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DOE OSTI2024

Complete quasilinear model for the acceleration-driven lower hybrid drift instability and a computational assessment of its validity

A complete quasilinear model is derived for the electrostatic acceleration-driven lower hybrid drift instability in a uniform two-species low-beta plasma in which current is perpendicular to the background magnetic field. The model consists of coupled nonlinear velocity space diffusion equations for the volume-averaged ion and electron distribution functions. Each species' diffusion coefficient depends on a time-evolving spectral density of the electric-field energy per unit volume and a time-evolving dispersion relation. The dispersion relation is expressed analytically in integral form without the use of asymptotic limits and applies to arbitrary distribution functions, so long as they can be expressed as a function of one velocity coordinate, e.g., f⁡(vy) or f⁡(v⊥). The quasilinear model conserves energy and is complete in that it fully describes the evolution of the distribution functions, including resonant and nonresonant particle-wave interactions, while accounting for distribution-function-dependent mixed-complex frequencies. Further, the quasilinear diffusion model is solved numerically and self-consistently using a Crank-Nicolson temporal discretization and a second-order finite-volume velocity-space discretization. Numerical solutions are compared to nonlinear fourth-order accurate continuum kinetic Vlasov-Poisson simulations. Evolution of electric-field energy, growth rates, distribution functions, and diffusion coefficients are shown to be in agreement with Vlasov simulations. The quasilinear model is shown to predict anomalous transport terms, like resistivity and heating, to within a factor of order unity. Discrepancies between the quasilinear model and Vlasov simulations are assessed and attributed primarily to lack of damping in the quasilinear description and to the use of unperturbed-orbit susceptibilities in the linear theory dispersion relation. The results illuminate the predictive accuracy of the quasilinear model, place approximate bounds on its validity, and provide much needed vetting of quasilinear theory's ability to predict the nonlinear state of a microturbulent plasma.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2023

Expanding-box Quasilinear Model of the Solar Wind

The expanding-box model of the solar wind has been adopted in the literature within the context of magnetohydrodynamics, hybrid, and full particle-in-cell simulations to investigate the dynamic evolution of the solar wind. The present paper extends such a method to the framework of self-consistent quasilinear kinetic theory. It is shown that the expanding-box quasilinear methodology is largely equivalent to the inhomogeneous steady-state quasilinear model discussed earlier in the literature, but a distinction regarding the description of wave dynamics between the two approaches is also found. The expanding-box quasilinear formalism is further extended to include the effects of a spiraling solar-wind magnetic field as well as collisional age effects. The present finding shows that the expanding-box quasilinear approach and the steady-state global-kinetic models may be employed interchangeably in order to address other more complex problems associated with the solar-wind dynamics.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2022

Quasilinear theory of general electromagnetic fluctuations including discrete particle effects for magnetized plasmas: General analysis

The general quasilinear Fokker–Planck kinetic equation for the gyrophase-averaged plasma particle distribution functions in magnetized plasmas is derived, making no restrictions on the energy of the particles and on the frequency of the electromagnetic fluctuations and avoiding the often made Coulomb approximation of the electromagnetic interactions. Here, the inclusion of discrete particle effects breaks the dichotomy of nonlinear kinetic plasma theory divided into the test particle and the test fluctuation approximation because it provides expression of both the non-collective and collective electromagnetic fluctuation spectra in terms of the plasma particle distribution functions. Within the validity of the quasilinear approach, the resulting full quasilinear transport equation can be regarded as a determining nonlinear equation for the time evolution of the plasma particle distribution functions.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2022

Quasilinear theory for inhomogeneous plasma

Here, 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 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 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.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2025

Quasilinear Analysis in the Source Region of Jovian Hectometric Emission Associated With Upward Electron Beams

Abstract Intense upward electron beams were measured by the Juno JADE instrument in the northern hemisphere, low‐latitude auroral zone source region. In this study we report on how these electron beams interact with plasma near and within the Jovian hectometric (HOM) emission (1 MHz 5 MHz) source region. Within the source region large upward loss cones are observed in the northern polar region at radial distances of 2Rj, magnetic latitude of . Intense, narrow electron beams ( 3 keV) are then observed, but within one second wave‐particle scattering is observed, filling the loss cone to energies 50 keV. These energies persist for several seconds before fading, leaving an empty loss cone again. The loss cone provides a free‐energy source for HOM emission resulting from the cyclotron maser instability. We use quasilinear analysis to examine the generation of HOM and the dynamics of wave‐particle interaction of the electron beams with HOM, and the generation via Landau interaction of whistler mode emission. The dynamic spectrum of the HOM emission generated by the loss‐cone electrons as well as that of the low‐frequency whistler‐mode waves generated by the up‐going electron beam can be constructed by quasilinear theory, which compare well with observation. The saturated state of the energetic electron velocity distribution function constructed via quasilinear theory also compare reasonably with observation.

Yoon, P. H. [Institute for Physical Science and Te↗

DOE OSTI2023

Comparison of Saturation Rules Used for Gyrokinetic Quasilinear Transport Modeling

Theory-based transport modeling has been widely successful and is built on the foundations of quasilinear theory. Specifically, the quasilinear expression of the flux can be used in combination with a saturation rule for the toroidal mode amplitude. Most transport models follow this approach. Saturation rules are heuristic and difficult to rigorously derive. We compare three common saturation rules using a fairly accurate quasilinear expression for the fluxes computed using local linear gyrokinetic simulation. We take plasma parameters from experimental H-mode profiles and magnetic equilibrium and include electrons, deuterium, and carbon species. We find that the various saturation rules provide qualitatively similar behavior. This may help to explain why the different theory-based transport models can all predict core tokamak profiles reasonably well. Comparisons with nonlinear local and global gyrokinetic simulations are discussed.

Parker, Scott E. (ORCID:000000030942630X)↗

DOE OSTI2024

Quasilinear theory: the lost ponderomotive effects and why they matter

Quasilinear theory (QLT) has been used for modeling wave–plasma interactions for decades but remains largely heuristic. Plasma inhomogeneity, ponderomotive effects, microscopic fluctuations, and collisions are not easily accommodated from first principles in QLT, and typically are ignored entirely, due to the limitations of the standard Fourier–Laplace global-mode approach. This results in inconsistencies, for example, violation of the action conservation for nonresonant waves. However, these issues can be avoided, and the theory can be substantially generalized and corrected, if QLT is formulated using more suitable analytical tools, particularly, the Weyl symbol calculus. Here, an attempt is made to deliver an accessible review of this modern formulation, provide intuitive calculations for special cases, and elaborate on the connection with the ‘oscillation-center QLT’ originally proposed by Dewar (Phys Fluids 16:1102, 1973). A Fokker–Planck equation for a ‘dressed’ distribution is derived from the Klimontovich equation and captures quasilinear diffusion, ponderomotive forces, and interactions with background fields for a generic Hamiltonian, so many known formulations of QLT for specific plasma models become corollaries of a single unifying theory. Also, waves are allowed to be off-shell (not constrained by a dispersion relation), which allows them to accommodate microscopic fluctuations. This leads to a collision integral of the Balescu–Lenard type that has all the usual properties but is not restricted to any specific plasma model. For on-shell waves, a generalized version of the classic oscillation-center QLT is obtained. Finally, combined with the wave-kinetic equation, this formulation not only conserves particles, momentum, and energy, like the classic QLT but also reinstates the action conservation for nonresonant waves.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2023

Resolution of a Few Problems in the Application of Quasilinear Theory to Calculating Diffusion Coefficients in Heliophysics

A theory of quasilinear diffusion for obliquely propagating electromagnetic waves was developed in the 1960's and applied in the 1970's to model scattering of relativistic electrons by a prescribed distribution of waves. In the latter work, a transformation of variables, from wavevector space to the temporal frequency and tangent of the wave normal angle, was used so that simple Gaussian functions of frequency and tangent of the wave normal angle could be multiplied together to define the distribution of wave power, although arbitrary distributions of power in these two variables is also permitted. Finally, in 2005, previous work was consolidated and has been widely used in heliophysics studies that require computation of quasilinear diffusion coefficients. Here it is shown that this transformation is inppropriate when the precise wave vector distribution is known. The correct transformation is derived and used to produce diffusion coefficients that can differ by orders of magnitude from those computed using the inappropriate transformation. The differences are largest when the distribution of wave power extends to wave normal angles near the resonance cone. When the ratio of the plasma frequency to the gyrofrequency is large, only low energies (keV) are affected, but as the ratio decreases higher energies (MeV) also show differences. It is also shown that the derivation from the 1960's uses a notation that results in the diffusion coefficients depending on the distribution of wave power with respect to the wave azimuthal angle whereas there should be no such dependence.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2024

Quasilinear theory and modelling of gyrokinetic turbulent transport in tokamaks

The theory, development, and validation of reduced quasilinear models of gyrokinetic turbulent transport in the closed flux surface core of tokamaks is reviewed. In combination with neoclassical collisional transport, these models are successful in accurately predicting core tokamak plasma temperature, density, rotation, and impurity profiles in a variety of confinement regimes. Refined experimental tests have been performed to validate the predictions of the quasilinear models, probing changes in the dominant gyrokinetic instabilities, as reflected in fluctuation measurements, cross-phases, and transport properties. These tests continue to produce a deeper understanding of the complex mix of instabilities at both electron and ion gyroradius scales.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2021

Reimagining full wave rf quasilinear theory in a tokamak

The velocity dependent resonant interaction of particles with applied radiofrequency (rf) waves during heating and current drive in the presence of pitch angle scattering collisions gives rise to narrow collisional velocity space boundary layers that dramatically enhance the role of collisions as recently shown by Catto ( J. Plasma Phys. , vol. 86, 2020, 815860302). The behaviour is a generalization of the narrow collisional boundary layer that forms during Landau damping as found by Johnston ( Phys. Fluids , vol. 14, 1971, pp. 2719–2726) and Auerbach ( Phys. Fluids , vol. 20, 1977, pp. 1836–1844). For a wave of parallel wave number ${k_{||}}$ interacting with weakly collisional plasma species of collision frequency $\nu$ and thermal speed ${v_{\textrm{th}}}$ , the effective collision frequency becomes of order $\nu {({k_{||}}{v_{th}}/\nu )^{2/3}} \gg \nu $ . The narrow boundary layers that arise because of the diffusive nature of the collisions allow a physically meaningful wave–particle interaction time to be defined that is the inverse of this effective collision frequency. The collisionality implied by the narrow boundary layer results in changes in the standard quasilinear treatment of applied rf fields in tokamaks while remaining consistent with causality. These changes occur because successive poloidal interactions with the rf are correlated in tokamak geometry and because the resonant velocity space dependent interactions are controlled by the spatial and temporal behaviour of the perturbed full wave fields rather than just the spatially local Landau and Doppler shifted cyclotron wave–particle resonance condition associated with unperturbed motion of the particles. The correlation of successive poloidal circuits of the tokamak leads to the appearance in the quasilinear operator of transit averaged resonance conditions localized in velocity space boundary layers that maintain negative definite entropy production.

Physics↗

DOE OSTI2021

Reimagining full wave rf quasilinear theory in a tokamak

The velocity dependent resonant interaction of particles with applied radio frequency (rf) waves during heating and current drive in the presence of pitch angle scattering collisions gives rise to narrow collisional velocity space boundary layers that dramatically enhance the role of collisions as recently shown by Catto (J. Plasma Phys., vol. 86, 815860302, 2020). The behavior is a generalization of the narrow collisional boundary layer that forms during Landau damping as found by Johnston (Phys. Fluids, vol. 14, 1971, pp. 2719-2726) and Auerbach (Phys. Fluids, vol. 20, 1977, pp. 1836-1844). For a wave of parallel wave number k|| interacting with weakly collisional plasma species of collision frequency ν and thermal speed vth , the effective collision frequency becomes of order ν(k_||v_th /ν)^2/3>> ν . The narrow boundary layers that arise because of the diffusive nature of the collisions allows a physically meaningful wave-particle interaction time to be defined that is the inverse of this effective collision frequency. The collisionality implied by the narrow boundary layer results in changes in the standard quasilinear treatment of applied rf fields in tokamaks while remaining consistent with causality. These changes occur because successive poloidal interactions with the rf are correlated in tokamak geometry and because the resonant velocity space dependent interactions are controlled by the spatial and temporal behavior of the perturbed full wave fields rather than just the spatially local Landau and Doppler shifted cyclotron wave-particle resonance condition associated with unperturbed motion of the particles. The correlation of successive poloidal circuits of the tokamak leads to the appearance in the quasilinear operator of transit averaged resonance conditions localized in velocity space boundary layers that maintain negative definite entropy production.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2021

Collisional effects on resonant particles in quasilinear theory

A careful examination of the effects of collisions on resonant wave-particle interactions leads to an alternate interpretation and deeper understanding of the quasilinear operator originally formulated by Kennel and Engelmann (Phys. Fluids vol. 9, 1966, pp. 2377- 2388) for collisionless, magnetized plasmas, and widely used to model radio frequency heating and current drive. The resonant and nearly resonant particles are particularly sensitive to collisions that pitch angle scatter them out of and into resonance. As a result, the resonant particle-wave interactions occur in the center of a narrow collisional boundary when the collision frequency nu is very small compared to the wave frequency omega. The diffusive nature of the pitch angle scattering combined with the wave-particle resonance condition enhances the collision frequency by (omega/nu)2/3 >>1, resulting in an effective resonant particle collision time of tau_int ~ (nu /omega)2/3 nu <<1/ nu . A rigorous collisional boundary layer analysis generalizes the standard quasilinear operator to a form that is fully consistent with Kennel-Englemann, but allows replacing the delta function appearing in the diffusivity with a simple integral (having the appropriate delta function limit) retaining the new physics associated with the narrow boundary layer, while preserving the entropy production principle. The limitations of the collisional boundary layer treatment are also estimated, and indicate that substantial departures from Maxwellian are not permitted.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NASA NTRS1985

Estimation and comparison of quasilinear electron heating in the shock foot at Jupiter and earth

A simple quasilinear model is developed to estimate the electron heating that occurs in the foot of a supercritical, quasiperpendicular shock through interactions with electrostatic waves generated by reflected ions. At earth the increase in electron thermal energy calculated using the measured wave amplitudes is negligible, while at Jupiter it is comparable with the observed temperature gain across the shock. The anisotropic quasilinear heating should destabilize whistler mode waves in the foot. These have been detected by the plasma wave instrument on Voyager with amplitudes sufficient to isotropize the electron distribution.

Moses, S. L.↗

NASA NTRS1989

Use of the quasilinearization algorithm for the simulation of LSS slewing

The use of the Maximum Principle for the large angle slewing of large space structures (LSS) usually results in the so-called two-point boundary-value problem, in which many requirements (e.g., minimum time, small amplitude, and limited control power, etc.) must be satisfied simultaneously. The successful solution of this problem depends largely on the use of an efficient numerical algorithm. There are many candidate algorithms available for this problem (e.g., quasilinearization, gradient, etc.). Here researchers discuss only the quasilinearization method which has been used for several cases of large angle slewing of LSS. The basic idea of this algorithm is to make a series of successive approximations of the solution from a particular solvable case (linear or nonlinear) to a more general practical case. For the rigid spacecraft slewing problem with no constraints on the controls, the solution procedure can be found in the literature. This procedure needs to be modified if a minimum time for the slewing problem is desired with control limits given. Recently, an indirect method for finding the minimum time was developed to meet all these requirements. For the general mixed (including both rigid and flexible parts) problem, an additional constraint of small vibrational amplitude on the flexible parts is imposed.

Li, Feiyue↗

NASA NTRS1990

Quasilinear saturation of forced current sheet tearing modes

Numerical studies of tearing modes in a nearly singular forced current sheet equilibrium (Liewer and Payne, 1990) show that the modes saturate quasilinearly when the width of the magnetic island formed by the reconnection is on the order of several times the linear mode width which scales as approximately (kS) exp -2/5, where S is the Lundquist number and k is the wavenumber. The modes saturate quasilinearly by flattening the current profile, converting magnetic energy into plasma energy. The longer wavelength modes, which saturate at higher levels, release the most energy. These modes may, nonlinearly, play a role in coronal heating when sharp current sheets form as a result of global magnetic stresses.

Liewer, Paulett C.↗

NASA NTRS1993

Control of nonlinear systems represented in quasilinear form

Methods to synthesize controllers for nonlinear systems are developed by exploiting the fact that under mild differentiability conditions, systems of the form: x-dot = f(x) + G(x)u can be represented in quasilinear form, viz: x-dot = A(x)x + B(x)u. Two classes of control methods are investigated. The first is zero-look-ahead control, where the control input depends only on the current values of A(x) and B(x). For this case the control input is computed by continuously solving a matrix Riccati equation as the system progresses along a trajectory. The second is controllers with look-ahead, where the control input depends on the future behavior of A(x) and B(x). These controllers use the similarity between quasilinear systems and linear time varying systems to find approximate solutions to optimal control type problems. The methods that are developed are not guaranteed to be globally stable. However in simulation studies they were found to be useful alternatives for synthesizing control laws for a general class of nonlinear systems.

Coetsee, Josef A.↗

NASA NTRS1995

Putting quasilinear theory to the test: A comparison of analytical theory with results from a trajectory code

Parallel and perpendicular diffusion coefficients were computed numerically by following particle orbits in a simulated magnetic field. The simulated field was chosen to have delta B/B(sub o) small, so as to provide a test of quasilinear theory in a regime where the theory should be most accurate. The simulation space is large enough to contain many magnetic field correlation lengths, so that effects of field line random walk can be studied. After presenting results for parallel diffusion, we will focus on two controversial issues relating to perpendicular diffusion: (1) Do quasilinear descriptions of perpendicular diffusion retain any validity for particles whose Larmor radius is smaller than a correlation length? (2) Does field line random walk lead to particle diffusion in the usual sense, or does it produce 'compound' diffusion for which particles spread out proportionally to t(exp 1/4) instead of t(exp 1/2)?

Bieber, J. W.↗

DOE OSTI2023

A conservative Galerkin solver for the quasilinear diffusion model in magnetized plasmas

We propose a conservative Galerkin scheme for the quasilinear model in three-dimensional momentum space and three-dimensional spectral space, with cylindrical symmetry. We construct an unconditionally conservative weak form and use a discretization that preserves conservation properties independent of the wave emission probability. The discrete operators, combined with a consistent quadrature rule, preserve all the conservation laws rigorously. The proposed scheme is quite general: it works for both relativistic and non-relativistic systems, for both magnetized and unmagnetized plasmas, and even for problems with time-dependent dispersion relations. We represent the particle distribution by continuous basis functions and use discontinuous basis functions for the wave spectral energy density, which enables the application of a positivity-preserving technique. We adopt the marching simplex algorithm, designed initially for computer graphics, for numerical integration on the resonance manifold. Furthermore, the numerical examples with a bump-on-tail initial configuration show how the unstable waves produce strong momentum space diffusion.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗