A&A 507, 589597 (2009)
Analytical description of nonlinear cosmic ray scattering: isotropic and quasilinear regimes of pitchangle diffusion
A. Shalchi  T. Skoda  R. C. Tautz  R. Schlickeiser
Institut für Theoretische Physik, Lehrstuhl IV: Weltraum und Astrophysik, RuhrUniversität Bochum, 44780 Bochum, Germany
Received 24 June 2009 / Accepted 21 August 2009
Abstract
Aims. We investigate pitchangle scattering, which is a fundamental process in the physics of cosmic rays.
Methods. By employing the secondorder quasilinear theory, the
pitchangle FokkerPlanck coefficient is calculated analytically for
the first time.
Results. We demonstrate that for sufficiently strong turbulence
the pitchangle FokkerPlanck coefficient is isotropic. The derived
results can be used to compute the parallel mean free path for all
forms of the turbulence spectrum. We also consider applications, namely
the transport of solar energetic particles and the propagation of
cosmic rays in the Galaxy.
Conclusions. The previously used assumption of isotropic
pitchangle diffusion is indeed correct for sufficiently strong
turbulence. An analytical description of nonlinear particle scattering
is possible.
Key words: acceleration of particles  diffusion  cosmic rays  Magnetohydrodynamics (MHD)  turbulence  interplanetary medium
1 Introduction
Here, we revisit the problem of charged particle transport in MHD turbulence. Particle transport is described by the diffusion tensor in the case of diffusive propagation. For certain parameters regimes (for a detailed discussion see Shalchi & Dosch 2009), one expects isotropic scattering. In this case the tensor is given by . In addition to the turbulent magnetic fields we expect the existence of a nonvanishing mean magnetic field . The latter parameter breaks the symmetry of the physical system leading to different diffusion coefficients along and across the mean magnetic field. For not too strong turbulent fields we expect . In this case the parallel spatial diffusion coefficient controls the particle motion.
An important example for the application of diffusion theory is the propagation and acceleration of charged cosmic rays (for a review see Shalchi 2009b). The investigation of these processes is relevant for different physical systems. Some examples are the solar corona (see, e.g., Fletcher 1997; Gkioulidou et al. 2007), the heliosphere (e.g., Dröge 2000; Shalchi et al. 2006; Alania & Wawrzynczak 2008), the interstellar medium (see, e.g., Yan & Lazarian 2002; Shalchi & Schlickeiser 2005), and shock waves (see, e.g., Zank et al. 2000; Li et al. 2003; Li et al. 2005; Zank et al. 2006).
The parallel mean free path
of the charged particle is related to the parallel spatial diffusion coefficient
via
and can be expressed as an integral over the inverse pitchangle FokkerPlanck coefficient
(see, e.g., Earl 1974)
with the pitchangle cosine /v and the particle velocity v.
The first approach to compute the parameter was the application of perturbation theory also known as quasilinear theory (QLT, Jokipii 1966). In the years after QLT had been developed, it was noticed that the theory is not able to describe pitchangle scattering at (corresponding to ) correctly. This problem, which is known as the scattering problem, was then investigated in numerous papers (see, e.g., Jones et al. 1973; Völk 1973; Owens 1974; Völk 1975; Goldstein 1976; Jones et al. 1978). In these articles, QLT has been improved by replacing unperturbed orbits by more appropriate models. However, some of these previous theories do not provide agreement with simulations or they are difficult to apply due to mathematical problems (see Shalchi 2009a,b). More recently, a second order quasilinear theory (SOQLT, Shalchi 2005a) was developed. This theory is in good agreement with testparticle simulations (for a detailed comparision between SOQLT and simulations we refer to Shalchi 2007) and is mathematically tractable. Furthermore, Tautz et al. (2008) have demonstrated that SOQLT can reproduce the simulations of Giacalone & Jokipii (1999) performed for isotropic turbulence.
In previous applications of nonlinear theories for pitchangle scattering and parallel spatial diffusion, only numerical results have been derived due to mathematical intractability. Analytical forms of these parameters are, however, very useful for different astrophysical applications. It is the purpose of the present article to investigate the SOQLT analytically for the first time. In previous articles, simple models have been used without justification. In the theory of diffusive shock acceleration, for instance, it was often assumed that pitchangle scattering is isotropic (see, e.g., Kirk & Schneider 1988; Schneider & Kirk 1989; Kirk & Schneider 1989), in disagreement with the quasilinear result. It is also the purpose of the present article to investigate the validity of the assumption of isotropic scattering.
In Sect. 8 we consider some applications of our analytical results, namely:
 1.
 cosmic rays from the Sun;
 2.
 interstellar transport and steep turbulence spectra;
 3.
 the Hillas limit and high energetic particles.
2 Standard quasilinear theory
The FokkerPlanck coefficient
used in Eq. (1) can be computed by employing the socalled TaylorGreenKubo formulation (TGK formulation, e.g. Taylor 1922; Green 1952; Kubo 1957)
(2) 
The acceleration parameter can be obtained from the NewtonLorentzequation whose parallel component reads
for purely magnetic fields^{}. Here we have used the unperturbed Larmor radius
(4) 
with the unperturbed gyrofrequency , the particle mass m and charge q, the mean magnetic field B_{0}, the speed of light c, and the particle momentum p.
The simplest method to compute the parameter
is the application of perturbation theory also known as quasilinear theory (Jokipii 1966). In this case the velocities v_{x} (t) and v_{y} (t) as well as the
particle trajectories
in Eq. (3)
are replaced by the unperturbed particle orbit. A further assumption
which is often used is that the stochastic magnetic fields are replaced
by the socalled magnetostatic
slab model for which we assume
.
For the slab model the magnetic correlation tensor is given by
(5) 
with the (symmetric) turbulence spectrum . The combination of quasilinear theory and the magnetostatic slab model is also known as standard quasilinear theory. In this case we find (see, e.g., Teufel & Schlickeiser 2002, 2003 for a detailed derivation of this formula)
with the quasilinear resonance function
All parameters used in the present article are explained in Table 1. The particle experiences only interaction with a certain wavenumber corresponding to . This scattering condition is known as gyroresonance. We find
In combination with Eq. (1) this formula can be used to compute the quasilinear parallel mean free path.
Table 1: Parameters used in the present article.
3 Second order quasilinear theory
3.1 The second order resonance function
In this section we employ the second order theory of Shalchi (2005) in
combination with the magnetostatic slab model. In the second order
theory we no longer assume unperturbed orbits. Instead,
quasilinear theory is employed in order to compute improved orbits. The
improved orbits are then combined with Eqs. (1)(3). Mathematically, the second order approach leads to
a modified (broadend) resonance function (for a detailed derivation see Shalchi 2005)^{}
Here the resonance function has the form of a Gaussian function with the width
The quasilinear resonance function (see Eq. (7)) can be recovered by considering the limit . The physics of a nonlinear particle motion is illustrated in Fig. 1.
Figure 1: The particle motion through the turbulent plasma. The turbulent magnetic field is represented by the dashed line. If there would be no interaction between the plasma and the cosmic rays, the particles would follow unperturbed orbits (dotted line). The latter trajectories are used in quasilinear theory. In reality, however, the particles experience scattering and, therefore, the true orbits decorrelate from the unperturbed motion (solid lines). 

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3.2 An approximation for the resonance function
The second order resonance function of Eq. (9) has the form
Please note that . In Eq. (11) we have used the resonance condition
(12) 
and the width as defined in Eq. (10). In principle the exponential (or Gaussian) function cuts off the wavenumber integral in Eq. (6). Furthermore, we have for the area under the resonance function
(13) 
To achieve a simplification we approximate the resonance function by
or in terms of the Heaviside stepfuntion H
Equations (14) and (15) have similar properties in comparison with the original function (11). Furthermore, the function is similar to the heuristic ansatz of Völk (1975). In Fig. 2 the resonance functions of quasilinear theory, secondorder theory (see Eq. (11)) and the approximation used in the present article (see Eqs. (14) and (15)) are visualized.
3.3 The pitchangle FokkerPlanck coefficient for the general case
The pitchangle FokkerPlanck coefficient from Eq. (6) has the form
with
(17) 
and
To solve the integral with the approximation of Eq. (14) we have to split the integral. It is convenient to introduce the parameters
=  
k_{0}  =  (19) 
By combining Eqs. (15) and (18) we find after straightforward algebra
=  
=  0.  (20) 
The function I_{} is more difficult to evaluate. We find
=  
(21) 
To proceed we have to distinguish between the cases and . We find after straighforward algebra
=  
=  (22) 
By using the parameter
Figure 2: The different resonance functions versus the resonance parameter /. The quasilinear resonance function (dashed line) is a function, whereas we find a Gaussian function within the second order theory (dotted line). In the present article we approximate the Gaussian function by a Heaviside stepfunction (solid line). 

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the total function I=I_{} + I_{+} can be written as
The case can be obtained from both formulas by employing a limiting process. We find
(25) 
The first term in Eq. (24) is a nonresonant term and can be called the quasiresonant term. The reason will become clearer in the next section where we will discuss special cases.
4 Special limits and cases
Here we explore Eq. (24) for special limits and cases to recover previous results.
4.1 The quasilinear limit
In this paragraph we investigate the limit
.
In this case we have to use the quasiresonant formula of Eq. (24). Therefore, we can approximate
=  
=  (26) 
With
(27) 
we find for
(28) 
and the pitchangle FokkerPlanck coefficient reads
=  
(29) 
in agreement with Eq. (8). Obviously, quasilinear theory is valid so long as the restriction holds. For pitchangles satisfying , quasilinear theory could be approximately correct.
4.2 Strong turbulence and 90scattering
Now we investigate the limit
.
In this case we have
(30) 
with the parameter defined in Eq. (23). Therefore the nonresonant term in Eq. (24) has the form
(31) 
which is a pitchangle independent result.
For strong turbulence (
)
we always have
.
In this case we find with Eq. (16) the form
with
Equation (32) corresponds to an isotropic form (see later discussions). The parameter D is the pitchangle FokkerPlanck coefficient at . The second order theory was already investigated analytically for in Shalchi (2005). In this paragraph we derived this limit from the general Eq. (24).
5 Results for a realistic turbulence spectrum
For the turbulence spectrum we employ the form introduced by Shalchi & Weinhorst (2009)
with the normalization constant
In Eq. (34) we have used the bendover scale of the turbulence^{} , the energy range spectral index q, and the inertial range spectral index s. The spectrum is correctly normalized for q>1 and s>1. In the following two paragraphs we investigate the quasiresonant as well as the nonresonant case analytically for the spectrum defined in Eq. (34).
5.1 The quasiresonant case
For the spectrum of Eq. (34) the quasiresonant function derived in Eq. (24) becomes
=  
(36) 
To proceed we employ the integral transformation and split the integral to derive
with
(38) 
The two integrals can be expressed by the hypergeometric function (see, e.g., Gradshteyn & Ryzhik 2000)
With this formula we find
with
(41) 
To proceed we consider . This case corresponds to the assumption of not too strong turbulence and low energetic particles. In this case the argument of the hypergeometric function is small ( ) and we can use (see, e.g., Abramowitz & Stegun 1974)
By combining Eqs. (37)(42) and by using R = R_{L}/ we find
Note that this result is only valid for . From this result we can derive the quasilinear limit by considering
(44) 
Figure 3: Shown are the results of QLT (dotted line), SOQLT within the LTA (solid line), and simulations (Qin 2002, dots) for . Also shown is the more accurate SOQLT result without LTA (star) which is closer to the simulations. Furthermore, the analytical results derived in the present paper are shown. The resonant case is correct for (bright grey line) and the nonresonant case is valid for (dark grey line). In this plot, the rigidity is R = 0.0363 and the strength of turbulence is assumed to be . 

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5.2 The nonresonant case
The calculations of the previous paragraph can be repeated for
.
In this case we have to use the nonresonant formula in Eq. (24) to derive
Except for the signs, Eq. (45) agrees with Eq. (43).
5.3 The general case
Equations (43) and (45) can be combined to find for arbitrary
the formula
With Eq. (16) the pitchangle FokkerPlanck coefficient becomes
This formula can be applied for . In the other case we have to consider the hypergeometric function of Eq. (40) in the limit of large arguments. In Figs. 3 and 4 we compare our analytical results with numerical results, QLT, and computer simulations.
Figure 4: Enlarge of Fig. 3 at small pitchangle cosines. 

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6 The parallel mean free path
By using Eq. (1) we can compute the parallel mean free path. It is useful to consider again two different cases.
6.1 The case
Here we can use Eq. (1) to find approximately
=  
(48) 
By employing Eq. (47) for the pitchangle FokkerPlanck coefficient in the limits , we derive
This formula provides the behavior in disagreement with the QLT result ( ).
6.2 The case
Here we can use Eq. (1) to find approximately
=  
=  
(50) 
By employing Eq. (47) for the pitchangle FokkerPlanck coefficient in the limits and , we derive
The first term corresponds to the wellknown quasilinear result. The second term is new and arises due to nonlinear effects. So long as the turbulent field is weak ( ) and the inertial range spectral index satisfies s < 2 the first term is dominant and quasilinear theory provides the correct result. For a weak turbulent field and the case s > 2, corresponding to a steep inertial range spectrum, the second (nonlinear) term controls the parallel mean free path. In the latter case the parallel mean free path scales like . For a strong turbulent field we have to employ Eq. (49). In Fig. 5 we have shown the parallel mean free path versus the turbulence strength .
Figure 5: The parallel mean free path computed by using QLT (dotted line) for s=5/3. Also shown are the analytical results of SOQLT, namely the weak turbulence solution (dashed line) of Eq. (51) and the strong turbulence solution (solid line) of Eq. (49). 

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7 Wave propagation effects
7.1 Parallel and antiparallel propagating waves
So far we have only discussed the magnetostatic case. In the current section we include plasma wave propagation effects by following the work of Schlickeiser (1989). We assume that there are only parallel and antiparallel propagating shear Alfvén waves.
The magnetostatic pitchangle FokkerPlanck coefficient has the form
(52) 
where we must distinguish the cases and . We include wave effects by employing the substitution
(53) 
where we have used the (energy dependent) parameter with the Alfvén velocity v_{A}. The parameter j is used to track the wave direction (j=+1 is used for forward and j=1 for backward to the ambient magnetic field propagating Alfvén waves). If we assume equal intensities of parallel and antiparallel propagating waves we can use
We notice that
(55) 
and
(56) 
For as well as we find a nonvanishing pitchangle FokkerPlanck coefficient. The question is whether wave propagation effects or nonlinear effects are more relevant. In the following paragraph we will investigate this issue.
7.2 Wave versus nonlinear effects
By combining Eqs. (46) and (54) we derive
=  
(57) 
Mathematically, plasma wave propagation effects enter the function in a similar way as the nonlinear effects. We can derive the parameter regimes for which wave effects are dominant and for which nonlinear effects have a stronger influence. It is convenient to define the characteristic velocity
(58) 
For low particles velocities we are in the wave dominated regime (since we have ) and for higher velocities we are in the nonlinear regime (since we have ). Magnetostatic quasilinear theory is valid for and (see Table 2).
Table 2: Plasmawave propagation versus nonlinearity.
8 Applications
8.1 Energetic particles from the sun
To describe the motion of charged particles along the magnetic field of
the Sun we can use the twodimensional FokkerPlanck equation:
(59) 
To proceed, we compute the spatial average
(60) 
and the FokkerPlanck equation becomes
(61) 
By assuming that and we can employ Eq. (32) and, therefore,
(62) 
This equation can be solved analytically without further assumptions. E.g., Shalchi (2006) has demonstrated that the solution of this equation can be expressed by Legendre polynomials :
(63) 
For sharp initial conditions ( ) this becomes
(64) 
An interesting property is the anisotropy A (t) which can be defined as
(65) 
With , , and
(66) 
we find
In the last step we have used Eq. (1) for the parallel mean free path. Analytical results such as Eq. (67) can be compared with measurements of solar particle events by spacecrafts such as Wind (see, e.g., Dröge & Kartavykh 2009).
8.2 Interstellar propagation and steep spectra
Figure 6: for pure slab geometry and steep spectra (s=2+5/3). The simulations (dots) from Shalchi (2007) are compared with numerical SOQLT results (dashed line) and the analytical SOQLT result (solid line) obtained by employing Eq. (51). For sake of comparison we have also shown the QLT result for s=5/3 (dotted line). The magnetostatic QLT result for s=2+5/3 is . 

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In Lazar et al. (2003) and Spanier & Schlickeiser (2005), the heating rate of the interstellar medium (ISM), especially the warm ionized medium, has been calculated. Within these two papers it was demonstrated that a steeper form of the turbulence spectrum (s > 2) could be reasonable. In this case we obtain by employing standard QLT an infinitely large parallel mean free path ( ). In Shalchi (2007) it has already been demonstrated the SOQLT is in agreement with simulations for such spectra. In the present section we compare our analytical finding with these previous results (see Fig. 6).
For the analytical results we can use Eq. (51)
since we assume not too strong stochastic fields. These analytical
forms are only valid for smaller rigidities. For high particle energies
the analytical results
deviate from the numerical finding. For weak turbulence and s>2 we can derive from Eq. (51) the formula
(68) 
For strong turbulence we have to use Eq. (49). The pitchangle FokkerPlanck coefficient for high particle rigidities can be derived from Eq. (40). In this case it is also straightforward to compute the parallel mean free path.
8.3 The Hillas limit and high energetic cosmic rays
In Shalchi et al. (2009a) we have investigated for the first time the propagation of ultrahigh energy particles within the framework of SOQLT. In this article we have computed numerically the pitchangle FokkerPlanck coefficient and the parallel mean free path. As shown there, the Hillas limit^{} is questionable.
It is the purpose of the present section to calculate analytically the
pitchangle FokkerPlanck coefficient of very high energetic particles.
For simplicity we assume
and we can use Eq. (33). By using a spectrum with sharp cutoff at short wavenumbers we find
(69) 
To evaluate this formula we assume corresponding to
(70) 
with the largest scale of the turbulence . With the spectrum defined in Eq. (34) we can derive
(71) 
With the integral transformation we derive
(72) 
with and the integral
(73) 
By using Gradshteyn & Ryzhik (2000) we can solve the integral to find
(74) 
Finally, we find for the pitchangle FokkerPlanck coefficient of ultrahigh energetic particles
(75) 
By using Eq. (1) it is a simple matter to calculate analytically the parallel mean free path
This result is valid for and strong stochastic fields. The quasilinear result for R_{L}/ is .
Quasilinear theory predicts that the parallel mean free path of particles with is infinity. Consequently, the motion of high energy particle is scatterfree (ballistic) and, therefore, such particles cannot be confined to the Galaxy. According to Eq. (76), however, we find finite scattering within the framework of SOQLT. Therefore, we expect a finite confinement time of real particles in the Galaxy. For details we refer to Shalchi et al. (2009).
9 Summary and conclusion
In the present article we have revisited the problem of pitchangle scattering and parallel diffusion of charged particles. Previous investigations are based on the quasilinear approximation or on nonlinear theories. In the latter case only numerical results were available for the pitchangle FokkerPlanck coefficient and the parallel mean free path. In Sect. 3 we have explored for the first time analytically the second order theory of Shalchi (2005a). By deriving general analytical expressions for the parameter , we have shown that the traditional quasilinear theory is correct for /B_{0} and the assumption of isotropic scattering is valid for strong turbulence. This result confirms previous articles about diffusive shock acceleration (see, e.g., Kirk & Schneider 1988; Schneider & Kirk 1989; Kirk & Schneider 1989). It should be noted, however, that for extremely strong turbulence one expects Bohmdiffusion (see Shalchi 2009a).
By employing the spectrum of Shalchi & Weinhorst (2009a) we derived analytical forms for the FokkerPlanck coefficient and the parallel mean free path . The formulae for the latter parameter can also be used for strong turbulence and for steep spectra (). Quasilinear theory for parallel spatial diffusion is only valid for weak turbulence ( ) and flat spectra (s < 2). For these spectra we have also introduced plasma wave propagation effects by following Schlickeiser (2002). For particle velocities satisfying the quasilinear results obtained for the plasma wave model are correct since nonlinear effect are supressed. For nonlinear effects are stronger and the magnetostatic model should provide a good approximation.
Analytical forms for the parameters and are very useful in the physics of cosmic rays. Therefore, we have presented some applications of our results (see Sect. 8). We have shown how energetic particles from the Sun can be described analytically be computing the anisotropy A. Such results can be compared with spacecraft observations such as Wind measurements (see, e.g., Dröge & Kartavykh 2009). As a second example we computed the parallel mean free path of particles in the ISM by assuming steep turbulence spectra as suggested by Lazar et al. (2003) and Spanier & Schlickeiser (2005). These analytical results complement the numerical work of Shalchi (2007). Standard quasilinear theory results in for such spectra. We have also considered the problem of ultrahigh energy cosmic rays. These results complement the numerical work of Shalchi et al. (2009a). We have derived for the first time a formula for the parallel mean free path of ultrahigh energy cosmic rays within SOQLT. According to this formula we have / if the particle Larmor radius exceeds the largest scale of turbulence ( ). We expect that our analytical results will lead to further interesting and important applications in astrophysics such as diffusive shock acceleration.
AcknowledgementsThis research was supported by the Deutsche Forschungsgemeinschaft (DFG) under the EmmyNoether Programm (grant SH 93/31) and project Schl 201/191. As a member of the Junges Kolleg, A. Shalchi also acknowledges support by the NordrheinWestfälische Akademie der Wissenschaften.
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Footnotes
 ... fields^{}
 In the current article we neglect electric fields since they are less important for spatial diffusion. If one is interested in stochastic acceleration, however, electric fields are relevant (see, e.g., Schlickeiser 2002).
 ...2005)^{}
 The resonance function of Eq. (9) was obtained by Shalchi (2005) by combining a second order approach in combination with two mathematical approximations, namely a LateTimeApproximation (LTA) and a approximation. These approximations were employed to ensure mathematical tractability. For a detailed explanation of these approximations and their justifications we refer to Shalchi (2005, 2009b).
 ... turbulence^{}
 The bendover or turnover scale denotes the frequency break between the large scales (energy range) and the intermediate scales (inertial range) of the turbulence. For the spectrum defined in Eq. (34) the bendover scale is directly proportional to the turbulence correlation length.
 ... limit^{}
 Within the framework of magnetostatic quasilinear theory, the resonance function is a sharp delta function. Since there exists a largest scale of turbulence , ultrahigh energy particles having a Larmor radius larger than this scale ( ) cannot be scattered and, therefore, they cannot be confined to the Galaxy. This limit is known as the Hillas limit (Hillas 1984).
All Tables
Table 1: Parameters used in the present article.
Table 2: Plasmawave propagation versus nonlinearity.
All Figures
Figure 1: The particle motion through the turbulent plasma. The turbulent magnetic field is represented by the dashed line. If there would be no interaction between the plasma and the cosmic rays, the particles would follow unperturbed orbits (dotted line). The latter trajectories are used in quasilinear theory. In reality, however, the particles experience scattering and, therefore, the true orbits decorrelate from the unperturbed motion (solid lines). 

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In the text 
Figure 2: The different resonance functions versus the resonance parameter /. The quasilinear resonance function (dashed line) is a function, whereas we find a Gaussian function within the second order theory (dotted line). In the present article we approximate the Gaussian function by a Heaviside stepfunction (solid line). 

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In the text 
Figure 3: Shown are the results of QLT (dotted line), SOQLT within the LTA (solid line), and simulations (Qin 2002, dots) for . Also shown is the more accurate SOQLT result without LTA (star) which is closer to the simulations. Furthermore, the analytical results derived in the present paper are shown. The resonant case is correct for (bright grey line) and the nonresonant case is valid for (dark grey line). In this plot, the rigidity is R = 0.0363 and the strength of turbulence is assumed to be . 

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In the text 
Figure 4: Enlarge of Fig. 3 at small pitchangle cosines. 

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In the text 
Figure 5: The parallel mean free path computed by using QLT (dotted line) for s=5/3. Also shown are the analytical results of SOQLT, namely the weak turbulence solution (dashed line) of Eq. (51) and the strong turbulence solution (solid line) of Eq. (49). 

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In the text 
Figure 6: for pure slab geometry and steep spectra (s=2+5/3). The simulations (dots) from Shalchi (2007) are compared with numerical SOQLT results (dashed line) and the analytical SOQLT result (solid line) obtained by employing Eq. (51). For sake of comparison we have also shown the QLT result for s=5/3 (dotted line). The magnetostatic QLT result for s=2+5/3 is . 

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In the text 
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