SCALING THEORY FOR CROSS-FIELD TRANSPORT OF COSMIC RAYS IN TURBULENT FIELDS

Understanding and predicting the transport of charged particles in turbulent media is a key challenge in a number of related disciplines, ranging from magnetic confinement fusion research to astrophysics. Often one is interested, in particular, in the interaction of fast particles with the background electrostatic or magnetic turbulence. In the fusion context, the interaction of alpha particles or accelerated beam ions with the background microturbulence in a tokamak, leading to diffusive losses, is a topic of great interest, whereas in astrophysics, it is, e.g., the diffusion of cosmic rays in intergalactic, interstellar, or interplanetary magnetic fields. In both cases, many open questions are left for a closer investigation. In the present work, we will concentrate on the transport of fast particles in cosmic magnetic fields, applying concepts which were developed recently in the context of tokamak physics (Hauff et al. 2009; Hauff & Jenko 2009). This can be done since although the turbulent scales are extremely different, the scattering mechanisms perpendicular to the background field are basically the same.

It is widely known that cosmic rays may exhibit a wide range of energies, reaching values larger than 10²⁰ eV. Interestingly, while the acceleration of particles to energies up to about 10¹⁴ eV can be explained by shock waves in supernovae, the origin of cosmic rays with larger energies remains unknown. Moreover, their propagation through the ubiquitous and dynamic plasma background in the universe is only partially understood. In the interstellar medium and solar wind, the magnetic fields are known to exhibit a turbulent nature (see, e.g., Sofue et al. 1986; Kronberg 1994; Tu & Marsch 1995; Grasso & Rubinstein 2001; Osman & Horbury 2007, 2009), and able to scatter cosmic rays across the mean magnetic field. The latter often exhibits a spiral-like shape in the arms of spiral galaxies or in the solar wind ("Parker spiral"), but it can also be characterized by a cell-type shape (as observed in intracluster gas) or a helical shape (like in synchrotron jets; Vallée 2004). The strength of these fields ranges from 10⁻¹⁴ T in the intergalactic medium to 10⁻⁸ T in the solar wind in the neighborhood of the Earth. Although the absolute value B₀ of these mean fields is rather small, the turbulent part $\tilde{B}$ may be relatively large. For example, for the solar wind, one finds $\tilde{B} / B_0 \approx$ 0.5–1 (Tu & Marsch 1995; Zimbardo et al. 2005). Moreover, satellite measurements find that the large-scale fluctuations in the solar wind are more or less isotropic, λ_∥ ∼ λ_⊥ ∼ 10⁶ km (Tu & Marsch 1995), where λ_∥ and λ_⊥ are, respectively, the parallel and perpendicular autocorrelation lengths of the fluctuating magnetic field component.

Perpendicular or cross-field transport of charged cosmic rays was investigated in numerous articles, studying, e.g., solar energetic particles or particles accelerated at interplanetary shocks (see, e.g., Schwenn 2006; Zank et al. 2007; Li et al. 2009, and references therein). One of the first approaches to this phenomenon was the application of perturbation theory also known as quasilinear theory (QLT; see Jokipii 1966). In QLT, it is assumed that the gyrocenters of the charged particles are tied to a single magnetic fieldline while the particle motion along the mean magnetic field occurs unperturbed corresponding to a constant parallel velocity v_∥ = const. Thus, the perpendicular diffusion coefficient depends linearly on the ratio $\tilde{B}^2 / B_0^2$. An improvement of the theoretical description of cross-field transport was achieved by taking into account that particles move diffusively along the mean magnetic field. In this case, parallel diffusion suppresses perpendicular transport to a subdiffusive level. The latter effect is also known as compound diffusion and has been described analytically in several papers (see, e.g., Kóta & Jokipii 2000; Shalchi 2005; Webb et al. 2006). If one takes into account that the particle can be scattered away from a single fieldline, diffusion can be recovered. Qin et al. (2002a) have confirmed subdiffusive transport by using computer simulations. According to Qin et al. (2002b) and Zimbardo et al. (2006), the question whether transport is diffusive or not depends on the assumed turbulence geometry. This conclusion is also in agreement with the theorem of reduced dimensionality proposed by Jokipii et al. (1993). In the past, several diffusion theories were developed such as the nonlinear guiding center theory (Matthaeus et al. 2003; Shalchi & Dosch 2008) and the weakly nonlinear theory (Shalchi et al. 2004). Both theories take into account the diffusive parallel motion of the charged particle. In some cases, however, one finds a result which is independent of the parallel diffusion coefficient (Shalchi et al. 2010) and, thus, parallel scattering can be neglected. This limit should be valid for particles with high parallel velocity.

In the present paper, we will try to shed new light on some fundamental transport mechanisms of fast particles in turbulent fields. While our choice of the nominal turbulence characteristics is inspired by the solar wind, we will vary the correlation lengths of the turbulence and the test particle velocities over a wide range of scales—such that one may also apply our results to other astrophysical scenarios. Moreover, we will employ dimensionless units throughout most of this paper which should make the transfer to other physical systems easy. Throughout this work, we will assume curvature and gradient-B drifts caused by the field inhomogeneities to be negligible. Moreover, we will focus primarily on particles with v_∥ ≳ v_⊥, for which pitch angle scattering effects are subdominant. The direction of the background field B₀ will be assumed to be e_z, with e_x and e_y as the perpendicular unit vectors.

The remainder of this paper is organized as follows. A brief review of particle fieldline diffusion is given in Section 2. Here, the Kubo number is identified as a critical parameter. In Section 3, the influence of finite gyroradii on the particle transport is discussed, and two distinct regimes are found. The numerical simulation scheme for the turbulence model and the particle motion is introduced in Section 4, and results for particles which follow magnetic fieldlines are shown in Section 5. In Section 6, simulation results for the diffusion of particles with large Larmor radii are presented, and the validity of a certain analytical approach is demonstrated. In Section 7, the results are extended to particle diffusion in two-dimensional (2D) turbulence. Finally, general analytical approaches—nonrelativistic and relativistic—are presented in Section 8. We end with a short summary and some conclusions in Section 9.

In the present section, we will assume that particles strictly follow the magnetic fieldlines—and we will use the expression "particle fieldline diffusion" for this scenario which applies to situations in which the Larmor radii are smaller than the fluctuation scales of the turbulence and in which pitch angle scattering effects are subdominant, such that the particles under consideration move more or less ballistically along the magnetic background field (in the simulations shown below, we will retain the effect of pitch angle scattering, but it will not matter much for particles with small pitch angles). In the literature, this scenario is also known as fieldline random walk limit. In this case, a particle diffusion coefficient can be related to a field line diffusion coefficient (for a review, see Shalchi 2009). Fast particles with larger Larmor radii will be treated in a second step in the following section.

To begin with, we shall discuss under which conditions diffusive behavior can occur in general. For the running diffusion coefficient D_x(t), the relation,

$$\begin{eqnarray} D_x(t) &\equiv & \frac{1}{2} \frac{d}{dt} \langle x(t)^2 \rangle \nonumber \\ &=& \int _0^t d\xi \,\langle \,v_x(0) v_x(\xi) \,\rangle \equiv \int _0^t d\xi \,L_{v_x}(\xi), \end{eqnarray} \tag{ 1 }$$

can easily be established, where the square brackets denote ensemble averaging. Equation (1) is the famous Taylor formula first derived in Taylor (1920). In the literature, it is sometimes referred to as the Green–Kubo formula, according to Green (1951) and Kubo (1957). However, since it can clearly be attributed to Taylor, his name shall be used hereafter. The importance of this formula lies in the connection between the diffusion coefficient and the Lagrangian autocorrelation function of the particle velocities, $L_{v_x}(t) \equiv \langle v_x(0) v_x(t) \rangle$. This means that the (running) diffusion coefficient becomes constant when the autocorrelation function goes to zero (sufficiently fast), i.e., when the particle decorrelates (looses its memory of the initial state). As long as a positive autocorrelation remains, the diffusion coefficient will be growing, and if $L_{v_x}$ becomes negative, D_x(t) will be decreasing with time. The importance of the Taylor formula lies in the connection between the concepts of decorrelation and diffusion which it establishes. The Lagrangian autocorrelation function is determined at points following the motion of single particles, i.e., concrete trajectories. However, the latter are unknown in general. What is often known instead are the statistical properties of the turbulent fields. In the case of a turbulent magnetic field with negligible parallel fluctuations (see below) and time dependence, the parallel magnetic potential $\tilde{A}_{\parallel }(\bf x)$ acts as a stream function, i.e., the particles follow contour lines of $\tilde{A}_{\parallel }$. One can define the respective Eulerian autocorrelation function as

$$\begin{equation} E(\mathbf x) \equiv \langle \tilde{A}_{\parallel }(0) \tilde{A}_{\parallel }(\mathbf x) \rangle \,. \end{equation} \tag{ 2 }$$

In contrast to Lagrangian autocorrelation functions, their Eulerian counterparts are defined as statistical averages evaluated at fixed spatial points in the laboratory frame. They can be calculated numerically by replacing the ensemble average by an average over space and time. Now, according to Vlad et al. (1998), "The analysis of turbulent diffusion in continuous velocity fields relies on the general problem of relating the Lagrangian and the Eulerian statistical quantities. [...] This is, in a sense, the fundamental problem of turbulence." Consequently, many attempts have been made to find a way to calculate Lagrangian quantities from Eulerian ones, including the Corrsin approximation (Corrsin 1959) and the decorrelation trajectory method (Vlad et al. 1998, 2004). However, it could be shown that all of these approaches are only valid in weak turbulence situations—not in cases when effects of vortex trapping become dominant (Hauff & Jenko 2006), which is often true both in laboratory and in astrophysical turbulence. So, a simple way to calculate particle diffusivities directly from the statistical properties of the stream function does not exist, although approximations exist for certain limiting cases (see below). Nevertheless, one can extract an important insight: in the case when $E(\bf x)$ decays to zero after a correlation time τ_c or over a correlation length λ_c, L_v(t) has to do the same, since the particles loose their correlation independent of their concrete trajectories. As we will see, such a loss of memory always occurs as long as the turbulence is three dimensional (3D) and the particles can move more or less freely along the mean field B₀.

The cross-field diffusion of particles in turbulent magnetic fields is dominated (in general) not by collisions as in classical molecular diffusion, but by the motion along fluctuating fieldlines. It is thus the randomness of the stream function which enables us to adopt the concept of diffusion. Denoting the turbulent parts with a tilde, one can write $\mathbf B(\mathbf x) = \mathbf B_0 + {\bf \tilde{B}}_{\perp }(\mathbf x)$. It is a common assumption that ${\bf \tilde{B}} = {\bf \tilde{B}}_{\perp }$, which is also used in this work. Moreover, we set $\mathbf B_0 = B_0 \mathbf e_z$ and assume that the fluctuations are not time dependent.

If we assume that guiding centers follow magnetic fieldlines, their perpendicular (turbulent) velocity, ${\bf \tilde{v}}_{0,\perp }$, can be expressed as (Liewer 1985)

$$\begin{equation} {\bf \tilde{v}}_{0,\perp } = v_{\parallel } {\bf \tilde{B}}_{\perp }/B_0\,. \end{equation} \tag{ 3 }$$

It describes the deviation of a particle from the unperturbed fieldline caused by the perpendicular turbulent component of B, i.e., it follows the perturbed fieldline. As simple geometric considerations show, Equation (3) is valid for $\tilde{B} \lesssim B_0$. For larger turbulence levels, a reduction factor has to be introduced, since v_∥ actually is the velocity along the perturbed fieldline, not the unperturbed. The validity of Equation (3) for different $\tilde{B} \lesssim B_0$ will be confirmed by means of numerical simulations in Section 5.

Since we assume that the perturbed magnetic field is perpendicular, it can be expressed via the vector potential ${\bf \tilde{A}}(\mathbf x) = {\tilde{A}}_{\parallel }(\mathbf x) \mathbf e_z$. In that case, the identity $\nabla \times {\bf \tilde{A}} \equiv \nabla \tilde{A}_{\parallel } \times {\bf e}_z$ holds. So Equation (3) can be rewritten as

$$\begin{equation} {\bf \tilde{v}}_{0,\perp } = v_{\parallel } \frac{\nabla \tilde{A}_{\parallel } \times {\bf e}_z}{B_0}\,, \end{equation} \tag{ 4 }$$

which has a Hamiltonian structure (as shown, e.g., in Isichenko 1992). Interestingly, this means that if $\tilde{A}_{\parallel }(\mathbf x) = \tilde{A}_{\parallel }(x,y)$, the field lines (and thus the particles in the case of particle fieldline diffusion) encircle vortices in the potential landscape of $\tilde{A}_{\parallel }$ in the x–y direction. Thus, they are trapped as shown in Figure 1 (left), and their trajectories are deterministic and nondiffusive. In the case of $\tilde{A}_{\parallel }(x,y,z)$ (Figure 1, right), however, their trajectories are chaotic and diffusive instead, although regions of vortex circulation can still be observed. Depending on the parallel and perpendicular scales, a particle may encircle its initial vortex several times. When the vortex decays (as felt by the particle moving along B₀), the particle escapes and follows an open fieldline (an open equipotential line), until a new vortex emerges and traps the particle again. We note in passing that the structure of the E × B drift motion in a turbulent electrostatic potential with a background magnetic field in the z-direction is ${\bf \tilde{v}}_{0,\perp } = -\nabla \phi \times {\bf e}_z/B_0$, where ϕ is the electrostatic potential. This means that mathematically, electrostatic and magnetic diffusion are closely connected.

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The question which now arises is: how can the diffusivity be determined from the scales of the stream function? It turns out that there are two distinct regimes, which can be distinguished by the so-called Kubo number (Kubo 1963; Vlad et al. 1998; Zimbardo et al. 2000):

$$\begin{equation} K \equiv \frac{V_B \tau _c}{\lambda _{\perp }} = \frac{\sqrt{\big\langle {\tilde{B}}_{\perp }^2\big\rangle }}{B_0}v_{\parallel } \frac{\tau _c}{\lambda _{\perp }} \,. \end{equation} \tag{ 5 }$$

Here, V_B denotes the rms of the perpendicular velocity $\tilde{v}_{0,\perp }$, τ_c is the decorrelation time of the particle (or the magnetic potential, in the present case), and λ_⊥ is its perpendicular correlation length, where we assume isotropy in the perpendicular directions. For particle fieldline diffusion, the only decorrelation mechanism is the parallel motion, so we can set τ_c = λ_∥/v_∥. This way, we obtain

$$\begin{equation} K = \frac{{\tilde{B}}_{\perp }}{B_0} \frac{\lambda _{\parallel }}{\lambda _{\perp }}\,, \end{equation} \tag{ 6 }$$

in accordance with Zimbardo et al. (2005). For simplicity, ${\tilde{B}}_{\perp }$ shall denote the respective rms value from now on. We want to emphasize that the perpendicular motion along the vortices of the stream function does not lead to a decorrelation, since the particle returns to a place equivalent to its initial place after one turn. In the following section, we will see that alternative decorrelation mechanisms are possible for particles with gyroradii exceeding the correlation length. The mean time of flight, τ_fl ≡ λ_⊥/V_B, is the average time it takes for a particle to travel the distance of one perpendicular correlation length, i.e., to "feel" the topology of the stream function. So, the Kubo number can also be written as

$$\begin{equation} K = \frac{\tau _c}{\tau _{\mathrm{fl}}}. \end{equation} \tag{ 7 }$$

2.1. Small Kubo Number Regime

If K ≲ 1, then τ_c ≲ τ_fl. This implies that a particle decorrelates before it "feels" the structure of the vortices. Consequently, the correlation length λ_⊥ is not able to influence the transport. In this case, we can assume $L_{v_x}(t) \propto E(z(t)\!=\!v_{\parallel }t)$. Choosing an exponential decrease $E(z) = \langle \tilde{A}_{\parallel }^2 \rangle e^{-z/\lambda _{\parallel }}$ leads to $L_{v_x}(t) = V_B^2 e^{-t /\tau _c}$ with τ_c = λ_∥/v_∥, and we obtain, solving Equation (1),

$$\begin{equation} D_{\perp }(t) = V_B^2 \tau _c (1-e^{-t/\tau _c}) \,, \end{equation} \tag{ 8 }$$

where we assume isotropy in x and y, such that D_x = D_y = D_⊥. So the saturation value (t → ∞) of the diffusion coefficient is

$$\begin{equation} D_{\perp } = V_B^2 \tau _c = (\tilde{B} / B_0)^2 \lambda _{\parallel } v_{\parallel } \,, \end{equation} \tag{ 9 }$$

whereas the running diffusion coefficient for t ≪ τ_c is

$$\begin{equation} D_{\perp }(t) = V_B^2 t = (\tilde{B} / B_0)^2 v_{\parallel }^2 t \,. \end{equation} \tag{ 10 }$$

Thus, the saturation value of Equation (9) is simply the value at t = τ_c. A D(t) curve for a small Kubo number is plotted in Figure 2 (red dashed curve). The "superdiffusive" (ballistic) regime directly transitions into the diffusive regime at t = τ_c. It should be mentioned that small deviations from this expressions may occur, depending on the concrete form of $L_{v_x}(t)$. If, for example, $L_{v_x}(t) = V_B^2 e^{-t^2/\tau _c^2}$ is chosen, one obtains $D_{\perp } = \sqrt{\pi }/2 \,V_B^2 \tau _c$.

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2.2. Large Kubo Number Regime

For K ≳ 1 (τ_fl ≲ τ_c), the particles are able to "feel" the vortex structures. As soon as t ≳ τ_fl, the particles are able to circle the vortices; they are trapped. This type of vortex trapping forces 〈x²(t)〉 to decrease, which means that D_⊥(t) is reduced, too. In a magnetic potential without z dependence (2D geometry), the particles would be trapped forever (see Figure 1 (left)), and D_⊥(t) would go to zero. However, given a finite z dependence, decorrelation and saturation of D_⊥(t) occur at t = τ_c. The three successive regimes of diffusion are shown in Figure 2 in black.

An obvious question is if it is possible to come up with a similar quantitative description for the saturation value D_⊥ as in the small Kubo number regime. A simple approach would assume particle trapping for a time τ_c, and, when they are released, the particles can travel an average distance of a correlation length λ_⊥. So, setting Δx = λ_⊥ and Δt = τ_c, one would obtain D = λ²_⊥/τ_c. However, reality turns out to be more complex. Magnetic fieldlines are sometimes bound to vortices (which are typically centered around large absolute values of $\tilde{A}_{\parallel }$), and sometimes meander between the vortices, without exhibiting a clear characteristic spatial scale (typically near $\tilde{A}_{\parallel } \approx 0$). For particles on the latter structures, the small Kubo number expression may be more appropriate, since the particles are not trapped. In reality, in the large Kubo number regime there is always a mixture of trapped and untrapped particles, with exchanges between these two species on the timescale of the particle decorrelation time.

Another widely used approach is the so-called Bohm scaling (see, e.g., Misguich et al. 1987). Here, the Corrsin approximation is used and leads to a scaling D_⊥ ∝ λ_⊥V_B. It is obvious that this scaling cannot be correct, since it would imply a finite diffusion coefficient even for a magnetic potential without z dependence, which is impossible due to the trapping effects. So, simple intuitive approaches do not work. In fact, finding a scaling law for D_⊥ with respect to the characteristic turbulence parameters τ_c (or λ_∥), λ_⊥, and V_B is quite difficult. It was first given in the beautiful theoretical work of Gruzinov, Isichenko, and Kalda (GIK; Gruzinov et al. 1990) using methods of percolation theory. A general review of percolation theory, including the GIK work, is reported in Isichenko (1992). The first application to magnetic transport can also be found in Isichenko (1992), and a first related numerical study in Zimbardo et al. (2000). The treatment of GIK starts with a simple model potential ψ₀(x, y) = sin xsin y, whose separatrices constitute a periodic square lattice. A weak time dependence (modeling very large Kubo numbers) is introduced, which allows for a connection of equipotential lines across the separatrices around the saddle points. An expression for the lifetime τ_h of a contour ψ = h ≪ 1 (with the maximum of the potential normalized to unity) is estimated by τ_h ≈ hτ_c. Finally, an expression for the diffusion coefficient is found, which is

$$\begin{equation} D_{\perp } \approx V_B^{0.7} \lambda _{\perp }^{1.3} \tau _c^{-0.3} \end{equation} \tag{ 11 }$$

for K ≫ 1. In contrast to the simple expressions presented above, there is a quite complex interaction of all three statistical values. Interestingly, although the GIK estimation was achieved using a simplified model, its validity for isotropic turbulence is excellent and has been confirmed in a number of numerical simulations for electrostatic (Reuss & Misguich 1996; Reuss et al. 1998; Hauff & Jenko 2006) as well as for magnetic (Zimbardo et al. 2000) turbulence.

2.3. General Expression

Since the dimension of D is m² s^-1, a general expression for its scaling can be given by D_⊥ ∝ K^γ λ²_⊥/τ_c, or

$$\begin{equation} D_{\perp } \approx V_B^{\gamma } \lambda _{\perp }^{2-\gamma }\tau _c^{\gamma -1}\,. \end{equation} \tag{ 12 }$$

In the case of particle fieldline diffusion, this is

$$\begin{equation} D_{\perp } \approx \left(\frac{\tilde{B}}{B_0}\right)^{\gamma } \lambda _{\perp }^{2-\gamma } \lambda _{\parallel }^{\gamma -1} v_{\parallel }\,. \end{equation} \tag{ 13 }$$

Hence, the above expressions are reproduced by setting γ = 2 for K ≲ 1 and γ = 0.7 for K ≳ 1. The wrong expression D_⊥ ∝ λ²_⊥/τ_c would be obtained by setting γ = 0, which indicates that the true high Kubo number scaling indeed lies between the strict trapping approach and the low Kubo number limit. For K ≈ 1, one finds γ = 1, since in that case, τ_c ≈ λ_⊥/V_B.

Once more, we want to stress some key features of the scaling laws we just obtained. In the low Kubo number limit, there are no trapping effects, since the decorrelation of the particles occurs before they are able to circle the turbulent vortices. Therefore, the diffusion coefficient does not scale with λ_⊥, but depends only on V_B and τ_c (thus in the case of particle fieldline diffusion, on λ_∥). The diffusivity increases with V_B ($\tilde{B}/B_0$) and τ_c (λ_∥), since the distance a particle can travel before decorrelating increases in both cases. In the high Kubo number regime, particle trapping becomes important, which makes the specification of a scaling law much more complicated. Because of trapping, the perpendicular correlation length λ_⊥ influences the diffusivity now, together with V_B and τ_c. The diffusivity increases with λ_⊥ and V_B, since in the former case, the distance a particle can travel while being trapped increases, and in the latter case, a particle that is not trapped moves a wider distance. In contrast to the low Kubo number case, the diffusivity decreases with growing τ_c, since particles are trapped for a longer time, which restricts their motion. Interestingly, the dependence of the perpendicular particle diffusivity on v_∥ is linear, independent of the Kubo number.

In the previous section, we have studied the transport of particles following the magnetic fieldlines while they move unperturbed along the mean field. Such investigations have already been done in the past, e.g., by Zimbardo et al. (2000). In the following, we would like to add one level of complexity and address the question: what happens if the finite Larmor radii of the particles become larger than the perpendicular correlation lengths of the fluctuations? (For a related study, see Pommois et al. 2007.) There are two possibilities which shall be discussed in the following.

3.1. Gyroaveraging

Figure 3 shows two particle trajectories plotted over one gyration period. For T_gyroV_B ≪ ρ_g (here, T_gyro denotes the gyration time and ρ_g the Larmor radius), the gyro-orbit can be approximated by a circle. The idea of "gyroaveraging" is to average the magnetic potential over one gyration period, and then to replace the potential in Equation (4) by new "gyroaveraged" values, so that the structure of the equations of motion stays the same. For the magnetic potential, averaged over one gyro-orbit, we write

$$\begin{equation} \langle \tilde{A}_{\parallel } \rangle (\mathbf x_0) = \frac{1}{2\pi }\oint d\varphi \, \tilde{A}_{\parallel }({\bf x}_0 + {\bf {\rho }_g}(\varphi))\,, \end{equation} \tag{ 14 }$$

where x₀ denotes the value of the center of gyration and ${\bf {\rho }_g}$ the gyroradius vector pointing from x₀ to the particle position, depending on the angle φ which runs from 0 to 2π. Rewriting this expression as a sum of discrete Fourier modes, we get

$$\begin{eqnarray} \langle \tilde{A}_{\parallel } \rangle (\mathbf x_0) &=& \frac{1}{2\pi }\oint d\varphi \sum _{\mathbf k} \tilde{A}_{\parallel, \mathbf k} e^{i \mathbf k \cdot (\mathbf x_0 + {\bf {\rho }_g}(\varphi))} \nonumber \\ & = & \sum _{\mathbf k} \tilde{A}_{\parallel, \mathbf k} e^{i \mathbf k \cdot \mathbf x_0} \frac{1}{2\pi }\oint d\varphi e^{i \mathbf k \cdot {\bf {\rho }_g}(\varphi)} \nonumber \\ & = & \sum _{\mathbf k} \tilde{A}_{\parallel, \mathbf k} \,e^{i \mathbf k \cdot \mathbf x_0} J_0(k \rho _g)\,. \nonumber \end{eqnarray}$$

Here, J₀ is the Bessel function of order zero. This so-called gyroaveraging approximation can be found in the literature, e.g., in Naitou et al. (1979) and Frieman & Chen (1982). The approximation is to replace the magnetic potential in Equation (4) by the new, gyroaveraged potential, which means that the turbulent perpendicular velocity V_B is changed. In all physically relevant cases, averaging leads to a reduction of V_B. The structure of the equations remains unchanged, however, and the decorrelation process still occurs at τ_c = λ_∥/v_∥. In Hauff & Jenko (2006), reduction factors of the perpendicular velocities are calculated for electrostatic turbulence. Applying these considerations to magnetic transport, one finds the gyroaveraged effective perpendicular velocity to be

$$\begin{equation} V_B^{\mathrm{eff}} = V_B (4\sqrt{\pi } \rho _g / \lambda _{\perp })^{-1/2}\,. \end{equation} \tag{ 15 }$$

Thus, according to Equations (10)–(12), the reduction of diffusivity due to gyroaveraging effects is larger for small Kubo numbers.

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We now have to ask under which circumstances the gyroaveraging approximation is valid. We can state three conditions.

• 1.  
  The Larmor orbit has to be almost circular, i.e., T_gyroV_B ≪ ρ_g.
• 2.  
  The structure of the magnetic potential must not vary significantly during one orbit turn due to the motion into the z-direction, i.e., T_gyrov_∥ ≪ λ_∥.
• 3.  
  Even if the gyro-orbit is almost circular, the potential the particle feels during one Larmor orbit must not be significantly distinct from the potential on the circle where the averaging is done. In other words, as can be seen in Figure 3, the particle must not leave the zone of correlation after one Larmor circulation. The condition for this is T_gyroV_B ≪ λ_⊥. Since gyroaveraging becomes relevant only for ρ_g ≳ λ_⊥, we may consider the first condition as included in the third one. For clarification and future reference, we define a new parameter

  $$\begin{equation} \Xi \equiv V_B\,\frac{T_{\mathrm{gyro}}}{\lambda _{\perp }}\,. \end{equation} \tag{ 16 }$$

  Gyroaveraging is valid for Ξ ≲ 1.

3.2. Gyro-decorrelation

Next, we would like to study what happens for Ξ ≳ 1. According to Figure 3 (solid curve), the particle does not return into the correlated zone after one orbit turn, which would be necessary for the orbit averaging approach. Instead, it decorrelates after a new "gyro-decorrelation time:"

$$\begin{equation} \tau _{\mathrm{gyro}} = T_{\mathrm{gyro}} \frac{\lambda _{\perp }}{2\pi \rho _g} = \frac{\lambda _{\perp }}{v_{\perp }}\,. \end{equation} \tag{ 17 }$$

So, the diffusion approach of Equation (12) has to be rewritten, identifying τ_c = τ_gyro:

$$\begin{equation} D_{\perp } \approx V_B^{\gamma } \lambda _{\perp }^{2-\gamma }\tau _{\mathrm{gyro}}^{\gamma -1} = \left(\frac{\tilde{B}}{B_0}\right)^{\gamma } v_{\parallel }^{\gamma } \lambda _{\perp } v_{\perp }^{1-\gamma }\,. \end{equation} \tag{ 18 }$$

If τ_c = τ_gyro is inserted in Equation (5), we obtain a new effective Kubo number in the case of gyro-decorrelation:

$$\begin{equation} K_{\mathrm{gyro}} = \frac{{\tilde{B}}_{\perp }}{B_0} \frac{v_{\parallel }}{v_{\perp }}\,. \end{equation} \tag{ 19 }$$

The correlation lengths have been replaced by the respective velocities. We want to emphasize again that for the gyro-decorrelation regime to be valid, Ξ ≳ 1 is necessary, but not sufficient. The condition ρ_g ≳ λ_⊥ has to be fulfilled, too, since the particle is of course not able to decorrelate perpendicularly if its gyroradius is smaller than the corresponding correlation length. Moreover, τ_gyro ≲ τ_∥ = λ_∥/v_∥ has to be satisfied, since the faster decorrelation mechanism determines the diffusion process. Therefore, if Ξ ≳ 1, but at the same time ρ_g ≲ λ_⊥ or τ_gyro ≳ τ_∥, we expect to obtain particle fieldline diffusion. This scenario will be confirmed by means of numerical simulations which will be shown in the following.

For the numerical simulations presented below, the relevant z component of the magnetic potential, $\tilde{A}_{\parallel }$, is created by superposing a sufficiently large number of random harmonic waves:

$$\begin{equation} \tilde{A}_{\parallel }({\bf x}) = \sum _{i=1}^N A_i \sin ({\bf k}_i\; {\bf \cdot }\; {\bf x} + \varphi _i)\,. \end{equation} \tag{ 20 }$$

In most simulations, we have chosen N = 1000. For convenience, the form of the spectrum is modeled as

$$\begin{equation} A_i = A_{\mathrm{max}} e^{-((k_x^2 + k_y^2)\lambda _{\perp }^2 + k_z^2 \lambda _{\parallel }^2) / 8}\,, \end{equation} \tag{ 21 }$$

so that the correlation lengths λ_⊥ and λ_∥ are identical to the e-folding lengths of the autocorrelation function in real space. The k_i values are chosen randomly inside an ellipsoid in k space for which A_i > 0.04 · A_max, and A_max is chosen to obtain the desired rms value of $\tilde{B}_{\perp }$. The fluctuation level of the magnetic field is chosen to be $\tilde{B}_{\perp } / B_0 \equiv 0.5$, reflecting the situation in the solar wind (Tu & Marsch 1995; Zimbardo et al. 2005). The Kubo number is varied by changing λ_∥ and λ_⊥, and the particles are inserted with distinct initial velocities v_∥ and v_⊥. For simplicity, the charge-to-mass ratio is chosen e/m ≡ 1, and B₀ ≡ 1, so that T_gyro = 2π and ρ_g = v_⊥. That way, the equation of motion for a gyrating particle is

$$\begin{equation} \dot{\mathbf {v}} = \mathbf {v} \times \mathbf {B} = \mathbf {v}\times \mathbf {e}_z + \mathbf {v}\times (\nabla \tilde{A}_{\parallel } \times \mathbf {e}_z)\,, \end{equation} \tag{ 22 }$$

which is solved numerically via a standard fourth-order Runge–Kutta method. For the following particle simulations, we will compare the exact particle diffusion coefficients obtained by Equation (22) with the ones obtained using particle fieldline diffusion as described by Equation (4).

Before we go on, we would like to briefly discuss two important details in the context of these simulations. First, the effect of pitch angle scattering is fully retained in all simulations. We will find, however, that it is a subdominant effect as long as v_∥ ≳ v_⊥. This is not at all an assumption, but a simulation result. Therefore, the theoretical description in this small pitch angle regime does not have to treat pitch angle scattering, and it will in fact be neglected. In studies that average over an isotropic distribution function of particles, it does play a role, however. Second, the choice of the amplitude distribution (for convenience, we choose a Gaussian distribution, since the corresponding spatial autocorrelation function is well behaved already for relatively small values of N) does not affect our results and conclusions as long as the pitch angles are sufficiently small, which is exactly the regime we are most interested in. This statement is based on careful tests. We have repeated some of the simulations shown below with a power-law spectrum as it is described in Pommois et al. (2005) instead of a Gaussian spectrum, and have gotten essentially the same answers for both running and saturated diffusion coefficients as long as the dimensionless parameters are the same. This finding is in line, e.g., with quasilinear expectations for the pitch angle diffusion coefficient D_μμ for a simple slab model (see, e.g., Shalchi 2009, Equation (3.36)) which vanishes for small pitch angles, independent of the choice of the amplitude spectrum. For a Gaussian k_∥ spectrum, in particular, one can show that D_μμ is not affected much for small pitch angles unless the perpendicular correlation length greatly exceeds the Larmor radius—and in this case, one expects to recover the well-known particle fieldline diffusion scenario, anyways, as discussed in the previous section. The more interesting new effect of gyro-decorrelation occurs in parameter regimes for which the choice of the amplitude spectrum is not critical.

We now turn to the numerical results. As a first test, we perform simulations of particles basically following the magnetic fieldlines for fields with both a small ($\tilde{B}_{\perp }/B_0 \equiv 0.001$) and a larger ($\tilde{B}_{\perp }/B_0 \equiv 0.5$) fluctuation level. While the field strength is kept fixed, the parallel and perpendicular correlation lengths are varied separately, and the diffusion coefficient is measured as a function of the Kubo number. The diffusion coefficient is calculated according to Equation (1) for a sufficiently large number of particles (typically of the order of 10⁴) in order to ensure smooth curves. From the curves D_⊥(λ_∥) and D_⊥(λ_⊥), the scaling exponent γ is then determined according to Equation (13). The resulting D_⊥(K) curves are plotted in Figures 4 and 5. For all curves, the scaling presented in Section 2 is reproduced with high accuracy (γ = 2 for K ≲ 1, γ = 0.7 for K ≳ 1). Moreover, the approach of Equation (4) is found to be justified not only for $\tilde{B}_{\perp }/B_0 \ll 1$, but also for $\tilde{B}_{\perp }/B_0 \lesssim 1$.

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We note that this scaling behavior for magnetic transport has already been found by Zimbardo et al. (2000). However, in contrast to Zimbardo et al. (2000, 2005), we did not find anomalous transport down to our lowest Kubo number K = 0.015. In our simulations, the transport became subdiffusive for K ≲ 0.1 only if the number of partial waves in Equation (20) was reduced to N = 100. We attribute this effect to the dominance of a very small number of waves, just shaking the particles sinusoidally instead of randomly. This can be avoided including more modes (this seems to be true for both Gaussian and power-law spectra). In the following section, we will examine how the results change if finite Larmor radii are taken into account.

We calculate the full particle trajectories according to Equation (22) and determine the saturated diffusion coefficient, just as before. This time, $\tilde{B}_{\perp }/B_0 \equiv 0.5$ for all simulations, and the Kubo number is varied again via varying either λ_∥ or λ_⊥. Now, the particle's initial velocity is a critical parameter, too. We present simulations for three initial conditions, choosing the pair (v_⊥, v_∥) to be (0.05, 0.1), (0.05, 1), or (5, 10). Since we want to keep pitch angle scattering effects negligible, v_∥ ≳ v_⊥ for all cases. The results of these simulations are shown in Figures 6–8. Here, the diffusion coefficient is plotted versus the parallel or perpendicular correlation length (depending on which one was varied). Moreover, the scaling exponent γ is determined from these curves according to Equation (13), assuming particle fieldline diffusion. (To avoid confusion, we remark that this is just done to quantify the deviations from particle fieldline diffusion. In the case of gyro-decorrelation, γ actually has to be determined via Equation (18). This is done in the following estimations of the diffusion coefficient.) Via this approach, the following features can be observed. First, for small Kubo numbers the particle fieldline diffusion scaling (γ = 2) discussed in the last section is reproduced. Second, for large Kubo numbers only one out of six curves exhibits the expected γ = 0.7 scaling, while the other five approach a value of γ ≈ 1. Since γ = 1 implies D_⊥ ∝ λ_⊥, it is possible that the gyro-decorrelation scaling according to Equation (18) is valid in that case, not the particle fieldline diffusion scaling. In order to check this conjecture and to pursue an analytical approach, we first have to clarify the behavior of some critical parameters defined in Section 3. In Figures 9–11, the parameters Ξ, ρ_g/λ_⊥, and τ_∥/τ_gyro, respectively, are plotted versus the variable correlation length (and versus the respective Kubo number) for the curves already presented in Figures 6–8. The Kubo number in the inset of the curve is calculated via Equation (6). We will go through all six cases in the following.

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Case 1. (v_⊥, v_∥) = (0.05, 0.1), λ_∥ varied (Figure 6, black curve). Ξ ≲ 1 for all parallel correlation lengths (Kubo numbers), so normal particle fieldline diffusion holds. Gyroaveraging is negligible, since ρ_g ≪ λ_⊥ for all cases. Inserting the constant parameters for $\tilde{B}_{\perp }/B_0$ and λ_⊥ into Equation (13), we obtain D ∼ 0.025 λ_∥ for K ≲ 1 and D ∼ 0.06 λ^−0.3_∥ for K ≳ 1, which fits the simulation data quite well.

Case 2. (v_⊥, v_∥) = (0.05, 0.1), λ_⊥ varied (Figure 6, red curve). For K ≳ 100 (λ_⊥ ≲ 0.05), all three conditions for a gyro-decorrelation are fulfilled, i.e., Ξ ≳ 1, ρ_g/λ_⊥ ≳ 1, τ_∥/τ_gyro ≳ 1. So below K = 100 (above λ_⊥ = 0.05), we expect particle fieldline diffusion to be valid, leading to D ∼ 0.25 for K ≲ 1. Above K = 100 (below λ_⊥ = 0.05), we have gyro-decorrelation. Here, we have to calculate a new effective Kubo number according to Equation (19), which gives K_gyro ≡ 1 for all values of λ_⊥. So we have to set γ = 1 in Equation (18) and obtain $D \sim (\tilde{B}_{\perp }/B_0) v_{\parallel } \lambda _{\perp } = 0.05\,\lambda _{\perp }$, which again fits the simulated curve quite well. We note in passing that our analytical scaling laws have the tendency to be too large by a factor of 2–3 for all the curves discussed here. This may well be due to the fact that correlation lengths and mean magnetic velocities can be defined in slightly different ways, leading to moderate deviations. However, the latter can be corrected easily by introducing a universal prefactor of the order of 0.4.

Case 3. (v_⊥, v_∥) = (5, 10), λ_∥ varied (Figure 8, black curve). Although Ξ ≳ 1 for all values of λ_∥, τ_∥/τ_gyro ≲ 1 for λ_∥ ≲ 2 (K ≲ 1). In this case, particle fieldline diffusion holds, leading to D ∼ 2.5 λ_∥. For λ_∥ ≳ 2, we have K_gyro ≡ 1; thus γ = 1 and according to Equation (18), D ∼ 5. Both analytical results fit the simulated curves quite well.

Case 4. (v_⊥, v_∥) = (5, 10), λ_⊥ varied (Figure 8, red curve). Ξ ≲ 1 for λ_⊥ ≳ 30 (K ≲ 0.2). For this case, we find D ∼ 25 (Equation (13)). The three conditions for gyro-decorrelation are fulfilled for λ_⊥ ≲ 5 (K ≳ 1), so for this case, we find (together with K_gyro ≡ 1) D ∼ 5 λ_⊥. Both values are again in good agreement with the simulated curve.

Case 5. (v_⊥, v_∥) = (0.05, 1), λ_∥ varied (Figure 7, black curve). Ξ ≳ 1 for all values of λ_∥, however, ρ_g/λ_⊥ ≲ 1, so that the particles still follow the fieldlines for all values. This means that Equation (13) holds, and depending on the Kubo number, we find D ∼ 0.25 λ_∥ for small λ_∥ (small K) and D ∼ 0.61 λ^−0.3_∥ for large λ_∥ (large K). Again, this fits the simulation results quite well.

Case 6. (v_⊥, v_∥) = (0.05, 1), λ_⊥ varied (Figure 7, red curve). Ξ ≲ 1 for λ_⊥ ≳ 3 (K ≲ 2). This gives D ∼ 2.5 according to Equation (13). The gyro-decorrelation condition is fulfilled only for λ_⊥ ≲ 0.05, since then λ_⊥ ≲ ρ_g. According to Equation (18), this gives D ∼ 0.25 λ_⊥, setting γ = 0.7, since K_gyro ≡ 10. Again, the correspondence to the numerical curves is quite good.

The preceding discussion of the simulation results presented in Figures 6–8 has shown that for a wide range of initial particle velocities and regimes, the findings can be explained by identifying the relevant decorrelation mechanism. One mainly has to determine whether the particles follow the magnetic fieldlines (Ξ ≲ 1 or—if at the same time ρ_g/λ_⊥ ≲ 1 and τ_∥/τ_gyro ≲ 1—Ξ ≳ 1), or if they decorrelate during their gyration motion (Ξ ≳ 1 and ρ_g/λ_⊥ ≳ 1 and τ_∥/τ_gyro ≳ 1). In the first case, Equation (13) holds, whereas in the latter, one must use Equation (18) instead. In these equations, γ has to be determined according to the Kubo number. Here, one may employ Equation (6) for the particle fieldline diffusion case, and Equation (19) for the gyro-decorrelation case.

Once more, we would like to note that we have found pitch angle scattering effects to be negligible for v_∥ ≳ v_⊥, and that this finding does not depend on our choice of the amplitude spectrum (as discussed above). Simulations with (v_⊥, v_∥) = (0.05, 0.01), on the other hand, lead to an irregular behavior of D_⊥(K), showing that the scaling laws presented here do not hold in the v_∥ ≲ v_⊥ regime. Thus, in studies that average over an isotropic distribution function of particles, pitch angle scattering effects do play a role, while they do not in cases where one wants to focus on particles with small pitch angles (this might be the case, e.g., for certain issues concerning solar energetic particles).

We now want to examine one particular aspect of particle diffusion in a 2D turbulent field, namely the existence of subdiffusive regimes. To this end, we create random fields as described in Section 4, but dropping the z dependence: $\tilde{A}_{\parallel }(x,y)$. As we have seen in Figure 1 (left), this means that fieldlines are trapped in vortices. Is this also true for particles? In Figure 12, the simulation results for the running diffusion coefficient are plotted for four different initial conditions (v_∥, v_⊥), together with the value of Ξ calculated according to Equation (16). As can be seen, the transport is subdiffusive for Ξ ≲ 1, as expected from the behavior of the magnetic fieldlines. For Ξ ≳ 1, however, the transport becomes diffusive, and the running diffusion coefficient saturates at the gyro-decorrelation time, τ_gyro ∼ λ_⊥/v_⊥. The particles are no more bound to the magnetic fieldlines, but have a stochastic motion on their own. This observation seems to be in contrast to the work of Jokipii and co-workers (Jokipii et al. 1993; Giacalone & Jokipii 1994; Jones et al. 1998), which states that "when the field turbulence is independent of one coordinate, the motion of particles across the magnetic field is essentially zero." We think, however, that the assumptions made in these works no longer hold for Ξ ≳ 1.

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Until now, we have studied the fundamental principles of particle diffusion in turbulent magnetic fields based on decorrelation models. In this section, more general expressions for the diffusion coefficients, depending on a number of parameters of the turbulent field as well as the particles' initial conditions, shall be developed. Here, we will again assume that pitch angle diffusion effects may be neglected—which is a good approximation for small pitch angles.

In the following, we drop the assumed ratio e/m = 1 and denote the particle mass with m and the absolute value of its charge with e. The cosine of the pitch angle of a particle is defined as

$$\begin{equation} \mu \equiv v_{\parallel } / v = \sqrt{E_{\parallel } / E}\,, \end{equation} \tag{ 23 }$$

and we will characterize the particle's initial conditions by (E, μ) instead of (v_∥, v_⊥). Consequently, we have

$$\begin{equation} V_B = \frac{\tilde{B}_{\perp }}{B_0}\mu \left(\frac{2E}{m}\right)^{1/2}\,. \end{equation} \tag{ 24 }$$

8.1. Nonrelativistic Case

The general expression for the perpendicular diffusion coefficient is (see Equation (12))

$$\begin{eqnarray} D_{\perp } & = & V_B^{\gamma }\lambda _{\perp }^{2-\gamma } \tau ^{\gamma -1} \nonumber \\ & = & \left(\frac{\tilde{B}_{\perp }}{B_0} \right)^{\gamma } \mu ^{\gamma } \left(\frac{2E}{m}\right)^{\gamma /2} \lambda _{\perp }^{2-\gamma } \tau ^{\gamma -1} \,. \end{eqnarray} \tag{ 25 }$$

γ depends on K or on K_gyro as described in Section 2.3 depending on whether particle fieldline diffusion or gyro-decorrelation dominates. For the decorrelation time τ, there are two possibilities. In the case of particle fieldline diffusion, we have

$$\begin{equation} \tau _{\parallel } = \frac{\lambda _{\parallel }}{v_{\parallel }} = \frac{\lambda _{\parallel }\sqrt{m}}{\mu \sqrt{2E}}\,, \end{equation} \tag{ 26 }$$

leading to

$$\begin{equation} D_{\perp } = \left(\frac{\tilde{B}_{\perp }}{B_0} \right)^{\gamma } \mu \sqrt{\frac{2}{m}} \lambda _{\parallel }^{\gamma -1} \lambda _{\perp }^{2-\gamma } \sqrt{E}\,. \end{equation} \tag{ 27 }$$

Including gyroaveraging effects as described in Section 3.1, we obtain (see Equation (15))

$$\begin{eqnarray} D_{\perp } &=& \left(\frac{\tilde{B}_{\perp }}{B_0} \right)^{\gamma } \mu \sqrt{\frac{2}{m}} \lambda _{\parallel }^{\gamma -1} \lambda _{\perp }^{2-\gamma } \sqrt{E} \nonumber \\ &&\times \left(\frac{\lambda _{\perp }}{4\sqrt{\pi }\rho _g} \right)^{\gamma /2} \,1.73^{2-\gamma } \nonumber \\ &=& 1.73^{2-\gamma } \left(\frac{\tilde{B}_{\perp }}{B_0} \right)^{\gamma } \frac{\mu }{(1-\mu ^2)^{\gamma /4}} m^{-1/2-\gamma /4} \nonumber \\ &&\times \left(\frac{eB}{4\sqrt{\pi }}\right)^{\gamma /2} \lambda _{\parallel }^{\gamma -1} \lambda _{\perp }^{2-\gamma /2} (2E)^{1/2-\gamma /4} \,. \end{eqnarray} \tag{ 28 }$$

The factor 1.73^2−γ comes from a gyroaveraging correction of the perpendicular correlation length (Hauff & Jenko 2006). Note that the gyroaveraging approximation is valid only for ρ_g ≳ λ_⊥. If Ξ ≳ 1 (and ρ_g/λ_⊥ ≳ 1, τ_∥/τ_gyro ≳ 1), the gyro-decorrelation mechanism is valid, and we have to chose the new decorrelation time

$$\begin{equation} \tau _{\mathrm{gyro}} = \frac{\lambda _{\perp }}{v_{\perp }} = \frac{\lambda _{\perp }\sqrt{m}}{\sqrt{1-\mu ^2}\sqrt{2E}}\, \end{equation} \tag{ 29 }$$

and thus obtain

$$\begin{equation} D_{\perp } = \left(\frac{\tilde{B}_{\perp }}{B_0} \right)^{\gamma } \frac{\mu ^{\gamma }}{(1-\mu ^2)^{\gamma /2-1/2}} \sqrt{\frac{2}{m}} \lambda _{\perp } \sqrt{E}\,. \end{equation} \tag{ 30 }$$

All three expressions for D_⊥ are distinct not only in their dependence on energy and pitch angle, but also in their dependence on the parallel or perpendicular correlation lengths. Interestingly, the dependence on the particle's energy is the same for particle fieldline diffusion and for gyro-decorrelation and does not depend on γ, i.e., on the Kubo number.

The critical parameter Ξ can be expressed generally via

$$\begin{equation} \Xi = V_B \frac{T_{\mathrm{gyro}}}{\lambda _{\perp }} = \frac{\tilde{B}_{\perp }}{B_0} \mu \frac{\sqrt{2 m E} 2 \pi }{eB \lambda _{\perp }}\,. \end{equation} \tag{ 31 }$$

8.2. Relativistic Case

For electrons with E ≳ 100 keV, relativistic effects become important. Since cosmic rays can have much larger energies, even in the solar wind, it is important to take them into account, too. We define the Lorentz factor

$$\begin{equation} \kappa \equiv \frac{1}{ \sqrt{ 1-\frac{v^2}{c^2}} } = \frac{E}{m_0 c^2} + 1\,, \end{equation} \tag{ 32 }$$

where c is the velocity of light and E is the kinetic energy of the particles. Since the relativistic mass is given by m = κ m₀ (here, m₀ is the rest mass), and the particle velocity by $v = c \sqrt{1-1/\kappa ^2}$, the quantities ρ_g and T_gyro have to be modified. Thus we get

$$\begin{equation} D_{\perp } = \left(\frac{\tilde{B}_{\perp }}{B_0} \right)^{\gamma }\mu ^{\gamma } (1-1/\kappa ^2)^{\gamma /2}c^{\gamma } \lambda _{\perp }^{2-\gamma } \tau ^{\gamma -1}\,, \end{equation} \tag{ 33 }$$

with the decorrelation times

$$\begin{equation} \tau _{\parallel } = \frac{\lambda _{\parallel }}{\mu \,c \sqrt{1-1/\kappa ^2}} \end{equation} \tag{ 34 }$$

and

$$\begin{equation} \tau _{\mathrm{gyro}} = \frac{\lambda _{\perp }}{\sqrt{1-\mu ^2}\,c\sqrt{1-1/\kappa ^2}}\,. \end{equation} \tag{ 35 }$$

Thus, in analogy to the nonrelativistic case, we obtain

$$\begin{equation} D_{\perp } = \left(\frac{\tilde{B}_{\perp }}{B_0} \right)^{\gamma } \mu \,c\, \lambda _{\parallel }^{\gamma -1} \lambda _{\perp }^{2-\gamma } \sqrt{1-1/\kappa ^2} \end{equation} \tag{ 36 }$$

for particle fieldline diffusion,

$$\begin{eqnarray} D_{\perp } &=& \left(\frac{\tilde{B}_{\perp }}{B_0} \right)^{\gamma } \frac{\mu }{(1-\mu ^2)^{\gamma /4}} c^{1-\gamma /2} \left(\frac{eB}{4\sqrt{\pi }m_0} \right)^{\gamma /2} \nonumber \\ &&\times 1.73^{2-\gamma } \lambda _{\parallel }^{\gamma -1} \lambda _{\perp }^{2-\gamma /2} \frac{(\kappa ^2-1)^{1/2-\gamma /4}}{\kappa } \end{eqnarray} \tag{ 37 }$$

for gyroaveraged particle fieldline diffusion, and

$$\begin{equation} D_{\perp } = \left(\frac{\tilde{B}_{\perp }}{B_0} \right)^{\gamma } \mu ^{\gamma }(1-\mu ^2)^{1/2-\gamma /2} c\, \lambda _{\perp } \sqrt{1-1/\kappa ^2} \end{equation} \tag{ 38 }$$

for gyro-decorrelation. The validity parameter Ξ from Equation (31) is modified to be

$$\begin{equation} \Xi = V_B \frac{T_{\mathrm{gyro}}}{\lambda _{\perp }} = \frac{\tilde{B}_{\perp }}{B_0} \mu \,c \sqrt{\kappa ^2-1} \frac{2\pi m_0}{e B \lambda _{\perp }} \,. \end{equation} \tag{ 39 }$$

Comparing Equations (36)–(38), one finds that, for particle fieldline diffusion as well as for gyro-decorrelation, the diffusivity is independent of the particle energy for large energies (κ → ∞), in contrast to the gyroaveraging approach. The dependence on the correlation lengths is different for all three cases. For very large energies, Ξ → ∞ and ρ_g → ∞; however, $\tau _{\parallel } / \tau _{\mathrm{gyro}} \rightarrow \frac{\lambda _{\parallel }}{\lambda _{\perp }} \frac{\sqrt{1-\mu ^2}}{\mu }$, which means that the gyro-decorrelation approach is expected to be valid for sufficiently large pitch angles and/or a large ratio λ_∥/λ_⊥.

In the present paper, we have studied some important aspects of the perpendicular transport of energetic charged particles (e.g., cosmic rays) in turbulent magnetic fields, focusing on regimes where pitch angle scattering effects are subdominant, i.e., on particles with v_∥/v_⊥ ≳ 1 (like solar energetic particles). It turned out that the respective physics can be qualitatively understood and quantitatively predicted via a careful analysis of the underlying decorrelation mechanisms. To this aim, both direct numerical simulations and scaling theories have been employed and found to be in good agreement. In particular, it was found that the particle diffusivity can be determined by identifying the fastest decorrelation mechanism, which is set by the particles' initial conditions and the statistical properties of the turbulence. While particles with small Larmor radii follow the magnetic fieldlines, the situation is more complicated if their Larmor radii exceed the perpendicular turbulence correlation lengths. If gyroaveraging is valid, the particles follow averaged magnetic fieldlines, leading to a reduction of the perpendicular diffusivity. If gyroaveraging is not valid, however, a newly identified perpendicular decorrelation mechanism applies, which leads to a distinct diffusive-type motion even if the turbulent fluctuations have a reduced dimensionality. This is the case if the characteristic parameters Ξ and τ_∥/τ_gyro both exceed unity. Explicit formulae—both in the nonrelativistic and relativistic cases—have been derived. Our approach provides an easy possibility of calculating diffusion coefficients of highly energetic, small pitch angle cosmic rays in turbulent magnetic fields which should prove useful in a large number of astrophysical situations.

This research was supported by Deutsche Forschungsgemeinschaft (DFG) under the Emmy–Noether program (grant SH 93/3-1) and grant Schl 201/19-1.