Heuristic Description of Perpendicular Diffusion of Energetic Particles in Astrophysical Plasmas

First, we assume that the random walk of magnetic field lines is diffusive in the scenario of interest²

$$\begin{eqnarray}&&\left\langle {\left({\rm{\Delta }}x\right)}^{2}\right\rangle =2{\kappa }_{\mathrm{FL}}| z| .\end{eqnarray} \tag{ 4 }$$

If we assume that there are no collisions and no pitch-angle scattering, we can set z = vμt where we used the pitch-angle cosine μ. Combining this with Equation (4) and averaging over μ yields $\langle {({\rm{\Delta }}x)}^{2}\rangle =v{\kappa }_{\mathrm{FL}}t$, and thus

$$\begin{eqnarray}&&{\kappa }_{\perp }=\displaystyle \frac{v}{2}{\kappa }_{\mathrm{FL}}\end{eqnarray} \tag{ 5 }$$

corresponding to the FLRW limit. This limit is stable because if condition (3) is met, it does not alter the transport. This case is highly relevant in the limit of long parallel mean free paths corresponding to high particle energies (see Figures 1 and 2).

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If there is strong pitch-angle scattering the parallel motion is diffusive meaning that

$$\begin{eqnarray}&&\left\langle {\left({\rm{\Delta }}z\right)}^{2}\right\rangle =2{\kappa }_{\parallel }t.\end{eqnarray} \tag{ 6 }$$

Assuming that field lines are diffusive and particles follow field lines, we can combine Equations (4) and (6) to find

$$\begin{eqnarray}&&\left\langle {\left({\rm{\Delta }}x\right)}^{2}\right\rangle \approx 2{\kappa }_{\mathrm{FL}}\sqrt{2{\kappa }_{\parallel }t}.\end{eqnarray} \tag{ 7 }$$

The running perpendicular diffusion coefficient is then

$$\begin{eqnarray}&&{d}_{\perp }(t)=\displaystyle \frac{1}{2}\displaystyle \frac{d}{{dt}}\left\langle {\left({\rm{\Delta }}x\right)}^{2}\right\rangle \approx {\kappa }_{\mathrm{FL}}\sqrt{\displaystyle \frac{{\kappa }_{\parallel }}{2t}}\end{eqnarray} \tag{ 8 }$$

corresponding to subdiffusive transport. However, diffusion will be restored as soon as condition (3) is satisfied as discussed in the next paragraph. For slab turbulence, on the other hand, this condition is never satisfied due to ${{\ell }}_{\perp }=\infty$, and thus we find compound subdiffusion as the final state of perpendicular transport.

We now assume that diffusion is restored as soon as the particles scatter away from their original field lines. This happens as soon as the condition (3) is satisfied. We also assume that this happens after the particles travel the distance L_K in the parallel direction leading to ${\kappa }_{\perp }/{\kappa }_{\parallel }=\langle {({\rm{\Delta }}x)}^{2}\rangle /\langle {({\rm{\Delta }}z)}^{2}\rangle ={{\ell }}_{\perp }^{2}/{L}_{K}^{2}$. In order to eliminate L_K we use the field line diffusion coefficient ${\kappa }_{\mathrm{FL}}=\langle {({\rm{\Delta }}x)}^{2}\rangle /(2| z| )={{\ell }}_{\perp }^{2}/{L}_{K}$ yielding

$$\begin{eqnarray}&&{\kappa }_{\perp }\approx {\left(\displaystyle \frac{{\kappa }_{\mathrm{FL}}}{{{\ell }}_{\perp }}\right)}^{2}{\kappa }_{\parallel }.\end{eqnarray} \tag{ 9 }$$

Alternatively, one can replace the scale ℓ_⊥ therein so that

$$\begin{eqnarray}&&{\kappa }_{\perp }\approx \displaystyle \frac{{\kappa }_{\mathrm{FL}}{\kappa }_{\parallel }}{{L}_{K}}\end{eqnarray} \tag{ 10 }$$

in agreement with Equation (8) of Rechester & Rosenbluth (1978) as well as Equation (4) of Krommes et al. (1983). The quantity L_K is either called the Kolmogorov–Lyapunov length or just the Kolmogorov length (see, e.g., Krommes et al. 1983). However, here L_K is not an exponentiation length but a characteristic distance along the mean field at which transverse complexity becomes significant. Furthermore, Equations (9) and (10) were obtained without assuming collisions, and thus we call this result the collisionless Rechester & Rosenbluth (CLRR) limit. One can also obtain this by using a slightly different derivation. We assume that we find compound subdiffusion until the particles satisfy condition (3), which happens at the diffusion time t_d so that Equations (7) and (8) become $2{{\ell }}_{\perp }^{2}=2{\kappa }_{\mathrm{FL}}\sqrt{2{\kappa }_{\parallel }{t}_{d}}$ as well as ${\kappa }_{\perp }={\kappa }_{\mathrm{FL}}\sqrt{{\kappa }_{\parallel }/(2{t}_{d})}$. Combining the latter two equations in order to eliminate t_d yields again Equation (9). In order to evaluate this further, we consider two subcases, namely, small and large values of the Kubo number, respectively. For small Kubo numbers the field line diffusion coefficient is given by the quasi-linear limit

$$\begin{eqnarray}&&{\kappa }_{\mathrm{FL}}={L}_{\parallel }\displaystyle \frac{\delta {B}_{x}^{2}}{{B}_{0}^{2}}.\end{eqnarray} \tag{ 11 }$$

With this form Equation (9) becomes

$$\begin{eqnarray}&&{\kappa }_{\perp }\approx {\left(\displaystyle \frac{{L}_{\parallel }}{{{\ell }}_{\perp }}\right)}^{2}\displaystyle \frac{\delta {B}_{x}^{4}}{{B}_{0}^{4}}{\kappa }_{\parallel }\propto \displaystyle \frac{{{\ell }}_{\parallel }^{2}}{{{\ell }}_{\perp }^{2}}\displaystyle \frac{\delta {B}_{x}^{4}}{{B}_{0}^{4}}{\kappa }_{\parallel }\end{eqnarray} \tag{ 12 }$$

in agreement with the scaling obtained from the diffusive UNLT theory in Shalchi (2015). Furthermore, we find

$$\begin{eqnarray}&&{L}_{K}\propto \displaystyle \frac{{{\ell }}_{\perp }^{2}}{{{\ell }}_{\parallel }}\displaystyle \frac{{B}_{0}^{2}}{\delta {B}_{x}^{2}}.\end{eqnarray} \tag{ 13 }$$

For large Kubo numbers, on the other hand, we have

$$\begin{eqnarray}&&{\kappa }_{\mathrm{FL}}={L}_{U}\displaystyle \frac{\delta {B}_{x}}{{B}_{0}}\end{eqnarray} \tag{ 14 }$$

with the ultra-scale L_U. Equation (14) is either called the nonlinear or Bohmian limit and is similar compared to the field line diffusion coefficient obtained by Kadomtsev & Pogutse (1979). Therewith, Equation (9) becomes

$$\begin{eqnarray}&&{\kappa }_{\perp }\approx {\left(\displaystyle \frac{{L}_{U}}{{{\ell }}_{\perp }}\right)}^{2}\displaystyle \frac{\delta {B}_{x}^{2}}{{B}_{0}^{2}}{\kappa }_{\parallel },\end{eqnarray} \tag{ 15 }$$

and the Kolmogorov scale is ${L}_{K}=({{\ell }}_{\perp }^{2}{B}_{0})/({L}_{U}\delta {B}_{x})$.

Let us now assume that parallel transport is diffusive but magnetic field lines are still ballistic when the particles start to satisfy condition (3). Then we can derive $\langle {({\rm{\Delta }}x)}^{2}\rangle =\langle {({\rm{\Delta }}z)}^{2}\rangle \delta {B}_{x}^{2}/{B}_{0}^{2}\,=\,2{\kappa }_{\parallel }t\delta {B}_{x}^{2}/{B}_{0}^{2}$, and thus

$$\begin{eqnarray}&&{\kappa }_{\perp }=\displaystyle \frac{\delta {B}_{x}^{2}}{{B}_{0}^{2}}{\kappa }_{\parallel },\end{eqnarray} \tag{ 16 }$$

which Krommes et al. (1983) called the fluid limit.

The simplest case is obtained for the early times when parallel and field line transport are ballistic. In this case

$$\begin{eqnarray}&&\left\langle {\left({\rm{\Delta }}x\right)}^{2}\right\rangle =\displaystyle \frac{\delta {B}_{x}^{2}}{{B}_{0}^{2}}\left\langle {\left({\rm{\Delta }}z\right)}^{2}\right\rangle =\displaystyle \frac{{v}^{2}}{3}\displaystyle \frac{\delta {B}_{x}^{2}}{{B}_{0}^{2}}{t}^{2}\end{eqnarray} \tag{ 17 }$$

so that

$$\begin{eqnarray}&&{d}_{\perp }(t)=\displaystyle \frac{{v}^{2}}{3}\displaystyle \frac{\delta {B}_{x}^{2}}{{B}_{0}^{2}}t\end{eqnarray} \tag{ 18 }$$

corresponding to ballistic perpendicular transport. However, this is not a stable regime since we only find this type of transport before condition (3) is met.

We now consider a scenario where the transport is still ballistic when the particles start to satisfy condition (3). Therefore, we use Equations (17) and (18) to derive $2{{\ell }}_{\perp }^{2}={v}^{2}\delta {B}_{x}^{2}{t}_{d}^{2}/(3{B}_{0}^{2})$ as well as ${\kappa }_{\perp }={v}^{2}\delta {B}_{x}^{2}{t}_{d}/(3{B}_{0}^{2})$. Combining the latter two equations leads to

$$\begin{eqnarray}&&{\kappa }_{\perp }=\sqrt{\displaystyle \frac{2}{3}}v{{\ell }}_{\perp }\displaystyle \frac{\delta {B}_{x}}{{B}_{0}}.\end{eqnarray} \tag{ 19 }$$

A similar result can be derived from Equation (2) by assuming a ballistic perpendicular motion.

In order to determine which case is valid for which scenario, one needs to explore at which time a certain process takes place. In the parallel direction particles need to travel a parallel mean free path in order to get diffusive, and thus t_∥ = λ_∥/v = 3κ_∥/v². In the following we focus on the case of short λ_∥. For small Kubo numbers the field lines become diffusive for $| z| \approx {{\ell }}_{\parallel }$ and the corresponding time is ${t}_{{\rm{}}{\rm{F}}{\rm{L}}}\approx {{\ell }}_{\parallel }^{2}/(2{\kappa }_{\parallel })$. Then, on the other hand, if we assume that condition (3) is satisfied while the field lines are still ballistic, we have $2{{\ell }}_{\perp }^{2}=2{\kappa }_{\parallel }{t}_{\mathrm{Fluid}}\delta {B}_{x}^{2}/{B}_{0}^{2}$. For ${t}_{\parallel }\lt {t}_{\mathrm{Fluid}}\lt {t}_{{\rm{}}{\rm{F}}{\rm{L}}}$ the final state is the fluid limit because then we find that parallel transport becomes diffusive first and then we meet condition (3). If, on the other hand, ${t}_{\parallel }\lt {t}_{{\rm{}}{\rm{F}}{\rm{L}}}\lt {t}_{\mathrm{Fluid}}$ the field lines become diffusive before condition (3) is met. This means that we find compound subdiffusion first. At even later times condition (3) is eventually met and diffusion is restored. The corresponding diffusion coefficient is then the CLRR limit. Using the formulas for the times discussed above, this means that we find CLRR diffusion for ${\lambda }_{\parallel }^{2}\ll {{\ell }}_{\parallel }^{2}\ll {{\ell }}_{\parallel }{L}_{K}$ where the Kolmogorov length L_K is given by Equation (13). Thus, for λ_∥ ≪ ℓ_∥ we either find the fluid limit or CLRR diffusion. If additionally ℓ_∥ ≫ L_K we find the fluid limit, but for ℓ_∥ ≪ L_K we get CLRR diffusion. It follows from Equation (13) that L_K/ℓ_∥ ≈ K⁻² ≫ 1 meaning that for small Kubo numbers we always find CLRR diffusion. For large Kubo numbers similar considerations can be made.

A problem of the heuristic approach is that the obtained formulas are only valid in asymptotic limits. Since the two most important cases are CLRR and FLRW limits, we propose for the perpendicular mean free path defined via λ_⊥ = 3κ_⊥/v the formula

$$\begin{eqnarray}&&\displaystyle \frac{{\lambda }_{\perp }}{{{\ell }}_{\perp }}=\displaystyle \frac{9{{\ell }}_{\perp }}{16{\lambda }_{\parallel }}{\left[\sqrt{1+\displaystyle \frac{8{\kappa }_{\mathrm{FL}}{\lambda }_{\parallel }}{3{{\ell }}_{\perp }^{2}}}-1\right]}^{2}.\end{eqnarray} \tag{ 20 }$$

Equation (20) was chosen so that for ${\lambda }_{\parallel }\to 0$ we obtain Equation (9) and for ${\lambda }_{\parallel }\to \infty$ we get Equation (5). Note that Equation (20) does not contain the fluid limit given by Equation (16), and thus it has some limitations.

The results obtained here are sometimes not comparable to previous results. First of all there are cases such as slab or two-dimensional (2D) turbulence. In the former case condition (3) is never satisfied leading to compound subdiffusion as the final state. In the 2D case parallel transport is not diffusive (see, e.g., Arendt & Shalchi 2018) violating the second rule. In some work (see, e.g., Matthaeus et al. 2003; Shalchi et al. 2004) a flat spectrum at large scales was used for the 2D modes. For this type of spectrum the ultra-scale is not finite, also violating the second rule. In order to determine the form of κ_FL, we have used the Kubo number. However, in some turbulence models (see, e.g., Goldreich & Sridhar 1995) there is only one scale, and thus the Kubo number becomes K = δB_x/B₀, often called the Alfvénic Mach number. The arguments presented above are still valid.