Parametric excitation of surface electron cyclotron X- and O- modes by alternating electric fields

Propagation of different electromagnetic surface waves (SW) is recognized as one of the most possible cause of undesirable plasma periphery heating, strengthening interaction between plasma column and walls of fusion device, impurity generation. This is found out clearly in experiments with ion cyclotron heating [1]. On the other hand in order to supply periodical operation of fusion devices, one can clean the devices inner surface. One of the prospective methods that can be used for this procedure is application both ion and electron cyclotron hardwares (without dismantling the whole device) for sustaining gas discharge, which can remove different contaminations from the chamber wall. The urgency of researches into parametric instability of SW at harmonics of cyclotron frequencies is caused as well by wide application of SW in various plasma technologies [2]. The present paper is devoted to development theory of parametric instability of slow surface Xand O-modes at electron cyclotron frequency for the model of semi-bounded plasma. Surface electron cyclotron O-modes (SECOM) were found theoretically [3] to propagate along plane plasmadielectric interface, when an external constant magnetic field is parallel to the plasma boundary and penetration depth of the modes is approximately equal to their wavelength. SECOM frequencies decrease if their wave number grows, their damping is determined by both collision (interaction between plasma particles) and kinetic (interaction between particles and plasma interface) mechanisms. Dispersion properties of surface electron cyclotron X-modes (SECXM) were studied theoretically in [4]. Unlike the SECOM they are unidirectional modes, their propagation coincides with direction of Larmor rotation of electrons on the plasma surface. They can propagate even along plasma-metal interface that makes them especially dangerous for destroying walls of fusion chambers. Sets of equations for these Xand O-modes are obtained by application of non-linear boundary conditions for their tangential magnetic field. Non-linear boundary conditions, which describe discontinuity of tangential component of their magnetic field are formulated by the aid of conception of existence of surface electric current, which is induced by the mentioned above external alternation electric field on the plasma interface. The reduced sets of equations for the Xand O-modes wave packets are obtained in approximation that the packets are composed by the main harmonic and two nearest satellite harmonics. EPJ Web of Conferences DOI: 10.1051/


Introduction
Propagation of different electromagnetic surface waves (SW) is recognized as one of the most possible cause of undesirable plasma periphery heating, strengthening interaction between plasma column and walls of fusion device, impurity generation. This is found out clearly in experiments with ion cyclotron heating [1]. On the other hand in order to supply periodical operation of fusion devices, one can clean the devices inner surface. One of the prospective methods that can be used for this procedure is application both ion and electron cyclotron hardwares (without dismantling the whole device) for sustaining gas discharge, which can remove different contaminations from the chamber wall. The urgency of researches into parametric instability of SW at harmonics of cyclotron frequencies is caused as well by wide application of SW in various plasma technologies [2].
The present paper is devoted to development theory of parametric instability of slow surface Xand O-modes at electron cyclotron frequency for the model of semi-bounded plasma. Surface electron cyclotron O-modes (SECOM) were found theoretically [3] to propagate along plane plasmadielectric interface, when an external constant magnetic field is parallel to the plasma boundary and penetration depth of the modes is approximately equal to their wavelength. SECOM frequencies decrease if their wave number grows, their damping is determined by both collision (interaction between plasma particles) and kinetic (interaction between particles and plasma interface) mechanisms. Dispersion properties of surface electron cyclotron X-modes (SECXM) were studied theoretically in [4]. Unlike the SECOM they are unidirectional modes, their propagation coincides with direction of Larmor rotation of electrons on the plasma surface. They can propagate even along plasma-metal interface that makes them especially dangerous for destroying walls of fusion chambers.
Sets of equations for these X-and O-modes are obtained by application of non-linear boundary conditions for their tangential magnetic field. Non-linear boundary conditions, which describe discontinuity of tangential component of their magnetic field are formulated by the aid of conception of existence of surface electric current, which is induced by the mentioned above external alternation electric field on the plasma interface. The reduced sets of equations for the X-and O-modes wave packets are obtained in approximation that the packets are composed by the main harmonic and two nearest satellite harmonics.  By the aid of reverse Fourier transform one can obtain equation for n -th harmonic of the X-and O-modes tangential electric field. This transform has been carried out using Jordan lemma. Then applying the residues theory one can obtain the equation for n -th harmonic of their tangential electric field on the plasma surface. There are two boundary conditions for these modes on the plasma-vacuum interface. The first of them is well-known linear condition for continuity of tangential electric field of the waves. The second boundary condition describes flowing of non-linear surface electric current along the plasma boundary. Discontinuity of tangential magnetic fields for the studies modes can be presented for the X-and O-modes in the following forms, respectively: .

EPJ Web of Conferences
Detailed description of j f is presented in [5]. Application of the conditions (2,3) is analogous to introducing a non-linear perturbations of surface charge density, which were applied in [6] for theoretical description of the process of solitary SW excitation.

Results of study of these parametric instabilities
Using boundary conditions one can obtain the following infinite set of equations for tangential component of the SECOM and SECXM electric field on the plasma-vacuum interface: here diagonal coefficients nearby tangential components of the X-and O-modes electric field are similar to dispersion functions of these modes: 0 ) , Results of the numerical analysis are presented on the following Figures 1-4.

Conclusions
Obtained infinite sets of equations for harmonics of these X-and O-modes tangential electric fields allow one to describe an initial stage of their parametric instability. The obtained results can be useful at first, for development of plasma technologies based on utilization surface electron cyclotron waves propagation. Secondly it can be useful for searching the regimes of weak activity of the edge localized modes in fusion devices, because it can allow one to decrease heating of the periphery plasma and impurity generation in fusion devices.