FAST MAGNETIC RECONNECTION AND SPONTANEOUS STOCHASTICITY

The standard theory for Alfvénic turbulence is currently the one suggested by GS95. The cornerstone of the theory is the so-called critical balance, which is the rough equality between the timescale of the eddy turnovers perpendicular to the local magnetic field and the periods of waves associated with the eddies.⁶ The predictions of the GS95 model are in rough agreement with numerical simulations (Cho & Vishniac 2000; Maron & Goldreich 2001; Cho et al. 2002; Beresnyak & Lazarian 2006), although some disagreement in terms of the measured spectral slope was noted. This disagreement produced a flow of papers with suggestions to improve the GS95 model by including additional effects like dynamical alignment (Boldyrev 2005, 2006), polarization intermittency (Beresnyak & Lazarian 2006), non-locality (Gogoberidze 2007). More recent studies in Beresnyak & Lazarian (2009, 2010) and Beresnyak (2011) indicate that all numerical simulations performed to date may not have sufficiently extended inertial range to get the actual spectral slope and therefore worries about the "inconsistency" of the GS95 model may be premature.

We shall add parenthetically that in a number of applications the empirical so-called composite 2D/slab model of magnetic fluctuations is used (see Bieber et al. 1994). In the latter model, which is also known as two-component model, it is assumed that interplanetary fluctuations can be described as a superposition of fluctuations with wavevectors parallel to the ambient large-scale magnetic field (the so-called slab modes) and perpendicular to the mean field (the so-called 2D modes). This model results in a Maltese cross structure of magnetic correlations. This model was developed to account for the solar wind observations, which it does well by adjusting the intensity of two components (see, e.g., Matthaeus et al. 1990). From the theoretical point of view, as well as from the point of numerical testing, this standard model is rather vulnerable. Indeed, 2D fluctuations are consistent with the theory of weak Alfvénic turbulence (see Ng & Bhattacharjee 1996; Lazarian & Vishniac 1999; Galtier et al. 2000), but this analytical theory predicts strengthening of the cascade with decrease of the turbulence scale. Therefore 2D fluctuations can describe Alfvénic turbulence only over a limited range of scales. In addition, slab modes do not arise naturally in MHD numerical simulations with large-scale driving (see Cho & Lazarian 2002, 2003). We do not discuss this model of MHD turbulence further, as it does not have physical motivation and is not supported by numerical modeling. It may be treated as a parameterization of a particular type of magnetic perturbation dominated by the peculiarities of driving. In contrast, we consider turbulence which has an extended universal inertial range over which the effects of driving and dissipation are negligible.

Below we shall adopt the GS95 model of turbulence to describe the Alfvénic part of MHD turbulent fluctuations. Note that the Alfvénic perturbations are most important for magnetic field wandering which, according to LV99, is the process that enables fast reconnection. The GS95 model can be generalized to compressible turbulence and even for supersonic motions numerical calculations show that the Alfvénic perturbations exhibit GS95 scaling (Cho & Lazarian 2002, 2003; Kowal & Lazarian 2010). In what follows we consider MHD turbulence where the flows of energy in the opposite directions are balanced. When this is not true, i.e., when the turbulence has non-zero cross-helicity, the properties of turbulence depart substantially from the GS95 model.⁷ Similarly, we shall not discuss MHD turbulence at high magnetic Prandtl numbers, when the field-parallel viscosity is much larger than resistivity (see Cho et al. 2002, 2003; Lazarian et al. 2004). This viscosity-dominated regime has important consequences for turbulence in partially ionized gas (Lazarian et al. 2004), while our present study deals mostly with conducting fluids and fully ionized plasmas. Our considerations do apply for large magnetic Prandtl numbers at scales greater than the viscous length, if the kinetic Reynolds number is also large.

The nature of Alfvénic cascade is expressed through the critical balance condition in the GS95 model of strong turbulence, namely,

$$\begin{equation} \ell _{\Vert }^{-1}v_A\sim \ell _{\bot }^{-1}\delta u_\ell, \end{equation} \tag{ 2 }$$

where δu_ℓ is the eddy velocity, while ℓ_|| and ℓ_⊥ are, respectively, eddy scales parallel and perpendicular to the local direction of magnetic field. The critical balance condition states that the parallel size of an eddy is determined by the distance Alfvénic perturbation can propagate during an eddy turnover. The qualifier "local" is important,⁸ as no universal relations exist if eddies are treated in respect to the global mean magnetic field (LV99; Cho & Vishniac 2000; Maron & Goldreich 2001; Lithwick & Goldreich 2001; Cho et al. 2002). Combining Equation (2) with the Kolmogorov cascade notion, i.e., that the energy transfer rate is δu²_ℓ/(ℓ_⊥/δu_ℓ) = const one gets δu_ℓ ∼ ℓ^1/3_⊥ and ℓ_|| ∼ ℓ^2/3_⊥. If the turbulence injection scale is L_i, then ℓ_||∝L^1/3_iℓ^2/3_⊥, which shows that the eddies get very anisotropic for small ℓ_⊥. Recent measurements of anisotropy in the solar wind are consistent with these predictions (Podesta 2010; Wicks et al. 2010, 2011).

Critical balance is a feature of strong turbulence, which is the case when the turbulence is injected isotropically with velocity amplitude u_L = v_A. If the turbulence is injected at velocities u_L ≪ v_A (or anisotropically with L_{i, ||} ≪ L_{i, ⊥}), then the cascade is weak, with ℓ_⊥ of the eddies decreasing but ℓ_|| = L_i unchanged by resonant wave interactions (Montgomery & Matthaeus 1995; Lazarian & Vishniac 1999; Galtier et al. 2000). In other words, as a result of the weak cascade the eddies get thinner, but preserve the same length along the local magnetic field. The energy cascade rate in the weak MHD turbulence regime is (Kraichnan 1965; Ng & Bhattacharjee 1996; Lazarian & Vishniac 1999)

$$\begin{equation} \varepsilon \approx \tau _{{\rm corr}} \delta u_\ell ^4/\ell _\bot ^2 \approx \delta u_\ell ^4 L_i/v_A \ell _\bot ^2, \end{equation} \tag{ 3 }$$

where τ_corr = L_i/v_A is the decorrelation time due to large-scale Alfvén waves. This implies that the scaling of weak turbulence is

$$\begin{equation} \delta u_\ell \sim u_L (\ell _{\bot }/L_i)^{1/2}. \end{equation} \tag{ 4 }$$

For weak turbulence initially the left-hand side (LHS) of Equation (2) is larger than the right-hand side (RHS), but as ℓ_⊥ decreases this eventually makes Equation (2) satisfied.

Comparing Equations (4) and (2) for $\ell _\Vert =L_i$, one can see that the transition to the strong MHD turbulence happens at the scale L_trans = L_iM²_A and the velocity at this scale is v_trans = u_LM_A, with M_A = u_L/v_A ≪ 1 the Alfvénic Mach number of the turbulence (Lazarian & Vishniac 1999; Lazarian 2006). Thus, weak turbulence has a limited, i.e., [L_i, L_iM²_A] inertial interval before it gets into the regime of strong turbulence at smaller scales. Note that weak and strong are not the characteristics of the amplitude of turbulent perturbations, but the strength of nonlinear interactions (see more discussion in Cho et al. 2003) and very weak Alfvénic perturbations can correspond to a strong Alfvénic cascade.

While GS95 assumed that the turbulent energy is isotropically injected with amplitude u_L = v_A at the scale L_i, LV99 provided general relations for the turbulent scaling at small scales for the case that the injection velocity u_L is less or equal to v_A. Combining Equations (2) and (3) at ℓ_⊥ = L_i, and the constant flux condition, we get the relations between the parallel and perpendicular scales of eddies in the strong GS95 cascade range, which can be written in terms of ℓ_|| and ℓ_⊥ (LV99):

$$\begin{equation} \ell _{\Vert }\approx L_i \left(\frac{\ell _{\bot }}{L_i}\right)^{2/3} M_A^{-4/3}, \end{equation} \tag{ 5 }$$

$$\begin{equation} \delta u_{\ell }\approx u_{L} \left(\frac{\ell _{\bot }}{L_i}\right)^{1/3} M_A^{1/3}. \end{equation} \tag{ 6 }$$

As we discuss later, the present-day debates of whether GS95 approach should be augmented by additional concepts like "dynamical alignment," "polarization," "non-locality" (Boldyrev 2006; Beresnyak & Lazarian 2006, 2009; Gogoberidze 2007) do not change the nature of the reconnection of the weakly turbulent magnetic field.

The above relations were exploited in LV99 to estimate magnetic field lateral diffusion. Moving a distance s along field lines, one finds that a pair of lines initially a distance ℓ⁽⁰⁾_⊥ apart at s = 0 separate at the rate

$$\begin{equation} \frac{d}{ds}\ell _\bot \simeq \frac{\delta b_\ell }{B_0}\simeq \frac{\delta u_\ell }{v_A}. \end{equation} \tag{ 7 }$$

Substituting the scaling given by Equation (6), one can solve to obtain

$$\begin{equation} \ell _\bot ^2 \simeq (s^3/L_i) M_A^4 \end{equation} \tag{ 8 }$$

when L_i > s ≫ ℓ⁽⁰⁾_⊥. This result applies only until s = L_i when ℓ_⊥(s) ≃ L_trans, the transitional scale between weak and strong turbulence. Since $\ell _\Vert =L_i$ is constant in weak turbulence, there are no eddies with parallel length scale greater than L_i. Thus, as s continues to increase, the lines encounter at every increment of s by L_i a new turbulent eddy and undergo a further separation by L_trans. The result is a diffusive random walk with

$$\begin{equation} \ell _\bot ^2 \simeq L_{{\rm trans}}^2 (s/L_i) \simeq s L_i M_A^4, \end{equation} \tag{ 9 }$$

for s > L_i. The results in Equations (8) and (9) are exact analogs for magnetic field lines of the two-particle diffusion of Lagrangian particle trajectories discussed in the next section and can be obtained more rigorously by the methods employed there. Since this discussion is somewhat out of logical order, we defer it to Appendix A.

A remarkable feature of Equations (8) and (9) is their complete lack of dependence on the initial separation ℓ⁽⁰⁾_⊥ of the lines. As we shall see below, it is this property which leads in the LV99 model to fast magnetic reconnection in MHD turbulence independent of the microscopic resistivity. The shear-Alfvénic component of turbulence is the most important for magnetic field-line wandering. Cho & Lazarian (2002, 2003) demonstrated the correspondence of the Alfvénic component of compressible MHD turbulence, obtained by decomposition into fundamental modes, to the Alfvénic turbulence in incompressible media. Thus we shall use the relations above to treat magnetic fields in realistically compressible astrophysical fluids.

It will be important for what follows to develop a theoretical understanding of the dynamics of Lagrangian particles in MHD turbulence. Here one means by "particles" not the microscopic charged constituents of the plasma (electrons and ions), but instead macroscopically small and microscopically large parcels of plasma fluid. The Lagrangian perspective is particularly important to understand turbulent processes and there has been an explosion of research in this area over the last decade (see Falkovich et al. 2001; Salazar & Collins 2009; Toschi & Bodenschatz 2009). There has been relatively little work done on the Lagrangian properties of MHD turbulence, however, where the effects of Alfvén waves on fluid particle motion can be significant. We shall here develop a theory of Lagrangian MHD turbulence at the level of GS95 phenomenology by a semi-quantitative use of the equations of motion.

We begin with the simplest case of one-particle or Taylor diffusion, which concerns the displacement δx(t) = x(t) − x₀ of a Lagrangian fluid particle from its initial position as it moves according to the advection equation

$$\begin{equation} \frac{d}{dt}{\bf x}(t)={\bf U}(t), \end{equation} \tag{ 10 }$$

with U(t) = u(x(t), t) the Lagrangian particle velocity. At early times t ≪ τ_L or short compared with the eddy turnover time τ_L = u_L/L_i, motion is ballistic with 〈x²(t)〉 ∼ u²_Lt². It was shown by Taylor (1921) that the motion is diffusive 〈δx_i(t)δx_j(t)〉 ∼ 2D_ijt at times t ≫ τ_L, with the eddy diffusivity

$$\begin{equation} D_{ij}=\frac{1}{2}\int _{-\infty }^{+\infty } dt\,\langle U_i(t) U_j(0)\rangle. \end{equation} \tag{ 11 }$$

These results carry over unchanged to MHD turbulence. For the weak magnetic field case with M_A ⩾ 1 (including hydrodynamic turbulence with M_A = ∞), $\langle U(t)U(0)\rangle \sim u_L^2 e^{-t/\tau _L}$ (Mordant et al. 2001; Busse & Müller 2008). Thus,

$$\begin{equation} D\sim u_L^2\tau _L \sim u_L L_i, \end{equation} \tag{ 12 }$$

exactly as for hydrodynamics of neutral fluids. When M_A ≃ 1, one expects slight differences from the hydrodynamic case due to anisotropy of MHD turbulence. For example, the velocity components $U_\Vert$ and U_⊥ ought to have timescales τ_L || > τ_L ⊥, since the parallel components are pseudo-Alfvénic modes with essentially passive scalar dynamics, due to reduced nonlinear interactions. Thus, one expects that $D_\Vert >D_\perp$, as indeed observed in simulations of Busse & Müller (2008). For M_A ≪ 1, there are more profound effects due to the strong field. Weak MHD turbulence becomes relevant, for which the nonlinear timescale is τ_L = M⁻²_Aω⁻¹_A with ω_A = v_A/L_i the forcing-scale Alfvén wave frequency (see Kraichnan 1965; Sridhar & Goldreich 1994; Galtier et al. 2000; cf. also Equation (25) below for ℓ_⊥ = L_i). Furthermore, fluid particles in the Lagrangian frame experience fast shear-Alfvénic oscillations in velocity, so that

$$\begin{equation} \langle U_\perp (t)U_\perp (0)\rangle \sim u_L^2 {\rm Re}(e^{i\omega _At-t/\tau _L}). \end{equation} \tag{ 13 }$$

Thus, Equation (11) gives

$$\begin{equation} D_\perp \sim u_L^2 \frac{\tau _L}{(\omega _A\tau _L)^2+1}. \end{equation} \tag{ 14 }$$

Since ω_Aτ_L = M⁻²_A ≫ 1,

$$\begin{equation} D_\perp \sim u_L^2 \frac{1}{\omega _A^2\tau _L} \sim u_L L M_A^3. \end{equation} \tag{ 15 }$$

There is a large reduction in the turbulent one-particle diffusivity due to inefficient advection by Alfvén wave modes. Physically, the velocity of a particle transported by Alfvén wave turbulence experiences rapid oscillations in sign that lead to large cancellations in the net displacement.

We note as an aside that the Taylor one-particle diffusivity given by Equation (11) coincides with the effective turbulent diffusivity of an advected scalar field (e.g., temperature), whenever the "eddy-diffusivity" concept is valid. In order for such a description of the turbulence effects to be accurate, the scalar fields must vary slowly on spatial scales of order L_i and, in that case, they experience an augmentation of their molecular diffusivity by a "turbulent diffusivity" precisely given by Equation (11). Our results in Equations (12) and (15) thus reproduce those of Lazarian (2006) for the eddy thermal diffusivity in MHD turbulence. Of course, in general, when there is no separation between the scale of variation of the scalar field and the scale L_i of the turbulence, then the gradient-transport assumption breaks down (e.g., Tennekes & Lumley 1972, Section 2.3). In such cases, eddy-diffusivity models can produce erroneous results and may suffice only for order-of-magnitude estimates of scalar transport.

Two-particle turbulent diffusion or Richardson diffusion concerns instead the separation $\hbox{\boldmath $\ell $}(t)={\bf x}(t)-{\bf x}^{\prime }(t)$ between a pair of Lagrangian fluid particles. It was proposed by Richardson (1926) that this separation grows in turbulent flow according to the formula

$$\begin{equation} \frac{d}{dt}\langle \ell _i(t)\ell _j(t)\rangle =\langle D_{ij}(\hbox{\boldmath $\ell $})\rangle, \end{equation} \tag{ 16 }$$

with a scale-dependent eddy-diffusivity D(ℓ). In hydrodynamic turbulence Richardson deduced that D(ℓ) ∼ ε^1/3ℓ^4/3 (see also Obukhov 1941) and thus ℓ²(t) ∼ εt³. An analytical formula for the two-particle eddy diffusivity was later derived by Batchelor (1950); Kraichnan (1966):

$$\begin{equation} D_{ij}(\hbox{\boldmath $\ell $})=\int _{-\infty }^0 dt \langle \delta U_i(\hbox{\boldmath $\ell $},0) \delta U_j(\hbox{\boldmath $\ell $},t)\rangle, \end{equation} \tag{ 17 }$$

with $\delta U_i(\hbox{\boldmath $\ell $},t)\equiv U_i({\bf x}+\hbox{\boldmath $\ell $},t)-U_i({\bf x},t)$ the relative velocity at time t of a pair of fluid particles which were at positions x and ${\bf x}+\hbox{\boldmath $\ell $}$ at time 0. This formula is the analog for two-particle eddy diffusivity of Taylor's formula (11) for one-particle diffusivity and it is valid in MHD turbulence just as in hydrodynamic turbulence. We can assume similarly as before that

$$\begin{equation} \langle \delta U_\perp (\ell,0)\delta U_\perp (\ell,t)\rangle \sim \delta u^2_\ell {\rm Re}(e^{i\omega _{A\,\ell }t-|t|/\tau _\ell }), \end{equation} \tag{ 18 }$$

where ω_{A, ℓ} is the Alfvén wave frequency and τ_ℓ is the nonlinear interaction time both at length scale ℓ. The result for the two-particle diffusivity analogous to Equation (14) is

$$\begin{equation} D_\perp (\ell) \sim \delta u^2_\ell \frac{\tau _\ell }{(\omega _{A\,\ell }\tau _\ell)^2+1}. \end{equation} \tag{ 19 }$$

We consider here only the case M_A < 1, for which weak turbulence holds when ℓ_⊥ > L_trans and strong turbulence when ℓ_⊥ < L_trans. We shall also treat particle dispersion only in the direction perpendicular to the background magnetic field, since this is what is required later for the application to reconnection. See Busse & Müller (2008) and Eyink (2011) for some discussion of particle dispersion in the field-parallel direction.

We first consider the strong-turbulence regime when ℓ_⊥(t) < L_trans. In this case, τ_ℓ = ε^−1/3ℓ^2/3_⊥ and $\omega _{A \,\ell }=v_A/\ell _\Vert$, where $\ell _\Vert$ is the parallel scale of the turbulent eddy with perpendicular scale ℓ_⊥(t) (and not the particle separation $\ell _\Vert (t)$ in the field-parallel direction). Because of the critical balance condition in strong MHD turbulence,

$$\begin{equation} \omega _{A\,\ell } \tau _\ell ={\rm (const.)}. \end{equation} \tag{ 20 }$$

In that case

$$\begin{equation} \frac{\tau _\ell }{(\omega _{A\,\ell }\tau _\ell)^2+1}= f \tau _\ell, \end{equation} \tag{ 21 }$$

with f = [(ω_A ℓτ_ℓ)² + 1]⁻¹ < 1 a constant factor which represents the reduced efficiency of particle transport due to wave oscillations. Equation (19) gives

$$\begin{equation} D_\perp (\ell) \sim f \delta u^2_\ell \tau _\ell \sim f \varepsilon ^{1/3}\ell ^{4/3}_\perp, \end{equation} \tag{ 22 }$$

with the same scaling as the original Richardson (1926) two-particle diffusivity. Thus,

$$\begin{equation} \ell ^2_\perp (t) \sim f^3 \varepsilon t^3, \end{equation} \tag{ 23 }$$

showing the same t³-law as in the hydrodynamic case, with only a reduced coefficient.

We now consider the weak-turbulence regime with ℓ_⊥(t) > L_trans. The pair of particles are advected apart by a turbulent eddy with corresponding perpendicular length scale but with $\ell _\Vert =L_i$ for all eddies, so that ω_A ℓ = ω_A = v_A/L_i, independent of ℓ_⊥. The nonlinear interaction time τ_ℓ is estimated from

$$\begin{equation} \frac{\delta u^2_\ell }{\tau _\ell }=\varepsilon =\frac{u_L^4}{v_AL_i}, \end{equation} \tag{ 24 }$$

together with Equation (4) to give

$$\begin{equation} \tau _\ell = (\ell _\perp /v_A) M_A^{-2}. \end{equation} \tag{ 25 }$$

Since

$$\begin{equation} \omega _A\tau _\ell =\ell _\perp /L_{{\rm trans}}\gg 1, \end{equation} \tag{ 26 }$$

it follows from Equation (19) that

$$\begin{equation} D_\perp (\ell) \sim \frac{\delta u^2_\ell }{\omega _A^2\tau _\ell } \sim \frac{\varepsilon }{\omega _A^2}= u_LL_iM_A^{3}. \end{equation} \tag{ 27 }$$

Note that D_⊥(ℓ) is identical with the one-particle diffusivity D_⊥ given by Equation (15) and, in particular, is independent of ℓ_⊥. It thus follows that in the weak-turbulence regime

$$\begin{equation} \ell ^2_\perp (t) \sim u_LL_iM_A^3 t, \end{equation} \tag{ 28 }$$

and two-particle diffusion is identical to one-particle diffusion. It should be possible to verify this interesting prediction by analytical methods of weak-turbulence theory for Lagrangian particle motion (see Balk 2002). The crossover between the dispersion laws in Equations (23) and (28) occurs at the time t_c ∼ L_i/v_A = M_At_L.

Magnetic field lines do not move. Magnetic fields evolve under the Maxwell equations, of course, but single field lines do not possess an identity over time. As has long been understood, magnetic field-line motion is a "metaphysical" concept, defined by convention and not testable by experiment (see Newcomb 1958; Vasyliunas 1972; Alfvén 1976). Any line-velocity which leads to agreement with the Maxwell equations for the magnetic field evolution is equally acceptable, e.g., the bulk plasma velocity u in ideal MHD, the ${\bf E}\hbox{\boldmath $\times $}{\bf B}$ drift velocity in low-energy plasmas, etc. Flux-freezing is such a powerful heuristic tool, however, that it is often forgotten that the concept of magnetic line-motion rests on certain conventions and is not a direct, physical reality. In particular, real physical plasmas always possess a Spitzer plasma resistivity (along with other possible forms of non-ideality, such as electron inertia, electron pressure anisotropy, etc.). The validity of flux-freezing in the standard sense depends upon careful consideration of the limit when such non-ideal effects are taken to be small.

Since field lines do not move in reality, one is free to adopt any convention for their motion which is consistent with the dynamical equations. Let us for the moment assume the validity of the resistive induction equation

$$\begin{equation} \partial _t{\bf B}= \hbox{\boldmath $\nabla $}\hbox{\boldmath $\times $}({\bf u}\hbox{\boldmath $\times $}{\bf B}) +\lambda \triangle {\bf B}, \end{equation} \tag{ 29 }$$

with λ = ηc²/4π the magnetic diffusivity for a simple scalar resistivity η.⁹ A smooth, deterministic velocity field u_* is flux preserving if the magnetic field obeys

$$\begin{equation} \partial _t{\bf B}= \hbox{\boldmath $\nabla $}\hbox{\boldmath $\times $}({\bf u}_*\hbox{\boldmath $\times $}{\bf B}). \end{equation} \tag{ 30 }$$

Unfortunately, for the resistive MHD model in three space dimensions, there is generally no flux-preserving velocity whatsoever. For example, Wilmot-Smith et al. (2005) provide a counterexample with a closed magnetic field line through which the flux is changing in time, so that no deterministic flux-preserving velocity can exist. This seems to leave one without any consistent and meaningful approach to discuss field-line "motion" in the presence of Ohmic resistivity.

On the other hand, magnetic line-motion in a resistive MHD plasma may be very naturally regarded as stochastic (Eyink 2009). Let the plasma fluid velocity be u(x, t), assumed here to be divergence-free for an incompressible plasma.¹⁰ Then we may define stochastic Lagrangian trajectories $\widetilde{{\bf x}}({\bf a},t)$ by the initial-value problem

$$\begin{equation} \left\lbrace \begin{array}{@{}l}\dsty{d\widetilde{{\bf x}}\over {dt}}({\bf a},t) ={\bf u}(\widetilde{{\bf x}}({\bf a},t),t) +\sqrt{2\lambda } \widetilde{\hbox{\boldmath $\eta $}}(t), \\[10pt] \widetilde{{\bf x}}({\bf a},t_0) = {\bf a}, \end{array} \right. \end{equation} \tag{ 31 }$$

where $\widetilde{\hbox{\boldmath $\eta $}}(t)$ is a 3D vector white-noise process, delta correlated in time. For any initial magnetic field B₀(x) one can form the random field

$$\begin{equation} \widetilde{{\bf B}}({\bf x},t)= \left. {\bf B}_0({\bf a})\hbox{\boldmath $\cdot $}\hbox{\boldmath $\nabla $}_a\widetilde{{\bf x}}({\bf a},t)\right|_{\widetilde{{\bf a}}({\bf x},t)}, \end{equation} \tag{ 32 }$$

where $\widetilde{{\bf a}}({\bf x},t)$ is the spatial inverse map to $\widetilde{{\bf x}}({\bf a},t)$. If the flow map were deterministic, then Equation (32) would be the standard Lundquist formula for the magnetic field. What is true in the present case with λ > 0 is that the ensemble-average field

$$\begin{equation} {\bf B}({\bf x},t) = \overline{\,\,\widetilde{{\bf B}}({\bf x},t)\,\,}, \end{equation} \tag{ 33 }$$

over all independent realizations of the white noise is the exact solution of the resistive induction Equation (29) with initial condition

$$\begin{equation} {\bf B}({\bf x},t_0)={\bf B}_0({\bf x}). \end{equation} \tag{ 34 }$$

In fact, Equations (31)–(33) are mathematically equivalent to Equations (29) and (34). It should not be too surprising that averaging over Brownian motions will reproduce the Laplacian diffusion term λ▵B.

Equations (31)–(33) are equivalent to the following path-integral formula, derived in Eyink (2011):

$$\begin{eqnarray} {\bf B}({\bf x},t) &=& \int _{{\bf a}(t)={\bf x}} {\cal D}{\bf a}\,\, {\bf B}_0({\bf a}(t_0))\hbox{\boldmath $\cdot $}{\bf {\cal J}}({\bf a},t) \nonumber\\ && \times \exp \left(-{{1}\over {4\lambda }}\int _{t_0}^t d\tau \, |\dot{{\bf a}}(\tau)-{\bf u}^\nu ({\bf a}(\tau),\tau)|^2\right), \quad\ \ \ \end{eqnarray} \tag{ 35 }$$

where B is interpreted as a 3D row vector and ${\bf {\cal J}}({\bf a},\tau)$ is a 3 × 3 matrix satisfying the following ordinary differential equation along the trajectory a(τ):

$$\begin{equation} \left\lbrace \begin{array}{@{}l}\dsty{d \over {d\tau }}{\bf {\cal J}}({\bf a},\tau)={\bf {\cal J}}({\bf a},\tau)\hbox{\boldmath $\nabla $}_x{\bf u}({\bf a}(\tau),\tau) \\[10pt] {\bf {\cal J}}({\bf a},t_0)={\bf I}. \end{array} \right. \end{equation} \tag{ 36 }$$

By applying $\hbox{\boldmath $\nabla $}_a$ to Equation (31) it is easy to see that ${\bf {\cal J}}({\bf a},\tau)= \hbox{\boldmath $\nabla $}_a\widetilde{{\bf x}}({\bf a}(t_0),\tau)$ and thus we may identify

$$\begin{equation} \widetilde{{\bf B}}({\bf a}(\tau),\tau)={\bf B}_0({\bf a}(t_0))\hbox{\boldmath $\cdot $}{\bf {\cal J}}({\bf a},\tau). \end{equation} \tag{ 37 }$$

As often with such path-integration techniques, only the initial field B₀(x) and the final field B(x, t) have real physical meaning and the intermediate random fields $\widetilde{{\bf B}}({\bf x},t)$ are merely calculational devices. We shall refer to the latter as the "virtual" magnetic fields, since they enter as non-real, intermediate states between two physical fields at different times. Equation (33) (or Equation (35)) shows that summing over all of the "virtual" fields at any instant reproduces the true, physical magnetic field. It follows from Equation (32) (or Equation (37)) that each "virtual" field $\widetilde{{\bf B}}$ is topologically equivalent to the initial field B₀ and is simply stretched, deformed, and advected by its own stochastic flow $\widetilde{{\bf x}}$. Field lines for two different "virtual" fields that arise from statistically independent flows may pass directly through each other as they move, since they correspond to completely different random realizations. The topology of the field lines is only changed by the final averaging step, which corresponds to resistive gluing of all the "virtual" magnetic field vectors which arrive to the same point x at time t. It is this resistive averaging which reconnects the lines of the initial magnetic field B₀. See Figure 1 for a pictorial representation of this process.

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We shall refer to either relations (31)–(33) or the equivalent path-integral formulae (35)–(36) as the SFF relations. They provide a consistent approach to describe the "motion" of magnetic field lines in the presence of resistivity. Setting λ = 0 one formally recovers the standard deterministic flux-freezing relations of ideal MHD, but this conclusion is not rigorous. A careful discussion is required of the limiting process involved.

Standard flux-freezing arguments are invoked in the limit of high conductivity or vanishingly small magnetic diffusivity. It is usually assumed that flux-freezing in the conventional sense will hold better and better as λ → 0. This is not true, however, when the plasma flow is turbulent. Instead, flux-freezing in this circumstance breaks down in a completely novel and unexpected way: it remains stochastic (Eyink 2007, 2011).

Consider first the simpler situation of a smooth, laminar solution of the resistive MHD equations in the limit of very small λ. Define the standard (deterministic) Lagrangian flow map by

$$\begin{equation} \left\lbrace \begin{array}{@{}l}\dsty{{d \bf x} \over {dt}}({\bf a},t)={\bf u}({\bf x}({\bf a},t),t), \\[10pt] {\bf x}({\bf a},0)= {\bf a}.\end{array} \right. \end{equation} \tag{ 38 }$$

It is not hard to see under the stated assumptions that¹¹

$$\begin{equation} |\widetilde{{\bf x}}({\bf a},t)-{\bf x}({\bf a},t)|=O(\sqrt{\lambda t}), \end{equation} \tag{ 39 }$$

for typical realizations of the "virtual" flows $\widetilde{{\bf x}}$. The real magnetic field is also given to very good accuracy by the standard Lundquist formula, with errors that vanish as λ → 0. Thus, each "virtual" field line is wiggling a small distance ${\sim} \sqrt{\lambda t}$ around a particular physical field line for very small λ (and not too large times t). For this reason, one does not need to make use of the concept of "virtual" field lines for smooth, laminar solutions of the MHD equations. The lines of force of the real, physical magnetic field are themselves "nearly" frozen-in for very small resistivities and one does not need to make an essential distinction between real and "virtual" fields.

This is not the case for rough, turbulent solutions of the resistive MHD equations, even for vanishingly small λ. By "rough" fields we mean here more precisely that u, B which solve the MHD equations have power-law energy spectra ∝k⁻ⁿ, with 1 < n < 3, for a long range of wavenumbers k with kinematic viscosity ν and magnetic diffusivity λ both small. For example, in the GS95 theory of MHD turbulence the 3D (anisotropic) energy spectra for both u, B can be written in the form

$$\begin{equation} E(k_\perp,k_\Vert)\sim v_A^2 M_A^{4/3} L_i^{-1/3} k_\perp ^{-10/3} f\left({{k_\Vert L_i^{1/3}}\over {k_\perp ^{2/3}M_A^{4/3}}}\right), \end{equation} \tag{ 40 }$$

where we added, compared to the original expressions, the dependence on M_A following LV99. The original GS95 theory was formulated for M_A ≡ 1, as we mentioned earlier. Note that the above expression is expected to be valid for large enough $k_\Vert$ and k_⊥. The function f is not specified in GS95, but Cho et al. (2002) showed that it can be fitted by a Castaing function (see Castaing et al. 1990), which, in its turn, outside the vicinity of k_|| = 0, can be approximated by an exponent. As we mentioned earlier, k_|| and k_⊥ in the expression are not the wavevectors in the conventional sense as the corresponding scales should be measured in the local system of reference.¹²

One thus finds that E(k_⊥)∝k^−5/3_⊥ and $E(k_\Vert)\propto k_\Vert ^{-2}$, so that the fields predicted by GS95 are "rough" in our sense. In this case, Equation (39) is invalid except for very short times, due to the properties of two-particle Richardson diffusion. For example, in the transverse direction perpendicular to the local magnetic field, Equation (39) holds only for times less than $t_\lambda =O(\sqrt{\lambda /\varepsilon })$. If one considers a pair of independent "virtual" flows $\widetilde{{\bf x}},\widetilde{{\bf x}}^{\prime }$, then for times t ≫ t_λ in the root mean square (rms) sense

$$\begin{equation} |\widetilde{{\bf x}}_\perp ({\bf a},t)-\widetilde{{\bf x}}_\perp ^{\prime }({\bf a},t)| \sim (\varepsilon t^3)^{1/2}. \end{equation} \tag{ 41 }$$

See Equation (23). This result is completely independent of magnetic diffusivity, with very remarkable consequences. As λ, ν → 0, and thus also t_λ → 0, the two realizations stay far apart for all times t and do not collapse to a single real particle trajectory. This phenomenon has been called spontaneous stochasticity because the distance between "virtual" particle trajectories stays finite (and random) in the limit, similar to the way that a spontaneous magnetization develops in a ferromagnet below the Curie temperature when an external magnetic field is taken to zero (Bernard et al. 1998; Gawe¸dzki & Vergassola 2000; E & vanden Eijnden 2000, 2001; Chaves et al. 2003; Kupiainen 2003). In the limit λ, ν → 0 all of the "virtual" flows $\widetilde{{\bf x}}({\bf a},t)$ solve the deterministic initial-value problem in Equation (38), which then possesses infinitely many solutions! For more discussion of spontaneous stochasticity and for a review of experimental and numerical evidence of this remarkable phenomenon, see Eyink (2011).

It is an immediate consequence that in a "rough" turbulent velocity field, the conventional notion of flux-freezing must break down. The violations are not small. Taking the square of Equation (39) as the conventional estimate for line-slippage, 〈ℓ²_⊥〉_frozen ∼ λt, and comparing with the predicted amount in Equation (41), 〈ℓ²_⊥〉 ∼ εt³, one finds that

$$\begin{equation} \frac{\langle \ell ^2(t)\rangle }{\langle \ell ^2(t)\rangle }_{{\rm frozen}}\sim M_A^2 \left(\frac{t}{t_L}\right)^2 S_L, \end{equation} \tag{ 42 }$$

where t_L = L_i/u_L is the eddy turnover time and S_L = v_AL_i/λ is the Lundquist number based on the turbulence injection scale. For S_L ≫ 1 typical of astrophysical systems, the violations of standard flux-freezing are enormous and increase with time. It is also not true, however, that flux-freezing is completely violated. SFF in the sense of Equations (31)–(33) remains valid for the "virtual" particle trajectories in a plasma flow which corresponds to a high (kinetic and magnetic) Reynolds number, turbulent MHD solution.

There is another way to understand spontaneous stochasticity by considering the real trajectories of actual plasma fluid elements, which obey Equation (38). Suppose that one considers two-fluid elements located initially at locations a, a' which are displaced from each other by a small amount $\hbox{\boldmath $\rho $}={\bf a}^{\prime }-{\bf a}$. At early times the separation of the particles is ballistic (Batchelor 1950). For example, GS95 theory implies that in the transverse direction

$$\begin{equation} |{\bf x}_\perp ({\bf a}^{\prime },t)-{\bf x}_\perp ({\bf a},t)|\sim |{\bf u}_\perp ({\bf a}^{\prime })-{\bf u}_\perp ({\bf a})|t \sim (\varepsilon \rho _\perp)^{1/3} t, \end{equation} \tag{ 43 }$$

in the rms sense, for times t < t₀ = O((ρ²_⊥/ε)^1/3) and separations ρ_⊥ > ρ_ν = O((ν³/ε)^1/4). However, for t ≫ t₀ the Richardson-type result

$$\begin{equation} |{\bf x}_\perp ({\bf a}^{\prime },t)-{\bf x}_\perp ({\bf a},t)| \sim (\varepsilon t^3)^{1/2}, \end{equation} \tag{ 44 }$$

again holds in the rms sense. Note that the initial particle separation $\hbox{\boldmath $\rho $}$ is completely "forgotten" at sufficiently long times! See Figure 2. In the limit taking first ν → 0 and then ρ → 0, one obtains infinitely many solutions of the deterministic initial-value problem in Equation (38). In fact, one can see by taking $\widetilde{{\bf a}}^{\prime }={\bf a}+\widetilde{\hbox{\boldmath $\rho $}}$ for a random displacement $\widetilde{\hbox{\boldmath $\rho $}}$ that the real fluid particle trajectories also become random in this limit. More precisely, consider an ensemble of random displacements inside the ball $|\widetilde{\hbox{\boldmath $\rho $}}| <\rho _0.$ Then one obtains a random ensemble of real particle trajectories by solving Equation (38) for ${\bf x}({\bf a}+\widetilde{\hbox{\boldmath $\rho $}},t)$ and taking the limits first ν → 0 and then ρ₀ → 0. In fact, it can be shown using Richardson's diffusion approximation for two-particle dispersion that this is the same as the random ensemble of "virtual" particle trajectories obtained by solving Equation (31) and taking the limits ν, λ → 0 together (E & vanden Eijnden 2000). Put another way, for every "virtual" particle trajectory $\widetilde{{\bf x}}({\bf a},t)$ there is a real fluid particle trajectory ${\bf x}({\bf a}+\widetilde{\hbox{\boldmath $\rho $}},t)$ such that $|\widetilde{\hbox{\boldmath $\rho $}}|\rightarrow 0$ and $|\widetilde{{\bf x}}({\bf a},t)-{\bf x}({\bf a}+\widetilde{\hbox{\boldmath $\rho $}},t)|\rightarrow 0$ in the limit ν, λ → 0. "Virtual" particle trajectories coincide with real fluid particle trajectories in this statistical sense.

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The breakdown of conventional flux-freezing due to "roughness" of the fields and spontaneous stochasticity of the Lagrangian flows means that magnetic line-topology is no longer preserved in time for the limit λ → 0. The infinite ensemble of "virtual magnetic fields" frozen-in to the stochastic flows all have exactly the same line-topology as the initial magnetic field. However, the average over the ensemble of virtual field lines that arrive to the same final point which gives the resultant physical magnetic field will, in general, change the topology. In particular, this permits fast topology change (magnetic reconnection) independent of the precise value of the resistivity. It is worth noting, however, that not all topological conservation laws are vitiated. Magnetic helicity conservation, for example, is particularly robust, even when velocity and magnetic fields are extremely "rough." It has been proved by Caflisch et al. (1997, Theorem 4.2) that magnetic helicity is conserved in the limit as λ → 0 even for such "rough" fields, as long as the scaling exponents of velocity and magnetic field third-order structure functions remain positive. This result is consistent with very extreme singularities such as sparsely distributed tangential discontinuities (current and vortex sheets). Thus, fast reconnection due to the mechanism of spontaneous stochasticity will generally conserve magnetic helicity.

One last important remark: everything that we have stated about Lagrangian particle trajectories applies just as well to the magnetic field lines $\hbox{\boldmath $\xi $}({\bf x},\sigma)$, defined by solving (at each fixed time t)

$$\begin{equation} \left\lbrace \begin{array}{@{}l}\dsty{{d{\bm \xi}}\over {d\sigma }}({\bf x},\sigma) ={\bf B}(\hbox{\boldmath $\xi $}({\bf x},\sigma),t), \\[10pt] \hbox{\boldmath $\xi $}({\bf x},0)= {\bf x}.\end{array} \right. \end{equation} \tag{ 45 }$$

Here $\hbox{\boldmath $\xi $}({\bf x},\sigma)$ is the field line which passes through point x (at time t). The parameter σ is related to arc length s along the field line by $ds=|{\bf B}(\hbox{\boldmath $\xi $}({\bf x},\sigma),t)|d\sigma$. In GS95 theory, field lines which correspond to two nearby points x, x' with r = x' − x will separate in the transverse direction as

$$\begin{equation} |\hbox{\boldmath $\xi $}_\perp ({\bf x}^{\prime },s)-\hbox{\boldmath $\xi $}_\perp ({\bf x},s)|\sim (s^3/L_i)^{1/2}M_A^2, \end{equation} \tag{ 46 }$$

in the rms sense, for s ≫ s₀ = O(v_A(r²_⊥/ε)^1/3) and for r_⊥ > r_λ = O((λ³/ε)^1/4); see Equation (8). This is exactly the "stochastic line-wandering" which was invoked in the LV99 theory of fast magnetic reconnection and, later, in the theory of thermal conduction in a turbulent MHD plasma (see Narayan & Medvedev 2001 for M_A = 1 and Lazarian 2006 for an arbitrary M_A). Just as for Lagrangian trajectories, the initial separation r between field lines is "forgotten" after proceeding a large enough distance along them. This is the reason that the aforementioned results for the thermal conductivity do not depend upon the electron Larmor radius ρ_e. In fact, magnetic field lines become stochastic in the limit λ → 0 in precisely the same sense as do Lagrangian particle trajectories in the limit ν → 0.¹³ Consider a random bundle of field lines $\hbox{\boldmath $\xi $}({\bf x}+\widetilde{{\bf r}},s)$ for a stochastic displacement vector $\widetilde{{\bf r}}$ distributed over the ball $|\widetilde{{\bf r}}|<r_0$ of radius r₀. By taking the limits first λ → 0 and then r₀ → 0, one obtains an infinite ensemble of magnetic field lines all passing through the same point x!

Our arguments for the essential stochasticity of turbulent flux-freezing in Sections 4.1 and 4.2 may have given the impression that the origin of the randomness is necessarily connected with Ohmic resistivity. If our conclusions really did depend upon the resistive MHD model in Equation (29), then they could be criticized as physically unrealistic. Indeed, there are in actuality many non-ideal terms that appear in the magnetic equations of motion for an ionized plasma, which may be summarized in the generalized Ohm's law

$$\begin{eqnarray} {\bf E}&=&-\frac{1}{c}{\bf u}\hbox{\boldmath $\times $}{\bf B}+\eta _\perp {\bf J}_\perp +\eta _\Vert {\bf J}_\Vert +\frac{{\bf J}\hbox{\boldmath $\times $}{\bf B}}{nec} -\frac{\hbox{\boldmath $\nabla $}\hbox{\boldmath $\cdot $}{\bf P}_e}{nec} \nonumber \\ && + \frac{m_e}{ne^2}\left(\frac{\partial {\bf J}}{\partial t} + \hbox{\boldmath $\nabla $}\hbox{\boldmath $\cdot $}\left({\bf u}{\bf J}+{\bf J}{\bf u}-\frac{1}{ne}{\bf J}{\bf J}\right)\right), \end{eqnarray} \tag{ 47 }$$

when quasineutrality and m_e ≪ m_i are assumed (see Vasyliunas 1975; Priest & Forbes 2000; Bhattacharjee et al. 1999). The electric fields appearing on the RHS are, respectively, the motional field, Ohmic fields associated with perpendicular and parallel resistivities, the Hall field, a contribution from the electron pressure tensor, and electron inertial contributions. All of these terms have been invoked in various theories of fast magnetic reconnection. However, we shall argue that in the turbulent environments that we consider none of these terms alters our fundamental conclusions. In the first place, the stochasticity of flux-freezing in turbulent plasmas is due not to resistivity (or to other terms in the generalized Ohm's law) but instead to the "roughness" of the MHD solutions in a long inertial range. In the second place, the precise microscopic plasma mechanism of "line-breaking" that acts at small scales (below the ion or electron gyroradii) is irrelevant to the inertial-range turbulence dynamics, which will be fundamentally the same for any such mechanism.

To demonstrate the first point, we show that flux-freezing is effectively stochastic in the turbulent inertial range even for the ideal induction equation

$$\begin{equation} \partial _t{\bf B}=\hbox{\boldmath $\nabla $}\hbox{\boldmath $\times $}({\bf u}\hbox{\boldmath $\times $}{\bf B}), \end{equation} \tag{ 48 }$$

with all non-ideal terms in Equation (47) set to zero! Here we assume that u, B are smooth at length scales below some cutoff <ℓ_d but rough (turbulent) at larger scales. Note that the above Equation (48) is realistic at leading order for collisionless, magnetized plasmas at scales larger than the ion gyroradius ρ_i (Kulsrud 1983), while the smoothness scale ℓ_d set by field-perpendicular viscosity and resistivity is often of the same order or smaller than ρ_i. Thus, our assumptions are quite realistic. Because of the cutoff ℓ_d, flux-freezing in the standard sense must, in fact, be valid. How then can we claim that it becomes stochastic? To see this, consider the magnetic field observed at some finite space resolution ρ:

$$\begin{equation} \overline{{\bf B}}_\rho ({\bf x},t)=\int d^3r \,\,G_\rho ({\bf r}) {\bf B}({\bf x}+{\bf r},t), \end{equation} \tag{ 49 }$$

where we have introduced a coarse-graining kernel G_ρ to represent the smearing effect of the observation over a ball of radius ρ around the space point x. We shall assume below that ℓ_d ≪ ρ ≪ L_i, the scale of the largest turbulent eddies. Thus, ρ ≳ ρ_i satisfies these conditions. Applying the standard Lundquist formula for the frozen-in magnetic field, one obtains

$$\begin{equation} \overline{{\bf B}}_\rho ({\bf x},t)= \int \left.\frac{{\bf B}({\bf a},t_0)\hbox{\boldmath $\cdot $}\hbox{\boldmath $\nabla $}_a {\bf x}_{t,t_0}({\bf a})}{{\rm det}(\hbox{\boldmath $\nabla $}_a{\bf x}_{t,t_0}({\bf a}))} \right|_{{\bf x}_{t,t_0}({\bf a})={\bf x}+{\bf r}}G_\rho ({\bf r}). \,d^3r. \end{equation} \tag{ 50 }$$

The Lagrangian particle trajectories that appear in this formula start in the ball of radius ρ around x at time t and then follow the flow velocity u backward in time to t₀, as illustrated in Figure 3. When ρ lies in a GS95 turbulent inertial range, then the trajectories explosively separate to a perpendicular distance Δx_⊥ ∼ (ε|t − t₀|³)^1/2, independent of ρ at times |t − t₀| ≫ (ρ²/ε)^1/3.

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The result is indistinguishable from the stochastic Lundquist formula (35) which was derived in Section 4.1 using the stochastic representation of Laplacian resistivity. In fact, in a formal mathematical limit taking first ℓ_d → 0, then ρ → 0, the Lagrangian trajectories in Equation (50) remain stochastic and the two formulae coincide. This is a rigorous theorem for the Kazantsev–Kraichnan dynamo model, where it has been proved that the ensemble of stochastic Lagrangian trajectories as constructed above is precisely the same as that obtained for the λ → 0 limit (E & vanden Eijnden 2000). Stochasticity of flux-freezing in not due intrinsically to resistivity or other microscopic plasma mechanisms that "break" field lines but is, instead, a fundamental consequence of turbulent Richardson diffusion.

Higher-order terms in the generalized Ohm's law Equation (47) that do not appear in the ideal Equation (48) will lead to melding and merging of field lines at scales <ρ_i. However, the above argument strongly suggests that these details of the microscopic plasma processes do not affect the dynamics at scales larger than ρ_i. In some cases this can be shown more analytically by defining a suitable "motion" of field lines consistent with the induction equation. For example, the formulation in Section 4.1 based on addition of a Brownian motion to the Lagrangian particle dynamics, Equation (31), can be carried over to certain instances of the generalized Ohm's law (see in particular Eyink 2009 for the HMHD equations). This approach is used in Appendix B to argue that neither the Hall effect nor Ohmic resistivity will have any significant influence on the inertial-range turbulence dynamics at large enough scales. The Hall term, for example, does not affect the dynamics at scales much greater than $\delta _i=\rho _i/\sqrt{\beta _i}$, the ion skin depth. Unfortunately, it is difficult to extend this type of argument to all cases of the generalized Ohm's law because it is not known how to define a "motion" of field lines consistent with the induction equation for the general case.

On the other hand, there is a different argument which applies in general and leads to the same conclusion that flux-freezing must be intrinsically stochastic in turbulent plasmas. While the "motion" of magnetic field lines is a conventional and somewhat arbitrary concept, the motion of plasma is perfectly well defined within the validity of an MHD description. Plasma fluid moves with the bulk velocity u. Thus, field lines may be tracked by "tagging" the lines with plasma fluid elements and then following these as Lagrangian fluid particles (Newcomb 1958; Axford 1984). In the case of a smooth, laminar solution of the ideal MHD equations, this is unambiguous because of Alfvén's theorem: two plasma particles which start on a certain field line must share a field line for all times. One can then, by convention, consider this as the "same" field line as the initial one. This approach fails for a non-ideal Ohm's law,

$$\begin{equation} {\bf E}+\frac{1}{c}{\bf u}\hbox{\boldmath $\times $}{\bf B}={\bf R}, \end{equation} \tag{ 51 }$$

where R represents all of the terms on the RHS of Equation (47) other than the motional term. Clearly, R is just the electric field in the rest frame of the plasma flowing with the bulk velocity u. EMF due to these non-ideal terms leads to time-dependent magnetic flux in the rest frame, corresponding to a slippage of field lines. This vitiates the usual method to assign an identity to individual field lines over time, because plasma elements shift their attachments to lines. Charged particles move along magnetic field lines, but two plasma elements that start on one field line will sit on distinct field lines at later times.

Now consider a turbulent plasma where the non-ideal term is numerically "small" but the plasma has a turbulent inertial range in which the velocity field u is rough, with a power-law energy spectrum extending down to a smallest length scale ρ₀ ≈ ρ_i. The slight shifts in line-attachments are enormously amplified by explosive relative advection, as illustrated in Figure 4. Consider a single magnetic field line in this plasma and, along it, two plasma fluid particles at initial locations a and a'. Due to a combination of the non-ideal field R and advection by sub-inertial-range eddies, the two plasma particles will end up on distinct field lines displaced a distance |x(a', t) − x(a, t)| = ρ₀ apart in a time τ₀ which is generally microscopically small compared with the eddy turnover time t_L. Because of Equation (44), the two plasma elements will subsequently end up on field lines displaced a macroscopic distance (εt³)^1/2 apart independent of ρ₀ in a time t ≫ τ₀. ¹⁴ In effect, the original field line has split apart into two lines separated by macroscopic distance in a finite time t. Of course, this same argument works if we take $\tilde{{\bf a}}^{\prime }={\bf a}+\tilde{\hbox{\boldmath $\rho $}}$ for some small random displacement $\tilde{\hbox{\boldmath $\rho $}}$ along the initial field line with $|\tilde{\hbox{\boldmath $\rho $}}|<\rho _0$. We have already argued in Section 4.2 that the random ensemble of Lagrangian trajectories ${\bf x}({\bf a}+\tilde{\hbox{\boldmath $\rho $}},t)$ will coincide for t ≫ τ₀ with the random ensemble $\tilde{{\bf x}}({\bf a},t)$ obtained from the perturbation by Brownian motion in Equation (19). The initial field line will be stochastically "frozen-in" to this ensemble of random flows.

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Turbulent advection accelerates the separation of field lines so effectively that the microscopic plasma process of line-slippage, whatever its origin, is rapidly "forgotten." The non-ideal terms in the generalized Ohm's law become insignificant in further separating lines which have exceeded the distance ρ₀ apart. When the plasma flow velocity u is "rough," it is the motional term in the generalized Ohm's law which, after a very short time, dominates all of the other terms in the separation of field lines.

The constraints on steady-state reconnection from the MHD balance equations are well known. For a reconnection layer of length L_x and thickness Δ, the constraint of energy conservation is

$$\begin{equation} {{1}\over {2}} \rho v_{{\rm out}}^2v_{{\rm out}}\Delta = {{1}\over {8\pi }}B^2_xv_{{\rm rec}}L_x, \end{equation} \tag{ 52 }$$

where v_rec is the inflow plasma speed or reconnection velocity and v_out is the outflow velocity from the sides of the reconnection layer. Here it is assumed that the energy in the outflow jets is primarily kinetic, since the B_y component of the outgoing field lines at the outer edges of the reconnection layer is still relatively small. The critical constraint of mass conservation is

$$\begin{equation} v_{{\rm out}} \Delta = v_{{\rm rec}}L_x. \end{equation} \tag{ 53 }$$

Combining Equations (52) and (53) yields

$$\begin{equation} v_{{\rm out}} = {{B_x}\over {\sqrt{4\pi \rho }}}=v_A, \end{equation} \tag{ 54 }$$

so that the outflow velocity is at the Alfvén speed of the in-coming magnetic fields. The momentum equation enforces pressure balance along and across the reconnection layer and does not yield a new constraint in our setting. From Equations (53) and (54) we see that

$$\begin{equation} v_{{\rm rec}}= v_A {{\Delta }\over {L_x}}. \end{equation} \tag{ 55 }$$

Our discussion to this point has been completely general and no particular mechanism of line-slippage has been assumed. To proceed further, a specific reconnection mechanism must be invoked which permits one to estimate the thickness Δ of the layer.

The classical Sweet–Parker model assumes resistive reconnection in a laminar MHD solution. There are then several ways to estimate the thickness Δ, which are all consistent. The most common approach is to use the steady-state Faraday's law, which implies the constancy in the vertical y-direction of the reconnection electric field E_z. Outside the reconnection layer E_z = v_recB_x/c, while inside E_z = ηJ_z = ηcB_x/4πΔ. Equating these gives

$$\begin{equation} v_{{\rm rec}} = {{\lambda }\over {\Delta }}, \end{equation} \tag{ 56 }$$

where λ = ηc²/4π is the magnetic diffusivity. Combining Equations (55) and (56) gives the well-known Sweet–Parker results that

$$\begin{equation} \Delta = L_x/\sqrt{S}, \,\,\,\,v_{{\rm rec}} = v_A/\sqrt{S}, \end{equation} \tag{ 57 }$$

where S = v_AL_x/λ is the Lundquist number. Another way to obtain this result is based on standard flux-freezing ideas for a laminar plasma flow (Kulsrud 2005). The mean-square vertical distance that a magnetic field line can diffuse by resistivity in time t is

$$\begin{equation} \langle y^2(t)\rangle \sim \lambda t. \end{equation} \tag{ 58 }$$

Each reconnected field line is the result of resistive gluing of field lines that diffuse vertically across the reconnection layer. At the same time, the field lines are advected out of the sides of the reconnection layer of length L_x at a velocity of order v_A. Thus, the time that the lines can spend in the resistive layer is the Alfvén crossing time t_A = L_x/v_A. Thus, field lines can only be merged that are separated by a distance

$$\begin{equation} \Delta = \sqrt{\langle y^2(t_A)\rangle } \sim \sqrt{\lambda t_A} = L_x/\sqrt{S}. \end{equation} \tag{ 59 }$$

We thus recover Equation (57). This is an important dynamical consistency check on the Sweet–Parker model of laminar resistive reconnection.

For large Lundquist numbers, the Sweet–Parker reconnection speed Equation (57) is too small to account for observed reconnection rates in most astrophysical settings. However, the Sweet–Parker laminar MHD solution is not appropriate for turbulent plasmas at large S. For simplicity, we shall consider reconnection in the presence of a steady-state background turbulence with an integral length scale L_i and rms velocity u_L. We shall consider sub- and trans-Alfvénic turbulence with u_L ⩽ v_A or M_A ⩽ 1, under the assumption that the large-scale magnetic-flux tubes are strong and well defined in the presence of turbulence. In some situations the reconnection may be initially laminar and then turbulence is initiated by instabilities triggered by small fluctuations from the environment. Initial fast reconnection by the Hall effect (Shay et al. 1998; Wang et al. 2000; Birn et al. 2001; Drake 2001) or other collisionless processes may itself provide the energy release to generate small-scale MHD turbulence. We shall consider only the final stage of steady-state reconnection in the presence of turbulence with the given characteristic length and velocity scales.

The key issue is to estimate Δ. We shall give two different methods of determining the width of the turbulent reconnection layer.

Our first estimate is based on SFF (Section 4) and two-particle turbulent diffusivity (Section 3.2). We have seen that GS95 theory implies that the turbulent separation of pairs of "virtual" field lines backward in time, in the y-direction perpendicular to the mean magnetic field, is given by Equations (23) and (28), or

$$\begin{equation} \langle y^2(t)\rangle \sim \left\lbrace \begin{array}{@{}ll}\varepsilon t^3, & t_\eta \ll t<t_c \\[6pt] u_LL_i M_A^3 t, & t>t_c, \cr \end{array} \right. \end{equation} \tag{ 60 }$$

for the strong and weak MHD turbulence regimes, respectively, with t_c = L_i/v_A. Reconnected field lines that emerge from the turbulent reconnection layer are constituted from field lines obtained by following "virtual" field lines backward in time over the Alfvén crossing period t_A = L_x/v_A. We thus obtain $\Delta = \sqrt{\langle y^2(t_A)\rangle }$ as the width of the turbulent reconnection layer. There are two cases to consider, depending upon whether t_A < t_c or t_A > t_c. Using ε ∼ u⁴_L/v_AL_i, we rewrite Equation (60) as

$$\begin{equation} \langle y^2(t)\rangle \sim \left\lbrace \begin{array}{@{}ll}(L_x^3/L_i) M_A^4 (t/t_A)^3 &{\rm if}\ t<t_c \\ \ms L_iL_x M_A^4 (t/t_A), &{\rm if}\ t>t_c. \end{array}\right. \end{equation} \tag{ 61 }$$

Since t_c≶t_A exactly when L_i≶L_x, respectively, we get finally that

$$\begin{equation} \Delta = L_x M_A^2 \min \left\lbrace \left({{L_x}\over {L_i}}\right)^{1/2}, \left({{L_i}\over {L_x}}\right)^{1/2}\right\rbrace. \end{equation} \tag{ 62 }$$

This is precisely the result of Lazarian & Vishniac (1999). The present derivation avoids many complications in their original argument due to the fact that "virtual" field lines pass directly through each other. LV99 had to consider the limit to the reconnection rate set by the speed with which reconnected magnetic field elements can pass through each other on the way out of the reconnection zone, employing a plausible but heuristic self-consistency argument. This problem is avoided completely and rigorously by using SFF (which is mathematically equivalent to the resistive MHD equations).

The thickness Δ of the turbulent reconnection layer was originally estimated in LV99 by a geometric argument, based upon the spontaneous stochasticity of the magnetic field lines themselves. The plasma in the reconnection zone can only be expelled along the lines of the (real, physical) magnetic field. As we have seen in Section 3.1, the lateral wandering as one moves a distance s along a field line is given by Equations (8), (9), or

$$\begin{equation} \langle y^2(s)\rangle \sim \left\lbrace \begin{array}{@{}ll}(s^3/L_i)M_A^4 &{\rm if}\ s<L_i \\ \ms L_i s M_A^4, &{\rm if}\ s>L_i. \end{array}\right. \end{equation} \tag{ 63 }$$

Since the plasma must move a distance s ∼ L_x along the field line in order to escape the reconnection layer, we can estimate the thickness of the layer by $\Delta \sim \sqrt{\langle y^2(L_x)\rangle }$. Using Equation (63) we immediately recover Equation (62). The agreement of these two arguments demonstrates some basic dynamical consistency of the LV99 theory.

We obtain from both arguments the LV99 estimate for the reconnection speed

$$\begin{equation} v_{{\rm rec}} = v_A M_A^2 \min \left\lbrace \left({{L_x}\over {L_i}}\right)^{1/2}, \left({{L_i}\over {L_x}}\right)^{1/2}\right\rbrace. \end{equation} \tag{ 64 }$$

Note that v_trans = v_AM²_A is the amplitude of velocity fluctuations at the transition between weak and strong MHD regimes. This can be a sizable fraction of the Alfvén velocity. Furthermore, Equation (64) is "fast" in the strictest sense of being independent of the microscopic resistivity. The physical mechanism of this independence is spontaneous stochasticity, both of the Lagrangian particle trajectories and of the magnetic field lines. In both cases, initial separations by a microscopic resistive length are amplified to macroscopic distances as one traverses along the wandering curves by an amount (of time or of path length, respectively) which is independent of the microscopic electrical resistivity.

The above discussion has approximated the reconnection layer as a zone of homogeneous MHD turbulence, with a constant Alfvénic Mach number M_A = u_L/v_A set by the reconnecting x-component of the upstream magnetic field, $v_A=B_x/\sqrt{4\pi \rho }$. In reality, there will be a "magnetic shear layer" with a continuous change of the mean magnetic field $\overline{B}_x(y)$ across the layer. Likewise, there are nontrivial profiles $\overline{u}_y(x,y)$ and $\overline{u}_x(x,y)$ of both incoming and outgoing velocities, which are known to have nontrivial effects on two-particle dispersion (Shen & Yeung 1997; Pope 2002). An interesting refinement of the original LV99 theory would be to consider the effect of variation of the mean velocity and magnetic fields across the reconnection layer. One plausible effect of magnetic shear will be to increase M_A within the layer, permitting greater lateral wandering of field lines and increasing the thickness Δ. Less easy to guess are the effects of the change in orientations of magnetic fields for flux tubes reconnecting at an angle of less than 180° (Linton et al. 2001). The simulation results of Kowal et al. (2009) on tubes crossed at various angles give support to the idea that the effects are minimal, but further theoretical and numerical study of this would be desirable. Another effect which has been neglected in our derivation above is the turbulent energy dissipation that exists in the reconnection layer. This can be estimated as ρεΔL_x with ε = u⁴_L/v_AL_i. If this is added into the energy balance Equation (52) one obtains a result for the outflow velocity

$$\begin{equation} v_{{\rm out}}^2=v_A^2\left(1-2M_A^4\frac{L_x}{L_i}\right), \end{equation} \tag{ 65 }$$

which can be substantially reduced from the Alfvén speed. By the mass conservation constraint Equation (53), there is a similar reduction in the reconnection speed. However, these effects are ignorable under the condition M⁴_A ≪ L_i/L_x or, equivalently, Δ ≪ L_i. This is generally well satisfied. Other modifications and refinements of the LV99 model can be envisaged and are worth pursuing (see the opening paragraph of this section and also Section 6.4).

It is important to emphasize that the original LV99 argument made no essential use of averaging over turbulent ensembles. The stochastic line-wandering which is the essence of their argument holds in every realization of the flow, at each instant of time. The "spontaneous stochasticity" of field lines is not a statistical result in the usual sense of turbulence theory and does not arise from ensemble averaging. The only use of ensembles in LV99 is to get a measure of the "typical" wandering distance Δ of the field lines (in an rms sense). If one looks at different ensemble members of the turbulent flow, or at different single-time snapshots of the steady reconnection state, then Δ will fluctuate. Thus, the reconnection rate will also fluctuate a considerable amount over ensemble members or over time (e.g., see Figures 12–14 in Kowal et al. 2009). But it will be "fast" in each realization and at each instant of time because the mass outflow constraint is lifted by the large wandering of field lines.

Exactly the same remarks apply to the rederivation of LV99 using SFF and Richardson diffusion. SFF is also not "stochastic" in the usual sense of turbulence theory. The new derivation uses ensemble averaging just to get an estimate of the "typical" wandering of stochastic Lagrangian trajectories in an rms sense. Numerical studies of Richardson's t³ law in hydrodynamic turbulence show that it is a very robust property. At most points in a turbulent flow, a "cloud" of stochastic particles spreads with a power law very close to t³. There is some intermittency effect, with certain rarer, high-intensity points showing a much faster dispersion or low-intensity a slower dispersion, but it is not a huge effect. Only the higher-order moments of the relative separation distance are strongly affected (Boffetta & Sokolov 2002).

The LV99 theory, therefore, does not involve "turbulent resistivity" or "turbulent magnetic diffusivity" as this is usually understood. This is ordinarily meant to be an enhanced diffusivity experienced by the ensemble-averaged magnetic field 〈B〉. However, it is only if one assumes some scale separation between the mean and the fluctuations that the effect of the fluctuations can be legitimately described as an enhanced diffusivity (Moffatt 1983). In realistic turbulent flows, with no scale separation, this phenomenological description as an effective diffusivity can be wildly inaccurate. LV99 make no appeal to such concepts and, indeed, never considers the ensemble-average field 〈B〉 at all. Neither does Nature. The Sun does not average over an ensemble of coronal loops in order to get fast reconnection!

The focus of this paper has been on the reconnection of large scale, oppositely directed magnetic-flux tubes in a turbulent plasma, since it is this phenomenon which has the most direct interest for astrophysics. On the other hand, a similar process of reconnection of local magnetic field lines must take place at small scales in homogeneous MHD turbulence, with or without a background magnetic field. It has been argued that such small-scale magnetic reconnection is an essential feature of MHD turbulence (Matthaeus & Lamkin 1986; Lazarian & Vishniac 1999). A substantial literature has since arisen on the relation between MHD turbulence and magnetic reconnection (e.g., see Lazarian 2005; Eyink & Aluie 2006; Servidio et al. 2010; Mininni & Pouquet 2009). We cannot review this entire subject here, but it is important to discuss briefly the implications of "spontaneous stochasticity" for the problem.

It was observed already in LV99 that nearly parallel field lines in adjacent turbulent eddies must frequently intersect and reconnect according to GS95 theory. The local geometry is similar to that in large-scale reconnection "writ small," with the local mean magnetic field acting as a guide field and the field lines with anti-parallel transverse components undergoing reconnection. The crossing-points and knots in the local magnetic field lines, if unresolved, would create topological obstructions to the plasma flow which would render hydrodynamic motion impossible. LV99 argued instead that line-intersections at perpendicular length scale ℓ_⊥ are resolved by magnetic reconnection in the eddy turnover time ℓ_⊥/δu(ℓ), independent of the resistivity. Fast magnetic reconnection is thus crucial for the very existence of MHD turbulence of the kind supposed in GS95.

The stochastic "line-wandering" implied by GS95 theory (Section 3.1) suggests that a very complex line-topology must exist in turbulent flows with magnetic fields that are non-smooth on inertial-range scales. This complexity is made particularly evident by the "spontaneous stochasticity" phenomenon (Section 4.2), according to which there are infinitely many field lines passing though each point in the limit of zero resistivity! This result suggests that reconnection sites are distributed densely in space for MHD turbulence at very high conductivity. Turbulent plasma flows with rough magnetic and velocity fields have a very strange and intricate geometry when compared with the smooth, laminar flows that have often been the focus of theoretical and empirical studies of reconnection. Another indication of this complexity is the fractal distribution of clustered magnetic nulls for a magnetic field with a turbulent, power-law energy spectrum (Albright 1999). This latter fact is very intimately connected with the stochastic "line-wandering" phenomenon deduced in LV99, since the magnetic nulls form natural sites of separation of adjacent magnetic field lines. See Davila & Vassilicos (2003) and Goto & Vassilicos (2004) for a discussion of the closely related hydrodynamic phenomenon associated with Lagrangian particle trajectories and stagnation points of the turbulent velocity field.

The above facts strongly suggest that magnetic reconnection must be a process that is active always and everywhere in homogeneous MHD turbulence. As one measure of the local reconnection rate one may take the estimates in Equations (23) and (28) of the lateral diffusion of magnetic field lines (for strong and weak turbulence, respectively). For example, in the strong-turbulence regime, field lines diffuse in an eddy turnover time τ_ℓ = ε^−1/3ℓ^2/3_⊥ at scale ℓ_⊥ by a mean-square perpendicular distance τ³_ℓ = ℓ_⊥². Thus, reconnection rates are exactly large enough to resolve knots of field lines at length scale ℓ_⊥ in the natural evolution time at that scale, as argued in LV99. Note that dℓ_⊥/dt ≃ δu(ℓ) and thus the reconnection speed of pairs of field lines separated by inertial-scale separations ℓ_⊥ is equal to the relative velocity δu(ℓ) between the two lines.

Another way to quantify small-scale reconnection in MHD turbulence is by violations of Alfvén's theorem at length scale ℓ. This was the approach adopted by Eyink & Aluie (2006), who studied the MHD induction equation in a coarse-grained description at the (isotropic) length scale ℓ¹⁵:

$$\begin{equation} \partial _t\overline{{\bf B}}_\ell =\hbox{\boldmath $\nabla $}\hbox{\boldmath $\times $}[\overline{{\bf u}}_\ell \hbox{\boldmath $\times $}\overline{{\bf B}}_\ell +c\,\hbox{\boldmath $\varepsilon $}_\ell -\lambda \hbox{\boldmath $\nabla $}\hbox{\boldmath $\times $}\overline{{\bf B}}_\ell ]. \end{equation} \tag{ 66 }$$

Simple estimates show that the direct resistive contribution proportional to λ is negligible at high magnetic Reynolds numbers whenever the magnetic energy stays finite in that limit (see Eyink 2005). It is the turbulent subscale EMF $\hbox{\boldmath $\varepsilon $}_\ell =\overline{({\bf u}\hbox{\boldmath $\times $}{\bf B})}_\ell -\overline{{\bf u}}_\ell \hbox{\boldmath $\times $}\overline{{\bf B}}_\ell$ which provides the necessary electric field for reconnection at the scale ℓ. The magnetic flux through a material loop $\overline{C}_\ell (t)$ advected by the coarse-grained velocity $\overline{{\bf u}}_\ell$ satisfies

$$\begin{equation} \frac{d}{dt}\oint _{\overline{C}_\ell (t)} \overline{{\bf A}}_\ell \hbox{\boldmath $\cdot $}d{\bf x}= c \oint _{\overline{C}_\ell (t)} \overline{\hbox{\boldmath $\varepsilon $}}_\ell \hbox{\boldmath $\cdot $}d{\bf x}. \end{equation} \tag{ 67 }$$

Eyink & Aluie (2006) established necessary conditions for a non-vanishing line-voltage on the RHS of the above equation, independent of ℓ. These conditions are: either (1) a non-rectifiable (fractal) loop $\overline{C}_\ell (t)$, or (2) a blowup of u or B along the loop, or (3) tangential discontinuities of both u and B intersecting the loop $\overline{C}_\ell (t)$ over a finite length. One or more of these conditions is plausibly satisfied in turbulent flow, implying ubiquitous reconnection. An important remark is that MHD turbulence is dominated by scale–local interactions under very general assumptions (see Eyink 2005; Aluie & Eyink 2010). In particular, the turbulent EMF $\hbox{\boldmath $\varepsilon $}_\ell$ is dominated by contributions from velocity and magnetic field modes around the scale ℓ. It is the EMF from these modes which is responsible for reconnection of the lines of the coarse-grained magnetic fields at length scale ℓ and microscopic plasma processes at smaller scales play no direct role.

While astrophysical reality is 3D, there is a scholarly and intellectual value to considering MHD turbulence in 2D space. Comparison of toy models with reality is often a good way to test and sharpen one's understanding. Numerical simulations can also be better resolved in 2D than in 3D, permitting study of higher kinetic and magnetic Reynolds number flows. Furthermore, the conventional view is that MHD turbulence in 2D and in 3D are very similar, much more so than for hydrodynamics of neutral fluids. 3D MHD possesses the three ideal, quadratic invariants of total energy, cross-helicity, and magnetic helicity and 2D MHD has a nearly identical set of ideal invariants, with magnetic helicity replaced by the mean-square magnetic potential. Absolute equilibrium spectra in 3D suggest that energy and cross-helicity cascade to high wavenumbers and magnetic helicity to low wavenumbers (Frisch et al. 1975), while the same argument in 2D leads to an identical conclusion, with an inverse cascade of magnetic potential replacing that of magnetic helicity (Fyfe & Montgomery 1976). Numerical simulations have corroborated these predictions, with the most recent 2D MHD simulations giving energy spectra close to k^−5/3 or k^−3/2 in the forward cascade range (Gomez et al. 1999; Biskamp & Schwarz 2001; Ng et al. 2003). On the other hand, from the point of view of GS95 theory, 2D MHD turbulence appears as a very degenerate case. In particular, shear-Alfvén waves that play the dominant role in 3D MHD turbulence according to GS95 are entirely lacking in 2D, where only pseudo-Alfvén wave modes exist. This fact suggests that the character of MHD turbulence in 2D must be quite different than in 3D and certainly not well described by GS95.

Because of the numerical advantages, many turbulent reconnection studies have also been performed in 2D, beginning with the important work of Matthaeus & Lamkin (1985, 1986). The numerical evidence is rather controversial, however. Matthaeus & Lamkin (1985) reported the reconnection of magnetic islands at rates nearly independent of resistivity in the early stages of their simulations. Servidio et al. (2010) have more recently made a similar study of Ohmic electric fields at X-points in homogeneous, decaying 2D MHD turbulence. This is really a case of small-scale magnetic reconnection, as considered in our previous Section 5.3, and not directly relevant to the issue of reconnection of large-scale flux tubes. Two other numerical studies have recently been made of large-scale magnetic reconnection in 2D, by Loureiro et al. (2009) and Kulpa-Dybeł et al. (2010), which reach different conclusions. On the one hand, Loureiro et al. (2009) had a better resolution but used periodic boundary conditions, which strongly constrain the ability to do averaging of the reconnection rate and the attainment of the steady state for reconnection. They inferred from their data that the 2D turbulent reconnection rate may be independent of resistivity. On the other hand, Kulpa-Dybeł et al. (2010) used smaller data cubes but longer averaging, which is enabled by their outflow boundary conditions. They concluded that the reconnection does depend on resistivity and therefore is slow. The raw data of the two groups are actually rather similar and higher Lundquist-number simulations are probably necessary to resolve the matter.

What does theory say? In particular, can LV99 or some LV99-like theory apply in 2D¹⁶? At first thought, one might doubt that LV99 is relevant at all since its central element—stochastic line-wandering—seems ruled out in 2D on the simple grounds that magnetic field lines may not cross. However, closer consideration shows that this argument is fallacious. Magnetic field lines can cross in 2D, at very special points, the magnetic nulls. Furthermore, in a rough magnetic field, the magnetic nulls and null–null lines will form a very complex, fractal web (Albright 1999) which may permit quite unconstrained wandering of field lines. Indeed, the phenomenon of spontaneous stochasticity of field lines ought to occur in 2D as well as in 3D, if the magnetic field is rough. There are rigorous examples of this type in 2D, where integral curves of a non-smooth vector field are non-unique at every point in space (e.g., see Section II.5 and Figure 2 in Hartman 2002). From the Lagrangian point of view of Richardson diffusion and SFF, there is also little difference between 2D and 3D. The Richardson t³-law has been observed, for example, in the inverse-energy cascade range of 2D hydrodynamic turbulence (Celani & Vergassola 2001; Goto & Vassilicos 2004) and its analog could be expected in 2D MHD turbulence. Indeed, if velocity fields are sufficiently rough in 2D MHD turbulence, then spontaneous stochasticity of Lagrangian trajectories ought to hold in 2D just as in 3D, and an LV99-like theory ought to apply.

The above arguments are, however, quite delicate. As a counterexample, we may consider weak MHD turbulence in 2D. The most important difference from 3D is that only pseudo-Alfvén wave modes exist in 2D. Because the weak-turbulence cascade preserves $k_\Vert =1/L_i$ while increasing k_⊥, the total wavevector is nearly perpendicular and the pseudo-Alfvén polarization vector nearly parallel to the magnetic field. Our estimate from Section 3.2 for the two-particle diffusivity of weak MHD turbulence is thus reduced in the field-perpendicular direction by a factor of $(\ell _\perp /\ell _\Vert)^2$. The result is that

$$\begin{equation} D_\perp (\ell) \sim \left(\frac{\ell _\perp }{\ell _\Vert }\right)^2 \frac{\varepsilon }{\omega _A^2} = \ell _\perp ^2 \frac{\varepsilon }{v_A^2}. \end{equation} \tag{ 68 }$$

Solving for the perpendicular particle separation from dℓ²_⊥/dt = D_⊥(ℓ) gives an exponential growth proportional to the initial separation

$$\begin{equation} \ell ^2_\perp (t) \simeq \ell ^2_\perp (0)\exp ({\rm (const.)}\varepsilon t/v_A^2), \end{equation} \tag{ 69 }$$

i.e., no spontaneous stochasticity! There will be no 2D fast reconnection if such a weak-turbulence cascade persists down to dissipation scales. Naively, one would expect this to be true in 2D from a simple timescale argument. The interaction time of a pair of pseudo-Alfvén waves with parallel length scale $\ell _\Vert$ and fluctuation amplitude δu_ℓ is $\ell _\Vert /\delta u_\ell$, which will in general be longer than the Alfvén wave period $\ell _\Vert /v_A$. This argument suggests that there is no strong MHD turbulence at all in 2D and thus no fast reconnection.

The only way to avoid this conclusion that we can see is that the 2D turbulence develops very singular structures or "tangential discontinuities" with δu_ℓ ≃ v_A, independent of ℓ over the inertial range. Note that the rigorous result of Eyink & Aluie (2006) for the 2D case requires such discontinuities and/or blowup in the magnitudes of the velocity or magnetic fields. There seems also to be support for this scenario from numerical simulations of 2D MHD turbulence, which show very intermittent current and vortex sheets in a background of weaker turbulence. It is possible that this is one of those instances where the weak-turbulence cascade is unstable to the formation of localized, singular structures (cf. Majda et al. 1997; Rumpf et al. 2009). The crucial issue seems to be whether these intermittent structures occur sufficiently densely in spacetime. We shall not attempt to resolve this issue here, but only conclude that the existence of strong MHD turbulence and fast turbulent reconnection in 2D are quite open questions.

As we mentioned in the introduction, various ideas how turbulence can increase the reconnection rate were discussed as far back as 40 years ago. However, these ideas fell short of solving the problem. For instance, some papers have concentrated on the effects that turbulence induces on the microphysical level. In particular, Speiser (1970) showed that in collisionless plasmas the electron collision time should be replaced with the electron retention time in the current sheet. Also Jacobson & Moses (1984) proposed that the current diffusivity should be modified to include the diffusion of electrons across the mean field due to small-scale stochasticity. All these effects can only marginally change the reconnection rates.

The closest predecessor to LV99 was the important work of Matthaeus & Lamkin (1985, 1986, ML85,86). Those authors studied 2D magnetic reconnection in the presence of external turbulence, both theoretically and numerically. They emphasized the very close analogies between the magnetic reconnection layer at high Lundquist numbers and homogeneous MHD turbulence. They also pointed out various turbulence mechanisms that would enhance reconnection rates, including multiple X-points as reconnection sites and motional EMF of magnetic bubbles advecting out of the reconnection zone. However, ML85,86 did not understand the importance of "spontaneous stochasticity" of field lines and of Lagrangian trajectories and they did not arrive at an analytical prediction for the reconnection speed. An enhancement of the reconnection rate was reported in their numerical study, but the setup precluded the calculation of a long-term average reconnection rate. A more recent study along the lines of the approach in Matthaeus & Lamkin (1985) is that of Watson et al. (2007), where the effects of small-scale turbulence on 2D reconnection were studied and no significant effects of turbulence on reconnection were found. However, this new work studies flow-driven merging in the super-Alfvénic limit, with very strong flows ramming magnetic field together. In this situation, turbulence-induced X-points that might provide additional reconnection sites are usually rapidly ejected from the sheet. As discussed in the previous section, it is still a wide open question whether fast reconnection can occur in 2D MHD.

Some other numerical simulation papers have claimed to see evidence for fast reconnection in resistive MHD simulations. An important study of tearing instability of current sheets in the presence of background 2D turbulence and the formation of large-scale, long-lived magnetic islands has been performed in Politano et al. (1989). They present evidence for "fast energy dissipation" in 2D MHD turbulence and show that this result does not change as they change the resolution. A more recent work of Mininni & Pouquet (2009) provides evidence for "fast dissipation" also in 3D MHD turbulence. This phenomenon is consistent with the idea of fast reconnection, but cannot be treated as a direct evidence of the process. Evidently, fast energy dissipation and fast magnetic reconnection are rather different physical processes. A series of papers by Galsgaard and Nordlund, in particular Galsgaard & Nordlund (1997b), might also be interpreted as providing indirect evidence for fast reconnection. The authors noted that in their simulations they could not produce highly twisted magnetic fields. One of the interpretations of this finding could be the relaxation of magnetic field via reconnection. In this case, this observations could be related to the numerical finding of Lapenta & Bettarini (2011) which shows that reconnecting magnetic configurations spontaneously get chaotic and dissipate, which in turn may be related to the predictions in LV99. However, in view of many uncertainties of the numerical studies, we are not confident of this connection. The highest resolution simulations of Galsgaard & Nordlund (1997b) were only 136³ and at modest Lundquist numbers that could not permit a turbulent inertial range.

A number of papers have attempted to treat turbulent magnetic reconnection by the formal methods of mean-field electrodynamics (MFE), e.g., Strauss 1988 and Kim & Diamond 2001. Before we address these works in any detail, let us briefly discuss the relationships of LV99 itself to MFE. LV99 theory as originally formulated does not consider ensemble- or time-averaged magnetic fields, does not employ MFE and, in fact, makes only very limited use of probability and averaging. The key observation to derive LV99 predictions using the exact formula (35) for the magnetic field—the "stochastic Lundquist formula"—is that it does not reduce to the conventional Lundquist formula as λ → 0 in a turbulent plasma. The typical lateral spread of Lagrangian trajectories that contribute in that limit is estimated from GS95 turbulence theory to be ℓ²_⊥ ∼ ε|t − t₀|³ for t < L_i/v_A and to be ℓ²_⊥ ∼ u_LL_iM³_A|t − t₀| for t > L_i/v_A. This is a turbulent mixing effect but not in the usual sense of an enhanced "eddy diffusivity" or "β-effect" experienced by a mean magnetic field. Within the latter concept (see Parker 1979), magnetic fields are presumed to be mixed up passively by turbulence and the rate of reconnection thereby accelerated. This solution is, however, not tenable for any dynamical important magnetic field which would resist bending by turbulent motions at small scales. The concept in its original formulation is therefore not applicable to any astrophysical fields, apart from unrealistically weak fields, which are of marginal astrophysical importance, anyhow.

Of course, it is always possible to define in a purely formal manner an "effective diffusivity" which, when substituted into the MFE equations, will reproduce the predictions of any theory for the reconnection velocity v_rec and width Δ of the reconnection layer. All that one needs to do is set

$$\begin{equation} \lambda _{{\rm turb}}\equiv v_{{\rm rec}}\Delta =\frac{L_x}{v_A}v_{{\rm rec}}^2, \end{equation} \tag{ 70 }$$

for the desired v_rec and for the corresponding layer width Δ ≡ L_x(v_rec/v_A) imposed by mass balance. In the case of LV99, this leads to a "turbulent magnetic diffusivity"

$$\begin{equation} \lambda _{\perp \,{\rm turb}}= u_LL_iM_A^3 \min \left\lbrace \left(\frac{L_x}{L_i}\right)^2,1\right\rbrace, \end{equation} \tag{ 71 }$$

where the subscript "⊥" denotes that this diffusivity is in the direction transverse to the background field. There is not necessarily any real physics associated with this purely formal definition of a diffusivity.

In the weak-turbulence regime for L_x > L_i, however, this "turbulent magnetic diffusivity" reduces to the turbulent thermal diffusivity derived by Lazarian (2006). This is not an accident. As we have discussed in deriving Equation (60), the nearly straight field lines in the weak-turbulence regime experience a Brownian motion in the plane perpendicular to their direction, with a diffusion constant λ_⊥ turb = u_LL_iM³_A, as in Equation (71). This implies a term in the induction equation for the mean field $\overline{{\bf B}}({\bf x},t)$ of the form $\lambda _{\perp \,{\rm turb}}\triangle _\perp \overline{{\bf B}}$. There is therefore a consistent MFE approach to recover the predictions of the LV99 theory for reconnection in the weak-turbulence regime, by means of an appropriate anisotropic turbulent magnetic diffusivity.

This is not true for the strong-turbulence regime with L_x < L_i. Recall from the discussion around Equation (60) that field lines in strong turbulence do not separate diffusively in the transverse direction, as ℓ²_⊥(t) ∼ λ_⊥ turbt, but instead undergo a Richardson diffusion ℓ²_⊥(t) ∼ εt³. This superdiffusive line motion cannot be consistently described by a Laplacian term in the MFE equations. Note also that strict accuracy of the MFE description requires a slow variation of mean fields on the integral scale L_i of the turbulence, whereas the width of the reconnection zone according to Equation (62) always satisfies Δ < L_i. One would therefore not expect MFE with a simple Laplacian diffusion term to consistently and accurately describe the reconnection layer in strong turbulence predicted by LV99.

With this discussion as backdrop, let us review some of the attempts to discuss magnetic reconnection using MFE, beginning with Kim & Diamond (2001, KD01). In the first place, it must be emphasized that they consider a rather unconventional set up with a very strong guide field 〈B_z〉 and with fluctuating components B_rms also much larger than the reconnecting component 〈B_H〉, i.e., 〈B_z〉 ≫ B_rms ≫ 〈|B_H|〉. Because of the first assumption, reconnecting lines of the mean field are almost exactly parallel, with only tiny components of opposing sign. Because of the second assumption M_A ≫ 1 and there is no recognizable reconnection geometry for actual lines of the unaveraged magnetic field: the anti-parallel horizontal components are just a small mean effect. These conditions were adopted for analytical convenience and do not generally reflect astrophysical reality (particularly the second). Then KD01 develop closure approximations for the turbulent magnetic diffusivity using the 3D RMHD equations. Assuming that the small-scale fluctuations are statistically stationary and appealing to boundary conditions (e.g., periodic) which permit one to neglect surface flux terms, they derive a strongly quenched turbulent diffusivity with λ_turb = O(λ), the plasma diffusivity. They therefore conclude that magnetic reconnection proceeds at the slow Sweet–Parker rate, in contradiction to LV99. However, we find their conclusions of little relevance to LV99. In the first place, they consider a quite different problem, with M_A ≫ 1. In the second place, their main finding of a quenched turbulent magnetic diffusivity is due to the conservation of magnetic potential by the nonlinear terms of the RMHD equations. This quenching is unlikely to persist with the open boundary conditions required for stationary reconnection or to be a feature of turbulent reconnection governed by the full 3D MHD equations.

Finally, one should not mix up the concepts we discuss here with the so-called hyper-resistivity (Strauss 1986; Bhattacharjee & Hameiri 1986; Hameiri & Bhattacharjee 1987; Diamond & Malkov 2003), which is another attempt to derive fast reconnection from turbulence within the context of mean-field resistive MHD. The form of the parallel electric field can be derived from magnetic helicity conservation. Integrating by parts one obtains a term which looks like an effective resistivity proportional to the magnetic helicity current. There are several assumptions implicit in this derivation. Fundamental to the hyper-resistive approach is the assumption that the magnetic helicity of mean fields and of small scale, statistically stationary turbulent fields are separately conserved, up to tiny resistivity effects. However, this ignores magnetic helicity fluxes through open boundaries, essential for stationary reconnection, that vitiate the conservation constraint.

There is a general objection to all mean-field approaches which, to "explain" fast reconnection, make appeal to some effective dissipation (resistivity, hyper-resistivity, etc.) experienced by the fields once averaged over ensembles or over small space-time scales. The difficulty is that it is the lines of the full magnetic field that must be rapidly reconnected, not just the lines of the mean field. The former implies the latter, but not conversely. Nature does not average over ensembles of small-scale turbulence to get fast reconnection! No mean-field approach can claim to have explained the observed rapid pace of magnetic reconnection unless it is shown that the reconnection rates obtained in the theory are strictly independent of the length and timescales of the averaging (see Eyink & Aluie 2006 for further discussion).

The most popular alternative to LV99 is currently not a competing turbulence theory of reconnection but instead non-MHD approaches based on the Hall effect in a two-fluid model (Shay et al. 1998, 1999; Wang et al. 2000; Birn et al. 2001; Drake 2001; Malakit et al. 2009; Cassak et al. 2010), or on full kinetic Vlasov dynamics (Daughton et al. 2006, 2008, 2011; Che et al. 2011). We have argued in Section 4.3 and at length in Appendix B that collisionless effects are, in most cases, irrelevant to determining reconnection speeds in the presence of turbulence. We do believe that there are some situations in certain astrophysical systems where kinetic effects will influence the rate of reconnection. However, formulating a criterion to determine if kinetic effects are important requires some care in the presence of turbulence. A popular criterion for "collisionless reconnection" is the condition that the Sweet–Parker thickness δ_SP be smaller than the ion skin depth δ_i (Yamada et al. 2006; Zweibel & Yamada 2009). This condition has been checked to correctly signal "collisionless reconnection" in laminar simulations and experiments. However, as we now discuss, this condition in turbulent plasmas is not relevant to the validity of LV99.

There is even some question of how to define the "Sweet–Parker thickness" δ_SP in turbulent reconnection. In the original resistive MHD model of LV99, local, small-scale reconnection events were argued to have a Sweet–Parker Y-type structure and thickness

$$\begin{equation} \ell _\eta ^\perp \simeq L_i M_A^{-1}S_{L}^{-3/4}, \end{equation} \tag{ 72 }$$

with ℓ^⊥_η the resistive cutoff length of GS95 turbulence theory (see LV99; Equation (14)). However, this may not be the most relevant length scale for determining the importance of collisionless effects. Within GS95 phenomenology one can estimate the pointwise ratio of the Hall electric field to the MHD motional field as

$$\begin{equation} \frac{J/en}{u} \simeq \frac{c\delta B(\ell _\eta ^\perp)/4\pi ne\ell _\eta ^\perp }{u_L} \simeq \frac{\delta _i}{L_i} M_A S_L^{1/2}, \end{equation} \tag{ 73 }$$

where S_L = v_AL_i/λ is the Lundquist number based on the forcing length scale of the turbulence and M_A = u_L/v_A is the Alfvénic Mach number. This suggests the definition of the "Sweet–Parker thickness" as

$$\begin{equation} \delta _{{\rm SP}} =L_i M_A^{-1} S_L^{-1/2}, \end{equation} \tag{ 74 }$$

so that (J/en)/u_L ≃ δ_i/δ_SP. The length scale δ_SP is the analog of the Taylor microscale in hydrodynamic turbulence, whereas the distance ℓ^⊥_η is the GS95 analog of the Kolmogorov scale. If the magnetic diffusivity in the definition of the Lundquist number is assumed to be that based on the Spitzer resistivity, given by λ = δ²_ev_{th, e}/ℓ_ei where δ_e is the electron skin depth, v_{th, e} is the electron thermal velocity, and ℓ_ei is the electron mean-free-path length for collisions with ions, then $S_L=(\frac{m_e}{m_i})^{1/2} \beta ^{-1/2} (\frac{\ell _{ei}}{\delta _e})^2 (\frac{L_i}{\ell _{ei}}),$ with β = v²_{th, i}/v_A² the plasma beta. Substituting into Equation (73) and defining M = u_L/v_{th, i} as the Mach number gives

$$\begin{equation} \frac{\delta _i}{\delta _{{\rm SP}}}\simeq \left(\frac{m_i}{m_e}\right)^{1/4} M \beta ^{1/4} \left(\frac{\ell _{ei}}{L_i}\right)^{1/2}, \end{equation} \tag{ 75 }$$

which coincides precisely with the ratio defined by Yamada et al. (2006), Equation (6).

With this definition of δ_SP, it is easy to see from the data in our Table 1 that the ratio δ_i/δ_SP ≃ 10⁻³ for the warm ISM, ≃ 1 for post-CME current sheets and ≃ 10³ for solar wind impinging on the magnetosphere. However, this has nothing to do with the validity of LV99 for those systems. What it does say is that Ohmic resistivity may not be the mechanism for line-breaking in the latter two cases and the original resistive model of LV99 is not an adequate description at sufficiently small scales. Thus, the structure of local, small-scale reconnection events should be strongly modified by Hall or other collisionless effects, with possibly an X-type structure, an ion layer thickness ∼δ_i, quadrupolar magnetic fields, etc. However, these local effects are irrelevant to determining the global rate of large-scale reconnection (see Appendix B).

What is the correct criterion to determine a breakdown of LV99 and the importance of collisionless effects? The LV99 model assumes that the thickness Δ of the reconnection layer is set by turbulent MHD dynamics (line-wandering and Richardson diffusion). Thus, self-consistency requires that the length scale Δ must lie within the range of scales where shear-Alfvén modes are correctly described by incompressible MHD. This implies a criterion for "turbulent collisionless reconnection"

$$\begin{equation} \rho _i\gtrsim \Delta, \end{equation} \tag{ 76 }$$

with Δ calculated from Equation (62) and ρ_i the ion cyclotron radius (see Section 2). Since Δ∝L_x, the large length scale of the reconnecting flux structures, this criterion is far from being satisfied in most astrophysical settings. For example, in the three cases of Table 1, one finds using Δ = LM²_A that ρ_i/Δ ≃ 10⁻¹³ for the warm ISM, ≃ 10⁻⁶ for post-CME sheets, and ≃ 0.1 for the magnetosphere. Only in the latter case is the condition in Equation (76) close to being satisfied. Reconnection events in the magnetosphere probably do typically involve collisionless effects in an essential way. It must be appreciated, however, that this is a highly nongeneric situation and turbulent fluid effects will generally predominate in astrophysical reconnection.

Even in magnetospheric reconnection the presence of background turbulence in the environment should not be ignored. Like nearly every cosmic plasma, the magnetosphere commonly exhibits turbulence, either statistically steady or in sporadic bursts (Zimbardo et al. 2010). Energy spectra are generally similar to those in the solar wind, with spectral exponents close to −5/3 and −7/3 at scales above and below the ion gyroradius/ion skin depth (since β ≃ 0.1–10, these are nearly the same), respectively (see Table 1 in Zimbardo et al. 2010). The relation between observed turbulence in the magnetosphere and enhanced reconnection speeds is still under debate in the literature (Eastwood et al. 2009). The original LV99 theory does not apply in the situation that ρ_i ≃ Δ > ρ_e, but a version of LV99 based on EMHD might provide some insight. Of course, since ρ_e is only 43 times smaller than ρ_i for a hydrogen plasma, any EMHD inertial range must be of limited extent. Kinetic effects enter at electron scales, not accurately described by a fluid model, and approaches based on solving the Vlasov equation (see Section 6.3) become important. Some insights of LV99 should still carry over, such as the importance of field-line wandering in enhancing reconnection rates (see Section 6.3 for more discussion).

Let us discuss briefly the support for LV99 from observations and simulations.

The solar corona is a system proximate to Earth where LV99 theory should be applicable. Recent observations of a greatly thickened current sheet associated with CMEs give broad support to LV99 (Ciaravella & Raymond 2008; Bemporad et al. 2008). For example, the width Δ of the reconnection zone from Equation (62) was found by Ciaravella & Raymond (2008) to be just 10 times smaller than the observed thickness. This is quite good agreement, given that the formula has an unknown prefactor (which probably depends upon global geometry) and that some of the inputs (particularly the turbulence integral scale L_i) were unknown from observations and had to be crudely estimated. LV99 predicted also the phenomenon of triggering of magnetic reconnection. Indeed, as the reconnection speed depends on the level of magnetic field stochasticity, the stochasticity induced by Alfvén wave packages can enhance the reconnection speed. This is consistent with the recent observations by Sych et al. (2009) who reported a phenomenological relationship between oscillations in a sunspot and flaring energy release above an active region above the sunspot. The authors proposed that the pulsations in the flaring energy release are triggered by wave packages arising from sunspots. The LV99 mechanism presents a very natural explanation for the phenomenon.

While the Sun presents one of the best studied examples of magnetic reconnection, ISM has the advantage of testing LV99 in a collisional environment (see Yamada 2007). Similarly, collisional reconnection was observed in the solar photosphere (Park et al. 2009). For both types of environments effects of partial ionization may be important.¹⁷ The generalization of the LV99 model for a partially ionized gas is presented in Lazarian et al. (2004). The direct numerical testing of the regimes of reconnection described in that paper requires the use of a two-fluid code and has not been done so far. Given the importance of magnetic fields in ISM, such a testing is very appealing.

The most convincing tests of LV99 would be of its predicted scaling laws for the width in Equation (62) and for the reconnection velocity in Equation (64) in terms of the parameters u_L, v_A, L_i, and L_x. These scalings are most easily checked in numerical MHD studies, which achieve only moderate Reynolds numbers but which permit controlled experiments. Kowal et al. (2009) used the relation ε ≃ u⁴_L/v_AL_i to rewrite formula (64) as

$$\begin{equation} v_{{\rm rec}} = \left({{\varepsilon }\over {v_A L_x}}\right)^{1/2}\min \lbrace L_x,L_i\rbrace, \end{equation} \tag{ 77 }$$

and tested the scalings in ε and L_i by means of numerical simulations. Kowal et al. (2009) studied in particular the weak-turbulence regime with L_x > L_i, investigating the predicted power-law scalings v_rec∝ε^1/2L_i. The dependence on ε^1/2 was confirmed, but the predicted linear relation ∝L_i was not well verified and replaced with a weaker dependence closer to L^3/4_i for larger L_i ≲ L_x. This behavior is possibly associated with a crossover to the strong-turbulence regime with L_i > L_x. Note that formula (77) for strong turbulence predicts independence of the reconnection rate from L_i, except through the dependence on L_i of the mean energy dissipation ε.

The independence of reconnection rate on L_i for strong turbulence is quite convenient for observational tests of LV99, since L_i is much harder to reliably estimate in natural reconnection phenomena than are v_A, L_x and ε. Indeed, the source of turbulence may not be so easily identified or localized in wavenumber. One approach is to obtain u_L from Doppler line-broadening (e.g., Bemporad et al. 2008) and then to estimate L_i = u⁴_L/v_Aε. Of course, if spectral information is available, then one can obtain L_i directly from the peak of the energy spectrum. The mean energy dissipation rate ε is a source term for plasma heating, which can be estimated from observations of electromagnetic radiation. To make such an estimation reliably requires not only a radiation model but also some understanding of where the energy from the cascade is deposited at scales smaller than the ion gyroradius (Schekochihin et al. 2009). It may be preferable to estimate ε from coarse-grained measurements of the energy-flux rate Π_ℓ, which is equal, on average, to the dissipation rate. For example, the flux of magnetic energy is given by $\Pi _\ell ^B=\overline{{\bf J}}_\ell \hbox{\boldmath $\cdot $}\hbox{\boldmath $\varepsilon $}_\ell$ where $\overline{{\bf J}}_\ell =(c/4\pi)\hbox{\boldmath $\nabla $}\hbox{\boldmath $\times $}\overline{{\bf B}}_\ell$ is the coarse-grained current and $\hbox{\boldmath $\varepsilon $}_\ell =\overline{({\bf u}\hbox{\boldmath $\times $}{\bf B})_\ell }-\overline{{\bf u}}_\ell \hbox{\boldmath $\times $}\overline{{\bf B}}_\ell$ is the subscale EMF. Because of the scale locality of the MHD energy cascade it is possible to develop very quantitatively accurate approximations to the subscale EMF which depend only upon the coarse-grained gradients $\hbox{\boldmath $\nabla $}\overline{{\bf u}}_\ell$ and $\hbox{\boldmath $\nabla $}\overline{{\bf B}}_\ell$. For discussion of the approximation, see Eyink (2006).

Additional testing of magnetic reconnection can be produced through indirect means. Indeed, the process of reconnection may be responsible for a wide variety of processes. For instance, the LV99 model of reconnection was invoked to explain gamma-ray bursts (Lazarian et al. 2002; Zhang & Yan 2011), acceleration of cosmic rays (de Gouveia Dal Pino & Lazarian 2003; Lazarian 2005; Lazarian & Opher 2009; Lazarian & Desiati 2010), removal of magnetic field during star formation (Lazarian 2005; Santos-Lima et al. 2010). As the processes are quantified and elaborated, it is possible to get insight into the magnetic reconnection that is driving them.

Plasma conductivity is high for most astrophysical circumstances. It has therefore generally been assumed that "flux-freezing" is an excellent approximation for astrophysical plasmas. The principle of "frozen-in" field lines provides a powerful heuristic which allows simple, back-of-the-envelope estimates in place of full solutions (analytical or numerical) of the MHD equations. As such, the "flux-freezing" principle has been applied to gain insight into diverse processes, such as star formation, stellar collapse, magnetic dynamo, solar wind–magnetospheric interactions, etc. The predictions of this simple principle often accord very well with observations. Flux-freezing is used to explain, for example, the magnetization of white dwarfs and neutron stars, the low angular momentum of stars, and the spiral structure of lines of force in the solar wind.

However, for every success of the "flux-freezing" principle, there is an equally stark failure. If the standard "flux-frozenness" property held to good accuracy, then topology changes of magnetic field could not occur at the very fast rates observed in solar flares and CMEs. During star formation, the magnetic pressure of in-falling field lines would be so great as to prevent gravitational collapse altogether. The tangled line-structure in small-scale dynamos would quench the exponential growth of magnetic field. The most common attempts to explain such contrary observations invoke additional physical effects that violate conventional "flux-freezing," e.g., additional terms in the generalized Ohm's law besides collisional resistivity. Unfortunately, most of these microscopic plasma effects seem to be too small to have much effect in astrophysical MHD plasmas at very high kinetic and magnetic Reynolds numbers.

Turbulence is a natural suspect to explain apparent breakdowns of "flux-freezing" in astrophysical environments. Most attempts to use turbulence invoke it as just another effective non-ideal term in the generalized Ohm's law. Indeed, any averaging of Ohm's law (e.g., over ensembles, space, or time) produces an additional "turbulent EMF" that leads to a possible resistivity-independent violation of "flux-freezing" for the mean magnetic field. This is a correct observation as far as it goes, but not a fundamental solution to the problem. "Flux-frozenness" in the conventional sense must be violated for the full magnetic field and not just for some averaged field.

Our solution is quite different. We have argued that "flux-freezing" in the standard sense is violated in a turbulent inertial range at high kinetic and magnetic Reynolds numbers, but continues to hold in a novel stochastic sense. This SFF law in turbulent plasmas is a consequence of the remarkable turbulence phenomenon of "spontaneous stochasticity" of Lagrangian particle trajectories. As we have shown in detail, it is this stochastic form of "flux-freezing" which underlies the LV99 theory of fast magnetic reconnection. In any turbulent astrophysical system, "SFF" is the natural replacement for conventional flux-freezing. We believe that it will provide a powerful heuristic tool to explain many astrophysical phenomena. To use this tool with confidence requires a sound understanding of MHD plasma turbulence, in particular Lagrangian particle statistics. This is a clear focus of further theoretical, experimental, and numerical work.

Spontaneous stochasticity is a property not only of Lagrangian trajectories, but also of magnetic field lines themselves. The identity of "the" field line passing through a point becomes blurred in a high-magnetic-Reynolds-number turbulent plasma. Certainly, at any moment in time there is a unique set of field lines that describe the field, one through each point. However, there are an infinite number of field lines that can be drawn through a small volume, and if we follow for some parallel distance along these field lines, then they diverge explosively away from one another. In a turbulent plasma with a rough magnetic field, the distance that these lines diverge apart is independent of the dimensions of this initial volume, when the parallel distance traveled greatly exceeds this dimension. As a limiting situation, for vanishingly small initial volumes, there are infinitely many field lines that emerge and meander randomly out of one point! At sufficiently large scales, that is, above the dissipation scale, this line-wandering is independent of the value of resistivity, and, within limits, to the details of the microscopic physics (see, for example, Appendix B).

This was the original insight of Lazarian & Vishniac (1999). The important implication is that plasma, flowing a finite distance along magnetic fields lines, will diffuse away from the mean magnetic field direction. This effect can accelerate reconnection speeds not only in the presence of inertial-range turbulence, but whenever field lines exhibit some disordered, random character. If we use line-wandering in a turbulent reconnection zone to calculate the width of the outflow region around the current sheet, then we find that the classic Sweet–Parker rate becomes the fast reconnection rate given in LV99, independent of resistivity. As we have shown in the present paper, this same reconnection rate can also be obtained by an alternative approach in which an ensemble of field lines move stochastically in time. This agreement follows from a formal equivalence between the two approaches.

The precise quantitative details of magnetic line-wandering depend upon the scaling laws of MHD turbulence. We have adopted here the predictions of phenomenological GS95 theory, and these should surely be verified and possibly refined in future numerical and experimental studies. However, a parameter study in LV99 shows that the conclusion of fast reconnection in a 3D turbulent flow holds for a range of spectral slopes and spectrum anisotropies that encompass the existing suggestions of modifying the GS95 model. Our work neglects also some features of large-scale reconnection. By treating the reconnection zone as a typical part of a turbulent plasma we are neglecting the effects of a strongly sheared large-scale field and the fast outflow along magnetic fields that will accompany reconnection events. These features are naturally included in numerical simulations (Kowal et al. 2009) and do not seem to result in any significant modification of the reconnection rate, but further work along this line is certainly desirable.

Since our conclusions are consistent with the work of LV99 it is worth considering recent work on alternative approaches to calculating reconnection rates. Over the last decade, more traditional approaches to reconnection have changed considerably. At the time of its introduction, the models competing with LV99 were modifications of the single X-point collisionless reconnection scheme first introduced by Petschek (1964). Those models had pointwise localized reconnection regions which were stabilized via plasma effects so that the outflow opened up on larger scales (see Figure 5). Such configurations would be difficult to realize in the presence of random forcing, which would be expected to collapse the reconnection layer. Moreover, Ciaravella & Raymond (2008) argued that observations of solar flares were inconsistent with single X-point reconnection.

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In response to these objections, more recent models of collisionless reconnection have acquired several features in common with the LV99 model. In particular, they have moved to consideration of volume-filling reconnection (although it is not clear how this volume filling is achieved in the presence of a single reconnection layer; see Drake et al. 2006). While much of the discussion still centers around magnetic islands produced by reconnection, in three dimensions these islands are expected to evolve into contracting 3D loops or ropes due to tearing-type instabilities in electron current layers (Daughton et al. 2008, 2011). This is broadly similar to what is depicted in Figure 5, at least in the sense of introducing stochasticity to the reconnection zone. The 3D PIC simulation studies reported in these works should help, in particular, to interpret magnetospheric reconnection phenomena. However, although the simulations are described as "turbulent," they do not exhibit the inertial-range power-law spectra observed in the magnetosphere and do not take into account either the pre-existing turbulence found in many of its regions (due to temperature anisotropy, velocity shear, Kelvin–Helmholtz instability, etc.) or inertial-range turbulence generated as a consequence of reconnection itself (Zimbardo et al. 2010). An EMHD analog of LV99 and Kowal et al. (2009) studies may help to estimate such effects.

Similar remarks apply to the recent 3D PIC study of Che et al. (2011), which observes microturbulence in the electron current layer during reconnection. The authors identify the source of this "turbulence" as a filamentation instability driven by current gradients, very similar to a related instability in the EMHD model. The key aim of this work was to identify the term in the generalized Ohm's law which supplies the reconnection electric field to break the "frozen-in" condition. However, this study ignores the ambient inertial-range turbulence observed in the magnetosphere and other astrophysical plasmas, which may strongly modify laminar instabilities. Also, while there is interest in understanding the origin and character of reconnection electric fields (e.g., for particle acceleration), we have argued at length in Section 4.3 that the precise mechanism of "line-breaking" is irrelevant for the rate of reconnection in the presence of high-Reynolds-number inertial-range turbulence.

Departure from the concept of laminar reconnection and the introduction of magnetic stochasticity is also apparent in a number of the recent papers appealing to the tearing mode instability to drive fast MHD reconnection (Loureiro et al. 2009; Bhattacharjee et al. 2009).¹⁸ Several 2D numerical studies (Samtaney et al. 2009; Cassak et al. 2009; Bhattacharjee et al. 2009; Huang & Bhattacharjee 2010) have provided evidence that reconnection in the plasmoid-unstable region is independent of resistivity and a simple, heuristic picture of a multi-level plasmoid hierarchy has been proposed for this regime by Uzdensky et al. (2010). A recent very high resolution study of Ng & Ragunathan (2011), however, has found that plasmoids do not form even when the Lundquist number exceeds the putative critical value, except when insufficient numerical resolution is used or when a small amount of noise is added externally. Thus, this alternative MHD mechanism of fast reconnection may require some level of environmental noise, similar to LV99. In any case, since tearing modes exist even in a collisional fluid, this may open another channel of reconnection in such fluids. As we discuss below, this reconnection should not be "too fast" to account for the observational data.

A fundamental consideration for all reconnection models is that they must explain fast reconnection in collisional and collisionless plasmas. At the same time it should be possible for reconnection to be delayed for significant amounts of time. Otherwise the accumulation of magnetic flux prior to a solar flare would be inexplicable. Fast reconnection as a feature of turbulence sets a minimum reconnection speed which allows the topology of the magnetic field to evolve on dynamical timescales. Thus the LV99 model can explain the accumulation of flux provided that there are no generically fast reconnection processes at work.¹⁹ An alternative explanation based on collisionless reconnection in a laminar environment has been suggested by Cassak et al. (2006) and Uzdensky (2007). However, it is not clear whether this explanation will account also for the large observed thickness of the macroscopic current sheets (Ciaravella & Raymond 2008; Bemporad et al. 2008).

If local turbulence is driven by release of energy from the magnetic field, it may result in a runaway turbulent reconnection process which may be relevant to some numerical simulations(Lapenta 2008; Bettarini & Lapenta 2010). Alternatively, if tearing modes begin by driving relatively slow reconnection, but by some process destabilize the current sheet, then a similar runaway might result. Turbulence looks like a natural candidate for such process, but one should be open to alternative suggestions. For instance, we note that while X-point reconnection is clearly unstable for an isolated current sheet in a collisional fluid, interacting current sheets can produce bursts of fast X-point reconnection, separated by periods of slow evolution (Pang et al. 2010). Formally, in this case we might not need to invoke turbulent feedback,²⁰ but simply rely on the relatively slow pace of stochastic reconnection to evolve the magnetic field from one outburst to the next.

In any case, in most astrophysical situations one has to deal with the pre-existing turbulence, which is the inevitable consequence of the high Reynolds number of astrophysical fluids and for which abundant empirical evidence exists. Such turbulence may modify or suppress instabilities, including the tearing mode instability. In this paper we have shown that it, by itself, induces fast reconnection on dynamical timescales.

Much remains to be done to fully clarify the effects of MHD plasma turbulence on astrophysical reconnection. Without attempting to be exhaustive, let us mention some of the issues that we regard as most pressing.

Of the possible extensions and refinements of LV99 mentioned in Section 5, we think one of the most important is including the effects of magnetic and velocity shear. A related issue not addressed in the current paper is the detailed structure of the turbulent reconnection zone. We have invoked here the GS95 phenomenology of MHD turbulence, but intermittency effects not included in GS95 could be relevant. In principle, an independent theoretical derivation of the reconnection speed can be based on the line-voltage studied in Kowal et al. (2009). The arguments of LV99 suggest that the reconnection voltage should be contributed primarily by the motional EMF of already reconnected field lines, but a detailed analysis is required. Addressing these issues will be the subject of future work.

In its original formulation, LV99 is a model of non-relativistic reconnection. However, it is clear that the idea of enabling fast reconnection by extending the thickness of the outflow region through magnetic field wandering should be applicable to relativistic flows. This extension was implicitly used in some models of gamma-ray bursts (see Lazarian et al. 2002; Zhang & Yan 2011). Formal and detailed studies of relativistic turbulent reconnection are therefore important.

Similarly, inertial-range turbulence with power-law spectra is an idealization adopted in LV99 to get analytically tractable results. In many cases, however, wandering of field lines and of Lagrangian trajectories is significantly affected by larger or smaller scales outside the inertial range. Deviations from pure power-law spectra may occur due to multiple turbulent energy injection scales or to the effects of Ohmic dissipation. These effects of inhomogeneity of turbulence driving are important subjects for further quantitative studies.

Last, but not least, effects of turbulence dissipation via viscosity is another issue where more studies are required. Lazarian et al. (2004) extended LV99 to the case of partially ionized gas where the dissipation of turbulence arises mostly through viscosity. The concept of Richardson diffusion and its relation to turbulent reconnection in this case needs additional work.

Consider the problem of the growth in separation of a pair of magnetic field lines, starting at points displaced by vector $\hbox{\boldmath $\ell $}$, as one moves a distance s in arc length along the field lines passing through the two points (see Jokipii 1973). The exact equation for the change in separation can be obtained from Equation (45) to be

$$\begin{equation} {{d}\over {ds}} \hbox{\boldmath $\ell $}(s) =\hat{{\bf b}}(\hbox{\boldmath $\xi $}^{\prime }(s))-\hat{{\bf b}}(\hbox{\boldmath $\xi $}(s)), \end{equation} \tag{ A1 }$$

where s is the arc length, $\hbox{\boldmath $\ell $}(s) =\hbox{\boldmath $\xi $}^{\prime }(s)-\hbox{\boldmath $\xi $}(s)$, and $\hat{{\bf b}}={\bf B}/|{\bf B}|$ is the unit tangent vector along the magnetic field line. By analogy with Richardson diffusion for Lagrangian particle trajectories, one can define a "two-line diffusivity"

$$\begin{equation} D_{ij}^B(\hbox{\boldmath $\ell $}) = \int _{-\infty }^0 ds\,\langle \delta \hat{b}_i(\hbox{\boldmath $\ell $},0) \delta \hat{b}_j(\hbox{\boldmath $\ell $},s)\rangle, \end{equation} \tag{ A2 }$$

with units of length, where $\delta \hat{b}_i(\hbox{\boldmath $\ell $},s)=\hat{b}_i(\hbox{\boldmath $\xi $}^{\prime }(s))-\hat{b}_i(\hbox{\boldmath $\xi $}(s))$ with $\hbox{\boldmath $\ell $}=\hbox{\boldmath $\xi $}^{\prime }(0)-\hbox{\boldmath $\xi $}(0)$, so that

$$\begin{equation} {{d}\over {ds}}\langle \ell _i(s)\ell _j(s)\rangle =\langle D_{ij}^B(\ell)\rangle. \end{equation} \tag{ A3 }$$

The above equations are formally exact (Batchelor 1950; Kraichnan 1966). Note that Equation (A2) allows one to write

$$\begin{equation} D_{ij}^B(\hbox{\boldmath $\ell $})\sim \delta \hat{b}_i(\hbox{\boldmath $\ell $})\delta \hat{b}_j(\hbox{\boldmath $\ell $})s_{{\rm int}}(\hbox{\boldmath $\ell $}), \end{equation} \tag{ A4 }$$

where $s_{{\rm int}}(\hbox{\boldmath $\ell $})$ is an integral correlation length of the increment in the tangent vector moving along the lines. (This is properly a tensor quantity, but here written as a scalar.) To determine the line-wandering, one must have a model for this integral correlation length.

The LV99 theory is obtained by modeling the integrand in Equation (A2) for field-perpendicular increments as

$$\begin{equation} \langle \delta \hat{b}_\perp (\hbox{\boldmath $\ell $},0)\delta \hat{b}_\perp (\hbox{\boldmath $\ell $},s)\rangle \sim \frac{\delta u^2(\ell)}{v_A^2}{{\rm Re}}(e^{is/\ell _\Vert -|s|/\lambda (\ell)}). \end{equation} \tag{ A5 }$$

Note that $|\delta \hat{b}_\perp (\ell)|\sim \delta B(\ell)/B_0\sim \delta u(\ell)/v_A$. There are two effects in the s-dependent part. First, there is a correlation length λ(ℓ) of tangent-vector increments along the field line, which gives an exponential decay. Within LV99 theory, it is assumed that this length is the distance traveled by a random Alfvénic disturbance propagating along the field line with velocity v_A in one energy cascade time at scale ℓ_⊥. That is,

$$\begin{equation} \lambda (\ell) = v_A\tau _\ell =v_A \frac{\delta u^2(\ell)}{\varepsilon }. \end{equation} \tag{ A6 }$$

The second effect arises from the periodic variation along the field lines due to regular Alfvén wave trains. It is assumed that, at perpendicular scale ℓ_⊥, the Alfvén wave trains that are dominant are those with wavelength $\ell _\Vert$, the parallel length scale corresponding to perpendicular scale ℓ_⊥. Integrating Equation (A5) in s gives the result

$$\begin{equation} s_{{\rm int}}(\ell) \sim \frac{1/\lambda (\ell)}{1/\lambda ^2(\ell) + 1/\ell _\Vert ^2}\sim \frac{\ell _\Vert ^2}{\lambda (\ell)}= \frac{\varepsilon }{v_A} \frac{\ell _\Vert ^2}{\delta u^2(\ell)}, \end{equation} \tag{ A7 }$$

for $\lambda (\ell)\ge \ell _\Vert$. Substituting into Equation (A4) gives

$$\begin{equation} D^B_\perp (\ell) \sim \frac{\varepsilon \ell _\Vert ^2}{v_A^3}=\frac{\ell _\Vert ^2}{L_i}M_A^4, \end{equation} \tag{ A8 }$$

where note that a factor of δu²(ℓ) has canceled between numerator and denominator. From this last formula, various specific cases can be considered.

In the strong GS95 turbulence regime, critical balance implies that $\lambda (\ell) \sim \ell _\Vert$ and thus, from Equation (A7), $s_{{\rm int}}(\ell)\sim \ell _\Vert$. This gives the intuitive result that

$$\begin{equation} D_\perp ^B(\ell)\sim (\delta u_\ell /v_A)^2 \ell _\Vert \sim L_i\left({{\ell _\perp }\over {L_i}}\right)^{4/3} M_A^{4/3}, \end{equation} \tag{ A9 }$$

where we have substituted from Equations (5) and (6) for $\ell _\Vert$ and δu_ℓ/v_A. Alternatively, one can just substitute for $\ell _\Vert$ from Equation (5) into Equation (A8). The equation

$$\begin{equation} {{d}\over {ds}}\ell _\perp ^2\sim D_\perp ^B(\ell) \sim L_i\left({{\ell _\perp }\over {L_i}}\right)^{4/3} M_A^{4/3}, \end{equation} \tag{ A10 }$$

is then easily solved to give Equation (8) in the text for L_i > s ≫ ℓ⁽⁰⁾_⊥. This result was already obtained for M_A = 1 by Skilling et al. (1974) in a discussion of cosmic-ray diffusion, assuming a simple isotropic Kolmogorov hydrodynamic model of magnetized turbulence. In the weak-turbulence regime one has instead $\ell _\Vert =L_i$ a constant, which substituted into Equation (A8) gives

$$\begin{equation} {{d}\over {ds}}\ell _\perp ^2\sim D_\perp ^B(\ell) \sim L_i M_A^4, \end{equation} \tag{ A11 }$$

and this is solved to give the diffusive result (9) in the text.

All of the non-ideal terms in the generalized Ohm's law (47)—Ohmic resistivity, Hall term, pressure tensor, electron inertia, etc.—are expected to be insignificant in a long turbulent inertial range, at sufficiently large length scales. Thus, they do not alter the predictions of the LV99 theory for the reconnection rates of large-scale flux structures. We shall here present more detailed analytical arguments for this thesis, in the specific example of the incompressible HMHD equations. As remarked in the text, the HMHD dynamics is valid in a literal sense for almost no astrophysical plasmas. However, it has been used in many reconnection studies as a simple fluid model that exemplifies the Hall effect (Shay et al. 1998, 1999; Wang et al. 2000; Birn et al. 2001; Drake 2001; Malakit et al. 2009; Cassak et al. 2010), so that it is a useful example to consider in this respect.

The incompressible HMHD equations can be written as

$$\begin{equation} (\partial _t {\bf u}+{\bf u}\hbox{\boldmath $\cdot $}\hbox{\boldmath $\nabla $}){\bf u}=-\hbox{\boldmath $\nabla $}p + {\bf j}\hbox{\boldmath $\times $}{\bf b}+ \nu \triangle {\bf u}, \end{equation} \tag{ B1 }$$

$$\begin{equation} \partial _t{\bf b}= \hbox{\boldmath $\nabla $}\hbox{\boldmath $\times $}[({\bf u}-\delta _i{\bf j})\hbox{\boldmath $\times $}{\bf b}] +\lambda \triangle {\bf b}, \end{equation} \tag{ B2 }$$

with $\hbox{\boldmath $\nabla $}\hbox{\boldmath $\cdot $}{\bf u}=\hbox{\boldmath $\nabla $}\hbox{\boldmath $\cdot $}{\bf b}=0$ and ${\bf j}=\hbox{\boldmath $\nabla $}\hbox{\boldmath $\times $}{\bf b}$. Here we have introduced the Alfvén variable ${\bf b}={\bf B}/\sqrt{4\pi \rho }$ with units of velocity and δ_i = c/ω_{p, i} is the ion skin depth. The above system has two inviscid constants of motion that cascade to small scales, the total energy

$$\begin{equation} E=\frac{1}{2}\int d^3x\, [|{\bf u}({\bf x})|^2+|{\bf b}({\bf x})|^2], \end{equation} \tag{ B3 }$$

and the cross-helicity

$$\begin{equation} H_C=\int d^3x\, {\bf u}({\bf x})\cdot [{\bf b}({\bf x})+\frac{1}{2}\delta _i \hbox{\boldmath $\nabla $}\hbox{\boldmath $\times $}{\bf u}({\bf x})], \end{equation} \tag{ B4 }$$

as well as the magnetic helicity which cascades to large scales. It is not our purpose here to give an exhaustive account of the turbulent cascade phenomenology of the HMHD model. Instead, we just wish to establish the simple fact that the Hall term modifies turbulence properties only at length scales ℓ ≲ δ_i, whereas length scales ℓ ≳ δ_i will behave as in standard MHD turbulence without the Hall term.

Our first argument is based on a spatial coarse-graining approach in an Eulerian formulation (e.g., see Eyink & Aluie 2006). The HMHD induction equation coarse-grained (low-pass filtered) at length scale ℓ becomes

$$\begin{equation} \partial _t\overline{{\bf b}}_\ell = \hbox{\boldmath $\nabla $}\hbox{\boldmath $\times $}[(\overline{{\bf u}}_\ell -\delta _i{\overline{\xyz}}_\ell)\hbox{\boldmath $\times $}\overline{{\bf b}}_\ell +\hbox{\boldmath $\varepsilon $}_\ell -\lambda {\overline{\xyz}}_\ell ], \end{equation} \tag{ B5 }$$

where the total contribution to the turbulent subscale EMF is

$$\begin{equation} \hbox{\boldmath $\varepsilon $}_\ell = \overline{({\bf u}_e\hbox{\boldmath $\times $}{\bf b})_\ell }-\overline{{\bf u}}_{e\,\ell }\hbox{\boldmath $\times $}\overline{{\bf b}}_\ell. \end{equation} \tag{ B6 }$$

and u_e ≡ u − δ_ij is the electron fluid velocity. It is easy to see that

$$\begin{equation} \frac{\delta _i{\overline{\xyz}}_\ell }{\overline{{\bf u}}_\ell }\simeq \frac{\delta _i \delta b(\ell)/\ell }{u_L} \simeq \frac{\delta _i}{\ell } M_A^{1/3}\left(\frac{\ell }{L_i}\right)^{1/3}, \end{equation} \tag{ B7 }$$

where the last expression uses GS95 theory to obtain the 1/3 powers of M_A and ℓ/L_i. However, a similar expression will hold in any theory of HMHD turbulence in which δb(ℓ) ∼ v_AM^q_A(ℓ/L_i)^p with q ⩾ 1 and p > 0. Hence $\delta _i{\overline{\xyz}}_\ell \ll \overline{u}_\ell$ for δ_i ≪ ℓ ≪ L_i, so that the coarse-grained dynamics in HMHD turbulence at those scales is identical in form to standard MHD. The only difference is that the EMF in Equation (B6) has an additional contribution from the Hall term

$$\begin{equation} \hbox{\boldmath $\varepsilon $}^{{\rm Hall}}_\ell = -\delta _i [\overline{({\bf j}\hbox{\boldmath $\times $}{\bf b})_\ell }-{\overline{\xyz}}_\ell \hbox{\boldmath $\times $}\overline{{\bf b}}_\ell ] = \frac{1}{2}\delta _i\hbox{\boldmath $\nabla $}[{\rm tr}(\hbox{\boldmath $\tau $}_{M\,\ell })]- \delta _i\hbox{\boldmath $\nabla $}\hbox{\boldmath $\cdot $}\hbox{\boldmath $\tau $}_{M\,\ell }, \end{equation} \tag{ B8 }$$

where $\hbox{\boldmath $\tau $}_{M\,\ell } = \overline{({\bf b}{\bf b})_\ell }-\overline{{\bf b}}_\ell \overline{{\bf b}}_\ell$ is the turbulent Maxwell stress. (Note incidentally that the gradient term in Equation (B8) does not contribute at all when substituted into the curl in Equation (B5).) The total EMF in Equation (B6) is thus $\hbox{\boldmath $\varepsilon $}_\ell =\hbox{\boldmath $\varepsilon $}_\ell ^{{\rm MHD}} +\hbox{\boldmath $\varepsilon $}_\ell ^{{\rm Hall}}$ including the standard MHD contribution $\hbox{\boldmath $\varepsilon $}^{{\rm MHD}}_\ell =\overline{({\bf u}\hbox{\boldmath $\times $}{\bf b})_\ell }-\overline{{\bf u}}_\ell \hbox{\boldmath $\times $}\overline{{\bf b}}_\ell$. It is easy to estimate the size of these various terms as

$$\begin{equation} {\bm \varepsilon}^{{\rm MHD}}\sim \delta u(\ell)\delta b(\ell),\,\, {\bm \varepsilon}^{{\rm Hall}}\sim \delta _i\frac{\delta b^2(\ell)}{\ell }, \,\, \overline{{\bf E}}^{{\rm Ohm}}_\ell \!=\lambda {\overline{\xyz}}_\ell \sim \lambda \frac{\delta b(\ell)}{\ell }. \end{equation} \tag{ B9 }$$

These estimates are rigorous upper bounds (cf. Eyink 2005) but they must be good order-of-magnitude estimates of the terms as well, in order to allow constant fluxes of energy and cross-helicity to small scales. In that case we must interpret ℓ in the upper bounds as ℓ_⊥, the scale perpendicular to the mean magnetic field.²¹ Instead at scales ℓ < δ_i the terms proportional to δ_i dominate, so that

$$\begin{eqnarray*} \Pi ^E_\ell &\sim & -{\overline{\xyz}}_\ell \hbox{\boldmath $\cdot $}\hbox{\boldmath $\varepsilon $}_\ell ^{{\rm Hall}}\sim \delta _i\frac{\delta b^3(\ell)}{\ell _\perp ^2}, \,\,\,\,\, \Pi ^C_\ell \sim -\delta _i \hbox{\boldmath $\nabla $}\overline{\hbox{\boldmath $\omega $}}_\ell \hbox{\boldmath $:$}\hbox{\boldmath $\tau $}_\ell - \overline{\hbox{\boldmath $\omega $}}_\ell \hbox{\boldmath $\cdot $}\hbox{\boldmath $\varepsilon $}_\ell ^{{\rm Hall}} \nonumber\\ &\sim & \delta _i\frac{\delta u(\ell)}{\ell _\perp ^2} [\delta u^2(\ell)+\delta b^2(\ell)]. \end{eqnarray*}$$

One thus gets constant fluxes with the scaling

$$\begin{equation*} \delta u(\ell)\sim \delta b(\ell) \sim (\varepsilon \ell _\perp ^2/\delta _i)^{1/3}, \end{equation*}$$

in agreement with the numerical results of Shaikh & Shukla (2009). As a matter of fact, the precise scaling at length scales ℓ < δ_i will not affect our conclusions. As usual, the large-scale Ohmic electric field is negligible compared with the MHD turbulent EMF whenever ℓδu(ℓ) ≳ λ, or ℓ ≳ ℓ^⊥_η = (λ³/ε)^1/4 assuming GS95.²² However, one can also see that $\hbox{\boldmath $\varepsilon $}^{{\rm Hall}}_\ell$ is negligible compared with $\hbox{\boldmath $\varepsilon $}^{{\rm MHD}}_\ell$ whenever ℓ ≳ δ_i. Here we have assumed that δb(ℓ) ≃ δu(ℓ), which is true in GS95 but also in any theory of MHD turbulence which predicts equipartition of kinetic and magnetic energies at small scales. These considerations show that the coarse-grained dynamics in HMHD turbulence reduce to those in standard MHD turbulence at length scales ℓ ≳ δ_i. This conclusion is in agreement with the numerical study of Dmitruk & Matthaeus (2006). They compared results of Hall and standard MHD simulations at kinetic and magnetic Reynolds numbers of 1000, with δ_i for the Hall simulation chosen about four times greater than the resistive dissipation scale. They found virtually identical results in the two simulations at scales greater than δ_i for all of the various quantities, such as velocities, magnetic fields, and electric fields.

Now let us consider the Lagrangian description of HMHD turbulence. SFF holds also in resistive HMHD, with magnetic field lines stochastically frozen-in to the electron fluid velocity u_e (Eyink 2009). Thus, the lines can be regarded to "move" according to the stochastic equation

$$\begin{equation} \frac{d{\bf x}}{dt}={\bf u}({\bf x},t)-\delta _i{\bf j}({\bf x},t)+\sqrt{2\lambda }\hbox{\boldmath $\eta $}(t). \end{equation} \tag{ B10 }$$

At separations ℓ ≫ ℓ^⊥_η where the direct effects of the resistive term $\sqrt{2\lambda }\hbox{\boldmath $\eta $}(t)$ can be neglected, the distance $\ell _\perp (t)=\sqrt{\langle y^2(t)\rangle }$ that field lines advect apart in the field-perpendicular direction is obtained from

$$\begin{equation} \frac{d}{dt}\ell ^2_\perp \sim D^e_\perp (\ell), \end{equation} \tag{ B11 }$$

with

$$\begin{equation} D^e_\perp (\ell)=\int _{-\infty }^0 dt\,\langle \delta {\bf u}_{e\,\perp }(\hbox{\boldmath $\ell $})\hbox{\boldmath $\cdot $}\delta {\bf u}_{e\,\perp }(\hbox{\boldmath $\ell $},t)\rangle. \end{equation} \tag{ B12 }$$

The electron velocity increment in Equation (B12) gets a contribution $\delta {\bf u}(\hbox{\boldmath $\ell $})$ from the bulk plasma velocity and another contribution $\delta _i\, \delta {\bf j}(\hbox{\boldmath $\ell $})$ from the Hall term. We would like to show that the Hall contribution is negligible whenever ℓ ≫ δ_i. This can be argued as follows.

First, we can estimate from energy balance alone without any other assumptions that ε ∼ λj²_rms and thus j ∼ j_rms = (ε/λ)^1/2. This allows us compare the typical size of the Hall velocity with the flow velocity as

$$\begin{equation} \frac{\delta _i j}{u}\sim \frac{\delta _i (\varepsilon /\lambda)^{1/2}}{u_L}= \frac{\delta _i}{\ell _\eta ^\perp }\cdot S_L^{-1/4}, \end{equation} \tag{ B13 }$$

where we have used ε = u⁴_L/v_AL_i and we have defined ℓ^⊥_η ≡ (λ³/ε)^1/4, without the assumption that it plays any dynamical role as a "dissipation scale." Now let the Lundquist number S_L increase by increasing L_i with ε fixed, and thus also λ, ℓ^⊥_η and the ratio

$$\begin{equation} \alpha \equiv \delta _i/\ell _\eta ^\perp, \end{equation} \tag{ B14 }$$

all fixed as well. It follows from Equation (B13) that δ_ij ≪ u always when α < 1 but even when α > 1 if S_L ≫ α⁴. We therefore expect that the Hall contribution to the two-particle diffusivity is negligible compared with the advective MHD contribution at high Lundquist numbers.

Physically, the Hall term is very short range correlated in both space and time, so that it acts on a pair of Lagrangian particles as two independent Brownian motions with an effective one-particle diffusivity. This can be shown more formally, by observing that

$$\begin{eqnarray} \langle \delta {\bf j}_\perp (\hbox{\boldmath $\ell $})\hbox{\boldmath $\cdot $}\delta {\bf j}_\perp (\hbox{\boldmath $\ell $},t)\rangle &=& 2[ \langle {\bf j}_\perp (\hbox{\boldmath $0$})\hbox{\boldmath $\cdot $}{\bf j}_\perp (\hbox{\boldmath $0$},t)\rangle - \langle {\bf j}_\perp (\hbox{\boldmath $0$})\hbox{\boldmath $\cdot $}{\bf j}_\perp (\hbox{\boldmath $\ell $},t)\rangle ] \nonumber\\ &\lesssim & 2 \langle {\bf j}_\perp (\hbox{\boldmath $0$})\hbox{\boldmath $\cdot $}{\bf j}_\perp (\hbox{\boldmath $0$},t)\rangle, \end{eqnarray} \tag{ B15 }$$

since the second term in the square brackets is a decaying term for increasing ℓ.²³ This upper bound is a considerable overestimate for smaller values of ℓ, but it suffices for our purpose. We can then estimate

$$\begin{equation} \int _{-\infty }^0 dt\, \langle {\bf j}_\perp (\hbox{\boldmath $0$})\hbox{\boldmath $\cdot $}{\bf j}_\perp (\hbox{\boldmath $0$},t)\rangle \sim j_{{\rm rms}}^2\tau _\lambda, \end{equation} \tag{ B16 }$$

where τ_λ = (λ/ε)^1/2 = 1/j_rms is the correlation time at the resistive dissipation scale. Multiplied by δ²_i, this is the Hall contribution to the one-particle diffusivity. We therefore obtain a conservative upper bound on the Hall contribution to the two-particle diffusivity, independent of ℓ, as

$$\begin{equation} D^{{\rm Hall}}_\perp (\ell) \lesssim \delta _i^2j_{{\rm rms}}^2\tau _\lambda =\alpha ^2 \lambda, \end{equation} \tag{ B17 }$$

where we have used Equations (B13) and (B14).

Finally, we compare with the advective contribution for ℓ > δ_i. From the Eulerian argument presented earlier in this Appendix, we expect that GS95 scaling works on those length scales. We can thus estimate for ℓ_⊥ > δ_i that

$$\begin{equation} D_\perp ^{{\rm MHD}}(\ell)\sim \ell _\perp \delta u(\ell)\sim (\varepsilon \ell _\perp ^4)^{1/3} \sim \alpha ^{4/3}\lambda \left(\frac{\ell _\perp }{\delta _i}\right)^{4/3}, \end{equation} \tag{ B18 }$$

where we have used λ = (εℓ^⊥ 4_η)^1/3 and the definition of α. We conclude that

$$\begin{equation} D_\perp ^{{\rm Hall}}(\ell)/D_\perp ^{{\rm MHD}}(\ell) \lesssim \alpha ^{2/3}\left(\frac{\delta _i}{\ell _\perp }\right)^{4/3} \ll 1, \end{equation} \tag{ B19 }$$

at least when $\ell _\perp \gg \sqrt{\alpha }\delta _i$. Thus, Richardson diffusion of field lines in HMHD turbulence is exactly the same as for MHD turbulence at perpendicular separations ℓ_⊥ much greater than the ion skin depth δ_i.

Now let us reconsider the problem of large-scale reconnection treated in Section 5, including the Hall effect within the HMHD model. Define Δ = L_iM²_A taking L_i ≃ L_x. We shall make the essential assumption that δ_i ≪ Δ. This assumption is almost always true in astrophysical reconnection problems. For example, using $\delta _i=\rho _i/\sqrt{\beta _i}$ and the data in Table 1, we can see that δ_i/Δ ∼ 10⁻¹³ for the warm ISM, ∼10⁻⁶ for post-CME current sheets and ∼10⁻¹ for solar wind impinging on the magnetosphere. Only in the latter case is the condition not extremely well satisfied. Let us then repeat the arguments in Section 5 taking into account the Hall term. Consider first the derivation of LV99 based on the stochastic wandering of magnetic field lines as one traces points of increasing arc length s along the lines. At small initial separations, the Hall term is important and the rms transverse distance $\ell _\perp (s)=\sqrt{\langle y^2(s)\rangle }$ grows as ℓ_⊥(s) ∼ (s³/δ_iL_i)M⁴_A.²⁴ As soon as neighboring lines have separated to ℓ_⊥ ≳ δ_i, then their further separation is the same as in standard MHD turbulence without the Hall term. However, to reach the separation ∼δ_i one must move along the field lines a distance of only $s\sim L_i^*\equiv L_i\left(\frac{\delta _i}{\Delta }\right)^{2/3}\ll L_i$ when δ_i ≪ Δ. In that case, formula (63) in Section 5.1 for 〈y²(s)〉 still holds, if s ≫ L*_i. It therefore follows that the LV99 estimate for the width of the reconnection layer and formula (64) for the reconnection velocity will hold, as long as δ_i ≪ Δ, since most of the separation of lines will be achieved for s ⩾ L*_i. The same conclusion can be derived from the Lagrangian perspective. For small initial separations ℓ_⊥ < δ_i the lines will separate laterally as ℓ_⊥(t) ∼ (δ²_iεt³)^1/4.²⁵ However, after a time $t\sim t_c^*=(\delta _i^2/\varepsilon)^{1/3}= \left(\frac{\delta _i}{\Delta }\right)^{2/3}(L_i/v_A)$ much shorter than t_c = L_i/v_A the pairs of lines will reach the separation ℓ_⊥(t) ∼ δ_i and GS95 scaling will take over. Formula (60) for 〈y²(t)〉 that follows from GS95 will hold as soon as t > t*_c and the bulk of the separation of lines will occur after this time if δ_i ≪ Δ. Thus, the estimate in Equation (62) for the width of the reconnection layer holds just as before, with only a negligible correction due to finite δ_i.

We thus reach the important conclusion that the Hall term is irrelevant to determining the rate of global reconnection in the presence of high-Reynolds-number plasma turbulence. At most a modest enhancement of the reconnection rate is obtained, but vanishingly small when δ_i ≪ Δ. We should mention that a similar conclusion was reached by Smith et al. (2004) on the basis of numerical simulations. However, that study can be fairly criticized because the simulations were performed in 2D and at such low Reynolds numbers that most of the measured reconnection rates (in a transient regime) were smaller than the steady-state Sweet–Parker rates. Our theoretical arguments, on the other hand, are valid in 3D and at arbitrarily high Reynolds numbers. This is not to say that the Hall term will have no effect whatsoever, but these will be confined to very small scales, e.g., modifying the structure of local reconnection events. However, these local effects are irrelevant to determining the global rate of large-scale reconnection.

Similar conclusions can be reached not only for the two-fluid Hall term, but also for other microscopic plasma mechanisms proposed to enhance reconnection speeds. For example, the MHD turbulent cascade may be terminated not by Spitzer resistivity but instead by some form of anomalous resistivity, e.g., due to ion acoustic instability. The latter produces a large effective magnetic diffusivity of order (m_i/m_e)^1/2δ²_eω_{c, e} at scales smaller than $\delta _i/\sqrt{\beta }$ (e.g., see Galeev & Sagdeev 1984; Kulsrud 2005, Section 14.7). But larger scales will behave as standard MHD turbulence and the LV99 predictions will again hold as long as δ_i ≪ Δ. This was verified in the simulations of Kowal et al. (2009), who found no effect of anomalous resistivity on reconnection speed. Turbulence is such a powerful accelerator of magnetic line-separation that it completely dominates microscopic plasma processes of line diffusion, as soon as the magnetic and kinetic Reynolds numbers are sufficiently high.