An equation of shear Alfvén eigenmodes ͑AE͒ in optimized stellarators of Wendelstein line ͑Helias configurations͒ is derived. The metric tensor coefficients, which are contained in this equation, are calculated analytically. Two numerical codes are developed: the first one, COBRA ͑COntinuum BRanches of Alfvén waves͒, is intended for the investigation of the structure of Alfvén continuum; the second, BOA ͑Branches Of Alfvén modes͒, solves the eigenvalue problem. The family of possible gaps in Alfvén continuum of a Helias configuration is obtained. It is predicted that there exist gaps which arise due to or are strongly affected by the variation of the shape of the plasma cross section along the large azimuth of the torus. In such gaps, discrete eigenmodes, namely, helicity-induced eigenmodes ͑HAE 21 ) and mirror-induced eigenmodes ͑MAE͒ are found. It is shown that plasma inhomogeneity may suppress the AEs with a wide region of localization.
Conditions of the existence of the Global Alfvén Eigenmodes (GAE) and Nonconventional Global Alfvén Eigenmodes (NGAE) predicted for stellarators by Ya. I. Kolesnichenko et al. [Phys. Rev. Lett. 94, 165004 (2005)] have been obtained. It is found that they depend on the nature of the rotational transform and that conditions for NGAE can be most easily satisfied in currentless stellarators. It is shown that the plasma compressibility may play an important role for the modes with the frequency about or less than that of the Toroidicity-induced Alfvén Eigenmodes (TAE). It is found that features of the Alfvén continuum in the vicinity of the k = 0 radius (k is the longitudinal wave number) can be very different, depending on a parameter which we refer to as "the sound parameter". Specific calculations modeling low-
It is shown that conventional expressions for the frequencies of the geodesic acoustic mode (GAM) and the gaps in the Alfvén continuum of toroidal plasmas are valid only when q 2 β 1, where q is the safety factor and β is the ratio of the plasma pressure to the magnetic field pressure. Expressions, which are relevant to plasmas with q 2 β 1 ( β = β s /(1 + β s ), where β s = c 2 s /v 2 A , c s is the sound velocity, v A is the Alfvén velocity) and convenient for practical use, are obtained for the GAM frequency and the characteristic values of the frequencies of gap modes (toroidicity-induced Alfvén eigenmodes, ellipticity-induced Alfvén eigenmodes and noncircularity-induced Alfvén eigenmodes). It is found that at any q 2 β the Alfvén continuum in the near-axis region is described by Mathieu's equation. Due to this, simple expressions are also obtained for the limit case of q 2 β 1.
Reduced linear equations of magnetohydrodynamics in high-aspect-ratio toroidal devices are derived, which are intended, first of all, for studying the Alfvén eigenmodes in stellarators and tokamaks. The equations take into account the effects of the plasma pressure and compressibility, which are known to be of importance for toroidicity-induced Alfvén eigenmodes, and are applicable to perturbations with arbitrary perpendicular wavelength. The reduction consists in eliminating high-frequency fast magnetoacoustic waves from the system and is shown not to affect the continuous spectrum of Alfvén and slow magnetoacoustic waves, which, to a large extent, determines the behavior of the waves of interest.
This work generalizes recent results [O. P. Fesenyuk et al., Plasma Phys. Controlled Fusion 54, 085014 (2012)] to plasmas with elongated cross section. It suggests new expressions for the frequencies of the geodesic acoustic mode and Alfvén gap modes in tokamaks, with a large ratio of the plasma pressure to the magnetic field pressure and a large safety factor (q≫1, which takes place in discharges with reversed-shear configuration and, especially, in hollow-current discharges).
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