Suprathermal Electron Transport in the Solar Wind: Effects of Coulomb Collisions and Whistler Turbulence

The nature and radial evolution of solar wind electrons in the suprathermal energy range are studied. A wave–particle interaction tensor and a Fokker–Planck Coulomb collision operator are introduced into the kinetic transport equation describing electron collisions and resonant interactions with whistler waves. The diffusion tensor includes diagonal and off-diagonal terms, and the Coulomb collision operator applies to arbitrary electron velocities describing collisions with both background protons and electrons. The background proton and electron densities and temperatures are based on previous turbulence models that mediate the supersonic solar wind. The electron velocity distribution functions and electron heat flux are calculated. Comparison and analysis of the numerical results with analytical solutions and observations in the near-Sun region are made. The numerical results reproduce well the creation of the sunward electron deficit observed in the near-Sun region. The deficit of the electron velocity distribution function below the core Maxwellian fit at low velocities results from Coulomb collisions, and the excess part above the core Maxwellian fit at high velocities is determined by strong wave–particle interactions.


Introduction
The electron velocity distribution function (eVDF) of the solar wind deviates significantly from thermal equilibrium (Feldman et al. 1974(Feldman et al. , 1975;;Marsch et al. 1982).Observations show that the solar wind eVDF often consists of three distinct components: a thermal core, a suprathermal halo, and a suprathermal strahl-a field-aligned beam of electrons traveling in the antisunward direction (Pilipp et al. 1987a(Pilipp et al. , 1987b;;Maksimovic et al. 1997;Ergun et al. 1998;Wang et al. 2012).The thermal core comprises ∼95% of the total electrons and commonly follows a Maxwellian distribution with an energy of ∼10 eV, and the suprathermal halo and strahl together form the remaining ∼5%, corresponding to an energy range of about 10 2 -10 3 eV.The halo and strahl electrons are believed to originate in the solar corona and escape into interplanetary space along open magnetic field lines (Pierrard et al. 1999;Štverák et al. 2008;Che & Goldstein 2014).It is widely held that strahl electrons are pitch-angle scattered into the halo as they propagate away from the solar corona (Horaites et al. 2015;Kim et al. 2015;Tang et al. 2020Tang et al. , 2022)).A solar wind electron component (sometimes called a proto-halo) has been observed that is thought to be related to the formation of the halo through the scattering of the strahl (Gurgiolo et al. 2012;Gurgiolo & Goldstein 2016).
Observations and theoretical studies of the eVDFs in the solar wind have been made for decades (Montgomery et al. 1968;McComas et al. 1992).Data analyses of the radial dependence of the eVDF between 0.3 and 4 au have found that the relative density of the halo portion radially increases while that of the strahl decreases with increasing heliocentric distance (Maksimovic et al. 2005;Stverák et al. 2015).Graham et al. (2017) find a constant rate at which the strahl pitch-angle distributions broaden with increasing heliocentric distance between 1 and 5.5 au.Beyond this distance until 9 au, the strahl is likely to be scattered entirely to form part of the halo.With the launch of the Parker Solar Probe (PSP), observations of solar wind electrons at less than 0.3 au are available (Fox et al. 2016).Halekas et al. (2020) found that the strahl is narrower and dominates the suprathermal fraction of the distribution near ∼0.17 au, and the halo is almost absent.Berčič et al. (2021b) analyzed eVDFs measured between ∼0.1 and ∼0.4 au by PSP and found that the strahl is distinct and its angular pitch-angle width decreases with increasing antisunward electron energy.PSP observations do not reveal substantial non-Maxwellian populations of the eVDFs in the near-Sun region, but reveal a decrease in the fractional abundance of suprathermal electrons close to the Sun (Berčič et al. 2020;Halekas et al. 2020;Abraham et al. 2022).These new findings, which suggest that there is a trend in the eVDFs such that the antisunward strahl component dominates the suprathermal electron population, indicate that there is little or even no halo present with decreasing heliocentric distance.
More importantly, in the near-Sun environment, a new feature of the solar wind eVDF, described as a sunward electron deficit, has been discovered (Halekas et al. 2020(Halekas et al. , 2021b)).The sunward electron deficit is identified by comparing the observed VDFs to a drifting bi-Maxwellian function fit to the core population in the antistrahl direction, i.e., in the sunward or anti-field-aligned direction.The observations match the Maxwellian core fit at low velocities.At moderate velocities, however, the sunward portion of the eVDF drops below the core fit and rises above it again at high velocities.The sunward electron deficit is evident within 0.2 au but is not generally observed beyond 0.3 au (Halekas et al. 2021a).The possible mechanisms responsible for creating the deficit are unclear.The deficit may occur as a result of the weakly collisional radial expansion of the solar wind (Berčič et al. 2021c) or from the scattering of the strahl population beyond a pitch angle of 90° (Micera et al. 2020).
Various theoretical studies of the formation of the suprathermal components (halo and strahl) in the solar wind eVDF have been presented in the past decades.Coulomb collisions between charged particles, mainly electron-electron (e-e) and electron-proton (e-p) collisions, were first considered.Scudder & Olbert (1979) demonstrated that the observed core + suprathermal feature of the eVDF is formed near 1 au by collisional processes and subsequent phase-space mapping along an inhomogeneous medium.Specifically, they numerically integrated the kinetic equation with a simplified Coulomb collision operator.This process is called "velocity filtration" (Scudder 1992(Scudder , 1994)).Canullo et al. (1996) derived analytical solutions to the same problem but employed a Chapman-Enskog-like perturbation expansion of the kinetic equation.Lie-Svendsen & Leer (2000) found that e-e collisions dominate e-p collisions in the formation of the high-energy tail of solar wind electrons by adding e-p scattering into their previous kinetic electron transport equation, which considered only e-e Coulomb collisions (Lie-Svendsen & Rees 1996;Lie-Svendsen et al. 1997).Multiple authors have shown that Coulomb collisions play a non-negligible role in regulating the eVDF and electron macroscopic parameters (e.g., see Pierrard et al. 1999Pierrard et al. , 2001;;Landi et al. 2012).Jeong et al. (2022) recently solved a gyro-averaged kinetic transport equation with Coulomb collisions along the Parker spiral magnetic field at heliocentric distances from ∼0.025 to ∼0.1 au.Their results quantify the parameters of the electron core and the strahl part of the electron velocity distributions.Specifically, since suprathermal electrons have a velocity much larger than the thermal velocity of background electrons and protons, a reduced Coulomb collision operator applied exclusively to energetic electron collisions is typically considered (Horaites et al. 2015;Tang et al. 2018).Horaites et al. (2018aHoraites et al. ( , 2018b) ) obtained an analytical solution for the strahl distribution in the fast solar wind.By contrast, Smith et al. (2012) adopted a more general Coulomb collision operator, valid for arbitrary electron velocities.
At large heliocentric distances and in the high-energy range, Coulomb collisions alone cannot scatter the suprathermal electrons into the observed width (Salem et al. 2003), and other scattering mechanisms (such as wave-particle interactions) were subsequently included.Ma & Summers (1998) were among the first to consider the influence of wave-particle resonance locally in the formation of the solar wind eVDF.Vocks & Mann (2003) and Vocks et al. (2005Vocks et al. ( , 2008) ) studied the formation of the two suprathermal components globally in the solar corona and solar wind due to whistler turbulence for electron energies up to more than 100 keV.Their studies showed that the quiet solar corona could produce suprathermal eVDFs by whistler turbulence wave-particle interactions and that such an electron population should be present in the solar wind.Adding a diffusion coefficient term to the exospheric solar wind model (Maksimovic et al. 1997), Pierrard et al. (2011) showed that nonthermal tails in the solar wind eVDF emerge from an initially Maxwellian distribution function, thanks to wave-particle interactions associated with whistler wave turbulence.Tang et al. (2020Tang et al. ( , 2022) ) developed a kinetic approach that includes a kinetic diffusion tensor.They showed that the off-diagonal terms of the diffusion tensor depress or balance the diffusion in both pitch-angle and velocity space caused by the diagonal diffusion terms.Recently, particle-incell simulations have shown that an initial core-strahl distribution triggers instabilities and scatters the strahl electrons toward larger pitch angles (Micera et al. 2020(Micera et al. , 2021)).On the other hand, Kim et al. (2015) and Yoon (2015) developed an asymptotic theory for solar wind electrons in local equilibrium with plasma wave turbulence (whistler and Langmuir wave turbulence), by treating three electron populations separately and assuming that the Maxwellian core does not experience any collisionless scattering, with the result that the halo electrons interact with whistler and Langmuir wave turbulence and the superhalo electrons only interact with Langmuir wave turbulence.
Instead of dealing with Coulomb collisions and turbulence scattering separately, it is advantageous to consider the two mechanisms together when describing electron transport in the solar wind.Boldyrev & Horaites (2019) presented an extended kinetic theory for strahl electrons scattered by both Coulomb collisions and wave-particle interactions of whistler waves along the Parker-spiral-shaped magnetic field lines.Although their theory incorporates the essential physics of the strahl formation, their work only considers the pitch-angle scattering of the two mechanisms.Jeong et al. (2020) developed a quasilinear model for the time evolution of the eVDF under the action of Coulomb collisions and the oblique fast-magnetosonic/whistler instabilities.
In the present paper, we focus on the nature and variation of the electron populations in the suprathermal energy range by combining Coulomb collisions and wave-particle interactions in the kinetic transport equation.The Coulomb collision operator describes collisions with Maxwellian backgrounds of electrons and protons and applies to arbitrary electron velocities.The wave-particle interaction results in a full-form diffusion tensor, including off-diagonal diffusion coefficients.By studying the radial evolution of the electron, we numerically analyze the effects of the Coulomb collisions (subscript "cc") and the resonant wave-particle interactions (subscript "wp") together.The shape of the solar wind eVDFs differs significantly in the presence of wave-particle interactions, especially in the suprathermal energy range.By comparing with the observed radial evolution of the solar wind eVDF from the near-Sun region, this suggests that waveparticle interactions do not significantly affect the eVDF, but become more effective after a certain distance, possibly once the core-strahl eVDF triggers whistler wave turbulence.In addition, our numerical results reproduce the sunward electron deficit observed in the solar eVDF and reveal a possible mechanism for its creation.Note that we do not attempt to model the full solar wind but instead simply prescribe a background flow field, density, and magnetic field.Our primary objective is to introduce an inner boundary where collisions are significant, to estimate the change in the whistler turbulence level with heliocentric distance, and to evaluate the response of the eVDF to both Coulomb collisions close to the Sun and wave-particle interactions farther from the Sun.
The structure of the paper is as follows.Section 2 reviews the electron transport model with scattering terms and briefly describes the numerical method.Section 3 presents the numerical results for the transport of electrons with Coulomb collisions.Comparison and analysis with an analytical solution are also provided in this section.We present extended numerical results of the transport of electrons with both Coulomb collisions and wave-particle interactions in Section 4. Section 5 discusses the properties of the numerical results and compares them with observations.Finally, our conclusions and future work are discussed in Section 6.

Transport Equation of Electrons
The gyrophase-averaged kinetic transport equation of electrons in the solar wind with Coulomb collisions and whistler turbulence is (Zank 2014;Tang et al. 2018Tang et al. , 2022) where (v, μ) is the magnitude of the velocity and pitch angle measured in the solar wind frame, r is the heliocentric distance measured in a "rest" frame, e is the electron charge, and E ∥ is the parallel electric field along the magnetic field line (i.e., the ambipolar electric field).(δf/δt) cc and (δf/δt) wp are scattering terms associated with Coulomb collisions and wave-particle interactions of whistler turbulence, respectively.The Coulomb collision operator describing collisions with Maxwellian backgrounds of electrons and protons can be expressed in the solar wind frame as (Helander & Sigmar 2005) where v the and v thp are the thermal velocities of the background Maxwellian electrons and protons, m e and m p are the electron and proton mass, and ( ) where n e and n p are the number densities of the background Maxwellian electrons and protons, and L ln is the Coulomb logarithm.The Coulomb collision operator Equation (2) is consistent with the collision operator used by Smith et al. (2012).The terms associated with c ν,e and c ν,p correspond to collisions with Maxwellian backgrounds of electrons and protons, respectively.
We choose the same wave-particle interaction scattering terms as we did in Tang et al. (2020Tang et al. ( , 2022)): The diffusion tensor for nonrelativistic electrons is expressed as (Steinacker & Miller 1992;Pierrard et al. 2011) 2 .s is the spectral index of the whistler waves, and A is the normalization constant related to the energy spectral densities of the whistler wave turbulence.Our knowledge of the intensity of the whistler turbulence in the solar wind is poor.Therefore, we can only estimate the approximate value of A based on the observational energy spectrum density of the whistler wave turbulence (Tang et al. 2020).

The Numerical Kinetic Transport Model
We numerically solve the transport equation including the scattering terms associated with Coulomb collisions and waveparticle interactions in specific plasma and magnetic field backgrounds.To this end, we combine our previous numerical model of Coulomb collisions (Tang et al. 2018) with our previous numerical model of the complete diffusion tensor for wave-particle interactions (Tang et al. 2022).The radial evolution of the solar wind electrons, especially the suprathermal portion, is investigated.
At the inner boundary r L = 0.2 au, solar wind electrons with a given velocity distribution function ( f L ) are continuously injected into the simulation box.The overall eVDF ( f L ) originating from the solar corona is assumed to be composed of two main parts: a Maxwellian core population (labeled by the subscript "c") and a suprathermal portion (labeled by the subscript "s"): The core Maxwellian distribution is not subject to waveparticle interactions with whistler turbulence and can be expressed as where T e is the background electron core temperature such that the electron thermal velocity . On the contrary, the suprathermal portion is subject to Coulomb collisions and wave-particle interactions of whistler turbulence.Observations show that the strahl dominates the suprathermal fraction of the eVDF near ∼0.17 au and the halo is absent (Halekas et al. 2020;Berčič et al. 2021b), making it safe to assume that the suprathermal portion f s only has strahl electrons at the inner boundary (0.2 au).Note that we do not claim that the halo component is entirely absent in the corona or outer corona, but this does suggest that our boundary conditions adopted at the inner boundary of 0.2 au in our calculation are well founded, like those of multiple authors (Horaites et al. 2018a;Cattell et al. 2021).We choose the analytical solution of the strahl obtained by Horaites et al. (2018aHoraites et al. ( , 2018b)): where C 0 ≡ 0.234, ò ≡ −2.14, and Ω ≡ −0.3, based on WIND data at 1 au.g is the "effective" Knudsen number and g = 0.75 in the fast solar wind.The numerical method is briefly summarized here.Letting f = Y/r 2 v 2 , we convert the transport Equation (1) of the overall solar wind electrons into an advection-diffusion-like equation of Y(r, v, μ) and then solve it numerically.The general form of the advection-diffusion-like equation of Y(r, v, μ) is where W and  are the advection vector and diffusion tensor, respectively.The specific forms of the components of the advection vector and the diffusion tensor vary with the scattering term, i.e., the specific forms of (δf/δt) cc and (δf/ δt) wp .In the present paper, we use blackboard bold  to refer to the diffusion tensor of the advection-diffusion-like equation of Y(r, v, μ) to distinguish the diffusion coefficients of distribution function f (r, v, μ).The three components of the advection vector W are (W r , W v , W μ ).In our numerical model, the original 3 × 3 diffusion tensor  has a reduced form: The numerical model of Tang et al. (2020) considered only the two diagonal terms of the diffusion tensor ( vv and  mm ).In the subsequent extended numerical model of Tang et al. (2022), the off-diagonal terms of the diffusion tensor ( m v and  mv ) are included.The extended numerical model enables us to study the effect of wave-particle interactions on the transport of solar wind electrons.

The Expanding Solar Wind Background
In our study, we use the background radial profiles of the solar wind electron and proton densities (n e and n p ) and temperatures (T e and T p ) derived by Adhikari et al. (2021).They developed a general theoretical model of nearly incompressible magnetohydrodynamic (NI MHD, 2D + slab) turbulence in the β ∼ 1 plasma (Zank et al. 2017) that is coupled to a two-fluid (proton and electron) solar wind model.Their turbulence-driven solar wind model describes the evolution of the solar wind speed U, the solar wind proton density n p , the proton temperature T p , the electron temperature T e , and the 2D slab turbulence components.In their model, turbulent heating is distributed between protons and electrons in a ratio of 60: 40 (Breech et al. 2010), and the Coulomb collisions between the protons and electrons and the heat conduction of the electrons.Adhikari et al. (2021) found that the proton temperature exhibits a radial profile of r −0.66 , implying that the proton temperature decreases gradually as a function of heliocentric distance.However, the electron temperature presents a radial profile of r −0.54 and r −0.24 between 0.2-0.7 au, and 0.7-1 au, respectively.This indicates that the electron temperature decreases rapidly near the Sun and more slowly at larger distance.Therefore, the electron temperature cannot be described by a single power law.The theoretical results of the solar wind proton and electron temperatures are very consistent with the observed proton and electron temperatures measured by PSP and Helios 2. Similarly, the theoretical solar wind proton density decreases as ∼r −2 and is consistent with the observed proton density measured by PSP and Helios 2. Note that the theoretical proton density of Adhikari et al. (2021) can be used as the electron density, since they assume that the proton density is equal to the electron density to maintain a charge neutrality.
The ambient magnetic field was chosen to have the form of Adhikari et al. (2017), which was adapted from Weber & Davis (1967): where r is in units of au, r a ∼ 0.05 au, ω a = 2.9 × 10 −6 rad s −1 , U = 400 km s −1 , and B a = 1.25 × 10 3 nT, so that B (r = 1) = 4.6 nT is the magnetic field at 1 au.This form of the magnetic field captures both the dominant radial magnetic field close to the Sun and the evolution to an interplanetary magnetic field that is predominantly transverse far from the Sun.A global parallel electric field, also known as the ambipolar electric field, accelerates the protons and decelerates the electrons in the solar wind.Berčič et al. (2021b) empirically estimated the ambipolar electric field and potential by analyzing eVDFs near the Sun measured by PSP.The electric field decreases with radial distance as a power-law function with an index of −1.69.Halekas et al. (2022) suggest that the electron pressure gradient and the associated electric field are necessary to explain the slowest solar wind streams.We keep the prescribed parallel electric field E ∥ in the transport Equation (1).For simplicity, in the present paper, we choose an expression for the parallel electric field as calculated from the electron momentum equation (Hollweg 1970;Jockers 1970):

Transport of Strahl Population with Coulomb Collisions: Analytical and Numerical Solutions
We first consider the effect of the Coulomb collisions.The gyrophase-averaged kinetic transport Equation (1) of the velocity distribution function f (r, v, μ) only includes the scattering term for Coulomb collisions: Many previous solar wind studies concerning Coulomb collisions (Horaites et al. 2015;Tang et al. 2018;Boldyrev & Horaites 2019) adopted a reduced Coulomb collision operator (δf/δt) cc on the right-hand side of Equation (12).The reduced Coulomb collision operator requires that the velocity of the test/target electron be much larger than the thermal velocity of the background electrons and protons, i.e., v ?v the and v ?v thp .In this case, the error function 1 and Chandrasekhar function G(x) → 1/2x 2 when x → ∞ , and the Coulomb collision operator Equation (2) has a simplified form.In addition, by further making assumptions about the transport equation and the background plasma (see Table 1), Horaites et al. (2018aHoraites et al. ( , 2018b) obtained an analytical model for the strahl electrons in the solar wind, i.e., Equation (8). Figure 1 shows the contour plots of the overall solar wind eVDF of Horaites et al. (2018aHoraites et al. ( , 2018b) ) at 0.2 and 1 au.Their overall eVDF is comprised of a Maxwellian core and the analytical strahl model Equation (8).
On the contrary, we can solve the electron transport Equation (12) using the numerical method provided by Tang et al. (2022), discussed in Section 2.1.This numerical method does not require the assumptions that Horaites et al. (2018aHoraites et al. ( , 2018b) ) made to obtain the analytical solution, hence providing a more general numerical solution to the transport Equation (12) valid for arbitrary velocity.Table 1 summarizes the assumptions of the analytical solution of Horaites et al. (2018aHoraites et al. ( , 2018b) ) and compares them with those of the numerical method.
Following the numerical treatment discussed in Section 2.1, the transport Equation (12) with the more general Coulomb collision operator Equation (2) on the right-hand side is  Horaites et al. (2018aHoraites et al. ( , 2018b)), normalized to f max at 0.2 and 1 au, respectively.Their overall eVDF is comprised of a Maxwellian core and the analytical strahl model Equation (8).The right panel is also the inner boundary condition in our calculation.The number density of the Maxwellian core is much larger than that of the suprathermal electrons.The same assumption is made.

3
The number density, temperature, magnetic field, and Knudsen number vary as a power law.Other forms of these parameters are allowed and used.
Not needed, and arbitrary pitch angles are allowed.

5
The electron bulk flow is subsonic and the parallel electric field is ignored.
The parallel electric field is included in the transport equation.

6
The velocity of the test electrons is much larger than the thermal velocity of the background electrons and protons.
The collisional scattering term is valid for arbitrary electron velocities.

7
The background electrons and ion core have similar temperatures.Other forms of these parameters are allowed.
converted to the advection-diffusion-like equation form: Figure 2 shows the numerical results of the contour levels of the overall eVDF in the plane (v ∥ , v ⊥ ) at 0.3, 0.5, 1, and 2 au.The overall eVDF does not have the elongated narrow tail seen in Figure 1 at high velocity in the antisunward direction, i.e., in the very large positive v ∥ region.On the contrary, the suprathermal portion has a finite width and does not extend very far into the suprathermal energy region.The numerical overall eVDFs are very similar to the observed eVDF near the Sun shown by Berčič et al. (2021b).

Transport of Electrons: Coulomb Collisions and Whistler Wave Turbulence
We now consider both Coulomb collisions and waveparticle interactions in the transport equation.In this case, the gyrophase-averaged kinetic transport Equation (1) of the velocity distribution function f (r, v, μ) includes the scattering term for Coulomb collisions (Equation ( 2)) and wave-particle interactions (Equation ( 4)), and the corresponding advection-diffusion-like equation of Y(r, v, μ) takes the form  In the present paper, we follow Tang et al. (2022) in choosing the normalization constant in the diffusion coefficients Equation (5a) as A 1 = 2.5 × 10 −7 (weak wave-particle interaction) and A 2 = 5 × 10 −7 (strong wave-particle interaction), and the spectral index of the whistler waves as s = 2.Such choices of the value of A allow the wave-particle interactions to increase from comparable to greater than the Coulomb collisions gradually.We can recognize the effect of wave-particle interactions by analyzing the radial evolution of the strahl eVDFs.Figure 3 shows the contour levels of the numerical results for the suprathermal eVDFs ( ) f log s 10 from 0.3 to 2 au.The first column (CCWP1) corresponds to the case of A 1 = 2.5 × 10 −7 , and the second column (CCWP2) to A 2 = 5 × 10 −7 .Compared with the numerical results for the Coulomb-collision-only case in Figure 2, the effect of waveparticle interactions is noticeable.At all positions, the eVDFs are more isotropic, which means more electrons with large velocities (high energy) are pitch-angle scattered by the whistler wave turbulence.

Discussions
Tang et al. (2020) used the method of characteristics to analyze the trajectories of electrons in the phase space (r, v, μ).By neglecting the scattering terms (diffusion coefficients) on the right-hand side of the transport equation, their electron trajectory calculation identified three electron categories: trapped, ballistic, and escaping electrons, which are consistent with the results of an exospheric model (Lamy et al. 2003).These characteristic equations for the electron trajectories are components of the advection vector W of Equation (9) in our method.Applying the method of characteristics to Equation (13), we can recognize the components of the advection vector and the diffusion tensor that come from Coulomb collisions: where the subscripts "ee" and "ep" indicate the effects from e-e and e-p collisions, respectively.The velocity dependence of the normalized advection vector components (W ee v and W ep v ) in an expanding solar wind background (Section 2.3) is shown in the top row of Figure 4. W ee v and W ep v decrease with the increasing velocity of the test/ target electrons, eventually decrease to less than zero (blue dashed lines), and remain negative after that.They act as drag terms in the Coulomb collisions operator Equation (2) when < W 0 v ee and < W 0 v ep , i.e., the regions below the blue dashed line.Previous research focused only on the drag effect of Coulomb collisions on solar wind electron transport and the formation of the solar wind eVDF (Tang et al. 2018;Horaites et al. 2019).This is valid for suprathermal electrons (halo and strahl).However, if the velocity of test/target electrons is small enough, W ee v and W ep v are positive, i.e., the regions above the blue dashed line, and they have the opposite effect than the drag term and act to provide a "push."Under the assumption of an analytical solution for strahl electrons (Equation ( 8)), the two components are always drag terms in the transport equation, since by assumption v ?v the , v ?v thp .The "push" effect of the Coulomb collisions in the case of v v the is ignored.On the contrary, the use of a general Coulomb collision operator is valid for arbitrary electron velocities, allowing our numerical result to cover the two effects of Coulomb collisions.
The bottom row of Figure 4 shows the normalized diffusion coefficients (two  vv and two  mm ) as a function of the target/test electron velocity in the numerical calculation.All diffusion coefficients decrease sharply as the target/test electron velocity increases.The velocity diffusion ( vv ; bottom left) for e-e collisions is more than 1 order of magnitude larger than that of ep collisions.With the increase of electron velocity,  vv ee gradually becomes larger than  vv ep by almost more than 3 orders of magnitude.On the other hand, the pitch-angle scattering ( mm ; bottom right) of the e-e and e-p collisions is comparable.The two graphs are reasonable and show that electron collisions with protons (e-p collisions) are essentially elastic, so there is no change in energy, but the pitch angle will change with collisions.Therefore, e-e collisions dominate e-p collisions with regard to energy in the formation of the suprathermal tail of solar wind electrons (Lie-Svendsen & Leer 2000), but completely ignoring the e-p collisions underestimates its pitch-angle scattering contribution to the suprathermal tail.In our present model, we keep the e-p collisions as well as the e-e collisions and investigate the pitch-angle properties of the suprathermal electrons, such as the strahl width.
We apply the same method to Equation (14) and recognize the additional components of the advection vector and the diffusion tensor that result from wave-particle interactions with whistler wave turbulence:  et al. 2022).This makes the target/test electrons accumulate at μ = 0, and the consequence can be seen from the two total eVDFs at 0.3 au in Figure 3.As A increases, m W wp becomes stronger, and more electrons congregate in the region of μ ∼ 0, i.e., along v ⊥ with v ∥ = 0.The eVDFs change from elongated along v ∥ to slightly elongated along v ⊥ .Moreover, with the accumulation of electrons along the perpendicular rather than the parallel direction, a larger perpendicular temperature than parallel temperature results.The off-diagonal diffusion coefficient D μv may cause the temperature anisotropy (T ⊥ / T ∥ ) to be larger than 1.
The sunward electron deficit discovered by PSP is a new property of the near-Sun solar wind eVDF, which is a departure from the Maxwellian fit for the core portion in the suprathermal energy range (Berčič et al. 2020(Berčič et al. , 2021a;;Halekas et al. 2020Halekas et al. , 2021a)).Berčič et al. (2021c) show that an instability driven by the sunward deficit can create the observed  5a)) A 1 = 2.5 × 10 −7 (weak waveparticle interaction) and A 2 = 5 × 10 −7 (strong wave-particle interaction), respectively.whistler waves and lower the heat flux stored in the solar wind eVDF.Figure 5 shows parallel cuts of the numerically calculated total solar wind eVDF at 0.3 au for three cases.The black dots are our numerical results, and the red dashed lines represent the corresponding fitted Maxwellian core plotted from Equation (7).Panel (c) represents a complete result that includes both Coulomb collisions and strong waveparticle interactions with A 2 = 5 × 10 −7 .The parallel cut of the computed solar wind eVDF for the Coulomb-collision-only and the combined Coulomb collision plus weak wave-particle interaction cases exhibit a property very similar to that of the observed sunward electron deficit (Halekas et al. 2021a(Halekas et al. , 2021b)).In these two cases, the antisunward portion of the eVDF extends to velocities greater than the Maxwellian core fit, whereas the sunward portion of the eVDF is deficient in particles compared to the sunward region of the fitted Maxwellian.
The possible mechanisms responsible for creating the deficit are still unclear.One explanation is that the deficit is a consequence of the weakly collisional radial expansion of the solar wind (Berčič et al. 2021c).Another explanation suggests that the deficit arises from the scattering of the strahl electrons beyond a pitch angle of θ > 90°, i.e., μ < 0 (Micera et al. 2020).Our numerical results show that the sunward electron deficit found by Halekas et al. (2021a) might result from a combination of Coulomb collisions and strong waveparticle interactions.In Figure 5, panel (a) represents the parallel cut of the overall eVDF with only Coulomb collisions.The sunward portion of the eVDF (v ∥ < 0) lies beneath the fitted Maxwellian core, and the difference increases as the electron velocity increases.Panel (b) shows a parallel cut of the overall eVDF with Coulomb collisions and weak wave-particle interactions.The sunward portion of the eVDF lies below the fitted Maxwellian core at low velocities and increases to gradually match the fitted Maxwellian at larger velocities, but does not exceed it.We can see the eVDF drops below the fitted Maxwellian core and exceeds it only in panel (c), for the case of both Coulomb collisions and strong wave-particle The components of the advection vector change sign as the electron velocity increases from positive to negative, showing that the two components act either as push or drag terms, changing smoothly from one to the other according to particle velocity.In the top row, v has the same interval (v = 1.5 × 10 8 −9 × 10 8 cm s −1 ), but is normalized by the electron and proton thermal velocity, respectively.The blue dashed line identifies the location of zero.interactions.Therefore, the part of the sunward deficit that is below the fitted Maxwellian core at low velocities results from Coulomb collisions.The part above the fitted Maxwellian core at high velocities results from sufficiently strong wave-particle interactions.
Electrons are the main carrier of heat flux away from the Sun.Using the latest PSP observations, Abraham et al. (2022) calculated the divergence of the electron heat flux (∇Q) over the heliocentric distance range of 0.15-0.5 au.They found that the divergence of the heat flux is positive at heliocentric distances below 0.33 au, while beyond 0.33 au it decreases to negative values.We calculate the radial evolution of the electron heat flux for the above three cases and show the results in Figure 6(a).The heat flux monotonically decreases with distance in the case of Coulomb collisions only (black dots).With the introduction of weak wave-particle scattering (red dots) and strong wave-particle scattering (blue dots), the indices for the radial decrease become smaller, changing from −2.6 to −2.27 to −2.1, which is slightly smaller than the observed index of ∼ − 2.9 (Scime et al. 2001).Perhaps more pertinent to the above discussion, the introduction of waveparticle interactions changes the monotonicity of the radial evolution of the electron heat flux.Figure 6(b) illustrates a plot of the divergence of the heat flux with heliocentric distance for the three cases.In the case of strong wave-particle interactions (blue line), the electron heat flux first increases with heliocentric distance (positive divergence) and then decreases with heliocentric distance (negative divergence).The location where the heat flux changes from increasing to decreasing (i.e., the divergence change sign) is about 0.3 au.The location from where the divergence of the heat flux changes sign from positive to negative is about 0.3 au.These properties are all consistent with the measurement of Abraham et al. (2022).
By comparing Figures 2 and 3, it can be seen that the shape of the solar wind eVDF, especially the halo and strahl in the suprathermal region, are significantly different for the cases with and without wave-particle interactions.Figure 7 compares the normalized diffusion coefficients for Coulomb collisions (red curves) and weak wave-particle scattering (black curves) as a function of electron velocity (energy).The pitch-angle scattering  mm cc and the velocity diffusion  vv cc terms for Coulomb collisions decrease with increasing electron velocity.However, the pitch-angle scattering terms for wave-particle interactions ( mm wp ) decrease much slower than  mm cc with increasing electron velocity and become larger than  mm cc at high velocities.By contrast, the velocity diffusion  mm wp for wave-particle interactions increases with increasing electron velocity and becomes much larger than  vv cc at high velocities.Consequently, wave-particle interactions can effectively scatter suprathermal electrons far more readily than Coulomb collisions.On the other hand, the latest near-Sun observations of the supersonic solar wind observed by PSP reveal the characteristic of the presence of halo and strahl electrons in the eVDF at distances less than 0.3 au.The strahl dominates the suprathermal fraction of the distribution function near ∼0.17 au (Halekas et al. 2020).Figure 2 shows that the strahl is noticeable and dominates the suprathermal portion of the eVDF  if only Coulomb collisions are considered, consistent with the observed near-Sun solar wind eVDF (Berčič et al. 2021b).Berčič et al. (2021b) showed that the strahl width in terms of v ⊥ appears almost constant, leading to a decreasing angular pitchangle width with increasing electron energy, consistent with the characteristic of  mm cc shown in Figure 7.These observations of the solar wind eVDF and strahl width suggest that waveparticle interactions are not yet particularly important close to the Sun, but begin to develop after a certain distance, which may be where the core-strahl eVDF triggers whistler turbulence.This scenario has been investigated on the basis of stability analyses of a core-strahl solar wind eVDF, showing that a core-strahl eVDF can trigger whistler heat flux instabilities as the solar wind expands and that these produce sunward whistler waves (Horaites et al. 2018a;Schroeder et al. 2021).The sunward whistler waves then scatter the initial core-strahl eVDF (Micera et al. 2020(Micera et al. , 2021)).Therefore, according to the observed radial evolution of the solar wind eVDF, it should be possible for us to determine this position, although this is beyond the scope of this paper.

Conclusions and Future Work
In the present paper, we combine Coulomb collisions and wave-particle interactions in the kinetic transport equation to study their joint impact on the transport of solar wind electron populations in the suprathermal energy range.Specifically, this allows us to investigate the radial evolution of the velocity distribution function and heat flux of the suprathermal electron population, by numerically analyzing and comparing the effects of the Coulomb collisions and the resonant waveparticle scattering.The Coulomb collision operator describes the collisions with electron and proton Maxwellian backgrounds, and we use a general form that applies to arbitrary electron velocities.The wave-particle interaction diffusion coefficients are obtained by assuming that the spectral densities of the whistler wave turbulence have equal intensities in both propagation directions and have a power-law spectrum.
We find that the velocity diffusion due to e-p collisions is 2 to 3 orders of magnitude weaker than that of e-e collisions (bottom left panel of Figure 4), which is consistent with the previous result that e-e collisions dominate e-p collisions in the formation of the suprathermal tail of solar wind electrons (Lie-Svendsen & Leer 2000).However, pitch-angle scattering from e-p collisions is of the same order of magnitude as that of e-e collisions in all energy ranges (bottom right pane of Figure 4).Therefore, e-p collisions cannot be neglected, especially when investigating the solar wind strahl width, which is likely due to pitch-angle scattering.Furthermore, we find an additional "push" term in the Coulomb collision operator that acts mainly in the low-energy range (the top panels of Figure 4), which has been neglected in previous studies of suprathermal electron transport.The "push" term results in the transport of lowenergy electrons toward the suprathermal energy range in phase space (r, v, μ), thereby affecting the radial evolution of the solar wind electrons and eVDF.
Our numerical solar wind eVDF is similar to that observed by PSP near the Sun at about 0.1-0.4au (Berčič et al. 2021b) when only Coulomb collisions are considered.However, the numerical solar wind eVDF differs significantly from previous numerical results and observations after the introduction of wave-particle scattering into the calculations.The change suggests that wave-particle interactions may not be an essential physical process in the near-Sun region, but only begin to become important in the solar wind after a certain distance from the Sun.One reason wave-particle interactions are not important in the near-Sun region may be due to the possible low occurrence of whistler waves in the near-Sun solar wind.Recent observations from PSP and STEREO may confirm this idea.Cattell et al. (2021Cattell et al. ( , 2022) ) find that whistler waves were not seen within a heliocentric distance 0.13 au.In more distant regions of the solar wind, beyond about 0.3 to 0.4 au, our numerical results that include Coulomb collisions and wave-particle scattering reproduce the observed characteristics of the radial evolution of the electron heat flux.The numerical electron heat flux first increases with distance (positive gradient) in the near-Sun region and then decreases at larger distances (negative gradient).The location where the transition occurs is about 0.3 au.This behavior of the electron heat flux is in excellent agreement with recent observations (Abraham et al. 2022).Finally, our numerical results provide a possible explanation for the sunward electron deficit discovered close to the Sun by PSP.The deficit at low velocities results from Coulomb collisions, and the excess part at high velocities is due to wave-particle interactions, since wave-particle interactions only affect suprathermal electrons.
Our results, consistent with observations in the near-Sun region, do have a drawback in the radial evolution of the eVDF.There are insufficient halo electrons in the sunward direction as the heliocentric distance increases, which is inconsistent with the widely held view that strahl electrons are gradually pitchangle scattered into halo electrons as they propagate away from the solar corona.This suggests that the scattering intensity, mainly wave-particle scattering, decreases too quickly with increasing distance in our current model and is insufficiently strong to continue to scatter strahl electrons into the halo at greater distances.This does not suggest that there is necessarily a problem with the kinetic eVDF transport model based on collisional and turbulence scattering, but that the problem may arise rather from an inadequate treatment of the transport and evolution of the turbulence responsible for electron scattering.In addition, in analyzing the numerical results, we have yet to divide the eVDF into three individual components, as done in the observations, since we treat the eVDF as a whole.In the present paper, we did not analyze the radial evolution of characteristics of the individual strahl, such as the pitch-angle width.We think that a more sophisticated turbulence model is required to provide us with the magnetic field fluctuations as a function of heliocentric distance, which will allow us to accurately calculate the ratio of the magnetic field fluctuations to the mean magnetic field (dB B 2 0 2 ) as a function of distance for the whistler turbulence.A subsequent paper will consider these extensions to the transport theory of thermal and suprathermal electrons in the solar wind.
Agreement OIA-2148653, and G.P.Z. the support of a NASA IMAP subaward under NASA contract 80GSFC19C0027.

Appendix Full Form of Transport Equation of Y Function
The electron transport equation with Coulomb collisions and whistler turbulence is where the scattering terms (δf/δt) cc and (δf/δt) wp are given by Equations ( 2) and (4).Let f = Y/r 2 v 2 , then we can convert the transport Equation (A1) into the advection-diffusion-like equation of function Y(r, v, μ): The three variables (r.v, μ) form a 3D Cartesian coordinate system.The components of the advection vector and the diffusion tensor are the following: A 2 erf and G(x) are the error function and the Chandrasekhar function.The collision frequencies c ν,e and c ν,p are given by

Figure 1 .
Figure 1.Contour plots of overall eVDF ofHoraites et al. (2018aHoraites et al. ( , 2018b)), normalized to f max at 0.2 and 1 au, respectively.Their overall eVDF is comprised of a Maxwellian core and the analytical strahl model Equation (8).The right panel is also the inner boundary condition in our calculation.

Figure 2 .
Figure 2. The numerical results showing the contour levels of the overall eVDF ( ) f log s

wp
We find that the off-diagonal term (D μv ) of the diffusion coefficient appears in the μ component of the advection velocity m W wp , which introduces additional transport in μ. < m W 0 wp for electrons with μ > 0 and vice versa (Tang

Figure 3 .
Figure 3.The numerical results of the contour levels of the overall eVDF ( ) f log s 10 from 0.3 to 2 au in the case of Coulomb collisions and wave-particle interactions are considered.The numbers after CCWP in the title refer to the normalization constant in the diffusion coefficients (Equation (5a)) A 1 = 2.5 × 10 −7 (weak waveparticle interaction) and A 2 = 5 × 10 −7 (strong wave-particle interaction), respectively.

Figure 4 .
Figure 4. Comparison of the normalized components of the advection vector (top row) and the diffusion tensor (lower row) of e-e and e-p collisions as a function of electron velocity.The components of the advection vector change sign as the electron velocity increases from positive to negative, showing that the two components act either as push or drag terms, changing smoothly from one to the other according to particle velocity.In the top row, v has the same interval (v = 1.5 × 10 8 −9 × 10 8 cm s −1 ), but is normalized by the electron and proton thermal velocity, respectively.The blue dashed line identifies the location of zero.

Figure 5 .
Figure 5. Parallel cuts of eVDF at 0.3 au of three cases.(a) Coulomb collisions.(b) Coulomb collisions and weak wave-particle interactions.(c) Coulomb collisions and strong wave-particle interactions.The red dashed lines represent the fitted Maxwellian core.The two vertical blue lines in (c) show the location of ±2v the .

Figure 6 .
Figure 6.Radial evolution of the electron heat flux for the three cases: the Coulomb-collision-only case (black dots), Coulomb collisions and weak wave-particle interactions (red dots; CCWP1), and Coulomb collisions and strong wave-particle interactions (blue dots; CCWP2).The numerical result of CCWP2 is well consistent with the observations.

Figure 7 .
Figure 7.Comparison of the normalized diffusion coefficients for Coulomb collisions and wave-particle interactions as a function of electron velocity (energy).The Coulomb collision term includes e-e and e-p collisions. vv wp and  mm wp are given by Equations (22) and (23), respectively.

Table 1
Assumptions of Transport Equation and Background PlasmaNo.