Direct prediction of saturated neoclassical tearing modes in slab using an equilibrium approach

Abstract We demonstrate for the first time that the nonlinear saturation of neoclassical tearing modes (NTMs) can be found directly using a variational principle based on Taylor relaxation, without needing to simulate the intermediate, resistivity-dependent dynamics. As in previous investigations of classical tearing mode saturation (Loizu et al 2020 Phys. Plasmas 27 070701; Loizu and Bonfiglio 2023 J. Plasma Phys. 89 905890507), we make use of Stepped Pressure Equilibrium Code (SPEC) (Hudson et al 2012 Phys. Plasmas 19 112502), an equilibrium solver based on the variational principle of the multi-region relaxed magnetohydrodynamics (MHDs), featuring stepped pressure profiles and arbitrary magnetic topology. We work in slab geometry and employ a simple bootstrap current model <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>J</mml:mi> <mml:mrow> <mml:mrow> <mml:mi>bs</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>C</mml:mi> <mml:mstyle scriptlevel="0"/> <mml:mi mathvariant="bold">∇</mml:mi> <mml:mi>p</mml:mi> </mml:mrow> </mml:math> to study the bootstrap-driven tearing modes, scanning over the asymptotic matching parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mrow> <mml:mi>′</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> and bootstrap current strength. Saturated island widths produced by SPEC agree well with the predictions of an initial value resistive MHDs code (Huang and Bhattacharjee 2016 Astrophys. J. 818 20) while being orders of magnitude faster to calculate. Additionally, we observe good agreement with a simple analytical modified Rutherford equation, without requiring any fitting coefficients. The match is obtained for both linearly unstable classical tearing modes in the presence of bootstrap current, and NTMs, which are linearly stable but nonlinear-unstable due to the effects of the bootstrap current.