On Kink Modes and Low-<i>n</i> Kink <i>β</i> Limits in HL-2A Sawtooth Discharges

Kink mode is one of the most important macroinstabilities in sawtooth discharge. In this article, the nature of kink modes was investigated through a combination of HL-2A experiments and simulations. Multiple <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>n</i></b> (toroidal mode number) kink branches were found in HL-2A sawtooth discharges. The distinct roles of different <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>n</i></b>-kink modes in plasma instability were identified. The <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>n</i></b> = 1 mode is identified as the dominant kink mode. The mode structure shrinks toward the center with an increase in <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</b> value. The mode linear growth rate is not necessarily related to whether the mode dominates plasma. The weakening effect of the shorter wavelength and its higher internal localization partially offset the advantage of the higher <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>n</i></b> mode in the global plasma instability. In practice, multiple <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>n</i></b>-kink branches tend to be stable if <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>n</i></b> = 1 mode is stable. Among the low-<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><b>n</b></i> modes, the stability limits for <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>n</i></b> <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\gt$ </tex-math></inline-formula> 1 modes would not be lower than <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>n</i></b> = 1 mode at <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {q}_{\mathbf {0}}=\mathbf {1.05}$ </tex-math></inline-formula> and its adjacent areas, and the stability limits of <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>n</i></b> = 1 and <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</b> kink modes decrease with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {q}_{\mathbf {0}}$ </tex-math></inline-formula> and exhibit some linear relationships if a normally parabolic current density profile is considered, which hints for the determination of the operational <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\boldsymbol {\beta }$ </tex-math></inline-formula> for HL-2A sawtooth discharges.