Conservation laws and connection formulae for resonant MHD waves

This paper reviews a recent development in the theory of resonant MHD waves in non-uniform plasmas. An asymptotic analysis of the equations for MHD waves in plasmas with high magnetic Reynolds numbers has shown that resonant slow and Alfven waves obey conservation laws and jump conditions across the dissipative layer. These conservation laws specify the dominant dynamics of the resonant MHD waves. In combination with the jump conditions they enable us to understand the basic physics of resonant MHD waves and also help us with the interpretation of results of large scale numerical simulations in resistive MHD. They also can be used to design an accurate and computationally simple methods for numerical studying resonant MHD waves in non-uniform plasmas. Conservation laws and jump conditions for resonant MHD waves are first discussed in linear MHD for 1-dimensional equilibrium states. Subsequently the generalization of these results to 2-dimensional equilibrium state in linear MHD and to nonlinear MHD is reviewed. The interaction of sound waves with an inhomogeneous plasma is discussed as an application of the theory. Firstly the results of linear theory are used to consider the interaction of sound waves with 1-dimensional magnetic tubes. The phenomenon of total resonant absorption is discussed. Secondly the nonlinear theory of cusp dissipative layers is used to study the interaction of sound waves with 1-dimensional inhomogeneous plasmas in planar geometry. New effects that owe their existence to nonlinearity in the cusp dissipative layer are reviewed.