A variational principle for surface waves in magnetohydrodynamics
A variational principle for the motion of the interface between an infinitely conducting fluid and a vacuum magnetic field is given. This variational principle not only gives the Laplace equation governing the velocity potential and the magnetic-field potential, but also provides all the boundary conditions appropriate to the interface between an infinitely conducting fluid and a vacuum magnetic field. However, unlike the hydrodynamical case, this variational principle does not have the simple physical interpretation of the stationarity of the fluid pressure + magnetic field pressure. An expression for the pressure was used by Bateman [1] as the Lagrangian and the equations of motion in an inviscid, incompressible fluid were derived from a variational principle. Luke [4] showed that a simple extension of this variational principle also provides the boundary conditions appropriate to a free surface. Debnath [2, 3] reformulated the full set of equations of motion for the surface-wave problem in magnetohydrodynamics, and made an initial-value investigation of the twoand three-dimensional dispersive surface waves in an electrically conducting fluid with a free surface. It is of special interest to consider a variational principle for the two-dimensional surface waves in magnetohydrodynamics, and to derive the full set of equations of motion used by Debnath [2] and Shivamoggi [5]. The purpose of this note is to give a variational principle for the motion of the interface between an infinitely conducting fluid and a vacuum magnetic field. This variational principle not only gives the Laplace equation governing the velocity potential and the magnetic-field potential, but also provides all the boundary conditions appropriate to the interface between an infinitely conducting fluid and a vacuum magnetic field. However, unlike the hydrodynamical case, this variational principle does not have the simple physical interpretation of the stationarity of the fluid pressure + magnetic field pressure. Consider an infinitely-conducting fluid of density p subjected to a gravitational field g = — giy and confined in the region 0 < y < h by a vacuum magnetic field H. Then the variational principle is