Integrating the finite-temperature wave equation across the plasma/vacuum interface

At the plasma/vacuum interface, the electromagnetic modes supported in vacuum connect to their finite-density counterparts as well as to supplementary finite-temperature modes supported by the plasma. To find the most general solution for a given plasma model containing M independent solutions inside the plasma, M boundary conditions have to be imposed. At each interface, two boundary conditions directly follow from Maxwell's equations. They require that the two components of the electric field tangential to the interface are continuous at the edge. The present paper proposes a method for finding the appropriate supplementary boundary conditions. Although the boundary conditions are derived to interface the TOMCAT wave code (Van Eester D and Koch R 1998 Plasma Phys. Control. Fusion 40 1949) with antenna coupling codes via the surface impedance matrix, the adopted philosophy can easily be extended to wave models other than the one used here. It is shown that, when formulating the problem in variational form (anticipating subsequent exploitation of the finite-element method), it suffices to impose the continuity of the surface terms (corresponding to the total flux) at the plasma/vacuum interface. When the test function in the variational is substituted for the electric field, the wave equation reduces to the power balance equation. The continuity of the surface terms guarantees that no power is lost at the interface where the vacuum modes (which carry their energy electromagnetically via the Poynting flux) pass on their energy to the plasma modes (which carry their energy both electromagnetically and via particles in coherent motion with the wave, i.e. as kinetic flux).