Diffusion Of Energetic Particles Research Articles

Anomalous Transport of Energetic Particles Interacting with Dynamic Small-Scale Magnetic Flux Rope Structures

le Roux and Zank [25] showed previously how one can derive from first principles a pitch-angle dependent fractional diffusion-advection kinetic equation to model the anomalous diffusion of energetic particles interacting with small-scale magnetic flux ropes (SMFRs) in the inner heliosphere on the basis of the standard focused transport equation. This equation has the following limitations: (1) The asymptotic power law of a Lévy distribution was specified to model the non-Gaussian statistics of the disturbed energetic particle trajectories generated during energetic particle interaction with numerous SMFRs. The second moment (variance) and higher moments of the Lévy distribution are infinite, indicating over-efficient non-local transport that is scale-free. (2) The theory does not naturally allow for a transition of anomalous transport to normal diffusion, or to a different anomalous diffusion state. An outline of a derivation is presented in which an exponentially truncated Lévy distribution was specified instead, resulting in a tempered fractional diffusion-advection kinetic equation that addresses these two concerns.

W fuzz layers: very high resistance to sputtering under fusion-relevant He + irradiations

In this study, we have modeled the sputtering process of energetic He+ ions colliding with W nano-fuzz materials, based on the physical processes, such as the collision and diffusion of energetic particles, sputtering and redeposition. Our modeling shows that the fuzzy nanomaterials with a large surface-to-volume ratio exhibit very high resistance to sputtering under fusion-relevant He+ irradiations, and their sputtering yields are mainly determined by the thickness of fuzzy nano-materials, the reflection coefficients and mean free paths of energetic particles, surface sputtering yields of a flat base material, and the geometry of nano-fuzz. Our measurements have confirmed that the surface sputtering yield of a W nano-fuzz layer with the columnar geometry of nano-fuzz in cross-section is about one magnitude of order lower than the one of smooth W substrates. This work provides a complete model for energetic particles colliding with the nano-fuzz layer and clarifies the fundamental sputtering process occurring in the nano-fuzz layer.

A Focused Transport-based Kinetic Fractional Diffusion-advection Equation for Energetic Particle Trapping and Reconnection-related Acceleration by Small-scale Magnetic Flux Ropes in the Solar Wind

Abstract Analysis of energetic particle inner heliospheric spacecraft data increasingly suggests the existence of anomalous diffusion phenomena that should be addressed to achieve a better understanding of energetic particle transport and acceleration in the expanding solar wind medium. Related to this is fast-growing observational evidence supporting the long-standing prediction from magnetohydrodynamic (MHD) theory and simulations of the presence of an inner heliospheric, dominant quasi-two-dimensional MHD turbulence component that contains coherent contracting and merging (reconnecting) small-scale magnetic flux rope (SMFR) structures. This suggests that energetic particle trapping in SMFRs should play a role in anomalous diffusion in the solar wind that warrants further investigation. However, progress in studying such anomalous energetic particle transport phenomena in the solar wind is hampered by the lack of a fundamental derivation of a general fractional kinetic transport equation linking macroscopic energetic particle fractional transport to the microscopic physics of energetic particle interaction with SMFR structures. Here, we outline details of how one can derive a closed ensemble-averaged focused transport equation in the form of a general kinetic fractional diffusion-advection equation from first principles following the nonlinear Eulerian correlation function closure approach of Sanchez et al. With this equation one can model the anomalous diffusion of energetic particles in ordinary, momentum, and pitch-angle space in response to particle trapping in numerous SMFRs advected with the solar wind flow.

Heuristic Description of Perpendicular Particle Transport in Turbulence with Super-diffusive Magnetic Field Lines

Abstract Recently a heuristic description of collisionless perpendicular diffusion of energetic particles was presented. The latter approach describes the transport of energetic particles across a mean magnetic field based on simple physical arguments. Although this approach was developed with the intention to improve our understanding of perpendicular diffusion, this heuristic approach also provided some interesting quantitative results such as an explanation of the factor a 2 used in the past to balance out inaccuracies of systematic analytical theories. However, the aforementioned heuristic approach is based on the assumption that magnetic field lines become diffusive after overcoming the initial free-streaming regime. In the current paper we alter the heuristic approach to make it applicable for turbulence spectra leading to super-diffusive magnetic fields lines. It is argued that particle diffusion is still restored in the late time limit. In the high-energy limit this recovery of diffusion is based on a hybrid model in which particles move half ballistically and half diffusively in the parallel direction. Furthermore, this leads to the relation between perpendicular and parallel mean free paths of the energetic particles. This type of transport was obtained in the past from test-particle simulations as well as systematic analytical theories. In the current paper we present the first time an explanation of this behavior based on simple physical arguments.

Heuristic Description of Perpendicular Diffusion of Energetic Particles in Astrophysical Plasmas

Abstract A heuristic approach for collisionless perpendicular diffusion of energetic particles is presented. Analytic forms for the corresponding diffusion coefficient are derived. The heuristic approach presented here explains the parameter a 2 used in previous theories in order to achieve agreement with simulations and its relation to collisionless Rechester & Rosenbluth diffusion. The obtained results are highly relevant for applications because previously used formulas are altered significantly in certain situations.

Particle Acceleration Due to Cosmic-ray Viscosity and Fluid Shear in Astrophysical Jets

Abstract An analytical model describing the acceleration of energetic charged particles due to cosmic-ray viscosity and fluid shear in radio jets based on the diffusive transport equation for cosmic rays is developed. The model indicates that laminar shear is not a significant mechanism for particle acceleration in nonrelativistic jets but may be important in relativistic jets. The model produces a power-law momentum spectrum of particles accelerated by the shear (i.e., as , where f 0 is the isotropic phase space distribution in the scattering frame). For ultra-relativistic jets with no back-flowing cocoon, for the energetic particle diffusion coefficient case κ ∝ p α . Increasing the shear in the background flow gives harder accelerated particle momentum spectra. Monoenergetic source solutions (Green’s functions) and Green’s formula are derived and used to obtain solutions for monoenergetic particle momentum spectra specified on the edge of the jet at radius r = r 2.

Decadal trends in the diurnal variation of galactic cosmic rays observed using neutron monitor data

Abstract. The diurnal variation (DV) in galactic cosmic ray (GCR) flux is a widely observed phenomenon in neutron monitor data. The background variation considered primarily in this study is due to the balance between the convection of energetic particles away from the Sun and the inward diffusion of energetic particles along magnetic field lines. However, there are also times of enhanced DV following geomagnetic disturbances caused by coronal mass ejections or corotating interaction regions. In this study we investigate changes in the DV over four solar cycles using ground-based neutron monitors at different magnetic latitudes and longitudes at Earth. We divide all of the hourly neutron monitor data into magnetic polarity cycles to investigate cycle-to-cycle variations in the phase and amplitude of the DV. The results show, in general, a similarity between each of the A < 0 cycles and A > 0 cycles, but with a phase change between the two. To investigate this further, we split the neutron monitor data by solar magnetic polarity between times when the dominant polarity was either directed outward (positive) or inward (negative) at the northern solar pole. We find that the maxima and minima of the DV changes by, typically, 1–2 h between the two polarity states for all non-polar neutron monitors. This difference between cycles becomes even larger in amplitude and phase with the removal of periods with enhanced DV caused by solar wind transients. The time difference between polarity cycles is found to vary in a 22-year cycle for both the maximum and minimum times of the DV. The times of the maximum and minimum in the DV do not always vary in the same manner between A > 0 and A < 0 polarity cycles, suggesting a slight change in the anisotropy vector of GCRs arriving at Earth between polarity cycles. Polar neutron monitors show differences in phase between polarity cycles which have asymptotic directions at mid-to-high latitudes. All neutron monitors show changes in the amplitude of the DV with solar polarity, with the amplitude of the DV being a factor of 2 greater in A < 0 cycles than A > 0 cycles. In most cases the change in timing of the maximum /minimum is greatest with the stations' geomagnetic cut-off rigidity shows little variation in the DV phase with latitude. We conclude that the change in the DV with the dominant solar polar polarity is not as simple as a phase change, but rather an asymmetric variation which is sensitive to the neutron monitor's asymptotic viewing direction.

Numerical Test of Analytical Theories for Perpendicular Diffusion in Small Kubo Number Turbulence

Abstract In the literature, one can find various analytical theories for perpendicular diffusion of energetic particles interacting with magnetic turbulence. Besides quasi-linear theory, there are different versions of the nonlinear guiding center (NLGC) theory and the unified nonlinear transport (UNLT) theory. For turbulence with high Kubo numbers, such as two-dimensional turbulence or noisy reduced magnetohydrodynamic turbulence, the aforementioned nonlinear theories provide similar results. For slab and small Kubo number turbulence, however, this is not the case. In the current paper, we compare different linear and nonlinear theories with each other and test-particle simulations for a noisy slab model corresponding to small Kubo number turbulence. We show that UNLT theory agrees very well with all performed test-particle simulations. In the limit of long parallel mean free paths, the perpendicular mean free path approaches asymptotically the quasi-linear limit as predicted by the UNLT theory. For short parallel mean free paths we find a Rechester & Rosenbluth type of scaling as predicted by UNLT theory as well. The original NLGC theory disagrees with all performed simulations regardless what the parallel mean free path is. The random ballistic interpretation of the NLGC theory agrees much better with the simulations, but compared to UNLT theory the agreement is inferior. We conclude that for this type of small Kubo number turbulence, only the latter theory allows for an accurate description of perpendicular diffusion.

Parallel diffusion of energetic particles interacting with noisy reduced MHD turbulence

Parallel diffusion of energetic particles interacting with noisy reduced MHD turbulence

Finite gyroradius corrections in the theory of perpendicular diffusion 2. Strong velocity diffusion

The current paper is a sequel to an article where we have started to incorporate finite gyroradius effects in the theory of perpendicular diffusion of energetic particles interacting with turbulent magnetic fields. In the previous paper we have focused on the case that velocity diffusion is suppressed and we derived corrections to the perpendicular diffusion coefficient. In the current article we focus on the limit of strong non-linear velocity diffusion. If finite gyroradius effects are not present, we derive the well-known limit where the perpendicular diffusion coefficient is directly proportional to the parallel diffusion coefficient. As in the previous paper, we employ different turbulence models as examples, namely the slab model, noisy slab turbulence, and the two-dimensional model. We show that finite gyroradius effects reduce the perpendicular mean free path in all considered cases except for the slab model where such effects do not occur.

Finite gyroradius corrections in the theory of perpendicular diffusion 1. Suppressed velocity diffusion

A fundamental problem in plasma physics, space science, and astrophysics is the transport of energetic particles interacting with stochastic magnetic fields. In particular the motion of particles across a large scale magnetic field is difficult to describe analytically. However, progress has been achieved in the recent years due to the development of the unified non-linear transport theory which can be used to describe magnetic field line diffusion as well as perpendicular diffusion of energetic particles. The latter theory agrees very well with different independently performed test-particle simulations. However, the theory is still based on different approximations and assumptions. In the current article we extend the theory by taking into account the finite gyroradius of the particle motion and calculate corrections in different asymptotic limits. We consider different turbulence models as examples such as the slab model, noisy slab turbulence, and the two-dimensional model. Whereas there are no finite gyroradius corrections for slab turbulence, the perpendicular diffusion coefficient is reduced in the other two cases. The matter investigated in this article is also related to the parameter “a2” occurring in non-linear diffusion theories.

Perpendicular diffusion of energetic particles: Numerical test of the theorem on reduced dimensionality

A fundamental statement in diffusion theory is provided by the so-called theorem on reduced dimensionality. The latter theorem is saying that if the dimensionality of the turbulence is reduced, charged particles are tied to a single magnetic field line. If there is pitch-angle scattering and therewith parallel diffusion, this usually means that perpendicular transport is subdiffusive. Subdiffusive transport was found in numerous simulations for slab turbulence. However, it was unclear whether the theorem is valid for other models with reduced dimensionality such as the two-dimensional model. In the current paper, we simultaneously trace magnetic field lines and energetic particles and we compute the distance between the particle and the initial field line. We confirm the aforementioned theorem for slab turbulence but we cannot confirm it for two-dimensional turbulence. We also show that particles are not tied to field lines for two-component turbulence.

PERPENDICULAR DIFFUSION OF ENERGETIC PARTICLES IN NOISY REDUCED MAGNETOHYDRODYNAMIC TURBULENCE

Recently a model for noisy reduced magnetohydrodynamic turbulence was proposed. The latter model was already used to study the random walk of magnetic field lines. In the current article we use the same model to investigate the diffusion of energetic particles across the mean magnetic field. To compute the perpendicular diffusion coefficient two analytical theories are used, namely the Non- Linear Guiding Center (NLGC) theory and the Unified Non-Linear Transport (UNLT) theory. It is shown that the two theories provide different results for the perpendicular diffusion coefficient. We also perform test-particle simulations for the aforementioned turbulence model. We show that only the UNLT theory describes perpendicular transport accurately confirming that the latter theory is a powerful tool in diffusion theory.

Estimates of the power per mode number of broadband ULF waves at geosynchronous orbit

AbstractIn studies of radial diffusion processes in the magnetosphere it is well known that ultralow frequency (ULF) waves of frequency mωd can resonantly interact with particles of drift frequency ωd, where m is the waves' azimuthal mode number. Due to difficulties in estimating m, an oversimplifying assumption is often made in simulations, namely that all ULF wave power is located at a single mode number. In this paper a technique is presented for extracting information on the distribution of ULF power in a range of azimuthal mode numbers. As a first step, the cross power and phase differences between time series from azimuthally aligned magnetometers are calculated. Subsequently, through integrating the ULF power at particular ranges of phase differences that correspond to particular mode numbers, estimates of the fraction of the total power at each phase difference range or mode number are provided. Albeit entwined with many ambiguities, this technique offers critical information that is currently missing when estimating radial diffusion of energetic particles. As proof‐of‐concept, the technique is first tested successfully for a well‐studied case of narrowband ULF Field Line Resonances (FLR) for which the mode number was calculated simultaneously through ground‐based and space measurements. Subsequently, the technique is demonstrated for the broadband ULF waves that accompanied the 2003 “Halloween” magnetospheric storms. The temporal evolution of power at each mode number gives insight into the evolution of ULF waves during a storm as well as more accurate characterization of broadband ULF waves that can be used in radial diffusion simulations.

TRANSPORT OF COSMIC-RAY PROTONS IN INTERMITTENT HELIOSPHERIC TURBULENCE: MODEL AND SIMULATIONS

The transport of charged energetic particles in the presence of strong intermittent heliospheric turbulence is computationally analyzed based on known properties of the interplanetary magnetic field and solar wind plasma at 1 Astronomical Unit (AU). The turbulence is assumed to be static, composite, and quasi-three-dimensional with a varying energy distribution between a one-dimensional Alfv\'enic (slab) and a structured two-dimensional component. The spatial fluctuations of the turbulent magnetic field are modeled either as homogeneous with a Gaussian probability distribution function (PDF), or as intermittent on large and small scales with a q-Gaussian PDF. Simulations showed that energetic particle diffusion coefficients both parallel and perpendicular to the background magnetic field are significantly affected by intermittency in the turbulence. This effect is especially strong for parallel transport where for large-scale intermittency results show an extended phase of subdiffusive parallel transport during which cross-field transport diffusion dominates. The effects of intermittency are found to depend on particle rigidity and the fraction of slab energy in the turbulence, yielding a perpendicular to parallel mean free path ratio close to 1 for large-scale intermittency. Investigation of higher order transport moments (kurtosis) indicates that non-Gaussian statistical properties of the intermittent turbulent magnetic field are present in the parallel transport, especially for low rigidity particles at all times.

Pitch-Angle Dependent Perpendicular Diffusion of Energetic Particles Interacting With Magnetic Turbulence

We employ a test-particle code to explore diffusion of energetic particles interacting with the solar wind plasma.The first time we investigate the pitch-angle dependence of perpendicular diffusion for different parameter regimes.These results are important for numerical solutions of the pitch-angle dependent Cosmic Ray Fokker-Planck equa-tion. We compare our finding also with predictions made by analytical theory.

Cosmic-ray diffusion modeling: Solutions using variational methods

The diffusion of energetic particles in turbulent magnetic fields is usually described via the two-point, two-time velocity correlation function. A variational principle is used to determine the characteristic function that results from the Fourier-transformed correlation function. Both for a linear approximation and for the wave vector set to zero, explicit solutions are derived that depend on the Fokker-Planck coefficient of pitch-angle scattering. It is shown that, for an isotropic form of the Fokker-Planck coefficient, the characteristic function is divergent, which can be remedied only by using a Fokker-Planck coefficient that is finite at all pitch angles.

Quasilinear model for energetic particle diffusion in radial and velocity space

A quasilinear model for passive energetic particle (EP) turbulent diffusion in radial and velocity space is fitted and tested against nonlinear gyrokinetic tokamak simulations with the GYRO code [J. Candy and R. E. Waltz, Phys. Rev. Lett. 91, 045001 (2003)]. Off diagonal elements of a symmetric positive definite 2×2 EP diffusion matrix account for fluxes up radial (energy) gradients driven by energy (radial) gradients of the EP velocity space distribution function. The quasilinear ratio kernel of the model is provided by a simple analytic formula for the EP radial and velocity space EP diffusivity relative to radial thermal ion energy diffusivity at each linear mode of the turbulence driven by the thermal plasma. The TGLF [G. M. Staebler, J. E. Kinsey, and R. E. Waltz, Phys. Plasmas 14, 0055909 (2007); ibid. 15, 0055908 (2008)] tokamak transport model provides the linear mode frequency and growth rates to the kernel as well as the nonlinear spectral weight for each mode.

Anomalous diffusion of energetic particles: connecting experiment and simulations

Recent experimental and theoretical studies highlight the need to predict the turbulence-induced radial diffusion of energetic particles quantitatively in present-day and future fusion devices. Gyrokinetic simulations with passive fast ion species in realistic tokamak equilibria are used to investigate the influence of the particle energy on diffusion due to electrostatic and electromagnetic turbulence. To facilitate more straightforward comparisons between theory and experiments, scaling relations are derived that allow for writing the relevant diffusivities as functions of experimentally readily accessible quantities. In this context, an improved method for estimating the magnetic fluctuation level is described.

Compound diffusion of energetic particles: a Kappa model for the parallel distribution function

One can assume that energetic particles follow magnetic field lines while they propagate through a magnetized plasma. The latter scenario is usually described by the so-called field line random walk limit. This limit, however, is only valid if parallel diffusion is suppressed. As soon as the latter effect is taken into account, perpendicular transport becomes subdiffusive. This physical scenario is usually called compound diffusion or compound subdiffusion and can be described by a Chapman-Kolmogorov equation. In the latter equation the parallel distribution function is an essential ingredient. In the present paper we replace the standard Gaussian model by a Kappa distribution to compute distribution functions and mean square displacements across the field.