Perpendicular Mean Free Path Research Articles

The Diffusion Tensor of Protons at 1 au: Comparing Simulation, Observation, and Theory

The natural variation in plasma parameters observed at 1 au can lead to a variation in transport parameters, such as diffusion and drift coefficients, for energetic charged particles of solar and galactic origin. Given the importance of these parameters to particle transport studies, this variation is investigated through test particle simulations over a range of energies in the presence of simulated turbulence with properties corresponding to an ensemble of observed turbulence conditions at Earth. The resulting transport coefficients are then compared with observational estimates from the literature, as well as the predictions of several scattering theories. Parallel and perpendicular mean free paths are shown to vary widely, for the former in agreement with prior observational estimates, but not for the latter. Furthermore, a large disparity between the predictions of theory and the simulation results is noted for the perpendicular mean free path. As such, these results indicate that particle transport studies, particularly predictive ones, need to take into account this natural variation in transport coefficients.

Modulation of Cosmic-Ray Antiprotons in the Heliosphere: Simulations for a Solar Cycle

The precision measurements of galactic cosmic-ray protons from the Payload for Antimatter Matter Exploration and Light-nuclei Astrophysics and the Alpha Magnetic Spectrometer are reproduced using a well-established three-dimensional numerical model for the period 2006 July–2019 November. The resulting modulation parameters are applied to simulate the modulation for cosmic antiprotons over the same period, which includes the times of minimum modulation before and after 2009, the maximum modulation from 2012 to 2015, including the reversal of the Sun’s magnetic field polarity, and the approach to new minimum modulation in 2020. Apart from their local interstellar spectra, the modulation of protons and antiprotons differ in their charge sign and consequent drift pattern. The lowest proton flux was in 2014 February–March, but the lowest simulated antiproton flux is found to have been in 2015 March–April. These simulated fluxes are used to predict the proton-to-antiproton ratios as a function of rigidity. The trends in these ratios contribute to clarifying, to a large extent, the phenomenon of charge-sign dependence of heliospheric modulation during vastly different phases of the solar activity cycle. This is reiterated and emphasized by displaying so-called hysteresis loops. It is also illustrated how the values of the parallel and perpendicular mean free paths, as well as the drift scale, vary with rigidity over this extensive period. The drift scale is found to be at its lowest level during the polarity reversal period, while the lowest level of the mean free paths is found to be in 2015 March–April.

Revisiting the Revisited Palmer Consensus: New Insights from Jovian Electron Transport

Abstract Novel insights into the behavior of the diffusion coefficients of charged particles in the inner heliosphere are of great importance to any study of the transport of these particles and are especially relevant with regard to the transport of low-energy electrons. The present study undertakes an exhaustive investigation into the diffusion parameters needed to reproduce low-energy electron intensities as observed at Earth, using a state-of-the-art 3D cosmic ray transport code. To this end, the transport of Jovian electrons is considered, as Jupiter represents the predominant source of these particles in the inner heliosphere, and because a careful comparison of model results with observations taken during periods of good and poor magnetic connectivity between Earth and Jupiter allows for conclusions to be drawn as to both parallel and perpendicular diffusion coefficients. This study then compares these results with the predictions made by various scattering theories. Best-fit parameters for parallel and perpendicular mean free paths at 1 au fall reasonably well within the span of observational values reported by previous studies, but best-fit radial and rigidity dependences vary widely. However, a large number of diffusion parameters lead to reasonable to-good fits to observations, and it is argued that considerable caution must be exercised when comparing theoretical results for diffusion coefficients with diffusion parameters calculated from particle transport studies.

The Transport and Evolution of MHD Turbulence throughout the Heliosphere: Models and Observations

A detailed study of solar wind turbulence throughout the heliosphere in both the upwind and downwind directions is presented. We use an incompressible magnetohydrodynamic (MHD) turbulence model that includes the effects of electrons, the separation of turbulence energy into proton and electron heating, the electron heat flux, and Coulomb collisions between protons and electrons. We derive expressions for the turbulence cascade rate corresponding to the energy in forward and backward propagating modes, the fluctuating kinetic and magnetic energy, the normalized cross-helicity, and the normalized residual energy, and calculate the turbulence cascade rate from 0.17 to 75 au in the upwind and downwind directions. Finally, we use the turbulence transport models to derive cosmic ray (CR) parallel and perpendicular mean free paths (mfps) in the upwind and downwind heliocentric directions. We find that turbulence in the upwind and downwind directions is different, in part because of the asymmetric distribution of new born pickup ions in the two directions, which results in the CR mfps being different in the two directions. This is important for models that describe the modulation of cosmic rays by the solar wind.

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.

Spreading protons in the heliosphere: a note on cross-field diffusion effects

We study how a high-speed solar wind stream embedded in a slow solar wind affects the transport and energy changes of solar energetic protons in interplanetary space, assuming different levels of cross-field diffusion. This is done using a particle transport model that computes directional particle intensities and first order parallel anisotropies in a background solar wind generated by the magnetohydrodynamic model EUHFORIA. In particular, we consider a mono-energetic 4 MeV proton injection over an extended region located at a heliographic radial distance of 0.1 AU. By using different values for the perpendicular proton mean free path, we study how cross-field diffusion may affect the energetic particle spread and intensity profiles near a high-speed solar wind stream and a corotating interaction region (CIR). We find that both a strong cross-field diffusion and a solar wind rarefaction region are capable of dispersing SEPs efficiently, producing overall low particle intensities which can in some cases prevent the SEPs from being detected in-situ, since their intensity may drop below the detected preevent intensity levels. We also discuss how accelerated particle populations form on the reverse and forward shock waves, separated by the stream interface inside the CIR. Under strong levels of cross-field diffusion, particles cross the SI and hence both accelerated particle populations merge together.

Correlation of Long-term Cosmic-Ray Modulation with Solar Activity Parameters

Abstract In this work, we analyze the long-term cosmic-ray modulation observed by the Hermanus neutron monitor, which is the detector with the longest cosmic-ray record, from 1957 July. For our study we use the force-field approximation to the cosmic-ray transport equation, and the newest results on the mean free paths from the scattering theory. We compare the modulation parameter (ϕ) with different rigidity (P) dependences: P, P 2, and P 2/3. We correlate them with solar and interplanetary parameters. We found that (1) these rigidity dependences properly describe the modulation, (2) long-term cosmic-ray variations are better correlated with the magnitude of the heliospheric magnetic field (HMF) than the sunspot number, solar wind speed, and tilt angle of the HMF, and (3) the theoretical dependence of the parallel mean free path on the magnetic field variance is in agreement with the modulation parameter and therefore with the neutron monitor record. We also found that the force-field approximation is not able to take into account the effects of three-dimensional particle transport, showing a poor correlation with the perpendicular mean free path.

Galactic Cosmic Rays Modulation in the Vicinity of Corotating Interaction Regions: Observations During the Last Two Solar Minima

Abstract Corotating interaction regions (CIRs) are responsible for short-term recurrent cosmic-ray modulation, prominent near solar minima. Using the OMNI data sets for two periods of low solar activity near the beginning and end of solar cycle 24, superposed epoch analysis was performed on the solar wind plasma features for 53 and 43 events during periods 2007–2008 and 2017–2018, respectively. Turbulent properties of the solar wind were studied using the variance method for each CIR. Power spectra have been constructed for overlapped subintervals in the vicinity of stream interfaces (SIs). Using measured correlation lengths and turbulent energies, parallel and perpendicular diffusion mean free paths for cosmic-ray ions have been inferred based on two distinct theoretical formulations. For the two periods with opposite solar polarities, our results show that unlike solar wind speed, magnetic field strength, flow pressure, and proton density are relatively higher during the latest period. Increased turbulent energy and reduced parallel transport coefficients of energetic particles at the SIs are observed. The diffusion coefficients follow the same trends during both periods. The perpendicular diffusion starts increasing nearly a day before SIs and is higher in the fast wind. Superposed epoch analysis is performed on the >120 MeV proton count rate obtained from the CRIS instrument on board the ACE spacecraft for the same events. The recorded proton rates have peaks half a day before a SI and reach their minimum more than a day after a SI and have a high anticorrelation with the perpendicular diffusion coefficient.

Numerical Modeling of Galactic Cosmic-Ray Proton and Helium Observed by AMS-02 during the Solar Maximum of Solar Cycle 24

Abstract Galactic cosmic rays (GCRs) are affected by solar modulation while they propagate through the heliosphere. The study of the time variation of GCR spectra observed at Earth can shed light on the underlying physical processes, specifically diffusion and particle drifts. Recently, the AMS-02 experiment measured with very high accuracy the time variation of the cosmic-ray proton and helium flux between 2011 May and 2017 May in the rigidity range from 1 to 60 GV. In this work, a comprehensive three-dimensional steady-state numerical model is used to solve Parker’s transport equation and reproduce the monthly proton fluxes observed by AMS-02. We find that the rigidity slope of the perpendicular mean free path above 4 GV remains constant, while below 4 GV, it increases during solar maximum. Assuming the same mean free paths for helium and protons, the models are able to reproduce the time behavior of the p/He ratio observed by AMS-02. The dependence of the diffusion tensor on the particle mass-to-charge ratio, A/Z, is found to be the main cause of the time dependence of p/He below 3 GV.

Cosmic-Ray Diffusion Coefficients throughout the Inner Heliosphere from a Global Solar Wind Simulation

Abstract We use a three-dimensional magnetohydrodynamic simulation of the solar wind to calculate cosmic-ray diffusion coefficients throughout the inner heliosphere ( ). The simulation resolves large-scale solar wind flow, which is coupled to small-scale fluctuations through a turbulence model. Simulation results specify background solar wind fields and turbulence parameters, which are used to compute diffusion coefficients and study their behavior in the inner heliosphere. The parallel mean free path (mfp) is evaluated using quasi-linear theory, while the perpendicular mfp is determined from nonlinear guiding center theory with the random ballistic interpretation. Several runs examine varying turbulent energy and different solar source dipole tilts. We find that for most of the inner heliosphere, the radial mfp is dominated by diffusion parallel to the mean magnetic field; the parallel mfp remains at least an order of magnitude larger than the perpendicular mfp, except in the heliospheric current sheet, where the perpendicular mfp may be a few times larger than the parallel mfp. In the ecliptic region, the perpendicular mfp may influence the radial mfp at heliocentric distances larger than 1.5 au; our estimations of the parallel mfp in the ecliptic region at 1 au agree well with the Palmer “consensus” range of 0.08–0.3 au. Solar activity increases perpendicular diffusion and reduces parallel diffusion. The parallel mfp mostly varies with rigidity as , and the perpendicular mfp is weakly dependent on P. The mfps are weakly influenced by the choice of long-wavelength power spectra.

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.

MULTI-SPACECRAFT OBSERVATIONS AND TRANSPORT MODELING OF ENERGETIC ELECTRONS FOR A SERIES OF SOLAR PARTICLE EVENTS IN AUGUST 2010

ABSTRACT During 2010 August a series of solar particle events was observed by the two STEREO spacecraft as well as near-Earth spacecraft. The events, occurring on August 7, 14, and 18, originated from active regions 11093 and 11099. We combine in situ and remote-sensing observations with predictions from our model of three-dimensional anisotropic particle propagation in order to investigate the physical processes that caused the large angular spreads of energetic electrons during these events. In particular, we address the effects of the lateral transport of the electrons in the solar corona that is due to diffusion perpendicular to the average magnetic field in the interplanetary medium. We also study the influence of two coronal mass ejections and associated shock waves on the electron propagation, and a possible time variation of the transport conditions during the above period. For the August 18 event we also utilize electron observations from the MESSENGER spacecraft at a distance of 0.31 au from the Sun for an attempt to separate between radial and longitudinal dependencies in the transport process. Our modelings show that the parallel and perpendicular diffusion mean free paths of electrons can vary significantly not only as a function of the radial distance, but also of the heliospheric longitude. Normalized to a distance of 1 au, we derive values of λ ∥ in the range of 0.15–0.6 au, and values of λ ⊥ in the range of 0.005–0.01 au. We discuss how our results relate to various theoretical models for perpendicular diffusion, and whether there might be a functional relationship between the perpendicular and the parallel mean free path.

Cosmic ray propagation in sub-Alfvénic magnetohydrodynamic turbulence

This work has the main objective to provide a detailed investigation of cosmic ray propagation in magnetohydrodynamic turbulent fields generated by forcing the fluid velocity field at large scales. It provides a derivation of the particle mean free path dependences in terms of the turbulence level described by the Alfv\'enic Mach number and in terms of the particle rigidity. We use an upgrade version of the magnetohydrodynamic code {\tt RAMSES} which includes a forcing module and a kinetic module and solve the Lorentz equation for each particle. The simulations are performed using a 3 dimension periodical box in the test-particle and magnetostatic limits. The forcing module is implemented using an Ornstein-Uhlenbeck process. An ensemble average over a large number of particle trajectories is applied to reconstruct the particle mean free paths. We derive the cosmic ray mean free paths in terms of the Alfv\'enic Mach numbers and particle reduced rigidities in different turbulence forcing geometries. The reduced particle rigidity is $\rho=r_L/L$ where $r_L$ is the particle Larmor radius and $L$ is the simulation box length related to the turbulence coherence or injection scale $L_{inj}$ by $L \sim 5 L_{inj}$. We have investigated with a special attention compressible and solenoidal forcing geometries. We find that compressible forcing solutions are compatible with the quasi-linear theory or more advanced non-linear theories which predict a rigidity dependence as $\rho^{1/2}$ or $\rho^{1/3}$. Solenoidal forcing solutions at least at low or moderate Alfv\'enic numbers are not compatible with the above theoretical expectations and require more refined arguments to be interpreted. It appears especially for Alfv\'enic Mach numbers close to one that the wandering of field lines controls perpendicular mean free path solutions whatever the forcing geometry.

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.

A comparison of turbulence-reduced drift coefficients of importance for the modulation of galactic cosmic-ray protons in the supersonic solar wind

The study of the modulation of cosmic rays in the heliosphere relies heavily on a thorough understanding of the transport of these charged particles in the turbulent solar wind. Drift effects due to gradients and the curvature of the background magnetic field have long been known to be reduced in the presence of turbulence, and as such, several forms for the drift coefficient that include the effect of turbulence have been proposed. The present study aims to investigate the qualitative effects of various turbulence-reduced drift coefficients on cosmic ray intensities computed using an ab initio 3D steady-state cosmic-ray modulation code. Results from a two-component turbulence transport models are used as inputs for the basic turbulence quantities. Furthermore, an expression for the perpendicular mean free path is derived here from a modification of the non-linear guiding center theory of Matthaeus et al. (2003) assuming a 2D turbulence power spectrum with a k-1 energy range wavenumber dependence, and is used in conjunction with the various proposed turbulence-reduced drift coefficients. Cosmic-ray intensities computed using different drift coefficients but assuming the same turbulence conditions are found to differ widely. This study emphasises the need to gain a better understanding of the effect of turbulence on drifts in the heliosphere.

S.23.03 Anti-alcohol properties of the positive allosteric modulators of the GABAB receptor

The study of the modulation of cosmic rays in the heliosphere relies heavily on a thorough understanding of the transport of these charged particles in the turbulent solar wind. Drift effects due to gradients and the curvature of the background magnetic field have long been known to be reduced in the presence of turbulence, and as such, several forms for the drift coefficient that include the effect of turbulence have been proposed. The present study aims to investigate the qualitative effects of various turbulence-reduced drift coefficients on cosmic ray intensities computed using an ab initio 3D steady-state cosmic-ray modulation code. Results from a two-component turbulence transport models are used as inputs for the basic turbulence quantities. Furthermore, an expression for the perpendicular mean free path is derived here from a modification of the non-linear guiding center theory of Matthaeus et al. (2003) assuming a 2D turbulence power spectrum with a k-1 energy range wavenumber dependence, and is used in conjunction with the various proposed turbulence-reduced drift coefficients. Cosmic-ray intensities computed using different drift coefficients but assuming the same turbulence conditions are found to differ widely. This study emphasises the need to gain a better understanding of the effect of turbulence on drifts in the heliosphere.

Cosmic-Ray Modulation: an Ab Initio Approach

A better understanding of cosmic-ray modulation in the heliosphere can only be gained through a proper understanding of the effects of turbulence on the diffusion and drift of cosmic rays. We present an ab initio model for cosmic-ray modulation, incorporating for the first time the results yielded by a two-component turbulence transport model. This model is solved for periods of minimum solar activity, utilizing boundary values chosen so that model results are in fair to good agreement with spacecraft observations of turbulence quantities, not only in the solar ecliptic plane but also along the out-of-ecliptic trajectory of the Ulysses spacecraft. These results are employed as inputs for modelled slab and 2D turbulence energy spectra. The latter spectrum is chosen based on physical considerations, with a drop-off at the very lowest wavenumbers commencing at the 2D outerscale. There currently exist no models or observations for this quantity, and it is the only free parameter in this study. The modelled turbulence spectra are used as inputs for parallel mean free path expressions based on those derived from quasi-linear theory and perpendicular mean free paths from extended nonlinear guiding center theory. Furthermore, the effects of turbulence on cosmic-ray drifts are modelled in a self-consistent way, employing a recently developed model for drift along the wavy current sheet. The resulting diffusion coefficients and drift expressions are applied to the study of galactic cosmic-ray protons and antiprotons using a three-dimensional, steady-state cosmic-ray modulation code, and sample solutions in fair agreement with multiple spacecraft observations are presented.

The dependence of the parallel and perpendicular mean free paths on the rigidity of the solar energetic particles: Theoretical model versus observations

Context. The dependence of the parallel and perpendicular mean free paths on the particle rigidity is an important topic in the studies of the diffusion and propagation of charged energetic particles in a large-scale turbulent magnetic field.Aims. In this work, we investigate the dependence of the parallel and perpendicular mean free paths on the rigidity of solar energetic particles (SEPs) by means of both the theoretical model and spacecraft observations with regard to several typical SEP events.Methods. A direct method developed by previous studies and derived from the focused transport equation and Taylor-Green-Kubo (TGK) formulation is applied to explicitly determine the parallel and perpendicular mean free paths of SEPs in a turbulent and spatially varying magnetic field.Results. We find that the parallel and perpendicular mean free paths, λ ∥ and λ ⊥ , of energetic protons monotonically decrease with increasing particle rigidity, and the ratio λ ⊥ /λ ∥ monotonically increases with particle rigidity, when the magnetic turbulence is weak. Taking a series of typical SEP events together, it can also be seen that the ratio λ ⊥ /λ ∥ of the perpendicular to the parallel mean free paths remains in the range of 0.001−0.2.

SIMPLE ANALYTICAL FORMS OF THE PERPENDICULAR DIFFUSION COEFFICIENT FOR TWO-COMPONENT TURBULENCE. I. MAGNETOSTATIC TURBULENCE

We explore perpendicular diffusion based on the unified nonlinear transport theory. We derive simple analytical forms for the perpendicular mean free path and investigate the influence of different model spectra. We show that for cases where the field line random walk is normal diffusive, the perpendicular diffusion coefficient consists of only two transport regimes. Details of the spectral shape are less important, especially those of the inertial range. Only the macroscopic properties of the turbulence spectrum control the perpendicular diffusion coefficient. Simple formulae for the perpendicular diffusion coefficient are derived which can easily be implemented in solar modulation or shock acceleration codes.

AN AB INITIO MODEL FOR COSMIC-RAY MODULATION

A proper understanding of the effects of turbulence on the diffusion and drift of cosmic rays (CRs) is of vital importance for a better understanding of CR modulation in the heliosphere. This study presents an ab initio model for CR modulation, incorporating for the first time the results yielded by a two-component turbulence transport model. This model is solved for solar minimum heliospheric conditions, utilizing boundary values chosen so that model results are in reasonable agreement with spacecraft observations of turbulence quantities in the solar ecliptic plane and along the out-of-ecliptic trajectory of the Ulysses spacecraft. These results are employed as inputs for modeled slab and two-dimensional (2D) turbulence energy spectra. The modeled 2D spectrum is chosen based on physical considerations, with a drop-off at the very lowest wavenumbers. There currently exist no models or observations for the wavenumber where this drop-off occurs, and it is considered to be the only free parameter in this study. The modeled spectra are used as inputs for parallel mean free path expressions based on those derived from quasi-linear theory and perpendicular mean free paths from extended nonlinear guiding center theory. Furthermore, the effects of turbulence on CR drifts are modeled in a self-consistent way, also employing a recently developed model for wavy current sheet drift. The resulting diffusion and drift coefficients are applied to the study of galactic CR protons and antiprotons using a 3D, steady-state CR modulation code, and sample solutions in fair to good agreement with multiple spacecraft observations are presented.