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1. Quasilinear theory for inhomogeneous plasma

   https://doi.org/10.1017/S0022377822000502  

   Dodin, I.Y. (February 2022, Journal of Plasma Physics)

   This paper presents quasilinear theory (QLT) for a classical plasma interacting with inhomogeneous turbulence. The particle Hamiltonian is kept general; for example, relativistic, electromagnetic and gravitational effects are subsumed. A Fokker–Planck equation for the dressed ‘oscillation-centre’ distribution is derived from the Klimontovich equation and captures quasilinear diffusion, interaction with the background fields and ponderomotive effects simultaneously. The local diffusion coefficient is manifestly positive-semidefinite. Waves are allowed to be off-shell (i.e. not constrained by a dispersion relation), and a collision integral of the Balescu–Lenard type emerges in a form that is not restricted to any particular Hamiltonian. This operator conserves particles, momentum and energy, and it also satisfies the $$\smash {H}$$ -theorem, as usual. As a spin-off, a general expression for the spectrum of microscopic fluctuations is derived. For on-shell waves, which satisfy a quasilinear wave-kinetic equation, the theory conserves the momentum and energy of the wave–plasma system. The action of non-resonant waves is also conserved, unlike in the standard version of QLT. Dewar's oscillation-centre QLT of electrostatic turbulence ( Phys. Fluids , vol. 16, 1973, p. 1102) is proven formally as a particular case and given a concise formulation. Also discussed as examples are relativistic electromagnetic and gravitational interactions, and QLT for gravitational waves is proposed. 

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2. Noise-induced magnetic field saturation in kinetic simulations

   https://doi.org/10.1017/S0022377820000707  

   Juno, J.; Swisdak, M. M.; Tenbarge, J. M.; Skoutnev, V.; Hakim, A. (August 2020, Journal of Plasma Physics)

   null (Ed.)

   Monte Carlo methods are often employed to numerically integrate kinetic equations, such as the particle-in-cell method for the plasma kinetic equation, but these methods suffer from the introduction of counting noise to the solution. We report on a cautionary tale of counting noise modifying the nonlinear saturation of kinetic instabilities driven by unstable beams of plasma. We find a saturated magnetic field in under-resolved particle-in-cell simulations due to the sampling error in the current density. The noise-induced magnetic field is anomalous, as the magnetic field damps away in continuum kinetic and increased particle count particle-in-cell simulations. This modification of the saturated state has implications for a broad array of astrophysical phenomena beyond the simple plasma system considered here, and it stresses the care that must be taken when using particle methods for kinetic equations. 

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3. Diffusive and Nonlinear Scattering of Ring Current Protons by Electromagnetic Ion Cyclotron Waves in the Earth's Inner Magnetosphere

   https://doi.org/10.1029/2025JA034078  

   Ma, Q; Li, W; Bortnik, J; Hanzelka, M; Gan, L; Artemyev, A V; Shen, X‐C (August 2025, Journal of Geophysical Research: Space Physics)

   Abstract We evaluate the diffusive and nonlinear scattering of ring current protons by electromagnetic ion cyclotron (EMIC) waves in the Earth's inner magnetosphere using test particle simulations. EMIC waves are commonly observed inside and outside the plasmasphere with wave amplitudes ranging from 100 pT to several nT. Field‐aligned EMIC waves can scatter 1 keV–1 MeV protons counter‐streaming with respect to the waves through first order cyclotron resonance. Through the analyses of the proton equatorial pitch angle variations along the field line, our simulations reveal the typical interaction features including quasilinear diffusion for small wave amplitudes, phase trapping and bunching at intermediate and large pitch angles, anomalous phase trapping and positive phase bunching at small pitch angles, and non‐resonant scattering at pitch angles and energies outside the resonance regime. Using different wave amplitudes from 100 pT to 5 nT, we compared the modeling results of proton equatorial pitch angle variations between quasilinear and test particle simulations, and between diffusive scattering and advective effects. For monochromatic He‐band EMIC waves atL = 5, the interaction between protons and EMIC waves with amplitudes below 500 pT could be described as a diffusive process and quantified by quasilinear theory; nonlinear interactions and advection effects become important for wave amplitudes larger than 1 nT. The interactions between EMIC waves and ring current protons are analogous to the interactions between whistler‐mode chorus waves and radiation belt electrons described in previous studies, despite the quantitative differences in the wave amplitude threshold of quasilinear diffusion applicability. 

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4. Electrostatic upper-hybrid mode instability driven by a ring electron distribution

   https://doi.org/10.1063/5.0151710  

   Yoon, Peter H.; Omura, Yoshiharu (June 2023, Physics of Plasmas)

   Quasi electrostatic fluctuations in the upper-hybrid frequency range are commonly detected in the planetary magnetospheric environment. The origin of such phenomena may relate to the instability driven by a loss-cone feature associated with the electrons populating the dipole-like magnetic field. The present paper carries out a one-dimensional electrostatic particle-in-cell simulation accompanied by a reduced quasilinear kinetic theoretical analysis to investigate the dynamics of the upper-hybrid mode instability driven by an initial ring electron distribution function, which is a form of loss-cone distribution. A favorable comparison is found between the two approaches, which shows that the reduced quasilinear theory, which is grounded in the concept of a model of the particle distribution function that is assumed to maintain a fixed mathematical form except that the macroscopic parameters that define the distribution are allowed to evolve in time, can be an effective tool in the study of plasma instabilities, especially if it is guided by and validated against the more rigorous simulation result. 

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5. On the validity of quasilinear theory applied to the electron bump-on-tail instability

   https://doi.org/10.1063/5.0086442  

   Crews, D. W.; Shumlak, U. (April 2022, Physics of Plasmas)

   The accuracy of quasilinear theory applied to the electron bump-on-tail instability, a classic model problem, is explored with conservative high-order discontinuous Galerkin methods applied to both the quasilinear equations and to a direct simulation of the Vlasov–Poisson equations. The initial condition is chosen in the regime of beam parameters for which quasilinear theory should be applicable. Quasilinear diffusion is initially in good agreement with the direct simulation but later underestimates the turbulent momentum flux. The greater turbulent flux of the direct simulation leads to a correction from quasilinear evolution by quenching the instability in a finite time. Flux enhancement above quasilinear levels occurs as the phase space eddy turnover time in the largest amplitude wavepackets becomes comparable to the transit time of resonant phase fluid through wavepacket potentials. In this regime, eddies effectively turn over during wavepacket transit so that phase fluid predominantly disperses by eddy phase mixing rather than by randomly phased waves. The enhanced turbulent flux of resonant phase fluid leads, in turn, through energy conservation to an increase in non-resonant turbulent flux and, thus, to an enhanced heating of the main thermal body above quasilinear predictions. These findings shed light on the kinetic turbulence fluctuation spectrum and support the theory that collisionless momentum diffusion beyond the quasilinear approximation can be understood through the dynamics of phase space eddies (or clumps and granulations). 

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