PERPENDICULAR DIFFUSION OF COSMIC RAYS FOR A GOLDREICH–SRIDHAR SPECTRUM

Wave vectors are usually defined in the system of mean or global magnetic field. This can be a confusing point when one deals with the modern understanding of the GS95 model of turbulence. There the wave vectors should be understood more in the sense of wavelets rather than normal Fourier transforms, as the parallel and perpendicular wavenumbers are given with respect to the local direction of the magnetic field. Incidentally, such local perturbations are actually sampled by cosmic rays streaming along the actual turbulent magnetic field.

Various approaches for defining the parallel and perpendicular scales have been employed (see a comprehensive discussion of various techniques in Beresnyak & Lazarian 2009b). The essence of those techniques is defining the local direction of the magnetic field and calculating the correlations with respect to this direction. This does not involve the separation of scales assumption, as the local magnetic field can be defined at any scale. It is important that different techniques provide the same answer for the relation of the parallel and perpendicular scales of eddies (see Beresnyak & Lazarian 2009b and references therein), which, as we mentioned earlier, for the sake of historical continuity we identify with the inverse of wavenumbers k_|| and k_⊥.

At the same time, in the earlier theories (and even in the original GS95 publication) this distinction was not considered. Therefore, this issue is of potential confusion. In what follows, we consider k_|| and k_⊥ in the local system of reference whenever the GS95 model is mentioned.

The time-dependent correlation tensor of magnetic fluctuations is defined as

$$\begin{equation} P_{lm} (\vec{k},t) = \langle\delta B_l (\vec{k},t) \delta B_m^{*} (\vec{k},0)\rangle. \end{equation} \tag{ 1 }$$

A common assumption is that all tensor components of the magnetic fluctuations have the same temporal behavior, corresponding to $P_{lm} (\vec{k},t) = \Gamma (\vec{k},t) P_{lm} (\vec{k},0)$. The function $\Gamma (\vec{k},t)$ is usually called the dynamical correlation function. For this function, different models have been discussed previously. Exponential decay of perturbations is something that has been suggested by many authors and which is roughly consistent with turbulence studies. Bieber et al. (1994), for instance, have proposed the so-called damping model of dynamical turbulence in which the dynamical correlation function is given by

$$\begin{eqnarray} &&\Gamma (\vec{k},t) = e^{-t / \tau } \end{eqnarray} \tag{ 2 }$$

with the wave vector dependent correlation time $\tau = \tau (\vec{k})$. This form was also used by Shalchi et al. (2006), where the so-called nonlinear anisotropic dynamical turbulence (NADT) model has been discussed. In the GS95 model the time τ is different, thus, for instance, in Chandran (2000) the time

$$\begin{equation} \tau = \frac{l}{v_A} \left(l k_{\perp } \right)^{-2/3} \end{equation} \tag{ 3 }$$

is used, with the Alfvén speed v_A and the characteristic length l. Such dynamical correlation effects could be important in describing the propagation of charged particles along the mean magnetic field. In particular, for reproducing parallel mean free paths of low energetic electrons in the solar system, dynamical turbulence effects are essential (see, e.g., Bieber et al. 1994; Shalchi et al. 2006). The situation is different if one investigates diffusion of particles across the mean magnetic field. As shown by Shalchi et al. (2006), perpendicular diffusion coefficients are only weakly influenced by the turbulence dynamics. Since we focus on particles which propagate much faster than the Alfvén speed (v ≫ v_A) we apply the magnetostatic approximation by setting $\Gamma (\vec{k},t) =1$ and, therefore, $P_{lm} (\vec{k},t) = P_{lm} (\vec{k})$. In the following, we discuss the static tensor $P_{lm} (\vec{k})$.

According to Matthaeus & Smith (1981), the correlation tensor for arbitrary but axisymmetric turbulence is given by

$$\begin{eqnarray} &&P_{lm} (\vec{k}) = A(k_{\parallel },k_{\perp }) \left[ \delta _{lm} - {k_l k_m \over k^2} \right]. \end{eqnarray} \tag{ 4 }$$

Here we have neglected magnetic helicity. The function A(k_∥, k_⊥) describes the turbulence spectrum. In the following paragraphs, we discuss two analytical forms that have been proposed previously. Those forms have to satisfy the normalization condition

$$\begin{eqnarray} \delta B^2 & = & \int d^3 k \; \left[ P_{xx} (\vec{k}) + P_{yy} (\vec{k}) + P_{zz} (\vec{k}) \right] \nonumber \\ & = & 8 \pi \int _{0}^{\infty } d k_{\perp } \int _{0}^{\infty } d k_{\parallel } \; k_{\perp } A(k_{\parallel },k_{\perp }). \end{eqnarray} \tag{ 5 }$$

Here we have used the wave vector components along and across the mean magnetic field k_∥ and k_⊥, respectively. Equation (4) takes into account the solenoidal constraint$\nabla \cdot \vec{B} =0$. Furthermore, we have assumed axisymmetric turbulence and a symmetric spectrum with A(k_∥, k_⊥) = A(−k_∥, k_⊥).

The GS95 model does not predict the exact form of the tensor of Alfvénic perturbations. It only states that most energy will be in the state of critical balance. In what follows we shall adopt the tensor of magnetic field empirically obtained in CLV02. These authors argued that the actual tensor contains so-called Castang functions, but in a simplified form can be written as

$$\begin{eqnarray} &&\hspace{-3pt}A(k_{\parallel },k_{\perp }) = \frac{\delta B^2}{12 \pi } E_B^{-4/3} l^{-1/3} k_{\perp }^{-10/3} e^{-(l^{1/3} \left| k_{\parallel } \right|)/(E_B^{4/3} k_{\perp }^{2/3})} \Theta {\left(k_{\perp } l - 1 \right)}.\nonumber \\ \end{eqnarray} \tag{ 6 }$$

The factors in this model were chosen to satisfy the normalization constraint (Equation (5)). Furthermore, we have used the ratio of the turbulence and the mean field E_B ≡ δB/B₀ and the characteristic length scale l. The form of tensor given by Equation (6) was adopted for calculations in Yan & Lazarian (2002, 2004, 2008). In contrast, Chandran (2000) adopted a steep cutoff of the modes beyond the critical balance k_|| ∼ k^2/3_⊥ and thus underestimated the efficiency of the cosmic ray scattering.

A lot of work has been done to improve our understanding of turbulence and charged particle transport in the solar system. Here, one needs to take into account the limited inertial range of turbulent motions over which the turbulence is present. While usually in talking about turbulence one disregards the peculiarities induced by the injection scale and the dissipation scale and concentrates on the inertial range, it is not clear whether this is possible to do for the solar wind.

While numerical simulations provide evidence in favor of the Goldreich–Sridhar spectrum, other models may be applicable to describe the transient state of the solar wind fluctuations. In particular, an empirical model of slab and 2D fluctuations was proposed (see Matthaeus et al. 1990) and was widely used for describing the properties of turbulence. In this model, the 2D fluctuations may be associated with the theoretically motivated fluctuations of the weak Alfvénic turbulence (see Lazarian & Vishniac 1999; Galtier et al. 2000), while the origin of the slab Alfvén modes may be due to the particular driving of turbulence.⁶

Matthaeus et al. (1990) have shown that magnetic correlations in the solar wind have the form of a maltese cross. These observations support the assumption that there are slab modes (defined as $\delta \vec{B} (\vec{x}) = \delta \vec{B} (z)$) superposed with dominant 2D modes (defined as $\delta \vec{B} (\vec{x}) = \delta \vec{B} (x,y)$). The latter model is also known as the slab/2D composite model. In this case, the spectrum A(k_∥, k_⊥) is defined as

$$\begin{equation} A(k_{\parallel },k_{\perp }) = g^{\rm slab} (k_{\parallel }) \frac{\delta (k_{\perp })}{k_{\perp }} + g^{\rm 2D} (k_{\perp }) \frac{\delta (k_{\parallel })}{k_{\perp }} \end{equation} \tag{ 7 }$$

with the one-dimensional (1D) spectra of the slab modes g^slab(k_∥) and the 2D modes g^2D(k_⊥), respectively. Later, Weinhorst & Shalchi (2010) used an enhanced slab/2D model to reproduce the observations presented by Matthaeus et al. (1990). Although observations in the solar wind support the analytical form (Equation (7)), there has also been some theoretical work showing that the slab/2D model is indeed a good approximation for interplanetary turbulence. The theory of nearly incompressible MHD, for instance, predicts a collapse in dimensionality making turbulence in the solar wind a superposition of dominant 2D modes and a slab component (see Zank & Matthaeus 1993). The theory of plasma instabilities also predicts the existence of 2D modes (see Stockem et al. 2006).

Recently, Hunana & Zank (2010) have formulated a nearly incompressible theory of MHD in the presence of a static large-scale inhomogeneous background. This theory was primarily developed to describe solar wind turbulence where the assumption of slab/2D turbulence with the dominance of the 2D component has often been used. Those authors have shown that, compared to the homogeneous description of Zank & Matthaeus (1993), the reduction of dimensionality from three-dimensional (3D) to 2D turbulence is only weak. Close to the Sun, however, where the large-scale magnetic field can be considered as purely radial, the collapse of dimensionality to 2D is complete. It seems that in the solar system the heliocentric distance controls the structure of turbulence.

For a complete analytical description of 2D turbulence, one has to specify the two 1D spectra in Equation (7). However, we will demonstrate below that the slab modes do not contribute to the perpendicular diffusion coefficient. Thus, we only have to know the spectrum of the 2D modes. In the following, we use the model spectrum suggested by Shalchi & Weinhorst (2009). The latter authors have suggested to use

$$\begin{equation} g^{\rm 2D}(k_\perp) = \frac{2C(s,q)}{\pi } \delta B_{\rm 2D}^2 l_{\rm 2D} \frac{(k_\perp l_{\rm 2D})^{q}}{ [1 + (k_\perp l_{\rm 2D})^2]^{(s+q)/2}}. \end{equation} \tag{ 8 }$$

The parameter q is the energy range spectral index and controls the behavior of turbulence at large scales. For the inertial range spectral index s, we assume s = 5/3 corresponding to a Kolmogorov (1941) spectrum. In Equation (8), we have used the normalization function

$$\begin{equation} C(s, q) = \frac{\Gamma \left(\frac{s+q}{2} \right)}{2 \Gamma \left(\frac{s-1}{2} \right) \Gamma \left(\frac{q+1}{2} \right)} \end{equation} \tag{ 9 }$$

with the Gamma function Γ(z). In the spectrum, we have used the bendover scale l_2D and magnetic energy of the 2D modes δB²_2D. Analogously, one can define the parameters l_slab and δB²_slab to describe the slab fluctuations. There are indications from interplanetary studies (see, e.g., Bieber et al. 1996) that δB²_slab ≈ 0.2δB² and δB²_2D ≈ 0.8δB². Although these values are only valid for solar wind turbulence, we use the same values for our interstellar studies.

To derive an analytical theory for perpendicular diffusion is a difficult task (see Shalchi 2009b for a review). A certain breakthrough has been achieved by Matthaeus et al. (2003) who derived the so-called nonlinear guiding center (NLGC) theory. Although the theory provides analytical diffusion coefficients which agree with computer simulations and solar wind observations (see, e.g., Matthaeus et al. 2003; Bieber et al. 2004; Shalchi et al. 2006) the theory is problematic for the following reasons.

• 1.  
  For magnetostatic slab turbulence, it is well known that cross-field scattering is subdiffusive. One of the earlier papers which described this kind of transport is the article of Kóta & Jokipii (2000) where a compound (sub)diffusion model had been used.⁷ The original NLGC theory, however, provides a diffusive result which clearly disagrees with our understanding of particle transport.
• 2.  
  The NLGC theory is based on the following equation of motion: v_x = av_zδB_x/B₀ where a is a parameter which has to be determined by comparing results of the theory with computer simulations. At least for a = 1 this equation corresponds to the assumption that guiding centers follow magnetic field lines. In this case, however, we expect subdiffusive transport if the particle motion along the mean field is diffusive (see Shalchi 2009b for a review) and if the turbulence is assumed to be static. This is again in disagreement with the NLGC theory which predicts diffusion even for the case a = 1.
• 3.  
  If we set a = 1 corresponding to the assumption that particles follow field lines and if we suppress parallel diffusion by setting λ_∥ = ∞, one should obtain the so-called field line random walk (FLRW) limit. This limit, however, cannot be derived from the original NLGC theory (except for very special cases).

The inability of the NLGC theory to provide the correct results for those limits is problematic, because a reliable theory should provide the correct results for all cases and limits. Therefore, the NLGC theory has been improved recently. Shalchi & Dosch (2008), for instance, have combined the Newton–Lorentz equation with some of the approximations used in the original theory. The improved theory of Shalchi & Dosch (2008) is no longer based on the assumption v_x = av_zδB_x/B₀, nor does their approach contain any unknown parameters such as a. The resulting theory, however, does not explain the subdiffusive behavior in the appropriate limits.

Thus, Shalchi (2010) has revisited the problem of perpendicular diffusion by presenting a further improvement of the NLGC theory. In his work, the author has no longer assumed that fourth-order correlations can be approximated by a product of two second-order correlations as suggested by Matthaeus et al. (2003). Shalchi (2010) has shown that this approximation is, in general, incorrect and he computed fourth-order correlations from the Fokker–Planck equation. As a consequence the following integral equation for the perpendicular diffusion coefficient has been found⁸

$$\begin{equation} \kappa _{\perp } = \frac{a^2 v^2}{3 B_0^2} \int d^3 k \; \frac{P_{xx} (\vec{k})}{F(\vec{k}) + (4/3) \kappa _{\perp } k_{\perp }^2 + v/\lambda _{\parallel }} \end{equation} \tag{ 10 }$$

with

$$\begin{equation} F(\vec{k}) = (v k_{\parallel })^2 /\big(3 \kappa _{\perp } k_{\perp }^2\big). \end{equation} \tag{ 11 }$$

Here the parameter a has been used as in the original NLGC theory (see the discussion above). This result is different from the Matthaeus et al. (2003) theory, since these authors derived F(k_∥, k_⊥) = κ_∥k²_∥ and the factor 4/3 in Equation (10) does not appear. Equation (10) with Equation (11) is valid for arbitrary but magnetostatic and axisymmetric turbulence described by the tensor component $P_{xx} (\vec{k})$. As shown in Shalchi (2010), Equation (10) provides some interesting limits. If we suppress parallel scattering corresponding to v/λ_∥ = 0 and by setting a² = 1 (corresponding to the case that the particle is tied to a single magnetic field line), we find the well-known FLRW limit with

$$\begin{equation} \kappa _{\perp } = \frac{v}{2} \kappa _{\rm FL}, \end{equation} \tag{ 12 }$$

where we have used the diffusion coefficient of FLRW given by

$$\begin{equation} \kappa _{\rm FL} = \frac{1}{B_0^2} \int d^3 k \; P_{xx} (\vec{k}) \frac{\kappa _{\rm FL} k_{\perp }^2}{\big(\kappa _{\rm FL} k_{\perp }^2\big)^2 + k_{\parallel }^2}. \end{equation} \tag{ 13 }$$

This limit, which should be provided by any nonlinear theory of perpendicular diffusion, cannot be derived from the original NLGC theory. Equation (13) agrees with the nonlinear formula derived by Matthaeus et al. (1995) to describe (diffusive) FLRW.

For finite v/λ_∥ and magnetostatic slab turbulence, the new theory represented by Equations (10) and (11) provides κ_⊥ = 0 corresponding to subdiffusion. Furthermore, the theory contains the solution κ_⊥ = 0 for other turbulence models. Subdiffusive transport in real physical scenarios has been discussed in the literature (see, e.g., Zimbardo et al. 2006). Standard quasilinear theory (SQLT) can also be obtained from the Shalchi (2010) theory as special limit. A major difference between the original theory and the new theory represented by Equations (10) and (11) can be obtained for full 3D turbulence. This case is investigated in the present paper.

Equations (10) and (11) can be combined with Equation (4) to deduce

$$\begin{eqnarray} \kappa _{\perp } &=& \frac{2 \pi a^2 v^2}{3 B_0^2} \int _{0}^{\infty } d k_{\perp }\nonumber \\ &&\times \int _{0}^{\infty } d k_{\parallel } \; \frac{2 k_{\parallel }^2 + k_{\perp }^2}{k_{\parallel }^2 + k_{\perp }^2} \frac{k_{\perp } A(k_{\parallel },k_{\perp })}{F(k_{\parallel },k_{\perp }) + (4/3) \kappa _{\perp } k_{\perp }^2 + v/\lambda _{\parallel }}\nonumber \\ \end{eqnarray} \tag{ 14 }$$

with the function F(k_∥, k_⊥) defined in Equation (11).

The theory used here describes the perpendicular diffusion due to the interaction with a magnetic field which consists of a mean magnetic field superposed by turbulence. The mean field in the theory is assumed to be constant. In a spatially non-uniform mean magnetic field perpendicular diffusion is caused by the gradient and curvature drift of the guiding center (see Schlickeiser & Jenko 2010) implying that the ratio κ_⊥/κ_∥ ∝ R²_L should increase quadratically with the Larmor radius. In the uniform mean magnetic field considered in the present paper this effect does not occur and is, therefore, neglected henceforth.

By combining Equations (14) and (11) with Equation (6), one can describe perpendicular diffusion for the magnetic correlation tensor proposed by CLV02. By using the integral transformations x_⊥ = lk_⊥ and x_∥ = lk_∥, one finds after straightforward algebra

$$\begin{eqnarray} \frac{\lambda _{\perp }}{\lambda _{\parallel }} & = & \frac{a^2}{6} \left(\frac{\delta B}{B_0} \right)^{2/3} \int _{1}^{\infty } d x_{\perp } \; \int _{0}^{\infty } d x_{\parallel } \; e^{-\left(x_{\parallel } x_{\perp }^{-2/3} E_B^{-4/3}\right)} \nonumber \\ & &\times \frac{2 x_{\parallel }^2 + x_{\perp }^2}{x_{\parallel }^2 + x_{\perp }^2} \frac{x_{\perp }^{-7/3}}{\big(\lambda _{\parallel } x_{\parallel }^2\big)/\big(\lambda _{\perp } x_{\perp }^2\big) + \big(4 \lambda _{\parallel } \lambda _{\perp } x_{\perp }^2\big)/(9 l^2)+1}.\nonumber \\ \end{eqnarray} \tag{ 15 }$$

Here, we have used the relation λ_⊥ = 3κ_⊥/v.

For numerical reasons, we employ the integral transformation y = 1/x_⊥. By using x ≡ x_∥, we can easily derive

$$\begin{eqnarray} \hspace{-15pt}\frac{\lambda _{\perp }}{\lambda _{\parallel }} & = & \frac{a^2}{6} \left(\frac{\delta B}{B_0} \right)^{2/3} \int _{0}^{1} d y \; \int _{0}^{\infty } d x \; e^{-\left(x y^{2/3} E_B^{-4/3}\right)} \nonumber \\ \hspace{-15pt} & &\times \frac{2 x^2 y^2 + 1}{x^2 y^2 + 1} \frac{y^{7/3}}{x^2 y^4 \lambda _{\parallel }/\lambda _{\perp } + y^2 + 4 \lambda _{\parallel } \lambda _{\perp } / (3 l)^2}. \end{eqnarray} \tag{ 16 }$$

The perpendicular diffusion coefficient derived in Equation (16) depends on three parameters, namely, E_B ≡ δB/B₀, a², and λ_∥/l. Whereas the first two parameters can be considered as external parameters which have to be specified, the parallel mean free path has to be computed by employing a further transport theory. Different models for λ_∥ are discussed in Section 4. In Section 5, we show the results for the perpendicular diffusion coefficient.

A well-known model to approximate interplanetary turbulence is the slab/2D model as discussed in Section 2.3. In this case, the function A(k_∥, k_⊥) in Equation (14) has the form given by Equation (7). Therefore, the perpendicular diffusion coefficient can be written as a sum of slab and a 2D contribution

$$\begin{eqnarray} \hspace{-25pt} \kappa _{\perp } & = & \frac{\pi a^2 v^2}{3 B_0^2} \left[ 4 \int _{0}^{\infty } d k_{\parallel } \; \frac{g^{\rm slab} (k_{\parallel })}{F(k_{\parallel },k_{\perp }=0) + v/\lambda _{\parallel }} \right. \nonumber \\ \hspace{-25pt} && + \left. \int _{0}^{\infty } d k_{\perp } \; \frac{g^{\rm 2D} (k_{\perp })}{F(k_{\parallel }=0,k_{\perp }) + (4/3) \kappa _{\perp } k_{\perp }^2 + v/\lambda _{\parallel }} \right]. \end{eqnarray} \tag{ 17 }$$

Because we have F(k_∥, k_⊥ = 0) = ∞ we find a vanishing slab contribution corresponding to subdiffusion. Thus, the perpendicular diffusion coefficient for the two-component model is entirely controlled by the 2D modes. With F(k_∥ = 0, k_⊥) = 0 we find

$$\begin{eqnarray} &&\kappa _{\perp } = \frac{\pi a^2 v^2}{3 B_0^2} \int _{0}^{\infty } d k_{\perp } \; \frac{g^{\rm 2D} (k_{\perp })}{(4/3) \kappa _{\perp } k_{\perp }^2 + v/\lambda _{\parallel }}. \end{eqnarray} \tag{ 18 }$$

Apart from the factor 4/3 this result agrees with the formula provided by the extended NLGC theory discussed by Shalchi (2006). Therefore, the analytical results for the perpendicular diffusion coefficient derived in Shalchi et al. (2010) are also confirmed. With the spectrum of the 2D modes discussed in Equation (8), we find

$$\begin{eqnarray} \hspace{-23pt}\frac{\lambda _{\perp }}{\lambda _{\parallel }} &=& 2 a^2 C (s,q) \left(\frac{\delta B_{\rm 2D}}{B_0} \right)^2\nonumber \\ \hspace{-23pt} &&\times \int _{0}^{\infty } d x_{\perp } \; \frac{x_{\perp }^{q}}{\left(1+x_{\perp }^2 \right)^{(s+q)/2}} \frac{1}{\big(4 \lambda _{\parallel } \lambda _{\perp } x_{\perp }^2\big)/(9 l^2) +1} \end{eqnarray} \tag{ 19 }$$

with l ≡ l_2D.

In both models discussed above, we have to know λ_∥ = λ_∥(R) to solve the corresponding integral equation and to compute the perpendicular mean free path versus the magnetic rigidity R. In the following section, we discuss cosmic ray propagation along the mean magnetic field.

This simplest model for the parallel mean free path is so-called Bohm diffusion. In this approach, it is assumed that the parallel mean free path cannot become smaller than the Larmor radius of the charged particle R_L. For strong turbulence, one could assume that the parallel mean free path gets close to the smallest possible value. Therefore, we have

$$\begin{equation} \lambda _{\parallel }^{\rm Bohm} = R_L \quad \Rightarrow \quad \lambda _{\parallel }^{\rm Bohm}/l = R. \end{equation} \tag{ 20 }$$

Shalchi (2009a) has derived the Bohm limit from a nonlinear diffusion theory by assuming strong turbulent magnetic fields, i.e., δB ≫ B₀. Bohm diffusion is often used to model the particle diffusion coefficients in the theoretical description of particle acceleration at interstellar shock waves such as supernova remnants (see, e.g., Duffy 1992; Berezhko & Völk 2007).

In Figure 1, we compare the Bohm diffusion coefficient (Equation (20)) with the other models used in this paper to replace the parallel diffusion coefficient in our equations for the perpendicular mean free path.

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The formula which emerges from SQLT (Jokipii 1966) reads (see Shalchi 2009b for a detailed derivation)

$$\begin{equation} \lambda _{\parallel }^{\rm SQLT} \approx \frac{3 l_{\rm slab}}{16 \pi C(s)} \left(\frac{B_{0}}{\delta B_{\rm slab}} \right)^2 R^{2-s} \left[ \frac{8}{(2-s)(4-s)} + R^{s} \right]. \end{equation} \tag{ 21 }$$

This formula has been derived for a magnetostatic slab model. Here we have used the bendover scale of the slab modes l_slab, the strength of the turbulent magnetic field of the slab modes δB_slab, the mean magnetic field B₀, and the inertial range spectral index s. In the present paper, we assume that the characteristic length scales of the different modes are equal, i.e., l_slab ≡ l. Thus, we have R = R_L/l_slab ≡ R_L/l in Equation (21). Equation (21) has been derived for the following spectrum

$$\begin{equation} g^{\rm slab}(k_{\parallel }) = \frac{C(s)}{2 \pi } \delta B_{\rm slab}^2 l_{\rm slab} \frac{1}{[ 1 + (k_{\parallel } l_{\rm slab})^2 ]^{s/2}}. \end{equation} \tag{ 22 }$$

For the normalization function, we have used C(s) := C(s, q = 0) with C(s, q) from Equation (9). This spectrum was originally introduced by Bieber et al. (1994).

The mean free path based on quasilinear theory is compared with those derived by employing the other models for the parallel diffusion coefficients in Figure 1. Note that Equation (21) is also correct for parallel propagating Shear Alfvén waves if the restriction v ≫ v_A holds. For the turbulence and particle parameters we have employed the values shown in Table 1.

Another possibility to obtain diffusion coefficients in the ISM is to fit models to the observed data, the boron-to-carbon ratio in the Galactic cosmic ray flux. The latter procedure suggests for particle rigidities ζ = c p/q larger ζ₀≈3–4 GV a diffusion coefficient of the form

$$\begin{equation} \kappa _{\parallel }^{\rm Pheno} = \kappa _0 \left(\frac{\zeta }{\zeta _0} \right)^{0.6}. \end{equation} \tag{ 23 }$$

The normalization κ₀ depends on the particular model used to fit the observational data, e.g., Büsching & Potgieter (2008) obtain a value of κ₀ = 0.027(kpc)²(Myr)⁻¹ for ζ₀ = 4 GV, whereas Ptuskin et al. (2006) obtain a value of κ₀ = 0.073(kpc)²(Myr)⁻¹ for ζ₀ = 3 GV for their plain diffusion model.

Therefore, we have for the parallel mean free path

$$\begin{equation} \lambda _{\parallel }^{\rm Pheno} = \frac{3}{v} \kappa _{\parallel }^{\rm Pheno} = \frac{3 \kappa _0}{v} \left(\frac{\zeta }{\zeta _0} \right)^{0.6}. \end{equation} \tag{ 24 }$$

The phenomenological parallel diffusion coefficient as given in Equation (23) is only valid for ζ>ζ₀≈ 3–4 GV for the plain diffusion models (i.e., no re-acceleration, no Galactic wind) we discuss here. For these rigidities, the particles can be considered relativistic and we may approximate their speed by the speed of light, v ≈ c. As mentioned above, the rigidity ζ is defined as ζ = cp/q, whereas we have used R_L = v/Ω = pc/(qB₀). Therefore, we have ζ = B₀R_L or ζ = lB₀R. The rigidity dependence derived from the cosmic ray boron-to-carbon ratio is in agreement with results derived theoretically by Shalchi & Schlickeiser (2005) who found λ_∥ ∼ R^0.6. The phenomenological model for κ₀ = 0.027(kpc)²(Myr)⁻¹ and ζ₀ = 4 GV is shown in Figure 1 in comparison with the other models.

To compute the ratio λ_∥/l, we have to specify the bendover scale l. In the following we assume 1 pc and, consequently, we derive

$$\begin{equation} \frac{\lambda _{\parallel }^{\rm Pheno}}{l} = \frac{3 \kappa _0}{c l} \left(\frac{R}{R_0} \right)^{0.6} \end{equation} \tag{ 25 }$$

with

$$\begin{equation} R_0 = \frac{\zeta _0}{l B_0}. \end{equation} \tag{ 26 }$$

If we use l = 1 pc we have 3κ₀/(cl) ≈ 0.26. For protons and for B₀ = 0.6nT we have B₀l ≈ 3.6 × 10⁶ GV and, therefore, R₀ ≈ 10⁻⁶. Finally, we deduce

$$\begin{equation} \frac{\lambda _{\parallel }^{\rm Pheno}}{l} = 0.26 \cdot (R \cdot 10^6)^{0.6}. \end{equation} \tag{ 27 }$$

In Figure 1, this formula is compared to the other models used for the parallel mean free path for interstellar turbulence parameter values.

Here, we compute the perpendicular mean free path by solving Equation (16) numerically corresponding to the application of the turbulence tensor proposed by CLV02. Furthermore, we apply the two-component turbulence model to compute the perpendicular diffusion coefficient. In Figures 2 and 3, we show the perpendicular diffusion coefficient and the ratio λ_⊥/λ_∥ versus the parallel mean free path λ_∥/l. The latter parameter is normalized to the characteristic length scale (bendover scale) of the turbulence l. For the parameters in the theory we used a² = 1 and δB²/B²₀ = 0.5. According to our results, the tensor of CLV02 provides a perpendicular diffusion coefficient which is about a factor 5 smaller compared to the result obtained by the slab/2D model. Furthermore, we find for λ_∥ ≪ l that the ratio λ_⊥/λ_∥ is constant. The latter behavior is in agreement with previous results (see, e.g., Equation (27) in Yan & Lazarian 2008). We can also see by considering Figure 2 that the perpendicular mean free path becomes constant for λ_∥ ≫ l. This behavior is in agreement with Equation (26) of Yan & Lazarian (2008).⁹ A more detailed comparison of our new results and the limits discussed in Yan & Lazarian (2008) will be subject of future work.

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Here, we compute the diffusion coefficient across the mean magnetic field for the tensor suggested by CLV02. To replace the parallel mean free path in this formula, we employ the three different models discussed in Section 4, namely Equations (20), (21), and (27). For the turbulence parameters, we use the values shown in Table 1 for the ISM. The results for the perpendicular diffusion coefficient are shown in Figure 4 and the results for the ratio λ_⊥/λ_∥ in Figure 5. If the perpendicular mean free path is plotted versus the parameter R_L, we do not see a big difference between the results obtained using the quasilinear parallel diffusion coefficients and those obtained by using the phenomenological model. If we plot the ratio λ_⊥/λ_∥ we find a difference, especially for higher energies. The results obtained for Bohm diffusion deviate considerably from the other results. It seems that the assumption of Bohm diffusion is not appropriate for particle propagation studies in the interplanetary or interstellar system. The situation could be different in interstellar shock waves where it is common to assume Bohm diffusion (see, e.g., Berezhko & Völk 2007).

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In the previous section, we have computed the mean free paths along and across the mean magnetic field versus the parameter R which is sometimes called the dimensionless rigidity. To use parameters without units is convenient in theoretical and computational physics. If one compares theoretical results with observations, however, it is better to compute diffusion coefficients in (kpc)² Myr⁻¹ (here we used kpc = kiloparsecs and Myr = megayears) and rigidities in GV (here we used GV = Gigavolts).

With the parameter

$$\begin{equation} \alpha \,{=}\, {|q|} B_0 l_{\rm slab}, \end{equation} \tag{ 28 }$$

we can relate the magnetic rigidities ζ to the parameter R via

$$\begin{equation} \zeta = \frac{\alpha R}{|q|}. \end{equation} \tag{ 29 }$$

For the parameter values shown in Table 1, we have for galactic protons α(p⁺) ≈ 3.6 × 10⁶ GeV. Therefore, the diffusion coefficient can be converted by using

$$\begin{equation} \kappa _{\parallel } (\zeta) = 0.1 \frac{(\rm kpc)^2}{\rm Myr} \; \frac{\lambda _{\parallel }}{l_{\rm slab}}\; \frac{\zeta }{\sqrt{\zeta ^2 + \zeta _0^2}} \end{equation} \tag{ 30 }$$

with the parameter ζ₀ = E₀/|q| and the rest energy E₀.

In Figure 6, we have shown the parallel and perpendicular diffusion coefficients for the phenomenological model by using the units described above and by employing the turbulence correlation tensor of CLV02. In Figure 7, we have plotted the ratio κ_⊥/κ_∥ for the same models and parameters. As shown, the perpendicular diffusion coefficient becomes approximately constant for high particle energies. For all considered energies/rigidities, the perpendicular diffusion coefficient is much smaller than the parallel diffusion coefficient.

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Above we have investigated cross-field diffusion of charged particles in the ISM. Here we revisit the problem of perpendicular diffusion in the solar system which has been explored previously by employing standard NLGC theory and by assuming two-component turbulence (see, e.g., Bieber et al. 2004; Shalchi et al. 2006, 2010).

In solar wind physics, we usually measure the magnetic rigidity in MV = Megavolts instead of GV. Furthermore, one usually plots the mean free paths of the charged particles in AU instead of the diffusion coefficient in (kpc)² Myr⁻¹. By using the same notation as used in the previous section, we now have for interplanetary protons α(p⁺) ≈ 4 × 10³ MeV.

In Figure 8, we have shown the parallel and perpendicular mean free path for the quasilinear parallel diffusion coefficient by using the units described above and by employing the turbulence correlation tensor of CLV02. In Figure 9, we have plotted the ratio λ_⊥/λ_∥ for the same models and parameters.

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The application of SQLT to compute the diffusion coefficient is questionable. Although for the considered particle energies/rigidities the magnetostatic approximation can be used, it is unclear whether quasilinear theory is valid. There are indications that 2D modes can scatter particles due to nonlinear effects (see Shalchi et al. 2004); there are newer investigations showing that this scattering could be subdiffusive (see Shalchi et al. 2008). In any case, SQLT as used here agrees approximately with solar wind observations and can, therefore, be used for our estimations of perpendicular diffusion coefficients.

In Figure 8, we have compared our theoretical results with two values for λ_⊥ obtained from solar wind observations. One value was obtained from Ulysses measurements of Galactic protons (Burger et al. 2000). The second value had been suggested by Palmer (1982) who concluded that the average perpendicular diffusion coefficient at 1 AU heliocentric distance is κ_⊥ c/v ≈ 10²¹ cm² s⁻¹ corresponding to λ_⊥ ≈ 0.0067 AU. Although this quite good agreement between theory and observations is impressive, one should note that it has been achieved previously using standard NLGC theory in combination with the slab/2D model (see Bieber et al. 2004; Shalchi et al. 2006). The new results shown here, however, are based on an improved theory for cross-field scattering and a magnetic correlation tensor which is based on the Goldreich–Sridhar spectrum.