Quasilinear Theory Research Articles

Quasi-linear theory of forced magnetic reconnection for the transition from the linear to the Rutherford regime

Using the in-viscid two-field reduced magneto-hydrodynamic (MHD) model, a new analytical theory is developed to unify the Hahm–Kulsrud–Taylor (HKT) linear solution and the Rutherford quasi-linear regime. Adopting a quasi-linear approach, we obtain a closed system of equations for plasma response in the Taylor problem. An integral form of an analytical solution is obtained for the forced magnetic reconnection, uniformly valid throughout the entire regimes from the HKT linear solution to the Rutherford quasi-linear solution. In particular, the quasi-linear effect can be described by a single coefficient , where and ψ c are the Lundquist number and amplitude of external magnetic perturbation, respectively. The HKT linear solution for the response can be recovered when the index K s → 0. On the other hand, the quasi-linear effect plays a key role in the island growth when K s ∼ 1. Our new analytical solution has also been compared with reduced MHD simulations with agreement.

Acceleration of Ring Current Protons Driven by Magnetosonic Waves: Comparisons of Test Particle Simulations with Quasilinear Calculations

Abstract Wave–particle interactions play important roles in local acceleration and pitch angle scattering loss of charged particle dynamics in astrophysics and space physics, and energy transfer between plasma waves and particles is a key problem in plasma physics. Using full relativistic test particle simulations, we quantitatively evaluate the magnetosonic wave-driven acceleration of Earth's ring current protons both inside and outside the plasmapause at L = 4.5. Analysis of resonance conditions indicates that the broadband magnetosonic waves can affect protons by multiple harmonic cyclotron resonances. With a representative amplitude ∼200 pT, magnetosonic waves can produce energy diffusion coefficients, obtained from test particle simulations, at rates of ∼10−4 s−1 for protons ∼1–3 keV inside the plasmapause and ∼3–30 keV outside the plasmapause, both at equatorial pitch angles ∼75°–90°. Compared with the quasilinear theory calculations, the test particle diffusion coefficients exhibit a difference due to the nonresonant transit time effect outside the plasmapause. Subsequent 2D Fokker–Planck simulations indicate that on a timescale of ∼2 days, magnetosonic waves can effectively enhance the phase space density of the ring current protons ∼10 keV inside the plasmapause and ∼100 keV outside the plasmapause at equatorial pitch angles ∼60°–90°, thereby leading to the evolvement of the ring current proton distribution from a Kappa-type to a 90°-peaked profile. Our results demonstrate that the local acceleration of the ring current protons by magnetosonic waves is an important process of energy transfer between different populations of magnetospheric protons and hence contributes to the dynamic evolution of Earth's ring current.

Scattering effect of very low frequency transmitter signals on energetic electrons in Earth’s inner belt and slot region

Whistler mode very low frequency (VLF) waves from man-made ground-based transmitters in a frequency range of 10–30 kHz are mainly used for submarine communication, and they propagate primarily in the Earth-lower ionosphere waveguide and part of their energy can leak into the inner magnetosphere, leading the energetic electrons in inner radiation belt and slot region to precipitate into atmosphere and then affect the energetic electron dynamics in the near-Earth space. The scattering effects of artificial VLF signals from NWC, NAA and DHO38 transmitters on energetic electrons in Earth’s inner belt and slot region are investigated in detail in this work. Based on the quasi-linear theory and the Full Diffusion Code, we calculate the bounce-average pitch angle diffusion coefficients induced by NWC, NAA and DHO38 VLF transmitter signals, for which the resonance harmonics |<i>N</i>| ≤ 10 are considered, respectively. We further implement the one-dimensional Fokker-Planck diffusion simulations by using the available pitch angle diffusion rates to model the dynamic evolutions of energetic electrons caused by the scattering of the VLF transmitter signals in the inner belt and slot region in 200 d. The simulation results indicate that the NWC VLF transmitter signals are dominant in scattering ~100 keV electrons with pitch angles less than 60° at <i>L</i> ≤ 1.8, and the mainly scattered electron energy values increase with <i>L</i>-shell decreasing , from <i>L</i> = 1.8 to <i>L </i>= 1.5, the mainly scattered electron energy increases from 90–120 keV to 550–650 keV. The NAA and DHO38 VLF transmitter signals are important in scattering < 20 keV electrons with pitch angles less than 70° at higher <i>L</i>-shells (2.2 ≤ <i>L</i> ≤ 2.7), from <i>L</i> = 2.2 to <i>L</i> = 2.7, the mainly scattered electron energy decreases from 10–20 keV to several keV. The VLF transmitter signals are found to have a slight influence on the loss of energetic electrons with pitch angles larger than 80°.

Stress relaxation of porcine tendon under simulated biological environment: experiment and modeling

The aim of the study was to investigate the viscoelastic response in the low and high physiological strain with the use of experimental and modeling approach. Viscoelastic response in the low, transition and high physiologic strain (3, 6 and 9%) with consideration of simulated biological environment (0.9% saline solution, 37 °C) was measured in relaxation tests. Preconditioning of tendons was considered in the testing protocol and the applied range of load was obtained from tensile testing. The quasi-linear viscoelasticity theory was used to fit experimental data to obtain constants (moduli and times of relaxation), which can be used for description of the viscoelastic behavior of tendons. The exponential non-linear elastic representation of the stress response in ramp strain was also estimated. Differences between stress relaxation process can be seen between tendons stretched to the physiological strain range (3%) and exceeding this range (6 and 9%). The strains of 6% and 9% showed a similar stress relaxation trend displaying relatively rapid relaxation for the first 70 seconds, whereas the lowest strain of 3% displayed relatively slow relaxation. Results of the model fitting showed that the quasi-linear viscoelastic model gives the best fit in the range of low physiological strain level.

Viscoelastic characterization of porcine brain tissue mechanical properties under indentation loading

The goal of this study was to measure the mechanical properties of porcine brain tissue and determine if they were dependent on anatomical region or direction. Multistep stress relaxation indentations with a cylindrical probe were performed at 10, 20, and 30% nominal strain on multiple regions in the sagittal, horizontal, and coronal planes. Linear and nonlinear (using the quasilinear theory of viscoelasticity [QLV]) constitutive formulations were applied to extract parameters to capture the mechanical behavior of brain tissue. The linear viscoelastic analytic approach provided the best fit to the experimental data of the models tested. Within each directional plane there were region-dependent differences. The cerebellum was the softest region within each loading direction. Although the majority of the regions were isotropic, the cerebellum white matter and thalamus were anisotropic. The characterization of these mechanical properties can be used to inform finite element models of the pig brain to help predict a more biofidelic response in animal models of traumatic brain injury. Statement of SignificanceFinite element models been developed to predict brain tissue response to traumatic brain injury (TBI) to advance protective and preventative strategies. In order to improve the accuracy of these computational models, appropriate mechanical experimentation is required to identify brain viscoelasticity, heterogeneity, and anisotropy. Our custom indentation design allows for high spatial resolution to characterize mechanical properties based on anatomical region and loading direction. Due to the challenges in procuring human brain tissue, porcine brain models are a suitable substitute to study TBI based on its structural similarities to that of human brains. This study will further illuminate the complexity of brain tissue mechanics in response to injury loading.

Study on electron stochastic motions in the magnetosonic wave field: Test particle simulations

Using the test particle simulation method, we investigate the stochastic motion of electrons with energy of 300 keV in a monochromatic magnetosonic (MS) wave field. This study is motivated by the violation of the quasi-linear theory assumption, when strong MS waves (amplitude up to ~1 nT) are present in the Earth’s magnetosphere. First, electron motion can become stochastic when the wave amplitude exceeds a certain threshold. If an electron initially resonates with the MS wave via bounce resonance, as the bounce resonance order increases, the amplitude threshold of electron stochastic motion increases until it reaches the peak at about the 11th order in our study, then the amplitude threshold slowly declines. Further, we find that the coexistence of bounce and Landau resonances between electrons and MS waves will significantly reduce the amplitude threshold. In some cases, the electron motion can become stochastic in the field of an MS wave with amplitudes below 1 nT. Regardless, if neither the bounce nor Landau resonance condition is satisfied initially, then the amplitude threshold of stochastic motion shows an increasing trend for lower frequencies and a decreasing trend for higher frequencies, even though the amplitude threshold is always very large (> 5 nT). Our study suggests that electron stochastic motion should also be considered when modeling electron dynamics regulated by intense MS waves in the Earth’s magnetosphere.

Testing physical models for cosmic ray transport coefficients on galactic scales: self-confinement and extrinsic turbulence at ∼GeV energies

ABSTRACT The microphysics of ∼ GeV cosmic ray (CR) transport on galactic scales remain deeply uncertain, with almost all studies adopting simple prescriptions (e.g. constant diffusivity). We explore different physically motivated, anisotropic, dynamical CR transport scalings in high-resolution cosmological Feedback In Realistic Environment (FIRE) simulations of dwarf and ∼L* galaxies where scattering rates vary with local plasma properties motivated by extrinsic turbulence (ET) or self-confinement (SC) scenarios, with varying assumptions about e.g. turbulent power spectra on un-resolved scales, Alfvén-wave damping, etc. We self-consistently predict observables including γ-rays (Lγ), grammage, residence times, and CR energy densities to constrain the models. We demonstrate many non-linear dynamical effects (not captured in simpler models) tend to enhance confinement. For example, in multiphase media, even allowing arbitrary fast transport in neutral gas does not substantially reduce CR residence times (or Lγ), as transport is rate-limited by the ionized WIM and ‘inner CGM’ gaseous halo (104–106 K gas within $\lesssim 10\!-\!30\,$ kpc), and Lγ can be dominated by trapping in small ‘patches’. Most physical ET models contribute negligible scattering of ∼1–10 GeV CRs, but it is crucial to account for anisotropy and damping (especially of fast modes) or else scattering rates would violate observations. We show that the most widely assumed scalings for SC models produce excessive confinement by factors ≳100 in the warm ionized medium (WIM) and inner CGM, where turbulent and Landau damping dominate. This suggests either a breakdown of quasi-linear theory used to derive the CR transport parameters in SC, or that other novel damping mechanisms dominate in intermediate-density ionized gas.

Momentum-space-resolved measurements using oblique electron cyclotron emission for the validation of the quasi-linear theory of electron cyclotron current drive

Electron cyclotron resonance heating (ECRH) can drive large current densities through electron cyclotron current drive (ECCD). ECCD is expected to be crucial for high-performance plasmas in future fusion reactors like ITER and DEMO, making the current drive efficiency of ECCD a critical design parameter for future reactors. In present-day devices, good agreement between measured and predicted current drive efficiency has been found. However, to ensure the reliability in future machines, a direct validation of the electron momentum distribution function is needed.As a first step towards this goal, we present in this paper oblique electron cyclotron emission (ECE) measurements of a low-density plasma in the ASDEX Upgrade tokamak. Two oblique ECE diagnostics are used to allow the simultaneous measurements of electrons streaming co- and counter-directionally with the plasma current. Predictions for the distribution function are computed with the bounce-averaged Fokker-Planck code RELAX. 9 9E Westerhof et al, Rijnhuizen report,1992. To allow direct comparison with the measurements, synthetic radiation temperatures are computed with the code ECRad. 10 10S Denk et al, Computer Physics Communications, p. 107175, 2020. Good agreement is found if radial transport occurring predominantly at low electron energies is included.We demonstrate that oblique ECE diagnostics measure the electron distribution function directly at the ECRH deposition site in phase space. Furthermore, they are sensitive to the abundance of pitch-angle scattered electrons that reduce the ECCD efficiency. Limitations and uncertainties of the measurements and the modeling are discussed.

Quasilinear dynamics of ordinary mode electromagnetic cyclotron instability driven by the interaction of rotating electron beam with magnetized plasma

The quasilinear theory is employed to study the temporal evolution of the cyclotron instability near the ordinary mode in a magnetized plasma in the presence of the rotating electron beam. The positive gradient region of the initial distribution function is eroded by the diffusion process driven by the wave fields. It is found that the wave non-linear effects modify the electron distribution function by transferring resonance particles to the valley between the plasma bulk and the bump on tail. This movement ultimately leads to flattening the distribution function in the interaction region, which is a saturation state.

Combined electron firehose and electromagnetic ion cyclotron instabilities: quasilinear approach

ABSTRACT Various plasma waves and instabilities are abundantly present in the solar wind plasma, as evidenced by spacecraft observations. Among these, propagating modes and instabilities driven by temperature anisotropies are known to play a significant role in the solar wind dynamics. In situ measurements reveal that the threshold conditions for these instabilities adequately explain the solar wind conditions at large heliocentric distances. This paper pays attention to the combined effects of electron firehose instability driven by excessive parallel electron temperature anisotropy (T⊥e < T∥e) at high beta conditions, and electromagnetic ion cyclotron instability driven by excessive perpendicular proton temperature anisotropy (T⊥i > T∥i). By employing quasilinear kinetic theory based upon the assumption of bi-Maxwellian velocity distribution functions for protons and electrons, the dynamical evolution of the combined instabilities and their mutual interactions mediated by the particles is explored in depth. It is found that while in some cases, the two unstable modes are excited and saturated at distinct spatial and temporal scales, in other cases, the two unstable modes are intermingled such that a straightforward interpretation is not so easy. This shows that when the dynamics of protons and electrons are mutually coupled and when multiple unstable modes are excited in the system, the dynamical consequences can be quite complex.

On the Generation of Compressible Mirror-mode Fluctuations in the Inner Heliosheath

Abstract Measurements made with the Voyager 1 spacecraft indicate that significant levels of compressive fluctuations exist in the inner heliosheath. Some studies have already been performed with respect to the mirror-mode instability in the downstream region close to the solar wind termination shock, and here we extend the investigation to the whole inner heliosheath. We employ quasilinear theory and results from a global magnetohydrodynamic model of the heliosphere to compute the time evolution of both the temperature anisotropy and the energy density of the corresponding magnetic fluctuations, and we demonstrate their likely presence in the inner heliosheath. Furthermore, we compute the associated, locally generated density fluctuations. The results can serve as inputs for future models of the transport of compressible turbulence in the inner heliosheath.

Learning how structures form in drift-wave turbulence

Drift-wave turbulence produces anomalous transport via cross-correlations between fluctuations. This transport has profound implications for confinement, structure formation, and virtually all aspects of the non-linear turbulent dynamics. In this work, we use a data-driven method based on deep learning in order to study turbulent transport in the 2D Hasegawa–Wakatani system and infer a reduced mean-field model from numerical solution. In addition to the usual turbulent diffusion, we find an effect which couples the particle flux to the local gradient of vorticity, which tends to modulate the density profile. The direct coupling to the shear is relatively weak. In addition, the deep learning method finds a model for spontaneous zonal flow generation by negative viscosity, stabilized by non-linear and hyperviscous terms. We compare these results to analytic calculations using quasilinear theory and wave kinetics, finding qualitative agreement, though the calculations miss certain higher-order effects. A simplified, 1-D model for the evolution of the profile, flow, and intensity based on the deep learning results is solved numerically and compared to previous models for staircasing based on bistability. We see that the physics uncovered by the deep learning method provided simple explanations for the formation of zonal structures in the density, flow, and turbulence fields. We highlight the important role of symmetry in the deep learning method and speculate on the portability of the method to other applications.

Turbulence-level dependence of cosmic ray parallel diffusion

ABSTRACT Understanding the transport of energetic cosmic rays belongs to the most challenging topics in astrophysics. Diffusion due to scattering by electromagnetic fluctuations is a key process in cosmic ray transport. The transition from a ballistic to a diffusive-propagation regime is presented in direct numerical calculations of diffusion coefficients for homogeneous magnetic field lines subject to turbulent perturbations. Simulation results are compared with theoretical derivations of the parallel diffusion coefficient’s dependences on the energy and the fluctuation amplitudes in the limit of weak turbulence. The present study shows that the widely used extrapolation of the energy scaling for the parallel diffusion coefficient to high turbulence levels predicted by quasi-linear theory does not provide a universally accurate description in the resonant-scattering regime. It is highlighted here that the numerically calculated diffusion coefficients can be polluted for low energies due to missing resonant interaction possibilities of the particles with the turbulence. Five reduced-rigidity regimes are established, which are separated by analytical boundaries derived in this work. Consequently, a proper description of cosmic ray propagation can only be achieved by using a turbulence-level-dependent diffusion coefficient and can contribute to solving the Galactic cosmic ray gradient problem.

Impurity pinch generated by trapped particle driven turbulence

Impurity turbulent transport in tokamak plasmas is investigated in the case of trapped ion mode (TIM) and trapped electron mode (TEM) using a non-linear bounce-averaged gyrokinetic model. In particular, we focus on the thermo-diffusive pinch and the curvature-driven pinch observed in non-linear simulations. Thermodiffusion is analyzed independently of curvature pinch by disabling the latter artificially. A parameter scan shows that the direction of thermodiffusion depends mainly on two factors. First, it depends on the sign of the turbulence phase velocity, and the resulting transport is inward in the case of TEM whereas it is outward in the case of TIM. Second, it depends on the sign of the impurity temperature gradient. Moreover, the magnitude of the thermodiffusion tends to decrease with respect to the impurity charge Z, while it is proportional to the intensity of background turbulence. Lastly, the direction of the curvature pinch is investigated as a function of the magnetic shear. The results of non-linear simulations are compared with quasi-linear theory. The Kubo number is calculated for our non-linear simulations to justify this comparison. Quasi-linear theory is found to be in qualitative agreement with our gyrokinetic numerical observations.

Global Survey of Plasma Sheet Electron Precipitation due to Whistler Mode Chorus Waves in Earth's Magnetosphere

AbstractWhistler mode chorus waves can scatter plasma sheet electrons into the loss cone and produce the Earth's diffuse aurora. Van Allen Probes observed plasma sheet electron injections and intense chorus waves on 24 November 2012. We use quasilinear theory to calculate the precipitating electron fluxes, demonstrating that the chorus waves could lead to high differential energy fluxes of precipitating electrons with characteristic energies of 10–30 keV. Using this method, we calculate the precipitating electron flux from 2012 to 2019 when the Van Allen Probes were near the magnetic equator and perform global surveys of electron precipitation under different geomagnetic conditions. The most significant electron precipitation due to chorus is found from the nightside to dawn sectors over 4 < L < 6.5. The average total precipitating energy flux is enhanced during disturbed conditions, with time‐averaged values reaching ~3–10 erg/cm2/s when AE ≥ 500 nT.

Particle acceleration in relativistic turbulence: A theoretical appraisal

The authors provide a comprehensive study, both analytical and numerical, of cosmic ray acceleration in relativistic magnetohydrodynamic (MHD) turbulence. Effects beyond the magnetostatic quasi-linear theory, such as resonance broadening due to large-scale turbulence modes, as well as a non-MHD effect, are incorporated.

Wave-driven non-classical electron transport in a low temperature magnetically expanding plasma

The presence of instabilities in a low density, low temperature plasma expanding through an axially symmetric magnetic nozzle is investigated in the context of non-classical electron cross field transport. Electrostatic probes are used to characterize the background plasma properties and instabilities. The measurements show a primarily azimuthally propagating mode with a broad, incoherent power spectrum that appears linear at low frequencies. It is demonstrated that the observed dispersion is consistent with the lower hybrid drift instability. The energy and linear growth rate of this wave are related through quasilinear theory to an effective electron collision frequency that is shown to be dominant over classical collisions.

A comparison between kinetic theory and particle-in-cell simulations of anomalous electron transport in E×B plasma discharges

Understanding anomalous electron transport in E×B discharges remains a key challenge in the development of self-consistent models of these systems. It has been shown that short-wavelength, high-frequency instabilities in the azimuthal E×B direction may be responsible for increased electron transport due to an enhanced electron-ion friction force. Although a theoretical model based on quasi-linear kinetic theory has previously been proposed to describe this friction force, it has so far only undergone limited validation testing. Here, we rigorously assess this theoretical model by comparison with the friction force self-consistently obtained from 2D axial-azimuthal particle-in-cell simulations. The simulation geometry is based on a recently established benchmark configuration for E×B discharges, and a broad parametric study is performed by varying the magnetic field strength, the discharge current density, and the presence of different neutral collisional processes. Overall, the theory is found to be in very good agreement with the simulation results for all cases studied, verifying the underlying physical mechanisms leading to enhanced electron transport. We demonstrate, however, that the friction force depends sensitively on the shape of the electron velocity distribution function, thus posing significant challenges to fully self-consistent, first principles modeling of anomalous transport in fluid simulations.

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 & 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 scatter them out of and into resonance, as for Landau damping as shown by Johnston (Phys. Fluids, vol. 14, 1971, pp. 2719–2726) and Auerbach (Phys. Fluids, vol. 20, 1977, pp. 1836–1844). As a result, the resonant particle–wave interactions occur in the centre of a narrow collisional boundary when the collision frequency $\unicode[STIX]{x1D708}$ is very small compared to the wave frequency $\unicode[STIX]{x1D714}$ . The diffusive nature of the pitch angle scattering combined with the wave–particle resonance condition enhances the collision frequency by $(\unicode[STIX]{x1D714}/\unicode[STIX]{x1D708})^{2/3}\gg 1$ , resulting in an effective resonant particle collisional interaction time of $\unicode[STIX]{x1D70F}_{\text{int}}\sim (\unicode[STIX]{x1D708}/\unicode[STIX]{x1D714})^{2/3}/\unicode[STIX]{x1D708}\ll 1/\unicode[STIX]{x1D708}$ . A 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.

Thermal ions heat transport induced by reversed shear Alfvén eigenmode on the HL-2A tokamak

Experimental investigations of thermal ions heat transport induced by reversed shear Alfvén eigenmodes (RSAEs) have been carried out on the HL-2A tokamak. It is found that the RSAEs are driven unstable by passing particles during the second half of sawtooth periods, and can not only degrade the bremsstrahlung radiations, but also cause obvious reduction of the ion temperature in the core plasma. Interestingly, the RSAEs excited by energetic ions can resonant with the thermal ions, and then lead to a heat transport process. Statistical results suggest there is a quadratic dependence between thermal ion heat flux perturbation and mode amplitude, which indicates a diffusive mechanism of plasma transport and is well explained by the theoretical interpretations derived from quasi-linear transport theory.