The Adjoint Problem in the Presence of a Deformed Surface: The Example of the Rosensweig Instability on Magnetic Fluids

The Rosensweig instability is the phenomenon that above a certain threshold of a vertical magnetic field peaks appear on the free surface of a horizontal layer of magnetic fluid. In contrast to almost all classical hydrodynamical systems, the nonlinearities of the Rosensweig instability are entirely triggered by the properties of a deformed and a priori unknown surface. The resulting problems in defining an adjoint operator for such nonlinearities are illustrated. The implications concerning amplitude equations for pattern forming systems with a deformed surface are discussed.