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

As the present study seeks to derive insights as to low-energy electron parallel and perpendicular MFPs, great care needs to be taken as to the spacecraft data with which the results of the above model need to be compared. Specifically, one would need to distinguish between observations taken during periods of good and poor magnetic connectivity. This has been done by Vogt et al. (2020, 2022), and the present study employs the same data sets considered in those papers, and shown in Figure 1 as function of kinetic energy, alongside the flyby data used to construct the source spectrum employed in the transport model (Equation (5)), and fits to data corresponding to periods of good and bad magnetic connectivity (dotted lines) with which computed Jovian intensities are to be compared (see below). For periods of good magnetic connectivity, observations taken by Ulysses (KET) during 1991 January (Heber et al. 2005), ISEE 3 from 1979–1984 (Moses 1987) and Voyager 1 in 1977 (Nndanganeni & Potgieter 2018) were considered. For periods of poor magnetic connectivity, ISEE 3 data from 1979–1984 (Moses 1987) as well as SOHO-EPHIN observations taken from 2007 June to 2008 July (Kühl et al. 2013) were considered. It should be noted that the requirement of finding sufficient data corresponding to periods of good and poor magnetic connection leads to model comparisons with observations taken over a relatively broad span of time, corresponding to significant variations in transport conditions in the heliosphere. These variations can be due to solar-cycle-dependent changes in, for example, the HMF; solar cycle dependent changes in the turbulence conditions, which would influence Jovian electron transport parameters (see, e.g., Zhao et al. 2018; Caballero-Lopez et al. 2019; Engelbrecht & Wolmarans 2020); or transient structures like corotating interaction regions, which could also affect the transport of these particles (Jokipii 1999). Given these varying conditions, the approach taken in this study is to model background plasma quantities in the Jovian electron transport model (described below) in as general way as possible, to broadly reproduce average conditions in the solar ecliptic plane. Then, to avoid a particular bias, model diffusion coefficients are specified with general, varying power-law dependences on particle rigidity and radial distance. Such an approach would then imply that radial and rigidity dependences of transport coefficients that yield good fits to the data sets listed above would to a reasonable degree reflect the influence of the average conditions Jovian electrons encountered during this period.

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The Jovian electron transport model as used in this study is based on the Compute Unified Device Architecture (CUDA) stochastic differential equation (SDE) solver introduced by Dunzlaff et al. (2015), as applied by Vogt et al. (2020) and Vogt et al. (2022). The equivalent SDE representation of the Parker (1965) transport equation (TPE) employed here is derived by Strauss et al. (2011), and given by

$$\begin{eqnarray}\begin{array}{rcl}{dr} & = & \left[\displaystyle \frac{1}{{r}^{2}}\displaystyle \frac{\partial }{\partial r}({r}^{2}{\kappa }_{{rr}})+\displaystyle \frac{1}{r\sin \theta }\displaystyle \frac{\partial {\kappa }_{\phi r}}{\partial \phi }-{u}_{\mathrm{SW}}\right]{dt}\\ & & +\sqrt{2{\kappa }_{{rr}}-\displaystyle \frac{2{\kappa }_{r\phi }^{2}}{{\kappa }_{\phi \phi }}}d{\omega }_{r}+\displaystyle \frac{\sqrt{2}{\kappa }_{\phi r}}{\sqrt{{\kappa }_{\phi \phi }}}d{\omega }_{\phi }\\ d\theta & = & \left[\displaystyle \frac{1}{{r}^{2}\sin \theta }\displaystyle \frac{\partial }{\partial \theta }(\sin \theta {\kappa }_{\theta \theta })\right]{dt}+\displaystyle \frac{\sqrt{2{\kappa }_{\theta \theta }}}{r}d{\omega }_{\theta }\\ d\phi & = & \left[\displaystyle \frac{1}{{r}^{2}{\sin }^{2}\theta }\displaystyle \frac{\partial {\kappa }_{\phi \phi }}{\partial \phi }+\displaystyle \frac{1}{{r}^{2}\sin \theta }\displaystyle \frac{\partial }{\partial r}(r{\kappa }_{r\phi })\right]{dt}\\ & & +\displaystyle \frac{\sqrt{2{\kappa }_{\phi \phi }}}{r\sin \theta }d{\omega }_{\phi }\\ {dE} & = & \left[\displaystyle \frac{1}{3{r}^{2}}\displaystyle \frac{\partial }{\partial r}({r}^{2}{u}_{\mathrm{SW}}){\rm{\Gamma }}E\right]{dt},\end{array}\end{eqnarray} \tag{ 1 }$$

with Γ = (E + 2E₀)/(E + E₀), and $d\omega =\zeta \sqrt{{dt}}$ a Wiener process, with ζ a stochastic element as a vector of Gaussian-distributed random numbers. Note that dt = 0.0001 day denotes an increment in time, and that a time-backward approach is taken to solving the above equations (see, e.g., Pei et al. 2010; Kopp et al. 2012; Engelbrecht & Burger 2015; Strauss & Effenberger 2017; Moloto et al. 2019) with a constant radial solar wind speed, which is chosen to be u_SW = 400 km s⁻¹, a reasonable assumption in the region of interest to this study (Köhnlein 1996). Elements of the diffusion tensor in spherical coordinates are denoted by κ, and are transformed to field-aligned coordinates according to the approach of Burger et al. (2008), under the assumption of a Parker (1958) HMF geometry. Note that the (solar-cycle-varying) magnitude of the HMF does not directly enter into the solution for Equation (1), unless the diffusion coefficients directly depend on this quantity. Diffusion coefficients in spherical coordinates are then related to coefficients in field-aligned coordinates by

$$\begin{eqnarray}\begin{array}{rcl}{\kappa }_{{rr}} & = & {\kappa }_{\parallel }{\cos }^{2}{\rm{\Psi }}+{\kappa }_{\perp ,r}{\sin }^{2}{\rm{\Psi }}\\ {\kappa }_{r\phi } & = & ({\kappa }_{\perp ,r}-{\kappa }_{\parallel })\cos {\rm{\Psi }}\sin {\rm{\Psi }}\\ {\kappa }_{\theta \theta } & = & {\kappa }_{\perp ,\theta }\\ {\kappa }_{\phi r} & = & ({\kappa }_{\perp ,r}-{\kappa }_{\parallel })\cos {\rm{\Psi }}\sin {\rm{\Psi }}\\ {\kappa }_{\phi \phi } & = & {\kappa }_{\parallel }{\sin }^{2}{\rm{\Psi }}+{\kappa }_{\perp ,r}{\cos }^{2}{\rm{\Psi }},\end{array}\end{eqnarray} \tag{ 2 }$$

where Ψ denotes the HMF winding angle. Although the winding angle has been shown to vary by ∼±5° at 1 au by Smith & Bieber (1991), this variation is relatively small, and therefore no solar cycle dependence is assumed here for this quantity. The diffusion tensor in field-aligned coordinates being given by

$$\begin{eqnarray}{\boldsymbol{K}}^{\prime} =\left[\begin{array}{ccc}{\kappa }_{\parallel } & 0 & 0\\ 0 & {\kappa }_{\perp ,\theta } & 0\\ 0 & 0 & {\kappa }_{\perp ,r}\end{array}\right],\end{eqnarray} \tag{ 3 }$$

where the quantities κ_∥ and κ_⊥ denote diffusion coefficients parallel and perpendicular to the HMF, respectively, and are related to particle mean free paths via κ = v λ/3 (e.g., Shalchi 2009), with v the particle speed. In the present study it is assumed that κ_⊥,r = κ_⊥,θ = κ_⊥. Some studies assume an enhancement in κ_⊥,θ relative to κ_⊥,r (see, e.g., Burger et al. 2000; Ferreira et al. 2001). Such an enhancement was originally assumed to explain lower-than-expected observed galactic cosmic ray latitude gradients (see, e.g., Heber et al. 1996). However, this mechanism is not the only way in which computed latitude gradients may be reduced so as to be more in line with observations. The assumption of a Fisk (1996)-type heliospheric magnetic field (see, e.g., Burger et al. 2008; Sternal et al. 2011), or the natural increase of isotropic perpendicular diffusion coefficients due to the observed (e.g., Forsyth et al. 1996) increase in magnetic variance with latitude (see, e.g., Engelbrecht & Burger 2013a; Moloto et al. 2019; Shen et al. 2021) can also lead to a reduction in latitude gradients. Theoretically, such an enhancement in the perpendicular diffusion coefficient κ_⊥,θ relative to κ_⊥,r may be the result of an underlying nonaxisymmetry in 2D HMF turbulence (see, e.g., Ruffolo et al. 2008; Strauss et al. 2016), but, as a strong nonaxisymmetry has not been reported in observations of HMF turbulence, the assumption of anisotropic perpendicular diffusion is not made in this study. Note also that the above description does not include the influence of drift effects, as these are known to have a negligible effect on Jovian transport (e.g., Ferreira et al. 2001), and would also be reduced by the enhanced turbulence levels in the very inner heliosphere (see, e.g., Minnie et al. 2007b; Engelbrecht et al. 2017).

Equation (1) is solved iteratively, employing an Euler–Maruyama scheme (see Strauss & Effenberger 2017, and references therein), for N = 50,000 pseudoparticles per energy bin released at Earth, which represent various phase-space trajectories (e.g., Zhang 1999). Pseudoparticles of a specified initial energy are traced iteratively, backwards in time according to Equation (1), until they reach a particular boundary. If these pseudoparticle trajectories reach either the Sun, or the location of the heliopause (here assumed to be at 120 au, see Gurnett et al. 2013), they are discarded, and new pseudoparticle trajectories are initiated at the source, as the aim of this study is only to study electrons of Jovian origin. If they reach the location of the Jovian magnetosphere at 5.2 au, the trajectories are weighted according to their exit energies E ^_exit , as well as the number of pseudoparticle trajectories, by the source spectrum j^s _jov, to give the Jovian differential intensity at Earth:

$$\begin{eqnarray}&&{j}_{\mathrm{jov}}=\displaystyle \frac{\sum _{i=1}^{N}{j}_{\mathrm{jov}}^{s}({E}_{i}^{\mathrm{exit}})}{N},\end{eqnarray} \tag{ 4 }$$

where i denotes a particular pseudoparticle trajectory. The Jovian source spectrum employed here is that proposed by Vogt et al. (2018) based on flyby measurements by the Pioneer 10 spacecraft (Teegarden et al. 1974) and Ulysses data (Heber et al. 2005), and given by

$$\begin{eqnarray*}&&{j}_{\mathrm{jov}}^{s}=1.029\times {10}^{4}{\left(E/{E}_{0}\right)}^{-1.63}{e}^{-E/{E}_{b}},\end{eqnarray*}$$

with E_b = 9.4 MeV, and in units of particles m⁻² s⁻¹ sr⁻¹ MeV⁻¹. The Jovian magnetosphere is here modeled as was done by Strauss et al. (2011), in that it is modeled as a volume enclosed by a solid angle in spherical coordinates. The latitudinal and azimuthal extent of this volume are equal to 200R_j , where R_j = 71,492 km is the Jovian radius, while its radial extents in the sunward and tail directions are 100 and 200R_j , respectively. Strauss et al. (2011) report that the assumed size of the Jovian magnetosphere has little influence on their simulation results, and subsequently these parameters are not varied in this study. For more specific detail as to the solution of SDE pertaining to the study of Jovian electron transport, the interested reader is invited to consult Strauss et al. (2011) and Vogt et al. (2020).

In contrast to previous studies, the MFP expressions employed here are defined to include independent rigidity and nonlinear radial dependences. Furthermore, this approach does not make the simplifying assumption that λ_⊥ is proportional to λ_∥ (see, e.g., Ferreira et al. 2001). In order to study the radial and rigidity dependences of MFPs required to fit observations, as well as the values of these quantities at 1 au so as to compare these with theoretical predictions of the same, this study still implements simplified parametric expressions for these MFPs, motivated by the fact that the transport code here employed is written to perform on graphic processing units GPUs in CUDA, which allows for the vast performance increase required of such a study, but limits the number of definable variables due to memory constraints (see Dunzlaff et al. 2015, for additional information). Subsequently, λ_∥ and λ_⊥ are defined as

$$\begin{eqnarray}&&{\lambda }_{\parallel }={\lambda }_{\parallel ,0}{\left(\displaystyle \frac{P}{{P}_{0}}\right)}^{\alpha }{\left(\displaystyle \frac{r}{{r}_{e}}\right)}^{\beta },\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{\lambda }_{\perp }={\lambda }_{\perp ,0}{\left(\displaystyle \frac{P}{{P}_{0}}\right)}^{\gamma }{\left(\displaystyle \frac{r}{{r}_{e}}\right)}^{\delta },\end{eqnarray} \tag{ 6 }$$

where r_e = 1 au, and P₀ = 1 GV, in an approach similar to that employed by Engelbrecht & Di Felice (2020). Here, parameters α and β specify the respective rigidity and radial dependences of the parallel MFP, which assumes a value of λ_0,∥ (in astronomical units) at Earth, with γ, δ, and λ_0,⊥ playing the same roles with respect to the perpendicular MFP. In the parameter studies employed here, these quantities are varied over a broad range of values, chosen so as to include, as well as extend beyond, various theoretical predictions for these quantities. In terms of MFP values at 1 au, the ranges λ_0,∥ ∈ [0.05, 1.0] au and λ_0,⊥ ∈ [0.001, 0.5] au are chosen, so as to include values reported in previous such studies for the parallel MFP (e.g., Palmer 1982; Bieber et al. 1994; Dröge et al. 2014) as well as for the perpendicular MFP (e.g., Ferrando et al. 1993; Giacalone 1998; Burger et al. 2000; Zhang et al. 2007; Vogt et al. 2020), while taking into account various theoretical predictions and expressions previously used in modeling studies for these MFPs (e.g., Ferreira et al. 2003; Engelbrecht & Burger 2013b; Nndanganeni & Potgieter 2018; Engelbrecht 2019). For the rigidity dependences, it is assumed that α ∈ [−0.1, 0.67] and γ ∈ [−0.1, 0.33], incorporating the flat rigidity dependence expected of λ_∥ for low-energy electrons (e.g., Dröge 1994; Potgieter 1996; Evenson 1998) within a range that includes dependences expected at higher rigidities as well as from different scattering theories (e.g., Dröge 2000; Teufel & Schlickeiser 2002; Shalchi & Schlickeiser 2004; Burger et al. 2008; Engelbrecht & Burger 2010). The range of rigidity dependences for λ_⊥ was chosen following similar reasoning, with the aim of incorporating a flat rigidity dependence within a range that incorporates moderate rigidity dependences expected from various theories (e.g., Shalchi et al. 2004a; Gammon & Shalchi 2017; Dempers & Engelbrecht 2020) as well as prior particle transport studies (e.g., Zhang et al. 2007; Dröge & Kartavykh 2009) and numerical test particle simulations of diffusion coefficients (e.g., Minnie et al. 2007a; Dundovic et al. 2020). Lastly, radial dependences were varied such that β ∈ [0.5, 1.5] and δ ∈ [0.5, 1.5], taking into account various possible radial dependences for these MFPs reported by studies employing turbulence quantities yielded by various turbulence transport models as inputs for several theoretical expressions (see, e.g., Engelbrecht & Burger 2013b; Wiengarten et al. 2016; Chhiber et al. 2017; Zhao et al. 2017; Adhikari et al. 2021). In order to study the various dependencies in Equations (5) and (6), four initial electron energies are chosen, motivated by the fact that for these energies spacecraft data is available for periods of good and bad magnetic connection to the Jovian source as well as at the Jovian source itself. The three higher energies furthermore correspond to the three lowest-energy channels of the ISEE 3 data by Moses (1987). As the results of the parameter studies have to be compared to spectral fits (given by the dotted lines in Figure 1), this choice allows for the most transparent analysis of the deviation of the simulation results from in situ data. The six parameters governing the MFP expressions used in this study, along with the relevant initial energies, are

$$\begin{eqnarray}\begin{array}{rcl}E & = & [0.0025,0.006,0.0085,0.0125]\,\mathrm{GeV}\\ {\lambda }_{\parallel ,0} & = & [0.05,0.106,0.224,0.473,1.0]\,\mathrm{au}\\ {\lambda }_{\perp ,0} & = & [0.001,0.00473,0.02236,0.1057,0.5]\,\mathrm{au}\\ \alpha & = & [-0.1,-0.01,0.01,0.04,0.165,0.67]\\ \gamma & = & [-0.1,-0.01,0.01,0.03,0.1,0.33]\\ \beta & = & [0.5,0.658,0.866,1.397,1.5]\\ \delta & = & [0.5,0.658,0.866,1.397,1.5],\end{array}\end{eqnarray} \tag{ 7 }$$

which form a parameter space of consisting of 90,000 grid points, resulting in 180,000 simulations for comparison with observational points both well and badly connected to the Jovian source as shown in Figure 1.

In order to evaluate our results, the mean deviation from the data fits in the case of good and bad magnetic connection as displayed by Figure 1 was calculated. Specifically, percentage deviations of model results from the fits to Jovian electron observations for both good and bad magnetic connections for each of the parameter configurations were taken for each of the energies considered. These deviations were then averaged to acquire an average percentage deviation for a particular parameter set. This was to establish whether a particular fit fell within a specific percentage deviation of the data across all energies considered, thereby reducing bias toward a particular energy. Note that, for any particular parameter set, comparisons were made to data fits for both good and poor magnetic connection, so that the goodness-of-fit condition applied to parameters corresponding to both parallel and perpendicular MFPs simultaneously. We found that 26 parameter configurations showed a mean deviation from the eight data points at Earth orbit below 1% (being 0.0289% of the configurations), and 720 configurations showed less than 5% deviation, which corresponds to 0.8% of the configurations we considered. Scatter plots of these parameter sets are displayed in Figure 2 for parameters pertaining to the parallel MFP, and Figure 3 for the perpendicular MFP. Three aspects of these parameters become readily apparent: first, that the number of solutions corresponding to a reasonable fit to Jovian intensity spectra at Earth depends strongly on the assumed fitting criteria; second, that a considerable range of parameters (in combination) can in principle lead to reasonable fits to observed intensities; and third, that there is considerable overlap between best-fit parameters, as evinced by the number of points on Figures 2 and 3 being somewhat less than the total number of points found to satisfy the 1% and 5% fitting criteria.

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The question naturally arises as to whether the distribution of best-fit parameter sets may yield some insight into a preferred radial or rigidity dependence in either the parallel or perpendicular MFPs. To this end, Figures 4 and 5 show histograms constructed from the parameters shown in Figures 2 and 3, for the 1% and 5% fit criteria, respectively. For the 5% fit criteria, it is not readily apparent whether any particular values for the radial and rigidity parameters in Equations (5) and (6) are favored. As to values for the parallel and perpendicular MFPs at Earth, values less than 0.22 au appear to be favored in the case of the former quantity, and less than 0.00473 au in the case of the latter. Parameter distributions change somewhat when the 1% fit criterion is applied. Now there appears to be little in the way of a clear preference for λ_∥,0, but the inclination toward the smaller values for λ_⊥,0 remains the same. However, care should be taken when this latter inclination is considered, given that no clear conclusions can be drawn here as to the rigidity and radial dependences of the perpendicular MFP from this histogram, with the implication that a broad range of radial and rigidity dependences for the perpendicular MFP yield good fits to observations, in conjunction with a fairly broad range of parameters for the parallel MFP. For the case of the parallel MFP there appears to be a preference for a P^0.17 dependence, roughly in line with what is expected from QLT electron parallel MFPs. However, it should be noted that a fair number of points on this particular histogram correspond to all rigidity dependencies considered in this study. Things are somewhat less clear-cut for the parallel MFP radial dependence, with two dependencies, namely, r^0.66 and r^1.14, are clearly favored. These are relatively close to the ∼ r scaling expected of the electron QLT parallel MFP (see, e.g., Engelbrecht 2019), but it must again be noted that a fair number of best-fit parameters correspond to the other scalings considered for the radial dependence of λ_∥.

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For comparison with the results of prior studies, Figure 6 shows the parallel and perpendicular MFPs corresponding to the 10 best-fit solutions presented in this study as a function of rigidity. These 10 best-fitting configurations are listed in Table 1 and still show a degeneracy with the exception of the values for λ_⊥, which are confined to the two lowest possible choices. Also shown are corresponding MFP values reported by Chenette et al. (1977), Bieber et al. (1994), Zhang et al. (2007), Dröge et al. (2014), and Vogt et al. (2020), with the Palmer consensus range. Note that the relatively small number of points reported here is due to the fact that there are overlaps in best-fit values for both the parallel and perpendicular MFPs, something particularly apparent when perpendicular MFPs are considered (bottom panel). In terms of λ_∥, the points reported here vary with roughly an order of magnitude across all four energies considered here but still remain within the range of values reported by Bieber et al. (1994). However, due to this variation, a clear rigidity dependence cannot be easily discerned. In terms of λ_⊥, two values appear to be favored, with the larger value being in fair agreement with the values reported by Chenette et al. (1977) and Zhang et al. (2007). For all energies considered, the perpendicular MFP appears to display a relatively flat rigidity dependence.

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For the electron MFP parallel to the background HMF, various studies have generally favored expressions derived from the QLT of Jokipii (1966) derived under the assumption of composite turbulence (Matthaeus et al. 1990; Bieber et al. 1996), as parallel MFPs calculated from this theory are in reasonable agreement with numerical test particle simulations for relatively low levels of turbulence (e.g., Minnie et al. 2007a), provide results in reasonable agreement with the Palmer (1982) consensus range (Bieber et al. 1994; Vogt et al. 2020), and due to the fact that the use of these expressions in cosmic ray transport codes has been shown to lead to computed galactic cosmic ray proton and electron intensities in reasonable agreement with spacecraft observations (e.g., Engelbrecht & Burger 2013a, 2013b; Engelbrecht 2019; Moloto & Engelbrecht 2020). Comparisons are here made with electron parallel MFPs constructed by Engelbrecht & Burger (2010, 2013b) from the QLT results derived by Teufel & Schlickeiser (2003), assuming the random sweeping (RS) and damping (DT) models of dynamical turbulence. These models were proposed by, e.g., Bieber & Matthaeus (1991), Bieber et al. (1994) to model the time-dependent decorrelation effects on the correlation function from which the spectral tensor, and hence the turbulence power spectrum that serves as a key input to the scattering theories used to derive particle transport coefficients, is derived (see Batchelor 1970), given by

$$\begin{eqnarray}&&{S}_{{ij}}\left({\boldsymbol{k}},t\right)={S}_{{ij}}({\boldsymbol{k}}){\rm{\Gamma }}({\boldsymbol{k}},t),\end{eqnarray} \tag{ 8 }$$

where ${S}_{{ij}}\left({\boldsymbol{k}}\right)$ is the homogeneous spectral tensor, and Γ( k , t) is known as the dynamical correlation function. For the damping model of dynamical turbulence, the latter quantity is assumed to have the form (Bieber & Matthaeus 1991; Bieber et al. 1994)

$$\begin{eqnarray}&&{\rm{\Gamma }}({\boldsymbol{k}},t)=\exp \left(-\alpha \left|{\boldsymbol{k}}\right|{V}_{{\rm{A}}}t\right),\end{eqnarray} \tag{ 9 }$$

with V_A denoting the Alfvén speed. This choice allows for several physical timescales in the turbulence to be modeled indirectly through the quantity $\alpha \in \left[0,1\right]$, a parameter that adjusts the relative strength of the dynamical effects, with a value of unity corresponding to the strongest dynamical effect. An example of this is that if α is interpreted as the ratio of the fluctuating to background magnetic fields, the timescale $\alpha \left|{\boldsymbol{k}}\right|{V}_{{\rm{A}}}$ would correspond to a mean eddy turnover time τ = l/u (Bieber et al. 1994), associated with a turbulent eddy with a characteristic length l and speed u (Batchelor 1970). Bieber et al. (1994) also argue that the exponential decay rate in Equation (9) could be taken to be equivalent to the Kolmogorov decay rate at the appropriate scales. The random sweeping model assumes a Gaussian form for the dynamical correlation function, motivated by the fact that this form appears to fit the observed temporal behavior of decorrelation functions found in various studies (e.g., Chen & Kraichnan 1989; Katul et al. 1995; He & Tong 2011), and is given by (Bieber et al. 1994)

$$\begin{eqnarray}&&{\rm{\Gamma }}({\boldsymbol{k}},t)=\exp \left(-{\alpha }^{2}{k}^{2}{V}_{{\rm{A}}}^{2}{t}^{2}\right).\end{eqnarray} \tag{ 10 }$$

These models have been extensively used in the derivation of cosmic ray MFPs (e.g., Teufel & Schlickeiser 2002, 2003; Shalchi 2014; Shalchi et al. 2004b; Gammon & Shalchi 2017; Dempers & Engelbrecht 2020). The slab modal spectral form employed by Teufel & Schlickeiser (2003) to derive the approximate analytical expressions used by Engelbrecht & Burger (2010, 2013b) to construct the parallel MFP expressions used in those studies is given in terms of the wavenumber parallel to the background HMF by

$$\begin{eqnarray}&&{S}^{\mathrm{slab}}({k}_{\parallel })=\left\{\begin{array}{l}{g}_{o}{k}_{m}^{-s},| {k}_{\parallel }| \leqslant {k}_{m};\\ {g}_{o}| {k}_{\parallel }{| }^{-s},{k}_{m}\leqslant | {k}_{\parallel }| \leqslant {k}_{d};\\ {g}_{1}| {k}_{\parallel }{| }^{-p},| {k}_{\parallel }| \geqslant {k}_{d},\end{array}\right.\end{eqnarray} \tag{ 11 }$$

with s and p the spectral indices of the inertial and dissipation ranges, and k_m and k_d the wavenumbers at which these ranges commence, respectively. Furthermore,

$$\begin{eqnarray}&&{g}_{o}={\left[s+\displaystyle \frac{s-p}{p-1}{\left(\displaystyle \frac{{k}_{m}}{{k}_{d}}\right)}^{s-1}\right]}^{-1}\displaystyle \frac{\delta {B}_{\mathrm{sl}}^{2}{k}_{m}^{s-1}(s-1)}{8\pi },\end{eqnarray} \tag{ 12 }$$

with the subscript "sl" denoting a slab turbulence quantity, such as the magnetic variance δ B², and ${g}_{1}={g}_{0}{k}_{d}^{p-s}$. Assuming the random sweeping model of dynamical turbulence, the electron parallel MFP is given by (Engelbrecht & Burger 2010)

$$\begin{eqnarray}\begin{array}{rcl}{\lambda }_{\parallel } & = & \displaystyle \frac{3s}{\sqrt{\pi }(s-1)}\displaystyle \frac{{R}^{2}}{{k}_{\min }}{\left(\displaystyle \frac{{B}_{o}}{\delta {B}_{\mathrm{slab}}}\right)}^{2}\\ & & \cdot \left[\displaystyle \frac{1}{4\sqrt{\pi }}+\left(\displaystyle \frac{1}{{\rm{\Gamma }}(p/2)}+\displaystyle \frac{1}{\sqrt{\pi }(p-2)}\right)\displaystyle \frac{{b}^{p-2}}{{Q}^{p-s}{R}^{s}}\right.\\ & & +\left.\displaystyle \frac{2}{\sqrt{\pi }(2-s)(4-s)}\displaystyle \frac{1}{{R}^{s}}\right],\end{array}\end{eqnarray} \tag{ 13 }$$

with B_o denoting the background HMF, usually assumed to be a Parker (1958) field. For the damping model of dynamical turbulence, the electron parallel MFP is given by (Engelbrecht & Burger 2013b)

$$\begin{eqnarray}\begin{array}{rcl}{\lambda }_{\parallel } & = & \displaystyle \frac{3s}{\sqrt{\pi }(s-1)}\displaystyle \frac{{R}^{2}}{{k}_{\min }}{\left(\displaystyle \frac{{B}_{o}}{\delta {B}_{\mathrm{slab}}}\right)}^{2}\\ & & \cdot \left[\displaystyle \frac{1}{4\sqrt{\pi }}{+}_{2}{F}_{1}\left(1,\displaystyle \frac{1}{p-1},\displaystyle \frac{p}{p-1};-\displaystyle \frac{\pi a}{{f}_{1}}{Q}^{p-2}\right)\right.\\ & & \times \displaystyle \frac{\sqrt{\pi }a}{{f}_{1}{R}^{s}{Q}^{p-s}}\\ & & \left.+\displaystyle \frac{2}{\sqrt{\pi }(2-s)(4-s)}\displaystyle \frac{1}{{R}^{s}}\right],\end{array}\end{eqnarray} \tag{ 14 }$$

following the notation introduced by Teufel & Schlickeiser (2003), so that

$$\begin{eqnarray}\begin{array}{rcl}a & = & \displaystyle \frac{v}{{\alpha }_{d}{V}_{{\rm{A}}}},\\ b & = & a/2,\\ R & = & {R}_{{\rm{L}}}{k}_{\min },\\ Q & = & {R}_{{\rm{L}}}{k}_{d},\\ {f}_{1} & = & \displaystyle \frac{2}{p-2}+\displaystyle \frac{2}{2-s},\end{array}\end{eqnarray} \tag{ 15 }$$

with R_L the maximal particle Larmor radius, and v its speed.

As with the parallel MFP, many expressions have been proposed in the literature to describe the MFP of charged particles perpendicular to the background HMF (see, e.g., Zank et al. 2004; Shalchi 2009, 2020; Ruffolo et al. 2012; Qin & Zhang 2014). As motivated above, three expressions for this quantity in the present study will be considered. These are derived from the unified nonlinear theory (UNLT) proposed by Shalchi (2010), the random ballistic decorrelation (RBD) interpretation proposed by Ruffolo et al. (2012) of the nonlinear guiding center (NLGC) theory of Matthaeus et al. (2003), and the standard NLGC. The expression derived from the standard NLGC is included in this analysis as it has been used successfully in many cosmic ray modulation studies (e.g., Engelbrecht & Burger 2010; Moloto et al. 2018; Moloto & Engelbrecht 2020). The RBD expression used here is derived assuming as a basic input a theoretically (Matthaeus et al. 2007) and observationally (Bieber et al. 1993) motivated form for the 2D modal fluctuation spectrum, as employed by Engelbrecht & Burger (2015). This spectral form is given by

$$\begin{eqnarray*}&&{S}_{2{\rm{D}}}({k}_{\perp })=\displaystyle \frac{{C}_{0}{\lambda }_{2{\rm{D}}}\delta {B}_{2{\rm{D}}}^{2}}{2\pi {k}_{\perp }}\left\{\begin{array}{l}\displaystyle \frac{{\lambda }_{2T}}{{\lambda }_{2{\rm{D}}}}{\left({\lambda }_{2T}{k}_{\perp }\right)}^{-p},{k}_{\perp }\lt {\lambda }_{2T}^{-1}\\ {\left({\lambda }_{2{\rm{D}}}{k}_{\perp }\right)}^{-1},{\lambda }_{2T}^{-1}\leqslant {k}_{\perp }\lt {\lambda }_{2{\rm{D}}}^{-1}\\ {\left({\lambda }_{2{\rm{D}}}{k}_{\perp }\right)}^{-s},{k}_{\perp }\geqslant {\lambda }_{2{\rm{D}}}^{-1}\end{array}\right.,\end{eqnarray*}$$

with λ_2T and λ_2D the respective length scales at which the energy-containing and inertial ranges commence (see, e.g., Matthaeus et al. 2007), with a ${k}_{\perp }^{-1}$ wavenumber dependence in the energy-containing range, and a ν spectral index in the inertial range, here assumed to equal the often-observed (e.g., Smith et al. 2006; Bruno & Carbone 2016) Kolmogorov (1941) value of 5/3. The quantity q denotes the spectral index of an inner range proposed by Matthaeus et al. (2007) on theoretical grounds, and is here, as in Engelbrecht & Burger (2013a), assumed to equal 3. Note that dynamical effects and the effects of the dissipation range are not considered for the 2D spectrum, as these have been demonstrated to be negligible for the resulting derived perpendicular MFP expressions (Gammon & Shalchi 2017; Dempers & Engelbrecht 2020), due to the greater sensitivity of these theories to the behavior of quantities pertaining to the energy-containing range of the assumed spectral form (Engelbrecht & Burger 2015). Note that dynamical effects from the slab turbulence power spectrum still affect perpendicular MFP expressions derived from these theories due to their dependence on the parallel MFP (Engelbrecht & Burger 2010; Dempers & Engelbrecht 2020). The length scale λ_2T is not well constrained observationally, and the present study will employ the modulation study-based scaling of λ_2T = 12.5λ_2D proposed by Engelbrecht & Burger (2013a), who found that this scaling leads differential intensities, computed using an ab initio galactic cosmic ray modulation code, in good agreement with spacecraft observations. This scaling also falls within the range of estimates calculated for this quantity from observations of magnetic island sizes in the solar wind by Engelbrecht (2019). The normalization constant for the 2D spectrum used here is given by

$$\begin{eqnarray*}&&{C}_{0}={\left[\mathrm{ln}\left(\displaystyle \frac{{\lambda }_{\mathrm{out}}}{{\lambda }_{2{\rm{D}}}}\right)+\displaystyle \frac{1}{1+q}+\displaystyle \frac{1}{\nu -1}\right]}^{-1},\end{eqnarray*}$$

and is calculated from the relation ${\int }_{0}^{\infty }{{dk}}_{\perp }{S}^{2{\rm{D}}}\left({k}_{\perp }\right)=\delta {B}_{2{\rm{D}}}^{2}$. The RBD expression considered here is that derived by Dempers & Engelbrecht (2020). Given that those authors find that dynamical effects do not significantly contribute to the perpendicular MFP derived from this theory and that the contribution to λ_⊥ due to the energy-containing range is predominant, only the expression derived for magnetostatic 2D turbulence derived by those authors, given by

$$\begin{eqnarray}\begin{array}{rcl}{\lambda }_{\perp } & \approx & \displaystyle \frac{\sqrt{3}{{aC}}_{0}\delta {B}_{2{\rm{D}}}^{2}}{{B}_{0}\sqrt{\delta {B}_{T}^{2}}}\\ & & \times \left[\displaystyle \frac{a\sqrt{\delta {B}_{T}^{2}}{\lambda }_{\parallel }}{{B}_{0}\sqrt{3}}({e}^{-{x}^{2}{\lambda }_{2{\rm{D}}}^{2}}-{e}^{-{x}^{2}{\lambda }_{2T}^{2}})\right]\\ & & -\displaystyle \frac{\sqrt{3}{{aC}}_{0}\delta {B}_{2{\rm{D}}}^{2}}{{B}_{0}\sqrt{\delta {B}_{T}^{2}}}\left[\sqrt{\pi }\left({\lambda }_{2{\rm{D}}}\mathrm{erfc}[x{\lambda }_{2{\rm{D}}}]\right.\right.\\ & & -\left.\left.{\lambda }_{2T}\mathrm{erfc}[x{\lambda }_{2T}]\right)\right],\end{array}\end{eqnarray} \tag{ 16 }$$

with

$$\begin{eqnarray*}&&x=\displaystyle \frac{\sqrt{3}{B}_{0}}{a\sqrt{\delta {B}_{T}^{2}}{\lambda }_{\parallel }},\end{eqnarray*}$$

where $\mathrm{erfc}$ denotes the complimentary error function considered here. Note that $\delta {B}_{T}^{2}$ refers to the total (slab+2D) transverse variance.

The expression for the UNLT perpendicular MFP derived by Shalchi (2018) will be used here. This expression is derived using as input the Shalchi & Weinhorst (2009) expression for the 2D power spectrum with an energy range with spectral index q_s and an inertial range with spectral index s, given by

$$\begin{eqnarray}\begin{array}{rcl}{g}^{2{\rm{D}}}({k}_{\perp }) & = & \displaystyle \frac{2D(s,{q}_{s})}{\pi }\delta {B}_{2{\rm{D}}}^{2}{\lambda }_{2{\rm{D}}}\\ & & \times \displaystyle \frac{{\left({k}_{\perp }{\lambda }_{2{\rm{D}}}\right)}^{{q}_{s}}}{{\left[1+{\left({k}_{\perp }{\lambda }_{2{\rm{D}}}\right)}^{2}\right]}^{(s+{q}_{s})/2}},\end{array}\end{eqnarray} \tag{ 17 }$$

where

$$\begin{eqnarray}&&D(s,{q}_{s})=\displaystyle \frac{{\rm{\Gamma }}\left(\tfrac{s+{q}_{s}}{2}\right)}{{\rm{\Gamma }}\left(\tfrac{s-1}{2}\right){\rm{\Gamma }}\left(\tfrac{1+{q}_{s}}{2}\right)},\end{eqnarray} \tag{ 18 }$$

where Γ denotes the standard Gamma function, which yields, under the assumption that −1 < q_s < 1 and $9{\lambda }_{2{\rm{D}}}^{2}\ll 4{\lambda }_{\parallel }{\lambda }_{\perp }$ (a condition that should be met at the energies relevant to this study; see also Shalchi 2020),

$$\begin{eqnarray}\begin{array}{rcl}{\lambda }_{\perp } & = & {\left[{\left(\displaystyle \frac{9}{4}\right)}^{({q}_{s}+1)/2}{a}^{2}D(s,{q}_{s})\displaystyle \frac{\delta {B}_{2{\rm{D}}}^{2}}{{B}_{0}^{2}}{\lambda }_{2{\rm{D}}}^{{q}_{s}+1}\right]}^{2/({q}_{s}+3)},\\ & & \times {\left[{\rm{\Gamma }}\left(\displaystyle \frac{1+{q}_{s}}{2}\right){\rm{\Gamma }}\left(\displaystyle \frac{1-{q}_{s}}{2}\right)\right]}^{2/({q}_{s}+3)}{\lambda }_{\parallel }^{(1+{q}_{s})/(3+{q}_{s})}.\end{array}\end{eqnarray} \tag{ 19 }$$

When evaluating this expression here, it is assumed that s = 5/3 and q_s = −0.99, to make the form of Equation (17) more compatible with Equation (3.2) for the purposes of direct comparison. The last expression to be compared with is derived from the standard NLGC of Matthaeus et al. (2003) by Shalchi et al. (2004a) and modified by Burger et al. (2008). This expression, given by

$$\begin{eqnarray}&&{\lambda }_{\perp }={\left[{\alpha }^{2}\sqrt{3\pi }\displaystyle \frac{2\nu -1}{\nu }\displaystyle \frac{{\rm{\Gamma }}(\nu )}{{\rm{\Gamma }}(\nu -1/2)}{\lambda }_{2{\rm{D}}}\displaystyle \frac{\delta {B}_{2{\rm{D}}}^{2}}{{B}_{0}^{2}}\right]}^{2/3}{\lambda }_{\parallel }^{1/3},\end{eqnarray} \tag{ 20 }$$

derived using a 2D turbulence power spectrum with a wavenumber-independent energy-containing range and an inertial range with a 2ν spectral index, and is similar to that derived from the same theory by Zank et al. (2004). Although these assumptions do not reflect the behavior of observed power spectra (see, e.g., Bieber et al. 1993; Goldstein & Roberts 1999), this expression is considered due to its relatively frequent use in various cosmic ray modulation studies (see, e.g., Oughton & Engelbrecht 2021, and references therein).

In order to achieve a meaningful comparison between the theoretical predictions for Jovian electron parallel and perpendicular MFPs as discussed above and the results of the simulations presented here, some care needs to be taken when choosing values for turbulence quantities used as inputs for the theoretical expressions. To this end, we take an observation-based approach. Given that the Jovian electron observations with which model differential intensities are compared were taken over an extended period, covering several solar cycles, such a comparison can become complicated, due to the known solar cycle dependence of the relevant turbulence quantities (see, e.g., Zhao et al. 2018). This latter phenomenon can be seen in Figure 7, which displays yearly averaged values at 1 au for the total magnetic variance, as well as the square of the HMF magnitude, over the period of interest to this study. These magnetic variances are calculated from ACE and OMNI data, using the partial variance technique discussed by Burger et al. (2021). For the purposes of this study, we employ values averaged over this interval for both the total variance and HMF magnitude, as indicated by the solid lines in the figure, such that ${\overline{B}}_{o}=6.60$ nT, and ${\overline{\delta B}}_{T}^{2}=11.98$ nT². Slab and 2D variances are calculated from this total variance according to the 20:80 anisotropy ratio of Bieber et al. (1994), although it should be noted that this quantity varies considerably (see, e.g., Dasso et al. 2005; Oughton et al. 2015). These variance values are then assumed to scale radially as ∼ r^−2.4, a scaling consistent with Voyager observations (Zank et al. 1996), and consistent with radial scalings for magnetic variances yielded in the region of interest to this study by several turbulence transport models (e.g., Wiengarten et al. 2016; Adhikari et al. 2017). For the slab and 2D correlation scales, we employ at 1 au the observational values computed, for the slow solar wind, by Weygand et al. (2011). These values are scaled as ∼ r^0.5, after the Voyager observations of Smith et al. (2001).

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Figure 8 shows the various parallel and perpendicular MFP expressions discussed above (left and right panels, respectively) as a function of rigidity at three radial distances. These plots include values for these quantities as reported above, calculated using Equations (5) and (6), at the energies of 2.5, 6, 8.5, and 12.5 MeV considered in this study. These points are shown along with values reported by prior studies, where applicable, as shown also in Figure 6. It is immediately apparent that the damping turbulence parallel MFP consistently yields results within the range of the simulation results, both from this study and from others, as well as at all radial distances considered, as opposed to the random sweeping parallel MFP. Values reported here for λ_∥ also appear to favor the flatter rigidity dependence displayed by the DT parallel MFP, especially at 5 au. In the right panels of Figure 8, the various theoretical expressions for λ_⊥ are evaluated using both the random sweeping (red lines) and damping turbulence (blue lines) parallel MFP expressions. At 1 au, both the RBD and UNLT perpendicular MFP expressions yield results consistently larger than observations, although only nominally so for the RS RBD and the DT UNLT results. The NLGC results, for both the RS and DT parallel MFPs, fall within the range of observations of the current study and those reported by others, and remain below the Palmer consensus range. At 2.5 and 5 au, the RS RBD result and the DT UNLT result fall within the range of the largest values for λ_⊥ reported here, this also being true for the DT RBD perpendicular MFP at 5 au, due to the larger spread in observational values reported here for λ_⊥. At all radial distances shown, the NLGC expression for λ_⊥ yields values well within the ranges of values reported here for this quantity.

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