Does Magnetic Reconnection Change Topology?

In astrophysics, magnetic reconnection has been invoked both as a main mechanism that governs the underlying dynamics at large scales, e.g., in launching outflows in stars and accretion disks, and also as an accompanying process working in the background. It has been suggested, as an example for the latter, that reconnection regulates the structure of a strongly magnetized corona [1]. Reconnection is widely thought to be intimately related to the magneto-rotational instability (MRI) ([2, 3]) and Parker-Rayleigh-Taylor instability [4], which might in turn interfere or enhance the reconnection rate [5]. Reconnection may also affect the saturation rate of the MRI and also the generation of nonthermal particles [6]. It is now widely believed that reconection (i) is a ubiquitous process occurring in different systems from the solar surface to highly conducting accretion disks; (ii) is fast, i.e., with a rate orders of magnitude faster than the resistive time; (iii) is responsible for a variety of other phenomena such as particle acceleration and plasma heating; and in astrophysical systems (iv) proceeds in turbulence. For a recent review covering these aspects see e.g., [7].

Magnetic reconnection seems to be pervasive in astrophysics with a vast literature (see also [8, 9] for a recent review). Yet there seems to be no consensus in its definition and its relation to magnetic topology-change. In fact, sometimes hand-waving arguments based on magnetic field lines in $2D$ setups seem to be generalized to $3D$ without any mathematical or physical justification. \colorblack For example, reconnection is sometimes understood or defined as magnetic topology-change. However, this notion which seems to originate from 2D setups where the field line connecting plasma element A to another element B at time $t_{0}$, connects A to C but not B at a later time $t_{1}$ is inadequate to describe 3D configurations. In 2D there is a discontinuous change in magnetic connectivity implying topology-change (leading to different regions with different ”topologies” separated by separatrices). In 3D, on the other hand, this process can proceed without any discontinuous change in magnetic connectivity (thus no topology-change and no separatrices) because there is ”enough” space for field lines to flip [10].

Our aim in this paper is to present a formalism to understand reconnection, magnetic topology and topology-change without invoking the concept of magnetic field lines, their motion through plasma or the notion of flux-freezing in laminar [11] or turbulent flows [12]. These concepts have of course been greatly reformulated and refined in the last decades, in particular for turbulent flows (e.g., see [12, 13, 14, 15, 16]). However, a much simpler picture seems to arise using magnetic path-lines, instead of field lines, which is what we will do in this paper. We can imagine small magnetic disturbances or wave-packets moving along the local magnetic field with Alfvén velocity ${\bf V}_{A}$.¹¹1 This is motivated by the concept of quasiparticles in quantum field theories, where e.g., vibrational modes in a crystal are taken as \colorblackquasiparticles (phonons). Our classical description is much simpler here as we treat these entities as Lagrangian particles in a fluid rather than taking them as \colorblackentities that e.g., can couple to gauge potentials. \colorblackThese magnetic excitations or ”Alfvénic wave-packets” will be taken as fluid particles in a Lagrangian description, i.e., we will consider wave-packets as particles moving with Alfvén velocity and study their trajectories and the corresponding topology.

Magnetic field lines provide a powerful notion in many problems, nevertheless, their behavior in real plasmas seem to be complicated enough that other simpler approaches could be appreciated. At any given time $t_{0}$, magnetic field lines are defined as parametric curves, e.g., $\xi(s;t_{0})$ with arc-length parameter $s$, which provide a ”pattern” for the field in real space at a given time, i.e, a ”snapshot” of the field. \colorblackIn real astrophysical fluids, which are turbulent, field lines become stochastic and the notion of a single field line loses its meaning unless a proper coarse-graining is applied, i.e., field lines of the average, large-scale field are considered [15, 7]. Also, the pattern of field lines in $3D$ may change abruptly at a later time $t_{0}+\delta t$ as these curves do not evolve smoothly in time; see the middle panel in Fig.(1). Magnetic path-lines, i.e., trajectories of Alfvénic wave-packets, provide an alternative tool (see below).

If astrophysical magnetic fields can indeed undergo sudden changes in real space, plasma outflows can be considered as a secondary effect, observed as reconnection events. Alfvén wave-packets follow the local field, hence their trajectories provide a footprint of these changes in the field. Mathematically, the study of these trajectories, i.e., solutions of $\dot{\bf x}(t)={\bf V}_{A}({\bf x}(t),t)$, is analogous to the Lagrangian dynamics in hydrodynamics. For example, we are interested in the separation of two such trajectories, i.e., $|{\bf x}(t)-{\bf y}(t)|$ at time $t$, which is related to Lyapunov exponents of the dynamical system $\dot{\bf x}={\bf V}_{A}$, or assuming incompressibility and absorbing density to magnetic field’s definition, $\dot{\bf x}(t)={\bf B(x}(t),t)$ where magnetic field $\bf B$ satisfies the induction equation.²²2Throughout this paper, we assume a suitable non-dimensionalization, e.g., using an integral length scale $L$ and large-scale field $B_{0}$. The corresponding phase space ${\bf(x,B)}$ contains all possible states of this dynamical system \colorblackwhich describes the motion of a single wave-packet. For $N$ wave-packets, we deal with an $N$-body system with $6N$ dimensional phase space and thus in the fluid approximation, the phase space would be infinite-dimensional.

We will employ a physically intuitive and mathematically careful approach to magnetic topology in this paper. Topology is concerned with those properties of spaces that remain invariant under any continuous deformation, i.e, stretching and bending without cutting. Two objects (spaces) A and B (e.g., a solid ball and a cube) have the same topology if nearby points on A are mapped onto nearby points on B and vise versa: i.e., nearby points do not discontinuously mapped to points far away from each other. This means that there is a continuous, one-to-one and onto map from A to B with a continuous inverse (i.e., a homeomorphism). Thus if we deform object A (magnetic field at time $t$) to make object B (magnetic field at time $t\pm\delta t$), topology is preserved if nearby points are mapped onto nearby points, i.e., the distance between points is continuous in time. If the map from A to B is not onto and one-to-one, then points are ”destroyed” during deformation thus topology will change. It means that the time evolution of A (magnetic field at time $t$) to B (magnetic field at time $t\pm\delta t$) does not respect time reversal symmetry. For example, dissipation in a magnetized plasma annihilates wave-packets and breaks time-reversal symmetry, thus changing magnetic topology.

Roughly speaking, for a magnetic field continuous in space and time, we expect ”smooth deformation” in time—no discontinuous jump in its values or abrupt change in its direction. Therefore, wave-packets should move with a continuous velocity ${\bf V}_{A}({\bf x},t)$ on continuous trajectories³³3Mathematically, this is because Alfvénic trajectories ${\bf x}(t)$ solve $\dot{\bf x}(t)={\bf B}$ with smooth $\bf B$., otherwise, we would expect a ”topology-change”.

Hence, for magnetic topology to be preserved in time, we expect that nearby wave-packets, at time $t$, should remain nearby at a slightly different time $t\pm\delta t$. In fact, because of Lipschitz continuity of magnetic field (see §II), this intuitive notion is equivalent to the following statement: nearby wave-packets moving with almost the same velocity, at time $t$, should remain nearby, moving with almost the same velocity, at a slightly different time $t\pm\delta t$⁴⁴4This argument can be easily made mathematically precise using the $\epsilon\delta$ definition.. This simply means that the metric topology of the phase space, at any time $t$, is the same as its topology at a slightly different time $t\pm\delta t$, i.e, the topology is preserved (see §II.1). The aim of this paper is to make these statements mathematically precise, and show how the emerging formalism can help gain a deeper and simpler picture for both reconnection and topology-change. Incidentally, from now on, we will assume incompressibility, absorbing density to the redefinition of magnetic field, with an appropriate non-dimensionalization as ${\bf B}={\bf V}_{A}$; see also Fig.(2).

blackThis Lagrangian formalism based on wave-packets, briefly described above, also provides a natural way to define and estimate the rate at which magnetic topology changes. As we will see, dissipation causes the volume of the phase space to contract. Eventually, over a time $\tau_{T}$, magnetic field completely diffuses away, and the phase-space dimension approaches zero. Thus the dissipation rate $1/\tau_{T}$ can be taken as the rate of change of dimension. On the other hand, dimension is a topological invariant, i.e., its change means topology-change, thus the rate of topology change can be similarly defined as $1/\tau_{T}$. We will see that reconnection corresponds to rapid but continuous separation of these trajectories. Discontinuous divergence, which is not allowed if $\bf B$ is uniformly Lipschitz continuous, would lead to topology-change (nearby point not mapped onto nearby points). Dissipation, which breaks time-reversal invariance by annihilating wave-packets, can cause topology change in real dissipative systems; see also Fig.(1). Mathematically, dissipation destroys the onto and one-to-one properties of the mapping, i.e., time evolution.

As for the detailed plan of the present work, in the following section, we first consider the divergence of Alfvénic wave-packets, recovering the fast Lazarian-Vishniac reconnection rate [17, 15, 7] in turbulent plasmas. Then, we will focus on magnetic topology, using a dynamical systems theory, and estimate its rate of change in both laminar and turbulent flows. The physical implications and connections to previous work will be discussed in Discussion.

Astrophysical reconnection is understood to involve rapid changes in magnetic field configuration, thus trajectories of Alfvénic wave-packets moving along the field can rapidly diverge. The dynamics of the solutions, path-lines or Alfvénic trajectories, is in fact similar to Lagrangian dynamics in fluid mechanics—magnetic field plays the role of the velocity field. One important quantity is what is known as $2$-particle diffusion in fluid dynamics, i.e., the separation (divergence) of any pair of trajectories over time; see e.g., [18]. In the following, we will see how this simple notion, when applied to wave-packets, explains fast reconnection in turbulence and also clarifies the distinction between topology-change and reconnection.

The Lipschitz continuity of magnetic field means that $|{\bf B}({\bf x},t)-{\bf B}({\bf y},t)|\leq C|{\bf x}-{\bf y}|^{h}$ for some real $C\geq 0$ and $0<h\leq 1$. Consider the spatial separation of two arbitrary wave-packets ${\bf x}(t)$ and ${\bf y}(t)$ at time $t$, which were initially separated by $\Delta(t=0):=\Delta_{0}$.⁵⁵5Several notations exist in Lagrangian dynamics to denote the Lagrangian flow map, i.e, the map from particle’s initial point ${\bf x}_{0}:={\bf x}(t=0)$ to its final point ${\bf x}_{t_{0}}^{t}({\bf x}_{0})$ at time $t$. We will simply use ${\bf x}(t)$ for the position of the particle at time $t$, implicitly assuming that its initial position ${\bf x}(t=0)={\bf x}_{0}$ is given. Using this and taking the time derivative of $\Delta(t)$, we arrive at $d\Delta(t)/dt\leq C[\Delta(t)]^{h}$, with simple solution

In non-turbulent flows, $h\rightarrow 1$, thus $\Delta(t)\leq\Delta_{0}e^{C(t-t_{0})}$. At long times, assuming a near Hölder equality, we get $\Delta(t)\simeq\Delta_{0}e^{C(t-t_{0})}$ for a chaotic flow with Lyapunov exponent $C$; implying that the initial conditions (i.e., the value of $\Delta_{0}$) are never forgotten. The important point is that in the limit, when the initial separation goes to zero, even for a chaotic flow, the final separation vanishes:

On the other hand, for $0<h<1$, we find $\Delta(t)\simeq\Big{[}C(1-h)(t-t_{0})\Big{]}^{1\over 1-h}$, which implies that the information about initial conditions is lost! In other words, no matter how small the initial separation is, the wave-packets separate super-linearly with time:

For $h=1/3$, corresponding to the Kolmogorov scaling for velocity field [19], we get Richardson law [20]. The Hölder continuity for the magnetic field, i.e., $0<h<1$, results from the well-known effect of anomalous dissipation in turbulent plasmas; see e.g., [21]. In a turbulent flow with velocity field $\bf u$, the kinetic energy is viscously dissipated at a rate $\epsilon_{\nu}\equiv\nu|\nabla{\bf u}|^{2}$ while the magnetic energy is dissipated at a rate $\epsilon_{\eta}\equiv\eta|\nabla{\bf B}|^{2}$ (with viscosity $\nu$ and resistivity $\eta$). In fully developed turbulence, the Reynolds number $R_{e}\equiv LU/\nu$ and magnetic Reynolds number $R_{m}\equiv LU/\eta$, with characteristic length and velocity $L,U$, are high, i.e., one can take the limit of vanishing viscosity and resistivity; see also eqs.(15) & (16) in §II.4. Thus one may naively expect that the viscous or resistive energy dissipation rate should vanish in these limits, i.e., $\lim_{\nu\rightarrow 0}\nu|\nabla{\bf u}|^{2}\rightarrow 0$ and $\lim_{\eta\rightarrow 0}\eta|\nabla{\bf B}|^{2}\rightarrow 0$. However, in turbulence, experiments and numerical simulations indicate otherwise (see [22] and references therein). In a fully turbulent fluid, no matter how small we take the viscosity or resistivity, the viscous dissipation rate as well as magnetic dissipation rate remain finite: $\lim_{\nu\rightarrow 0}\nu|\nabla{\bf u}|^{2}\nrightarrow 0$ and $\lim_{\eta\rightarrow 0}\eta|\nabla{\bf B}|^{2}\nrightarrow 0$. These dissipative anomalies indicate that the spatial derivatives of velocity and magnetic fields should blow up, i.e., the field become Hölder singular⁶⁶6The real vector field ${\bf B}({\bf x})$ is Hölder continuous in ${{\bf x}}\in{\mathbb{R}}^{n}$ if $|{\bf B}(x)-{\bf B}(y)|\leq C|x-y|^{h}$ for some $C>0$ and $h>0$. If $h=1$, for any $x,y$, ${\bf B}$ is uniformly Lipschitz continuous. Also ${\bf B}$ is called Hölder singular if $0<h<1$. A uniformly Lipschitz function $f(x)$, i.e., one which satisfies $|f(x)-f(y)|\leq C_{f}|x-y|^{h_{f}}$ for some $C_{f}>0$ with $h_{f}=1$, has a bounded derivative, i.e., $|f^{\prime}(x)|<M$ for some $M>0$. In contrast, the derivative of a Hölder singular function $f$, i.e., one which satisfies $|f(x)-f(y)|\leq C_{f}|x-y|^{h_{f}}$ for some $C_{f}>0$ with $0<h_{f}<1$, can blow up; $|f^{\prime}(x)|>\infty$. which implies ill-defined spatial derivatives and hence ill-defined MHD equations (see e.g., [23, 14, 21, 22, 24]). In order to remove such singularities, one may use the coarse-grained field ${\bf B}_{l}$, defined by eq.(10), and look at the ”average” field at larger scales; see §II.3.

Eq.(2) implies that the mean square separation of wave-packets scales as $\langle|\Delta(t)|^{2}\rangle\sim t^{2\over 1-h}$. Thus averaging over, in a reconnection zone of scale $L$, it will take one Alfvén time, $t_{A}=L/V_{A}$, for wave-packets to leave the reconnection zone, during which they will on average separate by the distance

The reconnection speed $V_{rec}\sim{V_{A}\Delta/L}$ (which results from mass conservation in a zone of length $L$ and width $\Delta$) is thus given by

Assuming $h=1/3$, similar to the turbulent velocity field (see e.g., [21], Sec.IV), we arrive at the Lazarian-Vishniac reconnection speed [17] (see also [15] Sec.3). Contrast the above estimate with reconnection rate due to Ohmic diffusion of wave-packets, $\langle|\Delta(t_{A})|^{2}\rangle\sim\eta t_{A}$, i.e.,

where $S:=V_{A}L/\eta$ is the Lundquist number. This result, the Sweet-Parker speed [25, 26], is extremely slow in any astrophysical setting due to typically very large Lundquist numbers; see e.g., [27, 7]. These results, of course, are not new: we have simply recovered them here without appealing to the notion of field line diffusion. In the following subsections, we will also see that our approach not only clarifies the concept of magnetic topology and distinguishes it from reconnection, but it also provides an estimate for the rate of topology-change. In fact, it turns out that magnetic topology in laminar flows changes with a rate proportional to resistivity; see eq.(9).

Rapidly diverging trajectories even in the limit of their vanishing initial separation, associated to spontaneous stochasticity or the ”real” butterfly effect [28], differs from simple chaos (the butterfly effect). In the former case, the dynamics is singular (e.g., magnetic field is Hölder so $\dot{\bf x}={\bf B}$ has non-unique solutions) and randomness of trajectories persists at finite times even in the limit of vanishing noise. This is distinct from simple chaos in which predictability times can be lengthened arbitrarily by reducing noise since final solution always remains proportional to the initial conditions. This is, incidentally, the primary reason why the weather (governed by Navier-Stokes equations) cannot be forecasted for more than almost two weeks: this time cannot be lengthened even with more advanced technology (the real butterfly effect).⁷⁷7A nice discussion of this effect in weather forecasting, along with historical notes regarding E. Lorenz pioneering work, can be found in the popular book by T. Palmer: The Primacy of Doubt [29].On the other hand, in simple chaos (the butterfly effect), such predictability times can be lengthened by reducing noise (e.g., by making initial separation of trajectories arbitrarily smaller). Spontaneous stochasticity can be regarded as “super-chaos” associated with the formation of singularities in the dynamics and consequent divergence of Lyapunov exponents to infinity. Vanishingly small random perturbations can then be propagated to large scales in a finite amount of time. Extremely small but unavoidable sources of noise such as thermal fluctuations have been shown to easily trigger spontaneous stochasticity in turbulence [30]. These statements explain why, in the above calculations for a turbulent flow, we employed mean square separation: the very notion of a Lagrangian trajectory breaks down in turbulence since the system is spontaneously stochastic and trajectories remain intrinsically random in fully developed turbulence. This effect is in fact intimately related to the old, well-known notion of Richardson (2-particle) diffusion (for a brief and nice discussion of Richardson diffusion and its connection to spontaneous stochasticity, see [12]). At large (inertial) scales, the singularity can be removed by a coarse-graining procedure which leads to a smooth, large-scale magnetic field on any inertial scale $l$; see §II.3.

In this paper, we have argued that reconnection is associated with continuous divergence of Alfvénic wave-packets (magnetic path-lines) over time, i.e., roughly speaking, rapid but smooth change in magnetic pattern over time. Topology-change is associated with discontinuous divergence of trajectories or their annihilation by dissipative effects; resistivity in laminar flows and turbulent dissipation in turbulence. In laminar flows, reconnection and topology-change are thus completely different phenomena. Even in chaotic flows, where the Alfvénic trajectories diverge exponentially, their separation at later times remains proportional to the initial separation, unlike turbulent flows. In turbulence, the picture is somehow complicated and both reconnection and topology-change originate from spontaneous stochasticity or super-chaos—persistent random behavior of trajectories independent of their initial separation. Specifically, in turbulence, the non-linear dynamics governing magnetic field becomes singular and randomness in wave-packet trajectories survives even in the limit of vanishing noise, i.e., fast separation of trajectories at later times even when the initial separation of trajectories tends to zero. Fast turbulent reconnection results from enhanced turbulent mixing of trajectories while topology-change results from turbulence’s time-reversal symmetry breaking. These effects are deeply related to spontaneous stochasticity.

The simple approach presented in this paper provides a clear and intuitive topology for magnetic fields independent of the notion of magnetic field lines and their complicated motion through plasma. In fact, versatility of a powerful tool, such as Feynman’s diagrams; concept of a gravitational field or the notion of magnetic field lines, can make it look more physical than it is in reality leading to misuse. Feynman used to interpret his diagrams in a more physical way than what is understood today. Gravitational field around a mass became only part of the metric in general relativity. Similarly, the powerful notion of magnetic field lines, since their introduction by Faraday, can be replaced with other more appropriate tools such as magnetic path-lines for gaining a deeper and simpler picture. In a real magnetized plasma in three dimensions, for example, magnetic field lines might become quite inadequate to describe processes such as reconnection or magnetic topology change. Our approach in this paper provides an alternative way of looking at reconnection and magnetic topology-change, in terms of magnetic path-lines ${\bf x}(t)$ solving $\dot{\bf x}(t)={\bf B(x}(t),t)$ instead of field lines, which are parametric curves depicting magnetic field pattern only at a given time.

Alfvénic trajectories or magnetic path-lines are associated with a dynamical system whose phase space has a natural metric topology for magnetic field. This magnetic topology evolves with time and it is easy to see what conditions should be satisfied for the topology to be preserved. In laminar flows, magnetic topology can change with a rate proportional to resistivity, hence if reconnection is fast, i.e., it occurs on time scales much shorter than resistive time, then reconnection and topology change should be distinct phenomena. In turbulence, time-reversal symmetry is broken and topology cannot be preserved. In fact, magnetic topology in a turbulent plasma can change by enhanced turbulent diffusion on any inertial scale.

Our simple calculations also suggest connections to other approaches to magnetic reconnection. For example, one can use the coarse-grained induction equation, eq.(11), to study the time evolution of the unit tangent vector, $\hat{\bf B}_{l}={{\bf B}_{l}/B_{l}}$. In fact, the coarse-grained induction equation implies

where $(.)^{\perp}$ indicates the perpendicular direction with respect to the large-scale field ${\bf B}_{l}$ and

The term ${\bm{\Sigma}}_{l}$ (${\bm{\sigma}}_{l}$) has been shown to govern magnetic reconnection in turbulence (laminar flows) [13]. On the other hand, we have shown in this paper, eq.(18), that the rate of magnetic topology change depends on the functional derivative of $B_{l}{\bm{\Sigma}}_{l}$ (and $B_{l}{\bm{\sigma}}_{l}$ on small scales in laminar flows):

This estimate for turbulent flows implies that magnetic topology changes on all turbulent inertial scales $l$ and unlike laminar flows, it is independent of small scale (plasma) effects. These are the distinguishing characteristics of turbulent (stochastic) reconnection as well [17, 13, 7]. In fact, ”naive” dimensional analysis of the above estimate suggests a fast rate of order $\tau_{T}^{-1}\sim\Delta u_{l}/l$, i.e., eddy turn-over rate on scale $l$. In laminar flows, this rate is proportional to (typically negligible) resistivity; eq.(9). Spontaneous stochasticity seems to enhance both topology-change and reconnection in turbulence.

Finally, our approach also emphasizes the crucial roles scale and turbulence play in the study of magnetic topology change as well as its connection to turbulent reconnection. Any physical measurement can be performed only in a finite region of space (and during a finite interval in time). This somehow resembles the uncertainty principle in the sense that we can never measure the ”real” mathematical magnetic field ${\bf B(x},t)$ no matter how technologically advanced our measuring instruments become over time. In fact, although electromagnetic fields are in most applications assumed to be well-defined at small scales, this is not true for general physical fields. In condensed matter physics, for example, it is common for a general field theory to have a natural cut-off at small length scales (or high energies), e.g., the spacing between atoms in a lattice crystal. In most quantum field theories, there exist (ultra-violet) infinities and the field is not well-defined at very small scales. Hence, a cut-off is usually introduced to regularize the theory (regularization). This is of course part of the strong methodology known as (Wilsonian) Renormalization Group (RG) theory which is based on integrating out the small degrees of freedom, i.e. coarse-graining. What we ”observe” as a magnetic field is the big picture, i.e., large-scale field ${\bf B}_{l}$ at larger scales $l$ not tiny details, i.e., field fluctuations, which can be summed over on much smaller scales. This is the gist of RG methodology which plays also a crucial role in our presentation here. As for the role of turbulence, it is well known that in both plasma physics and astrophysics, magnetized fluids of interest are usually also turbulent due to external forcing or different internal instabilities or in fact even due to reconnection itself [7]. Even if the system is initially non-turbulent, reconnection can make the flow turbulent. Unlike laminar flows in ideal magnetized fluids, where magnetic field is approximately frozen into the flow (Alfvén flux-freezing; [11]), in turbulence, magnetic field follows the flow only in a statistical sense: this is stochastic flux-freezing formulated by Eyink [12]. Turbulent flows have non-trivial features like unpredictability, enhanced mixing and spontaneous stochasticity which tend to tangle the threading magnetic field stochastically, increasing its spatial complexity in a geometric sense [22]. Because of strong magnetic tension forces, at some point, the field may relax to a smoother configuration which in turn launches eruptive fluid motions [33, 34, 35], potentially observable as ”reconnection” events, e.g., on the solar surface.