Relation of wave energy and momentum with the plasma dispersion relation in an inhomogeneous plasma

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Abstract
The expressions for wave energy and angular momentum commonly used in homogeneous and near‐homogeneous media is generalized to inhomogeneous media governed by a nonlocal conductivity tensor. The expression for wave energy applies to linear excitations in an arbitrary three‐dimensional equilibrium, while the expression for angular momentum applies to linear excitations of azimuthally symmetric equilibria. The wave energy Ewave is interpreted as the energy transferred from linear external sources to the plasma if there is no dissipation. With dissipation, such a simple interpretation is lacking as energy is also thermally absorbed. However, for azimuthally symmetric equilibria, the expression for the wave energy in a frame rotating with a frequency ω can be unambiguously separated from thermal energy. This expression is given by Ewave −ωLwave/ l, where Lwave is the wave angular momentum defined in the text and l the azimuthal wavenumber and it is closely related to the real part of a dispersion relation for marginal stability. The imaginary part of the dispersion is closely related to the energy input into a system. Another useful quantity discussed is the impedance form, which can be used for three‐dimensional equilibrium without an ignorable coordinate and the expression is closely related to the wave impedance used in antenna theory. Applications to stability theory are also discussed.