density. We further investigate the case in which the flow along magnetic lines is not ideal.
Highlights
Variational principles for magnetohydrodynamics (MHD) were introduced previously in both Lagrangian and Eulerian approaches
Vladimirov and Moffatt [2] in a list of works studied a Eulerian variational approach for incompressible MHD. Their variational approach contained three more variables in addition to the seven functions that appear in the standard equations of incompressible MHD, which are the magnetic field ~B, velocity field ~v, and pressure P
Yahalom and Lynden-Bell [8] combined the work of Sturrock [1] with the work of Sakurai [6] to obtain a Eulerian variational principle for barotropic MHD, which depends on only six variables
Summary
Variational principles for magnetohydrodynamics (MHD) were introduced previously in both Lagrangian and Eulerian approaches. A subsequent paper [18] was concerned with stationary flows and introduced and eight-variable stationary variational principle; here, we shall attempt to improve on this and obtain a three-variable stationary variational principle for nonbarotropic MHD This will be done for a general case in which the magnetic field lines need not lie on entropy surfaces; for the restricted case in which the magnetic field lines lie on entropy surfaces, see [19]. Applications of this paper may arise for both linear and nonlinear stability analysis of stationary nonbarotropic MHD flows [20,21] and for designing numerical algorithms for integrating the equations of MHD [22,23,24] Another possible application is connected to obtaining new analytic solutions in terms of the variational functions [25], as will be described below.
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