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Cosmic Ray Transport with Magnetic Focusing and the"Telegraph"model

Cosmic rays (CR), constrained by scattering on magnetic irregularities, are believed to propagate diffusively. But a well-known defect of diffusive approximation whereby some of the particles propagate unrealistically fast has directed interest towards an alternative CR transport model based on the"telegraph"equation. However, its derivations often lack rigor and transparency leading to inconsistent results. We apply the classic Chapman-Enskog method to the CR transport problem. We show that no"telegraph"(second order time derivative) term emerges in any order of a proper asymptotic expansion with systematically eliminated short time scales. Nevertheless, this term may formally be\emph{ converted} from the \emph{fourth} order hyper-diffusive term of the expansion. But, both the telegraph and hyperdiffusive terms may only be important for a short relaxation period associated with either strong pitch-angle anisotropy or spatial inhomogeneity of the initial CR distribution. Beyond this period the system evolves diffusively in both cases. The term conversion, that makes the telegraph and Chapman-Enskog approaches reasonably equivalent, is possible only after this relaxation period. During this period, the telegraph solution is argued to be unphysical. Unlike the hyperdiffusion correction, it is not uniformly valid and introduces implausible singular components to the solution. These dominate the solution during the relaxation period. As they are shown not to be inherent in the underlying scattering problem, we argue that the telegraph term is involuntarily acquired in an asymptotic reduction of the problem.


Preliminary Considerations
The problem addressed here is fundamental but not new to the cosmic ray (CR) transport studies. It can be formulated very plainly: How to describe CR transport by only their isotropic component, after the anisotropic one has been suppressed by scattering on magnetic irregularities? Suppose the angular distribution of CRs is given by the function f (µ,t, z) obeying an equation from which the rapid gyro-phase rotation is already removed (drift approximation, e.g., Vedenov et al. 1962;Kulsrud 2005) (1) Here z is the local coordinate along the ambient magnetic field, µ is the cosine of the particle pitch angle, and D is the pitch angle diffusion coefficient. Now, we make the next step in simplifying the transport description and seek an equation for the pitch-angle averaged distribution The basic solution to this problem has been known for at least half a century (e.g., Jokipii 1966 and references therein). To the leading order in 1/D (assuming the characteristic scale and time of the problem being longer than particle mean free path and collision time) it can be obtained straightforwardly by averaging eq.(1) and substituting ∂ f /∂ µ ≪ f 0 from eq.(1) as: Thus, the following diffusion equation for f 0 results The only questionable step in this derivation was our neglect of ∂ f /∂t compared to v∂ f /∂ z in eq.(2). It is somewhat justified a posteriori by the small parameter1/D ≪ 1 in the final result given by eq.(3), making ∂ f /∂t hopefully small. On the other hand this is true for ∂ f 0 /∂t but not necessarily for ∂ f /∂t since it may contain also the rapidly decaying anisotropic partf = f − f 0 in the initial distribution. For Dt 1, however,f must die out and neglecting ∂ f /∂t appears plausible for the long-time evolution of CR distribution.
However convincing the justification, the CR diffusion model encounters the problem of a superluminal, or simply too fast particle propagation. Although rather common for diffusive models, the problem is not important if the number of such particles is small. There are cases, however, such as that of the propagation of ultra high-energy cosmic rays, where this problem must be addressed (Aloisio et al. 2009). Various attempts, starting as early as in 60s, e.g., (Axford 1965), have been made to devise a better transport equation for CRs. Unfortunately, in our view, they lack mathematical rigor and clarity and often lead to incorrect results.
In the most recent such treatment, due to Litvinenko & Schlickeiser (2013), a higher order in 1/D ≪ 1 term was included by retaining ∂ f /∂t, dropped in the simplest derivation above. This strategy gave rise to an additional ∂ 2 f 0 /∂t 2 -term in the "master" equation. This additional term transforms eq.(3) into a "telegraph" equation: From a mathematical perspective, asymptotic reduction schemes should be continuable to higher orders. Proceeding to higher orders using the above approach will apparently generate an infinite series of higher time derivatives, thus introducing a series of shorter time scales into the master equation. Note that the goal should be the opposite, i.e., to eliminate short-time and transient phenomena in favor of the long-time evolution. At a minimum, the treatment of Litvinenko & Schlickeiser (2013) appears to encounter a truncation problem which is also present in the earlier work due to Earl (1973). The approach there was based on the decomposition of eq.(3) using the eigenfunctions of the angular operator on its r.h.s. This approach has been extended and described in detail by Schwadron & Gombosi (1994).
Leaving aside the validity of its derivation for a moment, the telegraph equation has both advantages and disadvantages compared to the plain diffusion equation introduced, e.g., by Jokipii (1966). One obvious advantage is that if the initial conditions make the second and the third term dominate over the first one (at least in the early phase of evolution), CRs must propagate nearly ballistically and if τ has a proper value, their speed may also be realistic. For example, this speed was derived in (Earl 1973) to be v/ √ 3. This is just the rms velocity projection of an isotropic CR distribution on z, which appears to be a potentially plausible solution to the problem of superluminal propagation.
The disadvantage of eq.(4) is that it is no longer an evolution equation and requires the time derivative ∂ t f 0 as an initial condition. Although this can be inferred from the angular distribution at t = 0 (if available) with recourse to the primary equation (1), the description of CR transport using eq.(4), by contrast with eq.(3), is not self-contained. We will also show below, that the transport coefficient τ in this equation, that was iteratively obtained from eq.(1) in (Litvinenko & Schlickeiser 2013) and by truncation of eigenfunction expansion in (Earl 1973), is not consistent with the regular asymptotic treatment and incorrect. The difference is due to the ∂ 4 f 0 /∂ z 4 -term, neglected in eq.(4) but contributing to the same order as the τ-term. Being converted into ∂ 2 f 0 /∂t 2 -form by using the leading order approximation given by eq.(3), it comes out with a markedly different coefficient τ. In addition, the τ-term in eq.(4) is sub-dominant compared to the other two terms for Dt 1, thus representing a transient process in the CR transport. Strictly speaking, it should be omitted in the asymptotic transport description along with the small hyper-diffusion term ∼ ∂ 4 f 0 /∂ z 4 , particularly if the term ∼ ∂ 3 f 0 /∂ z 3 does not vanish or when the magnetic focusing is present. These effects contribute to the CR transport at a lower order of approximation.
There are at least two reasons why we undertake a derivation of master equation to higher (fourth) order approximation using the Chapman-Enskog method. First, as we explained above, it is necessary to clarify the role of the telegraph τ-term entertained in the literature as an allegedly viable alternative to the standard diffusion model. Second, it is necessary to obtain the transport coefficients valid for arbitrary D (µ), that is for an arbitrary spectrum of the scattering magnetic fluctuations. In particular, the diffusion equation in eq.(3) supplemented by a convective term u (z) ∂ f 0 /∂ z for the case of the bulk fluid motion with velocity u, has been the main tool in studies of diffusive shock acceleration (DSA). In most DSA applications, it is crucial to allow for an arbitrary fluctuation spectrum and its dependence upon f 0 . This dependence directly affects the particle spectrum and acceleration time.
In the next section, the basic transport equation with magnetic focusing is introduced and the shortcomings of the reduction scheme based on direct iteration are demonstrated. The appropriate asymptotic method is elaborated in Sec.3. Apart from what we discussed above regarding the telegraph equation, the objective of Sec.3 is to create a framework suitable also for nonlinear (e.g., Ptuskin et al. 2008;Malkov et al. 2010b) and quasi-linear (Fujita et al. 2011;Malkov et al. 2013) versions of CR transport which are important for both the DSA and subsequent escape of CR. In these settings, the CR pressure is high enough to strongly modify at least the pitch-angle diffusion coefficient D and possibly the shock structure itself (Malkov et al. 2010b). In Sec.3.1 the implications of our results for the telegraph model are discussed and Sec.4 concludes the paper.

CR Transport Equation and its Asymptotic Reduction
Energetic particles (e.g., CRs) in a magnetic field, slowly varying on the particle gyro-scale, are transported according to the following gyro-phase averaged equation, e.g. (Vedenov et al. 1962;Jokipii 1966;Kulsrud 2005) Here v and µ are the particle velocity and pitch angle, z points in the local field direction, σ = −B −1 ∂ B/∂ z is the magnetic mirror inverse scale and ν is the (constant) pitch angle scattering rate, while D (µ) ∼ 1 depends on the spectrum of magnetic fluctuations. As the fastest transport is assumed to be in µ, we introduce the following small parameter where l is the characteristic scale of the problem. It may be determined by B (z) variation, CR source, or propagation distance. By measuring time in ν −1 , z in l, and simply replacing σ l → σ ∼ 1, the above equation is transformed as follows A suitable scheme for asymptotic reduction of the above equation using ε ≪ 1 is due to Chapman and Enskog, suggested in development of the earlier ideas by Hilbert (a good discussion of the history of this method with mathematical details is given by Cercignani 1988). Originally, it was applied to Boltzmann equation in a strongly collisional regime. Similar approaches have been used in plasma physics, e.g., in regards to the hydrodynamic description of collisional magnetized plasmas (Braginskii 1965) and the problem of run-away electrons (Gurevich 1961;Kruskal & Bernstein 1964).
Regardless of the asymptotic scheme, eq. (7) suggests to seek f as a series in ε so that˜ f = f n>0 = 0. The equation for f 0 , which is the main ("master") equation of the method, takes the following form We see from this equation that, similarly to the case of Lorentz's gas in an electric field (Gurevich 1961;Kruskal & Bernstein 1964), f 0 depends on the "slow time" t 2 = ε 2 t rather than on t. Indeed, the two problems are similar in that they describe a secular expansion of particles in phase space. The expansion occurs in z-direction for CR diffusion and in energy for runaway electrons. It is driven by a rapid isotropization in pitch angle plus the convection in z-direction, or acceleration in the electric field direction, for the CR transport and electron runaway, respectively.
The slow dependence of f 0 on time in eq.(10) may suggest to attribute the time derivative term in eq.(7) to a higher order approximation (thus moving it to the r.h.s.). Such ordering has been employed by Litvinenko & Schlickeiser (2013) and the term ∝ ∂ 2 f 0 /∂t 2 has been produced in eq.(10). Note, that a series of terms with progressively higher time derivatives, multiplied by small parameters, would have been generated on the r.h.s of eq.(10), had this scheme been continued to higher orders. These terms would be responsible for the initial transport relaxation occurring in progressively shorter times. They would be associated with small scales in the initial angular distribution. These transient phenomena will be removed using the asymptotic reduction scheme in the next section.
Unlike f 0 ,f in eq.(8) does depend on t as on a "fast" time. Therefore, it is illegitimate to attribute the first term on the l.h.s of eq.(7) to any order of approximation different from that of the second term, notwithstanding its fast decay for t 1. Thus, using eq.(7) we must apply the following ordering The above expansion scheme is sufficient to recover the leading order of f 0 evolution from eq.(10) by substituting there ∂ f 1 /∂ µ ≈ − (2D) −1 ∂ f 0 /∂ z, obtained from the last equation for t 1. However, this scheme is not suitable for determining f n for n ≥ 2 to submit to eq.(10). Indeed, as it may be seen from eq.(11), the solubility condition for f 2 at t ≫ 1 is . This is clearly too strong a restriction. The reason for this inconsistency of the direct asymptotic expansion is that f 0 depends on the slow time t 2 = ε 2 t, so that the multiple time scales are involved in the problem. The Chapman-Enskog method is known to be the right way to treat this situation, on which we elaborate in the next section.

Chapman-Enskog Expansion
As we have seen, the reduction of the original CR propagation problem given by eq.(7) to its isotropic part requires a multi-time asymptotic expansion. In the classic Chapman-Enskog method the operator ∂ /∂t is expanded instead. Perhaps more customary today and equivalently is to introduce a hierarchy of formally independent time variables (e.g., Nayfeh 1981) t → t 0 , t 1 , . . ., so that Instead of eq.(11), from eq. (7) we have where the conditions f n<0 = 0 are implied. The solution of this equation should be sought in the following form f n = f n (t 2 ,t 3 , . . . ; µ) +f n (t 0 ,t 1 , . . .; µ) wheref n and f n satisfy, respectively, the following two equations: and The solution forf n takes the formf and it can be easily found for an arbitrary n by expanding both sides of eq.(15) in a series of eigenfunctions of the diffusion operator on its l.h.s.: For D = 1, for example, ψ k are the Legendre polynomials with λ k = k (k + 1), k = 0, 1, . . .. The time dependent coefficients C (n) k are determined by the initial values off n (anisotropic part of the initial CR distribution) and the r.h.s. of eq.(15), that depends onf n−1 , obtained at the preceding step. It is seen, however, that allf n exponentially decay in time for t 1 and we may ignore them 1 as we are interested in evolving the system over times t ε −2 ≫ 1 and longer. Starting from n = 0 and using eq.(13), for the slowly varying part of f we have The solubility condition for f 1 (obtained by integrating both sides of eq.[13] in µ) also gives a trivial result so the last two conditions make the decomposition in eq.(14) consistent with eqs. (18-19), since from eq.(16) with n = 1 we havef and, thus bothf 0 andf 1 are, indeed, independent of t 0 and t 1 . We have introduced the function W (µ) here by the following two relations The solubility condition for f 2 yields the nontrivial and well-known (e.g., Jokipii 1966) result, which is actually the leading term of the ∂ f 0 /∂t expansion in ε ≪ 1 The solubility conditions for f 3 , f 4 , ... will generate the higher order terms of our expansion which, after some algebra, can be manipulated into the following expressions for the third and fourth orders of approximation We have denoted and U ′ = ∂U/∂ µ. The pitch-angle diffusion coefficient D (µ) and magnetic focusing σ are considered z-independent for simplicity, a limitation that can easily be relaxed by re-arranging the operators containing ∂ /∂ z in eq.(24). We can proceed to the higher orders of approximation ad infinitum since terms containing 1 − µ 2 ∂ f n /∂ µ can be expressed through f n−1 , f n−2 , .... According to eqs. (18-19), of interest is the evolution of f 0 on the time scales t 2 1 or t ε −2 so, as we already mentioned, the contributions off n (µ) to all the solubility conditions, similar to those given by eqs. (22-24), have to be dropped (as they become exponentially small) and onlyf n (µ)contributions should be retained. Using eqs. (18)(19)(22)(23)(24) to form the combinations ε n ∂ n f 0 /∂t n and summing up both sides, on the l.h.s. of the resulting equation we simply obtain ∂ f 0 /∂t (see eq. [12]). Therefore, the evolution of f 0 up to the fourth order in ε takes the following form where ∂ ′ z = ∂ z + σ and ∂ ′′ z = ∂ z + σ /2. The above algorithm allows one to obtain the master equation to an arbitrary order in ε. By construction, in no order of approximation will higher time derivatives emerge, as has been intentionally devised in the Chapman-Enskog method. We have truncated this process at the fourth order, ε 4 . As we show in the next subsection, this is the lowest order required to clarify the origin of the telegraph equation. Higher order terms can in principle be calculated at the expense of a more involved algebra. We argue below that such calculations would be of no avail.

Telegraph Term and Arguments against it
By contrast to the telegraph equation given by eqs.(4-5), that has been derived with no order control, eq.(25) is derived to the fourth order in ε. Yet, it has no second order time derivative and this challenges the derivation of eq.(4). Various versions of the telegraph equation have been obtained either without clear ordering, e.g., (Litvinenko & Schlickeiser 2013), or using a specific and simple D (µ), e.g. (Schwadron & Gombosi 1994). In an earlier treatment by (Earl 1973), an eigenfunction expansion was truncated, also with no order control. In most of these treatments care has not been exercised to systematically eliminate the short time scales which are irrelevant to the long-time evolution of the isotropic part of the CR distribution. In principle, this is acceptable if the reduction scheme is based on an exact solution of the original equation, to include all required orders of approximation into the master equation. Such approach appears to be pursued in (Schwadron & Gombosi 1994). The drawback here is a strongly limited choice of possible D (µ) and its f 0 -independence (see a brief discussion in Sec.4).
To understand the differences between the telegraph eq.(4) and eq.(25) obtained using the Chapman-Enskog multi-time decomposition, we simplify eq.(25) to remove noncritical terms. First, the telegraph coefficient τ does not contain magnetic focusing effect, so we may set σ = 0 in eq.(25) for the purpose of this section. Second, τ does not vanish in the case of symmetric pitch-angle scattering coefficient, that is D (−µ) = D (µ). This particular choice removes the term ∼ ε 3 containing ∂ 3 z in eq.(25). It is not included in the telegraph equation derived for magnetic focusing by Litvinenko & Schlickeiser (2013), anyway. Using these simplifications and the slow time T = ε 2 t/4, eq.(25) rewrites: where K is the hyper-diffusion coefficient To the same order in ε ≪ 1, the last equation can indeed be rewritten in form of a telegraph equation, cf. eq.(4): However, the comparison of eqs.(26-27) with eqs.
(4-5) shows that, apart from the ε 2 factor, originating from normalization, the telegraph coefficient in eq.(28) differs from the respective coefficient τ given in eq.(5). The reason is that eq.(4) has been obtained in, e.g., (Litvinenko & Schlickeiser 2013) by iterations not including all the fourth order terms which contribute to the telegraph term. A technically rigorous treatment by Schwadron & Gombosi (1994) gives an expression for τ, likely to be consistent with our result here but, as it was obtained for a power-law D (µ), a side by side comparison would require further steps, not worth taking here.
More importantly, the telegraph term in eq.(28) belongs to the type of terms with small parameter at highest derivative. The role of such terms is well known from other multi-scale problems, most notably the boundary layer problems. These terms correspond to corrections that are insignificant in a smooth part of the solution but become crucial near and inside boundary layers, thus determining their structure and scale. In the context of the telegraph equation, the boundary layer translates into the initial relaxation phase of the CR distribution 2 . However, as we have seen in Sec.3, this relaxation is primarily associated with the CR anisotropy, that is with f n , which we have shown to decay rapidly during the initial phase at t 1. This is exactly the characteristic time scale (T ∼ ε 2 ) when the telegraph term (second) on the r.h.s. of eq.(28) may come into play. To realize such scenario, one must start with an initial condition that either leads to a subsequent rapid relaxation in time or is strongly inhomogeneous. These two options become obvious after returning to the original time variable in eq.(28): Namely, in the limit ε → 0 there are two modes, of which the first being f 0 = f 0 (z). This is the main diffusion mode that slowly evolves in time when 0 < ε ≪ 1, and, as we are interested in the evolution over the time scales t ε −2 , the telegraph term becomes sub-dominant and can be discarded altogether. The second mode corresponds to a rapid decay of the initial distribution ∼ exp (−γt), at the rate γ = κ 2 /4K ∼ 1. It is associated with the decay of a strong initial anisotropy. The latter process cannot be adequately described by the equation for f 0 alone, and the telegraph term must be understood as a "ghost" term reflecting a rapid decay of an anisotropic part of the initial distribution. It follows that the rapidly changing partf in the decomposition in eq.(14) needs to be retained in the short-time analysis on an equal basis with f 0 .
The next possibility to make the telegraph term work is to start with a highly inhomogeneous initial distribution, so that the r.h.s. of eq.(29) is O (1) despite of small ε. The telegraph term becomes temporarily (t K/κ 2 ) dominant and the initial distribution indeed propagates ballistically rather than diffusively away from the CR source with the telegraph mode speeds, C = ±εκ 3/2 /4 √ K. However, the scale of initial profile must be shorter than the CR m.f.p. which, again, invalidates the equation in use. Besides, this case is not much different from the previous one as the sharp spatial inhomogeneity almost automatically results in strong angular anisotropy of the CR distribution. Therefore, the telegraph regime has virtually no validity range.
We conclude this section by adding yet another argument in disfavor of the telegraph equation. A consistent asymptotic reduction method must be continuable to infinity in powers of small ε. The Chapman-Enskog scheme clearly is. The outcome will be a series of terms ∼ ε n ∂ n z f 0 on the r.h.s. of eq.(25). To solve the resulting equation, only the initial distribution f 0 (0, z) is needed, as the equation remains evolutionary. Turning to the "telegraph" version of this equation we realize that progressively higher time derivatives, with higher powers of ε, will comprise the asymptotic series. The resulting equation will not be evolutionary and an arbitrarily large set of initial time derivatives of f 0 would then be needed to solve the initial value problem. These data can be extracted only from the full anisotropic distribution with recourse to the full (anisotropic) equation. Therefore, the telegraph equation is also not self-contained and unrealistic for practical use.

Summary and Conclusions
Using the Chapman-Enskog method, we have extended the CR diffusion equation with magnetic focusing to the fourth order in a small parameter ε = λ /l (CR mean free path to the problem scale). This clarifies the nature of the so called telegraph transport regime, widely publicized in the literature as a promising alternative to the diffusive CR propagation models. We have shown that the telegraph-term extension of the diffusion equation can be mapped from the hyper-diffusive term of the Chapman-Enskog expansion, although with a significantly different coefficient τ (cf. eqs.[4-5] with eqs. [27][28][29]). In addition, the telegraph equation (even with corrected coefficient τ), by contrast to the original Chapman-Enskog equation, is not self-contained and requires an initial condition for also the anisotropic part of the CR distribution. Furthermore, an attempt to proceed to higher orders in ε introduces progressively shorter time scales associated with the "ghost" terms reflecting a quick relaxation of the initial anisotropy or strong spatial inhomogeneity. Conversely, the classic Chapman-Enskog method is devised to eliminate the short time scales irrelevant to the long-term asymptotic evolution of the CR distribution. In particular, should an unacceptably short scale be present in the initial condition, the hyper-diffusive Chapman-Enskog term will quickly erase it, thus making the further evolution consistent with the asymptotic method used.
Apart from resolving the telegraph equation controversy, we have derived the CR transport equation for an arbitrary D (µ) in combination with magnetic focusing effect. This form of the transport equation, (25), is suitable for describing CR transport and acceleration problems where the phenomenon of self-confinement (D is a functional of f , D = D [ f ; µ,t]) is critical, e.g. (Ptuskin et al. 2008;Malkov et al. 2010bMalkov et al. , 2013Fujita et al. 2011). An accounting for the magnetic focusing effect is required for describing particle acceleration in CR-modified shocks with oblique magnetic fields, since the field increases towards such shocks due to the pressure exerted by the accelerated CRs (Malkov & Drury 2001), thus producing a mirror effect.
In conclusion, by comparison even with corrected telegraph equation, the Chapman-Enskog expansion is a considerably more useful and flexible tool to describe the long-time CR propagation. Efforts on improving the CR diffusion models, where their drawbacks are important, need to address the lower level transport, including anisotropic component of the CR distribution, directly. Recent treatments of this kind can be found in, e.g., (Aloisio et al. 2009;Malkov et al. 2010a). When the diffusive treatment is well within the method's validity range (weakly anisotropic spatially smooth CR distributions) neither the telegraph nor the hyper-diffusive term (∼ ε 4 ) is essential to the CR transport and can be omitted. Of interest are the magnetic mirror and ε 3 terms as they capture convective transport of the CRs.
The work of MM was supported by the NASA ATP-program under the Grant NNX14AH36G. He is also indebted to the University of Maryland for hospitality and partial support during this work. Partial support from US DoE is also appreciated.