# On the Turbulent Reduction of Drifts for Solar Energetic Particles

J.P. van den Berg11affiliation: Center for Space Research, North-West
University, Potchefstroom, 2522, South Africa<EMAIL_ADDRESS>22affiliation: South African National Space Agency, Hermanus, 7200, South
Africa , N.E. Engelbrecht11affiliation: Center for Space Research, North-West
University, Potchefstroom, 2522, South Africa<EMAIL_ADDRESS>, N.
Wijsen33affiliation: Department of Mathematics/Centre for mathematical Plasma
Astrophysics, KU Leuven, Belgium , R.D. Strauss11affiliation: Center for Space
Research, North-West University, Potchefstroom, 2522, South Africa;
<EMAIL_ADDRESS>
###### Abstract

Particle drifts perpendicular to the background magnetic field have been
proposed by some authors as an explanation for the very efficient
perpendicular transport of solar energetic particles (SEPs). This process,
however, competes with perpendicular diffusion caused by magnetic turbulence,
which can also disrupt the drift patterns and reduce the efficiency of drift
effects. The latter phenomenon is well known in cosmic ray studies, but not
yet considered in SEP models. Additionally, SEP models which do not include
drifts, especially for electrons, use turbulent drift reduction as a
justification of this omission, without critically evaluating or testing this
assumption. This article presents the first theoretical step for a theory of
drift suppression in SEP transport. This is done by deriving the turbulence-
dependent drift reduction function with a pitch-angle dependence, as
applicable for anisotropic particle distributions, and by investigating to
what extent drifts will be reduced in the inner heliosphere for realistic
turbulence conditions and different pitch-angle dependencies of the
perpendicular diffusion coefficient. The influence of the derived turbulent
drift reduction factors on the transport of SEPs are tested, using a state-of-
the-art SEP transport code, for several expressions of theoretically-derived
perpendicular diffusion coefficients. It is found, for realistic turbulence
conditions in the inner heliosphere, that cross-field diffusion will have the
largest influence on the perpendicular transport of SEPs, as opposed to
particle drifts.

cosmic rays – diffusion –- magnetic fields –- solar wind – Sun: heliosphere –
Sun: particle emmission – turbulence

=3

##  1 Introduction

The perpendicular transport of solar energetic particles (SEPs) has received
more attention in recent years in an attempt to explain observed widespread
events (e.g. Cliver et al., 1995; Dresing et al., 2014; Gómez-Herrero et al.,
2015). The two mechanisms responsible for transport perpendicular to the
background magnetic field are perpendicular diffusion and drifts. The latter
mechanism has received less attention in the context of SEPs and our
knowledge, therefore, is still lacking on this subject. One reason for its
neglect might be due to the lack of clear observational evidence of SEPs
experiencing drifts. As such, most work on the drifts experienced by SEPs is
based on single particle/full-orbit simulations (e.g. Kelly et al., 2012;
Marsh et al., 2013; Battarbee et al., 2017, 2018a), with some theoretical
investigations into the drift velocity (e.g. Burns & Halpern, 1968; Dalla et
al., 2013) and transport modelling (Wijsen et al., 2020). A summary of this
can be found in van den Berg et al. (2020), with the aspects most relevant to
the present study to be highlighted below.

As already mentioned, observational evidence pointing to SEPs experiencing
drifts is limited. Observations of the STEREO mission between October 2006 and
December 2013 reported by Richardson et al. (2014) seem to indicate that solar
energetic proton and electron events behave similarly. This follows from
comparing the onset or peak delay and the angular position of the peak
intensity as a function of the angular separation between the flare and the
spacecraft’s magnetic footpoint. If drifts were primarily responsible for
perpendicular transport, then it would be expected that the electrons and
protons should behave differently, although the data are not presented
according to the polarity of the magnetic field. The time span under
consideration did fall mainly in an $A<0$ polarity cycle ($A$ is the HMF
polarity, defined as having $A=+1$ when the magnetic field points outwards in
the Northern hemisphere), but the polarity seen by an observer depends on
their position relative to the heliospheric current sheet (HCS). Furthermore,
the considered electrons and protons have different rigidities, which
complicates direct comparison. More recently, Augusto et al. (2019) postulated
that SEP drifts along the HCS can help SEPs observed in ground level
enhancements to reach Earth from a magnetically unconnected event, if the
flare or coronal mass ejection and Earth are both close to the HCS. This is
based on an analysis of the 2017 September 10 event, when both the active
region and Earth were close to the HCS, but the transport time scales
(particle streaming along magnetic field to reach $1$ ${\rm au}$ and
perpendicular transport either through drifts or diffusion) were not
calculated to be compared to the time the particles had to travel to Earth
(i.e. the time between flare onset and in situ onset of SEPs).

Nonetheless, several studies have investigated the effect of drifts on SEPs.
Burns & Halpern (1968) derived expressions for the drift velocities of SEPs in
a Parker (1958) heliospheric magnetic field (HMF) and concluded that the
electric field drift is the dominant process, which shows that particles
follow the field line on which they started when this term is transformed into
the reference frame co-rotating with the Sun. Dalla et al. (2013) re-derived
these expressions and investigated the other drift terms as well, while Marsh
et al. (2013) verified these general predictions by integrating the Newton-
Lorentz equation in a uni-polar Parker HMF. These authors found that drifts
will be the most pronounced for high energy particles ($\sim 100$ ${\rm MeV}$
protons) or partially-ionised heavy ions. This investigation was extended by
Battarbee et al. (2017) to include neutral sheet drifts in a flat HCS, where
it was found that SEP drift patterns will be similar to galactic cosmic ray
(CR) drift patterns. Particles can be confined to the HCS in the $qA>0$
polarity cycle (here $q$ is the particle charge) and might have difficulty
reaching an observer on the other side of the HCS, while the energy dependence
of drift velocities will cause different observers to measure different
spectra. A wavy HCS was considered by Battarbee et al. (2018a) and it was
found that the HCS can efficiently transport particles to the poles where they
will drift more efficiently if they can escape the HCS through scattering.

Based on these models, Dalla et al. (2017a) used drifts to attribute the
observed energy dependent charge state of iron to the mass-to-charge-ratio
dependence of drifts and not the acceleration process (see e.g. Reames, 1999,
2013, and references therein). Similarly, Dalla et al. (2017b) attributed the
temporal evolution of the iron-to-oxygen-ratio to drifts and not the rigidity-
dependent mean free path (MFP) caused by turbulence generated by streaming
protons of similar rigidity (see again Reames, 1999, 2013, and references
therein). Battarbee et al. (2018b) used drifts to model the ground level
enhancement event of 2012 May 17 with mixed results: a source $80^{\circ}$
wider in longitude than the inferred coronal mass ejection was needed for
particles to reach the STEREO spacecraft and the flux at Mercury was
overestimated.

One aspect of the full-orbit simulations, discussed up to now, which is
potentially highly unrealistic is the pitch-angle scattering implemented as
random adjustments of the velocity vector at Poisson-distributed scattering
intervals (governed by the parallel MFP). A particle in these simulations
therefore follows an unperturbed motion between scatterings instead of being
continuously subjected to turbulent fluctuations perturbing the velocity
vector, as one would expect in a realistic scenario involving broadband
turbulence (see, e.g, Bruno & Carbone, 2016). This would explain why Marsh et
al. (2013) found that pitch-angle scattering and the parallel MFP have little
effect on drifts, because the particles experience maximum drift between the
occasional scatterings. The full-orbit simulations also do not contain
perpendicular diffusion, which will be present in turbulence with transverse
complexity (see e.g. Qin et al., 2002a, b), unlike the initial work of Kelly
et al. (2012), where a random magnetic foot point motion was added to a Parker
(1958) HMF in order to simulate the random walk of magnetic field lines, but
the velocity vector was also adjusted randomly at Poisson-distributed times.
In these simulations the particles show a diffusive spread perpendicular to
the background HMF with a slight asymmetry due to drifts.

The recent modelling efforts of Wijsen et al. (2020) used the focused
transport equation to investigate drifts in a uni-polar HMF with a pitch-angle
independent perpendicular diffusion coefficient. These authors verified, for
$3-36$ ${\rm MeV}$ protons, that different observers will see different
spectra, and found that perpendicular diffusion will diminish the effects of
drifts (i.e. break the latitudinal and longitudinal asymmetry with respect to
the injection site caused by drifts). Their investigation into an SEP
reservoir (different observers measuring the same intensity and spectra during
the fairly isotropic decay phase of an SEP event; see Reames, 2013, for a
summary) formed by a co-rotating interaction region, yielded an interesting
result: It is difficult to form a reservoir in the presence of drifts due to
the energy dependence of the drift speed and the preferred direction of the
drift velocity, causing different energy channels to decay at different rates,
different observers to measure different intensities, and the decay phase to
show significant anisotropies. Wijsen et al. (2020) concluded that if drifts
are present, then perpendicular diffusion must be strong enough to smear out
the effects of drifts in order to see the reservoir effect.

A major shortcoming of all these studies is the neglect of the turbulent
reduction of drifts. Drifts of cosmic rays, entering as a second order effect
compared to the first order diffusion processes, have been thoroughly
investigated and it has been found that drift effects must be suppressed in
models to fit observations (see, e.g., Burger et al., 2000). This is due to
turbulent fluctuations disrupting the large scale drifts and therefore
decreasing the drift velocity (see Engelbrecht et al., 2017, for a summary).
Numerical simulations (e.g. Minnie et al., 2007; Tautz & Shalchi, 2012) have
given clues to the circumstances leading to a reduction in drifts: 1) drift
reduction do not seem to occur in purely magnetostatic slab turbulence; 2) the
drifts also seem to be affected by the diffusion coefficients (probably
because different turbulence conditions will yield different diffusion and
drift conditions); 3) drifts decrease with an increase in the turbulence
strength, due to stronger turbulent drifts and 4) drifts decrease with a
decrease in the particle energy, due to low energy particles being perturbed
more than high energy particles (Burger & Visser, 2010; Engelbrecht et al.,
2017). It is important to note that all studies on this have only considered
isotropic particle distributions and do not consider pitch-angle dependencies.

It is then the aim of this paper to investigate the possible pitch-angle
dependence of the turbulent drift reduction factor and its implications for
SEP drifts. The investigation begins in Section 2 with a summary of Appendix
A, where the turbulence-reduced guiding centre velocity is derived
conceptually. This section also gives a summary of what is known about drift
suppression in the isotropic case and a derivation of the pitch-angle
dependent drift reduction factor. The drift reduction factor will be evaluated
for realistic turbulence and scattering conditions in Appendix B, while the
effect of this on the transport of solar energetic protons will be presented
in Section 3. Lastly, the implications of this investigation will be discussed
in Section 4.

##  2 Drift Reductions including a Pitch-angle Dependence

The derivation of the drift reduction factor with a pitch-angle dependence
will be given in this section, together with a discussion of this factor in
the isotropic limit and the origin of the drift reduction, based on the
derivations in Appendix A. Appendix A presents a conceptual derivation
(following Engelbrecht et al., 2017) to show that the gyro-phase averaged
guiding centre (GC) velocity in the solar wind (SW) frame,

$\displaystyle{\bf v}_{\rm gc}$ $\displaystyle=\sqrt{f_{s}}\,\mu v\,{\bf
b}_{0}+{\bf v}_{\rm sw}+f_{s}^{3/2}\frac{\mu p}{qB_{0}}\,{\bf
b}_{0}\times[({\bf b}_{0}\cdot{\bf\nabla}){\bf v}_{\rm sw}\,+$
$\displaystyle\;\;\;\;\;({\bf v}_{\rm sw}\cdot{\bf\nabla}){\bf
b}_{0}]\,+\frac{1+\mu^{2}}{4}\frac{vp}{qB_{0}}({\bf\nabla}f_{s}\times{\bf
b}_{0})\,+$
$\displaystyle\;\;\;\;\;f_{s}\frac{vp}{qB_{0}}\left\\{\mu^{2}({\bf\nabla}\times{\bf
b}_{0})_{\perp}+\frac{1-\mu^{2}}{2}\times\right.$
$\displaystyle\;\;\;\;\left.\left[({\bf\nabla}\times{\bf
b}_{0})_{\parallel}+\frac{{\bf
b}_{0}\times{\bf\nabla}B_{0}}{B_{0}}\right]\right\\},$ (1)

is modified by the effects of turbulence through a so-called drift reduction
factor, which can be written in terms of the gyrofrequency $\omega_{c}$ and a
perpendicular velocity correlation function’s decorrelation time as (Bieber &
Matthaeus, 1997)

$f_{s}=\frac{1}{1+(\omega_{c}\tau)^{-2}}.$ (2)

For Eq. 2 it was assumed that the SW velocity ${\bf v}_{\rm sw}$ is constant
and radial and that the background magnetic field ${\bf B}_{0}$ is stationary
(see Appendix A for the full expression in the general case). It was also
assumed that the turbulence is weak, since $f_{s}$ is derived from the
velocity correlation of a charged particle in a uniform magnetic field with
turbulence causing an exponential decorrelation of the perpendicular velocity
on a time scale $\tau$ (following Bieber & Matthaeus, 1997), while this form
of the GC velocity is calculated using a perturbation of the fields. Here,
$\mu$, $v$, $p$, and $q$ is the particle’s pitch-cosine, speed, momentum, and
charge in the SW frame, respectively, ${\bf b}_{0}={\bf B}_{0}/B_{0}$ is a
unit vector along the large scale average/background magnetic field, the
subscript $\parallel$ and $\perp$ refer to directions parallel and
perpendicular to ${\bf B}_{0}$, and $\omega_{c}=|q|B_{0}/m$ is the particle’s
cyclotron frequency, with $m$ the particle’s relativistic mass in the SW
frame. The terms in Eq. 2 describe the streaming of particles along ${\bf
b}_{0}$, advection by the SW, drifts caused by the streaming of particles into
different SW conditions and the advection of the magnetic field by the SW,
spatially varying turbulence conditions, curvature drift perpendicular and
parallel to the magnetic field (the parallel term will be absent if the fields
are evaluated at the GC’s position and not the particle’s position; Rossi &
Olbert, 1970; Burger et al., 1985; Wijsen, 2020), and gradient drift.

Notice that both the streaming and drifts in Eq. 2 are reduced in the presence
of turbulence. _The streaming terms_ ($\mu v\,{\bf
b}_{0}+(1-\mu^{2})vp({\bf\nabla}\times{\bf b}_{0})_{\parallel}/2qB_{0}$) _are
reduced because the particle is streaming along the total magnetic field and
not just the background magnetic field._ _The perpendicular drift terms are
reduced because the smooth drift pattern is disrupted by turbulent drifts._
Indeed, the perturbed part of the velocity (see Appendix A for details) can be
used to calculate the perpendicular diffusion coefficients (see e.g.
Fraschetti & Jokipii, 2011; Strauss et al., 2016) and the first term in the
perturbed component ($\mu v\,\delta{\bf B}/B_{0}$, where $\delta{\bf B}$ is
the fluctuating magnetic field) can be interpreted as particles following
random walking field lines. It can therefore be seen that _there is an
interplay between drifts and the perpendicular diffusion coefficients: in the
absence of turbulence, the particle will experience pure drift $(f_{s}=1)$,
while this drift velocity is reduced as soon as turbulence is present
$(f_{s}<1)$, but with the additional effect that the particle will then also
experience perpendicular diffusion._

Initially, parametric forms of the drift suppression factor based on fits to
results from numerical test particle simulations of drift coefficients in the
presence of simulated magnetic turbulence were presented, but the usefulness
of such expressions are limited to the simulation’s particular turbulence
conditions. Engelbrecht et al. (2017) derived a form which is able to explain
the various simulation results both qualitatively and quantitatively,

$f_{s}=\frac{1}{1+(\lambda_{\perp}^{0}/R_{L})^{2}(\delta B^{2}/B_{0}^{2})},$
(3)

where $\lambda_{\perp}^{0}$ is the perpendicular MFP for a nearly isotropic
distribution in the absence of focusing, $R_{L}=v/\omega_{c}$ is the
particle’s maximal Larmor radius, and $\delta B^{2}$ is the total variance of
the fluctuations. It is known from simulations that perpendicular diffusion in
pure magnetostatic slab turbulence is subdiffusive (Qin et al., 2002a) and
hence $\lambda_{\perp}^{0}\longrightarrow 0$ with $f_{s}\longrightarrow 1$. If
$\lambda_{\perp}^{0}/R_{L}$ is kept constant, then $f_{s}\longrightarrow 0$ as
the turbulence strength ($\delta B^{2}/B_{0}^{2}$) increases. Keeping $\delta
B^{2}/B_{0}^{2}$ constant and assuming that $\lambda_{\perp}^{0}$ is
independent of particle speed or only weakly dependent thereon, then
$\lambda_{\perp}^{0}/R_{L}\longrightarrow 0$ and $f_{s}\longrightarrow 1$ for
high energy particles (Engelbrecht et al., 2017).

Following Engelbrecht et al. (2017), it will be assumed that the drift
reduction factor is given by Eq. 2. Bieber & Matthaeus (1997) argued that
$\tau=\ell_{c}/v$ and introduced a perpendicular decorrelation length
$\ell_{c}=R_{L}^{2}/\kappa_{\rm FL}$, where $\kappa_{\rm FL}$ is the field
line random walk (FLRW) diffusion coefficient. Engelbrecht et al. (2017)
relaxed the assumption of decorrelation due to particles following random
walking field lines and proposed that
$\ell_{c}=R_{L}^{2}/\lambda_{\perp}^{0}$, and that $\tau$ is governed by the
particle’s perpendicular speed $v_{\perp}$ and not $v$. Engelbrecht et al.
(2017) also argued that the turbulence will cause the perpendicular speed to
change randomly and therefore interpreted $v_{\perp}\approx v\sqrt{\delta
B^{2}/B_{0}^{2}}$ as a root-mean-squared value.

A choice for $\tau$, however, must now be made which includes pitch-angle
information. Following Engelbrecht et al. (2017), it will be assumed that the
particle’s velocity decorrelates by diffusing across the magnetic field, such
that

$\ell_{c}=\frac{r_{L}^{2}}{\lambda_{\perp}(\mu)}=\frac{vr_{L}^{2}}{3D_{\perp}(\mu)},$
(4)

where $r_{L}$ is the particle’s (pitch-angle dependent) Larmor radius and
$D_{\perp}(\mu)$ is the pitch-angle dependent perpendicular diffusion
coefficient. A pitch-angle dependent perpendicular MFP,
$\lambda_{\perp}(\mu)=3D_{\perp}(\mu)/v$, has been introduced and related to
$D_{\perp}(\mu)$ in such a way that the usual definition of the perpendicular
MFP is recovered for an isotropic distribution in the absence of focusing,
that is (Shalchi, 2009)

$\lambda_{\perp}^{0}=\frac{1}{2}\int_{-1}^{1}\lambda_{\perp}(\mu)d\mu=\frac{3}{2v}\int_{-1}^{1}D_{\perp}(\mu)d\mu.$
(5)

This should be a reasonable estimate of the decorrelation time, since the
Larmor radius quantifies the extent to which the particle samples fluctuations
perpendicular to the background magnetic field and the perpendicular diffusion
coefficient quantifies how quickly the particle diffuses perpendicular to the
background magnetic field.

Incorporating also a pitch-angle dependence into the root-mean-squared value
of the perpendicular speed, $v_{\perp}\approx v\sqrt{1-\mu^{2}}\sqrt{\delta
B^{2}/B_{0}^{2}}$, the perpendicular decorrelation time is then proposed to be
$\tau=\ell_{c}/v_{\perp}\approx
vr_{L}^{2}/3D_{\perp}(\mu)v\sqrt{1-\mu^{2}}\sqrt{\delta B^{2}/B_{0}^{2}}$.
Lastly, the cyclotron frequency can be written in terms of the particle’s
maximal Larmor radius and speed, i.e. $\omega_{c}=v/R_{L}$. With this choice
of $\omega_{c}$ and $\tau$, the product in Eq. 2 is

$\omega_{c}\tau=\frac{\sqrt{1-\mu^{2}}}{D_{\perp}(\mu)}\frac{vR_{L}}{3}\sqrt{\frac{B_{0}^{2}}{\delta
B^{2}}},$

where $r_{L}=\sqrt{1-\mu^{2}}R_{L}$ was used. Substituting this into Eq. 2
gives an expression for the drift reduction factor,

$f_{s}(\mu)\approx\frac{1}{1+[D_{\perp}(\mu)/\sqrt{1-\mu^{2}}\kappa_{A}^{\rm
ws}]^{2}(\delta B^{2}/B_{0}^{2})},$ (6)

where $\kappa_{A}^{\rm ws}$ is the isotropic weak-scattering drift coefficient
of Eq. A3, so that the pitch-angle dependent, turbulence-reduced drift
coefficient then becomes

$D_{xy}(\mu)=f_{s}(\mu)\kappa_{A}^{\rm ws}(\mu)={\rm
sign}(q)f_{s}(\mu)\frac{1-\mu^{2}}{2}vR_{L}.$ (7)

This form of $f_{s}(\mu)$ will have significant consequences for focused SEPs,
as will be explored in the following paragraphs and Appendix B.

Note that Eq. 6 also encapsulates that which is already known about turbulent
drift suppression from computer simulations: 1) $f_{s}\longrightarrow 1$ as
$v$ becomes large (not taking into account the energy dependence of
$D_{\perp}$); 2) $f_{s}\longrightarrow 0$ as $\delta B^{2}/B_{0}^{2}$ becomes
large; 3) if $D_{\perp}$ is consistent with the Shalchi slab hypothesis
(Shalchi, 2006), then $D_{\perp}$ will be zero in pure slab turbulence and
hence $f_{s}=1$; 4) in most non-linear theories, $D_{\perp}$ is also dependent
on the parallel MFP or the pitch-angle diffusion coefficient, such that both
parallel and perpendicular diffusion will influence $f_{s}$. The greatest
uncertainty in Eq. 6 at this point, is the pitch-angle dependence of
$D_{\perp}(\mu)$. This is investigated for different theories of the
perpendicular diffusion coefficient in the next section and Appendix B.

Figure 1: Top panels: Parallel and perpendicular mean free paths, together
with the maximal Larmor radius. Bottom panels: The resulting drift reduction
factor (Eq. 3). Left panels: As a function of radius for $100$ $\mathrm{MeV}$
protons using the turbulence transport model results in Fig. 6. Right panels:
As a function of kinetic energy for protons at Earth using observed turbulence
quantities.

##  3 Consequences for Solar Energetic Particle Transport

The influence of the behaviour of the various diffusion coefficients and the
resulting drift reduction factors on SEP transport will be the subject of this
section. Turbulence quantities and the perpendicular diffusion coefficient
must be specified for Eq. 6. Appendix B discusses some key results of a
turbulence transport model (TTM) in the very inner heliosphere, but simplified
radial scalings will be used in this section for the turbulence quantities
needed. The magnetic variance and correlation lengths of the slab and 2D
turbulence are normalized to solar minimum observations at Earth, with the
assumption of a slab/2D ratio of 20:80. Following Burger et al. (2008), Moloto
et al. (2018), or Engelbrecht (2019b), the magnetic variance is modelled as
$\delta B^{2}=10~{}{\rm nT}^{2}~{}(1~{}{\rm au}/r)^{2.4}$ (Bieber et al.,
1994; Zank et al., 1996; Smith et al., 2001, 2006; Engelbrecht & Burger,
2013), while the slab correlation length is modelled as $\ell_{m}=2.5$
$\ell_{\rm 2D}$, with the 2D correlation length modelled by $\ell_{\rm
2D}=7.41\times 10^{-3}~{}{\rm au}~{}\sqrt{r/1~{}{\rm au}}$ (Smith et al.,
2001; Weygand et al., 2011). These scalings, illustrated in Fig. 6, are
consistent with observations and TTMs in the heliosphere beyond Earth’s orbit,
but differ in the very inner heliosphere; the interested reader should consult
Appendix B for these differences.

To investigate the turbulence-reduced drift coefficient, three different
expressions derived using different theoretical approaches are employed for
$D_{\perp}(\mu)$, which, according to Eq. 5, can be written as (e.g.
Engelbrecht, 2019b)

$D_{\perp}(\mu)=g(\mu)v\lambda_{\perp}^{0},$ (8)

where $g(\mu)$ governs the perpendicular diffusion coefficient’s pitch-angle
dependence. In most of what is to follow, it will be assumed that
$g(\mu)=\mu^{2}$, based on the numerical test particle simulations of Casse et
al. (2002) and the unified nonlinear theory (UNLT) of Shalchi (2010). However,
the consequences of assuming that $g(\mu)=|\mu|$, based on what is expected
from the FLRW perpendicular diffusion coefficient (see, e.g., Jokipii, 1966;
Matthaeus et al., 1995; Strauss et al., 2017), and an alternative form where
$g(\mu)=\alpha\mu^{2}+\beta$ (with $\alpha$ and $\beta$ constants), motivated
by the simulation results of Qin & Shalchi (2014) as discussed by Engelbrecht
(2019b), will also be considered. For the purposes of discussion, the choices
of $D_{\perp}$ made below are motivated by their relative simplicity and
tractability. More complicated expressions derived using, for example, more
physically motivated forms for the 2D turbulence power spectra do exist in the
literature (see, e.g., Engelbrecht & Burger, 2015; Dempers & Engelbrecht,
2020), but these are either nonlinear or depend on unusual functions (e.g.
exponential integral and hypergeometric functions) that may complicate their
characterization. Furthermore, such expressions often depend on physical
quantities, such as the energy-range onset length scale (see Matthaeus et al.,
2007), that are difficult to ascertain in the region of interest to this study
(see, e.g., Engelbrecht, 2019a).

Figure 2: Left: The pitch-angle dependent drift reduction factor (Eq. 6) for
$100$ $\mathrm{MeV}$ protons at Earth predicted by different functional forms
of the perpendicular diffusion coefficient. Right: The pitch-angle dependent
drift suppression factor (Eq. 6) for protons at Earth as a function of kinetic
energy. Perpendicular diffusion is assumed to be described by the non-linear
guiding centre theory (Eq. 9).

The first expression to be considered is based on an isotropic perpendicular
MFP expression from the nonlinear guiding centre (NLGC) theory of Matthaeus et
al. (2003) derived by Shalchi et al. (2004) and modified for a general ratio
of slab to 2D variance by Burger et al. (2008),

$\displaystyle D_{\perp}(\mu)$
$\displaystyle=\mu^{2}v\lambda_{\parallel}^{1/3}\times$ (9)
$\displaystyle\;\;\;\;\left[a^{2}\sqrt{3\pi}\frac{2\nu-1}{\nu}\frac{\Gamma(\nu)}{\Gamma(\nu-1/2)}\frac{\delta
B_{\rm 2D}^{2}}{B_{0}^{2}}\ell_{2D}\right]^{2/3},$

where $\lambda_{\parallel}$ is the particle’s parallel MFP and $a^{2}$ is
assumed to be equal to $1/3$ following Matthaeus et al. (2003). The isotropic
perpendicular MFP used in Eq. 9 is derived assuming a 2D turbulence power
spectrum with a wave number-independent energy-containing range and an
inertial range with spectral index $-2\nu$, which in this study is assumed to
take on the Kolmogorov $-5/3$ value. Observationally, such a spectral form is
not entirely realistic (see, e.g., Bieber et al., 1993; Matthaeus et al.,
2007), but its use here is motivated by its simplicity relative to expressions
derived using more realistic spectral forms (see, e.g., Engelbrecht & Burger,
2015; Dempers & Engelbrecht, 2020), as well as by the fact that its use in
galactic cosmic ray modulation results has lead to computed intensities in
good agreement with spacecraft observations (e.g. Moloto & Engelbrecht, 2020;
Engelbrecht & Moloto, 2021) and because it has also been applied to SEP
modelling (e.g. Qin et al., 2013; Wang et al., 2014).

The second expression for the pitch-angle dependent perpendicular diffusion
coefficient to be used here is that derived by Qin & Shalchi (2014) from the
Shalchi (2010) UNLT in the limit of dominant pitch-angle scattering. This
expression is given by

$D_{\perp}(\mu)=\frac{1}{2}a^{2}\mu^{2}v\frac{\delta B_{\rm
2D}^{2}}{B_{0}^{2}}\lambda_{\parallel}.$ (10)

The third pitch-angle dependent perpendicular diffusion coefficient to be
considered here is the FLRW result, which can also be derived from UNLT (e.g.
Matthaeus et al., 1995, 2007; Qin & Shalchi, 2014; Strauss et al., 2017), and
is given by

$D_{\perp}(\mu)=a|\mu|v\kappa_{\rm FL},$ (11)

where $\kappa_{\rm FL}$ denotes the FLRW diffusion coefficient. The expression
for this quantity employed here is that proposed by Shalchi & Wienhorst (2009)

$\kappa_{\rm FL}=\ell_{\rm 2D}\sqrt{\frac{2\nu-1}{2(q_{\rm 2D}-1)}\frac{\delta
B_{\rm 2D}^{2}}{B_{0}^{2}}},$ (12)

and is derived using the same 2D turbulence power spectrum form as used in Eq.
10, with $q_{2D}>1$ denoting the energy-containing range spectral index.

Lastly, both the NLGC and UNLT expressions used here require an isotropic
parallel MFP as input. To that end, an expression derived from quasi-linear
theory (QLT) assuming magnetostatic slab turbulence by Teufel & Schlickeiser
(2003) is employed, given by

$\lambda_{\parallel}=\frac{3s}{\pi(s-1)}\frac{R^{2}}{k_{m}}\frac{B_{0}^{2}}{\delta
B_{\rm sl}^{2}}\left[\frac{1}{4}+\frac{2R^{-s}}{(2-s)(4-s)}\right],$ (13)

where $R=R_{L}k_{m}$, with $k_{m}$ the wave number at which the inertial
range, here assumed to have a spectral index of $-5/3=-s$, of the slab
turbulence power spectrum commences.

The pitch-angle, energy, and radial dependencies of the turbulence-reduced
drift reduction factor are discussed in detail in Appendix B, together with
the radial and energy dependencies of the parallel and different perpendicular
MFPs discussed above. The bottom two panels of Fig. 1 show Eq. 3 for protons,
calculated using the MFPs in the top two panels, as function of radial
distance (left panel) and kinetic energy (right panel). Larger values for
$\lambda_{\perp}^{0}$ lead to smaller values for $f_{s}$ and hence, a greater
reduction in drift effects. As function of radial distance, the $100$ ${\rm
MeV}$ proton drift reduction function is relatively flat, only showing a
moderate upturn at the smallest radial distances considered here for the UNLT
perpendicular MFP. Almost no drift reduction can be seen at this energy for
FLRW, but this is not the case for all energies, as $f_{s}$ calculated using
the FLRW MFP does begin to drop below $1$ at energies below $\sim 100$ ${\rm
MeV}$. The UNLT result remains well below unity over the entire radial and
energy range considered here.

The left panel of Fig. 2 shows the pitch-angle dependence of Eq. 6 for $100$
${\rm MeV}$ protons and typical turbulence quantities measured at Earth. Field
aligned particles ($\mu=\pm 1$) will have a significant reduction in their
drift velocities for all of the different forms of $D_{\perp}(\mu)$
considered, while the behaviour of $f_{s}(\mu)$ towards $\mu=0$ depends
critically on the theory used for $D_{\perp}(\mu)$. Any perpendicular
diffusion coefficient which is zero at $\mu=0$ will have $f_{s}(0)=1$, and
hence particles will experience full drift effects at this pitch-angle, but
there does not seem to be any a priori reason that a particle with a
$90^{\circ}$ pitch-angle should not experience drift reductions. Both the
pitch-angle and energy dependence of Eq. 6 are shown in the right panel of
Fig. 2 at Earth for NLGC. For a fixed pitch-angle (except at $\mu=0$), $f_{s}$
becomes more pronounced at low energies. The main behaviour of $f_{s}(\mu)$ is
as follows: field aligned particles will experience more drift reductions, low
energy particles will experience less drifts, and drift reduction factors will
have a weak radial dependence.

Table 1:  Simulation naming Simulation number | Simulation label | Transport processes included in the simulation
---|---|---
1 | $\parallel$ | only parallel transport
2 | $\parallel,V_{d}$ | parallel transport and (non-reduced) drifts
3 |  $\parallel,V_{d}$, FLRW | parallel transport, drifts, and FLRW (see Eq. (11))
4 |  $\parallel,f_{s}V_{d}$, FLRW | parallel transport, reduced drifts, and FLRW
5 |  $\sqrt{f_{s}}\parallel,f_{s}V_{d}$, FLRW | reduced parallel transport, reduced drifts, and FLRW
6 |  $\parallel,V_{d}$, NLGC | idem to 3 but for NLGC (see Eq. (9))
7 |  $\parallel,f_{s}V_{d}$, NLGC | idem to 4 but for NLGC
8 |  $\sqrt{f_{s}}\parallel,f_{s}V_{d}$, NLGCT | idem to 5 but for NLGC
9 |  $\parallel,V_{d}$, UNLT | idem to 3 but for UNLT (see Eq. (10))
10 |  $\parallel,f_{s}V_{d}$, UNLT | idem to 4 but for UNLT
11 |  $\sqrt{f_{s}}\parallel,f_{s}V_{d}$, UNLT | idem to 5 but for UNLT

To illustrate the possible effect of the reduction factor $f_{s}(\mu)$ on SEP
events, Eq. 6 is incorporated into the energetic particle model PARADISE
(Wijsen et al., 2019; Wijsen, 2020). This model solves the focused transport
equation, including the effect of cross-field diffusion and particle drifts.
In this work, protons are simulated propagating trough a Parker SW of constant
radial speed $v_{\rm sw}=400$ ${\rm km}\cdot{\rm s}^{-1}$. Moreover, the HMF
is assumed to have a positive polarity ($A=+1$) and is scaled such that
$B(1~{}{\rm\,au})=5$ ${\rm nT}$. The positive HMF polarity implies that the
protons will drift in the southward direction (see e.g. Dalla et al., 2013).
The functional form of the pitch-angle diffusion coefficient is derived from
QLT and scaled such that the isotropic parallel MFP corresponds to Eq. (13).
For the simulations including cross-field diffusion, the three different
functional forms introduced in Eq. 9, Eq. 11, and Eq. 10 are considered.

As a boundary condition for the energetic particles, the isotropic injection
function

$\displaystyle f(r,\theta,\phi,E,\mu,t)\propto$
$\displaystyle\frac{1}{t}\exp{\left(-\frac{\tau_{a}}{t}-\frac{t}{\tau_{e}}\right)}\exp{\left[\frac{\phi^{2}+\theta^{2}}{2\sigma^{2}}\right]}$
$\displaystyle\times\delta(E/E_{0}-1)\delta(r/r_{0}-1)$

is prescribed at $r_{0}=0.05$ ${\rm au}$. A Gaussian injection profile with
$\sigma=5^{\circ}$ is assumed for the azimuthal $\phi$ and the latitudinal
$\theta$ angles, with the injection centred on the ecliptic plane. The
injection function follows a Reid–Axford temporal profile (Reid, 1964), with
$\tau_{a}=0.2$ ${\rm h}$ and $\tau_{e}=1$ ${\rm h}$. The kinetic energy $E$ of
the injected protons is assumed to be $E_{0}=100$ ${\rm MeV}$. The particles
are propagated for a total simulation time of $T=50$ ${\rm h}$.

An overview of the different simulations that were performed can be found in
Table 1. In what follows the different simulations will be referred to
according to their number in Table 1. For each of the simulations, the energy-
integrated omni-directional intensity,

$I(r,\theta,\phi,t)=\int_{0}^{E_{0}}\int_{-1}^{1}j(r,\theta,\phi,E,\mu,y)d\mu\,dE$

with $j=p^{2}f$ denoting the differential intensity, were calculated, together
with the ratio

$R(r,\theta,\phi):=\frac{\int_{0}^{T}I(r,\theta,\phi,t)dt}{\int_{0}^{T}I(r,-\vartheta,\phi,t)dt}.$
(14)

In particular, the ratios $R(\theta=15^{\circ})$ and $R(\theta=5^{\circ})$ for
observers located at $r=1$ ${\rm au}$ and magnetically connected to
$\theta=0^{\circ}$ and $\phi=0^{\circ}$ at the inner boundary are considered.
Fig. 3 shows the values obtained for these two ratios for the different
simulation. Without drifts, the particle distribution will remain symmetric
with respect to the injection site and hence
$R(\theta=15^{\circ})=R(\theta=5^{\circ})=1$. When drifts are included in the
simulations, protons drift southwards and as a consequence,
$R(\theta=15^{\circ})>R(\theta=5^{\circ})>1$. Fig. 3 illustrates that when no
perpendicular diffusion or drift reduction is present, the effect of drifts
can be significant, since $R(\theta=15^{\circ})=8.8$ and
$R(\theta=5^{\circ})=1.9$ for simulation 2. Note that the drift effects are
only seen very late in the event ($>2$ ${\rm h}$) and that these ratios use
the event integrated omni-directional intensity to clearly illustrate
differences. Fig. 3 also illustrates that these ratios are drastically reduced
when cross-field diffusion is switched on (simulations 3, 6 or 9). This
reduction is the largest for the simulations with UNLT and the smallest for
the simulations with FLRW. This ordering follows the ordering of the
associated perpendicular MFPs, i.e., $\lambda_{\perp}^{\rm
UNLT}>\lambda_{\perp}^{\rm NLGC}>\lambda_{\perp}^{\rm FLRW}$ (see Fig. 1). In
fact, under the assumed turbulence conditions, the UNLT cross-field diffusion
is strong enough to erase the drift-induced asymmetries almost entirely. When
the drift reduction factor $f_{s}$ is taken into account (simulations 4 or 7),
the remaining drift-induced asymmetries in the simulations with NLGC theory
and FLRW decrease further, as expected. Moreover, Fig. 3 illustrates that the
ratios decrease even further when also taking into account the reduction of
the particle streaming along the HMF (e.g., simulations 5 or 8). This is
because this reduced parallel streaming keeps the particles for a prolonged
amount of time at smaller heliocentric radial distances where the drifts are
weaker (e.g., Dalla et al., 2013).

Fig. 4 shows the omni-directional time-intensity profiles for simulations 3,
5, 6, 8, 9 and 11 for an observer located at $r=1$ ${\rm au}$ and magnetically
connected to the center of the injection region. It can be seen that the
reduction factor $\sqrt{f_{s}}$ on the streaming of the protons along the HMF
delays the onset times by $\sim 0.5$ ${\rm h}$, $\sim 2.8$ ${\rm h}$ and $\sim
3.5$ ${\rm h}$ for FLRW, NLGC, and UNLT, respectively. This is in accordance
with $\sqrt{f_{s}(\mu)}$ being smallest for UNLT and largest for FLRW (see
Figs. 1 and 2). Since $f_{s}(\mu)$ becomes smaller when $|\mu|$ approaches
unity, the spreading of the particles along the field lines will be reduced.
As a result, the injected distribution function will propagate as a more
coherent pulse along the HMF. This is illustrated in Fig. 5, which shows for
simulations 6 and 7 the omni-directional intensity at time $t=5$ ${\rm h}$
measured along the magnetic field lines with foot points at $\phi=0^{\circ}$
and projected on a meridional slice.

Figure 3: Event-integrated intensity ratios $R(\theta=15^{\circ})$ and
$R(\theta=5^{\circ})$ (Eq. 14) for observers located at $r=1$ ${\rm au}$ and
magnetically connected to $\theta=0^{\circ}$ and $\phi=0^{\circ}$ at the inner
boundary. Descriptions of the simulation numbers on the x-axis can be found in
Table 1. Figure 4: Omni-directional intensity profiles as a function of time
for an observer located at $r=1$ ${\rm au}$ and magnetically connected to the
center of the injection region. Figure 5: Omni-directional intensity at time
$t=5$ ${\rm h}$, measured along the magnetic field lines with foot points
located at $\phi=0^{\circ}$ and projected on a meridional slice. Left panel:
Without a reduction in the streaming term. Right panel: If the reduction of
the streaming term is included.

##  4 Summary and Discussion

This paper investigated the possible pitch-angle dependence of the turbulent
drift reduction factor and its implications for SEP drifts. Section 1
introduced the current application of drifts to SEPs and the major
shortcomings of these efforts by not considering drift reductions. The
investigation began in Section 2 with a summary of why drift reduction occurs
and what is known about drift suppression in the isotropic case, which was
supplemented by a conceptual derivation of the drift reduction in Appendix A.
Here it was emphasized that there is an interplay between drifts and
turbulence: in the absence of turbulence, particles experience drift motions,
but turbulence disrupts these drift patterns and cause the particles to rather
diffuse perpendicular to the background magnetic field.

A pitch-angle dependent drift reduction factor was also derived in Section 2.
The derived form is able to describe full orbit simulations qualitatively, but
is critically dependent on the assumed pitch-angle dependence of the
perpendicular diffusion coefficient. A perpendicular diffusion coefficient
which is zero for particles with $90^{\circ}$ pitch-angles will yield no drift
reductions for these particles, while perpendicular diffusion coefficients
which is a maximum for field aligned particles will yield no drifts for these
particles. Theories of cross-field diffusion predicting larger perpendicular
MFPs will also yield more drift reductions (in agreement with Engelbrecht et
al., 2017). The drift reduction factor was then evaluated for realistic
turbulence and scattering conditions in Appendix B to predict its effects on
SEP propagation. Here it was found that field aligned particles will
experience more drift reductions, that low energy particles will experience
less drifts, and that drift reduction factors will have a weak radial
dependence.

The drift reduction was investigated with the PARADISE model (Wijsen, 2020)
for protons subject to perpendicular diffusion coefficients with different
pitch-angle dependencies originating from the different FLRW, NLGC, and UNLT
theoretical approaches (see the discussion in Section 3). Although drifts
could potentially have a large effect, it was seen that the inclusion of
cross-field diffusion in a realistic scenario already diminishes the effect of
drifts (see Fig. 3). In the simulations, the strong perpendicular diffusion
predicted by UNLT, almost smeared out the signatures of drifts completely,
while the weaker perpendicular diffusion predicted by employing the smaller
FLRW perpendicular diffusion coefficient still showed signatures of drifts. By
including the proposed turbulence-reduced pitch-angle dependent drift
coefficient into the simulations, drift reduction then further diminishes the
effect of drifts, such that drifts can be considered to be a second order
effect. It was also seen that drifts are further reduced when the streaming of
particles, which will be discussed further below, is also included. This is
because particles then spend more time closer to the Sun where drift effects
are smaller. For realistic turbulence conditions in the heliosphere, as
considered here, it can therefore be expected that cross-field diffusion will
have the largest influence on the perpendicular transport of SEPs and not
particle drifts. Note that a Parker HMF was used in these modelling efforts
and that drifts in the geometry of magnetic field structures, like coronal
mass ejections or co-rotating interaction regions, might still be important
(according to the modelling of Wijsen et al., 2020), although it would also be
dependent on the turbulence conditions and levels, which could be
significantly enhanced in these structures (see, e.g., Wiengarten et al.,
2015).

It is interesting to note the prediction in Eq. 2 that the parallel streaming
term ($\mu v\,{\bf b}_{0}$) will also be reduced. Modelling showed that this
reduction in streaming will cause a delayed onset at an observer (Fig. 4),
with the delay larger for longer perpendicular MFPs. When combined with
perpendicular diffusion, this might solve the causality problem identified by
Strauss & Fichtner (2015), because particles with $\mu=\pm 1$ will stream at a
speed less than $v$, allowing them to also diffuse (if $D_{\perp}(\mu=\pm
1)\neq 0$) without moving faster than $v$ (depending on the value of
$D_{\perp}$). This factor is also comparable to the factor (the cosine of the
angle between the local mean magnetic field and the local random walking
magnetic field) used by Laitinen et al. (2013) to have particles following
their computed random walking field lines. Hence, this will have an influence
on the analysis of Laitinen & Dalla (2019), where the path length of a
diffusing particle is calculated (with the causality problem also present in
some approaches), and it would explain why these authors argue that the
streaming should be re-scaled. Lastly, this may also influence SEP path
lengths inferred from, e.g., PSP/IS$\odot$IS observations, as is done by
Chhiber et al. (2021).

A type of drift which has not yet been addressed in this work is that of
particle drifts along a current sheet. Burger (2012) derived an expression for
the neutral sheet drift velocity in terms of the gradient and curvature drift
for a nearly isotropic distribution (see also Engelbrecht et al., 2019).
Although this approach does not contain a pitch-angle dependence, it yields a
result dependent on the drift coefficient and one could therefore expect that
neutral sheet drifts will be reduced by the same factor as the gradient and
curvature drifts. The findings of Battarbee et al. (2017), that particles have
difficulty crossing a current sheet, seem to be verifiable with observations
(the observations of Klassen et al., 2018, might point to this). Notice,
however, that such an observation is still possible even if current sheet
drifts are suppressed, because the current sheet could act as a diffusion
barrier, as discussed by Strauss et al. (2016), due to, e.g., lower turbulence
conditions close to the heliospheric current sheet (as observed by PSP; Chen
et al., 2021). Furthermore, given the observed complexity of current sheets
and the plasma conditions in their vicinity (for a review, see Khabarova et
al., 2021), modelling the transport of charged particles in their vicinity
presents unique challenges (see, e.g., Pezzi et al., 2021) that are beyond the
scope of this work.

The new pitch-angle dependent drift reduction factor proposed here, as opposed
to the isotropic result of Engelbrecht et al. (2017), cannot as yet be
compared with the results of existing numerical test particle simulations of
the particle drift coefficients such as those performed by, e.g., Tautz &
Shalchi (2012), as no direct simulations investigating the pitch-angle
dependence of this drift reduction factor currently, to the best of our
knowledge, exist. Therefore, although the derived pitch-angle dependent drift
reduction factor, when averaged over pitch-angle, coincides nicely with the
isotropic drift reduction factor derived by Engelbrecht et al. (2017) for weak
turbulence conditions and small perpendicular MFPs, it is unclear whether this
reduction factor’s form would generally hold for all turbulence conditions
expected of the very inner heliosphere. This is evinced by the fact that for
large perpendicular MFPs it was found that the pitch-angle dependent drift
reduction factor is larger than the isotropic factor, depending on the
functional form used for the perpendicular diffusion coefficient (see Fig. 7).
It is possible that this functional form, derived as it is from the Bieber &
Matthaeus (1997) theory with its implicit assumption of low turbulence levels,
would simply not be applicable under the higher turbulence levels which would
theoretically (e.g. Shalchi, 2009) lead to larger perpendicular MFPs.
Furthermore, the consequences of the pitch-angle dependence derived here for
the drift reduction factor, such as the prediction that particles with
$90^{\circ}$ pitch-angles will experience no drift reductions if the
perpendicular diffusion coefficient is zero for that pitch-angle (e.g.
Engelbrecht, 2019b), would need to be investigated by means of direct
numerical test particle simulations. This will be the subject of future work.

This work is based on the research supported in part by the National Research
Foundation of South Africa (NRF grant numbers 120847, 120345, and 119424).
Opinions expressed and conclusions arrived at are those of the authors and are
not necessarily to be attributed to the NRF. JPvdB acknowledge support from
the South African National Space Agency. NW acknowledges funding from the
Research Foundation - Flanders (FWO – Vlaanderen, fellowship no. 1184319N).
Computational resources and services used in this work were provided by the
VSC (Flemish Supercomputer Centre), funded by the FWO and the Flemish
Government-Department EWI. Figures prepared with Matplotlib (Hunter, 2007) and
certain calculations done with NumPy (Harris et al., 2020) and SciPy (Virtanen
et al., 2020).

##  Appendix A Drift Reduction in the Isotropic and Anisotropic Limit

In galactic cosmic ray modulation studies it is customary to absorb the pitch-
angle averaged GC drift velocity,

$\left\langle{\bf
v}_{d}\right\rangle=\frac{vp}{3q}{\bf\nabla}\times\left(\frac{{\bf
b}_{0}}{B_{0}}\right),$ (A1)

into the diffusion tensor as a drift coefficient, i.e. $\left\langle{\bf
v}_{d}\right\rangle={\bf\nabla}\times(\kappa_{A}{\bf b}_{0})$, with the
subscript A referring to the fact that it forms the antisymmetric part of the
diffusion tensor (see e.g., Burger et al., 2008). This can be done if the GC
velocity is divergence free, which Eq. A1 is in the absence of electric fields
(Minnie et al., 2007; Burger & Visser, 2010; Engelbrecht et al., 2017). An
example of how the drift coefficient is calculated following Bieber &
Matthaeus (1997) and Burger & Visser (2010) will be briefly discussed. The
drift coefficient is calculated from a velocity correlation according to the
Taylor-Green-Kubo (TGK) formalism (see Shalchi, 2009, and references therein),

$D_{xy}(\mu)=\int_{0}^{\infty}\langle v_{x}(0)v_{y}(t)\rangle dt,$ (A2)

where $D_{xy}(\mu)$ is further averaged over pitch-angle to give
$\kappa_{A}=\int_{-1}^{1}D_{xy}(\mu)d\mu/2$.

Consider a homogeneous background magnetic field along the $z$-axis. It can be
showed that the particle’s velocity components perpendicular to the magnetic
field can be written as (Bieber & Matthaeus, 1997)

$\displaystyle v_{x}(t)$ $\displaystyle=v_{\perp}\cos(\omega_{c}t-\varphi)$
$\displaystyle v_{y}(t)$ $\displaystyle=-{\rm
sign}(q)v_{\perp}\sin(\omega_{c}t-\varphi),$

where $\varphi$ is the particle’s gyro-phase and $v_{\perp}=v\sqrt{1-\mu^{2}}$
is the particle’s speed perpendicular to the magnetic field. Now it can be
calculated that

$v_{x}(0)v_{y}(t)={\rm
sign}(q)v_{\perp}^{2}\left[\cos(\omega_{c}t)\sin\varphi\cos\varphi-\sin(\omega_{c}t)\cos^{2}\varphi\right]$

and this can be averaged over gyro-phase to obtain

$\langle v_{x}(0)v_{y}(t)\rangle_{\varphi}=-{\rm
sign}(q)\frac{1-\mu^{2}}{2}v^{2}\sin(\omega_{c}t)=-\langle
v_{y}(0)v_{x}(t)\rangle_{\varphi},$

which also shows why the drift coefficient forms the antisymmetric part of the
diffusion tensor (Burger & Visser, 2010).

Bieber & Matthaeus (1997) argued that turbulence (assumed to be stationary and
homogeneous) will decorrelate the unperturbed velocity components, such that
the correlation tends to zero in a finite time (i.e. as time
$t\longrightarrow\infty$, the velocity correlation functions approach zero
faster than $1/t$), and assumed that the correlation decreases exponentially
with a characteristic time $\tau$, such that the particle distribution becomes
nearly isotropic. The drift coefficient can then be calculated from Eq. A2 as

$D_{xy}(\mu)\approx-{\rm
sign}(q)\frac{1-\mu^{2}}{2}v^{2}\int_{0}^{\infty}\sin(\omega_{c}t)e^{-t/\tau}dt=-{\rm
sign}(q)\frac{1-\mu^{2}}{2}v^{2}\frac{\omega_{c}}{1/\tau^{2}+\omega_{c}^{2}}=-\frac{1-\mu^{2}}{2}3\kappa_{A}^{\rm
ws}f_{s},$

where

$\kappa_{A}^{\rm ws}=\frac{vp}{3qB_{0}}={\rm sign}(q)\frac{vR_{L}}{3}$ (A3)

is the weak scattering drift coefficient (see Forman et al., 1974) and

$f_{s}=\frac{(\omega_{c}\tau)^{2}}{1+(\omega_{c}\tau)^{2}}.$

is the so-called drift reduction factor. Since an unperturbed orbit is
assumed, the current derivation is strictly only valid for low turbulence
levels. Notice that $\kappa_{A}$ will be non-zero in a homogeneous field
without turbulence ($\tau\longrightarrow\infty$), due to the velocity
correlations in Eq. A2, but that $\left\langle{\bf v}_{d}\right\rangle$ will
then be zero because $\kappa_{A}^{\rm ws}{\bf b}_{0}$ will be constant (see
Burger & Visser, 2010).

Engelbrecht et al. (2017) present a conceptual explanation of the origins of
drift suppression which will be generalised here (see also Zhao et al., 2017).
Consider the gyro-phase averaged GC velocity in a flow frame (like the SW; see
Zhang, 2006; Wijsen, 2020, for the derivation)

$\displaystyle\langle{\bf v}_{\rm gc}\rangle_{\varphi}$ $\displaystyle=\mu
v\,{\bf b}+{\bf w}+\frac{m}{qB}{\bf b}\times\left[\frac{\partial{\bf
w}}{\partial t}+(\mu v\,{\bf b}+{\bf w})\cdot{\bf\nabla}{\bf
w}\right]+\frac{\mu p}{qB}{\bf b}\times\left(\frac{\partial{\bf b}}{\partial
t}+{\bf w}\cdot{\bf\nabla}{\bf b}\right)+$
$\displaystyle\;\;\;\;\frac{vp}{qB}\left\\{\mu^{2}({\bf\nabla}\times{\bf
b})_{\perp}+\frac{1-\mu^{2}}{2}\left[({\bf\nabla}\times{\bf
b})_{\parallel}+\frac{{\bf b}\times{\bf\nabla}B}{B}\right]\right\\}+$
$\displaystyle\;\;\;\;\frac{vp}{c^{2}qB}\left\\{\mu^{2}\left({\bf
w}\times\frac{\partial{\bf b}}{\partial
t}\right)_{\perp}+\frac{1-\mu^{2}}{2}\left[\left({\bf
w}\times\frac{\partial{\bf b}}{\partial t}\right)_{\parallel}+\frac{\partial
B}{\partial t}\frac{{\bf b}\times{\bf w}}{B}\right]\right\\}$ (A4)

where ${\bf w}$ is the flow velocity, the particle’s momentum, speed, and
pitch-angle are all measured in the flow frame, the fields are evaluated at
the particle’s position in the observer’s frame (note that if the fields are
evaluated at the GC’s position, then the $({\bf\nabla}\times{\bf
b})_{\parallel}$ and $({\bf w}\times\partial{\bf b}/\partial t)_{\parallel}$
terms will be absent; Rossi & Olbert, 1970; Burger et al., 1985; Wijsen,
2020), $m$ is the particle’s relativistic mass in the flow frame, and the
subscripts $\perp$ and $\parallel$ indicate directions perpendicular and
parallel to ${\bf b}$, respectively. It is important to notice that this
equation is derived assuming that the fields do not vary much over a Larmor
radius, that is $B/|\partial B_{i}/\partial x_{j}|\gg R_{L}$ and $w/|\partial
w_{i}/\partial x_{j}|\gg R_{L}$ (Rossi & Olbert, 1970; Burger et al., 1985;
Wijsen, 2020).

This equation is normally defined with respect to a smooth background magnetic
field, but consider what would happen if it were defined with respect to the
total (background and turbulent) magnetic field. This approach might raise
some problems: 1) the instantaneous GC position might not be well defined in a
turbulent magnetic field, especially if the turbulence is strong, and this
instantaneous GC velocity might not be valid (see the discussions of Burger et
al., 1985); 2) the pitch-angle might also be ill defined if it is with respect
to the total magnetic field. Nonetheless, one can hope that if suitable
averages are taken over the magnetic field and if the turbulence is not too
strong, it would yield well behaved quantities. For weak turbulence
conditions, it will then be explicitly assumed that the pitch-angle is defined
with respect to the background magnetic field.

Assume that the fields can be written as the sum of a background and
fluctuating component, i.e. ${\bf B}={\bf B}_{0}+\delta{\bf B}$, where
$\langle{\bf B}\rangle={\bf B}_{0}$ for a suitable average, and that the
fluctuations are transverse, such that ${\bf B}_{0}\cdot\delta{\bf B}=0$. The
total magnetic field magnitude is then $B=B_{0}\sqrt{1+\delta
B^{2}/B_{0}^{2}}$, where $\delta B^{2}\ll B_{0}^{2}$ for weak turbulence, and
with velocity fluctuations, the flow velocity would be ${\bf w}={\bf
w}_{0}+\delta{\bf w}$. Substituting this into the GC velocity, allows it to be
decomposed into an average and perturbed motion, i.e. $\langle{\bf v}_{\rm
gc}\rangle_{\varphi}={\bf v}_{\rm gc}+\delta{\bf v}_{\rm gc}$,

$\displaystyle{\bf v}_{\rm gc}$ $\displaystyle=\xi\mu v\,{\bf b}_{0}+{\bf
w}_{0}+\frac{\xi^{2}m}{qB_{0}}\,{\bf b}_{0}\times\left[\frac{\partial{\bf
w}_{0}}{\partial t}+(\xi\mu v\,{\bf b}_{0}+{\bf w}_{0})\cdot{\bf\nabla}{\bf
w}_{0}\right]+\frac{\xi^{3}\mu p}{qB_{0}}\,{\bf
b}_{0}\times\left(\frac{\partial{\bf b}_{0}}{\partial t}+{\bf
w}_{0}\cdot{\bf\nabla}{\bf b}_{0}\right)+$
$\displaystyle\;\;\;\;\;\frac{\xi^{2}vp}{qB_{0}}\left\\{\mu^{2}({\bf\nabla}\times{\bf
b}_{0})_{\perp}+\frac{1-\mu^{2}}{2}\left[({\bf\nabla}\times{\bf
b}_{0})_{\parallel}+\frac{{\bf
b}_{0}\times{\bf\nabla}B_{0}}{B_{0}}\right]\right\\}+$
$\displaystyle\;\;\;\;\;\frac{\xi^{2}vp}{c^{2}qB_{0}}\left\\{\mu^{2}\left({\bf
w}_{0}\times\frac{\partial{\bf b}_{0}}{\partial
t}\right)_{\perp}+\frac{1-\mu^{2}}{2}\left[\left({\bf
w}_{0}\times\frac{\partial{\bf b}_{0}}{\partial
t}\right)_{\parallel}+\frac{\partial B_{0}}{\partial t}\frac{{\bf
b}_{0}\times{\bf w}_{0}}{B_{0}}\right]\right\\}$ $\displaystyle\delta{\bf
v}_{\rm gc}$ $\displaystyle=\xi\mu v\,{\bf b}_{1}+\delta{\bf
w}+\frac{\xi^{2}m}{qB_{0}}{\bf b}_{0}\times\left[(\xi\mu v\,{\bf
b}_{1}+\delta{\bf w})\cdot{\bf\nabla}{\bf
w}_{0}\right]+\frac{\xi^{2}m}{qB_{0}}\,{\bf
b}_{1}\times\left\\{\frac{\partial{\bf w}_{0}}{\partial t}+\left[\xi\mu v({\bf
b}_{0}+{\bf b}_{1})+{\bf w}_{0}+\delta{\bf w}\right]\cdot{\bf\nabla}{\bf
w}_{0}\right\\}+$ $\displaystyle\;\;\;\;\;\frac{\xi^{2}m}{qB_{0}}({\bf
b}_{0}+{\bf b}_{1})\times\left\\{\frac{\partial\delta{\bf w}}{\partial
t}+\left[\xi\mu v({\bf b}_{0}+{\bf b}_{1})+{\bf w}_{0}+\delta{\bf
w}\right]\cdot{\bf\nabla}\delta{\bf w}\right\\}+\frac{\xi^{3}\mu
p}{qB_{0}}\,{\bf b}_{0}\times\left(\delta{\bf w}\cdot{\bf\nabla}{\bf
b}_{0}\right)+$ $\displaystyle\;\;\;\;\;\frac{\xi^{3}\mu p}{qB_{0}}\,{\bf
b}_{1}\times\left[\frac{\partial{\bf b}_{0}}{\partial t}+({\bf
w}_{0}+\delta{\bf w})\cdot{\bf\nabla}{\bf b}_{0}\right]+\frac{\xi^{3}\mu
p}{qB_{0}}({\bf b}_{0}+{\bf b}_{1})\times\left[\frac{\partial{\bf
b}_{1}}{\partial t}+({\bf w}_{0}+\delta{\bf w})\cdot{\bf\nabla}{\bf
b}_{1}\right]+$ $\displaystyle\;\;\;\;\;\frac{\xi\mu
p}{qB_{0}}\left[\frac{\partial\xi}{\partial t}+({\bf w}_{0}+\delta{\bf
w})\cdot{\bf\nabla}\xi\right]({\bf b}_{0}\times{\bf
b}_{1})+\frac{\xi^{2}vp}{qB_{0}}\left\\{\mu^{2}({\bf\nabla}\times{\bf
b}_{1})_{\perp}+\frac{1-\mu^{2}}{2}\left[({\bf\nabla}\times{\bf
b}_{1})_{\parallel}+\frac{{\bf
b}_{1}\times{\bf\nabla}B_{0}}{B_{0}}\right]\right\\}+$
$\displaystyle\;\;\;\;\;\frac{\xi
vp}{qB_{0}}\left\\{\mu^{2}\left[{\bf\nabla}\xi\times({\bf b}_{0}+{\bf
b}_{1})\right]_{\perp}+\frac{1-\mu^{2}}{2}\left[{\bf\nabla}\xi\times({\bf
b}_{0}+{\bf
b}_{1})\right]_{\parallel}+\frac{1-\mu^{2}}{2}\,{\bf\nabla}\xi\times({\bf
b}_{0}+{\bf b}_{1})\right\\}+$
$\displaystyle\;\;\;\;\;\frac{\xi^{2}vp}{c^{2}qB_{0}}\left\\{\mu^{2}\left(\delta{\bf
w}\times\frac{\partial{\bf b}_{0}}{\partial
t}\right)_{\perp}+\frac{1-\mu^{2}}{2}\left[\left(\delta{\bf
w}\times\frac{\partial{\bf b}_{0}}{\partial
t}\right)_{\parallel}+\frac{\partial B_{0}}{\partial t}\frac{{\bf
b}_{0}\times\delta{\bf w}}{B_{0}}\right]\right\\}+$
$\displaystyle\;\;\;\;\;\frac{\xi^{2}vp}{c^{2}qB_{0}}\left\\{\mu^{2}\left[({\bf
w}_{0}+\delta{\bf w})\times\frac{\partial{\bf b}_{1}}{\partial
t}\right]_{\perp}+\frac{1-\mu^{2}}{2}\left[({\bf w}_{0}+\delta{\bf
w})\times\frac{\partial{\bf b}_{1}}{\partial
t}\right]_{\parallel}+\frac{1-\mu^{2}}{2}\frac{\partial B_{0}}{\partial
t}\frac{{\bf b}_{1}\times({\bf w}_{0}+\delta{\bf w})}{B_{0}}\right\\}+$
$\displaystyle\;\;\;\;\;\frac{\xi vp}{c^{2}qB_{0}}\frac{\partial\xi}{\partial
t}\left\\{\mu^{2}\left[({\bf w}_{0}+\delta{\bf w})\times({\bf b}_{0}+{\bf
b}_{1})\right]_{\perp}+\frac{1-\mu^{2}}{2}\left[({\bf w}_{0}+\delta{\bf
w})\times({\bf b}_{0}+{\bf b}_{1})\right]_{\parallel}+\frac{1-\mu^{2}}{2}({\bf
w}_{0}+\delta{\bf w})\times({\bf b}_{0}+{\bf b}_{1})\right\\},$

where ${\bf b}_{0}={\bf B}_{0}/B_{0}$, ${\bf b}_{1}=\delta{\bf B}/B_{0}$, and
$\xi=1/\sqrt{1+\delta B^{2}/B_{0}^{2}}$. The average velocity is very similar
to Eq. A and reduce to its usual definition in the absence of turbulence
($\delta B^{2}/B_{0}^{2}=0$), while the perturbed velocity will vanish in the
absence of turbulence ($\delta{\bf B}=\delta{\bf w}={\bf 0}$).

Engelbrecht et al. (2017) defined the drift velocity, in the isotropic limit
and in the absence of electric fields, as a temporal average of the quantity
$pv{\bf\nabla}\times({\bf b}/B)/3q$. At this point it should be realised that
if the GC velocity is calculated with respect to the total magnetic field,
then the gyro- and/or pitch-angle-average will also imply that a suitable
average must be taken of the fluctuating fields: 1) the gyro-average for a
single particle can be interpreted as a temporal average over a gyration; 2)
the gyro-average and the pitch-angle average of a distribution of particles
imply an ensemble average because particles of different phases and pitch-
angles will sample different parts/realisations of the fluctuations. With an
ergodic assumption, the temporal average and ensemble average are equivalent.
The assumption that the fields do not vary much over a Larmor radius, also
implies, together with the averaging procedure, that turbulence with
scales/wavelengths less than the Larmor radius, should be averaged out.

Taking an appropriate average of the above equations then, such that
$\langle\delta{\bf B}\rangle=\langle\delta{\bf w}\rangle={\bf 0}$, yields

$\displaystyle{\bf V}_{\rm gc}$ $\displaystyle\approx\sqrt{f_{s}}\,\mu v\,{\bf
b}_{0}+{\bf w}_{0}+f_{s}\left[{\bf v}_{w}(\mu)+\sqrt{f_{s}}\,{\bf
v}_{t}(\mu)+{\bf v}_{d}(\mu)+\kappa_{A}^{\rm ws}(\mu)({\bf\nabla}\times{\bf
b}_{0})_{\parallel}+{\bf v}_{a}(\mu)+\frac{\kappa_{A}^{\rm
ws}(\mu)}{c^{2}}\left({\bf w}_{0}\times\frac{{\bf b}_{0}}{\partial
t}\right)_{\parallel}\right]+$
$\displaystyle\;\;\;\;\;\frac{3(1+\mu^{2})}{4}\frac{\kappa_{A}^{\rm
ws}}{c^{2}}\frac{\partial f_{s}}{\partial t}({\bf w}_{0}\times{\bf
b}_{0})+\frac{3(1+\mu^{2})}{4}({\bf\nabla}f_{s}\times\kappa_{A}^{\rm ws}{\bf
b}_{0}),$ (A5)

where higher order correlations, such as $\langle{\bf b}_{1}\times\delta{\bf
w}\rangle$, were neglected, $f_{s}=1/(1+\langle\delta B^{2}\rangle/B_{0}^{2})$
is identified as the drift suppression factor here,

$\displaystyle{\bf v}_{w}(\mu)$ $\displaystyle=\frac{3\kappa_{A}^{\rm
ws}}{v^{2}}{\bf b}_{0}\times\left[\frac{\partial{\bf w}_{0}}{\partial
t}+(\sqrt{f_{s}}\mu v\,{\bf b}_{0}+{\bf w}_{0})\cdot{\bf\nabla}{\bf
w}_{0}\right]$ $\displaystyle{\bf v}_{t}(\mu)$
$\displaystyle=3\frac{\mu}{v}\kappa_{A}^{\rm ws}{\bf
b}_{0}\times\left(\frac{\partial{\bf b}_{0}}{\partial t}+{\bf
w}\cdot{\bf\nabla}{\bf b}_{0}\right)$ $\displaystyle{\bf v}_{d}(\mu)$
$\displaystyle=3\mu^{2}\kappa_{A}^{\rm ws}({\bf\nabla}\times{\bf
b}_{0})_{\perp}+\kappa_{A}^{\rm ws}(\mu)\frac{{\bf
b}_{0}\times{\bf\nabla}B_{0}}{B_{0}}$ $\displaystyle{\bf v}_{a}(\mu)$
$\displaystyle=\frac{1}{c^{2}}\left[3\mu^{2}\kappa_{A}^{\rm ws}\left({\bf
w}_{0}\times\frac{\partial{\bf b}_{0}}{\partial
t}\right)_{\perp}+\kappa_{A}^{\rm ws}(\mu)\frac{\partial B_{0}}{\partial
t}\frac{{\bf b}_{0}\times{\bf w}}{B_{0}}\right]$

is the drift velocity caused by the temporal and spatial variation of the flow
velocity, the drift caused by the temporal and spatial variation of the
background magnetic field, the gradient and curvature drift, and the drift
velocity caused by the advection of the magnetic field by the flow,
respectively, the $({\bf\nabla}\times{\bf b}_{0})_{\parallel}$ and $({\bf
w}_{0}\times{\bf b}_{0}/\partial t)_{\parallel}$ terms will fall away if the
fields are evaluated at the GC’s position (Rossi & Olbert, 1970; Burger et
al., 1985; Wijsen, 2020), and

$\kappa_{A}^{\rm ws}(\mu)=\frac{1-\mu^{2}}{2}\frac{pv}{qB_{0}}$ (A6)

is the pitch-angle dependent weak scattering drift coefficient (notice that
this reduces to Eq. A3 when averaged over pitch-angle).

Notice that the last two terms in Eq. A arise due to spatially varying
turbulence conditions. Unfortunately, Eq. A cannot be simplified further, but
it reduces to the isotropic case,

$\left\langle{\bf V}_{\rm gc}\right\rangle_{\mu}={\bf
w}_{0}+f_{s}\frac{3\kappa_{A}^{\rm ws}}{v^{2}}{\bf
b}_{0}\times\left(\frac{\partial{\bf w}_{0}}{\partial t}+{\bf
w}_{0}\cdot{\bf\nabla}{\bf w}_{0}\right)+{\bf
w}_{0}\times\frac{\partial}{\partial t}\left[f_{s}\frac{\kappa_{A}^{\rm
ws}}{c^{2}}{\bf b}_{0}\right]+{\bf\nabla}\times(f_{s}\kappa_{A}^{\rm ws}{\bf
b}_{0}),$

if averaged over pitch-angle (with some additional terms compared to the usual
expressions in Rossi & Olbert, 1970; Burger et al., 1985, due to the
transformation to the flow frame). It is also interesting to note that a part
of the streaming term ($\mu v\,{\bf b}_{0}$) and the effect of a time
dependent magnetic field is zero for the pitch-angle averaged GC drift
velocity, as a stationary field is normally explicitly assumed in its
derivation (see e.g. Burger et al., 1985) although it need not be.

##  Appendix B Predictions for Solar Energetic Particles

To fully investigate the implications of the pitch-angle dependent turbulence-
reduced drift coefficient derived in Sec. 2 requires realistic inputs for
turbulence quantities such as the magnetic variance, which in turn also
influence the behaviour of the perpendicular diffusion coefficient required to
evaluate Eq. 6. In order to do so in the very inner heliosphere, the present
study utilizes values for the various required turbulence quantities
calculated by the Zank et al. (2017) TTM, as solved by Adhikari et al. (2020).
An in-depth discussion of this model is beyond the scope of this work, but the
interested reader is invited to consult, e.g., Zank et al. (2012, 2017, 2018)
and Adhikari et al. (2017) for more background on this and related models.
Exactly the same boundary values and assumptions as discussed by Adhikari et
al. (2020), reporting model outputs in good agreement with Parker Solar Probe
observations of several turbulence quantities at radial distances between
$35.55$ and $131.64$ solar radii, are employed here.

Figure 6: Top left: Variance of the magnetic turbulence as a function of
radius calculated by the turbulence transport model of Adhikari et al. (2020)
(solid lines), together with some scaling laws used in the outer heliosphere
for these quantities normalized to measured values at Earth (dashed lines).
Bottom left: Correlation lengths of the turbulence as a function of radius
calculated by the turbulence transport model of Adhikari et al. (2020) (blue
and red solid lines). The length scales at which the inertial range begins for
the slab (dashed cyan) and 2D (dashed orange) turbulence spectra are shown for
some assumed scaling laws of these quantities normalized to measured values at
Earth. Right panels: Various quantities used to calculate the drift reduction
factor in Fig. 1 for different theories of perpendicular diffusion, including
the parameter $\eta=(\lambda_{\perp}^{0}/R_{L})^{2}(\delta B^{2}/B_{0}^{2})$
(dashed lines), as a function of radius (top right) for $100~{}{\rm MeV}$
protons and energy (bottom right) for protons at Earth.

Fig. 6 shows magnetic variances (top panel) and correlation scales (bottom
panel) yielded by the Adhikari et al. (2020) model, as function of radial
distance. Note that, while this TTM yields several different turbulence
quantities, only those directly influencing the MFP expressions discussed in
Sec. 3 are shown in this figure. At the smallest radial distances considered,
both slab and 2D variances decrease steeply, before assuming an approximately
$r^{-3}$ radial scaling, as reported by Adhikari et al. (2020). Furthermore,
the values that these quantities assume relative to one another remain roughly
in proportion to each other. For the purposes of comparison in Sec. 3, the top
panel of Fig. 6 also shows simple radial scalings for the total, as well as
the slab and 2D variances, as employed in previous studies (see, e.g., Burger
et al., 2008; Moloto et al., 2018; Engelbrecht, 2019b), extrapolated to $0.05$
${\rm au}$, and with 2D and slab variances scaled following the 80:20 ratio of
the levels of these spectra reported by Bieber et al. (1994) for observations
at $1$ ${\rm au}$. A radial scaling of $r^{-2.4}$ is chosen here for the total
magnetic variance, normalised to a value of $10$ nT2 at Earth (see, e.g.,
Smith et al., 2006; Engelbrecht & Burger, 2013), as this roughly corresponds
to the radial dependence of variances calculated from Voyager observations
beyond $1$ ${\rm au}$ (see, e.g., Zank et al., 1996; Smith et al., 2001). It
should be noted that, due to the differing radial dependencies of the simple
scalings and the TTM, resulting variances are considerably different at small
radial distances ($<0.25$ ${\rm au}$), which would lead to potentially
significant differences in transport coefficients.

The slab and 2D correlation scales yielded by the Adhikari et al. (2020) TTM
show, at the smallest radial distances, an increase with radial distance,
which becomes less steep beyond $\sim 0.2$ ${\rm au}$, corresponding with the
decreasing behaviour seen in the magnetic variances. The bottom panel of Fig.
6 also shows simple radial scalings for these correlation lengths, normalised
to observational values reported by Weygand et al. (2011) at Earth, with the
radial dependence of $\sqrt{r}$ again chosen to correspond to that displayed
by Voyager observations beyond $1$ ${\rm au}$ (Smith et al., 2001). As with
the variances, the greatest discrepancies between the TTM outputs and the
simple scalings exists at the smallest radial distances considered. However,
approaching Earth, the scalings and TTM outputs are similar in value, and
display similar radial dependencies. In what is to follow in this appendix,
therefore, the outputs yielded by the Adhikari et al. (2020) TTM are employed
as inputs for Eq. 6 and the various expressions are used to model the
perpendicular diffusion coefficients in Sec. 3.

Figure 7: Comparison of the pitch-angle dependent drift coefficient of this
work (Eq. 7) when averaged over pitch-angle and the isotropic drift
coefficient of Engelbrecht et al. (2017) (Eq. 3; blue). Different functional
forms for the pitch-angle dependent perpendicular diffusion coefficient are
considered as a function of $\eta=(\lambda_{\perp}^{0}/R_{L})^{2}(\delta
B^{2}/B_{0}^{2})$. The shaded region indicates the range of $\eta$ values
expected from different theories in the inner heliosphere (see the right panel
of Fig. 6).

The top two panels of Fig. 1 show the MFPs corresponding to the various
expressions for $D_{\perp}$ discussed in Sec. 3 along with the parallel MFP
and Larmor radius, as function of radial distance for $100$ ${\rm MeV}$
protons (left panel), and as function of kinetic energy at Earth (right
panel), again for protons, and including the Palmer (1982) consensus range
values for parallel and perpendicular MFPs at $1$ ${\rm au}$. When evaluating
these length scales as function of radial distance, the turbulence quantities
computed using the Adhikari et al. (2020) TTM are used. Therefore, plots
extend only from $0.165$ ${\rm au}$ to $0.6$ ${\rm au}$, the radial extent
over which those authors solve that model. At Earth, observational values for
turbulence quantities reported by Weygand et al. (2011) and Smith et al.
(2006) are employed. These quantities are known to vary with solar activity
(see, e.g., Zhao et al., 2018; Engelbrecht & Wolmarans, 2020), and, as such,
solar minimum values are employed here. In order to discuss the behaviour of
these length scales, it is also shown how the ratio of total (slab+2D)
variance to HMF magnitude, the ratio of perpendicular MFP to maximal Larmor
radius, and the quantity $(\lambda_{\perp}^{0}/R_{L})^{2}(\delta
B^{2}/B_{0}^{2})$, vary as function of radial distance and kinetic energy in
the respective middle left and right panels of Fig. 1. As function of radial
distance, all $100$ ${\rm MeV}$ proton MFPs, as well as $R_{L}$, display a
relatively flat radial dependence, in line with that of $\delta B^{2}/B^{2}$,
except at the very smallest radial distances considered, where the steep
decrease in correlation scales seen in Fig. 6 strongly influences the
behaviour of these MFPs. At $100$ ${\rm MeV}$, all perpendicular MFPs remain
well below the parallel MFP, with the UNLT expression yielding the largest
values for $\lambda_{\perp}^{0}$ and the FLRW yielding the smallest values,
comparable to $R_{L}$.

At Earth, the parallel MFP remains larger than the Palmer (1982) consensus
range, due to the fact that solar minimum values for turbulence inputs were
chosen (for discussions of this, see Engelbrecht & Burger, 2013; Zhao et al.,
2018; Moloto et al., 2018). The NLGC expression for $\lambda_{\perp}^{0}$
remains closest to the Palmer (1982) consensus range for this quantity, with
the UNLT result being considerably larger, and displaying a steeper energy
dependence due to a stronger dependence on $\lambda_{\parallel}$. The FLRW
theory still yields the smallest perpendicular MFP, which is energy
independent, and smaller than the Larmor radius above $\sim 60$ ${\rm MeV}$.
From the middle panels of Fig. 1, the ratios of $\lambda_{\perp}^{0}$ to
maximal Larmor radius for all scattering theories, except the FLRW, remain
larger than unity as function of kinetic energy and radial distance, implying
that perpendicular diffusion would play a significant role in the transport of
these protons, relative to drift effects. For the FLRW result, this ratio is
$\sim 1$ for $100$ ${\rm MeV}$ protons, and decreases steadily as function of
radial distance. The quantity $(\lambda_{\perp}^{0}/R_{L})^{2}(\delta
B^{2}/B_{0}^{2})$ can also be seen to vary greatly in magnitude, depending on
the scattering theory from which $\lambda_{\perp}^{0}$ is derived, and
decreasing as function of kinetic energy, while remaining relatively constant
as function of radial distance. Incidentally, the relatively small ($<1$)
values for the ratio of $\delta B^{2}/B_{0}^{2}$ implies that the condition of
low turbulence levels imposed by the assumptions implicit in using the Bieber
& Matthaeus (1997) approach to deriving the turbulent drift reduction factor
is satisfied for the region of the very inner heliosphere considered here.

The bottom two panels of Fig. 1 show the drift reduction factors calculated
using the isotropic perpendicular MFPs discussed above as function of radial
distance (left panel) and kinetic energy (right panel). As expected from the
form of Eq. 3, larger values for $\lambda_{\perp}^{0}$ lead to smaller values
for $f_{s}$ and hence, a greater reduction in drift effects. As function of
radial distance, the $100$ ${\rm MeV}$ drift reduction function displays a
relatively flat radial dependence, only showing a moderate upturn at the
smallest radial distances considered here for the UNLT perpendicular MFP. For
the FLRW MFP, almost no drift reduction can be seen at this energy. This is
not, however, the case for all energies, as $f_{s}$ calculated using the FLRW
MFP does begin to drop below $1$ at energies below $\sim 100$ ${\rm MeV}$. The
drift reduction factors for the other MFP expressions considered here also
display clear energy dependencies, with the UNLT result remaining well below
unity over the entire energy range considered here.

Figure 8: Left: The pitch-angle dependent drift suppression factor (Eq. 6) for
$100\,\mathrm{MeV}$ protons as a function of radius using the quantities in
Fig. 6. Right: The isotorpic drift suppression factor (Eq. 3) for protons as a
function of radius and kinetic energy using the quantities in Fig. 6.
Perpendicular diffusion is assumed to be described by the non-linear guiding
centre theory (Eq. 9) in both panels.

The question naturally arises as to when the pitch angle average of Eq. 7
would correspond to what is expected from Eq. 3. Such a comparison can be done
by integrating Eq. 7, under the assumption of Eq. 8, where various pitch-angle
dependencies are assumed. Fig. 7 shows the results of these integrations
plotted alongside the isotropic result (Eq. 3), as function of
$\eta=(\lambda_{\perp}^{0}/R_{L})^{2}(\delta B^{2}/B_{0}^{2})$. The shaded
region on the plot indicates the expected range of values for $\eta$,
calculated using the perpendicular MFPs corresponding to the various
expressions for $D_{\perp}$ discussed above as function of position and energy
(see Fig. 1), as well as the outputs from the Adhikari et al. (2020) TTM. For
small values of $\eta$, and hence low levels of turbulence, the isotropic and
pitch-angle dependent drift reduction factors are very similar. Beyond
$\eta\approx 1$, however, results calculated with $g(\mu)=\mu^{2}$ and
$g(\mu)=|\mu|$ (orange and black lines, respectively) begin to deviate
considerably from the isotropic case (blue line). It can therefore be
concluded that the smaller perpendicular MFPs yielded by the NLGC and FLRW
theories (as employed here) will lead to smaller values of $\eta$ that, in
turn, implies greater agreement between the pitch-angle averaged approach to
the turbulent reduction of drift effects and the isotropic approach. Larger
perpendicular MFPs would by implication then lead to a drift reduction factor
that is larger than in the isotropic case, but still smaller than one,
implying that drift reduction would still occur. Intriguingly, should it be
assumed that $g(\mu)=\mu^{2}/3+2/9$, as motivated by Engelbrecht (2019b) based
on the numerical test particle simulations of Qin & Shalchi (2014), the pitch-
angle averaged drift reduction factor (cyan line) follows the isotropic result
closely.

Fig. 2 and Fig. 8 characterise the pitch-angle, energy, and radial dependence
of the turbulence-reduced drift reduction factor. The left panel of Fig. 2
shows the pitch-angle dependence of Eq. 6 for $100$ ${\rm MeV}$ protons and
typical turbulence quantities measured at Earth. It is clear that field
aligned particles ($\mu=\pm 1$) will have a significant reduction in their
drift velocities for all of the different pitch-angle dependencies considered
for $D_{\perp}(\mu)$. The behaviour of $f_{s}(\mu)$ towards $\mu=0$ depends
critically on the theory used for $D_{\perp}(\mu)$. FLRW yields a drift
reduction factor mostly $>0.1$, in agreement with Fig. 1, while $f_{s}(\mu)$
becomes narrower for NLGC and UNLT. Note that any perpendicular diffusion
coefficient which is zero at $\mu=0$ will have $f_{s}(0)=1$, and hence
particles will experience full drift effects at this pitch-angle. For the form
of $g(\mu)=\mu^{2}/3+2/9$, even particles with $90^{\circ}$ pitch-angles will
experience drift reductions. However, there does not seem to be any a priori
reason that a particle with a $90^{\circ}$ pitch-angle should not experience
drift reductions, because it is sampling the transverse turbulence regardless
of its pitch-angle. Both the pitch-angle and energy dependence of Eq. 6 are
shown in the right panel of Fig. 2 at Earth for NLGC. The drift reduction
factor $f_{s}$ becomes more pronounced at low energies, for a fixed pitch-
angle (except at $\mu=0$), causing $f_{s}(\mu)$ to become narrower in $\mu$ at
low energies.

The left panel of Fig. 8 shows the pitch-angle and radial dependence of the
drift suppression factor for $100$ ${\rm MeV}$ protons assuming the NLGC
result for the perpendicular diffusion (Eq. 9). The shape of $f_{s}(\mu)$ does
not change much with radius, although an upturn can be seen in the very inner
heliosphere, in agreement with what can be seen in Fig. 1. The isotropic drift
suppression factor’s energy and radial dependence is shown in the right panel
of Fig. 8, again for NLGC. The radial dependence is again weaker than the
energy dependence, but shows the upturn at small radial distances. It is clear
that low energy particles will experience much more drift reduction than high
energy particles. Similar plots can, of course, also be made for FLRW or UNLT,
but the qualitative behaviour of the drift reduction factor will still be the
same: field aligned particles will experience more drift reductions, low
energy particles will experience less drifts, and drift reduction factors will
have a weak radial dependence.

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