2.1 Propagation of cosmic rays and energetic particles

The attempt to understand and predict the propagation of cosmic rays through the interplanetary and interstellar magnetic fields has been a major driver of research into the effect of turbulence on the tangling of the magnetic field. In a seminal early paper, Jokipii (Jokipii Reference Jokipii1966) performed the first detailed quasilinear statistical calculation of the motion of charged particles in a spatially random magnetic field, establishing a quantitative connection to the turbulent power spectrum of magnetic field fluctuations. Subsequent work conceptually explained the observed spreading of solar energetic particles from an active region over $180^{\circ }$ in solar longitude as a consequence of a magnetic field line random walk due to turbulent magnetic field fluctuations (Jokipii & Parker Reference Jokipii and Parker1968).

The scattering and acceleration of cosmic rays by a spectrum of Alfvén waves with strictly parallel wavenumbers was treated analytically by Schlickeiser (Schlickeiser Reference Schlickeiser1989), and was later extended to include the interaction with fast mode waves in a low beta plasma (Schlickeiser & Miller Reference Schlickeiser and Miller1998). A nonlinear diffusion theory of the stochastic wandering of magnetic field lines, developed by Matthaeus et al. (Matthaeus et al. Reference Matthaeus, Gray, Pontius and Bieber1995), lead to the expectation of diffusive field line wandering in the perpendicular direction.

By the mid-1990s, numerical modelling of the wandering of magnetic field lines began to be widely used, including test particle calculations of energetic particle transport along those turbulent magnetic fields. These efforts require, as input, a model of the spectrum of magnetic fluctuations generated by the plasma turbulence, and a wide variety of such turbulent magnetic field models have been used. Sophisticated numerical field line following algorithms and complementary analytical approaches have been used to study realizations of slab turbulence models with magnetic field fluctuations $\unicode[STIX]{x1D6FF}\boldsymbol{B}(z)$ (Schlickeiser Reference Schlickeiser1989; Shalchi & Kourakis Reference Shalchi and Kourakis2007a ,Reference Shalchi and Kourakis b ; Shalchi Reference Shalchi2010b ), two-dimensional (2-D) turbulence models with $\unicode[STIX]{x1D6FF}\boldsymbol{B}(x,y)$ (Shalchi & Kourakis Reference Shalchi and Kourakis2007a ,Reference Shalchi and Kourakis b ; Guest & Shalchi Reference Guest and Shalchi2012), composite models including slab plus 2-D components with $\unicode[STIX]{x1D6FF}\boldsymbol{B}=\unicode[STIX]{x1D6FF}\boldsymbol{B}(z)+\unicode[STIX]{x1D6FF}\boldsymbol{B}(x,y)$ (Bieber, Wanner & Matthaeus Reference Bieber, Wanner and Matthaeus1996; Giacalone & Jokipii Reference Giacalone and Jokipii1999; Shalchi & Kourakis Reference Shalchi and Kourakis2007a ,Reference Shalchi and Kourakis b ; Qin & Shalchi Reference Qin and Shalchi2013) and full 3-D models with $\unicode[STIX]{x1D6FF}\boldsymbol{B}(x,y,z)$ , including both isotropic (Zimbardo et al. Reference Zimbardo, Veltri, Basile and Principato1995; Giacalone & Jokipii Reference Giacalone and Jokipii1999; Shalchi Reference Shalchi2010a ; Ragot Reference Ragot2011) and anisotropic distributions of magnetic fluctuations (Chandran Reference Chandran2000; Zimbardo, Veltri & Pommois Reference Zimbardo, Veltri and Pommois2000; Zimbardo, Pommois & Veltri Reference Zimbardo, Pommois and Veltri2006; Shalchi & Kolly Reference Shalchi and Kolly2013; Ruffolo & Matthaeus Reference Ruffolo and Matthaeus2013).

These various investigations, conducted by a wide range of researchers, have often found conflicting results. The mean square displacement $\langle (\unicode[STIX]{x1D6FF}r)^{2}\rangle$ between two magnetic field lines, as they are followed along the magnetic field line a distance $l$ , is often modelled by the power-law form

(2.1) $$\begin{eqnarray}\langle (\unicode[STIX]{x1D6FF}r)^{2}\rangle \propto |l|^{p}.\end{eqnarray}$$

For different turbulent magnetic field models, or even a variation of the parameters within a single model, the resulting magnetic field line wandering is sometimes found to be sub-diffusive ( $p<1$ ) and other times found to be super-diffusive ( $p>1$ ), and yet other studies recover a standard diffusive behaviour ( $p=1$ ). Based on a broad reading of the literature, the results of the analytical and numerical modelling appear to be rather sensitive to the parameters and properties of the turbulence model chosen. For example, analytical modelling of anisotropic 3-D turbulence (with $k_{\bot }\gg k_{\Vert }$ , where $k$ is the wavenumber and perpendicular and parallel are with respect to the direction of the local magnetic field) suggests the field line wandering is diffusive ( $p=1$ ) (Shalchi & Kolly Reference Shalchi and Kolly2013), in contrast to the anomalous diffusion ( $p\neq 1$ ) often found using other turbulent models. At present, the wandering of magnetic field lines, and its impact on the propagation of energetic particles, remains an active area of research.

Improved modelling of the propagation of energetic particles in turbulent magnetic fields can have a significant impact on our technological infrastructure, with serious implications for national security. Our society is increasingly dependent on space-borne assets, such as global positioning system (GPS) navigation and communication satellites, so the prediction of severe space weather events in near-Earth space has become critically important for the protection of these assets. Solar energetic particle (SEP) events, in which a violent event on the surface of the sun spews high-energy electrons and protons into the heliosphere, represent a threat to both robotic and human assets in space. This prompts an urgent need to develop the ability to predict whether high-energy particles from a particular SEP event will reach the position of a potentially susceptible spacecraft. A predictive capability requires understanding how the SEP particles propagate through the turbulent interplanetary magnetic field. For example, on 3 NOV 2011, an SEP event erupted on the far side of the solar surface, spewing out energetic protons and electrons that were measured at the STEREO A, SOHO and STEREO B spacecraft, covering more than $200^{\circ }$ of solar longitude (Richardson et al. Reference Richardson, von Rosenvinge, Cane, Christian, Cohen, Labrador, Leske, Mewaldt, Wiedenbeck and Stone2014). This wide longitudinal spread of significant SEP particle fluxes is not satisfactorily predicted by existing models of SEP propagation. An improved understanding of the magnetic field line wander in the turbulent interplanetary magnetic field, as a function of the plasma and turbulence parameters, is necessary to develop a reliable predictive capability.

In addition to studies of energetic particle propagation, thermal conduction in astrophysical plasmas, such as that occurring in galaxy cluster cooling flows, has also been found to be strongly affected by a stochastic magnetic field (Chandran & Cowley Reference Chandran and Cowley1998).

2.2 Magnetic confinement fusion

Coincident with the earliest studies on the propagation of energetic particles in space and astrophysical plasmas were complementary studies of anomalous electron heat transport in tokamaks of the magnetic confinement fusion program. In tokamak plasmas, gradient-driven instabilities generate turbulent fluctuations in the confining magnetic field. It has been proposed that the distortion of magnetic flux tubes as they are mapped along the turbulent confining magnetic field leads to a destruction of the magnetic flux surfaces (Rosenbluth et al. Reference Rosenbluth, Sagdeev, Taylor and Zaslavsky1966; Filonenko, Sagdeev & Zaslavsky Reference Filonenko, Sagdeev and Zaslavsky1967) that prevent radial mixing of hot central plasma with cold exterior plasma. It was recognized that collisional diffusion is unable to account for all of the electron heat transport measured in experiments, and that magnetic field line wander could potentially explain the additional, ‘anomalous’ transport. Further work quantitatively estimated the diffusion in collisional and collisionless regimes, suggesting that fluctuations of sufficient amplitude, caused by microinstabilities at the scale of the ion gyroradius, would consistently explain both the stochastic nature of the magnetic field and the observed electron heat transport (Rechester & Rosenbluth Reference Rechester and Rosenbluth1978).

Exploring the role of magnetic field line wander in enhancing electron heat transport in a tokamak plasma has continued over the years using analytical calculations and test particle modelling, and with the comparison of these results to experimental measurements (Galeev & Zeleny Reference Galeev and Zeleny1981; Krommes, Oberman & Kleva Reference Krommes, Oberman and Kleva1983; Haas & Thyagaraja Reference Haas and Thyagaraja1986; Laval Reference Laval1993; Spatschek Reference Spatschek2008). Recent advancements in the direct numerical simulation of weakly collisional plasma turbulence using nonlinear gyrokinetic simulations (Pueschel, Kammerer & Jenko Reference Pueschel, Kammerer and Jenko2008; Nevins, Wang & Candy Reference Nevins, Wang and Candy2011; Wang, Boldyrev & Perez Reference Wang, Boldyrev and Perez2011; Hatch et al. Reference Hatch, Pueschel, Jenko, Nevins, Terry and Doerk2012, Reference Hatch, Pueschel, Jenko, Nevins, Terry and Doerk2013) and other direct numerical approaches (del-Castillo-Negrete & Blazevski Reference del-Castillo-Negrete and Blazevski2014, Reference del-Castillo-Negrete and Blazevski2016) have enabled breakthrough studies of the cause of magnetic field line wander and its impact on confinement in fusion plasmas. Under fusion-relevant plasma conditions, gyrokinetic simulations showed that the magnetic field indeed rapidly becomes stochastic through gradient-instability-driven turbulence, but that this stochasticity does not always produce a significant enhancement in the electron heat flux (Nevins et al. Reference Nevins, Wang and Candy2011). The development of stochasticity appears to arise through nonlinear interactions among overlapping magnetic islands (Wang et al. Reference Wang, Boldyrev and Perez2011), supporting an idea proposed in early studies based on analytical considerations (Rechester & Rosenbluth Reference Rechester and Rosenbluth1978). By focusing on the properties of the dominant turbulent modes in ion temperature gradient and trapped electron mode instability-driven turbulence, this overlapping of magnetic islands arises through the nonlinear transfer of energy from the unstable ballooning modes of odd parity to stable tearing modes of even parity (Hatch et al. Reference Hatch, Pueschel, Jenko, Nevins, Terry and Doerk2012, Reference Hatch, Pueschel, Jenko, Nevins, Terry and Doerk2013).

2.3 The cascade of energy in plasma turbulence

It has been suggested that one can view the cascade of energy to small scales in magnetized plasma turbulence as due to the distortion of the perpendicular structure of Alfvén wavepackets as they propagating along a wandering, turbulent magnetic field (Similon & Sudan Reference Similon and Sudan1989), and this concept has been demonstrated numerically (Maron & Goldreich Reference Maron and Goldreich2001). An illustration of the distortion of a wavepacket with an initially circular cross-section perpendicular to the equilibrium magnetic field from a nonlinear gyrokinetic simulation of plasma turbulence is presented in figure 2. Recent work has adopted this framework to interpret current sheet formation in coronal loops (Rappazzo & Parker Reference Rappazzo and Parker2013), the nonlinear transfer of energy to smaller-scale Alfvén waves in basic laboratory experiments of plasma turbulence (Howes et al. Reference Howes, Drake, Nielson, Carter, Kletzing and Skiff2012; Drake et al. Reference Drake, Schroeder, Howes, Kletzing, Skiff, Carter and Auerbach2013; Howes & Nielson Reference Howes and Nielson2013; Howes et al. Reference Howes, Nielson, Drake, Schroeder, Skiff, Kletzing and Carter2013; Nielson et al. Reference Nielson, Howes and Dorland2013) and the evolution of magnetic flux surfaces in 3-D reduced MHD simulations (Servidio et al. Reference Servidio, Matthaeus, Wan, Ruffolo, Rappazzo and Oughton2014).

Figure 2. The distortion of a circular wavepacket as it propagates along the wandering magnetic field in a nonlinear gyrokinetic simulation of plasma turbulence.

Looking at this problem in more detail, the nonlinear physics underlying the turbulent cascade of energy from large to small scales is often described in Fourier space, where a nonlinear three-wave coupling mechanism has been identified that leads to a secular transfer of energy to modes with higher perpendicular wavenumber (Shebalin, Matthaeus & Montgomery Reference Shebalin, Matthaeus and Montgomery1983; Sridhar & Goldreich Reference Sridhar and Goldreich1994; Montgomery & Matthaeus Reference Montgomery and Matthaeus1995; Ng & Bhattacharjee Reference Ng and Bhattacharjee1996; Galtier et al. Reference Galtier, Nazarenko, Newell and Pouquet2000; Howes & Nielson Reference Howes and Nielson2013), resulting in an anisotropic cascade of energy in wavevector space. But the physical manifestation of the $k_{\Vert }=0$ mode that mediates this resonant three-wave interaction is obscured by the use of the Fourier (plane wave) decomposition. An analytical calculation modelling interactions between localized Alfvén wavepackets demonstrated that wavepackets involving a $k_{\Vert }=0$ component will lead to this lowest-order three-wave nonlinear coupling (Ng & Bhattacharjee Reference Ng and Bhattacharjee1996), and this mechanism has been demonstrated using laboratory experiments (Howes et al. Reference Howes, Drake, Nielson, Carter, Kletzing and Skiff2012, Reference Howes, Nielson, Drake, Schroeder, Skiff, Kletzing and Carter2013; Drake et al. Reference Drake, Schroeder, Howes, Kletzing, Skiff, Carter and Auerbach2013). Physically, the $k_{\Vert }=0$ component of a wavepacket represents a shear in the magnetic field, as depicted in figure 3. Investigation of the propagation of Alfvén waves in a wandering magnetic field demonstrates that the cascade of energy to small scales is represented in physical space (as opposed to Fourier space) as a shearing of the perpendicular structure of an Alfvén wave as it propagates along a wandering magnetic field, as depicted in figure 3(c,f). Thus, exploring the complementary picture of the plasma turbulent cascade as the distortion of Alfvén waves as they propagate along a wandering magnetic field may yield fresh insights into the nature of plasma turbulence.

Figure 3. Distortion of a magnetic flux tube in the perpendicular $(x,y)$ plane due to wandering of the magnetic field. (a) Magnetic field fluctuation $\unicode[STIX]{x1D6FF}B_{y}$ contains no $k_{z}=0$ component. (b) Three-dimensional tracing of magnetic field lines due to the magnetic fluctuation in (a) at positions $x=+\unicode[STIX]{x03C0}/2$ (red) and $x=-\unicode[STIX]{x03C0}/2$ (blue). The oscillating shear causes the magnetic field to become sheared and then subsequently to ‘unshear’. (c) The distortion of an Alfvén wave with an initially circular perpendicular structure (black) as it propagates along the wandering magnetic field. (d) For the case of a magnetic field fluctuation $\unicode[STIX]{x1D6FF}B_{y}$ with a non-zero $k_{z}=0$ component, the (e) 3-D magnetic field has a monotonic shear, leading to (f) the permanent distortion (red) of an initially circular Alfvén wave structure (black).

2.4 Particle acceleration

Models of diffusive shock acceleration at collisionless shocks require irregularities in the upstream magnetic field to return reflected particles back to the shock front repeatedly to achieve significant acceleration (Ragot Reference Ragot2001; Guo & Giacalone Reference Guo and Giacalone2010, Reference Guo and Giacalone2013, Reference Guo and Giacalone2015). The efficiency of such a shock acceleration mechanism is likely to be dependent on the nature of the upstream magnetic field irregularities, but existing investigations of these particle acceleration mechanisms often use unrealistic models of the turbulent plasma, such as slab turbulence with only a one-dimensional variation of the turbulent field, $\unicode[STIX]{x1D6FF}\boldsymbol{B}(z)$ (Guo & Giacalone Reference Guo and Giacalone2010). The development of a new empirical model of magnetic field line wander based on direct numerical simulations of plasma turbulence will enable more realistic modelling of the diffusive acceleration mechanism at collisionless shocks.

2.5 Magnetic reconnection

It has been proposed theoretically that magnetic field line wander can alter the physics of magnetic reconnection when the upstream plasma is turbulent (Lazarian & Vishniac Reference Lazarian and Vishniac1999), increasing the rate of reconnection and the thickness of the current sheets in the turbulent medium (Vishniac et al. Reference Vishniac, Pillsworth, Eyink, Kowal, Lazarian and Murray2012). Numerical simulations have played a key role in bearing out these ideas (Kowal et al. Reference Kowal, Lazarian, Vishniac and Otmianowska-Mazur2009, Reference Kowal, Lazarian, Vishniac and Otmianowska-Mazur2012). It has also been claimed that exponential field line separation in turbulent plasmas will lead to reconnection even in the absence of intense current sheets (Boozer Reference Boozer2014). It has been shown that stochasticity of the magnetic field in MHD turbulence simulations degrades the usual notion of flux freezing in an MHD plasma, potentially explaining fast reconnection of large-scale structures at MHD scales (Eyink et al. Reference Eyink, Vishniac, Lalescu, Aluie, Kanov, Bürger, Burns, Meneveau and Szalay2013). More recently, numerical simulations have been used to explore how breaking field line connectivity by stochasticity of the magnetic field can be a mechanism for fast reconnection (Huang, Bhattacharjee & Boozer Reference Huang, Bhattacharjee and Boozer2014). The properties of the turbulent upstream magnetic fields very likely influence how the reconnection mechanism is modified, so an improved model of magnetic field line wander in plasma turbulence will contribute to progress in the understanding of magnetic reconnection under the turbulent conditions of realistic space and astrophysical plasma environments.

3.1 Improving our understanding through direct numerical simulations

As detailed above, the nature of the magnetic field line wander in previous studies appears to be quite sensitive to the characteristics and parameters of the models describing the turbulent magnetic field. Many of the models of the turbulent magnetic field used in the literature exploring the phenomenon of magnetic field line wander and its implications are severely outdated, being inconsistent with the current understanding of plasma turbulence. Specifically, models employing slab, 2-D composite (slab plus two dimensions) and isotropic distributions of magnetic fluctuations, although more susceptible to theoretical analysis, are at odds with the anisotropic, 3-D nature of plasma turbulence that is now well established through decades of experimental, analytical and numerical research on magnetized plasma turbulence (Belcher & Davis Reference Belcher and Davis1971; Robinson & Rusbridge Reference Robinson and Rusbridge1971; Zweben, Menyuk & Taylor Reference Zweben, Menyuk and Taylor1979; Montgomery & Turner Reference Montgomery and Turner1981; Shebalin et al. Reference Shebalin, Matthaeus and Montgomery1983; Cho & Vishniac Reference Cho and Vishniac2000; Maron & Goldreich Reference Maron and Goldreich2001; Cho & Lazarian Reference Cho and Lazarian2004, Reference Cho and Lazarian2009; TenBarge & Howes Reference TenBarge and Howes2012). Recent direct multi-spacecraft measurements of turbulence in the solar wind confirm this anisotropic nature of the turbulent fluctuations (Sahraoui et al. Reference Sahraoui, Goldstein, Belmont, Canu and Rezeau2010; Narita et al. Reference Narita, Gary, Saito, Glassmeier and Motschmann2011; Roberts, Li & Li Reference Roberts, Li and Li2013; Roberts, Li & Jeska Reference Roberts, Li and Jeska2015).

In addition, the models employed in the studies reviewed above almost universally employ randomly phased magnetic field fluctuations, a characteristic inconsistent with any self-consistent realization of a turbulent magnetic field. Kolmogorov’s four-fifths law (Kolmogorov Reference Kolmogorov1941) is an exact statistical formula relating the mean energy dissipation rate to the third moment of the velocity fluctuations in hydrodynamic turbulence; subsequently, this third moment approach has been extended for the case of MHD turbulence (Chandrasekhar Reference Chandrasekhar1951; Politano & Pouquet Reference Politano and Pouquet1998; Yousef, Rincon & Schekochihin Reference Yousef, Rincon and Schekochihin2007). A spectrum of magnetic fluctuations with random phases yields an average third moment of zero, because it is the phase correlations among different magnetic fluctuations that are responsible for the nonlinear turbulent cascade of energy. Random phase models therefore lack some of the inherent attributes of a self-consistently determined turbulent magnetic field (Howes Reference Howes2015, Reference Howes2016). It remains an open question whether such correlations will indeed alter the nature of the magnetic field line wander resulting from plasma turbulence, but the apparent sensitivity of the results of previous studies to the characteristics of the magnetic field model suggests that a self-consistent turbulent magnetic field will yield the most well justified and physically relevant results.

The uncertainty in describing the turbulent magnetic field can be eliminated by using a turbulent magnetic field that is generated self-consistently by direct numerical simulation of the equations governing the turbulent plasma dynamics. Of the studies directly investigating magnetic field line wander reviewed above, only the recent studies of stochastic magnetic field development in fusion plasmas (Pueschel et al. Reference Pueschel, Kammerer and Jenko2008; Nevins et al. Reference Nevins, Wang and Candy2011; Wang et al. Reference Wang, Boldyrev and Perez2011; Hatch et al. Reference Hatch, Pueschel, Jenko, Nevins, Terry and Doerk2012, Reference Hatch, Pueschel, Jenko, Nevins, Terry and Doerk2013) employed direct numerical simulations, specifically using a nonlinear gyrokinetic code. A general approach to understand the development and saturation of the tangled magnetic field in general astrophysical plasma turbulence has not been attempted, and this provides a strong motivation for the present work.

3.2 The role of magnetic reconnection

Figure 4. Illustration of how magnetic reconnection can instantaneously change the magnetic connectivity in a turbulent plasma, thereby influencing the development of magnetic field line wander.

Another important ingredient in understanding how magnetic fields become dynamically tangled by plasma turbulence is the process of magnetic reconnection. Although the impact of pre-existing turbulent magnetic fields on the process of magnetic reconnection has been examined previously (Lazarian & Vishniac Reference Lazarian and Vishniac1999; Kowal et al. Reference Kowal, Lazarian, Vishniac and Otmianowska-Mazur2009, Reference Kowal, Lazarian, Vishniac and Otmianowska-Mazur2012; Vishniac et al. Reference Vishniac, Pillsworth, Eyink, Kowal, Lazarian and Murray2012; Boozer Reference Boozer2014; Huang et al. Reference Huang, Bhattacharjee and Boozer2014), the investigation of how reconnection mediates the tangling and untangling of magnetic fields in plasma turbulence has not been addressed by any existing study. For example, consider the two turbulent magnetic field lines depicted in figure 4. When a small reconnection event (denoted by the black cross) occurs between the two field lines, the connectivity of those two field lines changes instantaneously. Particles streaming along the red field line from the left end of the plasma, which previously had been connected to point A, will suddenly find themselves magnetically connected instead to point B. An open question is whether the impact of magnetic reconnection on the wandering of magnetic field lines can be described in the framework of a diffusion, or whether this possibility of sudden jumps in connectivity alter the nature of the resulting magnetic field line wander. Direct numerical simulations of the turbulent plasma that resolve the physics of magnetic reconnection, even in the collisionless limit relevant to many space and astrophysical systems of interest, are critical for answering this open question.

3.3 The turbulent solar wind as a fiducial example

The turbulent solar wind that pervades our heliosphere represents a fiducial system supporting a broad spectrum of turbulent plasma motions over more than seven orders of magnitude in scale (Kiyani, Osman & Chapman Reference Kiyani, Osman and Chapman2015). It is also the most thoroughly diagnosed turbulent astrophysical plasma in the universe, thereby providing unique opportunities for confronting any empirical description of magnetic field line wander with direct, in situ measurements of the turbulent interplanetary magnetic field. Thus, the solar wind provides an ideal case study for discussing the phenomenon of magnetic field line wander in a turbulent plasma. In addition, the tangling of the magnetic field in the solar wind has important consequences for the propagation of solar energetic particles erupting from violent solar activity, constituting an important space weather hazard for spaceborne human and technological assets. Improving the modelling of the tangled magnetic field in the heliosphere therefore has important societal implications.

Figure 5. Schematic of the magnetic energy wavenumber spectrum in the solar wind, showing the form of the spectrum in the energy containing, inertial and dissipation ranges. Ranges for the typical Larmor radius scales for protons $\unicode[STIX]{x1D70C}_{\text{EP},p}$ and electrons $\unicode[STIX]{x1D70C}_{\text{EP},e}$ from solar energetic particle events are depicted.

Figure 5 presents a diagram of the solar wind magnetic energy wavenumber spectrum – assuming the Taylor hypothesis (Taylor Reference Taylor1938) to convert the spacecraft-frame frequency to a corresponding wavenumber of spatial fluctuations in the super-Alfvénic solar wind (Howes, Klein & TenBarge Reference Howes, Klein and TenBarge2014; Klein, Howes & TenBarge Reference Klein, Howes and TenBarge2014) – for near-Earth space at heliocentric radius $R\sim 1$  AU. At the largest scales $l>10^{6}$  km (lowest wavenumbers) is the energy containing range (Matthaeus et al. Reference Matthaeus, Oughton, Pontius and Zhou1994; Tu & Marsch Reference Tu and Marsch1995; Bruno & Carbone Reference Bruno and Carbone2005), populated by large-scale plasma flow and magnetic field fluctuations. Through nonlinear interactions, these energy containing fluctuations feed their energy into the turbulent cascade at the outer scale, $l\sim 10^{6}$  km, where the steepening of the magnetic energy spectrum marks the beginning of the turbulent inertial range (Tu & Marsch Reference Tu and Marsch1995; Bruno & Carbone Reference Bruno and Carbone2005). Within the inertial range, energy is nonlinearly transferred scale by scale in a self-similar manner, leading to a single power-law spectrum down to the inner scale, which corresponds to one of the characteristic ion kinetic length scales at $l\sim 10^{2}$  km. At this ion scale, the magnetic energy spectrum breaks once again (Bourouaine et al. Reference Bourouaine, Alexandrova, Marsch and Maksimovic2012), marking the transition to the dissipation rangeFootnote ¹ (Alexandrova et al. Reference Alexandrova, Saur, Lacombe, Mangeney, Mitchell, Schwartz and Robert2009; Kiyani et al. Reference Kiyani, Chapman, Khotyaintsev, Dunlop and Sahraoui2009; Sahraoui et al. Reference Sahraoui, Goldstein, Robert and Khotyaintsev2009; Chen et al. Reference Chen, Horbury, Schekochihin, Wicks, Alexandrova and Mitchell2010; Sahraoui et al. Reference Sahraoui, Goldstein, Belmont, Canu and Rezeau2010, Reference Sahraoui, Huang, Belmont, Goldstein, Rétino, Robert and De Patoul2013). Finally, at the characteristic electron length scale of $l\sim 1$  km, the magnetic energy spectrum is often observed to exhibit an exponential roll off (Alexandrova et al. Reference Alexandrova, Lacombe, Mangeney, Grappin and Maksimovic2012; Sahraoui et al. Reference Sahraoui, Huang, Belmont, Goldstein, Rétino, Robert and De Patoul2013), interpreted to indicate the ultimate termination of the turbulent cascade. Also plotted in figure 5 is a representation of the Larmor radius scales for protons and electrons from SEP events, $\unicode[STIX]{x1D70C}_{\text{EP},p}$ and $\unicode[STIX]{x1D70C}_{\text{EP},e}$ ; collisionless wave–particle interactions of energetic particles with the turbulent magnetic fluctuations lead to scattering rates that peak at these Larmor scales. Simulations using the gyrokinetic code AstroGK reproduce quantitatively the features of the solar wind magnetic energy spectrum from the middle of the inertial range down to the sub-electron scales (TenBarge et al. Reference TenBarge, Podesta, Klein and Howes2012; TenBarge, Howes & Dorland Reference TenBarge, Howes and Dorland2013).

It is the turbulent magnetic field fluctuations over this broad range of scales that lead to the stochastic character of the magnetic field, so investigating how fluctuations in different scale ranges of this spectrum affect the magnetic field line wandering is an important long-term goal. To accomplish this goal, direct numerical simulations can be used to learn how magnetic field line wander develops and saturates in plasma turbulence.

3.4 A model for the development and saturation of magnetic field line wander

Here we propose a novel theoretical picture of the plasma physical processes involved in the development of magnetic field line wander and its saturation. The nonlinear interactions that underlie the turbulent cascade of energy to small scales also lead to a tangling of the magnetic field. In an ideal plasma with no magnetic reconnection, this tangling of the field would continue indefinitely, generating turbulent structures on ever smaller scales, leading to an ever more intricate wandering of the magnetic field. But, once the turbulent structures have reached sufficiently small scales that non-ideal physics breaks the frozen-in condition, magnetic reconnection may ensue, thereby untangling the magnetic field. We propose that the saturation of the magnetic field line wander represents a balance between the nonlinearly driven tangling and the magnetic-reconnection-mediated untangling, a physical picture that we aim to test thoroughly using direct numerical simulations.

A key observation is that these two mechanisms, nonlinear interactions (turbulence) and magnetic reconnection, depend differently on the fundamental parameters describing the turbulence and the plasma. For example, the physics of magnetic reconnection has a strong dependence on the plasma $\unicode[STIX]{x1D6FD}$ , but the physics of the turbulent nonlinear energy transfer appears to have very little dependence on this plasma parameter (Howes & Nielson Reference Howes and Nielson2013; Nielson et al. Reference Nielson, Howes and Dorland2013; Howes Reference Howes2015). What is entirely new here, compared to the body of literature on turbulent magnetic field line wander, is that most previous worksFootnote ² have completely ignored any role played by magnetic reconnection. From the theoretical picture above – a balance between turbulent tangling and reconnective untangling – we believe that it is inevitable that magnetic reconnection plays an important role in the physics of magnetic field line wander, possibly leading to an important, and as yet unrecognized, dependence on the plasma $\unicode[STIX]{x1D6FD}$ . A study based on direct numerical simulations that resolve the kinetic microphysics of magnetic reconnection is likely to break new ground on this important frontier in the study of magnetic field line wander and its implications for many important physical processes in the universe.

Discussions of the tangling of the magnetic field by plasma turbulence often employs the term ‘stochastic magnetic field’ as a generic label for any turbulently tangled magnetic field. Here we reserve the use of the term stochastic to cases in which the magnetic field indeed demonstrates a stochastic character, as demonstrated by an appropriate analysis, such as a Poincaré recurrence plot (see § 6). We choose to use the term magnetic field line wander to refer to the general case when the magnetic field does exhibit a topology in which the separation between two field lines increases (or decreases) as one moves a distance along the field, whether or not the magnetic field exhibits a stochastic character. How the physics of energetic particle propagation in astrophysical plasmas, heat and particle transport in fusion plasmas, magnetic reconnection and particle acceleration differs when the magnetic field lines merely wander, but are not fully stochastic, is interesting open question.

3.5 Key questions

To investigate and characterize the development and saturation of magnetic field line wander in turbulent plasmas, direct numerical simulations provide a valuable tool. Since magnetic reconnection may play an essential role in the physics of the field line tangling, a numerical method that resolves the kinetic microphysics of magnetic reconnection, particularly under the weakly collisional conditions relevant to many turbulent space and astrophysical plasma systems, is essential. This can be supplemented by fluid methods to explore how the nonlinear dynamics at large scales governs the tangling of the magnetic field. Such an approach, maintaining a strong connection to the nonlinear dynamics of plasma turbulence and magnetic reconnection, will enable important new questions about the nature of magnetic field line wander caused by plasma turbulence to be addressed:

1. (i) How does magnetic field line wander develop and saturate in plasma turbulence?

2. (ii) Do the properties of magnetic field line wander depend on the underlying physical mechanism that enables magnetic reconnection (collisionless versus resistive versus numerical reconnection)?

3. (iii) Is magnetic field line wander dominated by the large-scale or small-scale fluctuations of the turbulence?

4. (iv) Can we construct an empirical description of the magnetic field line wander in terms of the fundamental plasma and turbulence parameters?

3.6 Quantitative dependence on fundamental turbulence and plasma parameters

Turbulence in heliospheric and other astrophysical plasmas naturally leads to the tangling of the magnetic field, leading to the development of magnetic field line wander, a property that we would like to characterize quantitatively. To develop a quantitative measure of the wandering of field lines in such a plasma, consider choosing two points on different field lines separated by a distance $\unicode[STIX]{x1D6FF}r$ perpendicular to the magnetic field. For these two particular points, one may compute the perpendicular separation between the two field lines in the perpendicular plane as a function of the distance along the magnetic field line $l$ . Computing statistics of this quantity using direct numerical simulations of plasma turbulence enables the development of the quantitative characterization of the magnetic field line wander.

The statistics of this perpendicular separation $\unicode[STIX]{x1D6FF}r$ between two magnetic field lines can be quantitatively characterized in terms of a small number of important parameters. First are basic parameters to describe the wandering of field lines away from each other, including the distance travelled along the field line $l$ and the time for which turbulence has been dynamically tangling the field. Next are the dimensionless parameters that describe the turbulence itself. The amplitude, or strength, of the turbulence is characterized by the nonlinearity parameter, $\unicode[STIX]{x1D712}=k_{\bot }\unicode[STIX]{x1D6FF}B_{\bot }/(k_{\Vert }B_{0})$ (Goldreich & Sridhar Reference Goldreich and Sridhar1995; Howes & Nielson Reference Howes and Nielson2013), describing the amplitude of the turbulence at the driving scale (equivalent to the ‘Kubo’ number in some previous studies (Zimbardo et al. Reference Zimbardo, Veltri and Pommois2000)). For example, to explore the impact on particle diffusion by the amplitude of turbulent fluctuations, one study computed particle trajectories numerically in a tangled magnetic field consisting of 1000 randomly phased plane-wave modes (Hauff et al. Reference Hauff, Jenko, Shalchi and Schlickeiser2010), and another study used Ramaty High Energy Solar Spectroscopic Imager (RHESSI) observations of hard X-rays to estimate the magnitude of magnetic field line diffusion in flaring coronal loops under several assumptions (Bian, Kontar & MacKinnon Reference Bian, Kontar and MacKinnon2011). Both studies found that, in the large Kubo number limit ( $\unicode[STIX]{x1D712}\gg 1$ ), perpendicular diffusion becomes independent of the turbulence spectrum, in agreement with the predictions of percolation theory (Gruzinov, Isichenko & Kalda Reference Gruzinov, Isichenko and Kalda1990; Isichenko Reference Isichenko1992).

Another key dimensionless parameter is the isotropic driving wavenumber, $k_{0}\unicode[STIX]{x1D70C}_{i}$ (Howes et al. Reference Howes, Cowley, Dorland, Hammett, Quataert and Schekochihin2008a ; Howes, TenBarge & Dorland Reference Howes, TenBarge and Dorland2011), describing the driving scale, or energy injection scale, of the turbulence normalized to the ion Larmor radius, where it is assumed the turbulent fluctuations are isotropic with respect to the direction of the mean magnetic field at this scaleFootnote ³ . Finally, the key dimensionless parameters that describe the plasma are the ion plasma beta, $\unicode[STIX]{x1D6FD}_{i}=8\unicode[STIX]{x03C0}n_{0}T_{i}/B_{0}^{2}$ , and the ion-to-electron temperature ratio, $T_{i}/T_{e}$ . In summary, therefore, we expect that the separation between two magnetic field lines $\unicode[STIX]{x1D6FF}r$ as one follows the field lines through the plasma to have a dependence on these fundamental parameters given by $\unicode[STIX]{x1D6FF}r(l,t,\unicode[STIX]{x1D712},k_{0}\unicode[STIX]{x1D70C}_{i},\unicode[STIX]{x1D6FD}_{i},T_{i}/T_{e})$ .

For the most likely case of saturated turbulence in steady state, where the tangling of the magnetic field by the turbulence and untangling of the field by magnetic reconnection statistically balance, we conjecture that the $\unicode[STIX]{x1D6FF}r$ will not depend on time. A reasonable form to seek for the average squared field line separation in a turbulent steady state, $\langle (\unicode[STIX]{x1D6FF}r)^{2}\rangle$ , based on the previous analyses reviewed in § 2, can be written

(3.1) $$\begin{eqnarray}\langle (\unicode[STIX]{x1D6FF}r)^{2}\rangle =Al^{p}.\end{eqnarray}$$

Here $A(\unicode[STIX]{x1D712},k_{0}\unicode[STIX]{x1D70C}_{i},\unicode[STIX]{x1D6FD}_{i},T_{i}/T_{e})$ represents the amplitude of the average spreading of field lines, and we assume the average squared field line separation can be expressed as a power law of the distance along the field line $l$ , given by the exponent $p(\unicode[STIX]{x1D712},k_{0}\unicode[STIX]{x1D70C}_{i},\unicode[STIX]{x1D6FD}_{i},T_{i}/T_{e})$ . The long-term goal of this project is to determine empirically forms for $A$ and $p$ that can be used to characterize the magnetic field line wander in terms of the turbulence and plasma parameters.

Although the two turbulence parameters $\unicode[STIX]{x1D712}$ and $k_{0}\unicode[STIX]{x1D70C}_{i}$ together with the two plasma parameters $\unicode[STIX]{x1D6FD}_{i}$ and $T_{i}/T_{e}$ lead to a four-dimensional parameter space, the broad range of different turbulence models previously used to explore magnetic field line wander requires a much longer list of possible parameters. The most likely culprit responsible for the many conflicting results found in the literature is the tremendous variation among the different models chosen to describe the turbulent magnetic field. The use of direct numerical simulations of plasma turbulence enables us to eliminate the huge parameter space necessary to describe the plethora of different magnetic field models reviewed in § 2, at the expense of the necessarily limited dynamic range attainable by direct numerical simulations. But we believe that a more physically faithful characterization of the magnetic field line wander in plasma turbulence can be patched together through a judicious use of different simulation models appropriate to different ranges of the turbulent spectrum, as depicted in figure 5.

5.1 Development and saturation of turbulent magnetic energy spectrum

These driven turbulence simulations begin with a straight, uniform magnetic field and zero fluctuations. The oscillating Langevin antenna drives an external parallel current that generates perpendicular magnetic field fluctuations, each driven mode with a plane-wave pattern specified by the wavevector. Nonlinear interactions between the counterpropagating Alfvén waves generated by the antenna immediately begin to transfer energy to higher wavenumbers, and the magnetic energy spectrum begins to fill in as energy is continually injected into the driven modes. Figure 6 presents the perpendicular magnetic energy spectrum $E_{B_{\bot }}(k_{\bot })=\int _{-\infty }^{\infty }\,\text{d}k_{z}\int _{0}^{2\unicode[STIX]{x03C0}}\,\text{d}\unicode[STIX]{x1D703}k_{\bot }|\unicode[STIX]{x1D6FF}B(k_{\bot })|^{2}/8\unicode[STIX]{x03C0}$ at a number of times early in the development of the turbulent energy spectrum, where time is normalized using the linear kinetic Alfvén wave period for modes at the domain scale, $t/\unicode[STIX]{x1D70F}_{A}$ and $\unicode[STIX]{x1D714}_{A0}\unicode[STIX]{x1D70F}_{A}=1.74$ . Note that, due to the broad range of the logarithmic vertical axis, at the very early times $t/\unicode[STIX]{x1D70F}_{A}<0.1$ , very little energy has been injected into the turbulent cascade.

Figure 6. Development of the magnetic energy spectrum in the nonlinear gyrokinetic simulation of driven kinetic Alfvén wave turbulence. The normalized time for each spectrum $t/\unicode[STIX]{x1D70F}_{A}$ is labelled. The saturated spectrum for strong kinetic Alfvén wave cascade is expected to have a spectral index of $-2.8$ , plotted for comparison.

Figure 7 shows how the spectrum saturates to a statistically steady state. Ten perpendicular magnetic energy spectra (blue and black) are plotted at uniform linear time intervals $t_{j}/\unicode[STIX]{x1D70F}_{A}=j\unicode[STIX]{x0394}t/\unicode[STIX]{x1D70F}_{A}$ for $j=1,2,3,\ldots ,10$ and $\unicode[STIX]{x0394}t/\unicode[STIX]{x1D70F}_{A}=0.028$ . We also overplot all spectra over the time range $0.28\leqslant t/\unicode[STIX]{x1D70F}_{A}\leqslant 3.6$ (red). This plot shows that, although more energy is injected into the turbulent magnetic energy spectrum at $t/\unicode[STIX]{x1D70F}_{A}>0.28$ , the shape of the spectrum appears to be saturated at $t/\unicode[STIX]{x1D70F}_{A}=0.28$ – only the total energy content changes, but the shape of the spectrum does not. We also plot the exponentially cutoff magnetic energy spectrum (blue dashed) determined empirically from a sample of 100 Cluster spectra, $E_{B_{\bot }}(k_{\bot })\propto (k_{\bot }\unicode[STIX]{x1D70C}_{i})^{-2.8}\exp (-k_{\bot }\unicode[STIX]{x1D70C}_{e})$ (Alexandrova et al. Reference Alexandrova, Lacombe, Mangeney, Grappin and Maksimovic2012), a result reproduced previously using nonlinear gyrokinetic simulations (TenBarge & Howes Reference TenBarge and Howes2013; TenBarge et al. Reference TenBarge, Howes and Dorland2013). The present simulation also agrees well with this model magnetic energy spectrum.

Note that the spectra in figure 7 have been binned in bins of width $\unicode[STIX]{x0394}k_{\bot }\unicode[STIX]{x1D70C}_{i}=5$ , producing a smoother appearance than the spectra in figure 6, in which all possible values of $k_{\bot }\unicode[STIX]{x1D70C}_{i}$ from our perpendicular wavevector grid are represented.

Figure 7. Perpendicular magnetic energy spectra plotted at uniform time intervals of $\unicode[STIX]{x1D6FF}t/\unicode[STIX]{x1D70F}_{A}=0.028$ (blue and black), and all spectra overplotted over the range $0.28\leqslant t/\unicode[STIX]{x1D70F}_{A}\leqslant 3.6$ (red). For comparison, the slope for a power-law spectrum $E_{B_{\bot }}\propto k_{\bot }^{-2.8}$ (blue solid) and an exponentially cutoff spectrum $E_{B_{\bot }}\propto k_{\bot }^{-2.8}$ (blue dashed) are shown for comparison.

The time scales associated with the development and saturation of the turbulent cascade can be nicely illustrated by plotting the amplitude of the energy at a perpendicular wavenumber in the middle of the dynamic range at $k_{\bot }\unicode[STIX]{x1D70C}_{i}=20.6$ as a function of time, as shown in figure 8(a). At early times $t/\unicode[STIX]{x1D70F}_{A}<0.14$ , the perpendicular magnetic energy at $k_{\bot }\unicode[STIX]{x1D70C}_{i}=20.6$ increases as a steep power law with time, $E_{B_{\bot }}(k_{\bot })\propto t^{12}$ . At time $t_{1}/\unicode[STIX]{x1D70F}_{A}=0.14$ , the increase of energy with time at this wavenumber reduces to a less steep approximate power law with $E_{B_{\bot }}(k_{\bot })\propto t^{6}$ , before reaching a statistically steady value at $t_{2}/\unicode[STIX]{x1D70F}_{A}=0.28$ .Footnote ⁴ Beyond this time, the shape of the magnetic energy spectrum no longer evolves, as illustrated by the spectra plotted in figure 7, but the total amplitude of the energy varies over a factor of four, with the same spectral shape simply shifting up and down. This variation of the total energy in the spectrum is due to the fact that the finite-time-correlated driving of the oscillating Langevin antenna leads to a significant variation in the rate of energy injection (TenBarge et al. Reference TenBarge, Howes, Dorland and Hammett2014).

It is worthwhile pointing out how rapidly the magnetic energy spectrum saturates, requiring a length of time that is only a fraction of the period of the kinetic Alfvén wave at the domain scale, $0.28\unicode[STIX]{x1D70F}_{A}$ . This is likely due to the dispersive nature of the kinetic Alfvén wave fluctuations in the sub-ion length scale regime. $k_{\bot }\unicode[STIX]{x1D70C}_{i}\gg 1$ , where the parallel phase velocity of the waves increases linearly with $k_{\bot }\unicode[STIX]{x1D70C}_{i}$ , with an approximate scaling (Howes et al. Reference Howes, Klein and TenBarge2014)

(5.1) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D714}}{k_{\Vert }}=v_{A}\frac{k_{\bot }\unicode[STIX]{x1D70C}_{i}}{\sqrt{\unicode[STIX]{x1D6FD}_{i}+2/(1+T_{e}/T_{i})}}.\end{eqnarray}$$

Thus, the increasingly fast dynamics of the fluctuations with decreasing scale appears to saturate the energy spectrum more rapidly than the wave period of the domain-scale kinetic Alfvén waves that drive the cascade. It is also worthwhile noting that the turbulent cascade is very efficiently generated by driving counterpropagating kinetic Alfvén waves at the domain scale.

In figure 8(b), we plot the evolution of the spectral index $\unicode[STIX]{x1D702}$ of the perpendicular magnetic spectrum $E_{B_{\bot }}$ as a function of normalized time $t/\unicode[STIX]{x1D70F}_{A}$ . To determine the value of the spectral index, we perform a fit of the perpendicular magnetic spectrum at each time using the form

(5.2) $$\begin{eqnarray}E_{B_{\bot }}\propto (k_{\bot }\unicode[STIX]{x1D70C}_{i})^{\unicode[STIX]{x1D702}}\exp [-k_{\bot }\unicode[STIX]{x1D70C}_{e}]\end{eqnarray}$$

over the range of scales $10\leqslant k_{\bot }\unicode[STIX]{x1D70C}_{i}\leqslant 105$ . As shown in panel (b), the rate of increase of the spectral index decreases, particularly for $t>t_{1}$ , finally saturating to a constant value for $t\geqslant t_{2}$ .

Figure 8. (a) Evolution of the energy in modes with $k_{\bot }\unicode[STIX]{x1D70C}_{i}=20.6$ versus normalized time $t/\unicode[STIX]{x1D70F}_{A}$ , showing power-law growth until $t_{1}/\unicode[STIX]{x1D70F}_{A}=0.14$ , and a slower growth until $t_{2}/\unicode[STIX]{x1D70F}_{A}=0.28$ . (b) Evolution of the spectral index, $\unicode[STIX]{x1D702}$ , where the spectrum is fit by $(k_{\bot }\unicode[STIX]{x1D70C}_{i})^{\unicode[STIX]{x1D702}}\exp (-k_{\bot }\unicode[STIX]{x1D70C}_{e})$ , showing saturation at time $t_{2}/\unicode[STIX]{x1D70F}_{A}=0.28$ . (c) Evolution of the scalar expansion parameter $\unicode[STIX]{x1D70E}$ , showing that the separation of field lines saturates only at $t_{2}/\unicode[STIX]{x1D70F}_{A}=0.28$ , with little noticeable change at $t_{1}/\unicode[STIX]{x1D70F}_{A}=0.14$ . A second identical run (with a different pseudo-random number sequence governing the forcing) shows the evolution is statistically repeatable.

5.2 Development and saturation of magnetic field line wander

To estimate the development and saturation of magnetic field line wander, we devise a new diagnostic, the expansion parameter $\unicode[STIX]{x1D70E}$ , derived in appendix A. This diagnostic yields a scalar quantity that parameterizes the separation of magnetic field lines in the turbulent magnetic field throughout the simulation domain.

In figure 8(c), we plot the evolution of expansion parameter $\unicode[STIX]{x1D70E}$ as a function of normalized time $t/\unicode[STIX]{x1D70F}_{A}$ . We plot the results for two independent simulations (red and blue), with all of the same numerical and physical parameters, but different pseudo-random number sequences governing the driving by the oscillating Langevin antenna. The expansion parameter $\unicode[STIX]{x1D70E}$ increases as a power law right up to the time $t_{2}/\unicode[STIX]{x1D70F}_{A}=0.28$ , at which point $\unicode[STIX]{x1D70E}$ saturates to a statistically steady value. Note that unlike in panel (a), where the rate of increase of amplitude of turbulent cascade shows a marked decrease at time $t=t_{1}$ , the power-law increase of the expansion parameter $\unicode[STIX]{x1D70E}$ in panel (c) shows little change at $t=t_{1}$ . Note also that the two independent simulations generate statistically similar results for the evolution of this expansion parameter $\unicode[STIX]{x1D70E}$ . Therefore, it appears for these simulations with plasma parameters $\unicode[STIX]{x1D6FD}_{i}=1$ and $T_{i}/T_{e}=1$ and turbulent amplitude $\unicode[STIX]{x1D712}\simeq 1$ , the magnetic field line separation appears to saturate on the same time scale as the turbulent magnetic energy spectrum.