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Perpendicular diffusion coefficient of cosmic rays in the presence of weak adiabatic focusing

The influence of adiabatic focusing on particle diffusion is an important topic in astrophysics and plasma physics. In the past several authors have explored the influence of along-field adiabatic focusing on parallel diffusion of charged energetic particles. In this paper by using the Unified NonLinear Transport (UNLT) theory developed by Shalchi (SH2010) and the method of He and Schlickeiser (HS2014) we derive a new nonlinear perpendicular diffusion coefficient for non-uniform background magnetic field. This formula demonstrates that particle perpendicular diffusion coefficient is modified by along-field adiabatic focusing. For isotropic pitch-angle scattering and weak adiabatic focusing limit the derived perpendicular diffusion coefficient is independent of the sign of adiabatic focusing characteristic length. For two-component model we simplify the perpendicular diffusion coefficient up to second order of the power series of adiabatic focusing characteristic quantity. We find that the first order modifying factor is equal to zero and the sign of the second one is determined by the energy of particles.


ABSTRACT
The influence of adiabatic focusing on particle diffusion is an important topic in astrophysics and plasma physics. In the past several authors have explored the influence of along-field adiabatic focusing on parallel diffusion of charged energetic particles. In this paper by using the Unified NonLinear Transport (UNLT) theory developed by Shalchi (SH2010) and the method of He and Schlickeiser (HS2014) we derive a new nonlinear perpendicular diffusion coefficient for nonuniform background magnetic field. This formula demonstrates that particle perpendicular diffusion coefficient is modified by along-field adiabatic focusing.
For isotropic pitch-angle scattering and weak adiabatic focusing limit the derived perpendicular diffusion coefficient is independent of the sign of adiabatic focusing characteristic length. For two-component model we simplify the perpendicular diffusion coefficient up to second order of the power series of adiabatic focusing characteristic quantity. We find that the first order modifying factor is equal to zero and the sign of the second one is determined by the energy of particles.

INTRODUCTION
Energetic charged particles' propagating through magnetized plasmas is an important topic in the study of modulation of galactic cosmic rays (GCRs) in heliosphere (e.g., Parker 1963;Zhao et al. 2014), solar energetic particles (SEPs) transport in solar wind (e.g., Parker 1965;Roelof 1969;Ng & Reames 1994;Qin et al. 2005), energetic particles' shock acceleration in interplanetary space or from supernova remnants (e.g. Zank et al. 2000;Ferrand et al. 2014), etc. According to observations the magnetic field configuration of magnetized plasmas can be understood as the superposition of a large-scale background magnetic field B 0 = B 0êz and a turbulent component δ B in many scenarios. Both the mean magnetic field and the turbulent component can affect the properties of particles' propagation. Since the large-scale magnetic field breaks the symmetry of magnetized plasmas, particles' transport has to be distinguished between parallel and perpendicular components relative to the mean magnetic field direction. For simplicity purpose one usually assumes the mean magnetic fields uniform (e.g., Schlickeiser 2002;Shalchi 2009).
However, different observations, e.g., radio continuum surveys of interstellar space and direct in-situ measurements in solar system, show the spatially varying large scale magnetic fields for many scenarios, which leads to the adiabatic focusing effect of charged energetic particles and introduces modification to the particle diffusion coefficients (Roelof 1969;Earl 1976;Shalchi 2011a;Litvinenko 2012a,b;Shalchi & Danos 2013;He & Schlickeiser 2014;Wang & Qin 2016).
For weak adiabatic focusing limit and distribution function with small anisotropy, along-field adiabatic focusing gives rise to convective transport along and across the guide field, as well that as in momentum space (Schlickeiser & Shalchi 2008). In addition, by considering the spatial gradient and curvature of mean magnetic field, Schlickeiser & Jenko (2010) explored the influence of adiabatic focusing effects on particle perpendicular propagation. Since only weak limit is considered, the influence of along-field adiabatic focusing on particle parallel and perpendicular diffusion coefficient was not found.
By using the Unified NonLinear Transport (UNLT) theory (Shalchi 2010) for arbitrary anisotropic distribution function, Shalchi (2011a) derived a parallel diffusion coefficient formula with along-field adiabatic focusing. Employing the stochastic differential equations, completely equivalent to the Fokker-Planck equation, another version of parallel diffusion coefficient formula with adiabatic focusing was obtained (Litvinenko 2012a). This formula is also obtained in other papers (Beeck & Wibberenz 1986;Bieber & Burger 1990;Kóta 2000;Litvinenko 2012a;He & Schlickeiser 2014). Analytical and numerical studies demonstrated that the two versions are related by each other (Shalchi & Danos 2013).
Analogous to parallel diffusion, perpendicular diffusion is another important particle transport process. In the past, some approaches have been developed to derive the perpendicular diffusion coefficient, e.g., the NonLinear Guiding Center (NLGC) theory (Matthaeus et al. 2003), the Weakly NonLinear Theory (WNLT) (Shalchi et al. 2004), the Modification of NLGC , and the Unified NonLinear Transport (UNLT) theory (Shalchi 2010) and so on. However, these theories for perpendicular diffusion coefficient are all only applicable to the uniform mean magnetic field and adiabatic focusing effect is not considered.
In summary, a lot of studies have been done for perpendicular diffusion in uniform mean magnetic field, and parallel diffusion with along-field adiabatic focusing is also extensively investigated so far. But it is always an open topic whether along-field adiabatic focusing has influence on perpendicular diffusion. The perpendicular diffusion coefficient formula for uniform mean magnetic field derived in Shalchi (2010) (hereafter SH2010) is a well-known result and agrees well with simulations (e.g., Tautz & Shalchi 2011;Shalchi 2013;. In this paper by employing the UNLT theory and the method of He & Schlickeiser (2014) (hereafter HS2014) we explore the perpendicular diffusion coefficient with along-field adiabatic focusing effect.
The paper is organized as follows: In the Section 2 we introduce the definitions of perpendicular diffusion coefficient and along-field adiabatic focusing length, and from the standard Fokker-Planck equation by employing the UNLT theory we obtain the equations of the quantities which are related to spatial perpendicular diffusion coefficient. In Section 3 by using the method of HS2014 we derive the anisotropic distribution function and obtain the quantities related to spatial perpendicular diffusion coefficient which occur in equations in Section 2. And in Section 4 the spatial perpendicular diffusion coefficient with along-field adiabatic focusing is deduced, and for isotropic scattering and weak focusing limit the first and second-order modifying factors of perpendicular diffusion coefficient are derived. In Section 5 for two-component model the first and second modifying factors are explored.
Section 6 shows discussion and conclusions.

SIMPLIFICATION OF THE STANDARD FOKKER-PLANCK EQUATION
Similar as in SH2010, our starting point is the single particle motion equation (Matthaeus et al. 2003) is the large-scale magnetic field which are not necessarily uniform, δB x is the x-component of random magnetic field, v x (t) and v z (t) are the x and z components of particle speed v respectively. For convenience, we specialize to transverse fluctuations B 0 · δ B = 0 in this paper. Because the energetic charged particle feels the magnetic field at its own position, so variable z in the mean magnetic field B 0 (z) is a random variable with time. The parameter a 2 indicates the relation between single particle and the magnetic field lines, and the relevant details can be found in the papers of NLGC and UNLT theories and so on (Matthaeus et al. 2003;Shalchi 2015Shalchi , 2016. In this paper the well-known TGK (Taylor-Green-Kubo) formulation (see , Taylor 1922;Green 1951;Kubo 1957) is used to compute perpendicular diffusion coefficient 2.1. Along-field adiabatic focusing length The along-field adiabatic focusing length for spatially varying guiding field is shown as (Schlickeiser & Shalchi 2008) For the limit L → ∞ the spatially varying background magnetic field B 0 (z) tends to be uniform. In this paper, for mathematical tractability we assume the adiabatic focusing length is a constant, i.e., ∂L/∂z = 0. Then the mean magnetic field can be written as For convergent background magnetic field along z positive direction adiabatic focusing length satisfies L < 0, and for divergent case L > 0.

The formula of perpendicular diffusion coefficient
By combining Equations (1) and (2), perpendicular diffusion coefficient κ ⊥ can be with here µ is pitch-angle cosine and µ 0 is initial pitch-angle cosine, P xx ( k) is the magnetic field correlation tensor. In Equation (5) the Corrsin independence hypothesis (see , Corrsin 1959) is employed. And the quantity T ( k) is required to be real. Note that we also use the standard assumptions of magnetic turbulence, e.g., axisymmetry, homogeneity, magnetostatic, and so on. And f 0 [ x(t), µ, µ 0 , t] in Equation (6) is the distribution function of the standard Fokker-Planck equation with adiabatic focusing effect, which is shown as The latter equation includes the Fokker-Planck coefficients of pitch-angle diffusion D µµ and the perpendicular diffusion D ⊥ , along-field adiabatic focusing length L, and the differential operator ∆ ⊥ = ∂ 2 /∂x 2 + ∂ 2 /∂y 2 . However, the source term is neglected.
Equation (6) can also be written as with the quantity S( k, µ) as From Equations (5), (8) and (9) we can find that in order to get perpendicular diffusion coefficient κ ⊥ the formula S( k, µ) has to be obtained.

The governing equation of S( k, µ)
Let's set f [ x(t), µ, µ 0 , t] = e z(t)/L f 0 [ x(t), µ, µ 0 , t] and combining the standard Fokker-Planck equation (see, Equation (7)) we can obtain the following equation In what follows, from the latter equation we use the UNLT theory to derive the governing formula of quantity S( k, µ) (see, (9)) can be rewritten as After operating Fourier transformation on Equation (10) over spatial variable x we can where we assume that D µµ and D ⊥ are independent of x.
To multiply the time integral of Equation (13) from 0 to ∞ with the initial pitch-angle cosine µ 0 , and then integrate the results over µ 0 from −1 to 1, finally we can obtain The initial distribution can be defined as which denotes that the particles have a well-defined initial pitch-angle cosine, and h( k 0 ) is the distribution function in phase space for initial time t = t 0 . By assuming Γ(t → ∞) to be independent of the initial pitch-angle cosine µ 0 as in Shalchi (2010), we can obtain the following equation . And in the latter equation the following formula is used where the normalization condition d 3 k 0 h( k 0 ) = 1 must be satisfied for the initial time To proceed, by using the formula S( k, (11)) the governing equation of S( k, µ) can be got from Equation (16 Splitting S(µ, k) into an even function S + (µ, k) and an odd function S − (µ, k), i.e., , and then inserting this formula into Equation (18) yields To proceed, by integrating Equation (19) over µ from -1 to 1 and from 0 to 1 respectively, thereafter using the standard assumption D µµ (µ = ±1) = 0 we can get the following integral equations and If employing pitch-angle isotropic scattering model D µµ = D (1 − µ 2 ) with a constant D (e.g., Shalchi et al. 2009;Shalchi 2010) and the perpendicular diffusion coefficient model Qin & Shalchi 2014;Shalchi 2010Shalchi , 2017, Equation (21) can be simplified Since uniform mean magnetic field was considered in SH2010, only the two terms ∂S − /∂µ(µ = 0) and ∂S + /∂µ(µ = 0) occurred in the governing equations (see Equations (15) and (16) in SH2010). Since adiabatic focusing effect considered in this paper, we have to determine the four terms ∂S − /∂µ(µ = 0), ∂S + /∂µ(µ = 0), S + (µ = 0) and S − (µ = 0) in Equations (20) and (22). If the approximation formula ∂S − /∂µ ≈ ∂S − /∂µ(µ = 0) in SH2010 and the approximation method in Shalchi (2011b) are not used in Equations (20) and (22), the specific expression of S( k, µ) has to be obtained.
3.1. The anisotropic distribution function g( x, µ, µ 0 , t) By using the method of HS2014, from Equation (10) the exact anisotropic distribution function can be written as where R(µ) can be written as and the quantity M(µ) is introduced as in HS2014 For pitch-angle isotropic scattering D µµ = D(1 − µ 2 ) the quantity M(µ) can be simplified as (25)-(29) we can see that anisotropic distribution function g(µ) is an iteration function by itself. In other words, R(µ) is an iteration function by itself.

The formula of S( k, µ)
In Subsection 3.1 the anisotropic distribution function g(µ) is obtained by using the method of HS2014. In what follows, we derive the specific formula of S( k, µ), and then obtain the expressions of S + ( k, µ) and S − ( k, µ) which are the keys to express the perpendicular diffusion coefficient.
Inserting anisotropic distribution function (25) into Equation (24) we can get from which we can see that in order to get the expression for S( k, µ) we have to obtain the results of . By assuming the pitch-angle isotropic scattering D µµ = D(1 − µ 2 ) and using Equations (26)-(29) the following formula can be where the formula g( k) = d 3 xe i k· x g( x) and the original condition (15) are used.
By using formulas (38) and (39) we can get the following quantities From Equation (39) the following quantity can be obtained The formula Λ − ( k, k 0 , µ) is very complicated, and if ∂Λ − ( k, k 0 , µ)/∂µ can be neglected, the following approximation can be obtained here weak adiabatic focusing limit cosh(µξ) ≈ 1 is used. Note that the Equation (45) is precisely the approximation used in SH2010.

ALONG-FIELD ADIABATIC FOCUSING
Next, Equation (8) can be rewritten as By considering Equation (39), T ( k, k 0 ) becomes In fact, using Equations (5) and (47) the perpendicular diffusion coefficient can also be obtained, but we want to leave this for future work.
Integrating Equation (49) over k 0 for all values and then inserting it into Equation (5), the following equation can be obtained The latter equation can be rewritten as The real part is So far, Equations (50)-(53) are accurate except for some common assumptions.

The conditions of truncating R(µ)
From the above discussion we can see that R(µ) has to be truncated. Comparing to R 1 (µ) we find that R 2 (µ) and R 3 (µ) can be ignored under some conditions which can be satisfied by some special cases. In what follows, we explore these conditions. And in this paper we only explore the special cases which satisfy these conditions. Firstly, we compare the relative importance between R 1 (µ) and R 2 (µ). Let's set Z * g and T * g as the along-field spatial characteristic scale and temporal characteristic scale of the anisotropic distribution function g corresponding to the significant magnitude variation.
Here for the purpose of simplification we set the order of significant magnitude variation as O(1), i.e., ∆g = g ′ ∼ O(1). And similarly, T * F is set as the temporal characteristic scales of isotropic distribution function F corresponding to the significant magnitude variation. And we also approximate the order of significant magnitude variation as O(1), i.e., ∆F = F ′ ∼ O(1). Thus, R 1 (µ) and R 2 (µ) can be written as where the dimensionless time t ′ and dimensionless along-field distance z ′ are used. Then according to the above discussion derivatives terms in Equation (54) (55) as . Let's set T * equal to the maximum value of T * F and T * g . By comparing Equation (54) with Equation (55) we can obtain the relation vT * ≪ Z * g for the case L ≫ Z * g , and the relation vT * ≪ L for the case Z * g ≫ L. If the case L ∼ Z * g occurs, the relations vT * ≪ Z * g and vT * ≪ L have to be satisfied simultaneously. Therefore, if the condition vT * ≪ Z * g and vT * ≪ L are satisfied at the same time, we can find that R 2 (µ) can be neglected comparing to R 1 (µ) for any case. In general, if particle motion is completely passive in turbulence, stronger turbulence leads to faster change of the distribution function, i.e., stronger turbulence leads to smaller characteristic temporal scale T * . It also can be assumed that smaller gradient of anisotropic distribution function leads to larger spatial characteristic length Z * g . Therefore, for relatively strong turbulence, relatively small along-field gradient of anisotropic distribution function, and not too high energy of particles, the condition vT * ≪ Z * g might be satisfied. The condition vT * ≪ L denotes the adiabatic focusing characteristic length L is far greater than the characteristic distance vT * . Next, we only explore the cases which satisfy the requirements vT * ≪ Z * g and vT * ≪ L. In addition, we assume that perpendicular effects are not much larger than parallel effects, so we can ignore both R 2 (µ) and R 3 (µ) comparing with R 1 (µ). Then Equation (26) can be simplified as Combining Equations (25) and (56) gives Next, we operate Fourier transformation on Equation (57) where we assume that M(µ) and D µµ (µ) are all independent of the initial pitch-angle cosine µ 0 . Substituting Equation (58) into Equation (32) yields Equation (60) is the result of first-order iteration of R(µ). For more iterations of R(µ) we can get more complicated S( k, k 0 , µ) corresponding to more complicated Λ( k, µ). Therefore, if Equation (56) is used directly in Equation (32), the formula S( k, k 0 , µ) = S (0) ( k, k 0 , µ) can be obtained corresponding to Λ( k, µ) = Λ (0) ( k, µ) = 0.

Inserting Equations
with and Ω= v 2 It is noteworthy that Equation (79) is only applicable to the cases in which perpendicular diffusion is not considered and D µµ is independent of adiabatic focusing. So it is more appropriate to derive the formulas of D µµ and κ with adiabatic focusing and perpendicular diffusion effects, and then to re-deduce Equation (81). we would reserve this as our tasks in the future.

MODIFYING FACTOR a 2
For mathematical tractability we only explore the modifying factor a 2 for the two-component model (slab+2D, Matthaeus et al. 1990). For slab component contribution it is more convenient to start from Equation (71). We can easily find that the slab transport is subdiffusive, i.e., κ slab ⊥ = 0. So we only consider the 2D-component contribution. For pure two-dimensional model Equation (81) can be simplified as In this paper the following tensor of the two-dimensional (2D) magnetic turbulence is here we use the spectra of the two-dimensional modes g 2D (k ⊥ ) where q is the energy range index, s is the inertial range index, L 2D is box size, l 2D is the turnover scale, k cut is the inertial range cut-off wave number and D(s, q) is the normalized factor. The spectrum is correctly normalized for q > −1 and s > 1. The more detailed explanation can be found in Shalchi (2011c).
From Equation (85) we can find that the sign of a 2 is determined by formula W . In what follows, we explore the sign of formula W . From formulas (88) and (89) we can obtain ⊥ , which denotes the product of parallel and perpendicular mean free paths λ (0) and λ ⊥ is far greater than the turnover scale l 2D . And high energy charged particles satisfy this condition. For this case W becomes since k −s−3 max ≪ 1, the latter equation can be simplified as For q > 3 we can find η q−3 ≪ 1, then we can get For −1 < q < 3 Equation (92) can be simplified as In both special case (93) and (94) we find that W < 0, then a 2 < 0. Thus, we can see that for high energy particles the modified factor a 2 make perpendicular diffusion coefficient decrease.

The case
In this case the condition λ (0) λ (0) ⊥ ≪ l 2 2D corresponding to low energy particles should be satisfied. And we can derive the following equation from equation (90) since k −s−1 max ≪ 1, we can get from the latter equation For q > 1 we get η q−1 ≪ 1 and the latter formula can be simplified as For −1 < q < 1 we can obtain from formula (96) Clearly we find that for low energy particles the modified factor a 2 make perpendicular diffusion coefficient increase.
In summary, the specific modifying factor a 2 to perpendicular diffusion coefficient appears different forms for different particle energy. However, it is independent of energy range index q and the sign of adiabatic focusing length. Considering the discussion in Section 4.1, it is might only the results obtained in Subsection 5.2 that can satisfy the application scope of this paper.

DISCUSSION AND CONCLUSIONS
In this paper by employing the Unified NonLinear Transport (UNLT) theory in SH2010 and the method of HS2014 we explore the perpendicular diffusion of energetic charged particles in spatially varying background magnetic field, and the perpendicular diffusion coefficient with along-field adiabatic focusing is obtained. From this formula we find that along-field adiabatic focusing can introduce significant modification to the particle transport across the mean magnetic field.
Previous articles demonstrated that the parallel diffusion is decreased by along-field adiabatic focusing, and parallel diffusion coefficient is independent of the sign of adiabatic focusing (Beeck & Wibberenz 1986;Bieber & Burger 1990;Kóta 2000;Shalchi 2011a;Litvinenko 2012a,b;Shalchi & Danos 2013;He & Schlickeiser 2014). In this paper we find that for isotropic scattering and weak adiabatic limit the perpendicular diffusion coefficient for non-uniform background field is independent of the sign of adiabatic focusing length, which is similar to parallel diffusion coefficient. The perpendicular diffusion coefficient can be indicated as power series of adiabatic focusing characteristic quantity ξ and all of the modifying factors are shown as the functions of perpendicular and parallel diffusion coefficients for uniform background magnetic field. For two-component model we find that the first order modifying factor is equal to zero and the sign of the second one is determined by the energy of particles. For low energy particles the second order modifying factor is positive, and for high energy particles the modifying factor is negative.
But it must been emphasized that the perpendicular diffusion coefficient derived in this paper with three requirements: The first one is that the turbulence is relatively strong, the along-field gradient of anisotropic distribution function is small, and the energy of particles is not too high, so that vT * ≪ Z * g can be satisfied. The second one is that the perpendicular diffusion effect is not much higher than the parallel one. The third one is adiabatic focusing length is very large, i.e., vT * ≪ L. So the formulas derived in this paper might be only applicable to the special case of Section 5.2 corresponding to low energy particles. It is also noted that the specific formulas shown in this paper are only for the lowest order iteration results. By combining Equations (47) and (5) perpendicular diffusion coefficient κ ⊥ (ξ) with along-field adiabatic focusing can also be obtained. This is our future task. In fact, because we only explore the weak adiabatic focusing limit L → ∞ in this paper, formula B 0 (z) = B 0 (0)e −z/L ≈ B 0 (0) is established approximately. So the results derived by employing the approximation B 0 (z) ≈ B 0 (0) is also approximately valid.