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

TL;DR

This paper derives a new nonlinear perpendicular diffusion coefficient for cosmic rays in non-uniform magnetic fields, showing how adiabatic focusing influences particle diffusion in astrophysical plasmas.

Contribution

It introduces a novel formula for the perpendicular diffusion coefficient considering adiabatic focusing effects, extending previous models with a second-order approximation.

Findings

Perpendicular diffusion coefficient is modified by along-field adiabatic focusing.

In the weak focusing limit, the coefficient is independent of focusing sign.

The second-order correction's sign depends on particle energy.

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 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.

Equations

$$v_x(t)=a{v}_z(t)\frac{\delta B_x[\vec{x}(t),t]}{B_0[z(t)]},$$

$${\kappa }_{\perp }={\int }_0^{\infty }dt⟨v_x(t)v_x(0)⟩.$$

$$L^{-1}(z)=-\frac{1}{B_0(z)}\frac{d{B}_0(z)}{dz}.$$

$$B_0(z)=B_0(0)e^{-z/L}.$$

$${\kappa }_{\perp }=\frac{a^2{v}^2}{B_0^2(0)}\int d^{3}k{P}_{xx}(\vec{k})T(\vec{k})$$

$$T(\vec{k})$$

$$=\frac{1}{4}{\int }_0^{\infty }dt{\int }_{-1}^{+1}d\mu \mu {\int }_{-1}^{+1}d{\mu }_0{\mu }_0\int d^{3}x{e}^{i\vec{k}\cdot \vec{x}(t)}{e}^{z(t)/L}{f}_0[\vec{x}(t),\mu ,{\mu }_0,t],$$

$$\frac{\partial f_0}{\partial t}+v\mu \frac{\partial f_0}{\partial z}=\frac{\partial }{\partial \mu }\left(D_{\mu \mu }\frac{\partial f_0}{\partial \mu }\right)-\frac{v(1-{\mu }^2)}{2L}\frac{\partial f_0}{\partial \mu }+D_{\perp }{\Delta }_{\perp }{f}_0.$$

$$T(\vec{k})=\frac{1}{2}{\int }_{-1}^{1}d\mu \mu S(\vec{k},\mu ),$$

$$S(\vec{k},\mu )=\frac{1}{2}{\int }_0^{\infty }dt{\int }_{-1}^{1}d{\mu }_0{\mu }_0\int d^{3}x{e}^{i\vec{k}\cdot \vec{x}}{e}^{z(t)/L}{f}_0[\vec{x}(t),\mu ,{\mu }_0,t].$$

$$\frac{\partial f}{\partial t}+v\mu \frac{\partial f}{\partial z}=\frac{\partial }{\partial \mu }\left[D_{\mu \mu }\frac{\partial f}{\partial \mu }-\frac{v}{2L}(1-{\mu }^2)f\right]+\frac{v\mu }{L}f+D_{\perp }{\Delta }_{\perp }f,$$

$$S(\vec{k},\mu )=\frac{1}{2}{\int }_0^{\infty }dt{\int }_{-1}^{1}d{\mu }_0{\mu }_0\Gamma [\vec{k}(t),\mu ,{\mu }_0,t].$$

$$\Gamma [\vec{k}(t),\mu ,{\mu }_0,t]=\int d^{3}x{e}^{i\vec{k}\cdot \vec{x}}f[\vec{x}(t),\mu ,{\mu }_0,t].$$

$$\frac{\partial \Gamma }{\partial t}-i{k}_{\parallel }v\mu \Gamma =\frac{\partial }{\partial \mu }\left[D_{\mu \mu }\frac{\partial \Gamma }{\partial \mu }-\frac{v}{2L}\left(1-{\mu }^2\right)\Gamma \right]+\frac{v\mu }{L}\Gamma -k_{\perp }^2{D}_{\perp }\Gamma ,$$

$$\begin{array}{rl}& {\int }_{-1}^{1}d{\mu }_0{\mu }_0\Gamma (t=\infty )-{\int }_{-1}^{1}d{\mu }_0{\mu }_0\Gamma (t=0)-i{k}_{\parallel }v\mu {\int }_{-1}^{1}d{\mu }_0{\mu }_0{\int }_0^{\infty }dt\Gamma \\ & =\frac{\partial }{\partial \mu }\left[D_{\mu \mu }\frac{\partial }{\partial \mu }{\int }_{-1}^{1}d{\mu }_0{\mu }_0{\int }_0^{\infty }dt\Gamma -\frac{v}{2L}(1-{\mu }^2){\int }_{-1}^{1}d{\mu }_0{\mu }_0{\int }_0^{\infty }dt\Gamma \right]\\ & +\frac{v\mu }{L}{\int }_{-1}^{1}d{\mu }_0{\mu }_0{\int }_0^{\infty }dt\Gamma -k_{\perp }^2{D}_{\perp }{\int }_{-1}^{1}d{\mu }_0{\mu }_0{\int }_0^{\infty }dt\Gamma .\end{array}$$

$$\Gamma (t=0)=2\delta (\mu -{\mu }_0)h({\vec{k}}_0)=2\delta (\mu -{\mu }_0)\int d^3{x}_0{e}^{i{\vec{k}}_0\cdot {\vec{x}}_0}h({\vec{x}}_0),$$

$$\begin{array}{rl}& -2\mu h({\vec{k}}_0)-i{k}_{\parallel }v\mu {\int }_{-1}^{1}d{\mu }_0{\mu }_0{\int }_0^{\infty }dt\Gamma \\ & =\frac{\partial }{\partial \mu }\left[D_{\mu \mu }\frac{\partial }{\partial \mu }{\int }_{-1}^{1}d{\mu }_0{\mu }_0{\int }_0^{\infty }dt\Gamma -\frac{v}{2L}\left(1-{\mu }^2\right){\int }_{-1}^{1}d{\mu }_0{\mu }_0{\int }_0^{\infty }dt\Gamma \right]\\ & +\frac{v\mu }{L}{\int }_{-1}^{1}d{\mu }_0{\mu }_0{\int }_0^{\infty }dt\Gamma -k_{\perp }^2{D}_{\perp }{\int }_{-1}^{1}d{\mu }_0{\mu }_0{\int }_0^{\infty }dt\Gamma ,\end{array}$$

$$h({\vec{k}}_0)=\int d^3{x}_0{e}^{i{\vec{k}}_0\cdot {\vec{x}}_0}h({\vec{x}}_0),$$

$$-\mu h({\vec{k}}_0)-i{k}_{\parallel }v\mu S=\frac{\partial }{\partial \mu }\left[D_{\mu \mu }\frac{\partial S}{\partial \mu }-\frac{v}{2L}\left(1-{\mu }^2\right)S\right]+\frac{v\mu S}{L}-k_{\perp }^2{D}_{\perp }S.$$

$$\begin{array}{rl}& -\mu h({\vec{k}}_0)-i{k}_{\parallel }v\mu S_{+}-\frac{\partial }{\partial \mu }\left[D_{\mu \mu }\frac{\partial S_{-}}{\partial \mu }\right]-2\frac{v\mu }{L}{S}_{+}+\frac{v}{2L}\left(1-{\mu }^2\right)\frac{\partial S_{+}}{\partial \mu }+k_{\perp }^2{D}_{\perp }{S}_{-}\\ & =i{k}_{\parallel }v\mu S_{-}+\frac{\partial }{\partial \mu }\left[D_{\mu \mu }\frac{\partial S_{+}}{\partial \mu }\right]+2\frac{v\mu }{L}{S}_{-}-\frac{v}{2L}\left(1-{\mu }^2\right)\frac{\partial S_{-}}{\partial \mu }-k_{\perp }^2{D}_{\perp }{S}_{+}.\end{array}$$

$$(i{k}_{\parallel }v+\frac{v}{L}){\int }_{-1}^{1}d\mu \mu S_{-}=k_{\perp }^2{\int }_{-1}^{1}d\mu D_{\perp }{S}_{+}$$

$$-\frac{1}{2}h({\vec{k}}_0)-(i{k}_{\parallel }v+\frac{v}{L}){\int }_0^{1}d\mu \mu S_{+}+\left(D_{\mu \mu }\frac{\partial S_{-}}{\partial \mu }\right)(\mu =0)-\frac{v}{2L}{S}_{+}(\mu =0)+k_{\perp }^2{\int }_0^{1}d\mu D_{\perp }{S}_{-}$$

$$=-\left(D_{\mu \mu }\frac{\partial S_{+}}{\partial \mu }\right)(\mu =0)+\frac{v}{2L}{S}_{-}(\mu =0).$$

$$-\frac{1}{2}h({\vec{k}}_0)-(i{k}_{\parallel }v+\frac{v}{L}){\int }_0^{1}d\mu \mu S_{+}+D\frac{\partial S_{-}}{\partial \mu }(\mu =0)-\frac{v}{2L}{S}_{+}(\mu =0)+2{\kappa }_{\perp }{k}_{\perp }^2{\int }_0^{1}d\mu \mu S_{-}$$

$$=-D\frac{\partial S_{+}}{\partial \mu }(\mu =0)+\frac{v}{2L}{S}_{-}(\mu =0).$$

$$f(\vec{x},\mu ,t)=F(\vec{x},t)+g(\vec{x},\mu ,{\mu }_0,t).$$

$$S(\mu ,\vec{k})=\frac{1}{2}{\int }_{-1}^{1}d{\mu }_0{\mu }_0{\int }_0^{\infty }dt\int d^{3}x{e}^{i\vec{k}\cdot \vec{x}}g(\vec{x},\mu ,{\mu }_0,t).$$

$$g(\mu)=\left[1-\frac{2e^{M(\mu)}}{\int_{-1}^{1}d\mu e^{M(\mu)}}\right]L\left(\frac{\partial{F}}{\partial{z}}-2\frac{F}{L}\right)+e^{M(\mu)}\Bigg{[}\int_{-1}^{\mu}d\nu R(\nu)-\frac{\int_{-1}^{1}d\mu e^{M(\mu)}\int_{-1}^{\mu}d\nu R(\nu)}{\int_{-1}^{1}d\mu e^{M(\mu)}}\bigg{]},$$

$$R(\mu )=R_1(\mu )+R_2(\mu )+R_3(\mu )$$

$$R_1(\mu )$$

$$R_2(\mu )$$

$$R_3(\mu )$$

$$M(\mu )=\frac{v}{2L}{\int }_{-1}^{\mu }d\nu \frac{1-{\nu }^2}{D_{\nu \nu }(\nu )}.$$

Full text

perpendicular diffusion coefficient, UNLT, adiabatic focusing

1 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 $\vec{B}_{0}=B_{0}\hat{e}_{z}$ and a turbulent component $\delta\vec{B}$ 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 (Qin & Zhang, 2014), 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; Hussein & Shalchi, 2014; Shalchi & Hussein, 2014). 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.

2 SIMPLIFICATION OF THE STANDARD FOKKER-PLANCK EQUATION

Similar as in SH2010, our starting point is the single particle motion equation (Matthaeus et al., 2003)

[TABLE]

here $B_{0}[z(t)]$ is the large-scale magnetic field which are not necessarily uniform, $\delta B_{x}$ 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 $\vec{B_{0}}\cdot\delta\vec{B}=0$ 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, 2015, 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

[TABLE]

2.1 Along-field adiabatic focusing length

The along-field adiabatic focusing length for spatially varying guiding field is shown as (Schlickeiser & Shalchi, 2008)

[TABLE]

For the limit $L\rightarrow{\infty}$ 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., $\partial{L}/\partial{z}=0$. Then the mean magnetic field can be written as

[TABLE]

For convergent background magnetic field along $z$ positive direction adiabatic focusing length satisfies $L<0$, and for divergent case $L>0$.

2.2 The formula of perpendicular diffusion coefficient

By combining Equations (1) and (2), perpendicular diffusion coefficient $\kappa_{\bot}$ can be obtained

[TABLE]

with

[TABLE]

here $\mu$ is pitch-angle cosine and $\mu_{0}$ is initial pitch-angle cosine, $P_{xx}(\vec{k})$ is the magnetic field correlation tensor. In Equation (5) the Corrsin independence hypothesis (see, Corrsin, 1959) is employed. And the quantity $T(\vec{k})$ 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 $f_{0}[\vec{x}(t),\mu,\mu_{0},t]$ in Equation (6) is the distribution function of the standard Fokker-Planck equation with adiabatic focusing effect, which is shown as follow

[TABLE]

The latter equation includes the Fokker-Planck coefficients of pitch-angle diffusion $D_{\mu\mu}$ and the perpendicular diffusion $D_{\bot}$, along-field adiabatic focusing length $L$, and the differential operator $\Delta_{\bot}=\partial^{2}/\partial x^{2}+\partial^{2}/\partial y^{2}$. However, the source term is neglected.

Equation (6) can also be written as

[TABLE]

with the quantity $S(\vec{k},\mu)$ as

[TABLE]

From Equations (5), (8) and (9) we can find that in order to get perpendicular diffusion coefficient $\kappa_{\bot}$ the formula $S(\vec{k},\mu)$ has to be obtained.

2.3 The governing equation of $\boldsymbol{S(\vec{k},\mu)}$

Let’s set $f[\vec{x}(t),\mu,\mu_{0},t]=e^{z(t)/L}f_{0}[\vec{x}(t),\mu,\mu_{0},t]$ and combining the standard Fokker-Planck equation (see, Equation (7)) we can obtain the following equation

[TABLE]

In what follows, from the latter equation we use the UNLT theory to derive the governing equation of $S(\vec{k},\mu)$. By employing the relation $f[\vec{x}(t),\mu,\mu_{0},t]=e^{z(t)/L}f_{0}[\vec{x}(t),\mu,\mu_{0},t]$ the formula of quantity $S(\vec{k},\mu)$ (see, (9)) can be rewritten as

[TABLE]

with

[TABLE]

After operating Fourier transformation on Equation (10) over spatial variable $\vec{x}$ we can get

[TABLE]

where we assume that $D_{\mu\mu}$ and $D_{\bot}$ are independent of $\vec{x}$.

To multiply the time integral of Equation (13) from 0 to $\infty$ with the initial pitch-angle cosine $\mu_{0}$, and then integrate the results over $\mu_{0}$ from $-1$ to $1$, finally we can obtain

[TABLE]

The initial distribution can be defined as

[TABLE]

which denotes that the particles have a well-defined initial pitch-angle cosine, and $h(\vec{k}_{0})$ is the distribution function in phase space for initial time $t=t_{0}$. By assuming $\Gamma(t\rightarrow\infty)$ to be independent of the initial pitch-angle cosine $\mu_{0}$ as in Shalchi (2010), we can obtain the following equation

[TABLE]

here, since $\vec{x}=\vec{x}_{0}$ for time $t=t_{0}$, Fourier transformation $\int d^{3}xe^{i\vec{k}\cdot\vec{x}}h(\vec{x})$ becomes $\int d^{3}x_{0}e^{i\vec{k}_{0}\cdot\vec{x}_{0}}h(\vec{x}_{0})$. And in the latter equation the following formula is used

[TABLE]

where the normalization condition $\int d^{3}k_{0}h(\vec{k}_{0})=1$ must be satisfied for the initial time $t=t_{0}$.

To proceed, by using the formula $S(\vec{k},\mu)=(1/2)\int_{0}^{\infty}dt\int_{-1}^{1}d\mu_{0}\mu_{0}\Gamma[\vec{k}(t),\mu,\mu_{0},t]$ (see Equation (11)) the governing equation of $S(\vec{k},\mu)$ can be got from Equation (16)

[TABLE]

Splitting $S(\mu,\vec{k})$ into an even function $S_{+}(\mu,\vec{k})$ and an odd function $S_{-}(\mu,\vec{k})$, i.e., $S(\mu,\vec{k})=S_{+}(\mu,\vec{k})+S_{-}(\mu,\vec{k})$, and then inserting this formula into Equation (18) yields

[TABLE]

To proceed, by integrating Equation (19) over $\mu$ from -1 to 1 and from 0 to 1 respectively, thereafter using the standard assumption $D_{\mu\mu}(\mu=\pm 1)=0$ we can get the following integral equations

[TABLE]

and

[TABLE]

If employing pitch-angle isotropic scattering model $D_{\mu\mu}=D\left(1-\mu^{2}\right)$ with a constant $D$ (e.g., Shalchi et al., 2009; Shalchi, 2010) and the perpendicular diffusion coefficient model $D_{\bot}=2\kappa_{\bot}|\mu|$ (e.g., Qin & Shalchi, 2014; Shalchi, 2010, 2017), Equation (21) can be simplified as

[TABLE]

Since uniform mean magnetic field was considered in SH2010, only the two terms $\partial{S_{-}}/\partial{\mu}(\mu=0)$ and $\partial{S_{+}}/\partial{\mu}(\mu=0)$ 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 $\partial{S_{-}}/\partial{\mu}(\mu=0)$, $\partial{S_{+}}/\partial{\mu}(\mu=0)$, $S_{+}(\mu=0)$ and $S_{-}(\mu=0)$ in Equations (20) and (22). If the approximation formula $\partial{S_{-}}/\partial{\mu}\approx\partial{S_{-}}/\partial{\mu}(\mu=0)$ in SH2010 and the approximation method in Shalchi (2011b) are not used in Equations (20) and (22), the specific expression of $S(\vec{k},\mu)$ has to be obtained.

3 THE FORMULA OF $\boldsymbol{S(\vec{k},\mu)}$

The distribution function $f(\vec{x},\mu,\mu_{0},t)$ of Equation (10) can be split into the isotropic distribution part $F(\vec{x},t)$ and anisotropic distribution part $g(\vec{x},\mu,\mu_{0},t)$

[TABLE]

By inserting Equation (23) into the formula of $S(\vec{k},\mu)$ (see Equation (11)), since $F(\vec{x},t)$ does not contain $\mu_{0}$, we can get

[TABLE]

If the anisotropic distribution function $g(\vec{x},\mu,\mu_{0},t)$ is obtained, the specific expressions of $S(\mu,\vec{k})$, $S_{+}(\vec{k},\mu)$, and $S_{-}(\vec{k},\mu)$ can be obtained from Equation (24). In the next Subsection we will employ the method of HS2014 to obtain $g(\vec{x},\mu,\mu_{0},t)$. Consequently, the terms $\partial{S_{-}}/\partial{\mu}(\mu=0)$, $\partial{S_{+}}/\partial{\mu}(\mu=0)$, $S_{+}(\mu=0)$, and $S_{-}(\mu=0)$ contained in Equations (20) and (22) will also be obtained.

3.1 The anisotropic distribution function $\boldsymbol{g(\vec{x},\mu,\mu_{0},t)}$

By using the method of HS2014, from Equation (10) the exact anisotropic distribution function can be written as

[TABLE]

where $R(\mu)$ can be written as

[TABLE]

with

[TABLE]

and the quantity $M(\mu)$ is introduced as in HS2014

[TABLE]

For pitch-angle isotropic scattering $D_{\mu\mu}=D(1-\mu^{2})$ the quantity $M(\mu)$ can be simplified as

[TABLE]

with $\xi=v/(2DL)$. From Equations (25)-(29) we can see that anisotropic distribution function $g(\mu)$ is an iteration function by itself. In other words, $R(\mu)$ is an iteration function by itself.

3.2 The formula of $\boldsymbol{S(\vec{k},\mu)}$

In Subsection 3.1 the anisotropic distribution function $g(\mu)$ is obtained by using the method of HS2014. In what follows, we derive the specific formula of $S(\vec{k},\mu)$, and then obtain the expressions of $S_{+}(\vec{k},\mu)$ and $S_{-}(\vec{k},\mu)$ which are the keys to express the perpendicular diffusion coefficient.

Inserting anisotropic distribution function (25) into Equation (24) we can get

[TABLE]

from which we can see that in order to get the expression for $S(\vec{k},\mu)$ we have to obtain the results of $\int_{-1}^{1}d\mu_{0}\mu_{0}\int_{0}^{\infty}dt\int d^{3}xe^{i\vec{k}\cdot\vec{x}}\int_{-1}^{\mu}d\nu R(\nu)$. By assuming the pitch-angle isotropic scattering $D_{\mu\mu}=D(1-\mu^{2})$ and using Equations (26)-(29) the following formula can be obtained

[TABLE]

with

[TABLE]

where the formula $g(\vec{k})=\int d^{3}xe^{i\vec{k}\cdot\vec{x}}g(\vec{x})$ and the original condition (15) are used.

Combining Equations (32) and (33) gives

[TABLE]

with

[TABLE]

here $\Phi(\vec{k},\vec{k}_{0},\mu)$ is Equation (34), and from Equations (35) and (36) we can find that $S(\vec{k},\mu)$ contains variable $\vec{k_{0}}$, i.e., $S(\vec{k},\mu)=S(\vec{k},\vec{k_{0}},\mu)$. From Equations (25), (34), (35) and (36) we can see that in order to get the formula of $S(\vec{k},\mu)$ we have to explore the expression of $R(\mu)$. Since $R(\mu)$ is an infinite iteration expression by itself, so $\Phi(\vec{k},\vec{k}_{0},\mu)$ is the infinite iteration function of $R(\mu)$. Therefore, $\Lambda(\vec{k},\vec{k}_{0},\mu)$ as well as $S(\vec{k},\vec{k}_{0},\mu)$ is an infinite iteration function of $R(\mu)$. Using the assumption $D_{\mu\mu}=D(1-\mu^{2})$ Equation (35) can be simplified as

[TABLE]

3.3 The formulas of $\boldsymbol{S_{+}(\vec{k},\vec{k}_{0},\mu)}$ and

$\boldsymbol{S_{-}(\vec{k},\vec{k}_{0},\mu)}$

The quantity $S(\vec{k},\mu)$ can be split as the sum of the even function $S_{+}(\vec{k},\mu)$ and the odd function $S_{-}(\vec{k},\mu)$ of $\mu$, and $S_{+}(\vec{k},\mu)$ and $S_{-}(\vec{k},\mu)$ is shown as follows

[TABLE]

where we use the even function $\Lambda_{+}(\vec{k},\vec{k}_{0},\mu)$ and the odd function $\Lambda_{-}(\vec{k},\vec{k}_{0},\mu)$ of $\mu$ which satisfy the formula $\Lambda(\vec{k},\vec{k}_{0},\mu)=\Lambda_{+}(\vec{k},\vec{k}_{0},\mu)+\Lambda_{-}(\vec{k},\vec{k}_{0},\mu)$. Obviously, $S_{+}(\vec{k},\vec{k}_{0},\mu)$ and $S_{-}(\vec{k},\vec{k}_{0},\mu)$ are all iteration function of $R(\mu)$.

By using formulas (38) and (39) we can get the following quantities

[TABLE]

From Equation (39) the following quantity can be obtained

[TABLE]

The formula $\Lambda_{-}(\vec{k},\vec{k}_{0},\mu)$ is very complicated, and if $\partial\Lambda_{-}(\vec{k},\vec{k}_{0},\mu)/\partial\mu$ can be neglected, the following approximation can be obtained

[TABLE]

here weak adiabatic focusing limit $\cosh(\mu\xi)\approx 1$ is used. Note that the Equation (45) is precisely the approximation used in SH2010.

4 THE PERPENDICULAR DIFFUSION COEFFICIENT WITH ALONG-FIELD ADIABATIC FOCUSING

Next, Equation (8) can be rewritten as

[TABLE]

By considering Equation (39), $T(\vec{k},\vec{k}_{0})$ becomes

[TABLE]

In fact, using Equations (5) and (47) the perpendicular diffusion coefficient can also be obtained, but we want to leave this for future work.

Equation (47) can be rewritten as

[TABLE]

Inserting Equations (40)-(43) and (48) into Equations (20) and (22) we can get

[TABLE]

with

[TABLE]

where the perpendicular diffusion coefficient model $D_{\bot}=2\kappa_{\bot}|\mu|$ and $2D\xi=v/L$ are employed.

Integrating Equation (49) over $\vec{k}_{0}$ for all values and then inserting it into Equation (5), the following equation can be obtained

[TABLE]

The latter equation can be rewritten as

[TABLE]

The real part is

[TABLE]

So far, Equations (50)-(53) are accurate except for some common assumptions.

From Equations (38), (39), (50) and (53) we can find that $\Xi(\vec{k},\vec{k}_{0},\xi)$ in Equation (53) is a complicated function of $\Lambda(\vec{k},\vec{k}_{0},\mu)$ which is the function of $R(\mu)$. However, $R(\mu)$ is a iteration function by itself. Therefore, in order to obtain specific expression of $\kappa_{\bot}$ we have to truncate $R(\mu)$.

4.1 The conditions of truncating $\boldsymbol{R(\mu)}$

From the above discussion we can see that $R(\mu)$ has to be truncated. Comparing to $R_{1}(\mu)$ we find that $R_{2}(\mu)$ and $R_{3}(\mu)$ 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 $R_{1}(\mu)$ and $R_{2}(\mu)$. 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., $\Delta g=g^{\prime}\sim O(1)$. 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., $\Delta F=F^{\prime}\sim O(1)$. Thus, $R_{1}(\mu)$ and $R_{2}(\mu)$ can be written as

[TABLE]

where the dimensionless time $t^{\prime}$ and dimensionless along-field distance $z^{\prime}$ are used. Then according to the above discussion derivatives terms in Equation (54) can be approximate as $\partial F^{\prime}/\partial t^{\prime}\sim\partial g^{\prime}/\partial t^{\prime}\sim O(1)$ (e.g., Fay, 1994). Since pitch-angle cosine $\mu$ belongs to the interval $[-1,1]$, so we can approximate the integral terms in Equation (55) as $[\int_{-1}^{\mu}d\nu\nu({\partial{g^{\prime}}}/{\partial{z^{\prime}}})-(1/2)\int_{-1}^{1}d\mu\mu({\partial{g^{\prime}}}/{\partial{z^{\prime}}})]\sim O(1)$ and $[\int_{-1}^{\mu}d\nu\nu g^{\prime}-(1/2)\int_{-1}^{1}d\mu\mu g^{\prime}]\sim O(1)$. 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 $vT^{*}\ll Z_{g}^{*}$ for the case $L\gg Z^{*}_{g}$, and the relation $vT^{*}\ll L$ for the case $Z^{*}_{g}\gg L$. If the case $L\sim Z^{*}_{g}$ occurs, the relations $vT^{*}\ll Z_{g}^{*}$ and $vT^{*}\ll L$ have to be satisfied simultaneously. Therefore, if the condition $vT^{*}\ll Z_{g}^{*}$ and $vT^{*}\ll L$ are satisfied at the same time, we can find that $R_{2}(\mu)$ can be neglected comparing to $R_{1}(\mu)$ 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 $vT^{*}\ll Z_{g}^{*}$ might be satisfied. The condition $vT^{*}\ll L$ 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 $vT^{*}\ll Z_{g}^{*}$ and $vT^{*}\ll L$. In addition, we assume that perpendicular effects are not much larger than parallel effects, so we can ignore both $R_{2}(\mu)$ and $R_{3}(\mu)$ comparing with $R_{1}(\mu)$. Then Equation (26) can be simplified as

[TABLE]

Combining Equations (25) and (56) gives

[TABLE]

Next, we operate Fourier transformation on Equation (57) and then insert the result into Equation (34) to yield

[TABLE]

with

[TABLE]

where we assume that $M(\mu)$ and $D_{\mu\mu}(\mu)$ are all independent of the initial pitch-angle cosine $\mu_{0}$. Substituting Equation (58) into Equation (32) yields

[TABLE]

with

[TABLE]

Equation (60) is the result of first-order iteration of $R(\mu)$. For more iterations of $R(\mu)$ we can get more complicated $S(\vec{k},\vec{k}_{0},\mu)$ corresponding to more complicated $\Lambda(\vec{k},\mu)$. Therefore, if Equation (56) is used directly in Equation (32), the formula $S(\vec{k},\vec{k}_{0},\mu)=S^{(0)}(\vec{k},\vec{k}_{0},\mu)$ can be obtained corresponding to $\Lambda(\vec{k},\mu)=\Lambda^{(0)}(\vec{k},\mu)=0$. Note that we call $S^{(0)}(\vec{k},\vec{k}_{0},\mu)$ the zeroth-order $S(\vec{k},\vec{k}_{0},\mu)$. After inserting Equation (25) into Equation (26) and using Equation (56), the first-order $S(\vec{k},\vec{k}_{0},\mu)$, or $S^{(1)}(\vec{k},\vec{k}_{0},\mu)$ can be found corresponding to the first-order $\Lambda^{(1)}(\vec{k},\mu)=0$ which is just Equation (61). If we perform n iterations of $R(\mu)$ and use Equation (56), the nth-order $S(\vec{k},\vec{k}_{0},\mu)$, i.e., $S^{(n)}(\vec{k},\vec{k}_{0},\mu)$, can be written as

[TABLE]

with

[TABLE]

Here $\Lambda^{(n-1)}(\vec{k},\mu)$ is a more complicated formula. Note that Equation (35) denotes the result for infinite iteration of $R(\mu)$.

4.2 Truncating $\boldsymbol{\Xi(\vec{k},\vec{k}_{0},\xi)}$

According to the truncation condition in above subsection, if the approximations

[TABLE]

are used, Equations (51) and (50) becomes

[TABLE]

with

[TABLE]

where the normalization condition $\int d^{3}k_{0}h(\vec{k}_{0})=1$ is used. Equation (67) with formula (68) is the nth-order result of $\kappa_{\bot}(\xi)$.

The expression $\Xi(\vec{k},\xi)$ is a very complicated function. For mathematical tractability in this paper we only consider the zeroth-order $\kappa_{\bot}(\xi)$ corresponding to $\Lambda^{(0)}(\vec{k},\mu=0)=0$ and $\partial{\Lambda_{-}^{(0)}(\vec{k},\mu)}/\partial{\mu})(\mu=0)=0$ in Equation (68). Then Equation (67) can be simplified as

[TABLE]

with

[TABLE]

From the latter equation we can see that $\kappa_{\bot}(\xi)$ is a complicated function of $\xi$. And for weak limit $|\xi|\ll 1$ Equation (69) can be written as

[TABLE]

The latter equation can be rewritten as series expansion

[TABLE]

with

[TABLE]

$\cdots\cdots$

[TABLE]

$\cdots\cdots$

Obviously, when parameter $\xi$ tends to zero Equation (71) tends to the result of Shalchi (2010). The perpendicular and parallel diffusion coefficient $\kappa_{\bot}^{(0)}$ and $\kappa_{\parallel}^{(0)}$ for uniform background magnetic field have been explored deeply (e.g., Shalchi, 2009). So it is convenient to explore the influence of adiabatic focusing on perpendicular diffusion if the modifying factors to perpendicular diffusion coefficient $\kappa_{\bot}(\xi)$, which is the function of $\kappa_{\bot}^{(0)}$ and $\kappa_{\parallel}^{(0)}$, can be obtained.

4.3 Power function expanding formula of $\boldsymbol{\kappa_{\bot}(\xi)}$

Perpendicular diffusion coefficient $\kappa_{\bot}(\xi)$ can be expanded as power function of $\xi$ as the following

[TABLE]

where $\kappa_{\bot}^{(0)}$ is the perpendicular diffusion coefficient for uniform background field, and $a_{1}$, $a_{2}$, $\cdots$ are the modifying factors to perpendicular diffusion coefficient introduced by adiabatic focusing. If $\xi\rightarrow 0$ the background magnetic field tends to be uniform. For $\xi=0$ only $\kappa_{\bot}^{(0)}$ is retained in formula (78). On the other hand, the following parallel diffusion coefficient with along-field adiabatic focusing was already derived in previous papers (see, e.g., Shalchi & Danos, 2013; Litvinenko, 2012a; He & Schlickeiser, 2014)

[TABLE]

where $\kappa_{\parallel}^{(0)}$ is the parallel diffusion coefficient for uniform background field, and $b_{2}=-1/15$.

Inserting Equations (78) and (79) into Equation (71) yields

[TABLE]

with

[TABLE]

and

[TABLE]

It is noteworthy that Equation (79) is only applicable to the cases in which perpendicular diffusion is not considered and $D_{\mu\mu}$ is independent of adiabatic focusing. So it is more appropriate to derive the formulas of $D_{\mu\mu}$ and $\kappa_{\parallel}$ with adiabatic focusing and perpendicular diffusion effects, and then to re-deduce Equation (81). we would reserve this as our tasks in the future.

5 MODIFYING FACTOR ${\bf a_{2}}$

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., $\kappa_{\bot}^{slab}=0$. So we only consider the 2D-component contribution. For pure two-dimensional model Equation (81) can be simplified as

[TABLE]

with

[TABLE]

In this paper the following tensor of the two-dimensional (2D) magnetic turbulence is used

[TABLE]

here we use the spectra of the two-dimensional modes $g^{2D}(k_{\bot})$

[TABLE]

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

[TABLE]

here formulas $\varepsilon=v^{2}l_{2D}^{2}/(3\kappa_{\parallel}^{(0)}\kappa_{\bot}^{(0)})=3l_{2D}^{2}/(\lambda_{\parallel}^{(0)}\lambda_{\bot}^{(0)})$, $x=k_{\bot}l_{2D}$, $\eta=l_{2D}/L_{2D}$ and $\Omega=v^{2}/(3\kappa_{\parallel}^{(0)})+4k_{\bot}^{2}\kappa_{\bot}^{(0)}/3$ are used.

5.1 The case $\boldsymbol{\varepsilon=3l_{2D}^{2}/(\lambda_{\parallel}^{(0)}\lambda_{\bot}^{(0)})\rightarrow 0}$

This condition $\varepsilon\rightarrow 0$, i.e., $l_{2D}^{2}\ll\lambda_{\parallel}^{(0)}\lambda_{\bot}^{(0)}$, which denotes the product of parallel and perpendicular mean free paths $\lambda_{\parallel}^{(0)}$ and $\lambda_{\bot}^{(0)}$ is far greater than the turnover scale $l_{2D}$. And high energy charged particles satisfy this condition. For this case $W$ becomes

[TABLE]

since $k_{max}^{-s-3}\ll 1$, the latter equation can be simplified as

[TABLE]

For $q>3$ we can find $\eta^{q-3}\ll 1$, then we can get

[TABLE]

For $-1<q<3$ Equation (92) can be simplified as

[TABLE]

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.

5.2 The case $\boldsymbol{\varepsilon=3l_{2D}^{2}/(\lambda_{\parallel}^{(0)}\lambda_{\bot}^{(0)})\rightarrow\infty}$

In this case the condition $\lambda_{\parallel}^{(0)}\lambda_{\bot}^{(0)}\ll l_{2D}^{2}$ corresponding to low energy particles should be satisfied. And we can derive the following equation from equation (90)

[TABLE]

since $k_{max}^{-s-1}\ll 1$, we can get from the latter equation

[TABLE]

For $q>1$ we get $\eta^{q-1}\ll 1$ and the latter formula can be simplified as

[TABLE]

For $-1<q<1$ we can obtain from formula (96)

[TABLE]

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.

6 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 $\xi$ 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 $vT^{*}\ll Z_{g}^{*}$ 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., $vT^{*}\ll L$. 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 $\kappa_{\bot}(\xi)$ 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 $L\rightarrow\infty$ in this paper, formula $B_{0}(z)=B_{0}(0)e^{-z/L}\approx B_{0}(0)$ is established approximately. So the results derived by employing the approximation $B_{0}(z)\approx B_{0}(0)$ is also approximately valid.

We are partly supported by grants NNSFC 41125016, NNSFC 41574172, and NNSFC 41374177.