ASYMMETRIC DIFFUSION OF MAGNETIC FIELD LINES

Astrophysical plasmas feature a huge separation between the energy containing scale and the dissipation scale. Such a high-Reynolds number flows are necessarily turbulent (see, e.g., Armstrong et al. 1995). Conductive turbulent fluids generate their own magnetic fields by dynamo, as a result of which most of the astrophysical objects are magnetized to some degree, typically close to equipartition between kinetic and magnetic energies. One of the consequences of turbulence, whether in laboratory, astrophysical, or laboratory plasmas, is the magnetic field line stochasticity, which plays a crucial role in most key physical processes, such as thermal conduction, reconnection, particle transport, etc. Self-excited turbulence in tokamaks results in anomalous transport of particles perpendicular to the field and prevents reaching higher temperatures and densities. Recently, unusual features in the angular distribution of the arrival directions of cosmic rays (Abbasi et al. 2011) and spectral features for positrons (Adriani et al. 2009) revived studies of cosmic-ray propagation in stochastic magnetic fields (see, e.g., Beresnyak et al. 2011; Kistler et al. 2012). Indeed, carefully studying cosmic-ray diffusion could help discriminate between different models of positron excess, Fermi bubbles, and other currently unexplained cosmic-ray phenomena.

Turbulence is now understood as a multi-scale phenomenon, by and large owing to the pioneering paper of Richardson (1926), who studied turbulent diffusion and suggested that the diffusion coefficient depends on scale as D ∼ l^4/3, known as Richardson's law. Indeed, if two particles are separated by a distance l, the typical separation speed corresponds to the typical turbulent velocity on scale l, which is approximately δv ∼ l^1/3 (Kolmogorov 1941). This suggests that separation between particles grow as δl ∼ t^3/2. If turbulence is uniform and characterized only by the dissipation rate per unit mass, , which has units of cm² s⁻³, it is natural that the separation between two particles moving with the fluid, Δr, conforms to (Δr)² = g₀t³, where g₀ is a dimensionless number known as Richardson's constant. Richardson's diffusion has been studied extensively by experimental, theoretical, and numerical means (see, e.g., Sawford et al. 2008 and references therein). The turbulent diffusion of particles embedded in the MHD fluid received relatively less attention, however, the same type of diffusion is expected in the perpendicular direction due to the same scaling δv ∼ l^1/3 of strong MHD turbulence (see, e.g., Goldreich & Sridhar 1995; Beresnyak 2011, 2012a).

A different and very interesting question is how the magnetic field lines separate from each other in such an environment. This question is crucial because well-magnetized plasmas are often poorly collisional, with the ion Larmor radius being many orders of magnitude smaller that the mean free path from Coulomb collisions. In particular, the magnetized solar wind features mean free paths that are comparable to the distance from the Sun. In galaxy clusters the Coulomb mean free path is around 10–100 kpc. In most astrophysical environments there is also a high-energy component, called cosmic rays, for which Coulomb collisions are essentially negligible. Charged particles will therefore move along magnetic field lines for great distances and scatter mostly by magnetic perturbations. This will result in parallel diffusion being much larger than the perpendicular diffusion. In the absence of perpendicular momentum, particles will move along magnetic field lines and diffuse only due to the magnetic field line diffusion. The motion of the bulk of the plasma δv that causes ordinary diffusion could be neglected if the ion speed v_i is much larger than δv.¹ For cosmic rays this condition is also very well satisfied because they move along field lines with the speed comparable to the speed of light c. In other words, at least for short timescales, the fluid is frozen from most particles' perspective.

Despite being collisionless, plasmas in many circumstances can be described as fluids on scales larger than the ion Larmor radius (Schekochihin et al. 2009). The inertial range of MHD turbulence features strongly anisotropic perturbations which are much smaller in amplitude than the mean magnetic field. The key component of this turbulence is Alfvénic mode, which is why it is often called Alfvénic turbulence. Due to the fact that the Alfvén mode is driven by magnetic tension, not pressure, it is relatively unaffected by the lack of collisions. The presence of the slow mode in such highly anisotropic turbulence neither affects dynamics (Goldreich & Sridhar 1995; Beresnyak 2012a) nor influences magnetic field lines, as the anisotropic slow mode perturbation is mostly along the mean field. Therefore, the equations for the Alfvénic components, which are conventionally called reduced MHD (RMHD), are sufficient for studying field lines.

Perturbations in a strong mean magnetic field could be decomposed into backward and forward propagating components ${\bf w^\pm =v\pm b}/\sqrt{4\pi \rho }$ called Elsässer variables. Since perturbation sources are not uniform, MHD turbulence is naturally imbalanced, i.e., the amplitudes of w⁺ and w⁻ are not equal. This is verified by direct observations in the solar wind, where the dominant always propagates away from the Sun (see, e.g., Wicks et al. 2011). Other astrophysical sources are expected to have strong imbalance, for example, stellar winds and jets will emit predominantly outward-propagating component. Similarly, active galactic nuclei jets are expected to have Alfvén perturbations propagating away from the central engine, e.g., due to the black hole spin (Blandford & Znajek 1977). The theories of imbalanced Alfvénic turbulence are fairly young and have been verified mostly by comparison with simulations (Beresnyak & Lazarian 2009; Beresnyak 2011), although the solar wind measurements also show some promise. So far the model most consistent with the data is that of Beresnyak & Lazarian (2008b), which correctly explains the ratio of anisotropies and the ratio of energies, given a certain ratio of energy fluxes. Imbalanced relativistic force-free MHD turbulence, supposedly existing in such objects as parsec-scale jets and gamma-ray burst engines, has been simulated recently by J. Cho & A. Lazarian (2013, in preparation) and seems to exhibit properties consistent with the Beresnyak & Lazarian (2008b) model. Since we expect the Alfvén mode to survive in low-collisional environments such as jets, pulsar winds, etc., we are particularly interested in the properties of magnetic field line diffusion of Alfvénic turbulence. The study of magnetic field diffusion is also equivalent to the study of charged particle diffusion in the limit of negligible pitch angle scattering, e.g., due to zero perpendicular momentum.

Assuming a very strong mean field B₀ pointing in the x direction, the equation for the magnetic field line is

$$\begin{equation} \frac{d \bf r}{dx}= \frac{\bf b}{B_0}, \end{equation} \tag{ 1 }$$

where the magnetic perturbation b = B − B₀ is perpendicular to the mean field (Alfvén mode), so that the displacement vector r will only have perpendicular components. Similarly, particles moving along such a field in one direction will only experience perpendicular diffusion, as dx ≫ |dr|. Suppose we follow magnetic field lines that started from two points, separated by a small initial distance r₀. As the difference between B scales as δB_l ∼ l^1/3, we would expect a stochastic separation of the magnetic field lines to follow the law

$$\begin{equation} \langle (\Delta {\bf r})^2\rangle =g_{m} \epsilon v_A^{-3} |x|^3, \end{equation} \tag{ 2 }$$

where is the dissipation rate per unit mass, as defined above, and $v_A=B_0/\sqrt{4\pi \rho }$ is the Alfvén speed. This expression can be obtained by replacing t in Richardson's formula with the time variable for the Alfvén wave, x/v_A − t with t = 0. For the lack of a better term in the literature, we will designate this as Richardson–Alfvén diffusion and call the dimensionless constant g_m the Richardson–Alfvén constant. Although we call this diffusion by analogy with physical diffusion in time, this is rather a stochastic separation in space. Nevertheless, the term diffusion seems appropriate due to similarity with physical diffusion and the relevance of this problem to perpendicular diffusion of particles.

If magnetic field lines separate for a distance much larger than the outer scale of turbulence L, the δb becomes truly random and independent of the separation. In this limit the magnetic field lines experience random walk, i.e., ordinary diffusion 〈(Δr)²〉 ∼ |x|. This limit is known as field line random walk (FLRW) and has been used to describe perpendicular diffusion in Jokipii (1973). Note that the random walk must be symmetric with respect to the sign of x.

We used magnetic field snapshots obtained in simulations of Alfvénic turbulence. These simulations solved the RMHD equations with explicit dissipation and driving to achieve statistically stationary state. Further details behind the RMHD rationale, simulation setup, driving, numerical scheme, etc., can be found in Beresnyak (2012a). Each simulation represents stationary, strong MHD turbulence with strong mean field. The balanced simulation B1 has been previously reported in Beresnyak (2011) and imbalanced simulations I1–I6 have been reported in Beresnyak & Lazarian (2010). More details concerning these simulations can be found in the above references. The parameters of the simulations are summarized in Table 1, with the defining feature of each imbalance simulation being the ratio of the dissipation rates ^± for Elsässer components w^±.

Table 1. Simulation Parameters

Run	Resolution	+/−	w+/w−	$\ell _{\Vert *}^-/\ell _{\Vert *}^+$	$g_m^+/g_m^-$
B1	15363	1	1	1	1
I1	512 × 10242	1.19	1.16	1.07	1.03
I3	512 × 10242	1.41	1.37	1.15	1.15
I5	1024 × 15362	2.00	2.36	1.36	1.31
I6	1024 × 15362	4.50	6.70	1.78	1.71

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We were tracking the pairs of magnetic field lines started at random positions throughout the box and initially separated by distance r₀ by Equation (1). Figure 1 shows the tracking results for the B1 simulation. The transition to Richardson's diffusion happens when particles are separated by around 12 Kolmogorov (dissipation) scales η. We chose five initial separations, fractions of η. At large distances they seem to converge toward Richardson–Alfvén diffusion with g_m = 0.14. At sufficiently large distances the field lines started experiencing random walk, i.e., ordinary diffusion with diffusion coefficient of 0.3L_box. We typically used 4 × 10⁵ field line pairs for statistical averaging.

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The next tracking experiment involved simulations I1–I6. These also experienced Richardson–Alfvén diffusion, but now the diffusion speed was different depending on whether we track magnetic fields forward or backward (negative or positive dx). We plotted the ratio of forward to backward separations from the simulation I5 in Figure 2. It turned out that this ratio is fairly insensitive to the initial separation r₀, when r₀ was varied by a factor of around 16, with most difference being due to statistical error. Different snapshots of the same simulation showed more variation. We used this variation in time to estimate the error in the diffusion ratio. In the large x limit this ratio went to unity, consistent with the symmetry of random walk. We estimated the ratio of Richardson–Alfvén constants by taking the maximum of the ratio curves, which was also somewhere around the middle of the inertial range in terms of perpendicular separation. The measurements of the ratio $g_m^+/g_m^-$ are presented in Table 1 and Figure 3.

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Hydrodynamic turbulent diffusion forward and backward in time is known to be different by a factor of a ≈ 2 (see, e.g., Berg et al. 2006), which is due to fundamental non-reversibility of turbulence. In our case the diffusion of magnetic field lines comes from the w⁺ component, which propagates against mean field and w⁻ component which propagates along mean field. The diffusion of field lines along the field will be "forward in time" for w⁻ and "backward in time" for w⁺ and vice versa for the opposite direction. Assuming that the diffusion from b^± is proportional to the amplitude w^±, we can write for the diffusion asymmetry:

$$\begin{equation} \frac{g_m^+}{g_m^-}=\frac{w^-+a_mw^+}{a_mw^-+w^+}, \end{equation} \tag{ 3 }$$

where a_m is the time-asymmetry in MHD turbulence. Figure 3 indicates that this model agrees with data reasonably well, as long as a_m = 2.0–2.1, which is compatible with hydrodynamic time-asymmetry.

Equation (2) looks similar to the critical balance of Goldreich & Sridhar (1995). Introducing anisotropy constant C_A, critical balance can be written as $\ell _\Vert =C_A v_A \ell _\perp ^{2/3} \epsilon ^{-1/3}$ (Beresnyak 2012a), where $\ell _\Vert$ is the distance parallel to the field, i.e., analogous to x, and ℓ_⊥ is a perpendicular distance. This is indeed the same functional dependence as Equation (2). However, if we assume that these relations are identical, this would imply that $C_A=g_m^{-1/3}$. This is not satisfied, however. The difference is that the measurement of the diverging magnetic fields is quasi-Lagrangian, while the measurement of structure function that leads to anisotropy constant is Eulerian. This difference becomes even more pronounced in the imbalanced case, where each of the w^± components has its own anisotropy. The analogy between Richardson–Alfvén diffusion and critical balance would suggest that $g^+ (\ell _\Vert ^+)^3=g^- (\ell _\Vert ^-)^3$. We presented the ratio of parallel scales in the middle of the inertial range in Table 1. As we see, the expression above is not satisfied and the ratio of magnetic diffusions cannot be explained by the anisotropy difference. So, despite the similar functional form, Richardson–Alfvén diffusion has no direct relationship to Goldreich–Sridhar anisotropy.

Our measurement is the first clear demonstration of the x³ superdiffusion of magnetic field lines in simulations of MHD turbulence. Earlier Maron et al. (2004) tried to obtain X³ superdiffusion of magnetic field lines, but their results were inconclusive due to the limited size of the inertial range. This Letter is also the first observation of asymmetric superdiffusion. Superdiffusion of fast particles in the solar wind has been argued based on observational data (Perri & Zimbardo 2009). Superdiffusion of field lines has been discussed in Jokipii (1973). The reconnection model of Lazarian & Vishniac (1999) also uses perpendicular superdiffusion of field lines; see also Eyink et al. (2011). The superdiffusion of particles has been argued in Skilling et al. (1974), Narayan & Medvedev (2001), Lazarian (2006), and Yan & Lazarian (2008), however, the asymmetric superdiffusion has not been anticipated before. Our earlier measurement of perpendicular diffusion using MHD simulations (Beresnyak et al. 2011) has been made in the large separation limit and reproduced FLRW, which is symmetric. The measurements of cosmic-ray propagation in artificial random fields, such as Giacinti et al. (2012) can, in principle, reproduce superdiffusion, but since artificial fields lack the time-asymmetry of turbulent fields, they cannot reproduce asymmetric diffusion. Based on the similarity between Goldreich–Sridhar anisotropy and Richardson's diffusion, Narayan & Medvedev (2001) suggested that magnetic field lines separate within the Goldreich–Sridhar cone; however, according to the section above, this analogy is misleading, especially in the imbalanced case. Time-asymmetry of turbulence, which we confirmed in this Letter, has consequences for small-scale dynamo as well (Beresnyak 2012b).

One of the consequences of asymmetric perpendicular diffusion is an induced streaming. Indeed, if we consider two close magnetic field tubes, one of which is filled with isotropically distributed particles and another empty, the asymmetric diffusion into the empty tube will result in an average streaming ∼(1 − g⁺/g⁻) of particle's velocity. In particular, for relativistic particles, such as cosmic rays, this will result in a streaming velocity of 2c(1 − g⁺/g⁻)/π, which could easily exceed the threshold for streaming instability, v_A, as long as imbalance amplitude 1 − w⁻/w⁺ exceeds (3π/2)v_A/c, which is around 10⁻⁴ in the WISM. Therefore, the induced streaming will be counteracted by streaming instability (Kulsrud & Pearce 1969). Above the threshold for turbulent damping (Farmer & Goldreich 2004; Beresnyak & Lazarian 2008a) streaming instability will be suppressed and, according to the estimates in the above paper, the weak large-scale streaming should reappear at energies 10¹¹ eV and strong streaming is expected above 3 × 10¹³ eV, although such energies are already heavily influenced by pitch-angle scattering. The net effect of the streaming instability from particles with energies below 3 × 10¹³ eV will be a flux of slab waves which will increase the rate of pitch-angle scattering for these particles. The modeling of this effect will be subject of a future publication.

A.B. was supported by Los Alamos Director's Fellowship.