The Dispersion Relation for Electrostatic Fluctuations in Weakly Inhomogeneous Plasmas

Along the last few years we have conducted several investigations on waves in inhomogeneous plasmas, using the concept of effective dielectric tensor, which has been proposed as the correct dielectric tensor for the description of dielectric properties of inhomogeneous plasmas [1]. Among these investigations, we have considered cases where the magnetic field is homogeneous and other plasma parameters are inhomogeneous [2, 3, 4], cases where the magnetic field is inhomogeneous and inhomogeneities in the plasma parameters are neglected [5, 6], and cases where inhomogeneities are taken into account both in the plasma parameters and in the magnetic field [7]. For all these cases we case considered arbitrary direction of propagation relative to the ambient magnetic field, and we have taken into account relativistic effects. The expressions obtained for the dielectric tensor for all these cases satisfy Onsager symmetry, and as a consequence the anti-Hermitian part of the tensor only contains resonant terms, as required for proper description of the energy exchange between wave and particles. We have also applied the concept of effective dielectric tensor to the study of instabilities in the lower hybrid range, in a plasma featuring density and magnetic field inhomogeneities [8]. In the case of this application we have verified that the form of the dispersion relation conventionally used for electromagnetic waves was able to describe the so-called modified two stream instability (MTSI), an instability which occurs due to the existence of a relative drift between ions and electrons [9-13], as well as its purely growing limit for parallel propagation, known as ion Weibel instability (IWI) [14, 15]. However, the conventional form of the dispersion relation, along with the effective dielectric tensor, was not able to describe the lower hybrid drift instability (LHDI), which is expected to occur when inhomogeneities are taken into account in the description of the electron contribution to the dispersion relation [16-23]. The LHDI is known by its strong electrostatic character, and is frequently studied using the electrostatic approximation. The difficulty of the conventional form of the dispersion relation in describing the LHDI had already been noticed in an earlier analysis, in a formulation which did not use the effective dielectric tensor, adopting instead an ad hoc procedure to correct the lack of symmetry of the dielectric tensor [23]. We have therefore derived a new form of the dispersion relation, taking into account in the derivation the relationship between charge density and electric field expressed by Gauss law, which introduces a term featuring gradients of the inhomogeneous parameters, which is of the same order as other terms due to the inhomogeneity which were taken into account in the derivation of the dielectric tensor. This new form of the dispersion relation was able to describe in a local approximation both the LHDI and the MTSI(IWI), something which was not yet available in the literature up to that moment [8]. The fact that the LHDI was not described with use of the conventional form of the dispersion relation and required a new form of the dispersion relation to take into account all relevant inhomogeneity effects, and the strong electrostatic character of the instability, point out to the relevance of investigating the proper electrostatic limit of the dispersion relation. As it is known, for electrostatic fluctuations in homogeneous plasmas the dispersion relation may be written as εl = 0, where εl is the dielectric constant, obtained as follows, εl = kiεijkj k2 , (1)