Calculating the linear critical gradient for the ion-temperature-gradient mode in magnetically confined plasmas

Abstract

A first-principles method to calculate the critical temperature gradient for the onset of the ion-temperature-gradient mode (ITG) in linear gyrokinetics is presented. We find that conventional notions of the connection length previously invoked in tokamak research should be revised and replaced by a generalized correlation length to explain this onset in stellarators. Simple numerical experiments and gyrokinetic theory show that localized ‘spikes’ in shear, a hallmark of stellarator geometry, are generally insufficient to constrain the parallel correlation length of the mode. ITG modes that localize within bad drift curvature wells that have a critical gradient set by peak drift curvature are also observed. A case study of near-helical stellarators of increasing field period demonstrates that the critical gradient can indeed be controlled by manipulating the magnetic geometry, but underscores the need for a general framework to evaluate the critical gradient. We conclude that average curvature and global shear set the correlation length of resonant ITG modes near the absolute critical gradient, the physics of which is included through direct solution of the gyrokinetic equation. Our method, which handles the general geometry and is more efficient than conventional gyrokinetic solvers, could be applied to future studies of stellarator ITG turbulence optimization.

Coarse-grained gyrokinetics for the critical ion temperature gradient in stellarators

“…( 1)-( 3), is such that M < 1 is satisfied within the confining well (for | − | L) and M > 1 outside, so that the contribution to ϕ is negligible under velocity integration in Eq. ( 3) (see also the discussion in Roberg-Clark et al [26]). Furthermore, we assume J 0 1 within this region as its argument is small there.…”

Reduction of electrostatic turbulence in a quasi-helically symmetric stellarator via critical gradient optimization

“…The test particle orbits were simulated in vacuum with no coils included in the final equilibrium, which would presumably affect the results. Linear gyrokinetic simulations in local flux tube geometry using the GENE code (Jenko et al 2000) are performed to determine the critical gradient in the same manner as by Roberg-Clark et al (2021, i.e. by reducing the applied temperature gradient until a single, marginally unstable mode remains.…”

“…The input equilibrium for the optimization is a 'warm start' example file included in SIMSOPT with poor quasi-helical symmetry, n fp = 4 and aspect ratio A = R/a = 7, with R the major radius and a the minor radius, and an initial a/L T,crit = 0.50. The target A = 4.10 was chosen to enhance the drift curvature K d , and to take into account that pushing the aspect ratio below the number of field periods is difficult to achieve while maintaining finite rotational transform in a helical stellarator (Roberg-Clark et al 2021). We speculate a/L T,crit = 2.00 to be roughly the maximum achievable linear critical gradient in the absence of very large global shear stabilization (Roberg-Clark et al 2022).…”

Ion-temperature-gradient stability near the magnetic axis of quasisymmetric stellarators

“…As is characteristic of the ITG mode, the peak growth rate is negative for a range of values where η = L n /L T < η crit with η crit ∼ 1 [57], such as the condition for instability η > 2/3 of the toroidal branch of the ITG mode found in [58]. A more detailed understanding of the mechanisms that set the ITG critical gradient using first-principle approaches, such as the one in [59], are left for future work.…”