Numerical investigation of the influence of large turbulence scales on the parallel and perpendicular transport of cosmic rays

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1 Available online at Advances in Space Research 49 (1) Numerical investigation of the influence of large turbulence scales on the parallel and perpendicular transport of cosmic rays G. Qin a, A. Shalchi b, a State Key Laboratory of Space Weather, Center for Space Science and Applied Research, Chinese Academy of Sciences, Beijing 119, China b Department of Physics and Astronomy, University of Manitoba, Winnipeg, Manitoba, Canada R3T N Received 3 November 11; received in revised form 7 January 1; accepted 8 February 1 Available online 6 March 1 Abstract In recent analytical investigations it has been demonstrated that the turbulence behavior at large scales has a very strong influence on the perpendicular diffusion coefficient of charged particles. In the present paper we use computer simulations to investigate numerically cross field transport and particle propagation along the mean magnetic field for different turbulence models at large scales. Our results are compared with quasilinear theory and nonlinear diffusion theories. We show that for different forms of the turbulence spectrum at large scales, the perpendicular mean free paths obtained numerically are in agreement with recent predictions made by analytical theory. It is also shown that the parallel diffusion coefficient contains always a strong nonlinear contribution which is, however, independent of the assumed spectrum at large scales. Ó 1 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Magnetic fields; Turbulence; Energetic particles 1. Introduction 1.1. Particle transport theory The investigation of cosmic ray transport remains an interesting field of research. Transport theory connects two important and fundamental fields of modern physics, namely the physics of turbulence and the propagation and acceleration of cosmic particles. E.g., the particle diffusion coefficients enter the cosmic ray transport equation which is used to model the propagation of particles in the Milky Way (see, e.g., Ptuskin et al., 6; Büsching and Potgieter, 8 and references therein). In the physics of the heliosphere the knowledge of the diffusion coefficients are crucial in solar modulation studies (see, e.g., Hitge and Burger, 1; Burger and Visser, 1; Manuel et al., 11; Strauss et al., 11). Last but not least, the diffusion coefficients of charged particles enter the equations which are used to Corresponding author. Tel.: ; fax: address: andreasm4@yahoo.com (A. Shalchi). describe the mechanism of diffusive shock acceleration (see, e.g., Zank et al.,, 7; Dosch and Shalchi, 9, 1, Li et al., 1). The latter mechanism is responsible for the origin of cosmic rays. The diffusion coefficients, on the other hand, are sensitively controlled by our understanding of turbulence (see, e.g., Schlickeiser, and Shalchi, 9 for reviews). As described above our understanding of particle diffusion is crucial for an improved description of different physical scenarios in space science and astrophysics. Furthermore, particle-turbulence interactions are important in the physics of fusion devices (see, e.g., Hauff et al., 9; Hauff and Jenko, 9). 1.. Tools in particle diffusion theory There are, in principle, two different tools which can be used to explore the scattering of energetic particles due to turbulence, namely, 1. Analytical theory: In the analytical description of particle-plasma interactions one has to specify the magnetic correlation tensor describing the turbulence. Then one /$36. Ó 1 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:1.116/j.asr.1..35

2 1644 G. Qin, A. Shalchi / Advances in Space Research 49 (1) can try to compute the diffusion coefficients in the different directions by employing perturbation theory, extensions thereof, or full nonlinear theories (see again Schlickeiser, and Shalchi, 9 for reviews);. Computer simulations: In the recent years more and more test-particle codes were developed. In such simulations we still have to specify the turbulence structure. However, we can compute the diffusion coefficients by solving the Newton Lorentz equation without employing further approximations. Both approaches have their advantages and disadvantages. E.g., the simulations can only be easily performed for magnetostatic or undamped plasmawave turbulence models whereas analytical theories can be employed for arbitrary forms of the magnetic correlation tensor. For the applications discussed above (e.g., solar modulation, acceleration at shocks) one has to know analytical forms of the diffusion coefficients. Of course, such forms cannot be obtained from computer simulations. Analytical theories, however, are always based on assumptions. In quasilinear theory, for instance, one assumes unperturbed orbits which do not exist in reality. In nonlinear theories one has to employ different models and approximations (e.g., random phase approximation, diffusion approximation). Therefore, we believe that a complete understanding of transport phenomena can only be achieved if one uses a combination of numerical and analytical tools. Cosmic ray scattering has been investigated in numerous papers analytically (see, e.g., Jokipii, 1966; Völk, 1973; Bieber et al., 1994; Matthaeus et al., 3; Shalchi et al., 4; Shalchi and Kourakis, 7), see Shalchi, 9 for a review). The classical work of Jokipii (1966), for instance, employed a quasilinear theory (QLT) for particle transport. Matthaeus et al. (3) developed a nonlinear approach for cross field transport called the nonlinear guiding center (NLGC) theory. This approach was improved and extended later by Shalchi et al. (4), Shalchi (6b), Qin (7), Shalchi and Dosch (8), and Shalchi and Dosch (9). Shalchi (1) has developed an unified nonlinear transport (UNLT) theory for energetic particles and magnetic field lines. A compound diffusion model was used by Webb et al. (6) and later by Shalchi and Kourakis (7) to describe perpendicular scattering of charged particles. In this approach perpendicular diffusion is directly linked to the random walk of magnetic field lines. Numerical simulations were performed in the early years by Kaiser (1975), Kaiser et al. (1978) and later by Michałek et al. (1996), Giacalone and Jokipii (1999), Mace et al. (), Qin et al. (a,b, 6), Zimbardo et al. (6), Shalchi (7) and Tautz (9, 1) The influence of turbulence on the transport As described above, the turbulence structure influences the transport of energetic particles in the plasma. The most important transport mechanisms are diffusion of particles along and across the mean magnetic field, drifts, and stochastic acceleration. In the present paper we concentrate on the first two effects, namely parallel and perpendicular diffusion. Although the two effects are spatial diffusion effects, their physics is quite different The physics of parallel diffusion Parallel diffusion is strongly related to pitch-angle scattering via the famous relation (see, e.g., Jokipii, 1966; Hasselman and Wibberenz, 1968, Earl, 1974) k k ¼ 3j k v ¼ 3v 8 Z þ1 1 ð1 lþ dl D ll ðlþ : The latter formula is a consequence of a pitch-angle isotropization process as described by the cosmic ray Fokker Planck equation (see, e.g., Shalchi, 6a). The pitchangle Fokker Planck coefficient D ll ðlþ used here is mainly (but not only) controlled by gyro-resonant interactions lr L ¼ k 1 k. Here we used the parallel wavenumber of the turbulence k k, the pitch-angle cosine l ¼ v k =v, and the unperturbed Larmor radius R L (more details can be found in Shalchi, 9). The assumption of gyro-resonance, however, is only an approximation because real particles experience resonance-broadening due to the turbulence. In the recent years, resonance-broadening theories were developed to achieve more accurate description of parallel transport (see, e.g., Völk, 1973, 1975; Jones et al., 1973, 1978; Owens, 1974; Goldstein, 1976; Shalchi et al., 4; Shalchi, 5). Therefore, the statement above concerning gyro-resonance has to be extenuated: pitch-angle scattering is ð1þ mainly controlled by wavenumbers satisfying lr L k 1 k. Especially for particles with l, even this statement is no longer true The physics of perpendicular diffusion Whereas parallel transport can be described analytically by quasilinear theory and resonance-broadening theories, perpendicular transport has to be described by nonlinear theories. The perpendicular diffusion coefficient is strongly controlled by random walking magnetic field lines. The simplest description of transport across the mean magnetic field is, therefore, provided by the so-called Field Line Random Walk (FLRW) limit in which we have j? ¼ v j FL where we used the field line diffusion coefficient j FL. Within this model perpendicular diffusion is only caused by field line diffusion. However, Eq. () is only true if 1. Parallel scattering is suppressed;. The guiding centers of the particles are tied to a single magnetic field line. The first assumption is questionable, since we expect that pitch-angle scattering is a strong effect in real physical scenarios. Pitch-angle scattering and therewith parallel ðþ

3 G. Qin, A. Shalchi / Advances in Space Research 49 (1) diffusion, however, suppress perpendicular transport to a subdiffusive level (see, e.g., Shalchi, 9). In order to restore normal or Markovian diffusion, one has to allow the particle to scatter away from the magnetic field lines (see, e.g., Shalchi and Dosch, 8; Shalchi, 9; Dosch et al., 9) Scope of the present paper It is well-known that the perpendicular diffusion coefficient is mainly controlled by the largest scales of the turbulence (see, e.g., Shalchi, 1). However, it is not clear how important the largest scales are for parallel diffusion. The gyro-resonance model predicts that the largest scales should only influence parallel diffusion coefficients at very high energies. However, it is not clear how important the large scales are if parallel transport is nonlinear. For perpendicular transport we expect that the largest scales are important since this is a prediction of analytical theory. In the present paper we try to find an answer for the following two questions: 1. How strong is the nonlinear effect if the parallel diffusion coefficient is computed for different spectra at large scales?. Are the predictions made for j? by recent nonlinear diffusion theories for different spectra correct? The organization of the paper is as follows. In Section we discuss the turbulence model which is used in the present paper. In Section 3 we briefly discuss previous results provided by analytical theory and in Section 4 we discuss the simulation code used in the present paper and we show our new results. In Section 5 we summarize and conclude.. The turbulence model.1. The two-component model In the theory of field line random walk and charged particle transport one assumes that the total magnetic field is the superposition of a uniform mean field ~B ¼ B ~e z and fluctuations d~b. Here we have chosen a Cartesian system of coordinates so that the z-axis is aligned along the mean field. Such a configuration can be found in the solar wind or in the interstellar medium. In the first case the mean field has to be identified by the magnetic field of the Sun (e.g., B 4nT). For interplanetary studies, a common assumption is that the magnetic field fluctuations admit a strong component of nearly two-dimensional character with d~bð~xþ ¼d~Bðx; yþ, comprising perhaps 8 9% of the turbulent inertial range energy budget (see, e.g., Matthaeus et al., 199; Zank and Matthaeus, 1993; Bieber et al., 1996). The rest of the magnetic energy is contained in so-called slab modes with d~bð~xþ ¼d~BðzÞ. As a consequence the turbulent field can be written as d~bð~xþ ¼d~BðzÞþd~Bðx; yþ. In the literature this model is also known as slab/d composite or two-component model. The two-component model can be confirmed by using extensive analyses of solar wind data (see Matthaeus et al., 199). According to such observations magnetic correlations in the solar wind have the form of a so-called maltese cross. Similar measurements were done in the following years (see, e.g., Carbone et al., 1995; Bieber et al., 1996; Dasso et al., 5; Osman and Horbury, 7, 9a,b; Horbury et al., 8) which have confirmed this structure of interplanetary turbulence. A review of interplanetary turbulence can be found in Horbury et al. (5). Furthermore, numerical simulations suggest that two-dimensional dynamics is the leading order description of turbulence in the presence of a mean magnetic field (see, e.g., Oughton et al., 1994; Matthaeus et al., 1996; Shaikh and Zank, 7; Dmitruk and Matthaeus, 9). The theory of nearly incompressible MHD turbulence (see Zank and Matthaeus, 1993) predicts a collapse in dimensionality making turbulence in the solar wind a superposition of a dominant two-dimensional and a slab component. A more advanced discussion of this matter can be found in Hunana and Zank (1)... The spectrum of Shalchi and Weinhorst (9) For the composition model of pure slab and pure twodimensional modes, the magnetic correlation tensor in the wavevector space has the form P lm ¼ P slab lm þ P D lm. Here we have used the magnetic correlation tensor in the ~ k space which is defined as P lm ð ~ kþ¼hdb l ð ~ kþdb m ð~ kþi where h...i denotes the ensemble average. The tensor of the slab modes has the form P slab lm ð~ kþ¼g slab ðk k Þ dðk?þ d lm ; ð3þ k? with l; m ¼ x; y and the tensor of the two-dimensional modes has the form P D lm ð~ kþ¼g D ðk? Þ dðk kþ d lm k lk m ; ð4þ k? k with l; m ¼ x; y. Please note that tensor components with z are zero. For the slab modes the vanishing z-component results from the solenoidal constraint rd~bðzþ ¼. For the two-dimensional modes the vanishing parallel component is part of the model. For the two spectra g slab ðk k Þ and g D ðk? Þ we use the forms g slab ðk k Þ¼ CðsÞ p db slab l 1 slab h i s= ð5þ 1 þðk k l slab Þ and g D ðk? Þ¼ Dðs; qþ p db D l D h ðk? l D Þ q i ðsþqþ= : ð6þ 1 þðk? l D Þ

4 1646 G. Qin, A. Shalchi / Advances in Space Research 49 (1) The first model spectrum is in agreement with the one used by Bieber et al. (1994). The latter model for the two-dimensional spectrum has been developed by Shalchi and Weinhorst (9). For the inertial range spectral index s we assume the same values, namely s ¼ 5=3 corresponding to a Kolmogorov (1941) spectrum. For the energy range spectral index, however, we used different values. For the slab modes we employed a spectrum which is perfectly flat at large scales and for the two-dimensional modes we allow a general spectrum in the energy range which is controlled by the parameter q. The different physical consequences of the different values of q are discussed in Matthaeus et al. (7). In Eqs. (5) and (6) we have used the normalization functions Dðs; qþ ¼ and C s 1 C sþq CðsÞ Dðs; q ¼ Þ ¼ ð7þ C qþ1 C s p ffiffiffi : ð8þ p C s 1 Here we used the Gamma function p CðzÞ and we have employed the relation Cð1=Þ ¼ ffiffiffi p. The parameters used in the two spectra are listed in Table 1. The two spectra are correctly normalized for s > 1andq > Predictions of analytical theory In the present article we focus on numerical work to determine the parallel mean free path k k, the perpendicular mean free path k?, as well as the ratio k? =k k. However, we will frequently compare our numerical findings with different analytical results. In the following we briefly discuss results obtained by analytical theories Standard quasilinear theory Standard quasilinear theory can be used to compute the parallel mean free path. If one combines quasilinear theory with a magnetostatic slab/d model, we find for the pitchangle Fokker Planck coefficient D ll ¼ D slab ll þ DD ll. The parallel mean free path can then be computed by using the well-known formula (1). As shown in Shalchi et al. (8), however, there is no scattering contribution due Table 1 Turbulence parameters used in the present article. We have also shown the value of the corresponding parameter used in our simulations. Parameter Physical meaning Value s Inertial range spectral index 5/3 q D energy range spectral index Variable l slab Slab bendover scale Not used l D D bendover scale :1l slab db slab Magnetic energy of the slab modes :B db D Magnetic energy of the D modes :8B B Mean magnetic field Not used R Particle rigidity Variable to the two-dimensional modes in magnetostatic turbulence, i.e., D D ll ¼ and, thus, the quasilinear parallel mean free path is solely controlled by the slab modes. Therefore, there is no variation of the quasilinear parallel mean free path if we change the parameter (Shalchi et al., 8; Shalchi, 9) the quasilinear pitch-angle Fokker Planck coefficient is given by the formula D slab ll ¼ p X ð1 l Þ g slab k vjljb k ¼ X : ð9þ vjlj Here we have used the unperturbed gyro-frequency X ¼ v=r L. The other parameters are explained in the previous paragraphs. In combination with Eq. (1) we can easily compute the quasilinear parallel mean free path. Although a pure analytical solution can be derived (see, e.g., Shalchi, 9), we compute the parameter k k in the present paper by solving the l-integral in (1) numerically to obtain an accurate result. 3.. The FLRW limit The simplest theory for cross-field diffusion is the FLRW limit in which we assume that j? ¼ j FL v= where we used the diffusion coefficient of wandering magnetic field lines. In this limit the perpendicular mean free path is k? ¼ 3j FL =. Shalchi and Weinhorst (9) have computed the field line diffusion coefficient for the spectra defined above. They only found a finite diffusion coefficient for q > 1. In the other cases they found superdiffusive field line wandering which was later confirmed numerically by Shalchi and Qin (1). For the special case q > 1 the field line random walk limits yields sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3ðs 1Þ db D k? ¼ l D : ð1þ ðq 1Þ B For q < 1 the theory of field line wandering predicts superdiffusive transport and normal diffusion can no longer be described by the theory discussed here. The FLRW limit, however, provides a simplified description of the transport. Therefore, we don t compare this limit with numerical work Extended NLGC theory An important step in the analytical description of perpendicular transport has been achieved in Matthaeus et al. (3). In the latter paper a nonlinear integral equation has been derived which represent the so-called NLGC - see Eq. (7) of Matthaeus et al. (3). An extended version of the NLGC theory has been proposed by Shalchi (6b). Another important step has been made in Shalchi (1) who developed a unified nonlinear transport (UNLT) theory which can describe wandering magnetic field lines as well as perpendicular transport of energetic particles. For the slab/d model used in the present paper, however, the extended NLGC theory and the UNLT theory provide the same result. Analytical formulas have been

5 G. Qin, A. Shalchi / Advances in Space Research 49 (1) derived recently for the perpendicular mean free path (see Shalchi, 1). Those are based on the improved theory discussed above. According to Shalchi et al. (1) there are three different parameter regimes: k k k? 3l D and q < 1: k? ¼ 3 ðqþ1þ= a Dðs; qþ db D C 1 þ q B C 1 q l qþ1 D =ðqþ3þ =ðqþ3þ k ð1 qþ=ð3þqþ k ; ð11þ k k k? 3l D and q > 1: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3ðs 1Þ db D k? ¼ al D ; ð1þ ðq 1Þ B k k k? 3l D and arbitrary q: k? ¼ a db D k B k : ð13þ The latter result is independent of the energy range spectral index q. Here we used the dimensionless parameter a which was introduced in the original paper of Matthaeus et al. (3). According to a comparison with numerical data the latter parameter is a ¼ 1=3 which is, however, in disagreement with some analytical work (see, Shalchi and Dosch, 8) where it was shown that a is in the order of unity The weakly nonlinear theory Since it has been discovered that quasilinear theory is sometimes not correct if the parallel mean free path is computed for a slab/d model, Shalchi et al. (4) have developed a weakly nonlinear theory (WNLT). The latter theory is a resonance broadening theory and was confirmed by le Roux and Webb (7). The broadening described in the theory comes due to perpendicular diffusion itself. Therefore, WNLT describes parallel and perpendicular diffusion as a coupled process. More details and the fundamental equations of WNLT can be found in Shalchi et al. (4) and Shalchi (9). The main result of WNLT is that the parallel mean free path is reduced compared to the quasilinear result. In the present paper we solve Eq. (73) of Shalchi et al. (4) numerically to get the WNTL result. 4. Computer simulations In the following we compare the analytical predictions discussed in the previous section with simulations to explore the validity of analytical theories and to test the accuracy of the different theories for cross field transport The test-particle code The simulation code which is used in the present article is a modification of the code that has been used previously (see, e.g., Mace et al., ; Qin et al., a,b, 6 for details). In our simulations, we use exactly the same magnetic field configuration as described in Section (slab/d composite model with the model spectrum proposed by Shalchi and Weinhorst, 9). In order to generate the magnetic turbulence d~bð~xþ, we use a Fourier analysis with the settings described in the following. To create the slab component a periodic box of size 1l slab and N z ¼ points are used. The two-dimensional component is created by a periodic box of size 1l slab 1l slab and N x N x ¼ points are used. In addition, the parallel correlation scale of the slab component, l slab, is set to be 1 times as the two-dimensional correlation scale, l D. With the magnetic fields ~B created, we solve the Newton Lorentz equation by a fourth-order Runge Kutta method with adaptive step size control (see, e.g., Press et al., 199). After getting the trajectories of each particle, we have computed the running diffusion coefficients by using standard methods of statistical physics. 4.. Our results In the following we compute the parameters k k =l slab ; k? =l slab, and k? =k k. We perform our simulations for different values of R ¼ R L =l slab to explore the rigidity dependence of the diffusion coefficients. We compute the corresponding parameters for different values of the energy range index q of the two-dimensional modes, namely q ¼ :5; q ¼, and q ¼ 1:5. Our results are summarized in Tables 4. The other parameters which enter our simulations are shown in Table 1. Figs. 1 3 show the running diffusion coefficients for the different values of q as obtained from the simulations. In the considered cases, parallel transport is at least nearly diffusive, whereas perpendicular transport is diffusive or weakly subdiffusive. The latter statement agrees with some other recent numerical work (see, e.g., Shalchi and Kourakis, 7; Tautz and Shalchi, 1) Results for the parallel mean free path In Fig. 4 we show the parallel mean free path as obtained numerically in comparison with quasilinear Table Our numerical results for different values of the rigidity R ¼ R L =l slab. For the energy range spectral index of the two-dimensional modes we used q ¼ :5. R k k =l slab k? =l slab k? =k k

6 1648 G. Qin, A. Shalchi / Advances in Space Research 49 (1) Table 3 Our numerical results for different values of the rigidity R ¼ R L =l slab. For the energy range spectral index of the two-dimensional modes we used q ¼. R k k =l slab k? =l slab k? =k k Table 4 Our numerical results for different values of the rigidity R ¼ R L =l slab. For the energy range spectral index of the two-dimensional modes we used q ¼ 1:5. R k k =l slab k? =l slab k? =k k κ xx /(l slab Ω) κ zz /(l slab Ω) κ xx /(l slab Ω) t Ω Fig.. The simulated parallel j zz (lower panel) and perpendicular j xx (upper panel) diffusion coefficients versus the time for q ¼ and R L =l slab ¼ 1. The quantities shown here are normalized to the gyrofrequency X and the slab bendover scale l slab. κ zz /(l slab Ω) κ xx /(l slab Ω) tω κ zz /(l slab Ω) 6 4 Fig. 3. The simulated parallel j zz (lower panel) and perpendicular j xx (upper panel) diffusion coefficients versus the time for q ¼ 1:5 and R L =l slab ¼ 1. The quantities shown here are normalized to the gyrofrequency X and the slab bendover scale l slab t Ω Fig. 1. The simulated parallel j zz (lower panel) and perpendicular j xx (upper panel) diffusion coefficients versus the time for q ¼ :5 and R L =l slab ¼ 1. The quantities shown here are normalized to the gyrofrequency X and the slab bendover scale l slab. theory. As already described by Shalchi et al. (4), the true parallel diffusion coefficient is reduced compared to the quasilinear result. Previously it was unclear how the discrepancy between the quasilinear curve and the simulations depends on the spectrum of the two-dimensional modes. As shown in Fig. 4 the energy range spectral index q does not have a strong influence on the parallel diffusion coefficient. In Fig. 5 we have calculated the parallel mean free path by using the weakly nonlinear transport theory. The latter theory provides a result similar to the simulations: The energy range spectral index q does not have a strong influence on the parallel diffusion coefficient. For all values of q, the weakly nonlinear theory agrees well with the

7 G. Qin, A. Shalchi / Advances in Space Research 49 (1) λ 1 R=R / l L slab Fig. 4. The simulated parallel mean free path k k normalized to the slab bendover scale l slab versus the magnetic rigidity R ¼ R L =l slab for different values of the energy range spectral index q. Shown are the simulations for q ¼ :5 (stars), q ¼ (squares), and q ¼ 1:5 (dots). For comparison we have also shown the standard quasilinear parallel mean free path (dotted line) R=R L Fig. 6. The simulated perpendicular mean free path k? normalized to the slab bendover scale l slab versus the magnetic rigidity R ¼ R L =l slab for different values of the energy range spectral index q. Shown are the simulations for q ¼ :5 (stars), q ¼ (squares), and q ¼ 1:5 (dots). paper agree perfectly with the simulations of Giacalone and Jokipii (1999). Physically, the nonlinear effect shown in Figs. 4 and 5 comes due to resonance broadening caused by perpendicular diffusion as described by the weakly nonlinear theory of Shalchi et al. (4). Since quasilinear theory only describes gyro-resonant interactions, a linear description of parallel diffusion is incomplete. λ 1 R=R / l L slab 4... Results for the perpendicular mean free path In Fig. 6 we show the perpendicular mean free paths obtained numerically. We can see that the perpendicular diffusion coefficient is strongly controlled by the energy range spectral index q. This is in agreement with the prediction made by analytical theory - see below. The smallest perpendicular diffusion coefficient is obtained for the q ¼ 1:5. The values obtained for q ¼ and q ¼ :5 are an order of magnitude larger, showing how important the parameter q is. Fig. 5. The parallel mean free path k k normalized to the slab bendover scale l slab versus the magnetic rigidity R ¼ R L =l slab for different values of the energy range spectral index q. Shown is the standard quasilinear parallel mean free path (dotted line) and the results obtained by employing the weakly nonlinear theory. For the latter theory we have computed the parallel mean free path for q ¼ :5 (dashed line), q ¼ (dash-dotted line), and q ¼ 1:5 (solid line). simulations but is clearly in disagreement with quasilinear theory. Only for smaller rigidities we find that WNLT slightly overestimates the nonlinear effect. It has been shown previously that the bendover scale l D used in the model spectrum (6) has a strong influence on the strength of the nonlinear effect (see Shalchi, 7). For l D ¼ l slab, for instance, the nonlinear effect is weaker and the results provided by the code used in the current Other quantities In Figs. 7 and 8 we have computed the ratio k? =k k and the product k? k k =l D, respectively. The latter quantity is important since it controls the transport regime (see Section 3.3 of the present paper). As shown in the present paper, the ratio of perpendicular and parallel mean free path k? =k k depends strongly on the energy range of the turbulence spectrum. Therefore, we conclude, that the parameter q is a critical parameter in diffusion theory. We can also understand the latter results physically. It has been shown in previous work (see Shalchi and Weinhorst, 9) that the largest turbulence scales control the diffusion coefficient of wandering magnetic field lines. Field line wandering or meandering is one of the main contributors to the effect of particle diffusion across the mean magnetic

8 165 G. Qin, A. Shalchi / Advances in Space Research 49 (1) / λ R=R L Fig. 7. The simulated ratio k? =k k versus the magnetic rigidity R ¼ R L =l slab for different values of the energy range spectral index q. Shown are the simulations for q ¼ :5 (stars), q ¼ (squares), and q ¼ 1:5 (dots) R=R L Fig. 9. The simulated perpendicular mean free path k? normalized to the slab bendover scale l slab versus the magnetic rigidity R ¼ R L =l slab for q ¼ :5 (stars). We have compared the simulations with the results obtained by employing the original NLGC theory (dotted line) and the ENLGC (solid line). We have also shown the analytical result of Eq. (11) which is represented by the dashed line. λ / l D R=R L Fig. 8. The product k? k k =l slab versus the magnetic rigidity R ¼ R L=l slab for different values of the energy range spectral index q. Shown are the simulations for q ¼ :5 (stars), q ¼ (squares), and q ¼ 1:5 (dots). For comparison we have also shown the value k? k k =l D ¼ 3 indicating the critical value for the turnover of the analytical solutions see Eqs. (11) (13). field. Therefore, it is not a surprise that the ratio k? =k k depends on q. Furthermore, the latter effect was predicted by analytical theory as shown by Shalchi et al. (1) Comparison with analytical theory In Figs we compare our simulations with different analytical theories for perpendicular diffusion. We have shown the results provided by the original NLGC theory of Matthaeus et al. (3), the extende Matthaeus et al. (3), Shalchi (6b), and the analytical forms provided by Eqs. (11) (13). The latter equations are based on the extended NLGC theory, too. To evaluate the latter equations we have to know the parallel mean free path. In the present paper we employ the parallel mean free paths from the simulations. This is the reason why the theoretical lines fluctuate slightly. As already discussed in the aforementioned papers, there is only a small discrepancy between the different theories for the turbulence model used here. The reason for the difference between the NLGC theory and it s extended version is that in the latter theory there is no contribution due to the slab modes. For full threedimensional turbulence or a two-component model with dominant slab modes, the different versions of the NLGC theory provide different results (see, e.g., Tautz and Shalchi, 11). According to Figs our computer simulation confirm the different nonlinear guiding center theories. Only for some cases there is a factor 1.5 between analytical theory and the numerical findings of the present paper. This small discrepancy is acceptable if one takes into account the high inaccuracy of solar wind data. 5. Summary and conclusion In the present paper we have revisited the problem of cosmic ray diffusion. By using a well-established test-particle code (see, e.g., Qin et al., b, 6; Shalchi, 7)we have computed the parallel and perpendicular diffusion coefficients versus the magnetic rigidity. We derived the following new results: If we change the energy range spectral index q of the two-dimensional spectrum, we only find a small variation of the parallel mean free path in the simulations.

9 G. Qin, A. Shalchi / Advances in Space Research 49 (1) R=R L Fig. 1. The simulated perpendicular mean free path k? normalized to the slab bendover scale l slab versus the magnetic rigidity R ¼ R L =l slab for q ¼ (squares). We have compared the simulations with the results obtained by employing the original NLGC theory (dotted line) and the ENLGC (solid line). We have also shown the analytical result of Eq. (11) which is represented by the dashed line. 1 1 R=R / l L slab Fig. 11. The simulated perpendicular mean free path k? normalized to the slab bendover scale l slab versus the magnetic rigidity R ¼ R L =l slab for q ¼ 1:5 (dots). We have compared the simulations with the results obtained by employing the original NLGC theory (dotted line) and the ENLGC (solid line). We have also shown the analytical result of Eq. (1) which is represented by the dashed line. Quasilinear theory cannot reproduce the simulations due to it s inability to describe resonance broadening effects; The running parallel diffusion coefficients obtained for the different values of q, are at least nearly diffusive and not strongly sub- or superdiffusive. The weak superdiffusivity obtained by Shalchi and Kourakis (7) does not contradict the present results. The running perpendicular diffusion coefficients are weakly subdiffusive or diffusive in agreement with Shalchi and Kourakis (7). The predictions made by analytical theory concerning the influence of q are correct see Figs Improved analytical theories for perpendicular diffusion work very well, at least for the considered parameter regime. For high particle energies, the analytical theories slightly underestimate the perpendicular diffusion coefficient. The latter conclusion is in agreement with the numerical results obtained by Gao et al. (11). We conclude that the turbulence behavior in the energy range of the spectrum has a very strong influence on the perpendicular diffusion coefficient. This conclusion is in agreement with the predictions made by analytical theory (see Shalchi, 1). The intermediate scales of the inertial range of the spectrum are not important for diffusion across the mean magnetic field. Advanced nonlinear theories seem to agree very well with simulations apart from specific parameter regimes (e.g., very high energy particles). In the analytical work of Shalchi and Dosch (8), however, it was shown that the parameter a is increasing with increasing particle rigidity. The latter effects explains the difference discussed here. The parallel diffusion coefficient does not depend very much on the large scales. The latter result is new and has important implications for energetic particles in the solar wind: By choosing a specific value of q we cannot suppress the nonlinear effect. This conclusion is supported by the weakly nonlinear theory developed a few years ago (see Shalchi et al., 4). A certain problem in the context of this Shalchi and Dosch (8), Shalchi et al. (8) have shown that pitch-angle scattering in two-dimensional turbulence could be subdiffusive. Even for two-component turbulence there should not be a diffusive scattering contribution of the two-dimensional modes to the total pitchangle Fokker Planck coefficient. In order to compute the parallel diffusion coefficient, however, one has to rely in Eq. (1). This famous and very useful formula is based on the assumption of diffusive pitch-angle scattering and, therefore, it cannot be applied for subdiffusive cases. It has to be subject of future work to revisit the derivation of Eq. (1) for subdiffusive pitch-angle scattering. Then one can explore the relation between the present simulations, the weakly nonlinear transport theory, and the work done by Shalchi et al. (8). The problem discussed in the previous paragraph is also related to the so-called Palmer consensus (see, e.g., Bieber et al., 1994). In the latter paper it has been shown that quasilinear theory is able to reproduce observed interplanetary particle mean free path if a dynamical turbulence model is employed. In the present paper, however, we have shown that quasilinear theory is not correct in the general case. The only explanation for this problem could be that the two bendover scales l slab and l D are equal. In the latter case the nonlinear effect is suppressed if the parallel mean free path is computed (see Shalchi, 7).

10 165 G. Qin, A. Shalchi / Advances in Space Research 49 (1) Another question is whether there are parameter regimes for which nonlinear effects disagree with computer simulations. In the present paper we did not find such regimes apart from the small discrepancy found by Gao et al. (11). However, we expect that the nonlinear theories become invalid in the strong turbulence limit. E.g., the nonlinear theories are based on the assumption db z B if the perpendicular diffusion coefficient is calculated. Therefore, in the limit db z B, the theories discussed in the present paper are no longer reliable. Acknowledgements G. Qin acknowledges support by Grants NNSFC and NNSFC A. Shalchi acknowledges support by the Natural Sciences and Engineering Research Council (NSERC) of Canada. The computations were performed by Numerical Forecast Modeling R&D and VR System of State Key Laboratory of Space Weather and Special HPC work stand of Chinese Meridian Project. 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