A Primer on Focused Solar Energetic Particle Transport: Basic physics and recent modelling results
SSpace Science Reviews manuscript No. (will be inserted by the editor)
A Primer on Focused Solar Energetic Particle Transport
Basic physics and recent modelling results
Jabus van den Berg · Du Toit Strauss · Frederic Effenberger
Received: December 15, 2020/ Accepted: date
Abstract
The basics of focused transport as applied to solar energetic particlesare reviewed, paying special attention to areas of common misconception. Themicro-physics of charged particles interacting with slab turbulence are investi-gated to illustrate the concept of pitch-angle scattering, where after the distribu-tion function and focused transport equation are introduced as theoretical tools todescribe the transport processes and it is discussed how observable quantities canbe calculated from the distribution function. In particular, two approximations,the diffusion-advection and the telegraph equation, are compared in simplifiedsituations to the full solution of the focused transport equation describing par-ticle motion along a magnetic field line. It is shown that these approximationsare insufficient to capture the complexity of the physical processes involved. Toovercome such limitations, a finite-difference model, which is open for use by thepublic, is introduced to solve the focused transport equation. The use of the modelis briefly discussed and it is shown how the model can be applied to reproducean observed solar energetic electron event, providing insights into the accelerationand transport processes involved. Past work and literature on the application ofthese concepts are also reviewed, starting with the most basic models and buildingup to more complex models.
J.P. van den BergCentre for Space Research, North-West University, Potchefstroom, South AfricaSouth African National Space Agency, Hermanus, South AfricaE-mail: [email protected]://orcid.org/0000-0003-1170-1470R.D. StraussCentre for Space Research, North-West University, Potchefstroom, South AfricaE-mail: [email protected]://orcid.org/0000-0002-0205-0808F. EffenbergerGFZ German Research Centre For Geosciences, Potsdam, GermanyBay Area Environmental Research Institute, NASA Research Park, Moffett Field, CA, USAE-mail: feff[email protected] a r X i v : . [ phy s i c s . s p ace - ph ] D ec van den Berg et al. Keywords
Solar Energetic Particles · Particle Transport · Particle Acceleration · Focused Transport · Numerical Modelling · Review
Contents
Solar energetic particles (SEPs) are one of the key subjects in heliospheric physics,receiving even more interest in the last couple of years, mostly due to space mis-sions focusing on the Sun, such as the Parker Solar Probe (launched on 11 Au-gust 2018; http://parkersolarprobe.jhuapl.edu/index.php ; Fox et al. 2016) andthe Solar Orbiter mission (launched on 10 February 2020; http://sci.esa.int/solar-orbiter ; M¨uller et al. 2013). Their importance is not only related to theircharacter as highly energetic test particles, tracing the heliospheric plasma envi-ronment between their source close to the Sun and the observer, but also to theirpotential impact on space hardware and interplanetary travel by humans.A number of excellent and mostly up to date reviews on general SEP propertiesand their observational basis (Reames 1999, 2013, 2017; Ryan et al. 2000; Mewaldt2006; Klein and Dalla 2017) including specific topics such as scattering theoriesand perpendicular diffusion (Shalchi 2009, 2020) exist. However, a review focusingon the various aspects of the transport of SEPs, in a broad context and with a viewtowards applications and common misconceptions, is still somewhat lacking. Toour knowledge, the last review with a similar scope dates back to Dr¨oge (2000a),with a focus on pitch-angle scattering, so a fresh look appears justified. Focus willfall in particular on simulation work in the last decade or so, and how recent resultsfrom these can be reconciled with each other and the observational basis that hasbeen established over the last years. Although we aim for a comprehensive view ofthe subject, there will be important works falling through the cracks or which willbe left out due to space constraints and considerations of readability. We apologizeto all colleagues in advance, who feel that their favorite study is missing.The review of the subject begins by establishing the basics of focused trans-port for SEPs. Common misconceptions will be highlighted throughout and thecorrect interpretations will be explained. To build a conceptual understanding ofthe processes on a pitch-angle level, Section 2 will consider the microscopic physicsof a single charged particle interacting with electromagnetic slab turbulence. Theconcept of a distribution function, to model the macroscopic physics, will be intro-duced in Section 3, together with the focused transport equation. This section willalso investigate the applicability of analytical approximations to the full solutionof the focused transport equation. Here it will be emphasised that a numericalscheme is needed to solve the focused transport equation and in order to do so,a finite difference numerical scheme is presented in Appendix A with a link tothe source code. Many processes, requiring at least a 2D spatial geometry to becorrectly described, e.g. drift and perpendicular diffusion, are reviewed in Sec-tion 4. The review focuses specifically on modelling work, starting form the basic1D models and building up to the fully 3D models. Additional information arepresented in further appendices, which also provide a reference to more technicalaspects not fully discussed in the main text.We hope that this review encourages scientists, especially new to the subject,to investigate and apply the theory to actual problems in SEP research. The nu- merical tools, as described in the appendix with their source code freely available ,can be a starting point for such endeavours. https://github.com/RDStrauss/SEP_propagator van den Berg et al. Cosmic ray (CR) research usually deals with the isotropic limit (this refers toisotropy in momentum, a concept which will be clarified in this section), allowingCRs to be considered as a function of position, energy, and time. SEP transportis inherently time dependent, although certain event integrated distributions canbe considered to be in a steady state. Anisotropy, however, is ubiquitous in SEPtransport and isotropy is reached only during the decay phase of an SEP event.This is probably one of the most complicated aspects of SEP transport, as pitch-angle dependent transport must be considered and the processes must be describedon a more fundamental level than in the isotropic limit. Focused transport is, forthis reason, not well understood in general, as concepts well established in isotropictransport cannot be applied to anisotropic transport. It is, of course, possible toextend the anisotropic processes to the isotropic limit, but the reverse cannot bedone. In this section the Newton-Lorentz equation, some basic definitions, andthe process of magnetic focusing will be introduced. A slab turbulence model isintroduced in Appendix C and a particle is simulated in this turbulence fieldto illustrate the concept of pitch-angle scattering. The section concludes with asummary of the introduced concepts.2.1 The Newton-Lorentz EquationThe motion of a non-relativistic particle, with mass m and charge q , moving with avelocity (cid:126)v in an electric (cid:126)E and magnetic (cid:126)B field, is governed by the Newton-Lorentzequation (Rossi and Olbert 1970; Chen 1984)d (cid:126)p d t = q ( (cid:126)E + (cid:126)v × (cid:126)B ) , (1)where (cid:126)p = m(cid:126)v is the particle’s momentum, which is the most fundamental de-scription of charged particle transport in magnetized plasma, and the basis ofall transport equations. A non-relativistic description will be used here as an ap-proximation just to illustrate the basic concepts. Some analytical solutions of thisequation can be found in any plasma physics textbook (see e.g. Rossi and Olbert1970; Chen 1984; Choudhuri 1998). For electric and magnetic fields with spatialand temporal dependencies, it is relatively easily solvable with various numericalmethods (see e.g. Boris 1970; Birdsall and Langdon 1991). The effect of large scaleelectric fields will not be considered here and is only included to emphasize thata turbulent electric field will exert a force on the particle. Notice that since themagnetic force is perpendicular to the direction of motion, the magnetic field doesno work on the particle and cannot change its energy (Rossi and Olbert 1970;Chen 1984; Choudhuri 1998). For a particle moving in a constant and uniform magnetic field, with strength B , in the absence of electric fields, the vector product in Eq. 1 implies that theparticle experiences a centripetal acceleration and will gyrate around the magneticfield, with positive and negative particles gyrating in a left- and right-hand manner,respectively. The particle will gyrate around the magnetic field at the cyclotron Primer on Focused Solar Energetic Particle Transport 5 frequency ω c = | q | B m , (2)while tracing a circle with the Larmor radius (or gyro-radius) r L = mv ⊥ | q | B , (3)where v ⊥ is the speed of the particle perpendicular to the magnetic field (themaximal Larmor radius is defined as R L = mv/ | q | B ). The particle’s velocitycomponent parallel to the magnetic field, v (cid:107) , will cause the gyrating particle totrace a spiral trajectory (Rossi and Olbert 1970; Chen 1984; Choudhuri 1998).The particle’s pitch-angle is defined as the angle between the particle’s velocityvector and the magnetic field vector, α = arccos (cid:32) (cid:126)v · (cid:126)B vB (cid:33) = arcsin (cid:16) v ⊥ v (cid:17) = arccos (cid:18) v (cid:107) v (cid:19) = arctan (cid:32) v ⊥ v (cid:107) (cid:33) , (4)while the parallel and perpendicular speeds can be calculated from the pitch-angleby v (cid:107) = v cos α = vµ (5a) v ⊥ = v sin α = v (cid:112) − µ , (5b)respectively, where the so called pitch-cosine µ = cos α (6)is a quantity normally used in transport equations. Since the parallel and perpen-dicular speed is constant in a constant and uniform magnetic field, the pitch-anglewill also be constant in such a field (Rossi and Olbert 1970; Chen 1984; Choudhuri1998).The particle gyrate around an imaginary point called the guiding centre (GC)and its position can be found by subtracting a directional Larmor radius fromthe particle’s position. The directional Larmor radius can be interpreted as theinstantaneous radius of curvature projected onto the plane perpendicular to themagnetic field. This is illustrated in the left panel of Fig. 1 and can be written as (cid:126)r gc = (cid:126)r + mqB (cid:126)v × (cid:126)B , (7)where (cid:126)r is the particle’s position. The particle’s helical path can be decomposed,as a first approximation, into a gyration around the GC and the movement ofthe GC along the magnetic field (Northrop 1961; Rossi and Olbert 1970; Burgeret al. 1985), as illustrated in the right panel of Fig. 1. Notice however that theGC is a mathematical construct which is introduced as a tool to help describe theparticle’s motion. It is imperative to realise that the particle does not know that ithas a GC and is not affected by what happens to the GC. Furthermore, Burger (1987)points out that the concept of a GC is only well defined over a complete gyration.If the magnetic field change over a length (time) scale shorter than the Larmorradius (gyroperiod), the GC will be ill defined and might behave in an unexpectedmanner. van den Berg et al. ⃗ ⃗ ⃗ ⃗ ⃗ ⃗⃗ gc ⃗⃗ ⃗ ⃗ ( ⃗⃗ ⃗) ( ) ⃗⃗ ⃗ Fig. 1
Left:
Illustration of a proton’s position (blue vector), guiding centre (purple vector),and directional Larmor radius (green vector) during its gyration (black circle, with arrowsindicating the direction of rotation) around the background magnetic field line (red vector).This figure was adapted from Northrop (1961).
Right:
Simulation of a proton in a constant anduniform magnetic field performed with a fourth-order Runge-Kutta scheme. The trajectoriesof the particle (solid red) and its guiding centre (dotted blue: Eq. 7; dashed purple: runningaverage of particle’s position over a gyration) are shown, together with a single backgroundmagnetic field line (dashed black; coinciding with the guiding centre). (cid:126)F (cid:107) = − M ( ∂B /∂(cid:126)s ) = − M (cid:126) ∇ (cid:107) B , where d (cid:126)s is a line segment parallel to the magnetic field, (cid:126) ∇ (cid:107) denotes thegradient along the magnetic field, and M = mv ⊥ / B is the particle’s magneticmoment. Due to the invariance of the magnetic moment (d M/ d t = 0) in theabsence of magnetic turbulence and the conservation of kinetic energy, this forceis accompanied by an interchange between parallel and perpendicular energy: asthe particle moves into a region of larger magnetic field strength, its perpendicularspeed increases, with the effect that its parallel speed decreases. Ultimately thiscauses the particle’s motion to be reversed and the particle is mirrored. Not allparticles, however, will be mirrored. It can be shown that a particle starting outin a region with field strength B with | µ | > µ m = (cid:114) − BB m , (8)will not be able to penetrate a region of magnetic field strength B m (Rossi andOlbert 1970; Chen 1984; Choudhuri 1998). Due to the decrease of the heliospheric magnetic field (HMF) strength withheliocentric radius (Parker 1958, see also Appendix F), SEPs will experience mag-netic focusing. As a particle moves into regions of weaker parallel magnetic fields,the particle’s perpendicular speed will decrease while its parallel speed will in-crease, causing the particle’s motion to become increasingly ballistic. Focusing is
Primer on Focused Solar Energetic Particle Transport 7 the reason why SEP events are anisotropic, excluding the fact that the particles arepropagating away from their release at the Sun (Roelof 1969). Since focusing causesthe perpendicular speed to decrease, it might be incorrectly expected that the Larmorradius (which is dependent on the perpendicular speed) would also decrease . TheLarmor radius, however, is inversely proportional to the magnetic field strength,which decreases as 1 /r for the Parker (1958) HMF in the equatorial plane. Fromthe definition of the magnetic moment and its invariance, it can be seen that theperpendicular speed does not change at the same rate as the magnetic field, sincethe magnetic moment is dependent on the square of the perpendicular speed. TheHMF strength therefore decreases faster than the perpendicular speed close to theSun and this would cause the Larmor radius to increase as SEPs move away from theSun . Indeed a simple calculation of the maximal Larmor radius ( v ⊥ replaced by v in Eq. 3) for a 100 keV ( v ≈ . c ) electron close to the Sun (0 . ∼ .
868 km and ∼
186 km, respectively. Even if 99% of the electron’s speed isconverted to parallel speed ( v ⊥ = 0 . v ) by focusing, then the electron’s Larmorradius will still be equal to it’s initial Larmor radius in this example.2.3 The Effects of Slab TurbulenceCharged particles in the heliosphere would have followed the smooth motions de-scribed thus far, were it not for turbulence. Turbulence can be described as seem-ingly random fluctuations containing some level of correlations or structures (Gold-stein et al. 1995; Bruno and Carbone 2005). For the following discussion a non-relativistic proton is simulated in the turbulence model presented in Appendix C,with N +RH = N +LH = 1000, using a fourth-order Runge-Kutta method, with 21600steps per gyration for 5 gyrations (with respect to the background magnetic field; (cid:126)B = B ˆ (cid:126)z with B = 1 × − T). Keep in mind that these simulations will besimilar, except for the sense of gyration, for an electron with the same momentum.Unless otherwise stated, the particle was initialised at the origin with a velocityof (cid:126)v = (2 ˆ (cid:126)y + 3 ˆ (cid:126)z ) m · s − and an Alfv´en speed of V A = 0 . v was used sinceparticles normally move much faster than the waves. All quantities of interestwere calculated with respect to the background magnetic field, as this provides anatural and unchanging ‘reference’ field. To build a systematic understanding of the influence of turbulent fluctuations,consider a magnetostatic wave field with a single wavelength (not shown). If thewavelength is long enough, the magnetic field is changing slowly enough withposition such that the particle is able to follow the perturbed magnetic field line. Ifthe wavelength is short enough, the particle moves so quickly over the fluctuationsthat it does not have time to react to it and its trajectory is only slightly perturbed.
If the wavelength is on the order of the Larmor radius, the particle can resonatewith the wave and the particle’s trajectory is perturbed from the normal helix.Consider a moving resonant magnetic wave with a single wavelength, i.e. aparallel wavelength in the order of the particle’s Larmor radius, λ (cid:107) ∼ r L , but withno induced electric field. If the wave is moving much faster than the particle, the van den Berg et al. x [ R L ]0.750.500.250.000.250.500.75 y [ R L ] Particle trajectoryInstataneous guiding centreGuiding centre over gyrationMagnetic field line x [ R L ]0510152025 z [ R L ] y [ R L ] Fig. 2
Simulation of a proton in a constant and uniform background magnetic field with aspectrum of slab turbulence. The three-dimensional view ( top left ) is projected onto the xy -( top right ), xz - ( bottom left ), and yz -plane ( bottom right ). The trajectories of the particle (solidred) and its guiding centre (dotted blue: Eq. 7; dashed purple: running average of particle’sposition over a gyration) are shown, together with a single background magnetic field line. wave results in very large changes in both the perpendicular and parallel speedsand hence, in the pitch-angle. If the wave is moving much slower than the particle,the GC seems to jump to different regions of the slowly propagating magnetic fieldline over which it is moving. If the wave speed is equal to the particle’s parallelspeed (the Landau or Cherenkov resonance), a very strong resonance occur andthe GC seems to be bouncing between two turning points, reminiscent of classicalhard-sphere collisions. If the fluctuating electric field is also included, the onlysignificant result is that the particle’s energy then changes (see van den Berg2018, for illustrations of these discussions). Fig. 2 shows the trajectories of a proton and its GC when interacting with aspectrum of slab turbulence. The GC was calculated here in two different ways,firstly the ‘instantaneous GC’ was calculated from Eq. 7 and secondly the ‘aver-age GC over a gyration’ was calculated by performing a running average of theparticle’s position over a gyroperiod in the background magnetic field. Notice that
Primer on Focused Solar Energetic Particle Transport 9 c ]2101234 P o s i t i o n [ R L ] xyzx gc y gc z gc x c y c z c c ]0.750.500.250.000.250.500.751.00 V e l o c i t y [ v ] v x v y v z v gc x v gc y v gc z d x c / dtd y c / dtd z c / dt Fig. 3
Top:
Cartesian components of the position (solid) and instantaneous (dotted; Eq. 7)and gyro-averaged (dashed) guiding centre’s position vectors for the proton in Fig. 2.
Bottom:
Similar to the top panel, but for the velocity components. although the spiral trajectory of the particle is highly perturbed, it is still smoothand continuous. The motion of the instantaneous GC, however, is more irregularand reminiscent of classical hard-sphere collisions. The different behaviour of theparticle and the instantaneous GC can be understood if it is realized that anysmall changes in the particle’s velocity would be amplified when ‘projecting thedirectional Larmor radius to the distant position of the GC’. In contrast to this,the gyro-averaged GC follow a smoother trajectory.This qualitatively different behaviour between the particle and its GC canalso be seen in Fig. 3 where the Cartesian components of the particle’s and its
GC’s position and velocity vectors are shown. This behaviour is clearly seen inthe velocity components: the particle’s x - and y -velocity components still exhibita fairly regular oscillation, while the z -velocity component have irregular features;both the position and velocity components of the gyro-averaged GC are smoothaverages of the particle’s components; the instantaneous GC’s z -velocity compo- nent coincides with the particle’s z -velocity component, but its x - and y -velocitycomponents have discontinuous changes reminiscent of collisions.Although not shown here, the changes in the parallel (equal to the velocity’s z -component) and perpendicular speed components will cause the particle’s pitch-angle to change continuously in an irregular way. This is then pitch-angle scatteringand its effect can be seen as the particle is not moving at a constant speed along themagnetic field. Pitch-angle diffusion in velocity space therefore leads to parallel spatialdiffusion in configuration space (Shalchi 2009). It is also important to realise thatpitch-angle scattering is a continuous process and that the pitch-angle should notsimply be changed randomly according to some probability in simulations whichintegrate the Newton-Lorentz equation. Lastly notice that the GC stays close tothe background magnetic field line on which it started. It is expected, both fromtheoretical considerations and simulations (see Shalchi 2009, for a review), thatslab turbulence will lead to little or no perpendicular diffusion (mostly described asa random movement of the GC perpendicular to the background magnetic field).
In a reference frame moving with the wave, where the fluctuations are magne-tostatic with no induced electric field fluctuations, it is expected that the par-ticle’s energy should stay constant since the magnetic field alone cannot do anywork on the particle. Tsurutani and Lakhina (1997) gives the following proof:Consider only the magnetic forces exerted on the particle by the wave, assumethat the particle gains a quantum of energy ∆K = ¯ hω from the wave during aninteraction, and that the change in parallel momentum is m∆v (cid:107) = ¯ hk (cid:107) , where¯ h is Planck’s constant divided by 2 π . If the energy change is small comparedto the particle’s kinetic energy ( K = mv (cid:107) / mv ⊥ / ∆K = ωm∆v (cid:107) /k (cid:107) ≈ m ( v (cid:107) ∆v (cid:107) + v ⊥ ∆v ⊥ ), which gives12 m (cid:16) v (cid:107) − V A (cid:17) + 12 mv ⊥ = constantupon integration. This shows that the particle’s energy in the wave frame is con-served. In the observer’s frame, however, there exists an induced fluctuating electricfield, which can change the particle’s energy. Thus, the particle’s energy is conservedin the wave frame, but not in the observer’s frame .The trajectory of the simulated particle in velocity space ( v ⊥ as a function of v (cid:107) ) is shown in Fig. 4. The dashed semi-circles indicates constant speed, with thegreen and red vectors representing the particle’s initial and final velocity vectors,respectively. The dashed blue semi-circle indicates the particle’s initial speed inthe wave frame ( v pw = (cid:113) ( v (cid:107) − V A ) + v ⊥ ) and the blue dotted line is its initialvelocity vector in the wave frame. This figure clearly illustrates that the particle’senergy is conserved in the wave frame since the trajectory lies on the blue semi-circle, but that the particle’s energy is continuously changing in the observer’sframe. In this graph, the pitch-angle is the angle between the positive v (cid:107) -axis and the velocity vector. Pitch-angle scattering can therefore be seen here as thetrajectory moves on the semi-circle. Although this is the velocity space trajectoryfor only a single particle, the extent of the trajectory towards both 0 ◦ and 90 ◦ pitch-angles are indicative of turbulence trying to isotropise the distribution ofparticles (in this case a single particle) in the wave frame. Primer on Focused Solar Energetic Particle Transport 11 v [ v ]0.00.20.40.60.81.0 v [ v ] v pw =0.918 v [ V A =0.1 v ] v i =1 v v f =1.01 v v Fig. 4
Trajectory (solid black) of the proton in Fig. 2 in velocity space, where the perpen-dicular speed (Eq. 5b) is plotted as a function of the parallel speed (Eq. 5a). The dashedsemi-circles indicate constant speed and the pitch-angle is the angle between the positive v (cid:107) -axis and the velocity vector (dotted lines). The dashed blue semi-circle indicates the particle’sinitial speed in the wave frame. s (cid:107) -axis and the momentum vector (cid:126)p , is the pitch-angle α (Eq. 4). Using the definition of the pitch-cosine µ (Eq. 6), the parti-cle’s momentum can be decomposed into a parallel (cid:126)p (cid:107) = µp ˆ (cid:126)s (cid:107) (ˆ (cid:126)s (cid:107) is a unit vec-tor in the direction of the background magnetic field) and perpendicular (cid:126)p ⊥ = p (cid:112) − µ (cid:16) cos ϕ ˆ (cid:126)s ⊥ + sin ϕ ˆ (cid:126)s ⊥ (cid:17) component (ˆ (cid:126)s ⊥ and ˆ (cid:126)s ⊥ are two mutually per-pendicular unit vectors lying in the plane perpendicular to the background mag-netic field), similar to Eq. 5a and Eq. 5b, respectively. Here p is the magnitude ofthe momentum and can be thought of as a radius in momentum space. The par-allel momentum is the projection of the momentum vector onto the backgroundmagnetic field direction, while the perpendicular momentum is the projection ofthe momentum vector onto the plane perpendicular to the background magneticfield. The azimuthal angle, the angle between the s ⊥ -axis and the perpendicularmomentum, is the particle’s gyrophase ϕ and its rate of change is the cyclotronfrequency ω c (Eq. 2). The gyration of the particle around the magnetic field causesthe momentum vector to precess around the s (cid:107) -axis at the cyclotron frequency.Focusing will decrease the particle’s pitch-angle, while scattering will eitherincrease or decrease it, as indicated by the green and red arrows, respectively. As the number of particles under consideration in a real event is so large that there is,for all practical purposes, a particle in every phase of gyration, a gyrotropic distri-bution of particles are normally considered. This can be illustrated as a collectionof particles having the same pitch-angle, but different gyrophases and is indicatedby the grey circle (also referred to as a ring-distribution). This also represents a ⃗ ⃗ ̂ ̂ ̂ ⃗ Focusing Scattering Gyration Pitch-angle: ( )
Pitch-cosine: ( )
Gyrophase:
Cyclotron frequency: | |
Gyrotropic
Fig. 5
Illustration of the various processes and definitions introduced. Shown is the particle’smomentum space in a field-aligned coordinate system. See Section 2.4 for details. This picturewas inspired by Prinsloo et al. (2019). gyrotropic distribution of mono-energetic particles, since all of their momentumvectors have the same magnitude. If the particles were to have different energiesand gyrophases, but the same pitch-angle, then their momentum vectors will formthe shaded cone. The cone will then represent a possible anisotropic distributionas the particles have a preferred direction of motion along the background magneticfield . For a gyrotropic distribution of mono-energetic particles, pitch-angle scat-tering will cause the circle to change into a spherical shell (also referred to asa shell-distribution), assuming that the scattering does not change the particles’energy and that enough time has elapsed. Similarly, pitch-angle scattering willcause the cone of an anisotropic distribution to become a filled sphere. In such acase, the distribution will be called isotropic with particles of all energies movingin all directions. Turbulence can therefore drastically change the characteristics ofthe original particle distribution and will mostly act to isotropise an anisotropicdistribution.
Primer on Focused Solar Energetic Particle Transport 13
A macro-physical or ensemble averaged description of SEPs is needed for mostmodelling purposes and since SEPs are highly anisotropic, the Parker (1965) trans-port equation, used to model CRs, is inadequate for this purpose. The evolution ofthe anisotropic SEP distribution function can be described by a so called focusedtransport equation (FTE). The concept of a distribution function and the simplestform of the FTE will be introduced here.3.1 The Distribution FunctionThe distribution function f ( (cid:126)x ; (cid:126)p ; t ) = d N/ d x d p of a system is defined as thenumber density d N in a volume element d x d p of the 6-dimensional phase-spacespanned by the three spatial (cid:126)x and momentum (cid:126)p coordinates. It can be interpretedas the number of particles at time t having position vectors between (cid:126)x and (cid:126)x + d (cid:126)x with momentum vectors between (cid:126)p and (cid:126)p +d (cid:126)p . Integrating the distribution functionover all space and momentum would give the total number of particles in thesystem. Dividing the distribution function by the total number of particles, resultsin a probability distribution to find particles in the phase-space volume d x d p around ( (cid:126)x ; (cid:126)p ) at time t (Choudhuri 1998; Moraal 2013; Zank 2014).Plasma physics or transport theory textbooks (see e.g. Chen 1984; Choudhuri1998) usually defines the distribution function in terms of velocity and not mo-mentum. Such a distribution function is fine for non-relativistic particles, but forrelativistic particles, a distribution function defined in terms of momentum is pre-ferred. Consider an observer frame, where quantities are unprimed, and a framemoving with respect to the observer frame, where quantities are primed. It can beproven that the phase-space volume element is invariant, d x d p = d x (cid:48) d p (cid:48) (seee.g. Zank 2014), with the implication that the distribution function would also beinvariant, f ( (cid:126)x ; (cid:126)p ; t ) = f (cid:48) ( (cid:126)x (cid:48) ; (cid:126)p (cid:48) ; t (cid:48) ). This is expected since the distribution functionis related to the particle number density which is invariant between different ref-erence frames. It is here implicitly assumed that a non-relativistic transformationcan be made between the stationary and solar wind (SW) frames, so that t = t (cid:48) may be assumed. However, if the distribution function is defined in terms of ve-locity, then the phase-space volume element is not invariant and the distributionfunction also not (Moraal 2013; Zank 2014).For a plasma with a stationary background or large scale average magneticfield, the magnetic field can be used as a reference point. The distribution func-tion can then be defined in a field aligned coordinate system and a transformationfrom Cartesian to spherical coordinates can be made in momentum space (seeFig. 5), such that d N = f ( s (cid:107) ; (cid:126)s ⊥ ; p ; µ ; ϕ ; t )d s (cid:107) d s ⊥ d p d µ d ϕ . The dependence ofthe distribution function on ϕ can be averaged out to yield the gyrotropic distri-bution function f ( s ; p ; µ ; t ) = (cid:82) π f ( s (cid:107) ; (cid:126)s ⊥ ; p ; µ ; ϕ ; t )d ϕ/ π . By performing such anaverage, transport perpendicular to the magnetic field is removed (see e.g. Zank2014), hence the dependence on (cid:126)s ⊥ was neglected and s was written for s (cid:107) . Driftsor diffusion perpendicular to the magnetic field is therefore not described here andthis distribution function can be thought of as describing the number of particlesper phase space volume in a given flux tube (Ng and Wong 1979). Notice that the neglect of perpendicular transport implies that the intensity of an SEP eventmight be overestimated.The distribution function is a quantity of theoretical interest, but it can givea complete description of a system’s state and various useful quantities can becalculated from it (Chen 1984; Choudhuri 1998, e.g. show how the hydrodynamicequations can be derived from the distribution function and its governing equa-tion). The omni-directional intensity (ODI) F ( s ; p ; t ) = 12 (cid:90) − f ( s ; p ; µ (cid:48) ; t )d µ (cid:48) , is essentially the distribution function without a pitch-angle dependence and repre-sents the number of particles at time t within d s from s with a momentum between p and p + d p . It is related to the measured differential intensity in terms of kineticenergy by j = p F / A ( s ; p ; t ) = 3 (cid:82) − µ (cid:48) f ( s ; p ; µ (cid:48) ; t )d µ (cid:48) (cid:82) − f ( s ; p ; µ (cid:48) ; t )d µ (cid:48) , is a measure of how anisotropic the distribution is at a certain phase-space point( s ; p ) at a time t . Notice that the distribution function (phase-space density) ischanged to a probability by dividing with (cid:82) − f ( s ; p ; µ (cid:48) ; t )d µ (cid:48) , and the anisotropycan therefore be interpreted as essentially three times the average or expectedpitch-cosine. It has a value of 3 ( −
3) if all particles are moving along (in theopposite direction of) the magnetic field and a value of zero if there are equalnumber of particles moving in opposite directions (isotropic) or if all the particleshave no parallel speed (an unlikely case). The anisotropy is usually calculated inobservations from the pitch-angle distribution (PAD) F ( s ; p ; µ ; t ) = f ( s ; p ; µ ; t ) (cid:82) − f ( s ; p ; µ (cid:48) ; t )d µ (cid:48) , which is a probability distribution and is normally constructed from the sectoredmeasurements of detectors looking in different directions.3.2 The Focused Transport EquationThe distribution function’s evolution is in general governed by the Fokker-Planckequation, which is a generalisation of Liouville’s theorem for a distribution func-tion including the effects of random changes to the momentum coordinates by turbulence or collisions (Choudhuri 1998; Zank 2014). A transformation fromCartesian to spherical coordinates in momentum space is made and an averageover gyrophase is then preformed, as described in the previous paragraphs. Addi-tionally, a transformation can first be made from the observer’s frame to a waveframe, usually assumed to be the SW frame, because momentum diffusion can be Primer on Focused Solar Energetic Particle Transport 15 neglected in this frame. Alternatively, the Vlaslov equation, essentially the colli-sionless Boltzmann equation with the Lorentz force substituted, can be used as apoint of departure. The distribution function and the electric and magnetic fieldmust then be written as the sum of a large scale average and a rapid fluctuatingpart, with the fluctuating part acting as a perturbation on the average part. Suchderivations, as given by Zhang (2006) or Zank (2014), lead to the focused transportequation (FTE), but are lengthy and beyond the scope of the current discussion.Although the name “focused transport equation” might be a misnomer, as itdescribes the evolution of any anisotropic distribution, it is appropriate in the caseof SEPs since the anisotropy is caused primarily by focusing. The simplest formof the FTE, is that of Roelof (1969) without advection or energy losses ∂f∂t + ∂∂s [ µvf ] + ∂∂µ (cid:20) (1 − µ ) v L ( s ) f (cid:21) = ∂∂µ (cid:20) D µµ ∂f∂µ (cid:21) , (9)where L ( s ) is the focusing length of the magnetic field given by Eq. 22 and D µµ isthe pitch-angle diffusion coefficient (PADC) describing the random changes of thepitch-angle due to turbulence. This equation describes the evolution of the distri-bution function f ( s ; µ ; t ) for a constant particle speed v . The various terms, fromleft to right, describe temporal, spatial (the streaming of particles along the mag-netic field, since µv is their parallel speed), and pitch-angle changes (discussed inAppendix D) on the left hand side, and pitch-angle diffusion on the right hand side.It should be noticed that the FTE is a highly non-linear, second order, parabolicpartial differential equation. The different processes’ effects cannot be added lin-early because each process is dependent on quantities which are affected by theother processes. The various terms therefore affect one another and the dominat-ing process is ultimately determined by its relative strength. This non-linearityand competition between terms imply that none of the terms can be neglected tomodel SEPs realistically.The PADC must be specified and a variety of options are available from differ-ent theories. Three rather simple forms will be used here for illustrative proposes.A widely used PADC is that of Beeck and Wibberenz (1986), D BW µµ = D (1 − µ )( | µ | q − + H ) , (10)based on quasi-linear theory (QLT; Jokipii 1966; Shalchi 2009). Here D is thescattering amplitude, q is the spectral index of the magnetic turbulence’s inertialrange, and H is an arbitrary (in terms of its value) correction to describe theinclusion of dynamical effects. If q = 1 and H = 0, then D iso µµ ( µ ) = D (1 − µ ) (11)is called isotropic scattering. This PADC can be used in the presence of very strongturbulence, but if the turbulence is weaker and pitch-angle scattering is caused byresonances with a spectrum of waves, then anisotropic scattering must be used. If dynamical effects are neglected ( H = 0), then D QLT µµ ( µ ) = D (1 − µ ) | µ | q − (12)has the known problem of a resonance gap at µ = 0 ( D QLT µµ (0) = 0) (Dr¨oge 2000a).Fig. 6 shows the different PADCs and their derivatives. Care should be taken here D [ D ] Isotropic Scattering ( q = 1, H = 0)Kraichnan QLT ( q = 3/2, H = 0)Kolmogorov ( q = 5/3, H = 0.05) D / [ D ] Fig. 6
Pitch-angle diffusion coefficients ( top ) and their derivatives ( bottom ) of isotropic scat-tering (blue; Eq. 11), quasi-linear theory with a Kraichnan inertial range (red; Eq. 12 with q = 3 / q = 5 / H = 0 . not to confuse isotropic or anisotropic scattering with an isotropic or anisotropicdistribution.The scattering amplitude is usually calculated from the parallel mean free path(MFP). The MFP can be generally defined as the average distance moved by a par-ticle before its velocity is uncorrelated with its initial velocity . Based, however, on theresults of the previous section, the parallel MFP might be better interpreted as theaverage distance a particle would move in a turbulent plasma, being continuously sub-jected to small pitch-angle changes, before the pitch-angle is changed significantly andthe particle’s GC reverses its direction of motion parallel to the background magneticfield . The connecting formula between D and the parallel MFP, is λ (cid:107) = 38 v (cid:90) − (1 − µ (cid:48) ) D µµ ( µ (cid:48) ) d µ (cid:48) (13)for an isotropic distribution. Notice that this is not a formal definition, but rathera consequence of averaging Eq. 9 over pitch-cosine in the absence of focusing foran isotorpic distribution (Jokipii 1966; Hasselmann and Wibberenz 1970; Shalchi λ (cid:107) = 3 L (cid:82) − µ (cid:48) e G ( µ (cid:48) ) d µ (cid:48) (cid:82) − e G ( µ (cid:48) ) d µ (cid:48) , (14) Primer on Focused Solar Energetic Particle Transport 17 where G ( µ ) is given by Eq. 25. This expression reduces to the former in the absenceof focusing (Beeck and Wibberenz 1986; He and Schlickeiser 2014). From this itcan be seen that the interpretation of the parallel MFP is modified in the presence offocusing . It was already stated by Earl (1981) that the MFP would change due tothe focusing length, because focusing causes the distribution to have pitch-anglesclose to µ ∼ L ( s ) is position dependent, the parallel MFP would also change with position . Due to these reasons, it might bebetter to calculate D from observable turbulence properties instead (He and Wan2012, presents a spatially varying MFP based on these considerations).3.3 Comparison of the Diffusion and Telegraph Approximations to DescribeFocused TransportIn the isotropic limit, the transport could be well described by a diffusion equa-tion (see Parker 1965). The force field approximation could successfully be appliedto galactic CR spectra, even though all the complicated modulation processes(such as advection, diffusion, energy losses, and drifts) were absorbed into a singleparameter (i.e. the modulation potential; see Moraal 2013). Analytical approxi-mations also exist for the propagation time and average energy losses of CRs (seeagain Parker 1965). Within focused transport there is unfortunately no simplisticapproximation which give satisfactory results. The advection-diffusion and tele-graph approximations are introduced in Appendix E and it will be shown to whatextend these approximations can be used. The analytical approximations will becompared to two numerical solutions of the FTE. The first solution, revered to as‘the model’, uses a finite difference scheme and is given in Appendix A (includinga link to the source code). The second solution, used as synthetic data, uses astochastic differential equation approach and is discussed in Appendix B.Energy losses can be considered to be negligible for 100 keV electrons andwill be used here as an example. A constant parallel MFP and focusing lengthof λ (cid:107) = 0 . L = 0 . ξ = λ (cid:107) /L = 1 /
3, which is in theweak focusing limit necessary for the anisotropic case. The injection is located at s = 0 AU and an observer is assumed to be located at s = 1 . The temporal evolution of the probability density (comparable to the ODI througha proper scaling constant) as a function of position is shown in the top panelof Fig. 7, where the model, the diffusion approximation, and telegraph equationare compared to the synthetic data. Focusing causes a coherent pulse to form,propagating with speed ∼ u (see Appendix E) and composed mainly of particleswhich have not yet undergone significant scattering. The pulse spreads out with O m n i - d i r e c t i o n a l i n t e n s i t y Finite difference modelDiffusion approximationTelegraph equation t = 0.1 h t = 0.2 h t = 1.0 h t = 5.0 h0 2 4 6 8Time [h]0.000.050.100.150.200.250.300.35 O m n i - d i r e c t i o n a l i n t e n s i t y Finite difference modelDiffusion approximationTelegraph equationSynthetic data
Fig. 7
Top:
Temporal evolution of the normalised omni-directional intensity as a function ofposition for 100 keV electrons with isotropic pitch-angle scattering (Eq. 11), λ (cid:107) = 0 . L = 0 . Bottom:
Normalisedomni-directional intensity as a function of time as seen by an observer at s = 1 . time due to scattering, while the scattered particles, having smaller parallel speedsthan the focused particles in the pulse, form a wake behind the pulse. Both thediffusion approximation and telegraph equation is in good agreement with thesynthetic data at late times, while at early times the diffusion approximation is toodiffusive and the telegraph equation predicts a very sharp propagation front. TheGaussian shape of the diffusion approximation is clearly inadequate to describe thenon-symmetric density, except in the wake, and the causality violation is clearly visible ahead of the pulse. The model is in best agreement with the syntheticdata, although it is a little too diffusive at very early times (flux limiters are usedto reduce numerical diffusion which is, to some extend, always present in finitedifference models).The ODI as a function of time at the observer is shown in the bottom panelof Fig. 7. The intensity has a quick rise time up to a peak intensity, after which Primer on Focused Solar Energetic Particle Transport 19 A n i s o t r o p y Finite difference modelDiffusion approximationTelegraph equationSynthetic data1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.000.30.40.50.60.70.8 P i t c h - a n g l e d i s t r i b u t i o n Stationary global solutionModel: Close to peak intensityData: Peak intensityData: Event integrated
Fig. 8
Top:
Anisotropy as a function of time as seen by the observer in Fig. 7.
Bottom:
Pitch-angle distribution at peak intensity and time integrated pitch-angle distribution at theobservation point in Fig. 7 compared to the analytical stationary pitch-angle distribution(Eq. 24; black dashed line) and the model pitch-angle distribution at t = 1 h. the flux decreases slowly, characteristic of impulsive SEP events. This behaviourcan be understood by looking at the top panel: as the pulse propagates past theobserver, the event onset is seen followed by the peak intensity and the intensitydecrease in the wake of the pulse or the decay phase. Although both the diffusionapproximation and telegraph equation are comparable at late times, initially thesetwo solutions are too diffusive or restrictive, respectively, and it seems as if thetrue solution (i.e. the model) is an interpolation between the two approximations.The anisotropy as a function of time at the observer is shown in the top panel ofFig. 8. The distribution is initially highly anisotropic after which it becomes moreisotropic. Physically this is because the first particles to arrive at the observationpoint are particles with small pitch-angles that was focused and have experienced little scattering, while the particles in the wake have experienced more scatteringand are approaching diffusive behaviour. Some text refer to these first arrivingparticles, associated with large anisotropies, as particles undergoing scatter-freepropagation . The phrase scatter free is however a misnomer when considering parti-cle propagating in magnetic turbulence: all charged particles will experience these O m n i - d i r e c t i o n a l i n t e n s i t y Finite difference modelDiffusion approximationTelegraph equation t = 0.1 h t = 0.2 h t = 1.0 h t = 5.0 h0 2 4 6 8 10Time [h]0.000.050.100.150.200.250.300.35 O m n i - d i r e c t i o n a l i n t e n s i t y Finite difference modelDiffusion approximationTelegraph equationSynthetic data
Fig. 9
Similar to Fig. 7, but for anisotropic scattering (Eq. 12). turbulent fluctuations and will, to some extent, have their smooth gyro-motiondisturbed. The causality violation of the diffusion approximation can be seen asinfinite anisotropies before the event onset, while predicting a lower anisotropyduring the event’s onset. The telegraph equation is generally better at predictingthe anisotropy, but has a significantly delayed onset time.The bottom panel of Fig. 8 shows the event integrated PAD at the observercompared to the analytical stationary solution. The event integrated PAD cor-responds very well to the analytical solutions of the stationary PAD. It is moreinteresting to note the temporal behaviour of the PAD at the observer (a threepoint average in time and pitch-cosine was taken in the synthetic data to smoothout fluctuations). The distribution is beam-like at the event onset (not shown) with the PAD coinciding with the stationary solution just after the peak inten-sity (the model result), after which the distribution slowly approach isotropy (notshown). This seems to imply that the pulse has a quasi-stationary distribution setup by a balance between focusing and scattering, as suggested by Eq. 23 (Beeckand Wibberenz 1986).
Primer on Focused Solar Energetic Particle Transport 21 A n i s o t r o p y Finite difference modelDiffusion approximationTelegraph equationSynthetic data1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.000.30.40.50.60.7 P i t c h - a n g l e d i s t r i b u t i o n Stationary global solutionModel: Close to peak intensityData: Peak intensityData: Event integrated
Fig. 10
Similar to Fig. 8, but for the anisotropic scattering (Eq. 12) in Fig. 9
The temporal evolution of the probability density as a function of position is shownin the top panel of Fig. 9. The effect of anisotropic pitch-angle scattering is inter-esting and visible in the initial phase of the event. The delta injection seem to splitinto two coherent pulses propagating away from one another. This behaviour isdue to ineffective scattering across µ = 0 and the use of an isotropic injection. Theeffect of focusing can be seen as there are more particles in the pulse propagat-ing towards weaker magnetic fields. These two pulses, however, are combined intoone pulse by scattering some time after the injection. The diffusion approximationand telegraph equation is again in good agreement with the synthetic data at latetimes, but at early times even the model is too diffusive to replicate these twopulses (it is well known that finite difference models do not handle steep gradientswell). The ODI as a function of time at the observer is shown in the bottom panelof Fig. 9, while the anisotropy is shown in the top panel of Fig. 10. These resultsare comparable to the isotropic scattering scenario. It should be kept in mindthat the coefficients presented in Appendix E for the telegraph equation, are onlyappropriate if the weak focusing limit is considered. The model predicted intensity is a bit lower, probably due to numerical diffusion of the initial pulse. The eventintegrated PAD and PAD at peak intensity at the observer are shown in the bottompanel of Fig. 10. The effect of anisotropic scattering can be seen in the PAD as adecrease in crossing µ = 0 from positive to negative values. Also note here that thePAD at peak intensity (the ‘synthetic data’) does not coincide exactly with thestationary solution, but only shortly before or after the peak (the ‘model’). Thisshould be kept in mind if the PAD is used to extract parameters from data andit might be tempting to use the PAD at peak intensity because it is easier thancalculating the event integrated PAD. The 65 −
105 keV solar energetic electron event of 7 February 2010 observed bySTEREO-B (see Figure 9a in Dr¨oge et al. 2014) will be considered in this section.Electrons are injected at s = 0 .
05 AU with an energy of 80 keV and a reflectiveinner boundary assumed at s = 0 AU, supposedly caused by mirroring in theHMF. A constant radial MFP of 0 .
12 AU will be used with q = 5 / H = 0 . λ (cid:107) = λ r / cos ψ . The arc length, focusing length, radial and parallel MFP,and focusing parameter are shown in Fig. 11. These parameters have values of s = 1 .
139 AU, L = 0 .
936 AU, λ (cid:107) = 0 .
238 AU, and ξ = 0 .
253 at Earth. The parallelMFP formulation, which directly incorporates focusing, (Eq. 14) is also shown,assuming that D is calculated from Eq. 13. From the focusing parameter it canbe seen that focusing will have the largest effect within the first ∼ f ( s = s , t ) = Ct e − τ a /t − t/τ e (15)with C a normalisation constant and τ a = 0 . τ e = 1 hr the acceleration andescape time, respectively, will be assumed here. These best-fit model results arecompared to observations in Fig. 12. The top panel shows the assumed injectionfunction as a function of time, the two middle panels the calculated ODI andthe anisotropy. Appendix A illustrates how sensitive these results are to changingtransport parameters. Notice that there is a significant discrepancy between thefinite difference and stochastic differential equation (synthetic data) model duringthe decay phase of the event, even though the two models were run with thesame parameters. This is due to an implicitly assumed absorbing outer boundarycondition at s = 3 AU in the finite difference model. Calculating the Number of Particles Injected
The distribution function can also beused to calculate the average propagation time or energy losses suffered by parti-cles. Some analytical diffusion approximations can be found in Parker (1965), for example, while Strauss et al. (2011) show how easily a stochastic differential equa-tion model can be used to calculate these quantities for CRs and Jovian electrons,for example. Litvinenko et al. (2015) gives an approximation for the average prop-agation time of SEPs using the telegraph equation. Average energy losses havebeen investigated by Kocharov et al. (1998) and Zhang et al. (2009), for example,
Primer on Focused Solar Energetic Particle Transport 23 L e g n t h s c a l e s [ A U ] sL ( s ) r ( s )( s )0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 r [AU]10 F o c u s i n g p a r a m e t e r Fig. 11
Top:
Magnetic field arch length (black; Eq. 35), focusing length (blue; Eq. 36), andconstant radial (grey), ‘isotropic’ parallel (red, λ (cid:107) = λ r / cos ψ ), and ‘focusing included’ par-allel (dashed green; Eq. 14) mean free paths as a function of heliocentric radius. Bottom:
Focusing parameter (ratio of the parallel mean free path to the focusing length) as a functionof radius. the latter of which also briefly investigated propagation times. As a less obviousapplication, the problem of calculating the total number of particles released inan SEP event will be considered here.de Nolfo et al. (2019) compared the number of particles released from a longduration solar flare to the number of particles needed to produce the gamma-raysobserved by Fermi-LAT in order to access the possible acceleration mechanisms.These authors used PAMELA and STEREO A and B data to calculate the totalnumber of >
200 MeV protons observed in the heliosphere N obs during such flares.This number, however, is larger than the number of protons released in the flare N inj due to particle scattering causing particles to move past the observation pointmultiple times. A correction is therefore necessary as the PAMELA detector can- not discriminate between Sunwards and anti-Sunwards propagating particles, i.e.the pitch-angle dependence cannot be observed. To correct for this, N obs was di-vided by the average number of times a particle would cross the observation point¯ N cross ( N inj = N obs / ¯ N cross ). This number ¯ N cross , of course, depends on the under-lying turbulence and, as such, cannot be determined experimentally and must be I n j e c t i o n p r o f il e O m n i - d i r e c t i o n a l i n t e n s i t y = 1.17 AU; = 1.0 AU = 0.12 AU; = 0.08 MeV = 400.0 km/sparticle = 1.0(1 = electrons; 2 = protons)injection = 2.0(1 = delta-like; 2 = Reid-Axford) = 0.1 h; = 1.0 h A n i s o t r o p y Time [h]
Fig. 12
Best fit results of the finite difference model (solid green), together with the stochasticdifferential equation model (solid cyan), to the electron data from the 7 February 2010 event(black dots).
Top:
Injection function (Eq. 15) normalised to its peak value.
Middle:
Omni-directional intensity and anisotropy.
Bottom:
The ratio of forward to backward propagatingparticles as calculated by Eq. 16. estimated from simulations. In de Nolfo et al. (2019), this number was calculatedfrom two test particle simulations using an unspecified plasma turbulence field forthe scattering. These simulations yielded numbers varying between 3 . ¯ N cross = 8 was used for all fourteen events considered. Information regarding the propagation direction of the SEPs are, however,contained in the distribution function and if the pitch-angle dependence of thedistribution function can be determined (either experimentally or through simula-tions), the use of any ad-hoc corrections, such as implementing a rather arbitrary¯ N cross factor, is unnecessary. Formally, the ratio of forward (outwards) to backward Primer on Focused Solar Energetic Particle Transport 25 (inwards) propagating particles can be calculated as R = f out − f in f out + f in = (cid:82) +10 f d µ − (cid:82) − f d µ (cid:82) +10 f d µ + (cid:82) − f d µ = (cid:82) +10 f d µ − (cid:82) − f d µ (cid:82) +1 − f d µ . (16)This calculation was performed with the numerical model discussed in Section A,using the same parameters as in Fig. 12, and shown in the bottom panel of Fig. 12. R is, as expected, not constant for the entire duration of the event and roughlyfollows the temporal evolution of the anisotropy, A . There is, however, not a simplelinear relationship between these two quantities, with the ratio being A R = 3 (cid:82) +1 − µf d µ (cid:82) +10 f d µ − (cid:82) − f d µ , which should be integrated numerically. In order to estimate the total numberof SEPs passing e.g. a spacecraft position where the distribution function is notknown, a simple numerical model can be used, tuning the transport parametersto fit the ODI, and using the computed distribution function to calculate R andultimately use R to calculate the particle flux from the omni-directional particleintensity. In this section different models and/or applications of the FTE that were appliedto SEP transport will be reviewed. Models with increasing levels of complexitywill be discussed, starting from the spatial 1D version, the so-called Roelof (1969)equation, and extending to 3D full-orbit simulations.4.1 1D Simulations
The most simplistic view of SEP transport is that of ballistic motion along a singlesmooth and unperturbed Parker (1958) magnetic field line. Under this unrealisticassumption, the so-called onset time, t o , i.e. the time that a detector will startmeasuring an increase in SEP intensity in a given energy channel, is given by t o ( v ) = t i + sv , where t i is the injection time (i.e. when the particles are released from their accel-eration site) and s is their (magnetic) propagation length. This is generally referredto as a velocity dispersion analysis (VDA). The left panel of Fig. 13 shows suchan analysis from Lin et al. (1981), where a linear fit of t o against the inverse of β = v/c , gives an estimation of s . However, these analysis lead to seemingly con-tradictory results, e.g. the results presented in Fig. 13 indicate, for some energies, s < events, with results presented by Bian and Emslie (2019) and Li and Lee (2019),showing that the particle onset is delayed due to (weak) scattering. An improve-ment over the traditional VDA, using a so-called fractional VDA, which mitigatesthe uncertainties in detector onset determination, was recently established by Zhaoet al. (2019). Primer on Focused Solar Energetic Particle Transport 27
Fig. 13
Left panel:
An early velocity dispersion analysis from Lin et al. (1981).
Right panel:
Simulation results of a velocity dispersion analysis by Laitinen et al. (2015).
Fig. 14
Left panel:
A comparison between simulated and observed solar energetic electronfluxes, taken from Dr¨oge et al. (2006).
Right panel:
A selection of transport parameters obtainedby comparing simulated and observed solar energetic particle fluxes and anisotropies, fromDr¨oge (2000b).
Although simulations of the VDA can lead to insight regarding particle scatter-ing, a more thorough approach is to reproduce the observed temporal profiles ofboth the observed SEP intensity and anisotropy with simulation results. The left panel of Fig. 14 shows an example of such a data comparison from Dr¨oge et al.(2006): The top panel shows the assumed injection function (discussed more inthe next section), the middle panel the ODI, and the bottom panel the anisotropy.A careful comparison between simulations and observations, adjusting the correctcombination of parameters (predominantly D µµ and the injection function in the
1D approach), can lead to accurate estimations of e.g. the parallel MFP. These es-timations, for a large number of events and different energy channels, are presentedby Dr¨oge (2000b) and shown in the right panel of Fig. 14. Here, open and filledsymbols show results for electrons and protons, respectively, which are comparedto a theoretical prediction for protons. These results show that the SEP trans-port parameters can have very large (almost two orders of magnitude) inter-eventvariability, most likely related to the changing plasma conditions and levels ofmagnetic turbulence between the Sun and the observer. In addition, these resultsshow different behaviour for low energy protons and electrons, with the MFP forlow energy electrons increasing for decreasing energies (see also Dr¨oge and Kar-tavykh 2009). Such dependencies are expected from scattering theory (Teufel andSchlickeiser 2002) if a dissipation range is included in the assumed slab turbulencespectrum: low energy electrons resonate in this weak-turbulence regime, experi-encing very little pitch-angle scattering, leading to large MFPs. However, recentsimulations of very low energy electron transport by Kartavykh et al. (2013) haveshown rather large discrepancies with predicted theoretical results, and this is yetto be explained.Most of the 1D simulation results neglect SW effects, including SW convectionand adiabatic energy losses. Results from Ruffolo (1995) indicate that both ofthese effects could potentially be negligible for relativistic electrons while becomingincreasingly important for low energy particles. When both SW convection andenergy losses are considered, the event onset and peak time are slightly earlier,due to convection, while the peak intensity is lower and the decay is quicker, dueto energy losses (Ruffolo 1995; Kocharov et al. 1998). Moreover, Qin et al. (2006)found that a model without energy losses would generally overestimate the derivedparallel MFP.
In the previously discussed modelling work, the focus was on simulating the ef-fect of, specifically, pitch-angle scattering on the particle intensity observed atsome point far away from the SEP source. The acceleration process near the Sun(through flares and/or coronal mass ejections) are therefore mostly neglected andreduced to a so-called injection function: The temporal profile of SEPs releasedinto the interplanetary medium, which could also be energy, spatially, and pitch-angle dependent. Agueda et al. (2008) and Agueda et al. (2009) implement a 1DSEP model, where a series of short, impulsive bursts of particles are injected,allowing for a deconvolution of the transport and injection processes. The calcula-tion of these Green’s functions allow the modelling inversion to be done for a largenumber of events, forming a database of particle injection histories (e.g. Aguedaet al. 2012; Vainio et al. 2013). By comparing the simulated injection profiles toremote-sensing observations (e.g. soft- and hard-X-rays and radio-observations)can lead to insight regarding the source of these SEP electrons (Agueda et al.2014; Agueda and Lario 2016; Pacheco et al. 2019). An example of this inversion approach, taken from Pacheco et al. (2017), is shown in Fig. 15: The left panelcompares observations (thin lines) and modelling results (thick lines) for differ-ent pitch-angles (i.e. detector viewing directions). The best fit radial MFP, for45–65 keV electrons, is given at the top of the left panel. The right panel showsthe derived injection function (release history) of these electrons near the Sun, as
Primer on Focused Solar Energetic Particle Transport 29
Fig. 15
Left panel:
A comparison between pitch-angle dependent simulation results (thicklines) and STEREO B spacecraft observations (thin lines) for four different detector viewingdirections (Sun, Anti-sun, North, and South).
Right panel:
The derived injection function,this time for both STEREO A and B, for two different energy channels, along with radio-observations. Both figures are taken from Pacheco et al. (2017). compared with radio-observations. The inversion results presented here are con-sistent with the acceleration of electrons in a solar flare, with low-energy escapingelectrons producing the observed radio beams.Most modelling results, however, do not apply such a detailed deconvolutionmethod, and the standard approach is to adopt a Reid-Axford profile (Eq. 15;Reid 1964) for the injection function, characterized by an acceleration and a decaytimescale. Even these parameters can, of course, be constrained by observations(Ruffolo et al. 1998). Moreover, all of the injection functions discussed above as-sume an isotropic injection of particles, which might be an over-simplification.Results from Kocharov et al. (1998) suggest that the pitch-angle dependence ofthe injection function may affect the simulated intensities, especially close to theSEP source.
In addition to the results discussed above, Kartavykh et al. (2016) presented 1Dsimulation results where pitch-angle dependent shock acceleration at a movingshock is included, and hence also an energy coordinate. They show that particleacceleration occurs predominantly near the Sun ( r < .
05 AU), leading to a so-called “prompt” phase of the SEP event being observed at Earth, where after the propagation of the shock modulated the observed “gradual” phase of the event,potentially explaining the observation of so-called “mixed” particle events (e.g.Cane et al. 2003) showing a combination of both gradual and impulsive SEPcharacteristics. As a last application, Strauss et al. (2017) compared simulatedand observed temporal profiles of high-energy SEP events as measured by ground-
Fig. 16
Left panel:
An illustration of particle propagation without the inclusion of perpen-dicular diffusion (blue particle tied to the green magnetic field line) and with perpendiculardiffusion (blue particle able to decouple from the red magnetic field line).
Right panel:
Simu-lated SEP intensities in the equatorial plane, illustrating how particles diffuse perpendicularto the mean field. Both figures are taken from Dr¨oge et al. (2010). based neutron monitors (so-called ground level enhancements) to characterise therise and decay times of these events. These authors showed that the effect of pitch-angle scattering can be significant, even for high-energy ( >
100 MeV protons)ground level enhancement events.4.2 2D and 3D SimulationsThe observations of so-called widespread SEP events (see e.g. Dresing et al. 2012,amongst others) have shown that impulsively accelerated SEP electrons can beobserved up to ∼ ◦ away from their source. Possible mechanisms invoked toexplain this seemingly efficient cross-field transport are perpendicular diffusionand drift effects, in addition to a wide injection region. These effects cannot beincluded in a spatially 1D model, resulting in the development of SEP models withhigher dimensionality. These models are briefly discussed in this section.So far, in this work, the focus was only on discussing field-aligned transportand the perpendicular diffusion process was not touched on, for the most part.However, the discussion in Sec. 4.2.2 presents an illuminating picture of this pro-cess. Perpendicular transport can be retained in the FTE by first transforming tothe GC position before changing to spherical coordinates in momentum space andpreforming a gyro-averaging. Details of such derivations can be found in Zhang(2006), le Roux and Webb (2007), and Wijsen (2020). The FTE then includesparticle drifts and diffusion perpendicular to the magnetic field, and is given by ∂f∂t + ∂∂x i (cid:20) d x i d t f (cid:21) + ∂∂p (cid:20) d p d t f (cid:21) + ∂∂µ (cid:20) d µ d t f (cid:21) = ∂∂µ (cid:20) D µµ ∂f∂µ + D ⊥ µj ∂f∂x j (cid:21) + ∂∂x i (cid:20) D ⊥ iµ ∂f∂µ + D ⊥ ij ∂f∂x j (cid:21) , (17)where D ⊥ ij are the perpendicular diffusion coefficients, with the mixed terms D ⊥ iµ = D ⊥ µj usually neglected. Note that the distribution function f ( (cid:126)x ; p ; µ ; t ) is written Primer on Focused Solar Energetic Particle Transport 31 in a mixed coordinate system where the GC position (cid:126)x and time are measured inthe observer’s frame, and the momentum p and pitch-cosine µ are measured in theSW frame.In Eq. 17,d (cid:126)x d t = µv ˆ (cid:126)b + (cid:126)v sw + (cid:126)v d d p d t = p (cid:26) − µ (cid:126)b ˆ (cid:126)b : (cid:126) ∇ (cid:126)v sw − − µ (cid:126) ∇ · (cid:126)v sw − µv ˆ (cid:126)b · (cid:20) ∂(cid:126)v sw ∂t + ( (cid:126)v sw · (cid:126) ∇ ) (cid:126)v sw (cid:21)(cid:27) d µ d t = 1 − µ (cid:26) v (cid:126) ∇ · ˆ (cid:126)b + µ (cid:126) ∇ · (cid:126)v sw − µ ˆ (cid:126)b ˆ (cid:126)b : (cid:126) ∇ (cid:126)v sw − v ˆ (cid:126)b · (cid:20) ∂(cid:126)v sw ∂t + ( (cid:126)v sw · (cid:126) ∇ ) (cid:126)v sw (cid:21)(cid:27) , with ˆ (cid:126)b = (cid:126)B/B a unit vector in the direction of the background magnetic field, (cid:126)v sw the SW velocity, (cid:126)v d = µpqB ˆ (cid:126)b × (cid:34) ∂ ˆ (cid:126)b∂t + ( (cid:126)v sw · (cid:126) ∇ )ˆ (cid:126)b (cid:35) + pvqB (cid:34) µ ( (cid:126) ∇ × ˆ (cid:126)b ) ⊥ + 1 − µ (cid:126)b × (cid:126) ∇ BB (cid:35) + mqB (cid:126)v sw × (cid:34) ∂∂t (cid:32) ˆ (cid:126)bB (cid:33) + ( µv ˆ (cid:126)b + (cid:126)v sw ) · (cid:126) ∇ (cid:32) ˆ (cid:126)bB (cid:33)(cid:35) (18)the gyrophase averaged GC drift velocity perpendicular to the magnetic field(Northrop 1961; Rossi and Olbert 1970; Burger et al. 1985; Wijsen 2020), and (cid:126)a(cid:126)b : (cid:126)c(cid:126)d = a i b j c j d i a tensor contraction. The diffusion coefficients are also gyrophaseaveraged, although not explicitly indicated, and it is assumed that momentum dif-fusion is negligible in the SW. In the derivation it is additionally assumed thatthe SW is non-relativistic, and that v (cid:29) v sw . See Skilling (1971), Riffert (1986),Ruffolo (1995), Zhang (2006), le Roux and Webb (2012), le Roux et al. (2014),Zank (2014), and Wijsen (2020) for additional details and discussions about theFTE and its derivation.Perpendicular diffusion is still hotly debated and an aspect which is not en-tirely understood in SEP transport. This is mainly because the exact pitch-angledependence of the perpendicular diffusion coefficients are not yet known. The exactamount of perpendicular diffusion (i.e. the perpendicular MFP) is also currentlyuncertain. However, it should be emphasized that some level of perpendiculartransport must be present in SEP events: Any turbulent magnetic field with alevel of transversal complexity will lead to the perpendicular transport of chargedparticles. The SW has a significant transverse component (Matthaeus et al. 1990),with Bieber et al. (1996) suggesting a ratio of 80:20 for the ratio of 2D (transver-sal) to slab turbulence, suggesting that, at 1 AU, SW turbulence is predominantlytransversal. Therefore, the question should not be whether perpendicular diffusionoccurs in the SW, but rather how much perpendicular diffusion do SEPs experi-ence? The first model to simulate pitch-angle dependent SEP transport in a full 3Dgeometry, including adiabatic energy losses, was presented by Zhang et al. (2009),showing that the inclusion of perpendicular diffusion allows SEPs to propagate rather efficiently across magnetic field lines and thereby explaining the origin ofwidespread SEP events. In addition to this, they also showed that perpendiculardiffusion tends to smooth out any longitudinally dependent fine-structure in theSEP source. This was also later confirmed by Zhang and Zhao (2017) and Strausset al. (2017). A similar 3D model without energy losses was presented by Dr¨ogeet al. (2010), with selected results shown in Fig. 16. These authors pointed to oneof the major unsolved problems in multi-dimensional SEP transport modelling:The inclusion of perpendicular diffusion can explain the existence of widespreadSEP events, but the results are inconsistent with so-called drop-out events; SEPevents where very low levels of perpendicular diffusion is present and SEPs seemtightly tied to magnetic fieldlines (as inferred by e.g. Mazur et al. 2000). Thisapparent dichotomy is yet to be resolved (see also Wang et al. 2014). In later work,Dresing et al. (2012), Dr¨oge et al. (2014), and Dr¨oge et al. (2016) implemented aphenomenological description of the diffusion parameters, and through a detailedcomparison between simulation results and SEP electron observations from theSTEREO spacecraft, were able to constrain the level of perpendicular diffusionneeded. Similarly to the case of pitch-angle scattering, there is a rather largeinter-event variation in these parameters.Qin et al. (2011), investigating SEP transport in 3D, focusing especially on thesimulated anisotropy of SEP events, and similarly to Zhang et al. (2009), showsthat the level of the observed anisotropy depends strongly on the level of magneticconnectivity: When a virtual observer in a model is well connected to the SEPsource, it will observe a high anisotropy, whereas SEP particles that undergo sig-nificant perpendicular diffusion, and therefore are not magnetically well connectedto the source, show very low/insignificant anisotropies. This was later confirmedin the simulation of Strauss et al. (2017), using a more fundamental description ofthe transport coefficients, and is evident in the observations presented by Dresinget al. (2014). For impulsive SEP events, the observed anisotropy can therefore beused as a proxy for the level of magnetic connectivity to the SEP source, andthis could assist in estimating the size of the SEP source. Results indicate that,for impulsive electron events, an extended source alone cannot explain the ob-served longitudinal spread of SEPs; some level of perpendicular transport must bepresent.Many additional modelling studies are examining, amongst other effects, therole of perpendicular diffusion in influencing the energy spectrum of impulsive(Strauss et al. 2020) and gradual (Wang and Qin 2015b) SEP events, the effect ofperpendicular diffusion on SEPs released from a moving source (i.e. propagatinginterplanetary shock; Qin et al. 2013; Qin and Wang 2015), and whether theremay be simulated effects that can be tested against observations, such as possibleasymmetries and anisotropies in the simulated distribution (He et al. 2011; He2015; He and Wan 2017). Indeed, there are currently numerous simulation studieslooking at the effects of perpendicular diffusion and this has opened up a very richresearch sub-field.
Although the effects of perpendicular diffusion can be effectively studied intransport models, there are outstanding theoretical questions, including whether adiffusive description for perpendicular diffusion is valid (see Section 4.2.4), whetherperpendicular diffusion should rather be described as a field line meandering pro-cess (see Section 4.2.2), and the form of the perpendicular diffusion coefficient,which on the pitch-angle level is currently not well studied. Studies by e.g. Strauss
Primer on Focused Solar Energetic Particle Transport 33
Fig. 17
Top:
A sample path of a 10 MeV proton injected at the Sun (in red) superimposedon a nominal Parker spiral (in black; left ). The dashed blue line ( right ) shows a meanderingfield line and how the particle in this case follows the field line while scattering back and forthwhile decoupling slowly.
Bottom:
Difference in longitudinal spread between a standard Fokker-Planck simulation with pitch-angle independent perpendicular diffusion ( left ) and a combinedfield line random walk plus Fokker-Planck model ( right ). These contours are calculated for 3hafter the injection at φ = 0 at the sun. The figures are from Laitinen et al. (2016). and Fichtner (2014, 2015) has shown that the pitch-angle dependence of this coef-ficient is an important parameter, and implementing different forms lead to verydifferent simulation results. This is, of course, not unexpected for SEP transport,where a highly anisotropic particle distribution is formed. This remains an ongoingtopic of investigation, both theoretically (Strauss et al. 2016; Engelbrecht 2019;Shalchi 2020) and from numerical simulations (Qin and Shalchi 2009, 2014). Perpendicular transport is believed to be significantly controlled by the behaviourof the large-scale background magnetic field. Commonly, in modelling, this is as-sumed to follow a simple Parker spiral configuration. It is clear, however, thateven on larger scales, the SW is turbulent and field lines can deviate from such a simple geometry (e.g. Shalchi 2010). Thus, the field-line random walk (FLRW),in principle, has to be taken explicitly into account to model the perpendicularspread of particles correctly. One approach, discussed in Laitinen et al. (2016);Laitinen et al. (2017), is to calculate a random ensemble of meandering field linesand follow a stochastic trajectory of particles along those lines. Fig. 17 illustrates how particles follow the field line while slowly decoupling from it. In the bottompanel of the figure, the resulting differences for the spread of the distribution func-tion are shown. Together with the width of the injection region, this effect can bea significant contribution to the observed width of the particle distribution at 1AUand beyond. It can also impact on the actual path-length a particle experiences(Laitinen and Dalla 2019).Note, furthermore, that this modelling approach offers an explanation for thesimultaneous occurrence of wide-spread events and the observed drop-outs in parti-cle intensity (Mazur et al. 2000). Since in each actual event, a particular realizationof meandering field lines exist, empty and filled flux tubes can be close to eachother, while still spreading the particles to wide longitudes. Of course, with cur-rent modelling capabilities and limited knowledge of the interplanetary magneticfield configurations, these effects can only be understood statistically, i.e. for anensemble of events.The validity of the coupled FLRW and perpendicular diffusion approach hasbeen established to some degree through MHD and full-orbit simulations in syn-thetic turbulence (see the additional discussion in Section 4.2.4). Chuychai et al.(2007) discuss the field-line topology and trapping that can occur in a two-componentturbulence model. Ruffolo et al. (2012) show an explicit calculation of the perpen-dicular diffusion coefficient with a ‘random ballistic interpretation’ that can leadto a reduction in the coefficient because of the parallel or pitch-angle scattering ofthe particles. These are just a few examples that illustrate the need for a detailedlook at the interplay between parallel and perpendicular transport of particles andthe influence of different scales of turbulent fluctuations on the resulting particledistributions.
Drifts can also lead to transport of particles across magnetic fields and Dalla et al.(2013) recently derived expressions for the drift velocities of SEPs in a Parker(1958) HMF. The electric field drift due to the motional electric field will be inthe plane containing the Parker spiral and describe the co-rotation of particles withthe HMF as the Sun rotates. The gradient and curvature drift will furthermorebe smaller for low energy particles and in opposite directions for positively andnegatively charged particles. Marsh et al. (2013) verified these general predictionsby integrating the Newton-Lorentz equation (Eq. 1) in a uni-polar Parker HMF.These authors found that pitch-angle scattering and the MFP has little effect onthe drifts and that drifts will be the most pronounced for high energy particles( ∼
100 MeV protons) or partially ionised heavy ions. Dalla et al. (2017) and Dallaet al. (2017) used this model to attribute the observed energy dependent chargestate of iron and the temporal evolution of the iron-to-oxygen-ratio, respectively, tothe mass-to-charge-ratio dependence of drifts and not the usual rigidity dependentMFP with turbulence generated by streaming protons of similar rigidity and theacceleration process (see e.g. Reames 1999, and references therein).
Battarbee et al. (2017) extended the investigation of Marsh et al. (2013) toinclude neutral sheet drifts in a flat heliospheric current sheet (HCS). They foundthat SEP drift patterns will be similar to galactic CR drift patters: if A is thepolarity of the HMF in the Northern hemisphere ( A = +1 for outwards and A = − Primer on Focused Solar Energetic Particle Transport 35
Fig. 18
A simulation of proton drift patterns at 1 AU along a wavy heliospheric current sheetin two different polarity cycles (magnetic field pointing outwards, left row , or inwards, rightrow , in the Northern hemisphere) for different positions of the injection (above, top row , in, middle row , or below, bottom row , the current sheet). The drift of particles towards positivelongitudes are due to the co-rotation. These figures are from Battarbee et al. (2018a). the solar rotation in the inner heliosphere or outwards in the outer heliosphere if qA >
0, while particles will drift towards the poles and in the opposite directionof the solar rotation in the inner heliosphere or inwards in the outer heliosphereif qA <
0. Particles can therefore be confined to the HCS in the qA > ◦ wider in longitude thanthe inferred coronal mass ejection was needed for particles to reach the STEREOspacecrafts and the fluxes at Mercury was overestimated.Caution, however, should be taken with these results as strong pitch-anglescattering was implemented at Poisson-distributed scattering intervals (i.e. ran-dom adjustments of the pitch-angle and gyro-phase at random times), and theturbulent reduction of drifts has been neglected. Turbulent fluctuations disruptsthe large scale drifts and therefore decrease the drift velocity in Eq. 18. A review ofthe current knowledge and understanding of this subject is given by Engelbrecht et al. (2017). In summary, drift reduction do not seem to occur in purely magneto-static slab turbulence, the drift coefficient is reduced by the same factor in both ahomogeneous magnetic field and a magnetic field with large scale gradients for thesame turbulence conditions, and drifts decrease with an increase in the turbulencestrength or a decrease in the particle energy (Burger and Visser 2010; Engelbrecht Fig. 19
A simulation of proton intensity profiles ( bottom row ) at different observers of the 17May 2012 ground level enhancement event (observations shown in the top row ). Note that thedata from the Radiation Assessment Detector on the Mars Science Laboratory (MSL/RAD)is from measurements inside the protective flight shielding and does not represent the trueintensities. This figure is taken from Battarbee et al. (2018b). et al. 2017). It is important to note that the exact form of the drift suppressionfactor is not yet known and that all studies on this have only considered isotropicdistributions and do not include pitch-angle dependencies.Wijsen et al. (2020) included drifts in a uni-polar HMF into the FTE with apitch-angle independent perpendicular diffusion coefficient for 3 −
36 MeV protons.They verified that different observers will see different spectra and found thatperpendicular diffusion will diminish the effects of drifts. Richardson et al. (2014)investigated the 14 −
24 MeV proton events observed during the first seven yearsof the STEREO mission and compared this with the 0 . − One potential disadvantage of the diffusive SEP description mostly discussed aboveis that these models cannot capture the initial ballistic phase of SEP transportbefore pitch-angle scattering results in diffusive transport (see e.g. Laitinen andDalla 2017; Laitinen et al. 2017). The time, from SEP acceleration and release,until diffusive behaviour is reached, depends on the SW turbulence characteristicsnear the SEP source, and as such, is not well known. SEP transport models utiliz- ing full-orbit simulations (i.e. solving the Newton-Lorentz equations directly) doesnot have this limitation. However, such models are much more computationallyexpensive and are not always practically feasible.Kelly et al. (2012) present full-orbit simulations in a Parker HMF with superim-posed large scale magnetic fluctuations, based on earlier work by Pei et al. (2006).
Primer on Focused Solar Energetic Particle Transport 37
Fig. 20
A realization of a turbulent magnetic field ( left panel ) and the position of SEPsreleased into this structure ( right panel ). Both figures are taken from Guo and Giacalone(2014).
Similarly to model solutions described in Section 4.2.3 by Marsh et al. (2013) andlater co-workers, these simulations do not include small scale turbulence leadingto pitch-angle scattering, with particle scattering included in an ad-hoc fashion.However, even without scattering included, simulations from Kelly et al. (2012)show the role of field-line meandering in leading to cross-field particle transport.Full-orbit simulations by Guo and Giacalone (2014) implemented a more de-tailed turbulence model, covering both small and larger scales, so that both pitch-angle scattering by small-scale turbulence, and large scale meandering is included.Fig. 20 shows examples of these simulations: The left panel shows simulated mag-netic fieldlines originating from a small source region, while the right panel showsthe distribution of particles, a certain time after release into such as magnetic re-alization. These results shows that when SEP are released from compact sources,smaller than the turbulence correlations scale, particles can be confined to flux-tubes , forming a pattern of alternating empty and filled (with SEPs) regions ofspace. This has been put forward as an explanation for the observations of so-called drop-out events, where, presumably, the spacecraft moved through such a patchy region of filled and empty fluxtubes. These simulations do, however, dependon the structure and strength of the underlying turbulence, and continue to be anavenue of further research (Tooprakai et al. 2016; Ablaßmayer et al. 2016).4.3 Towards Predictive CapabilityWith the recent interest in crewed space travel, predicting, and thereby mitigat-ing the radiation risk posed by SEPs has become an ongoing problem is space science. In some sense the development of a physics-based model with real-timeSEP predictive capabilities has become the holy grail of SEP modelling studies.To understand how close we are to reaching this goal, it is useful to examine theapplication usability levels (AUL) as defined by Halford et al. (2019) which showthe natural progression from a basic research question (AUL 1) to validation and approval for use (AUL 9). Most, if not all, of the SEP modelling studies presentedup to here are concerned with basic research questions and trying to understandand characterize the underlying processes shaping SEP transport. Although thesefundamental studies will, in future, inform the predictive models, it is clear thatthere is much more work to be done before SEP models will have true predictivecapability. At the moment most SEP prediction algorithms are based on observedempirical relationships (e.g. Balch 2008). See also Anastasiadis et al. (2019) for arecent review on this topic.Most SEP models deal only with the transport of SEPs and neglect the ac-celeration thereof by pre-specifying the injection function discussed in previoussections. Such a 1D transport model, including only an energy coordinate, is givenby Kubo et al. (2015), while Marsh et al. (2015) uses a full 3D model also includingdrift effects. A different approach is used by Luhmann et al. (2017) where SEPsare back-tracked in an MHD simulated heliospheric background until the shock(source) region is reached, leading to a re-weighting of the SEP intensities basedon some analytical approximations for shock acceleration efficiency. While the bal-listic propagation assumption is an over-simplification for particle transport, moreinformation regarding the source region is possible. Aran et al. (2006) present alarge number of pre-computed 1D SEP modelling scenarios, including differentMHD generated shock scenarios, that could be applied to observed SEP events.Based on the relative success of the prediction models discussed above, it ap-pears that a physics-based SEP prediction model should have the following proper-ties: (i) In order to capture cross-field particle propagation, and the longitudinallydependent acceleration efficiency of shocks in the inner heliosphere, a spatially 2Dor 3D geometry must be used. (ii) A pitch-angle dependent (i.e. focused trans-port) approach is needed to capture the large particle anisotropies related to SEPevents. (iii) The model needs to treat the SEP source in a self-consistent manner,either through implicitly including shock acceleration for protons, or by specifyingthe appropriate remote-sensing observations for flare accelerated electrons. (iv) Ifshock acceleration is handled numerically, an energy coordinate is needed in themodel. (v) To account for large non-Parkerian magnetic field variations, an MHD(or equivalent) model must be used to simulate the underlying magnetic geome-try. (vi) Appropriate SEP transport parameters must be specified, although thesemay be based on a phenomenological description derived from the results of basicresearch models.SEP models that conform to most of these requirements, although sometimesusing simplifying assumptions, are EMMREM (Energetic particles, radial gra-dients, and coupling to MHD) model, described by Kozarev et al. (2010) andSchwadron et al. (2010), the iPATH (Particle Acceleration and Transport in theHeliosphere) model, described by Hu et al. (2017), and the PARADISE (PArticleRadiation Asset Directed at Interplanetary Space Exploration) model with initialdevelopments described by Wijsen et al. (2019). These models are being constantlyimproved, while some open research avenues remain, such as the details of SEP seedpopulation, whether the acceleration process is handled correctly, and whether the turbulent magnetic structures between the source and the observer are correctlydescribed. These questions remain unanswered as we do not have, and will mostlikely never have, sufficient in-situ measurements of these quantities.
Primer on Focused Solar Energetic Particle Transport 39
Section 2 started by illustrating SEP motion in a fluctuating magnetic field bydirectly solving the Newton-Lorentz equation. It shows that, even with turbulentfluctuations included, the particle trajectory remain relatively smooth, forming aso-called perturbed nearly-circular orbit. The guiding center, however, behaves asone would naively expect for a particle undergoing diffusion: the velocity compo-nents show random changes reminiscent of Brownian motion. It should always beremembered that, when describing SEP transport, one deals with so-called smallangle scattering which is a slow process where particle quantities (in this case, mostimportantly, the pitch-angle) undergoes many small changes, accumulating overseveral gyro-cycles into the particle changing it propagation direction. The particleis then scattered after moving an avrage distance λ (cid:107) through the slowly (slow withrespect to the particle’s gyro-motion) interacting turbulence.Also shown, by considering an anisotropic Alfv´enic turbulence wave field (mod-elled in Appendix C), is that particle scattering conserves energy in the wave frame.In reality, however, it is not possible in most cases to define a single wave frame, asturbulence can, at best, be approximated by a large number of waves propagatingboth along and perpendicular to the mean field at different propagation directions.In such a scenario, the particle can be considered to scattered from one wave frameto another, leading, after many such interactions, to energy (velocity/momentum)diffusion. In the solar wind, momentum diffusion is usually slow enough to beneglected in most applications.To specify the solar wind turbulence from the Sun to the Earth, and solvingthe Newton-Lorentz equation for each SEP particle, remains physically (in termsof knowing the exact turbulence structure) and computationally (in terms of thesimulation) impossible. Therefore, one rather evaluates the evolution of the SEP’sphase-space density, i.e. simulate the evolution of the SEP distribution function.This quantity was introduced in Section 3, along with the so-called focused trans-port equation which describes its evolution. A rigorous derivation of this equationwas not presented and the interested reader should refer to other, more completeworks in the references. In this macroscopic description of SEP transport, theparticle-turbulence interactions are incorporated into a diffusion coefficient. Forthe pitch-angle diffusion coefficient, for example, there is a large number of phe-nomenological descriptions, although based on theoretical arguments, and containsa number of free-parameter that can be adjusted in order to reproduce observationsby using a suitable model. These parameters can then be compared to theoreticalquantities and our fundamental knowledge of SEP transport can be improved.The focused transport equation cannot be solved analytically for the mostgeneral scenario, and only approximate solutions are available. In Section 3.3, twopopular analytical approximations (the Telegraph and Diffusion approximations,summarised in Appendix E for reference) were compared, which showed that, evenfor very simplistic modeling problems, these approximations do not give satisfac- tory solutions. Therefore, the focused transport equation must be solved numer-ically. This approach is still only an approximation of the true solution (someshortcomings of the finite-difference model were discussed, for instance, numericaldiffusion), but allows for the incorporation of all the required processes to give aconsistent description of an SEP event. As suitable numerical models to simulate SEP transport are not widely avail-able and difficult to construct, we have presented a finite difference model forsolving the focused transport equation in the dimension along the magnetic field.The model is briefly discussed in Appendix A and is available for use by the com-munity . We hope that this model might help scientists to have an alternative tothe limited analytical approximations.The last part of this manuscript, Section 4, review the last ∼
10 years of SEPsimulation studies, starting from very simplistic 1D ballistic approximations, andending at the most complex state-of-the-art 3D numerical models currently avail-able. Each subsection discussed the applicability of each of these approximationsand modelling approaches to specific SEP transport problems, and have outlinedpossible pitfalls associated with each. Depending on the aim of a study, it maybe possible to use a simplified approximation, as long as the user is aware of thephysical processes neglected and the assumptions made when using that approxi-mation. All of these models have aided in our current understanding of and insightinto the focused transport of SEPs, but have also raised many questions and high-lighted the aspects which are still not fully understood, i.e. the research questionswhich should be answered by current and upcoming researchers in the fields ofSEPs and turbulence.The attentive reader will notice that we have not reviewed the physics of per-pendicular diffusion in much detail in this manuscript and have focused more onfield-aligned transport and pitch-angle scattering. The reason for this is threefold:(1) A comprehensive review on the perpendicular diffusion coefficient was recentlypublished by Shalchi (2020). (2) The derivation of the correct focused transportequation that includes perpendicular diffusion is available (and presented here inEq. 17), but remains poorly understood: The most complete derivation of thisequation was recently published in the PhD thesis of Wijsen (2020) and has ledto some interesting implications, including that the perpendicular diffusion termsare only retained when the position vector is first transformed to the position ofthe guiding center where after averaging over gyro-phase is performed. In contrast,when specifying the SEP distribution’s position in terms of the particle position,assuming the distribution to be gyrotropic, and averaging over gryo-phase, allperpendicular diffusion terms disappear. The fact that the order of operations hassuch large implications (which has not been discussed in the literature in detail)implies that the due diligence on the transport equation has not been performed inenough detail to review in any sense and remain a (very) active research field. (3)The physics of perpendicular diffusion, on the pitch-angle level, also remains poorlyunderstood. It is now clear that perpendicular diffusion can be described as a com-bination of magnetic field wandering/meandering (where particles simply follow large scale (i.e. larger that the particles’ gyro-radius) turbulent fluctuations and small scale (on the scale of the particles’ gyro-radius) ‘scattering’ which displacesthe particle’s guiding center to different field lines. The second process allows theparticles to decouple from their field lines and to follow different meandering field lines. This ‘scattering’ process can be due to perpendicular propagating Alfv´enicfluctuations, or simply due to drift effects in a turbulent magnetic field. In addi-tion, it is not yet clear if a diffusive (perpendicular) description for SEP transportis applicable. With these processes unknown, we are not even confident, in this https://github.com/RDStrauss/SEP_propagator Primer on Focused Solar Energetic Particle Transport 41 manuscript, to propose a definition for the perpendicular mean free path, beyondthe most generic interpretation:
The perpendicular mean free path is the average dis-tance a particle propagates, perpendicular to the mean field, before it is decoupled fromit’s original field line . The ambiguity of this statement should convey the fact thatperpendicular diffusion is by no means well understood and that more researchneeds to be done.A brief account of physics-based SEP prediction models was also given. Thesemodels are able to deal with the transport of SEPs, from their acceleration site,to e.g. Earth, where the intensity is needed. However, the accuracy and applica-bility of these models are limited by our lack of understanding of where and how
SEPs are accelerated. In future, it is expected for these models to incorporateremote-sensing observations of flares and/or CMEs to better constrain the SEPacceleration site and, ultimately, the energy-dependent time profile of SEPs re-leased into the interplanetary medium. Once this is known, SEP transport modelscan propagate this injection profile to any region in the heliosphere. We express ourhope that the ongoing interest and new discoveries in SEP research will ultimatelylead also to better physics-informed predictive models that will be of value to ourspace-faring society.
Acknowledgements
This work is based on the research supported in part by the NationalResearch Foundation of South Africa (NRF grant numbers 120847, 120345, and 119424). Opin-ions expressed and conclusions arrived at are those of the authors and are not necessarily to beattributed to the NRF. JPvdB acknowledge support from the South African National SpaceAgency. FE acknowledges support from NASA grant NNX17AK25G. Additional support froman Alexander von Humboldt group linkage program is appreciated. We thank the InternationalSpace Science Institute (ISSI) for hosting our team on ‘Solar flare acceleration signatures andtheir connection to solar energetic particles’. We appreciate, as always, constructive researchdiscussions with our colleagues, in particular, we would like to thank Nicolas Wijsen, TimoLaitinen, Nina Dresing, and Kobus le Roux. Figures prepared with Matplotlib (Hunter 2007).2 van den Berg et al.
A A Finite Difference Solver
As shown in this work, analytical approximations of Eq. 9 have very severe limitations, andtherefore, it has to be integrated (solved) numerically to capture the transport processes in-volved. Such a numerical implementation, for this spatially 1D version of the transport equa-tion, is discussed by Strauss et al. (2017), which is based on the numerical techniques discussedin Strauss and Fichtner (2015). Details are also given in the dissertation of Heita (2018). Thismodel has subsequently been developed to be more user-friendly, and the source-code thereofcan be found at https://github.com/RDStrauss/SEP_propagator . The code is published un-der the Creative Commons license, but is not intended to be used for commercial applications.We ask anyone using this model to reference this paper in all research outputs and to contactthe authors when used extensively.The code contains a number of user-defined inputs, such as the particle species underconsideration (i.e. electrons or protons), the effective radial MFP, the SW speed, the kineticenergy of the particles, and different options regarding the injected SEP distribution at theinner boundary condition. Details can be found in the comments section of the source-code. InSection 3.3.3, this finite difference solver was applied to the 7 February 2010 electron event asobserved by STEREO B. Fig. 12 only showed a best fit scenario that can reproduce the observedparticle intensity and anisotropy very well. Here, the sensitivity of the code to parametervariation is illustrated with four cases in Fig. 21. The top row shows the slower rise for asmaller MFP, in the left panel, and a quicker rise and quicker decay for a larger MFP, in theright panel. The bottom row shows a similar variation for a longer acceleration time, in the leftpanel, and a longer escape time, in the right panel, in the injection function. These examplesolutions are also included in the online repository.
B A Stochastic Differential Equation Solver
Stochastic calculus is a study area with several works dealing with its mathematical formal-ism and application to a variety of problems, including Gardiner (1985), van Kampen (1992),Kloeden and Platen (1995), Øksendal (2000), Lemons (2002), and Strauss and Effenberger(2017). Of special interest is Gardiner (1985), Kloeden and Platen (1995), and Strauss and Ef-fenberger (2017), which gives an introduction of stochastic calculus specifically for the fields ofnatural sciences, an introduction focusing on numerical methods to solve stochastic differentialequations (SDEs), and a review of the application of this to CR modelling with toy models tointroduce the basic concepts, respectively. SDEs can be computationally expensive and thesetypes of models did not become feasible until the dawn of parallel-processing. Nonetheless,MacKinnon and Craig (1991) first applied SDEs in solving the FTE for binary collisions ofparticles with ‘cold’ hydrogen atoms in the chromosphere and Kocharov et al. (1998) first usedthem to solve the SEP model of Ruffolo (1995). A three dimensional focused transport modelfor SEPs with and without energy losses are presented by Qin et al. (2006) or Zhang et al.(2009) and Dr¨oge et al. (2010), respectively.If S and M represents the stochastic variables corresponding to s and µ , respectively, thenthe two first order SDEs equivalent to the Roelof equation (Eq. 9) ared S = µv d t d M = (cid:20) (1 − µ ) v L ( s ) + ∂D µµ ∂µ (cid:21) d t + (cid:112) D µµ d W µ ( t ) , where d W µ ( t ) is a Wiener process. These SDEs are solved using the Euler-Maruyama scheme, S ( t + ∆t ) = S ( t ) + M ( t ) v∆tM ( t + ∆t ) = M ( t ) + (cid:34) (1 − M ( t )) v L ( S ( t )) + ∂D µµ ∂µ (cid:12)(cid:12)(cid:12)(cid:12) µ = M ( t ) (cid:35) ∆t + (cid:113) D µµ ( M ( t )) ∆tΛ, from the initial values S ( t ) = s ( t ) and M ( t ) = µ ( t ) at initial time t (= 0 h), where ∆t (= 5 × − h) is the time step, and Λ is a pseudo-random number which is Normally distributedwith zero mean and unit variance (Kloeden and Platen 1995; Strauss and Effenberger 2017). Primer on Focused Solar Energetic Particle Transport 43 I n j e c t i o n P r o f il e I n t e n s i t y ( c m s r s M e V ) A n i s o t r o p y = 0.24 AU; E = 0.08 MeV V sw = 400.0 km/sparticle = 1(1 = electrons; 2 = protons) t a , t e = 0.1,1.0 hrs 01 I n j e c t i o n P r o f il e I n t e n s i t y ( c m s r s M e V ) A n i s o t r o p y = 0.06 AU; E = 0.08 MeV V sw = 400.0 km/sparticle = 1(1 = electrons; 2 = protons) t a , t e = 0.1,1.0 hrs01 I n j e c t i o n P r o f il e I n t e n s i t y ( c m s r s M e V ) A n i s o t r o p y = 0.12 AU; E = 0.08 MeV V sw = 400.0 km/sparticle = 1(1 = electrons; 2 = protons) t a , t e = 1.0,1.0 hrs 0.00.5 I n j e c t i o n P r o f il e I n t e n s i t y ( c m s r s M e V ) A n i s o t r o p y = 0.12 AU; E = 0.08 MeV V sw = 400.0 km/sparticle = 1(1 = electrons; 2 = protons) t a , t e = 0.1,10.0 hrs Fig. 21
Illustration of parameter sensitivity of the finite difference transport model in com-parison to the 7 February 2010 electron event.
Top left:
Smaller λ r of 0 .
06 AU.
Top right:
Larger λ r of 0 .
24 AU.
Bottom left:
Longer acceleration time of 1 h.
Bottom right:
Longerescape time of 10 h.A single solution of the SDEs represent only one possible realization of how a phase-spacedensity element, or pseudo-particle in SDE nomenclature, would evolve. In order to calculatequantities of interest, the SDE is solved 10 times. Temporal, spatial, and pitch-cosine binsare set up and the pseudo-particles are binned into the correct bin at each time step to createa phase-space density (Strauss and Effenberger 2017). To calculate, for example, the ODI atan observation point, only the spatial bin centred on the observation point is considered andfor each temporal bin the pitch-angle bins are added together. The spatial bin surroundingthe observer was chosen to have a volume of ∆s obs = v∆t , since a pseudo-particle withinthis distance from the observer, would probably cross the observer within the next time step.The anisotropy, however, is simply calculated from the average pitch-cosine of each particlefalling in the observer’s spatial bin within a temporal bin. This approach of binning also4 van den Berg et al.allows the calculation of uncertainties through the standard deviation of each bin, althoughthe uncertainties are mostly small due to the large number of pseudo-particles used.The isotropic injection is realised by giving each pseudo-particle a random pitch-cosinewhich is uniformly distributed between -1 and 1. The inner reflecting boundary, in the case ofa real SEP event, is handled similar to hard-sphere scattering of a planar surface, that is, if S < S → | S | and M → | M | . An additional reflective boundary condition is imposedon the pitch-cosine to ensure that it says within its allowed range, that is, if | M | > M → sign( M )2 − M (Strauss and Effenberger 2017). The Reid-Axford injection is realised by aconvolution of the delta injection solution with the Reid-Axford profile (following the approachof Dr¨oge et al. 2014), since the transport coefficients are not time-dependent. Notice that theinfinite derivatives of D µµ in the anisotropic scattering case, is problematic. If the derivativearound µ is too large (small), a dip (spike) will appear in the stationary PAD around µ = 0,because pseudo-particles are ‘advected’ away too efficiently (not ‘advected’ away efficientlyenough) from µ = 0 in µ -space by the derivative ( N. Wijsen , 2018, private communication). Inorder to avoid infinite derivatives, the derivative is limited to a maximum value (see van denBerg 2018, for an evaluation of the validity of this approach), that is,if (cid:12)(cid:12)(cid:12)(cid:12) ∂D µµ ∂µ (cid:12)(cid:12)(cid:12)(cid:12) > (cid:12)(cid:12)(cid:12)(cid:12) ∂D µµ ∂µ (cid:12)(cid:12)(cid:12)(cid:12) µ =1 then ∂D µµ ∂µ = sign( µ )2 (cid:12)(cid:12)(cid:12)(cid:12) ∂D µµ ∂µ (cid:12)(cid:12)(cid:12)(cid:12) µ =1 . C Model Slab Turbulence
Here a toy model for slab turbulence will be derived. It will be assumed that the total magneticfield can be written as the sum of a large-scale average/background magnetic field (cid:126)B anda fluctuating magnetic field δ (cid:126)B ; that the fluctuations are perpendicular to the backgroundmagnetic field, such that (cid:126)B · δ (cid:126)B = 0; that the fluctuations are random, such that (cid:104) δ (cid:126)B (cid:105) = (cid:126) (cid:104) (cid:126)B (cid:105) = (cid:126)B , where (cid:104)· · · (cid:105) indicates a suitable average; that the fluctuations are due toa superposition of different types of small-amplitude waves of different wave numbers andgyrophases with frequencies which are deterministically governed by the dispersion relationsof these waves, and that there are little to no interaction between the waves themselves (i.e. thewave viewpoint of turbulence); that only slab turbulence, which have wave vectors k (cid:107) parallelto the background magnetic field and is only dependent on the position along the backgroundmagnetic field, is the main contributor to pitch-angle scattering; that slab turbulence can bedescribed as circularly polarised (for how resonant wave-particle interactions can be describedusing circularly polarised waves, see e.g. Tsurutani and Lakhina 1997; Dr¨oge 2000a; Straussand le Roux 2019), non-dispersive Alfv´en waves, with angular frequency ω related to the wavenumber k by ω/k = V A , where V A is the Alfv´en speed; and that the background magnetic fieldis in the z -direction of the Cartesian coordinate system, so that (cid:126)B = B ˆ (cid:126)z (Goldstein et al.1995; Choudhuri 1998; Dr¨oge 2000a; Shalchi 2009; Bruno and Carbone 2005).With these assumptions and waves propagating along the z -direction, the fluctuating mag-netic field can have components δB x ( z ; t ) = N +RH (cid:88) i b i cos (cid:2) k (cid:107) i ( z − V A t ) + φ i (cid:3) + N − RH (cid:88) j b j cos (cid:2) k (cid:107) j ( z + V A t ) + φ j (cid:3) + N +LH (cid:88) m b m sin (cid:2) k (cid:107) m ( z − V A t ) + φ m (cid:3) + N − LH (cid:88) n b n sin (cid:2) k (cid:107) n ( z + V A t ) + φ n (cid:3) δB y ( z ; t ) = N +RH (cid:88) i b i sin (cid:2) k (cid:107) i ( z − V A t ) + φ i (cid:3) + N − RH (cid:88) j b j sin (cid:2) k (cid:107) j ( z + V A t ) + φ j (cid:3) + N +LH (cid:88) m b m cos (cid:2) k (cid:107) m ( z − V A t ) + φ m (cid:3) + N − LH (cid:88) n b n cos (cid:2) k (cid:107) n ( z + V A t ) + φ n (cid:3) , where b l are the amplitudes, φ l are random phase differences which are uniformly distributedbetween 0 and 2 π , and N l is the number of waves of a particular type. This model considers four Primer on Focused Solar Energetic Particle Transport 45 Fig. 22
Spectra of the sampled magnetic ( top panel ) and electric ( bottom panel ) fluctuationsof the toy slab turbulence model discussed in the text.types of waves: right (RH) and left (LH) hand polarised waves propagating in the positive (+)and negative (-) z -direction. These fluctuating magnetic fields will induce fluctuating electricfields of the form δE x ( z ; t ) = V A N +RH (cid:88) i b i sin (cid:2) k (cid:107) i ( z − V A t ) + φ i (cid:3) − N − RH (cid:88) j b j sin (cid:2) k (cid:107) j ( z + V A t ) + φ j (cid:3) + N +LH (cid:88) m b m cos (cid:2) k (cid:107) m ( z − V A t ) + φ m (cid:3) − N − LH (cid:88) n b n cos (cid:2) k (cid:107) n ( z + V A t ) + φ n (cid:3) δE y ( z ; t ) = V A − N +RH (cid:88) i b i cos (cid:2) k (cid:107) i ( z − V A t ) + φ i (cid:3) + N − RH (cid:88) j b j cos (cid:2) k (cid:107) j ( z + V A t ) + φ j (cid:3) − N +LH (cid:88) m b m sin (cid:2) k (cid:107) m ( z − V A t ) + φ m (cid:3) + N − LH (cid:88) n b n sin (cid:2) k (cid:107) n ( z + V A t ) + φ n (cid:3) . The induced electric field is also circularly polarised and 90 ◦ out of phase compared to themagnetic waves. With this form it can be verified that all of the Maxwell equations are satisfied.The fluctuations should form a spectrum when sampled. A slab spectrum (used for SEPmodelling by Strauss et al. 2017, among others) will be assumed to have the form g ( k (cid:107) ) = g k − s min if 0 ≤ k (cid:107) < k min k − s (cid:107) if k min ≤ k (cid:107) ≤ k d k p − sd k − p (cid:107) if k d < k (cid:107) , with a flat energy range below k min , an inertial range with spectral index s (assumed to beKolmogorov, s = 5 /
3) between k min and k d , and a dissipation range with spectral index p p = 3) above k d . The total variance of the slab fluctuations is related to thespectra by (Shalchi 2009; Zank 2014) δB = 8 π (cid:90) ∞ g ( k (cid:107) ) d k (cid:107) , (19)from which the proportionality constant can be calculated as g = s − π δB k s − (cid:34) s + s − pp − (cid:18) k min k d (cid:19) s − (cid:35) − . (20)The wave number at which the dissipation begins, follow a linear dependence on the protoncyclotron frequency in the SW (Duan et al. 2018; Woodham et al. 2018). For simplicity, it willtherefore be assumed that k d ≈ | q e | B m p V A and k min ≈ πk d , where q e is the elementary charge of an electron, m p is the mass of a proton, and such that k min (cid:28) k d . Discrete wave numbers are chosen such that log k (cid:107) l is equally spaced betweenlog( k min /
10) and log(10 k d ).By defining the ensemble average as (cid:104)· · · (cid:105) θ = 1(2 π ) (cid:90) π (cid:90) π (cid:90) π (cid:90) π · · · d θ i d θ j d θ m d θ n , where θ l = k (cid:107) l ( z − V A t ) + φ l for l = i, m and θ l = k (cid:107) l ( z + V A t ) + φ l for l = j, n , it canbe verified that (cid:104) (cid:126)B (cid:105) θ = B ˆ (cid:126)z and (cid:104) (cid:126)E (cid:105) θ = (cid:126)
0. The magnetic and electric variances can also becalculated as δB = N +RH (cid:88) i b i + N − RH (cid:88) j b j + N +LH (cid:88) m b m + N − LH (cid:88) n b n (21a) δE = V A δB , (21b)respectively. The summations in Eq. 21a would approximate Eq. 19 if the amplitudes arechosen as b l = (cid:114) N l N πg ( k (cid:107) l ) ∆k (cid:107) l , with δB = 0 . B in g (Eq. 20) to reflect the fact that the variance is some fraction ofthe magnetic field strength, ∆k (cid:107) l the difference between wave numbers centred on k (cid:107) l , N l = N +RH , N − RH , N +LH , N − LH for l = i, j, m, n , respectively, and N = N +RH + N − RH + N +LH + N − LH thetotal number of waves. It is assumed that each type of wave follows the same spectrum, butin reality the different types of waves might also have different spectra because the dissipationis set by different resonance conditions (see e.g. Engelbrecht and Strauss 2018; Strauss andle Roux 2019).This toy model for slab turbulence is verified in Fig. 22 where the spectra of both theelectric and magnetic field are shown. The fluctuations, with N +RH = N − RH = N +LH = N − LH =500 and V A = 10 m · s − , were sampled at the origin at a frequency of 20 f d for a durationof 1 / f min , where f d and f min is the frequency corresponding to k d and k min , respectively.Notice that similar results can be found in the magnetostatic case ( V A = 0), without theelectric field of course, if the fluctuations are sampled along the z -axis. It can be verified thatthe running average over time of the fluctuations go to zero (not shown), while the varianceapproach the correct values (keeping in mind that δB = δB x + δB y ). The spectrum becomesclearer if the power spectral density is binned and it can be seen that it has the same form asthe input spectrum. If the spectra of the two components are integrated and added together, avalue close to the correct variance is found. If the number of waves are increased, the discretewave numbers become less obvious in the spectra. Primer on Focused Solar Energetic Particle Transport 47 D Derivation of the Focusing Term and Steady State Pitch-angleDistribution
The pitch-angle transport term in the FTE, ∂∂µ (cid:20) (1 − µ ) v L ( s ) f (cid:21) , describes the mirroring or focusing of particles (Ruffolo 1995; Zank 2014). Following Ruffolo(1995), the focusing term can be calculated directly from the mirroring condition (Eq. 8). Theparticles’ pitch-angle change due to the movement of the particles into different regions of themagnetic field, d µ d t = d µ d B d B d s d s d t , where d s/ d t = v (cid:107) = µv . From the mirroring condition it follows thatd µ d B = − B m (cid:112) − B/B m = − − µ µB , where B m = B/ (1 − µ ) was used. Hence, the change in pitch-angle becomesd µ d t = − − µ µB d B d s µv = (1 − µ ) v L ( s ) , where 1 L ( s ) = − B ( s ) d B ( s )d s (22)relates the focusing length to the changing magnetic field.An expression can be derived for the steady state PAD, F ( µ ), by neglecting any spatialdependences ( ∂f/∂s = 0 and L constant), so that Eq. 9, in the steady state ( ∂f/∂t = 0),reduces to dd µ (cid:20) (1 − µ ) v L F ( µ ) (cid:21) = dd µ (cid:20) D µµ ( µ ) d F ( µ )d µ (cid:21) . (23)Integrating this twice with respect to µ and applying the normalisation condition ( (cid:82) − F d µ =1), yields (Earl 1981; Beeck and Wibberenz 1986; He and Schlickeiser 2014) F ( µ ) = e G ( µ ) (cid:82) − e G ( µ (cid:48) ) d µ (cid:48) , (24)where G ( µ ) = v L (cid:90) µ − µ (cid:48) D µµ ( µ (cid:48) ) d µ (cid:48) . (25)The fact that the stationary PAD is some exponential function of the pitch-cosine, illustratesthe fact that focusing causes the particles to be field aligned, with fewer particles movingopposite to the magnetic field. In the case of no focusing ( L → ∞ ), Eq. 23 reduces tod[ D µµ (d F/ d µ )] / d µ = 0, which yields F ( µ ) = 1 / F/ d µ = 0) condition and the normalisation condition. This states that theglobal distribution relaxes to isotropy, as expected for pitch-angle scattering in the absence offocusing. E The Diffusion-advection and Telegraph Approximation
If one is only interested in the local properties over which λ (cid:107) /L is approximately constant andif it is assumed that the distribution function can be written as f = F + F with F (cid:28) F (that is, small anisotropies), two analytical approximations are available for Eq. 9 with a deltainjection of isotropic particles, δ ( s − s ) δ ( t ), and a vanishing distribution function at infinity, f ( s → ±∞ ; t ) = 0. Note that in what follows the given expressions differ from those given insome of the references due to the use of unitless variables in the references.8 van den Berg et al. E.1 The Diffusion-advection Approximation
By assuming that the distribution is nearly isotropic, the evolution of the ODI is governed bya diffusion-advection equation (Litvinenko and Schlickeiser 2013; Effenberger and Litvinenko2014) ∂F ∂t = u ∂F ∂s + κ (cid:107) ∂ F ∂s , (26)where u = κ (cid:107) /L is the coherent advection speed caused by focusing and κ (cid:107) = vλ (cid:107) / λ (cid:107) given by Eq. 14. Thesolution of Eq. 26 is (Effenberger and Litvinenko 2014) F ( s ; t ) = 1 (cid:112) πκ (cid:107) t e − ( s − s − ut ) / κ (cid:107) t , (27)and the anisotropy can be calculated from this using A ( s ; t ) = 3 κ (cid:107) v (cid:18) L − F ∂F ∂s (cid:19) = 32 v (cid:18) s − s t + u (cid:19) . (28)A solution which might be more applicable to SEPs, is with a reflecting boundary at s = 0since SEPs are injected at the Sun and it can be assumed that the Sun’s magnetic field wouldmirror particles away from the Sun. In this case the solution of Eq. 26 is (Artmann et al. 2011), F ( s > t ) = 1 (cid:112) πκ (cid:107) t sinh (cid:32) ss κ (cid:107) t (cid:33) e − [( s − s − ut ) +2 ss ] / κ (cid:107) t , with an anisotropy given by A ( s > t ) = 32 v (cid:20) s − s coth( ss / κ (cid:107) t ) t + u (cid:21) . according to Eq. 28. Notice that both expressions for the anisotropy has the unphysical predic-tion of infinite anisotropies as t → t → ∞ due to focusing. A known problem of the diffusion approximation is that it is too diffu-sive and violates causality, predicting that particles will arrive at a point before the particlescould have physically propagated to that point (Litvinenko and Schlickeiser 2013; Effenbergerand Litvinenko 2014). E.2 The Telegraph Approximation
In an attempt to preserve causality, the evolution of the ODI can also be described by the tele-graph equation (Litvinenko and Schlickeiser 2013; Effenberger and Litvinenko 2014; Litvinenkoet al. 2015) ∂F ∂t + τ ∂ F ∂t = u ∂F ∂s + κ (cid:107) ∂ F ∂s , (29)where (Litvinenko and Noble 201) τ = κ (cid:107) − κ (cid:48)(cid:107) u , (30)with κ (cid:48)(cid:107) = v (cid:90) − Q ( µ (cid:48) )d µ (cid:48) , µ (cid:20) LD µµ (1 − µ ) v (cid:18) dd µ (cid:20) D µµ − µ Q (cid:21) + µ (cid:19)(cid:21) − vQ L , and Q ( µ = ±
1) = 0, which reduces to κ (cid:48)(cid:107) ≈ v (cid:40) λ (cid:107) + λ (cid:107) (cid:20) K (1) L (cid:21) + 6 L (cid:90) − µ (cid:48) (cid:20) K ( µ (cid:48) ) − K (1) K ( µ (cid:48) ) (cid:21) d µ (cid:48) (cid:41) (31) Primer on Focused Solar Energetic Particle Transport 49in the weak focusing limit ( λ (cid:107) /L (cid:28) K ( µ ) = ( v/ (cid:82) µ − (1 − µ (cid:48) ) /D µµ ( µ (cid:48) )d µ (cid:48) (Shalchi2011). The solution of Eq. 29 is (Litvinenko and Schlickeiser 2013; Effenberger and Litvinenko2014; Litvinenko et al. 2015) F ( s ; t ) = 12 e [( s − s ) /L − t/τ ] / √ κ (cid:107) τ (cid:104) I ( z ) + (cid:16) − κ (cid:107) τL (cid:17) t τz I ( z ) (cid:105) if | s − s | < t (cid:113) κ (cid:107) τ s = s ± t (cid:113) κ (cid:107) τ , (32)where I and I are modified Bessel functions of the first kind with argument z = 12 (cid:118)(cid:117)(cid:117)(cid:116)(cid:16) − κ (cid:107) τL (cid:17) (cid:34)(cid:18) tτ (cid:19) − ( s − s ) κ (cid:107) τ (cid:35) . For SEPs with an injection and reflecting boundary at s = 0, the solution of Eq. 29 isroughly twice that of Eq. 32 (Litvinenko et al. 2015), F ( s > t ) = e − t/ τ √ κ (cid:107) τ (cid:104) I ( z ) + t τz I ( z ) (cid:105) if s < t (cid:113) κ (cid:107) τ s = t (cid:113) κ (cid:107) τ , where z = 12 (cid:115)(cid:18) tτ (cid:19) − s κ (cid:107) τ . The anisotropy for the telegraph equation can be calculated numerically, for simplicity, from A ( s ; t ) = 3 κ (cid:107) v (cid:20) F (cid:18) τ ∂ F ∂t∂s − ∂F ∂s (cid:19) + 1 L (cid:18) − τF ∂F ∂t (cid:19)(cid:21) . (33)The expressions for κ (cid:107) and τ in the absence of focusing ( L → ∞ ) can be found in Litvinenkoand Schlickeiser (2013). Earl (1976, 1981) and Pauls (1993) (summarised by Pauls and Burger1991) derived and solved a modified telegraph equation. This solution yield the same ODI, butis dependent on coefficients which are more cumbersome to calculate. See Malkov and Sagdeev(2015) for a discussion on the validity of the telegraph equation. E.3 Transport Coefficients
From all the equations introduced in the previous two paragraphs, it follows for isotropicscattering (Eq. 11) that the various quantities are given by λ (cid:107) = v D G ( µ ) = µξF ( µ ) = ξ e µξ cosech( ξ ) κ (cid:107) = Lv (cid:20) coth( ξ ) − ξ (cid:21) = λ (cid:107) vξ (cid:20) coth( ξ ) − ξ (cid:21) κ (cid:48)(cid:107) = Lvξ (cid:20) − tanh( ξ ) ξ (cid:21) τ = Lv tanh( ξ ) = λ (cid:107) vξ tanh( ξ )0 van den Berg et al.where ξ = λ (cid:107) /L is the focusing parameter (Roelof 1969; Beeck and Wibberenz 1986; Shalchi2011; Litvinenko and Noble 201; Lasuik et al. 2017).For anisotropic scattering (Eq. 12) the various quantities are given by λ (cid:107) = 3 v D (2 − q )(4 − q ) G ( µ ) = sign( µ ) 4 − q ξ | µ | − q F q =3 / ( µ ) = e sign( µ )5 ξ √ | µ | / ÷ (cid:26) ξ sinh (cid:18) ξ (cid:19) + 14425 ξ (cid:20) − cosh (cid:18) ξ (cid:19)(cid:21)(cid:27) κ q =3 / (cid:107) = Lv (cid:26)(cid:20) ξ + 365 ξ (cid:21) cosh (cid:18) ξ (cid:19) − (cid:20) ξ (cid:21) sinh (cid:18) ξ (cid:19)(cid:27) ÷ (cid:26) ξ sinh (cid:18) ξ (cid:19) + 1 − cosh (cid:18) ξ (cid:19)(cid:27) κ (cid:48) ( ξ (cid:28) (cid:107) ≈ vλ (cid:107) (cid:20) − − q )(4 − q ) − q ) ξ (cid:21) where K ( µ ) = (4 − q ) λ (cid:107) (cid:2) sign( µ ) | µ | − q + 1 (cid:3) / τ is given by Eq. 30 (Earl 1981;Beeck and Wibberenz 1986; Shalchi 2011; Litvinenko and Noble 201). Notice that analyticalexpressions for F ( µ ) and κ (cid:107) are only easily available for a Kraichnan spectrum with q = 3 / q . F The Heliospheric Magnetic Field and its Focusing Length
The Parker (1958) HMF can be written as (cid:126)B
HMF = AB (cid:16) r r (cid:17) (cid:16) ˆ (cid:126)r − tan ψ ˆ (cid:126)φ (cid:17) , where A is the polarity, ˆ (cid:126)r and ˆ (cid:126)φ are unit vectors in the radial and azimuthal directions, respec-tively, and B is a normalization value, usually related to the HMF magnitude as observed atEarth, B ⊕ = 5 nT (for solar minimum conditions) at r = 1 AU, such that B = B ⊕ (cid:113) ω (cid:12) r /v sw ) , with ω (cid:12) ≈ π/
25 days = 2 . × − rad · s − the solar rotation rate and v sw = 400 km · s − the radially directed SW speed. The HMF spiral angle ψ is defined as the angle between theHMF line and the radial direction and is given bytan ψ = ω (cid:12) ( r − r (cid:12) ) v sw sin θ, (34)if it is assumed that the SW is immediately constant when leaving the solar surface, where r (cid:12) ≈ .
005 AU is the Sun’s radius and sin θ ≈ B HMF = B (cid:16) r r (cid:17) (cid:112) ψ, from which it is evident that B HMF decreases as 1 /r in the equatorial regions (Parker 1958;Owens and Forsyth 2013).The arc length of the Parker (1958) HMF line can be calculated by s = (cid:90) (cid:112) ψ d r. Primer on Focused Solar Energetic Particle Transport 51From the definition of the spiral angle (Eq. 34) it follows that d r = v sw d(tan ψ ) / ( ω (cid:12) sin θ ) andwith this change in variables, the previous equation can be integrated analytically to give s = v sw ω (cid:12) sin θ [tan ψ sec ψ + arcsinh (tan ψ )] , (35)where the integration constant must be zero to satisfy the condition s ( r (cid:12) ) = 0 (Lampa 2011).The focusing length (Eq. 22) can be calculated from Eq. F as1 L ( s ) = (cid:18) r (cid:12) + v sw tan ψ/ω (cid:12) sin θ − ω (cid:12) sin θv sw sin ψ cos ψ (cid:19) cos ψ, (36)where Eq. 34 was used to eliminate r . References
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