A reduced bump-on-tail plant for the steady operating region of a fast-ion-driven mode: amplitude equation, full-f Vlasov solver with Fokker-Planck collisions, regime map and a detection benchmark

Abstract

Version 2 of this paper adds a related-work section placing the work against the analytic quasi-steady theory of Lestz and Duarte (2021), the regime maps of Zhang et al. (2026) and the steady-or-chirping criterion of Duarte et al. (2017, 2018); places the near-marginal comparison against Lestz and Duarte's two validity constraints and withdraws version 1's reading of the 24 percent gap, which is now a candidate explanation and not an established one; relabels the drag ratio of the ITER-like slice as the collisional ratio of Lilley et al. and places the turbulence-corrected ITER ratio, an order of magnitude smaller, between the two steady slices tested; states the comparison with the cold-background regime map at the point where the reduced model fails; narrows three statements of version 1 to what was tested; and adds the experiments version 1 left open, a reverse ramp pair at the first operating point and the edge and a benchmark pair at a second operating point. An erratum corrects a sentence of version 1's Limits section that had the two near-threshold amplitudes reversed: the held level of the full-f solver is 24 percent above the cubic equation's steady state, not the other way round. The paper builds and verifies a two-level reduced model of a fast-ion-driven mode, the Berk–Breizman bump-on-tail problem, as a plant whose purpose is to locate and characterise the regime in which the mode saturates to a steady state, the regime a fusion power plant must stay in. Level 0 is the near-threshold cubic amplitude equation with Krook, diffusive and drag collisions; Level 1 is a full-f one-dimensional Vlasov solver with a Krook operator; Level 2 adds drag and diffusion acting on the departure from the initial distribution. Each level was verified against the literature before its results were used: the amplitude equation reproduces the critical Krook rate 4.4 and the critical diffusion rate 2.05; the Vlasov solver reproduces the linear growth rate and frequency to 0.01 percent, conserves energy to second order, and reproduces the Berk–Breizman saturation law within 4 percent across the steady region and the diffusive saturation formula within 4 to 7 percent away from marginal stability. For the warm distribution used, the external damping on the field is only 87 percent effective as an amplitude damping, so marginal stability lies at a damping of 1.15 times the collisionless drive rather than 1.0; with that correction the full-f solver agrees with the cubic equation on the steady side of the bifurcation near threshold, and saturates 24 percent above it at a finite distance from marginal stability, in the direction and of the size that other simulations have reported against the same theory. The steady region is mapped in the damping–collision plane: with Krook collisions the oscillating region is a tongue at collision rates below 0.3 to 0.4 of the drive, closed above it; with diffusion equal to the drive, with or without drag at 0.7 of diffusion, every damping from 0.1 to 0.9 of the drive is steady, one point still rising at the end of its run; with drag at 1.2 of diffusion no steady point was found, and at the strongest drives the reduced model leaves its validity where a cold-background resonant model saturates steadily. The edge of the steady region along the collision axis lies between diffusion rates of 0.4 and 0.5 of the drive at a damping of 0.5 of the drive, converged on two velocity grids, and within the same bracket at a damping of 0.3, where every steady level is 1.416 times larger. The detection benchmark is now stated across three ramp pairs at two operating points. Four plain baseline detectors computed from the mode's own emitted signal, calibrated on a quiet hold under a stated contract, were scored against a stated onset rule and its mirror, a retrospective clearance rule. On ramps across the edge and matched control ramps inside the steady region, at both operating points, the detrended envelope spread alone detects the regime change, 246 to 248 time units after the onset and never on a control; restoring the collisionality restores the endpoint level with no persistent difference, the modulation decaying on the collisional time scale so that the same detector clears, by the retrospective rule, about 1560 time units after the restoring ramp ends; the envelope and plain spread baselines report the operating point's drift, and the spectral peak sees nothing. This is the benchmark against which a regime monitor on this plant is to be judged: a latency of a few hundred time units after the operational onset at the two operating points tested, for the one baseline that passes its controls, not a general latency and not a false-alarm rate. The engineering reading throughout is that the steady region is wide, its margin is a measurable design parameter with the same bracket at two dampings, and the collisions ITER alphas feel leave the steady state in place.