Electrostatic-plasma-wave energy flux

VOLUME 29, NUMBER 3 PHYSICAL REVIE% A Electrostatic-plasma-wave P. Amendt Physics Department, MARCH 1984 energy flux and N. Rostoker Irvine, California 9271 7 (Received 29 August 1983) University of California, The direct loss of fusion plasma energy via electrostatic plasma waves propagating across field lines has previously been shown to be significant. We investigate the theoretical foundations of this phenomenon from a strictly transport perspective. The critical role assumed by the displacement current in sustaining cross-field transport of electrostatic energy is shown. An accompanying flux reveals a far larger involvement of the analysis of the composition of electrostatic-wave-energy field degrees of freedom than that associated with wave-energy density. A phenomenological degradation by general wave-particle interactions Fourier-type law describing wave-energy-current is shown to be the appropriate description of wave-energy conduction in the nonthermal regime. I. INTRODUCTION The phenomenon of thermal conduction by lattice waves in semiconductors has long been appreciated as sig- nificant within the solid-state physics community. As the field of fusion plasma physics is a comparatively young one, a similar recognition of the importance of plasma- With wave-energy transport has not been forthcoming. regard to energy confinement, this is an uncomfortable state of affairs. Present magnetic confinement approaches heavily emphasize the particle degrees of freedom over the field degrees of freedom. While the external magnetic fields may prove satisfactory in suppressing the cross-field particle component of energy transport, the consequences of unhindered transport of waves across the field have re- ceived only limited interest within the fusion community. The problem was first posed in a thermal equilibrium estimate by Rosenbluth and I. iu. ' They found that energy losses induced by cross-field transport of electrostatic waves can be comparable in magnitude to particle thermal diffusive losses. Subsequently, Molvig, Hers, and Tekula electron-plasma-wave found that runaway-induced energy transport was greater than neoclassical ion thermal con- duction and of the order of observed transport in inter- Recently, an estimate has mediate density tokamaks. losses by ion- been made of cross-field wave-energy cyclotron modes driven unstable by the Drummond- This estimate was Rosenbluth ion-cyclotron instability. distinguished by a departure from the previous use' of collision-induced wave-emission mechanisms in favor of large an induced emission version. A correspondingly enhancement over the thermal level was demonstrated. This succession of estimates underscores the timely flux need for an investigation of electrostatic-wave-energy from first principles. The study undertaken here will ad- dress the following fundamental questions: (1) What is the underlying physical basis for cross-field electrostatic-wave-energy transport in the strong-field lim- it? (2) How is the integrity of electrostatic-wave-energy transport influenced by a varying degree of plasma aniso- tropy? (3) Do the usual divisions of wave-energy density into particle and field components directly carry over into wave-energy flux? (4) Are the usual ambiguities in locally defining energy an and flux circumvented density by demanding equivalence between group velocity and velocity of energy transport? (5) How is wave thermal conduction nonthermal, collisionless regime? manifested in the The collective intent of these questions is to understand the nature of electrostatic plasma waves from a transport perspective. In Sec. II we begin by deriving an energy continuity equation for an electrostatic system with direct use of Joule s law. The electrostatic limit employed is dis- tinguished from previous conventions by the necessary re- quirement that the induced magnetic induction B also vanishes as the speed of light c approaches infinity. This allows for a tractable treatment of electrostatic waves even in anisotropic media. In Sec. III the role of the elec- trostatic flux vector in electrostatic-wave-energy transport is investigated. The key term responsible for unhindered cross-field transport of non-cyclotron-type wave energy is of electrostatic-wave- identified, and the constitution energy transport in an anisotropic plasma is examined. A discussion of the usual ambiguities associated with defin- ing local-energy flux within the context of an observable wave then follows. In Sec. IV, we investigate the role of weak wave dissipation mechanisms in degrading a wave- energy current by formulating a Fourier-type phenomeno- logical law that is valid in a nonthermal, weakly turbulent setting. II. A VLASOV DESCRIPTION OF ELECTROSTATIC ENERGY FLUX ed The rate at which particle kinetic energy 8& is dissipat- in Joule's law into field energy is embodied