A Primer on Focused Solar Energetic Particle Transport

Abstract

The basics of focused transport as applied to solar energetic particles are reviewed, paying special attention to areas of common misconception. The micro-physics of charged particles interacting with slab turbulence are investigated to illustrate the concept of pitch-angle scattering, where after the distribution function and focused transport equation are introduced as theoretical tools to describe the transport processes and it is discussed how observable quantities can be calculated from the distribution function. In particular, two approximations, the diffusion-advection and the telegraph equation, are compared in simplified situations to the full solution of the focused transport equation describing particle motion along a magnetic field line. It is shown that these approximations are insufficient to capture the complexity of the physical processes involved. To overcome such limitations, a finite-difference model, which is open for use by the public, is introduced to solve the focused transport equation. The use of the model is briefly discussed and it is shown how the model can be applied to reproduce an observed solar energetic electron event, providing insights into the acceleration and transport processes involved. Past work and literature on the application of these concepts are also reviewed, starting with the most basic models and building up to more complex models.

Appendices

Appendix A: Finite Difference Solver

As shown in this work, analytical approximations of Eq. (9) have very severe limitations, and therefore, it has to be integrated (solved) numerically to capture the transport processes involved. Such a numerical implementation, for this spatially 1D version of the transport equation, is discussed by Strauss et al. (2017b), which is based on the numerical techniques discussed in Strauss and Fichtner (2015). Details are also given in the dissertation of Heita (2018). This model has subsequently been developed to be more user-friendly, and the source code thereof can be found at https://github.com/RDStrauss/SEP_propagator. The code is published under 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 contact the authors when used extensively.

The code contains a number of user-defined inputs, such as the particle species under consideration (i.e. electrons or protons), the effective radial MFP, the SW speed, the kinetic energy of the particles, and different options regarding the injected SEP distribution at the inner boundary condition. Details can be found in the comments section of the source code. In Sect. 3.3.3, this finite difference solver was applied to the 7 February 2010 electron event as observed by STEREO B. Figure 13 only showed a best fit scenario that can reproduce the observed particle intensity and anisotropy very well. Here, the sensitivity of the code to parameter variation is illustrated with four cases in Fig. 22. The top row shows the slower rise for a smaller MFP, in the left panel, and a quicker rise and quicker decay for a larger MFP, in the right panel. The bottom row shows a similar variation for a longer acceleration time, in the left panel, and a longer escape time, in the right panel, in the injection function. These example solutions are also included in the online repository.

Fig. 22

Illustration of parameter sensitivity of the finite difference transport model in comparison to the 7 February 2010 electron event. Top left: Smaller $\lambda _{r}$ of $0.06~\mbox{AU}$. Top right: Larger $\lambda _{r}$ of $0.24~\mbox{AU}$. Bottom left: Longer acceleration time of $1~\mbox{h}$. Bottom right: Longer escape time of $10~\mbox{h}$

Appendix B: Stochastic Differential Equation Solver

Stochastic calculus is a study area with several works dealing with its mathematical formalism 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 Effenberger (2017), which give an introduction of stochastic calculus specifically for the fields of natural sciences, an introduction focusing on numerical methods to solve stochastic differential equations (SDEs), and a review of the application of this to CR modelling with toy models to introduce the basic concepts, respectively. The interested reader, especially for applications in CR and SEP modelling, is referred to the references in Strauss and Effenberger (2017). SDEs can be computationally expensive and these types 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 of particles with ‘cold’ hydrogen atoms in the chromosphere and Kocharov et al. (1998) first used them to solve the SEP model of Ruffolo (1995). Three dimensional focused transport models for SEPs with and without energy losses are presented by Qin et al. (2006) or Zhang et al. (2009) and Dröge et al. (2010), respectively.

If $S$ and $M$ represents the stochastic variables corresponding to $s$ and $\mu$, respectively, then the two first order SDEs equivalent to the Roelof equation (Eq. (9)) are

$$\begin{aligned} {\mathrm {d}}S & = \mu v \, {\mathrm {d}}t \\ {\mathrm {d}}M & = \left [ \frac{(1 - \mu ^{2})v}{2 L(s)} + \frac{\partial D_{\mu \mu }}{\partial \mu } \right ] {\mathrm {d}}t + \sqrt{2 D_{ \mu \mu }} {\mathrm {d}}W_{\mu }(t) , \end{aligned}$$

where ${\mathrm {d}}W_{\mu }(t)$ is a Wiener process. These SDEs are solved using the Euler-Maruyama scheme,

$$\begin{aligned} S(t + \Delta t) & = S(t) + M(t) v \Delta t \\ M(t + \Delta t) & = M(t) + \left [ \frac{(1 - M^{2}(t))v}{2 L(S(t))} + \left . \frac{\partial D_{\mu \mu }}{\partial \mu } \right |_{\mu = M(t)} \right ] \Delta t + \sqrt{2 D_{\mu \mu }(M(t)) \Delta t} \Lambda , \end{aligned}$$

from the initial values $S(t_{0}) = s(t_{0})$ and $M(t_{0}) = \mu (t_{0})$ at initial time $t_{0}$ ($= 0~\mbox{h}$), where $\Delta t$ (used as $5 \times 10^{-5}~\mbox{h}$ here) is the time step, and $\Lambda$ is a pseudo-random number which is Normally distributed with zero mean and unit variance (Kloeden and Platen 1995; Strauss and Effenberger 2017).

A single solution of the SDEs represent only one possible realization of how a phase-space density element, or pseudo-particle in SDE nomenclature, would evolve. In order to calculate quantities of interest, the SDE is solved $10^{6}$ times. Temporal, spatial, and pitch-cosine bins are set up and the pseudo-particles are binned into the correct bin at each time step to create a phase-space density (Strauss and Effenberger 2017). To calculate, for example, the ODI at an observation point, only the spatial bin centred on the observation point is considered and for each temporal bin the pitch-angle bins are added together. The spatial bin surrounding the observer was chosen to have a volume of $\Delta s_{\mathrm{obs}} = v \Delta t$, since a pseudo-particle within this 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 the particles falling in the observer’s spatial bin within a temporal bin. This approach of binning also allows the calculation of uncertainties through the standard deviation of each bin, although the 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-cosine which is uniformly distributed between -1 and 1. The inner reflecting boundary, in the case of a real SEP event, is handled similar to hard-sphere scattering of a planar surface, that is, if $S < 0~\mbox{AU}$ then $S \rightarrow |S|$ and $M \rightarrow |M|$. An additional reflective boundary condition is imposed on the pitch-cosine to ensure that it says within its allowed range, that is, if $|M| > 1$ then $M \rightarrow \mathrm{sign}(M) 2 - M$ (Strauss and Effenberger 2017). Notice that this might not be the correct implementation of this boundary condition, because the time step should be small enough when a pseudo-particle approaches the boundary such that the reflected solution will still fall in the bin containing the boundary (Wijsen 2020). The Reid-Axford injection is realised by a convolution of the delta injection solution with the Reid-Axford profile (following the approach of Dröge et al. 2014), since the transport coefficients are not time-dependent. Notice that the infinite derivatives of $D_{\mu \mu }$ in the anisotropic scattering case is problematic. If the derivative around $\mu$ is too large (small), a dip (spike) will appear in the stationary PAD around $\mu = 0$, because pseudo-particles are ‘advected’ away too efficiently (not ‘advected’ away efficiently enough) from $\mu = 0$ in $\mu$-space by the derivative (N. Wijsen, 2018, private communication). In order to avoid infinite derivatives, the derivative is limited to a maximum value (see van den Berg 2018, for an evaluation of the validity of this approach), that is,

$$ \mathrm{if} \;\;\;\;\;\;\;\; \left | \frac{\partial D_{\mu \mu }}{\partial \mu } \right | > 2 \left | \frac{\partial D_{\mu \mu }}{\partial \mu } \right |_{\mu = 1} \;\;\;\; \;\;\;\; \mathrm{then} \;\;\;\;\;\;\;\; \frac{\partial D_{\mu \mu }}{\partial \mu } = \mathrm{sign} (\mu ) 2 \left | \frac{\partial D_{\mu \mu }}{\partial \mu } \right |_{\mu = 1} . $$

Appendix C: Model Slab Turbulence

Here, a toy model for slab turbulence will be derived. It will be assumed that the total magnetic field can be written as the sum of a large-scale average/background magnetic field $\mathbf {B}_{0}$ and a fluctuating magnetic field $\delta \mathbf {B}$; that the fluctuations are perpendicular to the background magnetic field, such that $\mathbf {B}_{0} \cdot \delta \mathbf {B} = 0$; that the fluctuations are random, such that $\langle \delta \mathbf {B} \rangle = \mathbf {0}$ and $\langle \mathbf {B} \rangle = \mathbf {B}_{0}$, where $\langle \cdots \rangle$ indicates a suitable average; that the fluctuations are due to a superposition of different types of small-amplitude waves of different wave numbers and gyro-phases with frequencies which are deterministically governed by the dispersion relations of these waves, and that there are little to no interaction between the waves themselves (i.e. the wave viewpoint of turbulence); that only slab turbulence, which have wave vectors $k_{\parallel }$ parallel to the background magnetic field and is only dependent on the position along the background magnetic field, is the main contributor to pitch-angle scattering; that slab turbulence can be described as circularly polarised (for how resonant wave-particle interactions can be described using circularly polarised waves, see e.g. Tsurutani and Lakhina 1997; Dröge 2000a; Strauss and le Roux 2019), non-dispersive Alfvén waves, with angular frequency $\omega$ related to the wave number $k$ by $\omega /k = V_{A}$, where $V_{A}$ is the Alfvén speed; and that the background magnetic field is in the $z$-direction of the Cartesian coordinate system, so that $\mathbf {B}_{0} = B_{0} \, \hat{\mathbf {z}}$ (Goldstein et al. 1995; Choudhuri 1998; Dröge 2000a; Shalchi 2009; Bruno and Carbone 2005).

With these assumptions and waves propagating along the $z$-direction, the fluctuating magnetic field can have components

$$\begin{aligned} \delta B_{x} (z;t) & = \sum _{i}^{N_{\mathrm{RH}}^{+}} b_{0i} \cos \left [ k_{ \parallel i} (z - V_{A} t) + \phi _{i} \right ] + \sum _{j}^{N_{\mathrm{RH}}^{-}} b_{0j} \cos \left [ k_{\parallel j} (z + V_{A} t) + \phi _{j} \right ] + \\ & \;\;\;\; \sum _{m}^{N_{\mathrm{LH}}^{+}} b_{0m} \sin \left [ k_{ \parallel m} (z - V_{A} t) + \phi _{m} \right ] + \sum _{n}^{N_{\mathrm{LH}}^{-}} b_{0n} \sin \left [ k_{\parallel n} (z + V_{A} t) + \phi _{n} \right ] \\ \delta B_{y} (z;t) & = \sum _{i}^{N_{\mathrm{RH}}^{+}} b_{0i} \sin \left [ k_{ \parallel i} (z - V_{A} t) + \phi _{i} \right ] + \sum _{j}^{N_{\mathrm{RH}}^{-}} b_{0j} \sin \left [ k_{\parallel j} (z + V_{A} t) + \phi _{j} \right ] + \\ & \;\;\;\; \sum _{m}^{N_{\mathrm{LH}}^{+}} b_{0m} \cos \left [ k_{ \parallel m} (z - V_{A} t) + \phi _{m} \right ] + \sum _{n}^{N_{\mathrm{LH}}^{-}} b_{0n} \cos \left [ k_{\parallel n} (z + V_{A} t) + \phi _{n} \right ] , \end{aligned}$$

where $b_{0l}$ are the amplitudes, $\phi _{l}$ are random phase differences which are uniformly distributed between 0 and $2 \pi$, and $N_{l}$ is the number of waves of a particular type. This model considers four 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 electric fields of the form

$$\begin{aligned} \delta E_{x} (z;t) & = V_{A} \left \lbrace \sum _{i}^{N_{\mathrm{RH}}^{+}} b_{0i} \sin \left [ k_{\parallel i} (z - V_{A} t) + \phi _{i} \right ] - \sum _{j}^{N_{\mathrm{RH}}^{-}} b_{0j} \sin \left [ k_{\parallel j} (z + V_{A} t) + \phi _{j} \right ] \right . + \\ & \;\;\;\;\; \left . \sum _{m}^{N_{\mathrm{LH}}^{+}} b_{0m} \cos \left [ k_{ \parallel m} (z - V_{A} t) + \phi _{m} \right ] - \sum _{n}^{N_{\mathrm{LH}}^{-}} b_{0n} \cos \left [ k_{\parallel n} (z + V_{A} t) + \phi _{n} \right ] \right \rbrace \\ \delta E_{y} (z;t) & = V_{A} \left \lbrace - \sum _{i}^{N_{\mathrm{RH}}^{+}} b_{0i} \cos \left [ k_{\parallel i} (z - V_{A} t) + \phi _{i} \right ] + \sum _{j}^{N_{\mathrm{RH}}^{-}} b_{0j} \cos \left [ k_{\parallel j} (z + V_{A} t) + \phi _{j} \right ] \right . - \\ & \;\;\;\;\; \left . \sum _{m}^{N_{\mathrm{LH}}^{+}} b_{0m} \sin \left [ k_{ \parallel m} (z - V_{A} t) + \phi _{m} \right ] + \sum _{n}^{N_{\mathrm{LH}}^{-}} b_{0n} \sin \left [ k_{\parallel n} (z + V_{A} t) + \phi _{n} \right ] \right \rbrace . \end{aligned}$$

The induced electric field is also circularly polarised and $90^{\circ }$ out of phase compared to the magnetic 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 SEP modelling by Strauss et al. 2017a, among others) will be assumed to have the form

$$ g (k_{\parallel }) = g_{0} \left \lbrace \textstyle\begin{array}{l@{\quad }c@{\quad }l} k_{\mathrm{min}}^{-s} & \mathrm{if} & \;\;\;\;\, 0 \le k_{\parallel } < k_{ \mathrm{min}} \\ k_{\parallel }^{-s} & \mathrm{if} & k_{\mathrm{min}} \le k_{\parallel } \le k_{d} \\ k_{d}^{p-s} k_{\parallel }^{-p} & \mathrm{if} & \;\;\;\, k_{d} < k_{ \parallel } \end{array}\displaystyle \right . , $$

with a flat energy range below $k_{\mathrm{min}}$, an inertial range with spectral index $s$ (assumed to be Kolmogorov, $s = 5/3$) between $k_{\mathrm{min}}$ and $k_{d}$, and a dissipation range with spectral index $p$ (assumed to be $p = 3$) above $k_{d}$. The total variance of the slab fluctuations is related to the spectra by (Shalchi 2009; Zank 2014)

$$ \delta B^{2} = 8 \pi \int _{0}^{\infty } g (k_{\parallel }) \, {\mathrm {d}} k_{ \parallel } , $$

(19)

from which the proportionality constant can be calculated as

$$ g_{0} = \frac{s - 1}{8 \pi } \delta B^{2} k_{\mathrm{min}}^{s-1} \left [ s + \frac{s - p}{p - 1} \left ( \frac{k_{\mathrm{min}}}{k_{d}} \right )^{s-1} \right ]^{-1} . $$

(20)

The wave number at which the dissipation begins, follow a linear dependence on the proton cyclotron frequency in the SW (Duan et al. 2018; Woodham et al. 2018). For simplicity, it will therefore be assumed that

$$ k_{d} \approx \frac{|q_{e}| B_{0}}{m_{p} V_{A}} \;\;\;\;\;\;\;\; \mbox{and} \;\;\;\;\;\;\;\; k_{\mathrm{min}} \approx \frac{\pi k_{d}}{500} , $$

where $q_{e}$ is the elementary charge of an electron, $m_{p}$ is the mass of a proton, and such that $k_{\mathrm{min}} \ll k_{d}$. Discrete wave numbers are chosen such that $\log k_{\parallel l}$ is equally spaced between $\log (k_{\mathrm{min}} / 10)$ and $\log (10 k_{d})$.

By defining the ensemble average as

$$ \langle \cdots \rangle _{\theta } = \frac{1}{(2 \pi )^{4}} \int _{0}^{2 \pi } \!\!\! \int _{0}^{2 \pi } \!\!\! \int _{0}^{2 \pi } \!\!\! \int _{0}^{2 \pi } \!\!\! \cdots {\mathrm {d}}\theta _{i} \, {\mathrm {d}}\theta _{j} \, {\mathrm {d}} \theta _{m} \, {\mathrm {d}}\theta _{n} , $$

where $\theta _{l} = k_{\parallel l} (z - V_{A} t) + \phi _{l}$ for $l = i,m$ and $\theta _{l} = k_{\parallel l} (z + V_{A} t) + \phi _{l}$ for $l = j,n$, it can be verified that $\langle \mathbf {B} \rangle _{\theta } = B_{0} \, \hat{\mathbf {z}}$ and $\langle \mathbf {E} \rangle _{\theta } = \mathbf {0}$. The magnetic and electric variances can also be calculated as

$$\begin{aligned} \delta B^{2} & = \sum _{i}^{N_{\mathrm{RH}}^{+}} b_{0i}^{2} + \sum _{j}^{N_{ \mathrm{RH}}^{-}} b_{0j}^{2} + \sum _{m}^{N_{\mathrm{LH}}^{+}} b_{0m}^{2} + \sum _{n}^{N_{\mathrm{LH}}^{-}} b_{0n}^{2} \end{aligned}$$

(21a)

$$\begin{aligned} \delta E^{2} & = V_{A}^{2} \, \delta B^{2} , \end{aligned}$$

(21b)

respectively. The summations in Eq. (21a) would approximate Eq. (19) if the amplitudes are chosen as

$$ b_{0l} = \sqrt{\frac{N_{l}}{N} 8 \pi g (k_{\parallel l}) \Delta k_{ \parallel l}} , $$

with $\delta B^{2} = 0.1 \, B_{0}^{2}$ in $g_{0}$ (Eq. (20)) to reflect the fact that the variance is some fraction of the magnetic field strength, $\Delta k_{\parallel l}$ the difference between wave numbers centred on $k_{\parallel l}$, $N_{l} = N_{\mathrm{RH}}^{+}, N_{\mathrm{RH}}^{-}, N_{\mathrm{LH}}^{+}, N_{\mathrm{LH}}^{-}$ for $l=i,j,m,n$, respectively, and $N = N_{\mathrm{RH}}^{+} + N_{\mathrm{RH}}^{-} + N_{\mathrm{LH}}^{+} + N_{\mathrm{LH}}^{-}$ the total number of waves. It is assumed that each type of wave follows the same spectrum, but in reality the different types of waves might also have different spectra because the dissipation is set by different resonance conditions (see e.g. Engelbrecht and Strauss 2018; Strauss and le Roux 2019).

This toy model for slab turbulence is verified in Fig. 23 where the spectra of both the electric and magnetic fields are shown. The fluctuations, with $N_{\mathrm{RH}}^{+} = N_{\mathrm{RH}}^{-} = N_{\mathrm{LH}}^{+} = N_{\mathrm{LH}}^{-} = 500$ and $V_{A} = 10~\mbox{m} \cdot \mathrm{s}^{-1}$, were sampled at the origin at a frequency of $20 f_{d}$ for a duration of $1/20f_{\mathrm{min}}$, where $f_{d}$ and $f_{\mathrm{min}}$ is the frequency corresponding to $k_{d}$ and $k_{\mathrm{min}}$, respectively. Notice that similar results can be found in the magnetostatic case ($V_{A} = 0$), without the electric field of course, if the fluctuations are sampled along the $z$-axis. It can be verified that the running average over time of the fluctuations go to zero (not shown), while the variance approach the correct values (kee** in mind that $\delta B^{2} = \delta B_{x}^{2} + \delta B_{y}^{2}$). The spectrum becomes clearer if the power spectral density is binned and it can be seen that it has the same form as the input spectrum. If the spectra of the two components are integrated and added together, a value close to the correct variance is found. If the number of waves are increased, the discrete wave numbers become less obvious in the spectra.

Fig. 23

Spectra of the sampled magnetic (top panel) and electric (bottom panel) fluctuations of the toy slab turbulence model discussed in the text

Appendix D: Derivation of the Focusing Term and Steady State Pitch-Angle Distribution

The pitch-angle transport term in the FTE,

$$ \frac{\partial }{\partial \mu } \left [ \frac{(1 - \mu ^{2}) v}{2L(s)} f \right ] , $$

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)). The particles’ pitch-angle change due to the movement of the particles into different regions of the magnetic field,

$$ \frac{{\mathrm {d}} \mu }{{\mathrm {d}} t} = \frac{{\mathrm {d}} \mu }{{\mathrm {d}} B} \frac{{\mathrm {d}} B}{{\mathrm {d}} s} \frac{{\mathrm {d}} s}{{\mathrm {d}} t} , $$

where ${\mathrm {d}}s/{\mathrm {d}}t = v_{\parallel } = \mu v$. From the mirroring condition it follows that

$$ \frac{{\mathrm {d}} \mu }{{\mathrm {d}} B} = - \frac{1}{2 B_{m} \sqrt{1 - B / B_{m}}} = - \frac{1 - \mu ^{2}}{2 \mu B} , $$

where $B_{m} = B/(1 - \mu ^{2})$ was used. Hence, the change in pitch-angle becomes

$$ \frac{{\mathrm {d}} \mu }{{\mathrm {d}} t} = - \frac{1 - \mu ^{2}}{2 \mu B} \frac{{\mathrm {d}} B}{{\mathrm {d}} s} \mu v = \frac{(1 - \mu ^{2}) v}{2 L(s)} , $$

where

$$ \frac{1}{L(s)} = - \frac{1}{B(s)} \frac{{\mathrm {d}} B(s)}{{\mathrm {d}} s} $$

(22)

relates the focusing length to the changing magnetic field.

An expression can be derived for the steady state PAD, $F(\mu )$, by neglecting any spatial dependencies ($\partial f / \partial s = 0$ and $L$ constant), so that Eq. (9), in the steady state ($\partial f / \partial t = 0$), reduces to

$$ \frac{{\mathrm {d}}}{{\mathrm {d}} \mu } \left [ \frac{(1 - \mu ^{2}) v}{2L} F( \mu ) \right ] = \frac{{\mathrm {d}}}{{\mathrm {d}} \mu } \left [ D_{\mu \mu } ( \mu ) \frac{{\mathrm {d}} F(\mu )}{{\mathrm {d}} \mu } \right ] . $$

(23)

Integrating this twice with respect to $\mu$ and applying the normalisation condition ($\int _{-1}^{1} F {\mathrm {d}}\mu = 1$), yields (Earl 1981; Beeck and Wibberenz 1986; He and Schlickeiser 2014)

$$ F(\mu ) = \frac{e^{G(\mu )}}{\int _{-1}^{1} e^{G(\mu ')} {\mathrm {d}}\mu '} , $$

(24)

where

$$ G(\mu ) = \frac{v}{2L} \int _{0}^{\mu } \frac{1 - \mu ^{\prime \,2}}{D_{\mu \mu } (\mu ')} {\mathrm {d}}\mu ' . $$

(25)

The fact that the stationary PAD is some exponential function of the pitch-cosine, illustrates the fact that focusing causes the particles to be field aligned, with fewer particles moving opposite to the magnetic field. In the case of no focusing ($L \rightarrow \infty$), Eq. (23) reduces to ${\mathrm {d}}[ D_{\mu \mu } ({\mathrm {d}} F / {\mathrm {d}} \mu ) ]/{\mathrm {d}}\mu = 0$, which yields $F(\mu ) = 1/2$ upon integration and applying a reflective boundary (${\mathrm {d}}F/{\mathrm {d}}\mu = 0$) condition and the normalisation condition. This states that the global distribution relaxes to isotropy, as expected for pitch-angle scattering in the absence of focusing.

Appendix E: The Diffusion-Advection and Telegraph Approximations

If one is only interested in the local properties over which $\lambda _{\parallel }^{0} / L$ is approximately constant and if it is assumed that the distribution function can be written as $f(\mathbf {x}; p; t) = F_{0}(\mathbf {x}; p; t) + g(\mathbf {x}; p; \mu ; t)$ with $|g| \ll F_{0}$ (that is, small anisotropies), two analytical approximations are available for Eq. (9) with a delta injection of isotropic particles, $\delta (s - s_{0}) \delta (t)$, and a vanishing distribution function at infinity, $f(s \rightarrow \pm \infty ; t) = 0$. Note that, in what follows, the given expressions differ from those given in some of the references due to the use of unitless variables in the references.

5.1 E.1 The Diffusion-Advection Approximation

By assuming that the distribution is nearly isotropic, the evolution of the ODI is governed by a diffusion-advection equation (Litvinenko and Schlickeiser 2013; Effenberger and Litvinenko 2014)

$$ \frac{\partial F_{0}}{\partial t} = u \frac{\partial F_{0}}{\partial s} + \kappa _{\parallel } \frac{\partial ^{2} F_{0}}{\partial s^{2}} , $$

(26)

where $u = \kappa _{\parallel } / L$ is the coherent advection speed caused by focusing and $\kappa _{\parallel } = v \lambda _{\parallel } / 3$ is the spatial diffusion coefficient in the presence of focusing, with $\lambda _{\parallel }$ given by Eq. (14). The solution of Eq. (26) is (Effenberger and Litvinenko 2014)

$$ F_{0}(s;t) = \frac{1}{\sqrt{4 \pi \kappa _{\parallel } t}} e^{- (s-s_{0} - ut)^{2} / 4 \kappa _{\parallel } t} , $$

(27)

and the anisotropy can be calculated from this using

$$ A(s;t) = \frac{3 \kappa _{\parallel }}{v} \left ( \frac{1}{L} - \frac{1}{F_{0}} \frac{\partial F_{0}}{\partial s} \right ) = \frac{3}{2v} \left ( \frac{s-s_{0}}{t} + u \right ) . $$

(28)

A solution which might be more applicable to SEPs, is with a reflecting boundary at $s=0$ since it can be assumed that the Sun’s magnetic field would mirror particles away from the Sun. In this case the solution of Eq. (26) is (Artmann et al. 2011),

$$ F_{0}(s>0;t) = \frac{1}{\sqrt{\pi \kappa _{\parallel } t}} \sinh \left ( \frac{s s_{0}}{2 \kappa _{\parallel } t} \right ) e^{- [(s-s_{0} - ut)^{2} + 2 s s_{0}] / 4 \kappa _{\parallel } t} , $$

with an anisotropy given by

$$ A(s>0;t) = \frac{3}{2v} \left [ \frac{s-s_{0} \coth (s s_{0} / 2 \kappa _{\parallel } t)}{t} + u \right ] , $$

according to Eq. (28). Notice that both expressions for the anisotropy has the unphysical prediction of infinite anisotropies as $t \rightarrow 0$ and that there is a persistent anisotropy at late times as $t \rightarrow \infty$ due to focusing. A known problem of the diffusion approximation is that it is too diffusive and violates causality, i.e. predicting that particles will arrive at a point before the particles could have physically propagated to that point (Litvinenko and Schlickeiser 2013; Effenberger and Litvinenko 2014).

5.2 E.2 The Telegraph Approximation

In an attempt to preserve causality, the evolution of the ODI can also be described by the telegraph equation (Litvinenko and Schlickeiser 2013; Effenberger and Litvinenko 2014; Litvinenko et al. 2015)

$$ \frac{\partial F_{0}}{\partial t} + \tau \frac{\partial ^{2} F_{0}}{\partial t^{2}} = u \frac{\partial F_{0}}{\partial s} + \kappa _{\parallel } \frac{\partial ^{2} F_{0}}{\partial s^{2}} , $$

(29)

where (Litvinenko and Noble 2013)

$$ \tau = \frac{\kappa _{\parallel } - \kappa _{\parallel }'}{u^{2}} , $$

(30)

with

$$ \kappa _{\parallel }' = \frac{v^{2}}{4} \int _{-1}^{1} Q(\mu ') {\mathrm {d}} \mu ' \; , \;\;\;\;\;\; 0 = \frac{{\mathrm {d}}}{{\mathrm {d}}\mu } \left [ \frac{2 L D_{\mu \mu }}{(1 - \mu ^{2}) v} \left ( \frac{{\mathrm {d}}}{{\mathrm {d}}\mu } \left [ \frac{D_{\mu \mu }}{1 - \mu ^{2}} Q \right ] + \mu \right ) \right ] - \frac{vQ}{2L} , $$

and $Q(\mu = \pm 1) = 0$, which reduces to

$$ \kappa _{\parallel }' \approx \frac{v}{3} \left \lbrace \lambda _{ \parallel }^{0} + \lambda _{\parallel }^{0} \left [ \frac{K(1)}{L} \right ]^{2} + \frac{6}{L^{2}} \int _{-1}^{1} \mu ' \left [ \frac{1}{3} K^{3}(\mu ') - K(1) K^{2}(\mu ') \right ] {\mathrm {d}}\mu ' \right \rbrace $$

(31)

in the weak focusing limit ($\lambda _{\parallel }^{0} / L \ll 1$), with $K(\mu ) = (v/4) \int _{-1}^{\mu } (1 - \mu ^{\prime \,2}) / D_{\mu \mu } (\mu ') { \mathrm{d}}\mu '$ (Shalchi 2011). The solution of Eq. (29) is (Litvinenko and Schlickeiser 2013; Effenberger and Litvinenko 2014; Litvinenko et al. 2015)

$$ F_{0}(s;t) = \frac{1}{2} e^{[(s-s_{0})/L - t/\tau ]/2} \left \lbrace \textstyle\begin{array}{l@{\quad }l} \frac{1}{2 \sqrt{\kappa _{\parallel } \tau }} \left [ I_{0}(z) + \left ( 1 - \frac{\kappa _{\parallel } \tau }{L^{2}} \right ) \frac{t}{2 \tau z} I_{1}(z) \right ] & \mbox{if} \; |s-s_{0}| < t \sqrt{\frac{\kappa _{\parallel }}{\tau }} \\ 1 & \mbox{if} \; s = s_{0} \pm t \sqrt{ \frac{\kappa _{\parallel }}{\tau }} \\ 0 & \mbox{otherwise} \end{array}\displaystyle \right . , $$

(32)

where $I_{0}$ and $I_{1}$ are modified Bessel functions of the first kind with argument

$$ z = \frac{1}{2} \sqrt{\left ( 1 - \frac{\kappa _{\parallel } \tau }{L^{2}} \right ) \left [ \left ( \frac{t}{\tau } \right )^{2} - \frac{(s-s_{0})^{2}}{\kappa _{\parallel } \tau } \right ]} . $$

For SEPs with an injection and reflecting boundary at $s=0$, the solution of Eq. (29) is roughly twice that of Eq. (32) (Litvinenko et al. 2015),

$$ F_{0}(s>0;t) = e^{- t/2 \tau } \left \lbrace \textstyle\begin{array}{l@{\quad }l} \frac{1}{2 \sqrt{\kappa _{\parallel } \tau }} \left [ I_{0}(z_{0}) + \frac{t}{2 \tau z_{0}} I_{1}(z_{0}) \right ] & \mbox{if} \; s < t \sqrt{\frac{\kappa _{\parallel }}{\tau }} \\ 1 & \mbox{if} \; s = t \sqrt{\frac{\kappa _{\parallel }}{\tau }} \\ 0 & \mbox{otherwise} \end{array}\displaystyle \right . , $$

where

$$ z_{0} = \frac{1}{2} \sqrt{\left ( \frac{t}{\tau } \right )^{2} - \frac{s^{2}}{\kappa _{\parallel } \tau }} . $$

The anisotropy for the telegraph equation can be calculated numerically, for simplicity, from

$$ A(s;t) = \frac{3 \kappa _{\parallel }}{v} \left [ \frac{1}{F_{0}} \left ( \tau \frac{\partial ^{2} F_{0}}{\partial t \partial s} - \frac{\partial F_{0}}{\partial s} \right ) + \frac{1}{L} \left ( 1 - \frac{\tau }{F_{0}} \frac{\partial F_{0}}{\partial t} \right ) \right ] . $$

(33)

The expressions for $\kappa _{\parallel }$ and $\tau$ in the absence of focusing ($L \rightarrow \infty$) can be found in Litvinenko and Schlickeiser (2013). Earl (1976, 1981) and Pauls (1993) (summarised by Pauls and Burger 1991) derived and solved a modified telegraph equation. This solution yield the same ODI, but is dependent on coefficients which are more cumbersome to calculate. See Malkov and Sagdeev (2015) for a discussion on the validity of the telegraph equation.

5.3 E.3 Transport Coefficients

From all the equations introduced in the previous two paragraphs, it follows for isotropic scattering (Eq. (11)) that the various quantities are given by

$$\begin{aligned} \lambda _{\parallel }^{0} & = \frac{v}{2D_{0}} \\ G(\mu ) & = \mu \xi \\ F(\mu ) & = \frac{\xi }{2} e^{\mu \xi } \mathrm{cosech} (\xi ) \\ \kappa _{\parallel } & = L v \left [ \coth (\xi ) - \frac{1}{\xi } \right ] = \frac{\lambda _{\parallel }^{0} v}{\xi } \left [\coth (\xi ) - \frac{1}{\xi } \right ] \\ \kappa _{\parallel }' & = \frac{L v}{\xi } \left [ 1 - \frac{\tanh (\xi )}{\xi } \right ] \\ \tau & = \frac{L}{v} \tanh (\xi ) = \frac{\lambda _{\parallel }^{0}}{v \xi } \tanh (\xi ) \end{aligned}$$

where $\xi = \lambda _{\parallel }^{0} / L$ is the focusing parameter (Roelof 1969; Beeck and Wibberenz 1986; Shalchi 2011; Litvinenko and Noble 2013; Lasuik et al. 2017).

For anisotropic scattering (Eq. (12)) the various quantities are given by

$$\begin{aligned} \lambda _{\parallel }^{0} & = \frac{3v}{2 D_{0} (2-q) (4-q)} \\ G(\mu ) & = \mathrm{sign}(\mu ) \frac{4-q}{3} \xi |\mu |^{2-q} \\ F^{q=3/2}(\mu ) & = e^{\mathrm{sign}(\mu ) 5 \xi \sqrt{|\mu |} / 6} \div \left \lbrace \frac{24}{5 \xi } \sinh \left ( \frac{5}{6} \xi \right ) + \frac{144}{25 \xi ^{2}} \left [ 1 - \cosh \left ( \frac{5}{6} \xi \right ) \right ] \right \rbrace \\ \kappa _{\parallel }^{q=3/2} & = L v \left \lbrace \left [ \frac{5}{6} \xi + \frac{36}{5 \xi } \right ] \cosh \left ( \frac{5}{6} \xi \right ) - \left [ 3 + \frac{216}{25 \xi ^{2}} \right ] \sinh \left ( \frac{5}{6} \xi \right ) \right \rbrace \\ & \;\;\;\;\; \div \left \lbrace \frac{5}{6} \xi \sinh \left ( \frac{5}{6} \xi \right ) + 1 - \cosh \left ( \frac{5}{6} \xi \right ) \right \rbrace \\ \kappa _{\parallel }^{\prime \,(\xi \ll 1)} & \approx \frac{v \lambda _{\parallel }^{0}}{3} \left [ 1 - \frac{2(5-q) (4-q)^{2}}{27 (8-3q)} \xi ^{2} \right ] \end{aligned}$$

where $K(\mu ) = (4-q) \lambda _{\parallel }^{0} \left [ \mathrm{sign} (\mu ) |\mu |^{2-q} + 1 \right ] / 6$ in Eq. (31) and $\tau$ is given by Eq. (30) (Earl 1981; Beeck and Wibberenz 1986; Shalchi 2011; Litvinenko and Noble 2013). Notice that analytical expressions for $F(\mu )$ and $\kappa _{\parallel }$ are only easily available for a Kraichnan spectrum with $q=3/2$. Earl (1981) gives some expressions in the limit of weak and strong focusing which can be used to find approximate expressions for these two quantities with an arbitrary $q$.

Appendix F: The Heliospheric Magnetic Field and Its Focusing Length

The Parker (1958) HMF can be written as

$$ \mathbf {B}_{\mathrm{HMF}} = A B_{0} \left ( \frac{r_{0}}{r} \right )^{2} \left ( \hat{\mathbf {r}} - \tan \psi \, \hat{\boldsymbol{\phi }} \right ) , $$

where $A$ is the polarity, $\hat{\mathbf {r}}$ and $\hat{\boldsymbol{\phi }}$ are unit vectors in the radial and azimuthal directions, respectively, and $B_{0}$ is a normalization value, usually related to the HMF magnitude as observed at Earth, $B_{\oplus } = 5~\mbox{nT}$ (for solar minimum conditions) at $r_{0} = 1~\mbox{AU}$, such that

$$ B_{0} = \frac{B_{\oplus }}{\sqrt{1 + \left ( \omega _{\odot } r_{0}/v_{\mathrm{sw}} \right )^{2}}} , $$

with $\omega _{\odot } \approx 2 \pi / 25~\mbox{days} = 2.66 \times 10^{-6}~\mbox{rad} \cdot \mbox{s}^{-1}$ the solar rotation rate and $v_{\mathrm{sw}} = 400~\mbox{km} \cdot \mathrm{s}^{-1}$ the radially directed SW speed. The HMF spiral angle $\psi$ is defined as the angle between the HMF line and the radial direction and is given by

$$ \tan \psi = \frac{\omega _{\odot } (r - r_{\odot })}{v_{\mathrm{sw}}} \sin \theta , $$

(34)

if it is assumed that the SW is immediately constant when leaving the solar surface, where $r_{\odot } \approx 0.005~\mbox{AU}$ is the Sun’s radius and $\sin \theta \approx 1$ for an observer in the ecliptic. The magnitude of the HMF is given by

$$ B_{\mathrm{HMF}} = B_{0} \left ( \frac{r_{0}}{r} \right )^{2} \sqrt{1 + \tan ^{2} \psi } , $$

(35)

from which it is evident that $B_{\mathrm{HMF}}$ 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

$$ s = \int \!\! \sqrt{1 + \tan ^{2} \psi } \, {\mathrm {d}}r . $$

From the definition of the spiral angle (Eq. (34)) it follows that ${\mathrm {d}}r = v_{\mathrm{sw}} {\mathrm {d}} (\tan \psi ) / (\omega _{\odot } \sin \theta )$ and with this change in variables, the previous equation can be integrated analytically to give

$$ s = \frac{v_{\mathrm{sw}}}{2 \omega _{\odot } \sin \theta } \left [ \tan \psi \sec \psi + \mathrm{arcsinh} \left ( \tan \psi \right ) \right ] , $$

(36)

where the integration constant must be zero to satisfy the condition $s(r_{\odot }) = 0$ (Lampa 2011). The focusing length (Eq. (22)) can be calculated from Eq. (35) as

$$ \frac{1}{L(s)} = \left ( \frac{2}{r_{\odot } + v_{\mathrm{sw}} \tan \psi / \omega _{\odot } \sin \theta } - \frac{\omega _{\odot } \sin \theta }{v_{\mathrm{sw}}} \sin \psi \cos \psi \right ) \cos \psi , $$

(37)

where Eq. (34) was used to eliminate $r$.