2.1 Perturbations along field lines

Throughout, the particle gyroradius $r_{g}$ is regarded as small compared with the coherence scale $\ell _{c}$ of the turbulence and the turbulence is assumed magnetostatic, meaning that the characteristic eddy velocity $\langle \delta u^2\rangle ^{1/2}\ll v$, with $v=|\boldsymbol {v}\vert$ the particle velocity. To describe transport in a turbulence of structures, it is best to break the cascade into three different intervals of length scales $l$, as commonly done: the short-scale modes $l \ll r_{g}$, the large-scale ones $l \gg r_{g}$ and the resonant modes $l \sim r_{g}$. Short-scale modes will be found to exert a negligible influence, while large-scale modes both regulate adiabatically the evolution of the pitch angle of particles through mirroring effects (Cesarsky & Kulsrud Reference Cesarsky and Kulsrud1973; Chandran Reference Chandran2000a; Malyshkin & Kulsrud Reference Malyshkin and Kulsrud2001; Xu & Lazarian Reference Xu and Lazarian2020; Lazarian & Xu Reference Lazarian and Xu2021) and contribute to global transport through the random motion of field lines (e.g. Jokipii Reference Jokipii1971; Bykov & Toptygin Reference Bykov and Toptygin1993; Chuvilgin & Ptuskin Reference Chuvilgin and Ptuskin1993). The resonant modes with $l\sim r_{g}$ will provide a source of non-adiabaticity that leads to pitch-angle scattering. Here, the notion of ‘resonant’ means that the mode length scale $l\sim r_{g}$ and nothing else; in particular, no gyroresonance to a plane wave.

In general terms, as a particle propagates along a field line, it experiences magnetic perturbations of the form

(2.1)\begin{equation} \boldsymbol{b}\boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{B}\equiv m\boldsymbol{B} + \kappa B \boldsymbol{n}, \end{equation}

where $\boldsymbol {b}\equiv \boldsymbol {B}/B$ ($B\equiv \vert \boldsymbol {B}\vert$) represents the unit vector along the (total) magnetic field B. Together with $\boldsymbol {b}$, to which it is orthogonal, the unit vector $\boldsymbol {n}$ spans the osculating plane of the field line. Both scalars $\kappa$ and $m$ carry the dimension of an inverse length scale; $m\equiv B^{-1}\,\boldsymbol {b}\boldsymbol {\cdot }(\boldsymbol {b}\boldsymbol {\cdot } \boldsymbol {\nabla })\boldsymbol {B}$ characterizes the mirror force, while $\kappa \equiv B^{-1}\vert \boldsymbol {b}\times (\boldsymbol {b}\boldsymbol {\cdot } \boldsymbol {\nabla })\boldsymbol {B}\vert$ measures the local curvature of the magnetic field line.

The influence of large-scale modes $l\gg r_{g}$ can be followed in a guiding-centre description. It then reduces to the mirror force in a magnetostatic turbulence, $\partial _t \mu = - v (1-\mu ^2) m/2$, where $\mu \equiv \boldsymbol {v}\boldsymbol {\cdot }\boldsymbol {b}/v$ denotes the pitch-angle cosine of the particle. The particle momentum $p$ is exactly conserved in the magnetostatic approximation, while the magnetic moment $M\,\equiv \,(1-\mu ^2)p^2/2B$ is conserved to order $\mathcal {O}(r_{g}/l)$. The mirror forces influence $\mu$ in an adiabatic manner, decreasing it in regions of increasing magnetic field strength and vice versa. The overall process can lead to spatial diffusion, but it should not lead by itself to a scaling of the mean free path with $r_{g}$, given that the dominant effect is tied to the largest length scales on which most of the turbulent power is concentrated.

Unlike mirror-type fluctuations, field line curvature $\kappa$ is absent of both QLT and guiding-centre formalisms, at least for magnetostatic turbulence. Finite-$\kappa$ effects are contained to some degree in a wave description, yet the magnitude of $\kappa$ as predicted by QLT turns out too modest on the scales of interest (namely $l\sim r_{g}$) to play any role, see below. In the guiding-centre picture, $\kappa$ contributes to perpendicular drifts leaving the pitch angle unaffected because the corresponding modes on scales $l \gg r_{g}$ mostly renormalize the mean magnetic field that a particle then follows adiabatically, see § 2.2 below. However, it has long been recognized in the community of magnetospheric plasma physics that regions of sufficiently large curvature can lead to abrupt pitch-angle deflection (e.g. Speiser Reference Speiser1965; Gray & Lee Reference Gray and Lee1982; Birmingham Reference Birmingham1984; Büchner & Zelenyi Reference Büchner and Zelenyi1986, Reference Büchner and Zelenyi1989; Chen & Palmadesso Reference Chen and Palmadesso1986; Whipple et al. Reference Whipple, Northrop and Birmingham1986; Anderson et al. Reference Anderson, Decker, Paschalidis and Sarris1997; Delcourt et al. Reference Delcourt, Zelenyi and Sauvaud2000; Artemyev et al. Reference Artemyev, Orlova, Mourenas, Agapitov and Krasnoselskikh2013). That association with reconnecting current sheets provides an explicit connection to structures and intermittency.

Further below, it will be argued that such regions of large enough curvature exist in sufficient numbers to sustain parallel transport. Those regions emerge out of the non-Gaussian, power-law tails of the p.d.f. of the curvature strength and they are therefore directly related to the intermittent nature of the fluctuations. Being localized in space, they act sporadically on the particle trajectory. To capture their influence, one must therefore consider their p.d.f.s in their integrity, as measured in terms of strength and length scale; see Lemoine (Reference Lemoine2022) for similar developments in the context of particle acceleration. For the sake of simplicity, the discussion that follows focusses on such localized regions of high curvature, leaving aside the possible role of small-scale mirrors. As both mirrors and bends compose the perturbations seen along a field line – (2.1) above – the random fields $\kappa (\boldsymbol {x})$ and $m(\boldsymbol {x})$ are likely correlated, so that, by tracing the regions of large curvature, we will also trace those of intense, small-scale mirrors. As a matter of fact, regions of large $\kappa$ are commonly associated with low magnetic field strength, i.e. $\kappa \propto B^{-2}$ approximately (Yang et al. Reference Yang, Wan, Matthaeus, Shi, Parashar, Lu and Chen2019; Yuen & Lazarian Reference Yuen and Lazarian2020). The true physical cause responsible for jumps in the magnetic moment associated with a loss of adiabaticity is that the particle crosses a region in which the magnetic field is tangled on scales smaller than the particle gyroradius and this likely occurs in a region where both $\kappa$ and $m$ are significant. A key difference between $\kappa$ and $m$, however, is that the former is always positive, while the latter can take positive or negative values. As a particle crosses a wavepacket of mirror fluctuations, both positive and negative contributions tend to cancel each other, weakening the overall influence exerted on the particle. By contrast, a sharp bend of the magnetic field line will impart a net effect on the particle trajectory. The detailed contribution of magnetic mirrors on scales $l\sim r_{g}$ and their interplay with curved magnetic fields is thus deferred to a future study.

We make use of the short-hand notation $\kappa _l(\boldsymbol {x})$, which characterizes the value of the $\kappa$ field on scale $l$ at point $\boldsymbol {x}$ and which corresponds to the value that would be measured at $\boldsymbol {x}$ by coarse graining the turbulence on scale $l$, meaning filtering out the scales $< l$. In practice, we define $\boldsymbol {\kappa }(\boldsymbol {x})\equiv \boldsymbol {b}\boldsymbol {\cdot }\boldsymbol {\nabla }\boldsymbol {b}$ and $\boldsymbol {\kappa _l}(\boldsymbol {x})=G_l\star \boldsymbol {\kappa }$, where the $\star$ symbol stands for spatial convolution and $G_l(\boldsymbol {x})=(2{\rm \pi} l^2)^{-3/2}\exp (-x^2/2l^2)$ represents the coarse-graining kernel. In weak turbulence, the power spectrum $\mathcal {P}_{\kappa _l}$ of $\boldsymbol {\kappa _l}$ can then be written to first order in the perturbations in terms of the power spectrum of magnetic fluctuations $\mathcal {P}_B$ through

(2.2)\begin{equation} \mathcal{P}_{\kappa_l}(\boldsymbol{k}) \simeq \frac{1}{B^2}\vert{\tilde G}_l(\boldsymbol{k})\vert^2 k_\parallel^2 \mathcal{P}_{B}(\boldsymbol{k}). \end{equation}

Here, $B$ stands for the mean field strength on scales $\gg l$. The power spectra are normalized via $(2{\rm \pi} )^{-3}\int \,{\rm d}\boldsymbol {k} \mathcal {P}_B(\boldsymbol {k}) = \langle \delta B^2\rangle$, with $\langle \delta B^2\rangle$ the variance of magnetic field fluctuations, and $(2{\rm \pi} )^{-3}\int \,{\rm d}\boldsymbol {k} \mathcal {P}_{\kappa _l}(\boldsymbol {k})=\langle \boldsymbol {\kappa _l}^2\rangle$. The parallel wavenumber is defined as $k_\parallel = \boldsymbol {k}\boldsymbol {\cdot }\boldsymbol {B}/B$. In the following, we write $\langle \kappa _l\rangle \equiv \langle \boldsymbol {\kappa _l}^2\rangle ^{1/2}$ for simplicity. The general scaling of the characteristic curvature strength $\langle \kappa _l\rangle$ as a function of scale $l$ can then be obtained through direct integration of (2.2). For a Goldreich–Sridhar spectrum of the form $P_B(\boldsymbol {k}) \propto k_\perp ^{-10/3}g(k_\parallel /k_\perp ^{2/3}k_{\rm min}^{1/3})$, with $g(x)$ a function concentrated in $x\in [-1,+1]$ and of integral unity over ${\mathbb {R}}$ (Goldreich & Sridhar Reference Goldreich and Sridhar1995, Reference Goldreich and Sridhar1997), one obtains

(2.3)\begin{equation} \langle{\kappa_l}\rangle \sim \frac{\langle\delta B^2\rangle^{1/2}}{B} \left(\frac{l}{\ell_{c}}\right)^{{-}1/3}\,\ell_{c}^{{-}1}, \end{equation}

up to a prefactor of the order of unity. One would obtain a similar result for the mirror term on scale $l$, $\langle m_l\rangle$. Those quantities are here written to first order in the perturbations, neglecting intermittency effects. One nevertheless expects the above scaling $\langle \kappa _l\rangle \propto l^{-1/3}$ to remain approximately correct in the limit of large-amplitude turbulence, with the rescaling $B\rightarrow \langle B^2\rangle ^{1/2}$, the latter quantity representing the total mean field on large scales.

This definition of the perturbation introduces two length scales, one being $l$ of course, the other $\kappa _l^{-1}$ or $m_l^{-1}$ ($m_l$ mirror term on scale $l$). Here, $l$ is understood as the characteristic length scale over which those scalars depart from zero in some region of space. The dimensionless number $m_l l$ then characterizes the magnitude of the magnetic field perturbation at that point, while $\kappa _l l$ is related to the ratio of the curvature radius of the perturbed magnetic field line to the length scale over which the perturbation exists. A value $\kappa _l l \ll 1$ indicates a weak perturbation of the field line, while $\kappa _l l \gg 1$ rather corresponds to a sharp cusp. The outcome of an interaction of a particle of gyroradius $r_{g}$ with a bend of the magnetic field line thus depends on the relative hierarchy of the three length scales $r_{g}$, $\kappa _l^{-1}$ and $l$. Given that the characteristic fluctuation $\langle {\kappa _l}\rangle$ increases with decreasing scale, and that particles are sensitive to modes with $l\gtrsim r_{g}$ but insensitive to scales $l\ll r_{g}$, one can already anticipate that the maximum effect will result from interactions at $l\sim r_{g}$.

In the following, we focus on the evolution in time of the magnetic moment $M(t)$, rather than that of the pitch-angle cosine $\mu (t)$, because this allows us to isolate the contribution of small-scale structures, which affect both $M$ and $\mu$, from that of large-scale mirrors, which influence $\mu$ only (Kunz, Schekochihin & Stone Reference Kunz, Schekochihin and Stone2014). At constant $B$, $\Delta M/M = -\mu \,\Delta \mu /(1-\mu ^2)$, therefore magnetic moment violation evinces pitch-angle scattering, of course. Determining the mean free path to order-unity violation of $M$ thus provides a means to determine the mean free path to scattering by those regions of high curvature.

2.2 Interaction of a particle with a localized bend of the field line

To capture the role of curvature and coarse-graining scale, consider first the geometry of a magnetic reversal across a current sheet, characterized by a Harris profile $\boldsymbol {B}=B_0\{\kappa l_< {\rm tanh}(z/l_<),0,1\}$, including here a guide field along $z$ (e.g. Büchner & Zelenyi Reference Büchner and Zelenyi1986, Reference Büchner and Zelenyi1989; Chen & Palmadesso Reference Chen and Palmadesso1986). The curvature takes its maximum value $\kappa$ at $z=0$ and vanishes away from the current sheet on scales $\gg l_<$. Yet, in such a profile, the magnetic field line direction rotates by a finite angle $\theta$ across the sheet, with $\cos \theta = (1- \kappa ^2 l_<^2)/(1 + \kappa ^2 l_<^2)$, so that the influence of the curvature persists on large length scales $\gg l_<$, i.e. the perturbation takes the form of a kink rather than a cusp. In effect, coarse graining of the profile through a convolution with a Gaussian kernel of width $l\gg l_<$ – nothing changes if $l\ll l_<$ – does not modify the overall profile substantially, but renormalizes the length scales according to the substitutions $l_< \rightarrow l$ and $\kappa \rightarrow \kappa l_< /l $.

In § 3 below, we extract the statistics of field line curvature from a numerical simulation of incompressible MHD turbulence and categorize those statistics according to the coarse-graining scale $l$. The presence of current sheets of width $l_<$ with a profile similar to the above will be properly recorded on all scales $l \gg l_<$ with a coarse-grained curvature properly rescaled. For the time being, however, we focus on a bend laid on a single scale $l$. The above Harris profile can be modified to this effect, e.g. $\boldsymbol {B}=B_0\{1 + \kappa _l l({z}/{l} - 1) {\rm e}^{-({z^2}/{2 l^2})},0,1\}$. Here, the curvature vanishes on (parallel) length scales $\gg l$, while at $z=0$, it takes value $\kappa _l$. Integrating the motion of particles in such a magnetic geometry gives a behaviour illustrated in figure 1. We emphasize that this figure is a sketch, presented for illustrative purposes only. The form and the shape of the bend may vary, but to the extent that the physics is captured by the two scales $\kappa _l^{-1}$ and $l$, it captures the generic behaviour. Namely, for $l/r_{g}\ll 1$ (case a), the particle crosses the perturbation ballistically, without suffering significant deflection, while in the opposite limit $l\gg r_{g}$ (case c), the particle follows the field line adiabatically. In both cases, the normalized magnetic moment $\hat M(t)\equiv M(t)/M(0)$ remains approximately constant. On the contrary, when $l\sim r_{g}$ and $\kappa _l r_{g} \gtrsim 1$, the interaction becomes non-adiabatic and $\vert \Delta \hat M\vert \sim \mathcal {O}(1)$.

Figure 1. Simple cartoon illustrating the interaction of a particle (trajectory in blue) with a localized bend of the magnetic field line (in black), for three different cases: (a) small-scale mode $l\ll r_{g}$; (b) near-resonant mode $l\sim r_{g}$ and $\kappa _l r_{g}\gtrsim 1$; (c) large-scale bend $l\gg r_{g}$ and $\kappa _l r_{g}\ll 1$. In (a), the particle crosses the perturbation ballistically while, in (c), the particle follows the bend adiabatically; in both cases, the magnetic moment is approximately conserved, $\vert \Delta \hat M\vert \equiv \vert M(t)/M(0)-1\vert \ll 1$; in (b), the interaction gives rise to substantial non-adiabatic evolution of $M$, with $\vert \Delta \hat M\vert \sim \mathcal {O}(1)$.

The variation of the magnetic moment of a particle that crosses a region of high curvature $\kappa$, independently of the evolution of magnetic field lines on scales $\gg 1/\kappa$, is captured by the analysis of Birmingham (Reference Birmingham1984) in the limit $r_{g}< l$. The strength of the interaction depends critically on the parameter $\max (\kappa r_{g})$, because the magnetic moment changes by an amount

(2.4)\begin{equation} \frac{\Delta M}{M} \simeq \alpha \cos\varPhi \exp\left[-\frac{\beta}{\max\left(\kappa r_{g}\right)}\right], \end{equation}

with $\alpha$, $\beta$ coefficients determined in (21) of that reference, and $\varPhi$ the particle gyrophase at the location where $B$ finds its minimum. The interaction thus becomes non-adiabatic whenever $\max (\kappa r_{g})\gtrsim 0.1$ and it can either reduce or increase the magnetic moment, depending on the sign of the cosine factor. The exact value of the curvature does not play a significant role provided it exceeds that threshold.

As will be shown in § 3, regions with curvature $\kappa _l l \gtrsim 1$ are rare, all the more so at small scales $l\ll \ell _{c}$, since $\langle \kappa _l l \rangle \propto l^{2/3}$ (2.3). Ignoring the statistics beyond the root-mean-square (r.m.s.) $\langle \kappa _l \rangle$ as one would do in a quasilinear context, one would conclude that $\kappa _l r_{g} \lesssim (l/r_{g})^{-1/3}(r_{g}/\ell _{c})^{2/3}\ll 1$ for all $l\gtrsim r_{g}$, hence that curvature is everywhere weak and negligible. However, once we consider the full extent of the statistics of $\kappa _l$ (§ 3), we find a non-vanishing probability of observing $\kappa _l r_{g}\gtrsim 1$, with a maximum for $l\sim r_{g}$. Regarding small scales $l\ll r_{g}$, their contribution can be ignored, because the gyromotion of the particle leads to an effective coarse graining on scales $r_{g}$.

To model the influence of high-curvature regions, we consider in the following a simplified version of (2.4), namely $\Delta M/M = \pm 1$ for particles such that $\max (\kappa _l r_{g})\gtrsim 1$, and $\Delta M/M = 0$ otherwise. Since the local value of the gyroradius is what matters, in particular its maximum at maximum curvature, it proves important to distinguish $r_{g}$ from its mean value $\bar {r}_{g}$ measured in the r.m.s. magnetic field strength $\langle B^2\rangle ^{1/2}$. To this effect, we introduce a renormalized curvature that incorporates the dependence on the local strength of the magnetic field, as follows:

(2.5)\begin{equation} \hat\kappa_l(\boldsymbol{x})\equiv \kappa_l \frac{\langle B^2 \rangle^{1/2}}{B(\boldsymbol{x})}. \end{equation}

Hence, for a particle of average gyroradius $\bar {r}_{g}$, defined with respect to the r.m.s. $\langle B^2 \rangle ^{1/2}$, the product $\max (\kappa _l r_{g}) \simeq \bar {r}_{g} \max (\hat \kappa _l)$.

3.1 Particle trajectories

Figure 2 presents a selection of four different histories of $\mu (t)$, $\hat M(t)$ and $\hat \kappa (t)$ measured along particle trajectories. Those examples have been chosen because they are illustrative of the different types of histories that can be observed in that simulation. That sample is not meant to be representative, in the sense that one type of trajectories shown here may be encountered more frequently than another. Figure 2(a) shows one example in which $\mu (t)$ hardly evolves in time over 3 coherence lengths of the turbulence. The normalized curvature $\hat \kappa$ (orange line) does not take values larger than $\simeq 0.2$ here, and its variation takes place on scales of the order of the coherence scale $\ell _{c}$. The absence of noticeable variation of $\hat M$ is thus of no surprise. In panel (b), $\hat \kappa$ undergoes excursions up to values slightly larger than unity, yet on scales $\gtrsim 0.2\ell _{c}$ (judging from the full width at half-maximum), significantly larger than $\bar {r}_{g}$. Those excursions do not impact $\hat M$ significantly, up to a slight quiver during the interaction. Nonetheless, $\mu (t)$ evolves strongly, if adiabatically ($\Delta M/M\sim 0$) between the points of maximum $\hat \kappa$, which are likely associated with large-scale magnetic mirrors. Panels (c,d) show particles crossing more active regions. In (c), the magnetic moment undergoes two abrupt jumps, each of approximately ${\sim }20\,\%$, at respectively $ct/\ell _{c} = 1.1$ and $2.6$. Interestingly, the variation in magnetic moment occurs over a few gyroradii, as indicated by the oscillations, at locations where $\hat \kappa$ reaches values $\sim 10$. At those points, the conditions guaranteeing non-adiabaticity, see (2.4), $l\sim \bar {r}_{g}$, $\hat \kappa \bar {r}_{g}>1$ are fulfilled. Outside of those regions, the particle seems to be influenced, here as well, by large-scale mirrors. Finally, in panel (d), the particle undergoes one localized violent interaction that leads to a large jump in magnetic moment, $\Delta M/M\sim 1$. Not visible in this figure, the largest value of $\hat \kappa$ is here of the order of $3\times 10^3$ and the interaction takes place over a few gyroradii, ensuring a non-adiabatic transition.

Figure 2. Examples of histories of the pitch-angle cosine $\mu (t)$ (solid purple) as a function of time, drawn from the numerical simulation of incompressible MHD discussed in the text. In green solid line, the variation of the normalized magnetic moment $\hat M(t)-1\equiv M(t)/M(0)-1$; in orange, the $\log _{10}$ of the normalized curvature $\hat \kappa (\boldsymbol {x})\equiv \kappa (\boldsymbol {x}) \langle B^2\rangle ^{1/2}/B(\boldsymbol {x})$ measured at each point along the trajectories of those four particles. The gyroradius is such that $2{\rm \pi} \bar {r}_{g}/\ell _{c} = 0.1$.

3.2 Statistics of the field line curvature

In numerical simulations, the curvature $\kappa$ – as calculated without coarse graining – is observed to be distributed as a broken power law (Schekochihin et al. Reference Schekochihin, Cowley, Maron and Malyshkin2001; Yang et al. Reference Yang, Wan, Matthaeus, Shi, Parashar, Lu and Chen2019; Yuen & Lazarian Reference Yuen and Lazarian2020) with an index $s_\kappa \sim 2\rightarrow 2.5$ at large values of $\kappa$, for a p.d.f. ${ {\mathsf {p}}}_\kappa \propto \kappa ^{-s_\kappa }$. Interestingly, these studies have been performed in rather diverse conditions: Yang et al. (Reference Yang, Wan, Matthaeus, Shi, Parashar, Lu and Chen2019) discuss two- and three-dimensional incompressible MHD as well as two-dimensional kinetic simulations of large-amplitude turbulence ($\langle \delta B^2\rangle ^{1/2}/B\sim 1$), which yield $s_\kappa \simeq 2$ in two dimensions and $s_\kappa \simeq 2.5$ in three dimensions; Yuen & Lazarian (Reference Yuen and Lazarian2020) investigate compressible MHD turbulence with varying amplitudes to observe a trend of slightly softer spectra with decreasing turbulence level; finally, Schekochihin et al. (Reference Schekochihin, Cowley, Maron and Malyshkin2001) examines a wholly different configuration, i.e. the sub-viscous range of high-Pm turbulence, where the weak, small-scale magnetic field is stirred by a large-scale velocity field; the observed index, $s_\kappa \simeq 2$, agrees nicely with the theoretical model developed therein. Quite remarkably, recent in situ measurements of the statistics of field line curvature in the magnetosheath confirm those findings, in particular $s_\kappa \simeq 2.5$ at large curvature (Bandyopadhyay et al. Reference Bandyopadhyay, Yang, Matthaeus, Chasapis, Parashar, Russell, Strangeway, Torbert, Giles and Gershman2020; Huang et al. Reference Huang, Zhang, Sahraoui, Yuan, Deng, Jiang, Xu, Wei, He and Zhang2020). That property thus appears robust.

The coarse-grained variables $\kappa _l$ span a family of distributions ${ {\mathsf {p}}}_{\kappa _l}$, each characterized by the coarse-graining scale $l$. These distributions, more precisely the p.d.f.s of the dimensionless quantities $\kappa _l\,l$ (${ {\mathsf {p}}}_{\kappa _ll}$) and $\hat \kappa _ll$ (${ {\mathsf {p}}}_{\hat \kappa _ll}$) are displayed in figure 3. They have been obtained by direct sampling in the JHU-MHD simulation of incompressible turbulence, as follows: for a given coarse-graining scale $l$, we extract at most $(L_{\rm box}/l)^3$ data points and at each point $\boldsymbol {x}$, we compute $\kappa _l^{\rm num}(\boldsymbol {x}) = \vert \boldsymbol {b_l}\times \boldsymbol {b_l}\boldsymbol {\cdot }\boldsymbol {\nabla }\boldsymbol {B_l}\vert /\vert \boldsymbol {B_l}\vert$ where $\boldsymbol {B_l}(\boldsymbol {x})$ denotes the coarse-grained magnetic field on scale $l$ at $\boldsymbol {x}$; that quantity can be directly accessed using the numerical tools of the database. Sampling variance implies that data on $l\sim \ell _{c}$ (red points in figure 3) display a substantial level of shot noise. The grid size $\Delta x$ also affects the estimate, by introducing an effective maximal curvature scale $\kappa _l \sim 1/\Delta x$ at all values of $l$.

Figure 3. (a) Statistics of the curvature $\kappa _l$ coarse grained on scale $l$ (multiplied by $l$), as measured through direct sampling in the JHU-MHD simulation, for various coarse-graining scales, as indicated. Note that the $y-$axis shows $x { {\mathsf {p}}}_{\kappa _l l}(x)$ where $x\equiv \kappa _l l$. The dotted line shows a scaling ${ {\mathsf {p}}}_{\kappa _ll}(x)\propto x^{-2.5}$, for reference. (b) Same, for the normalized curvature $\hat \kappa _l \, l$. The dotted line shows a scaling ${ {\mathsf {p}}}_{\hat \kappa _ll}(x)\propto x^{-2.0}$, for reference. See text for details.

On scales $l\sim \ell _{c}$, the distribution of $p_{\kappa _l l}$ can be approximately described as Gaussian, with a mean value $\langle \kappa _l l \rangle \sim 1$, as could be expected on dimensional grounds. On smaller length scales, the p.d.f.s develop power-law tails, signalling intermittency. For such broken power-law distributions, the value of $x=\kappa _l l$, where $x\,{ {\mathsf {p}}}_{\kappa _ll}(x)$, finds its peak value provides a fair estimate of $\langle \kappa _l l\rangle$. That quantity is observed to scale approximately as predicted by (2.3), i.e. $(l/\ell _{c})^{2/3}$. That mean value would provide a faithful description of the p.d.f. if the latter were Gaussian, but it is not, and if one were to measure higher-order moments, they would depart sharply from Gaussian scalings. At large values, the statistics of $\kappa _l l$ indeed follow an approximate power law $p_{\kappa _ll}\propto (\kappa _ll)^{-2.5}$, while that of the normalized curvature $\hat \kappa _ll$ is harder, roughly $p_{\hat \kappa _ll}\propto (\hat \kappa _ll)^{-2.0}$. This is not surprising because $\hat \kappa _l \equiv \kappa _l \langle B^2\rangle ^{1/2}/B$, and $B$ and $\kappa _l$ are anti-correlated (Yang et al. Reference Yang, Wan, Matthaeus, Shi, Parashar, Lu and Chen2019; Yuen & Lazarian Reference Yuen and Lazarian2020). This anti-correlation between $\kappa _l$ and $B$ formalizes the intuition that magnetic field tension opposes the stretching and folding motions that would push the curvature to large values, i.e. that it is easier to bend a weak magnetic field than a strong one. More quantitatively, from $\kappa _l \propto B^{-2}$ and $p_{\kappa _ll}\propto (\kappa _ll)^{-2.5}$, one derives $\hat \kappa _l \propto \kappa _l/B \propto \kappa _l^{3/2}$, hence $p_{\hat \kappa _ll}\propto (\hat \kappa _ll)^{-2.0}$ indeed.

This observation finds a particular importance when one recalls that $\hat \kappa _l$, not $\kappa _l$, is the quantity that governs the strength of the interaction of a particle of mean gyroradius $\bar {r}_{g}$ with a sharp bend of the magnetic field line. The apparent anti-correlation between $\kappa$ and $B$ implies that particles see their gyroradius enlarged by a factor of a few or more when interacting with a localized bend of the magnetic field, which relaxes the constraint to achieve non-adiabatic interactions, $\hat \kappa _l l \gtrsim 1$ at $l\sim \bar {r}_{g}$, see (2.4).

Finally, those $\hat \kappa _l$ statistics allow us to calculate the mean free path to magnetic moment violation, by noting that the cumulative distribution function $P_{\hat \kappa _l l}(>x )$ provides the filling fraction of space where values $\hat \kappa _l l> x$ can be found. Accordingly, the quantity $l/P_{\hat \kappa _l}(>x )$ defines the mean free path to interaction with one such region. Hence

(3.1)\begin{equation} \lambda_{s} \equiv \frac{l}{\int_{1}^{+\infty}\,{\rm d}x {{\mathsf{p}}}_{\hat \kappa_l l}(x)}\quad \left(l\sim \bar{r}_{g}\right), \end{equation}

determines the mean free path $\lambda _{s}$ to interaction with order-of-unity variation of the magnetic moment according to (2.4). That mean free path, which is measured along the field line, neglects the influence of perpendicular drifts which, if sufficiently strong, might move the particle out of the region on a crossing time $\sim l/v$. This appears reasonable, as the magnitude of the drift velocity is $v_{D}\sim v r_{g}/(3L)$ for a mode on length scale $L> r_{g}$, so that the corresponding perpendicular displacement is of order $\sim (l/L)\,r_{g}/3$ with $l\sim r_{g}< L$. Such drifts nevertheless offer a potentially interesting source of perpendicular transport on long time scales.

In the present description, the global transport of the particles is controlled by a complex interplay of phenomena acting on different scales, in particular, confinement by large-scale mirrors, magnetic diffusivity associated with the meandering motion of field lines and scattering on highly curved regions. While the former two do not depend on particle rigidity, the latter becomes increasingly important at low rigidities, suggesting that those localized interactions that violate $M$ and thus regulate parallel transport will also regulate the general transport properties at sufficiently small rigidities. In effect, approximating ${ {\mathsf {p}}}_{\hat \kappa _l l}$ via

(3.2)\begin{equation} {{\mathsf{p}}}_{\hat \kappa_l l}(x)\,\sim\,\frac{1}{\langle\hat\kappa_l l \rangle}\left(\frac{x}{\langle\hat\kappa_l l \rangle}\right)^{-\alpha_{\hat\kappa}}, \end{equation}

with $\alpha _{\hat \kappa } \sim 2$ at $x\gg \langle \hat \kappa _l l \rangle$, one obtains

(3.3)\begin{equation} \lambda_{s}\simeq l\langle\hat\kappa_l l \rangle^{1-\alpha_{\hat\kappa}}\sim \bar{r}_{g}\left(\bar{r}_{g}/\ell_{c}\right)^{2(1-\alpha_{\hat\kappa})/3} \sim \ell_{c}^{0.7}\bar{r}_{g}^{0.3}. \end{equation}

The first equality derives from the power-law approximation of ${ {\mathsf {p}}}_{\hat \kappa _l l}$, while the second makes use of (2.3) and the third further assumes $\alpha \simeq 2$. This scaling is written as $r_{g}^{0.3}$ and not $r_{g}^{1/3}$ to emphasize the uncertainty related to the value of $\alpha$. A rigorous evaluation of $\lambda _{s}$, based on its definition (3.1) and the statistics measured in the JHU-MHD simulations (figure 3) confirms the above approximate scaling $\lambda _{s}\propto r_{g}^{0.3}$, see figure 4. It should be clear that the above result has nothing to do with the usual quasilinear result $\lambda _{s} \propto \bar {r}_{g} [k \langle \vert \delta B_k\vert ^2\rangle ]^{-1}$ at $k\sim \bar {r}_{g}^{-1}$ and $\langle \vert \delta B_k\vert ^2\rangle \propto k^{-5/3}$ (Berezinskii et al. Reference Berezinskii, Bulanov, Dogiel and Ptuskin1990), even though it shares a similar scaling with momentum.Footnote ³ Interestingly, in the conditions of incompressible, strong turbulence of the JHU-MHD simulation, the quasilinear calculation predicts a very mild scaling of $\lambda _{s}$ with momentum (Chandran Reference Chandran2000b; Yan & Lazarian Reference Yan and Lazarian2002).

Figure 4. Numerical evaluation of the mean free path for order-of-unity violations of the magnetic moment through scattering, as defined in (3.1) and using the statistics of $\hat \kappa _l l$ extracted from the JHU-MHD simulation. This mean free path, written here in units of $\ell _{c}$, is plotted as a function of rigidity $2{\rm \pi} r_{g}/\ell _{c}$, using the correspondence $l\sim r_{g}$ in defining the threshold beyond which $\hat M$ can change by an order-of-unity factor, as expressed by (2.4). The dotted grey line indicates a scaling $\propto \bar {r}_{g}^{0.3}$ for reference.

It must also be emphasized that figure 4 does not constitute a measurement of the mean free path to scattering vs rigidity by itself. It rather indicates what scaling one would expect on the basis of the theoretical model proposed in § 2.2, given the p.d.f. of the normalized curvature extracted from the JHU-MHD simulation. The following section extracts, however, this mean free path for one value of the rigidity through particle tracking in that same simulation.

3.3 Magnetic moment diffusion

Figure 5 presents the p.d.f. of $\hat M$, as measured from the sample of particles propagated through the turbulence volume. As indicated earlier, those particles have all been injected with a unique pitch-angle cosine $\mu (0)=0.5$, a unique rigidity $2{\rm \pi} \bar {r}_{g}/\ell _{c}=0.1$, albeit at different locations drawn at random; all particles are relativistic with $v\simeq c$. The trajectory of those particles has been followed for $3\ell _{c}/c$ by integrating the equation of motion using a Boris pusher, see Lemoine (Reference Lemoine2022) for details. Recalling that the length of the simulation volume is $L_{\rm box}\simeq 7 \ell _{c}$, the particles cannot cross the simulation box during the integration. Periodic boundary conditions are applied on all three sides of the simulation cube.

Figure 5. (a) The p.d.f. of the normalized magnetic moment $\hat M$ (times $\hat M$) at various times, as measured from the sample of particles propagated through the turbulence volume. All particles are injected with a same pitch-angle cosine $\mu (0)=0.5$ and rigidity $2{\rm \pi} r_{g}/\ell _{c}=0.1$. Although the sampling noise becomes substantial at large excursions of $\hat M$, the overall trend can be properly captured thanks to the large number of bins. (b) Cumulativedistribution function for $\vert \Delta \hat M\vert = \vert \hat M-1\vert$, emphasizing the deviations of $\hat M$ from its initial value ($=1$). This cumulative distribution function shows that, by $c t/\ell _{c}\simeq 2$, approximately 20 %–30 % of particles have seen their magnetic moment change by an order of unity or more. The slight glitch apparent at $\vert \Delta \hat M\vert =1$ results from the fact that $\hat M$ is a positive quantity, which makes $\Delta \hat M$ bounded from below by $1$.

The p.d.f. shown in figure 5(a) broadens in time through the development of a power-law tail. This signals encounters with intermittent, localized regions of high curvature. These power-law tails are indeed reminiscent of those observed in the momentum distribution of particles accelerated in strong turbulence, for which it was shown, on the basis of a large deviation argument, that rare interactions of substantial energy gain generically lead to power-law behaviour for the distribution, quite unlike a Brownian motion characterized by frequent interactions of modest energy change, which rather lead to Gaussian type distributions (Lemoine Reference Lemoine2021).

Figure 5(b) presents the cumulative distribution function of $\vert \Delta \hat M\vert \equiv \vert \hat M-1\vert$, to offer a closer look on the statistics of the deviations of $\hat M$. The connection between the p.d.f. shown in the left panel and the cumulative distribution function is not straightforward, because the p.d.f. derives from the sample of values of $\hat M$ at a given time, while the cumulative distribution measures the fraction of particles that have experienced a change of $\hat M$ by a given amount in the time interval $[0,t]$. This cumulative distribution function shows that, by $ct/\ell _{c}\simeq 1$, approximately $15\,\%$ of particles have suffered a order-of-unity change in $\hat M$; this fraction increases up to ${\simeq }25\,\%$ at $ct/\ell _{c}\simeq 2$.

Finally,figure 6 displays the evolution in time of the variance of the distribution shown in figure 5(a), to probe the possible diffusive regime of $\hat M$. While the values at early times are dominated by numerical noise, associated with the narrow core of the distribution of $\hat M$, a clear diffusive regime sets in at $ct/\ell _{c}\gtrsim 0.1$ with a scaling regime $\langle \Delta \hat M^2\rangle \simeq 0.8 ct/\ell _{c}\gtrsim 0.1$, corresponding to a diffusion coefficient $D_{\hat M}\simeq 0.4 c/\ell _{c}$. When translated into a mean free path to $\hat M$-violation, as $\lambda _{s}$, the corresponding value is $2.5\ell _{c}$, i.e. a factor of a few larger than the theoretical value shown in figure 4 for that rigidity $2{\rm \pi} \bar {r}_{g}/\ell _{c}=0.1$. This lends overall consistency to the picture presented here.

Figure 6. Evolution of the variance of $\Delta \hat M$ in time, as measured from the sample of tracked particles through the turbulence volume. The values at $ct/\ell _{c}\lesssim 0.1$ are dominated by numerical noise, whose magnitude is of the order of $\sim 0.1$, while the transition to a diffusive regime at $ct/\ell _{c}\gtrsim 0.1$ is manifest. The dotted line in that region indicates a linear (diffusive) scaling $\langle \Delta \hat M^2\rangle \simeq 0.8 ct/\ell _{c}$.