High-n helicity-induced shear Alfvén eigenmodes

The high-n helicity-induced shear Alfvén eigenmodes (HAE) are considered both analytically and numerically for the straight helical magnetic system, where n is the toroidal mode number. The eigenmode equation for the high-n HAE modes is derived along the field line and, with the aid of the averaging method, is shown to reduce to the Mathieu equation asymptotically. The discrete HAE modes are shown to exist inside the continuum spectrum gaps. The continuous spectrum gaps appear around ω² = ω_A²[N(lt-m)/2]² for N = 1,2,..., where ω_A is the toroidal Alfvén transit frequency, and l, m, and t are the polarity of helical coils, the toroidal pitch number of helical coils, and the rotational transform, respectively. For the same ω_A and t, the frequency of the helical continuum gap is larger than that of the continuum gap in tokamak plasmas by |l - t^-1m|. The polarity of helical coils l plays a crucial role in determining the spectrum gaps and the properties of the high-n HAE modes. The spectrum gaps near the magnetic axis are created by the helical ripple with circular flux surfaces for l = 1, and ≥ 3 helicals. For l = 2 helical systems, the spectrum gaps are created by the ellipticity of the flux surfaces. These analytical results for the continuum gaps and the existence of the high-n HAE modes in the continuum gaps are confirmed numerically for the 1 = 2 case, and it is found that the HAE modes exist for mode structures with the even and the odd parities.