Interaction between Hydromagnetic Waves and a Time-Dependent, Inhomogeneous Medium

Consider a system made up of a hydromagnetic wave and the slowly varying background fluid in which it propagates. It is shown that both the effect of the background on the wave and that of the wave on the background may be derived from Hamilton's principle using the averaged hydromagnetic Lagrangian density. The waves propagate adiabatically, conserving the wave action, and act on the background via a wave pressure term. Total momentum, angular momentum, and energy are conserved. When many waves are superimposed, as in weak turbulence, the wave kinetic equation replaces the adiabatic conservation equation. The accuracy of the averaging approximation is examined, and it is shown that it may be extended to all orders in the inhomogeneity. Also, Eulerian and Lagrangian averaging are discussed.