7th Asia-Pacific Conference on Plasma Physics, 12-17 Nov, 2023, Port Messe, Nagoya

Formulation of a self-consistent reduced transport theory for discrete plasmawaves near their instability threshold
V. N. Duarte1, J. B. Lestz2,3, N. N. Gorelenkov1, and R. B. White1
1Princeton Plasma Physics Laboratory, Princeton University, Princeton, NJ, USA,
2University of California, Irvine, CA, USA
3 General Atomics, San Diego, CA, USAe-mail: vduarte@pppl.gov
A quasilinear (QL) plasma transport theory
that incorporates Fokker-Planck dynamical friction (drag)and pitch angle scattering is self-consistently derived fromfirst principles for an isolated, marginally-unstable mode
resonating with an energetic minority species. It is foundthat drag fundamentally changes the structure of the waveparticle resonance, breaking its symmetry and leading to
the shifting and splitting of resonance lines [1]. In contrast,
scattering broadens the resonance in a symmetric fashion[2]. The work shows how to systematically calculate theresonance function, i.e. the envelope function that weights
the strength of the resonant interaction, in QL theory in thepresence of collisions. It is demonstrated that a QL systemthat employs the calculated broadening functions (Fig. 1)
recently reported [1] automatically recovers the nonlinear
growth rate and mode saturation levels for near-thresholdplasma instabilities previously calculated from asignificantly more complex nonlinear kinetic theory basedon solving a time delayed integro-differential equation[3,4]. Moreover, it is shown that a QL theory can beconstructed for a single resonance, provided that itexperiences enough background stochasticity. The results
enable realistic reduced modeling of diffusive transportobserved in fusion devices.
Comparison with fully nonlinear simulations
shows that the proposed quasilinear system preserves theexact instability saturation amplitude and thecorresponding particle redistribution of the fully nonlinear
theory. Even in situations in which drag leads to a
relatively small resonance shift, it still underpins majorchanges in the redistribution of resonant particles. This
novel influence of drag is of interest over a range of
disciplines that employ kinetic theory, being equallyimportant in plasmas and self-gravitating systems. Infusion plasmas, the effects are especially pronounced forfast-ion-driven instabilities in tokamaks with low aspectratio or negative triangularity, as evidenced by pastobservations. The same theory directly maps to theresonant dynamics of the rotating galactic bar and massive
bodies in its orbit, providing new techniques for analyzing
the bar deceleration and the radial migration in galactic
dynamics [5].
The present formulation gives simple
analytical answers to the shape of discrete-resonancecollisional functions in quasilinear theory, for both
Fokker-Planck scattering and drag collisions. Thesefunctions, that integrate to unity, replace a simple delta
function that appears in the diffusion coefficient for thecase of no broadening. It is shown that the knowledge ofthese functions removes a major arbitrariness with respect
to previous resonance broadening approaches, which
consisted of tuning broadening parameters to match theexpected saturation levels [6]. In addition, analyticalforms for the modification of the distribution function canbe naturally constructed. These forms are particularly
useful for code verification. This formulation is required
to model and verify the fast ion relaxation in tokamaks
upon interaction with Alfvénic eigenmodes being
implemented in the Resonance Broadened Quasilinear(RBQ) code [7], where particle diffusion occurs along thecanonical toroidal momentum and the energy of the beam
ion distribution function.

Figure 1: The resonance function is a measure of thestrength of resonant wave-particle energy exchange. It isshown for several ratios of collisional drag frequency (α)
over the diffusive scattering frequency (𝜈), illustrating
how scattering broadens the resonances while drag shifts
and splits them into multiple peaks.
Acknowledgments
This work was supported by the U.S. Department ofEnergy under contracts DE-AC02-09CH11466 and DESC0020337.
References[1] V. N. Duarte J. B. Lestz, N. N. Gorelenkov, and R. B.
White, Phys. Rev. Lett. 130, 105101 (2023).
[2] V. N. Duarte, N. N. Gorelenkov, R. B. White and H. L.
Berk, Phys. Plasmas 26, 120701 (2019).
[3] H. L. Berk, B. N. Breizman, and M. Pekker, Phys. Rev.
Lett. 76, 1256 (1996).
[4] J. B. Lestz and V. N. Duarte, Phys. Plasmas 28, 062102(2021).
[5] C. Hamilton, E. A. Tolman, L. Arzamasskiy, V. N.
Duarte, arXiv:2208.03855 (Astrophys. J., in press, 2023).
[6] H. Berk, B. Breizman, J. Fitzpatrick, and H. Wong,
Nucl. Fusion 35, 1661 (1995).
[7] N. N. Gorelenkov, V. N. Duarte, C. S. Collins, M.
Podesta, and R. B. White, Phys. Plasmas 26, 072507(2019).