Quasi-linear Diffusion Coefficients Research Articles

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

AbstractA 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.

Modeling the Simultaneous Dropout of Energetic Electrons and Protons by EMIC Wave Scattering

AbstractElectromagnetic ion cyclotron (EMIC) waves could cause a simultaneous dropout of radiation belt electrons and ring current protons. However, their effects on the dropout of both plasma populations have not been quantified in previous studies. In this paper, we model the simultaneous dropout of MeV electrons and hundreds of keV protons observed by Van Allen Probes within ∼40 min on 27 February 2014. The wave and particle measurements during the period of most intense EMIC waves at L ∼ 5.2 are used to calculate the quasilinear diffusion coefficients and simulate the evolution of both energetic electrons and protons. Our model well captures the dropout of electrons with energies >1 MeV and pitch angles <75°, and the concurrent dropout of protons with energies >200 keV and pitch angles >40°. This is the first modeling work quantitatively reproducing the simultaneous dropout of both populations due to EMIC wave scattering.

Temporal variability of quasi-linear pitch-angle diffusion

Kinetic wave-particle interactions in Earth’s outer radiation belt energize and scatter high-energy electrons, playing an important role in the dynamic variation of the extent and intensity of the outer belt. It is possible to model the effects of wave-particle interactions across long length and time scales using quasi-linear theory, leading to a Fokker-Planck equation to describe the effects of the waves on the high energy electrons. This powerful theory renders the efficacy of the wave-particle interaction in a diffusion coefficient that varies with energy or momentum and pitch angle. In this article we determine how the Fokker-Planck equation responds to the temporal variation of the quasi-linear diffusion coefficient in the case of pitch-angle diffusion due to plasmaspheric hiss. Guided by in-situ observations of how hiss wave activity and local number density change in time, we use stochastic parameterisation to describe the temporal evolution of hiss diffusion coefficients in ensemble numerical experiments. These experiments are informed by observations from three different example locations in near-Earth space, and a comparison of the results indicates that local differences in the distribution of diffusion coefficients can result in material differences to the ensemble solutions. We demonstrate that ensemble solutions of the Fokker-Planck equation depend both upon the timescale of variability (varied between minutes and hours), and the shape of the distribution of diffusion coefficients. Based upon theoretical construction of the diffusion coefficients and the results presented here, we argue that there is a useful maximum averaging timescale that should be used to construct a diffusion coefficient from observations, and that this timescale is likely less than the orbital period of most inner magnetospheric missions. We discuss time and length scales of wave-particle interactions relative to the drift velocity of high-energy electrons and confirm that arithmetic drift-averaging is can be appropriate in some cases. We show that in some locations, rare but large values of the diffusion coefficient occur during periods of relatively low number density. Ensemble solutions are sensitive to the presence of these rare values, supporting the need for accurate cold plasma density models in radiation belt descriptions.

Weak Turbulence and Quasilinear Diffusion for Relativistic Wave-Particle Interactions Via a Markov Approach

We derive weak turbulence and quasilinear models for relativistic charged particle dynamics in pitch-angle and energy space, due to interactions with electromagnetic waves propagating (anti-)parallel to a uniform background magnetic field. We use a Markovian approach that starts from the consideration of single particle motion in a prescribed electromagnetic field. This Markovian approach has a number of benefits, including: 1) the evident self-consistent relationship between a more general weak turbulence theory and the standard resonant diffusion quasilinear theory (as is commonly used in e.g. radiation belt and solar wind modeling); 2) the general nature of the Fokker-Planck equation that can be derived without any prior assumptions regarding its form; 3) the clear dependence of the form of the Fokker-Planck equation and the transport coefficients on given specific timescales. The quasilinear diffusion coefficients that we derive are not new in and of themselves, but this concise derivation and discussion of the weak turbulence and quasilinear theories using the Markovian framework is physically very instructive. The results presented herein form fundamental groundwork for future studies that consider phenomena for which some of the assumptions made in this manuscript may be relaxed.

Unusual high frequency EMIC waves: Detailed analysis of EMIC wave excitation and energy coupling between EMIC and magnetosonic waves

• Testing high frequency EMIC waves generation by anisotropic suprathermal protons. • Cyclotron resonance with magnetosonic waves is inefficient in thermal proton heating. • Quantitative understanding of the magnetosonic-EMIC energy coupling process is needed. A recent event study suggested a coupling process whereby suprathermal protons (10–100 eV), as a result of perpendicular heating by pre-existing magnetosonic waves, can excite electromagnetic ion cyclotron (EMIC) waves with frequency near the local proton cyclotron frequency ( f ~ 0.95 f cp ). The present study tests this coupling process. First, one-dimensional hybrid (kinetic ions/massless fluid electrons) simulations of parallel-propagating EMIC waves initialized with the plasma conditions derived from the event prove that high frequency EMIC waves can be generated by anisotropic suprathermal protons instead of more energetic (10s of keV) ring current protons. Calculation of the quasilinear pitch-angle diffusion coefficient for electrons using the simulated waves suggests that these high frequency EMIC waves outside the plasmasphere can play a role in the pitch-angle scattering of ~ MeV electrons due to the large wavenumber of the excited EMIC waves. Second, the role of the pre-existing magnetosonic waves in the suprathermal proton heating is examined using quasilinear diffusion theory and simple test particle calculation. It is found that the quasilinear diffusion becomes ineffective in energies relevant to the suprathermal protons, indicating that the low-energy ( < 10 eV) protons cannot be efficiently energized to the observed level through multi-harmonic cyclotron resonances with the magnetosonic waves observed. Rather, some kind of non-resonant process may have been at play, if the magnetosonic waves were actually involved in the heating. This result suggests that more quantitative understanding of the suggested energy coupling process through magnetosonic waves is needed.

Variability of Quasilinear Diffusion Coefficients for Plasmaspheric Hiss

AbstractIn the outer radiation belt, the acceleration and loss of high‐energy electrons is largely controlled by wave‐particle interactions. Quasilinear diffusion coefficients are an efficient way to capture the small‐scale physics of wave‐particle interactions due to magnetospheric wave modes such as plasmaspheric hiss. The strength of quasilinear diffusion coefficients as a function of energy and pitch angle depends on both wave parameters and plasma parameters such as ambient magnetic field strength, plasma number density, and composition. For plasmaspheric hiss in the magnetosphere, observations indicate large variations in the wave intensity and wave normal angle, but less is known about the simultaneous variability of the magnetic field and number density. We use in situ measurements from the Van Allen Probe mission to demonstrate the variability of selected factors that control the size and shape of pitch angle diffusion coefficients: wave intensity, magnetic field strength, and electron number density. We then compare with the variability of diffusion coefficients calculated individually from colocated and simultaneous groups of measurements. We show that the distribution of the plasmaspheric hiss diffusion coefficients is highly non‐Gaussian with large variance and that the distributions themselves vary strongly across the three phase space bins studied. In most bins studied, the plasmaspheric hiss diffusion coefficients tend to increase with geomagnetic activity, but our results indicate that new approaches that include natural variability may yield improved parameterizations. We suggest methods like stochastic parameterization of wave‐particle interactions could use variability information to improve modeling of the outer radiation belt.

A positive-definite form of bounce-averaged quasilinear velocity diffusion for the parallel inhomogeneity in a tokamak

In this paper, the analytical form of the quasilinear diffusion coefficients is modified from the Kennel–Engelmann diffusion coefficients to guarantee the positive definiteness of its bounce average in a toroidal geometry. By evaluating the parallel inhomogeneity of plasmas and magnetic fields in the trajectory integral, we can ensure the positive definiteness and help illuminate some non-resonant toroidal effects in the quasilinear diffusion. When the correlation length of the plasma-wave interaction is comparable to the magnetic field variation length, the variation becomes important and the parabolic variation at the outer-midplane, the inner-midplane, and trapping tips can be evaluated by Airy functions. The new form allows the coefficients to include both resonant and non-resonant contributions, and the correlations between the consecutive resonances and in many poloidal periods. The positive-definite form is implemented in a wave code TORIC and we present an example for ITER using this form.

Quasilinear diffusion coefficients in a finite Larmor radius expansion for ion cyclotron heated plasmas

In this paper, a reduced model of quasilinear velocity diffusion by a small Larmor radius approximation is derived to couple the Maxwell's equations and the Fokker Planck equation self-consistently for the ion cyclotron range of frequency waves in a tokamak. The reduced model ensures the important properties of the full model by Kennel-Engelmann diffusion, such as diffusion directions, wave polarizations, and H-theorem. The kinetic energy change (Ẇ) is used to derive the reduced model diffusion coefficients for the fundamental damping (n = 1) and the second harmonic damping (n = 2) to the lowest order of the finite Larmor radius expansion. The quasilinear diffusion coefficients are implemented in a coupled code (TORIC-CQL3D) with the equivalent reduced model of the dielectric tensor. We also present the simulations of the ITER minority heating scenario, in which the reduced model is verified within the allowable errors from the full model results.

A new quasilinear formulation for ICRF plasmas in a toroidal geometry

We present a new formulation for quasilinear velocity space diffusion for ICRF plasmas that considers two different aspects: (1) finite Larmor radius approximation and (2) includes the effect of toroidal geometry and constructs a positive definite form. In the first aspect, the Kennel-Engelmann (K-E) quasilinear diffusion coefficients are successfully approximated in a small Larmor radius limit and implemented for the numerical codes (TORIC-CQL3D). In the second aspect, the quasilinear diffusion is reformulated in a toroidal geometry in order to include the parallel dynamics in the inhomogeneous plasmas and magnetic fields. We use these two quasilinear formulations to simulate ITER plasmas with ICRF injection for minority fundamental heating and Tritium second harmonic cyclotron heating.

DE-1 versus Cluster spectral density observations: the effect on ion outflow above the polar cap

&lt;p&gt;Wave-particle interaction is a very important mechanism in describing the outflow of ions at high latitudes and high altitudes. Quasi-linear perpendicular velocity diffusion coefficients are used to describe the effect of wave-particle interactions, therefore it is essential to determine the correct diffusion coefficients that must be used to model the outflow of ions. In this study a Monte Carlo method is used to assess the role of different diffusion coefficients for O+ and H+ ions at high altitudes above the polar cap. Two different sets of diffusion coefficients obtained from Barghouthi [1997]; Barghouthi et al. [1998] and Nilsson et al. [2013] are used. Barghouthi [1997]; Barghouthi et al. [1998] used spectral density measurements from Dynamic Explorer 1 spacecraft (DE-1) observations to calculate the diffusion coefficients, while Nilsson et al. [2013] used spectral density measurements from the Cluster spacecraft to obtain the diffusion coefficients. It was found that diffusion coefficients from Barghouthi [1997]; Barghouthi et al. [1998] in the cusp (aurora) and central polar cap (polar wind) respectively, describe well the observations of ion outflow at altitudes lower than 5 RE, but yield unreasonably high parallel velocities and temperatures at higher altitudes. Also diffusion coefficients from Cluster spectral density measurements produce reasonable results for high altitudes and unreasonably low parallel velocities and temperatures for the low altitude region. Therefore it is suggested that a combination of these diffusion coefficients is used where the diffusion coefficients given by Barghouthi [1997]; Barghouthi et al. [1998] are used at low altitudes and the diffusion coefficients obtained from Cluster measurements are used at high altitudes.&lt;/p&gt;

The effects of phase decorrelation on the dynamics of the bump-on-tail instability

The dynamics of the bump-on-tail instability has been studied. The novelty of the work is the analysis of how the bump-on-tail dynamics is affected by an extrinsic stochastisation of the phase of the wave-particle interaction; here referred to as phase decorrelation. For this purpose, a nonlinear Monte Carlo model has been developed. When the characteristic time scale for macroscopic phase decorrelation becomes shorter than time scales of nonlinear wave-particle dynamics, the system may be described quasilinearly, with the phase decorrelation being replaced by a quasilinear diffusion coefficient in particle energy. A purely quasilinear Monte Carlo model, which is typically less computationally demanding than the fully nonlinear description due to the reduced dimensionality of phase space, has been developed for comparison. In this paper, parameter regimes, where the nonlinear and the quasilinear descriptions quantitatively agree on a macroscopic level, have been investigated, using combined theoretical and numerical analyses. Qualitative effects on the macroscopic dynamics by the presence of phase decorrelation and/or by structures of the energy distribution function in the proximity of the wave-particle resonance are also studied.

Self-consistent quasi-linear modelling of Lower Hybrid Current Drive in ITER and DEMO

First pass absorption of the Lower Hybrid waves in thermonuclear devices like ITER and DEMO is modelled by coupling the ray tracing equations for the wave phase and amplitude with the quasi-linear evolution of the electron distribution function. A system of coupled ordinary differential equations for each Fourier component of the spectrum radiated by the LH antenna is derived and solved when considering both 1D/2D Fokker-Planck model for the electron distribution function. This allows to reconstruct and to evolve the quasi-linear diffusion coefficient consistently with the wave propagation, to calculate the power deposition profile and the amount of current driven by the wave. As usually assumed, the Lower Hybrid Current Drive is not effective in a plasma of a tokamak fusion reactor like ITER or DEMO, because the high electron temperature would enhance the wave absorption and then limit the RF power deposition to the very periphery of the plasma column (near the separatrix). In this work by extensively using this self-consistent modelling for the propagation and absorption of the LH wave, a parametric study on the wave spectrum (and consequently on the antenna design) as spectrum width, peak value, secondary lobes etc. has been performed very accurately. Such a careful investigation aims at controlling the power deposition layer possibly in the external half radius of the plasma, thus providing a valuable aid to the solution of how to control the plasma current profile in a toroidal magnetic configuration. This analysis is useful not only for exploring the possibility of profile control of a pulsed operation reactor, but also in order to reconsider the feasibility of steady state regime.

Activity‐dependent global model of electron loss inside the plasmasphere

Using data from the CRRES plasma wave experiment, we develop quadratic fits to the mean of the wave amplitude squared for plasmaspheric hiss as a function of Kp, L, and magnetic latitude (λ) for the dayside (6 < magnetic local time (MLT) ≤ 21) and nightside (21 < MLT ≤ 6) magnetic local time sectors. The empirical model of hiss waves is used to compute quasi-linear pitch angle diffusion coefficients for energetic, relativistic, and ultrarelativistic electrons in the energy range of 1 keV to 10 MeV. In our calculations, we account for changes in hiss wave normal angle and plasma density with increasing λ. Electron lifetimes are then calculated from the diffusion coefficients and parameterized as a function of energy, Kp, and L. Coefficients for both the hiss model and the electron lifetimes are provided and can be easily incorporated into existing diffusion, convection, and particle tracing codes.

Resonant scattering of energetic electrons in the plasmasphere by monotonic whistler-mode waves artificially generated by ionospheric modification

Abstract. Modulated high-frequency (HF) heating of the ionosphere provides a feasible means of artificially generating extremely low-frequency (ELF)/very low-frequency (VLF) whistler waves, which can leak into the inner magnetosphere and contribute to resonant interactions with high-energy electrons in the plasmasphere. By ray tracing the magnetospheric propagation of ELF/VLF emissions artificially generated at low-invariant latitudes, we evaluate the relativistic electron resonant energies along the ray paths and show that propagating artificial ELF/VLF waves can resonate with electrons from ~ 100 keV to ~ 10 MeV. We further implement test particle simulations to investigate the effects of resonant scattering of energetic electrons due to triggered monotonic/single-frequency ELF/VLF waves. The results indicate that within the period of a resonance timescale, changes in electron pitch angle and kinetic energy are stochastic, and the overall effect is cumulative, that is, the changes averaged over all test electrons increase monotonically with time. The localized rates of wave-induced pitch-angle scattering and momentum diffusion in the plasmasphere are analyzed in detail for artificially generated ELF/VLF whistlers with an observable in situ amplitude of ~ 10 pT. While the local momentum diffusion of relativistic electrons is small, with a rate of &lt; 10−7 s−1, the local pitch-angle scattering can be intense near the loss cone with a rate of ~ 10−4 s−1. Our investigation further supports the feasibility of artificial triggering of ELF/VLF whistler waves for removal of high-energy electrons at lower L shells within the plasmasphere. Moreover, our test particle simulation results show quantitatively good agreement with quasi-linear diffusion coefficients, confirming the applicability of both methods to evaluate the resonant diffusion effect of artificial generated ELF/VLF whistlers.

Relationship between the quasi-linear diffusion coefficients and the key parameters of spatial energetic electrons

It has been proved that the ground-based electromagnetic wave can transfer into ionosphere and interact with high-energy particles. By changing the pitch angle and momentum, the particles are imposed to enter the bounce loss cone and drift loss cone, then electron precipitation takes place and the particle bursts form. In recent decades, the relationship has been observed among electromagnetic disturbance and particle bursts and seismic activity based on satellite data. Here, by wave-particle cyclotron resonant interaction combined with the observation range of LEO satellite (about 350–1000 km), the evolvement trend of the pitch angle quasi-linear diffusion coefficients induced by field-aligned electromagnetic waves, is studied with the change of VLF electromagnetic wave frequency, band width, energies of electron (0.1–20 MeV) and L shell (L=1.1–3). We also show the relationship between VLF electromagnetic wave frequency and minimum energy of precipitation electron induced by it, under certain pitch angle value. The relationship among these quantities may be used to provide theoretical explanation for satellite observations of energetic particle precipitation examples, to provide guidance for extracting information associated with earthquakes from the detection of high-energy particles on the satellite, and to lay the foundation on the data analysis of China seismo-electromagnetic satellite planned to launch at about the end of 2016.

Quasilinear Theory for Relativistic Particles

Quasilinear theory is developed by using canonical variables for relativistic particles. It is self-consistent, including momentum, pitch-angle, and spatial diffusions. By assuming the wave field is a superposition of known toroidal and poloidal Fourier modes, the quasilinear diffusion coefficients are written in a form which can be directly evaluated by using the output of a spectral full-wave solver of Maxwell equations in toroidal plasmas. The formalism is special for tokamaks which are axis-symmetric, therefore, it is simple and suitable for simulations of cyclotron heating, current drive and radio-frequency wave induced radial transport in ITER. PACS: 52.35.Py, 52.50.Sw, 52.35.Fa.

Normal-Angle Dependence of the Interaction between Radiation Belt Electrons and Fast Magnetosonic Waves

Fast magnetosonic (MS) waves have been suggested to contribute significantly to radiation belt electron dynamics via Landau resonance and transit time scattering. The corresponding quasi-linear diffusion coefficients in pitch angle, energy and particularly cross-pitch-angle-energy are calculated from the gyro-averaged test-particle simulations. It is found that the cross diffusion coefficient is an effective indicator to differentiate between the contributions of resonant and non-resonant mechanisms. Furthermore, the dependence of diffusion coefficients on the normal angle of MS waves are parametrically investigated. Numerical results show that the increasing normal angle can lead to the shrinking of the Landau resonance region and the expansion of the transit-time scattering region. With the increasing normal angle, the pitch angle diffusion coefficients decrease significantly (by about two orders of magnitude), while the other two diffusion coefficients have a relatively limited decrease (within one order of magnitude). For arbitrary normal angles, the magnitude of cross diffusion coefficients is comparable to the other diffusion coefficients, suggesting that the cross diffusion is indispensable in the kinetic simulation works.

Resonance zones and quasi-linear diffusion coefficients for radiation belt energetic electron interaction with oblique chorus waves in the Dungey magnetosphere

The resonance regions for resonant interactions of radiation belt electrons with obliquely propagating whistler-mode chorus waves are investigated in detail in the Dungey magnetic fields that are parameterized by the intensity of uniform southward interplanetary magnetic field (IMF) Bz or, equivalently, by the values of D=(M/Bz,0)1/3 (where M is the magnetic moment of the dipole and Bz,0 is the uniform southward IMF normal to the dipole’s equatorial plane). Adoption of background magnetic field model can considerably modify the determination of resonance regions. Compared to the results for the case of D = 50 (very close to the dipole field), the latitudinal coverage of resonance regions for 200 keV electrons interacting with chorus waves tends to become narrower for smaller D-values, regardless of equatorial pitch angle, resonance harmonics, and wave normal angle. In contrast, resonance regions for 1 MeV electrons tend to have very similar spatial lengths along the field line for various Dungey magnetic field models but cover different magnetic field intervals, indicative of a strong dependence on electron energy. For any given magnetic field line, the resonance regions where chorus-electron resonant interactions can take place rely closely on equatorial pitch angle, resonance harmonics, and kinetic energy. The resonance regions tend to cover broader latitudinal ranges for smaller equatorial pitch angles, higher resonance harmonics, and lower electron energies, consistent with the results in Ni and Summers [Phys. Plasmas 17, 042902, 042903 (2010)]. Calculations of quasi-linear bounce-averaged diffusion coefficients for radiation belt electrons due to nightside chorus waves indicate that the resultant scattering rates differ from using different Dungey magnetic field models, demonstrating a strong dependence of wave-induced electron scattering effect on the adoption of magnetic field model. Our results suggest that resonant wave-particle interaction processes should be implemented into a sophisticated, accurate global magnetic field model to pursue comprehensive and complete models of radiation belt electron dynamics.

On the quasi-linear diffusion in collisionless plasmas (to say nothing about Landau damping)

General quasi-linear diffusion coefficients for nonrelativistic collisionless plasmas are derived for unstable modes and analytically continued to damped modes. Properties of the resulting diffusion are investigated and discussed.

Velocity space diffusion of charged particles in weak magnetostatic fields: Nonlinear effects, model constraints, and implications for simulations

The velocity space diffusion of charged test particles in random magnetostatic fields is re-investigated from a semi-dynamical point of view. The dynamics of charged particles in resonance with parallel propagating electromagnetic waves is investigated numerically and compared with analytical results for the trapping width in velocity space, Δv∥, and the bounce frequency, ωb. It is demonstrated how an understanding of the basic resonance phenomenon can lead to a better understanding of the validity regions of the quasi-linear theory and their implications for numerical simulations. It is shown, using established analytical expressions for Δv∥ and ωb, that the quasi-linear diffusion coefficient can be written in a new physically illuminating form. The concept of an effective trapping width in velocity space for the turbulence modified resonance structure is introduced. It is shown how this effective resonance width implies a condition on the density of wave modes in Fourier space, in the vicinity of the resonant wave number. The implications of this condition for simulations utilizing discrete fields are discussed in detail and examples of simulations violating this condition are presented. Other issues pertinent to the simulation of velocity diffusion in turbulent electromagnetic fields are discussed, paying attention to the discretization of the fields and the temporal discretization of the dynamical equations.