Alfvénic gap eigenmode in a linear plasma with ending magnetic throats

Abstract

To guide the experimental design of a linear plasma device for studying the interaction between energetic ions and Alfvénic gap eigenmode (AGE), this work computes AGE referring to fusion conditions in a ultra-long large plasma cylinder ended with strong magnetic throats for axial confinement of charged particles. It is shown that:(i) for uniform equilibrium field between the ending throats, the dispersion relation of computed wave field agrees well with a simple analytical model for shear Alfvénic mode; (ii) for periodic equilibrium field with local defect, clear AGE is formed inside spectral gap for both low and high depths of magnetic throats, although lower depth yields easier observation. The strongest AGE can be in order of 3.1 × 10 −4 to equilibrium field, making it conveniently measurable in experiment. The AGE is a standing wave localized around the defect which is introduced to break the system's periodicity, and its wavelength is twice the system's period, consistent with Bragg's law. Parameter scan reveals that the AGE remains nearly the same when the number of magnetic ripples is reduced from 18 to 8, however, there occurs an upward frequency shift when the depth of magnetic ripples drops from 0.5 to 0.1, possibly due to a flute-like effect: shrinking resonant cavity of spectral gap. of energetic beam ions during tae instabilities. Nuclear Fusion, 33(5):749, 1993.

On the limit of minimum number and depth of periodic modulations for spectral gap formation via Bragg's reflection

“…4, in terms of normalized wave magnetic field at z = 0 m and r = 0 m. It can be seen that the center and width of the spectral gap remain unchanged when the number increases, implying the evidence of no frequency shift. This is consistent with the previous observation that the gap eigenmode (discrete eigenmode inside the spectral gap) changes negligibly for various magnetic mirrors as long as the number is above a certain value [12,13,23]. Then the variation of the spectral gap with mirror depth is illustrated in fig.…”

“…The underlying physics of this downward frequency shift is unclear, although a simple theoretical analysis has been proposed. It may be similar to a flute-like effect for which the resonant frequency decreases as the air column becomes longer [23]. Realistic conditions on the LAPD, including the limited number (≤4.5) and depth (≤0.39) of periodic modulations and the mixed feature of kinetic and inertial plasma regions, are also discussed and we found that the computed spectral gap agrees well with the experimental data measured previously [10].…”

“…5 is in the form of √ 1 − 0.92M 2 . The underlying mechanism is unknown and detailed exploration is left to future research, although we propose that it may be similar to a flute-like effect for which the resonant frequency decreases as the air column becomes longer [23]. Moreover, this shows that the analytical and numerical center frequencies agree better and the spatially dependent Bragg frequency becomes more constant for smaller depth (M ≤ 0.3), which is usually assumed in previous theoretical analyses [9,10,13,24].…”

“…As can be seen, when the depth increases, the center of the spectral gap shows a parabolic shape of descending frequency shift, besides a broadened width expected from Δf/f B = M [9,10,24]. Although a similar frequency shift was observed on the gap eigenmode of Alfvén waves for increased depth [13,23], this parabolic shape of the frequency shift on the spectral gap alone has not yet been revealed and is believed to be a new discovery.…”