Cosmic ray transport in non-uniform magnetic fields: consequences of gradient and curvature drifts

Transcription

1 J. Plasma Physics 2010), vol. 76, parts 3&4, pp c Cambridge University Press 2010 doi: /s Cosmic ray transport in non-uniform magnetic fields: consequences of gradient curvature drifts R. SCHLICKEISER 1 F. J E N K O 2 1 Institut für Theoretische Physik, Lehrstuhl IV: Weltraum- und Astrophysik, Ruhr-Universität ochum, D ochum, Germany rsch@tp4.ruhr-uni-bochum.de) 2 Max-Planck-Insitut für Plasmaphysik, EURATOM Association, Garching, Germany Received 2 October 2009 accepted 16 November 2009, first published online 8 January 2010) Abstract. Large-scale spatial variations of the guide magnetic field of interplanetary interstellar plasmas give rise to the mirror force p 2 /2mγ). The parallel component of this mirror force causes adiabatic focusing of the cosmic ray guiding center whereas the perpendicular component of the mirror force gives rise to the gradient curvature drifts of the cosmic ray guiding center. Adiabatic focusing the gradient curvature drift terms additionally enter the Fokker Planck transport equation for the gyrotropic cosmic ray particle phase space density in partially turbulent non-uniform magnetic fields. For magnetohydrodynamic turbulence with dominating magnetic fluctuations, the diffusion approximation is justified, which results in a modification of the diffusion convection transport equation for the isotropic part of the gyrotropic phase space density from the additional focusing drift terms. For axisymmetric undamped slab Alfvenic turbulence we show that all perpendicular spatial diffusion coefficients are caused by the non-vanishing gradient curvature drift terms. For a specific symmetric in μ) choice of the pitch-angle Fokker Planck coefficients we show that the ratio of the perpendicular to parallel spatial diffusion coefficients apart from a constant is determined by the spatial first derivatives of the non-constant cosmic ray Larmor radius in the non-uniform magnetic field. 1. Introduction Transport of cosmic ray particles in partially turbulent interstellar interplanetary electromagnetic fields is often treated by a Fokker Planck approach with a dominating uniform guide magnetic field with superposed electric magnetic fluctuations. However, it is well established by direct in-situ interplanetary magnetometer measurements by radio continuum surveys of our own external We dedicate this work to Prof. Dr. Padma K. Shukla at the occasion of his 60th birthday. For the last 12 years, Padma has been an esteemed valuable colleague of RS at the Institute of Theoretical Physics, Ruhr-University ochum, helping with his international contacts collaborations to establish maintain an enthusiastic stimulating research atmosphere at the institute.

2 318 R. Schlickeiser F. Jenko spiral galaxies that interplanetary interstellar guide magnetic fields are nonuniform on spatial scales are much larger than the Larmor radii of high-energy cosmic ray particles. It is well established that a spatially varying guide magnetic field gives rise to the adiabatic focusing of charged particles Roelof 1969; Earl 1974; Kunstmann 1979; Spangler asart 1981; ieber et al. 2002), which is closely related to the parallel component of the mirror force p 2 /2mγ) in the nonuniform field. In previous work, Schlickeiser Shalchi 2008) Schlickeiser 2009) investigated the consequences of the adiabatic focusing term in the Fokker Planck equation for the derivation of the diffusion convection transport equation for the isotropic part of the cosmic ray phase space distribution function. The adiabatic focusing term gives rise to new additional relevant terms in the diffusion convection transport equation as was demonstrated by applying the diffusion approximation Schlickeiser et al herafter referred to as paper I). Here we investigate the consequences of the perpendicular component of the mirror force which gives rise to the so called gradient curvature drifts oyd Serson 1969) of the cosmic ray guiding center. ecause these perpendicular drifts depend on the charge sign of the cosmic ray particle, we obtain charge-sign dependent transport acceleration terms in the diffusion convection transport equation of cosmic rays which may provide an alternative explanation of the recently observed dramatic rise in the positron fraction of local galactic cosmic ray electrons starting at 10 GeV to about 300 GeV by the PAMELA satellite experiment Adriani et al. 2009). We start in Section 2 with the guiding center approximation of cosmic ray transport in non-uniform magnetic fields leading to new transport terms in the cosmic ray Fokker Planck equation. In Section 3, we apply the diffusion approximation to the modified Fokker Planck equation implying new transport acceleration terms resulting from the guiding center drift effects. In Section 4, we investigate the consequences of the new transport equation for galactic positrons electrons. 2. Guiding center approximation When the characteristic scales of the spatial magnetic field variations are large compared to the Larmor radius of the charged cosmic ray particle of mass m a charge q a, one can solve the equation of motion of the cosmic ray particle using the adiabatic guiding center approximation oyd Serson 1969; Cap 1970). Here one is not interested in the gyromotion around the magnetic field, but considers only the motion of guiding center R =X, Y, Z) = x + c p /q a 2 ). In a weakly non-uniform magnetic field =0, y,), with y, 2.1) the energy conserving motion is governed by the constancy of the magnetic moment M = pv1 μ 2 ) /2 2,whereμ = p /p = p z /p is the cosine of the cosmic ray particle pitch-angle. The parallel component of the resulting equation of motion for the guiding center is where ṗ = F M = M ), 2.2) F M = M 2.3)

3 Cosmic ray transport in non-uniform magnetic fields 319 is the mirror force. Equation 2.2) readily yields for the change in pitch-angle the adiabatic focusing term μ = ṗ p = v1 μ2 ) ) = v1 μ2 ), 2.4) 2 2 where we introduce the focusing length along the guide magnetic field L 3 = z. 2.5) The perpendicular component of the mirror force 2.3) causes the gradient drift with the velocity d R dt G =Ẋ,Ẏ,0) F G = c M q a 2 = cm q a ) The curvature of the field lines causes another drift perpendicular to the field resulting from the perpendicular centrifuginal force acting on the particle oyd Serson 1969). The resulting curvature drift velocity is d R dt C =Ẋ,Ẏ,0) C = cvpμ2 q a 4 ). 2.7) The derivation of gyrokinetic plasma transport equations Antonsen Lane 1980; Catto et al. 1981) guarantees that no additional drift terms result. Combining the two drifts yields d R dt GC = d R dt G + d R dt pcv 1+μ 2 C =Ẋ,Ẏ,0) = q a μ 2 ) 1+μ 2 =ɛ a r L v μ 2 ), 2.8) where we have introduced the cosmic ray Larmor radius r L = pc/ q a ) charge-sign ɛ a = q a / q a. Consequently, we obtain for the change in perpendicular guiding center coordinates 1+μ 2 Ẋ = ɛ a r L v ) x μ 2 ) x, 2.9) 1+μ 2 Ẏ = ɛ a r L v ) y μ 2 ) y, 2.10) respectively. For the field configuration 2.1) we infer ) x 2 = y z y 2 y = 1 L 2, 2.11) ) y 2 = x 1 L 1, 2.12) ) x = y z y y = 1 L 2, 2.13)

4 320 R. Schlickeiser F. Jenko ) y = x = 1, 2.14) L 1 where we introduce the perpendicular magnetic field scale lengths L 1 = x, L 2 = y. 2.15) For the change in perpendicular guiding center coordinates 2.9) 2.10) we then obtain Ẋ ɛ ar L v 1 μ 2 ), Ẏ ɛ ar L v 1 μ 2 ). 2.16) 2L 2 2L 1 The additional adiabatic focusing 2.4) drift 2.16) terms enter the Fokker Planck equation for the gyrotropic particle phase space density fx, Y, z, p, μ, t) Schlickeiser 2002) which then reads f t + vμ f z + X Ẋf)+ Y Ẏf)+ μ μf) p 2 p p2ṗ loss f Sz,X,Y,p,t) =p 2 ν p 2 D νσ σ f. 2.17) From 2.15) we note the equivalence of ) f f X Y L 2 L 1 ) = 1 L 2 f X 1 L 1 f Y, 2.18) which both will be used in the following. The Fokker Planck 2.17) becomes f t + vμ f z + v 1 μ 2 ) f + ɛ avr L 1 μ 2 ) f 2 μ 2L 2 X ɛ avr L 1 μ 2 ) f 2L 1 Y p 2 p p2ṗ loss f Sz,X,Y,p,t) =p 2 ν p 2 D νσ σ f = μ D μμ μ f + p 2 μ p 2 D μj j f ) + p 2 k p 2 D kμ μ f ) + p 2 k p 2 D kj j f ). 2.19) Here we use the Einstein sum convention for indices the short notation ν = / x ν ). x ν μ, p, X, Y x σ μ, p, X, Y represent the four phase space variables μ, p, X, Y with non-vanishing Fokker Planck coefficient D νσ. We also introduce the Latin indices k, j representing x k,j p, X, y but not μ. Therefore the terms on the right h side of 2.19) in general represent 16 different Fokker Planck terms; however, depending on the specific type of turbulence considered, not all of them are non-zero, some of them are much larger than others, e.g. for magnetostatic turbulence only D μμ is not vanishing. The term p 2 p p 2 ṗ loss f with the positively counted loss rate ṗ loss p) accounts for additional continuous momentum loss processes resulting from the interactions of cosmic rays with ambient target photon matter fields such as Coulomb interactions, ionization, pion production, bremsstrahlung, synchrotron radiation inverse Compton interactions. The Fokker Planck coefficients account for the rom electromagnetic forces resulting from weak plasma turbulence. They can be calculated using quasilinear theory Schlickeiser 2002) or from nonlinear improvements of quasilinear theory such as nonlinear guiding center theory extended second-order quasilinear

5 Cosmic ray transport in non-uniform magnetic fields 321 theory for a recent review see Shalchi 2009) including also effects from magnetic field line wering Hauff et al. 2009). 3. Diffusion approximation For magnetohydrodynamic turbulence, the fluctuating electric fields are much smaller than the fluctuating magnetic fields δe δ). In this case the gyrotropic particle phase space distribution function fx, Y, z, p, μ, t) due to dominating pitchangle diffusion adjusts very quickly to a quasi-equilibrium through pitch-angle diffusion which is close to the isotropic equilibrium distribution F X, Y, z, p, t). Therefore the diffusion approximation Jokipii 1966; Hasselmann Wibberenz 1968; Skilling 1975; Schlickeiser 1989) to the particle motion is justified where one splits the phase space density into its average in pitch-angle cosine an anisotropic part where fx, Y, z, p, μ, t) =F X, Y, z, p, t)+gx, Y, z, p, μ, t), 3.1) F X, Y, z, p, t) = 1 2 dμ fx, Y, z, p, μ, t), 3.2) dμ gx, Y, z, p, μ, t) =0 3.3) where g F. We restrict our analysis to isotropic source terms Sz,X,Y,p,t) loss rates ṗ loss p). We now repeat the calculations of paper I but keeping the additional adiabatic focusing drift terms. As the individual steps of the calculations are almost identical, apart from the contributions from the additional focusing drift terms, we merely note the resulting modifications to the equations of paper I. Substituting 3.1) into 2.19), averaging over μ using g F yields instead of I-17) the pitch-angle averaged equation: F t S p 2 p p2ṗ loss F + v 1 dμμg + ɛ ) avr L F X ɛ ) avr L F Y 2 z 3 L 2 3 L 1 ɛ ) avr L X L 2 dμμ 2 g + ɛ avr L 4 4 = 1 ) 2p 2 k p 2 dμd kμ μ g Y L 1 dμμ 2 g + 1 2p 2 k p 2 dμd kj j F ) ). 3.4) 3.1. Cosmic ray anisotropy Subtracting 3.4) from 2.19) taking leading terms as in paper 1 we obtain for the cosmic ray anisotropy g μ D μμ μ g + D μj j F v 1 μ 2 )F 2 vμ F z + ɛ avr L 1 3μ 2 ) X F ɛ avr L 1 3μ 2 ) Y F. 3.5) 6L 2 6L 1

6 322 R. Schlickeiser F. Jenko Integrating 3.5) over μ we obtain D μμ g μ + D μj j F = c 1 + vμ2 2 F z + ɛ avr L μ 6L 2 v 2 1 μ 2 )F 1 μ 2 ) X F ɛ avr L μ 6L 1 1 μ 2 ) Y F. 3.6) The integration constant c 1 is determined from the requirement that the left h side of 3.6) vanishes for μ = ±1, yielding c 1 = v F 2 z, so that 3.6) becomes g μ = 1 μ2 )v 2D μμ F z F + ɛ avr L μ1 μ 2 ) X F 6L 2 D μμ ɛ avr L μ1 μ 2 ) Y F D μj j F. 3.7) 6L 1 D μμ D μμ Integrating 3.7) over μ results in g =c 2 v F 2 z F μ dx 1 x2 D μμ x) + ɛ μ avr L dx x1 x2 ) 6L 2 D μμ x) X F ɛ avr L 6L 1 μ dx x1 x2 ) μ D μμ x) Y F dx D μjx) D μμ x) j F. 3.8) The integration constant c 2 is determined by the condition 3.3) yielding c 2 = v F 4 z F dμ 1 μ)1 μ2 ) ɛ avr L dμ μ1 μ)1 μ2 ) X F D μμ x) + ɛ avr L dμ μ1 μ)1 μ2 ) Y F + 1 D μμ x) 2 Inserting 3.9) yields for the anisotropy 3.8) gz, X, Y, p, μ, t)= dμ 1 μ)1 μ2 ) μ 2 ɛ avr L dμ μ1 μ)1 μ2 ) D μμ x) + ɛ avr L + dμ μ1 μ)1 μ2 ) D μμ x) dμ 1 μ)d μjμ) dμ 1 μ)d μjμ) j F. 3.9) dx 1 x2 v F D μμ x) 4 z F μ 2 dx x1 x2 ) X F D μμ x) μ 2 dx x1 x2 ) Y F D μμ x) μ 2 dx D μjx) 1 D μμ x) 2 j F. 3.10) Obviously, the anisotropy 3.10) now consists of five contributions. The first is related to pitch-angle scattering the parallel spatial gradient of F. The second new contribution results from the gradient curvature drift motion perpendicular gradients of F. The last term sum over j = p, X, Y ) describes three

7 Cosmic ray transport in non-uniform magnetic fields 323 anisotropies from the momentum gradient of F, which has to be identified with the Compton Getting effect Compton Getting 1935), perpendicular gradients of F together with rom plasma turbulence forces, which agrees with the anisotropies derived in paper 1. Obviously, these last anisotropies compete with the gradient curvature drift anisotropy represented by the second third term in 3.10). The new anisotropy contributions modify the diffusion convection transport equation for the isotropic part F of the cosmic ray distribution function as we demonstrate next Modified diffusion convection equation With the anisotropy 3.10) we now calculate the integrals involving the anisotropy g appearing in the pitch-angle averaged 3.4). In terms of the pitch-angle integrals K 0 = K 2 = h k = we obtain from 3.7) Likewise, 3.8) yields dμμg = vk 0 F 4 z F Equation 3.10) provides dμμ 2 g = vk 1 F 6 z F dμ 1 μ2 ) 2,K 1 = dμ μ1 μ2 ) 2, 3.11) dμ μ2 1 μ 2 ) 2,a k = dμ 1 μ2 )D kμ, 3.12) D μμ dμ μ1 μ2 )D 1 kμ, c kj = dμ D kμd μj, 3.13) D μμ D μμ g dμd kμ μ = va k F 2 z F c kj j F + ɛ avr L h k X F ɛ avr L h k Y F. 3.14) 6L 2 6L 1 a j 2 j F + ɛ avr L K 1 X F ɛ avr L K 1 Y F. 3.15) h j 3 j F + ɛ avr L K 2 18L 2 X F ɛ avr L K 2 18L 1 Y F. 3.16) In the derivation we use the symmetry D νσ = D σν. Inserting 3.14) 3.16) into 3.4) we obtain the modified diffusion convection transport equation F t S p 2 p p2ṗ loss F + ɛ avr L X F ɛ avr L Y F vaj 3L 2 3L 1 z 4 j F v 2 K 0 F z 8 z F + ɛa v 2 r L K 1 X F ɛa v 2 r L K 1 Y F z 24L 2 z 24L 1 ɛa v 2 r L K 1 F + X 24L 2 z F ) ɛa v 2 r L K 1 F Y 24L 1 z F )

8 324 R. Schlickeiser F. Jenko ɛa vr L h j ɛa vr L h j v 2 rl 2 + X j F Y j F K 2 X 72L 2 2 v 2 rl 2 K 2 v 2 rl 2 Y 72L 2 Y F + K 2 X Y F 1 72L 1 L 2 = p 2 k p ) dμd kj c kj j F va k ɛ avr L h k X F ɛ avr L h k Y F X F + Y v 2 r 2 L K 2 72L 1 L 2 X F F z F 3.17) with k, j p, X, Y. In its most general form the modified diffusion convection transport equation is rather involved. In the following we therefore investigate a special class of plasma turbulence that simplifies the equation enormously. 4. Axisymmetric isospectral undamped slab Alfvenic turbulence As in Schlickeiser Shalchi 2008) we consider the isospectral case of undamped Alfvenic slab turbulence where the magnetic cross helicities are constants independent of wave number. Whereas the magnetic helicities σ ± indicate the polarization state of the forward backward moving Alfven waves, the cross helicity H c =2r refers to the relative abundance r = I + /I + + I ) of forward backward moving Alfven waves. According to Ch. 13 of Schlickeiser 2002) in this case the only non-vanishing Fokker Planck coefficients are D μμ, D μp,d pp. Due to the vanishing Fokker Planck coefficients D μx = D μy = D XX = D YY = D XY = D px = D py =0only the gradient curvature drift terms contribute to the perpendicular spatial diffusion of cosmic ray particles in contrast to the analysis in paper I. As a consequence, the only vanishing pitch-angle integrals in 3.13) are a p, h p c pp, so that the sum over k on the right h side of 3.17) provides non-zero contributions only for k = p. After straightforward but tedious algebra 3.17) reduces to the form F t S + 1 vk ) 1 ɛa vr L F 1 vk ) 1 ɛa vr L F + ) κzz F X 8 3L 2 Y 8 3L 1 z 1 p 2 p 2 ṗ loss + a zpp 2 ) F p X κ XX κ YX κ zx a Xp X F = Y κ YX κ YY κ zy a Yp Y F, 4.1) z κ zx κ zy κ zz a zp z F p 2 p p 2 a Xp a Yp a zp A p F with the pitch-angle averaged transport parameters A= 1 1 dμ D pp μ) D2 μpμ), 4.2) 2 κ zz = v2 K 0 8 = v2 8 dμ 1 μ2 ) 2, 4.3)

9 Cosmic ray transport in non-uniform magnetic fields 325 κ XX = v2 r 2 L K 2 72L 2 2 κ YY = v2 r 2 L K 2 72L 2 1 = v2 r 2 L 72L 2 2 = v2 rl 2 72L 2 1 dμ μ2 1 μ 2 ) 2, 4.4) dμ μ2 1 μ 2 ) 2, 4.5) κ YX = v2 r 2 L K 2 72L 1 L 2 = v2 rl 2 72L 1 L 2 dμ μ2 1 μ 2 ) 2, 4.6) κ zx = ɛ av 2 r L K 1 24L 2 = ɛ av 2 r L 24L 2 dμ μ1 μ2 ) 2, 4.7) κ zy = ɛ av 2 r L K 1 24L 1 = ɛ av 2 r L 24L 1 dμ μ1 μ2 ) 2, 4.8) a Xp = ɛ avr L h p = ɛ avr L dμ μ1 μ2 )D μp μ), 4.9) a Yp = ɛ avr L h p = ɛ avr L a zp = va p 4 = v 4 dμ μ1 μ2 )D μp μ), 4.10) dμ 1 μ2 )D μp μ), 4.11) respectively. We notice immediately that all perpendicular spatial diffusion coefficients are caused by the non-vanishing gradient curvature drift terms. For infinitely large perpendicular scale length L 1 = L 2 = ), 4.1) reduces to 17) of Schlickeiser Shalchi 2008) which refers to the linear phase space density Fz). Moreover, we note that the spatial diffusion coefficients κ zz, κ XX, κ YX κ YY are independent from the charge-sign of the cosmic ray particle. At the contrast, the convective transport terms a Xp a Yp as well as the spatial diffusion coefficients κ zx κ zy exhibit the cosmic ray charge-sign dependence. 5. Nonparallel spatial diffusion coefficients For a specific symmetric in μ) choice of the pitch-angle Fokker Planck coefficients we determine here the ratios of the perpendicular to parallel spatial diffusion coefficients. For a magnetic power spectrum of Alfvenic slab turbulene P k) k s with s<2, the quasilinear pitch-angle Fokker Planck coefficients is Jaekel Schlickeiser 1992) v 2 D μμ = μ s 1 μ 2 ), 5.1) 22 s)4 s)κ zz consistent with the parallel spatial diffusion coefficient 4.3). With 5.1) we immediately obtain for the two integrals K 1 = dμ μ1 μ2 ) 2 =0, 5.2)

10 326 R. Schlickeiser F. Jenko K 2 = dμ μ2 1 μ 2 ) 2 = 22 s)κ zz 6 s)v ) Consequently, we obtain κ zx = κ zy = 0, for the ratios of the remaining perpendicular to parallel spatial diffusion coefficients mean free paths κ XX = λ XX = 2 s ) 2 rl = 2 s ) 2 rl, κ zz λ zz 6 s 6L s) Y 5.4) κ YY = λ YY = 2 s ) 2 rl = 2 s ) 2 rl, κ zz λ zz 6 s 6L s) X 5.5) κ YX = λ YX = 2 s ) ) rl rl = 2 s r L r L κ zz λ zz 6 s 6L 1 6L s) X Y, 5.6) involving the spatial first derivatives of the non-constant cosmic ray Larmor radius in the non-uniform magnetic field. Due to the Larmor radius dependence, the ratios of perpendicular to parallel spatial diffusion coefficients increase proportional p/ q a ) 2 ) to the square of the cosmic ray particle rigidity. Moreover, cosmic rays with the same rigidity value have the same ratios of perpendicular to parallel spatial diffusion coefficients. 6. Summary discussion Large-scale spatial variations of the guide magnetic field of interplanetary interstellar plasmas give rise to the mirror force p 2 /2mγ). The parallel component of this mirror force causes adiabatic focusing of the cosmic ray guiding center whereas the perpendicular component of the mirror force gives rise to the gradient curvature drifts of the cosmic ray guiding center. Adiabatic focusing the gradient curvature drift terms additionally enter the Fokker Planck transport equation for the gyrotropic cosmic ray particle phase space density in partially turbulent non-uniform magnetic fields. For magnetohydrodynamic turbulence with dominating magnetic fluctuations δe δ) the diffusion approximation is justified, which results in a modification of the diffusion convection transport equation for the isotropic part of the gyrotropic phase space density from the additional focusing drift terms. Its transport parameters are pitch-angle averages of Fokker Planck coefficients which are obtained by using quasilinear theory or nonlinear improvements of quasilinear theory such as nonlinear guiding center theory extended second-order quasilinear theory including also effects from magnetic field line wering. For axisymmetric undamped slab Alfvenic turbulence we show that all perpendicular spatial diffusion coefficients are caused by the non-vanishing gradient curvature drift terms. While the spatial diffusion coefficients κ zz, κ XX, κ YX κ YY are independent from the charge-sign of the cosmic ray particle, we find that the convective transport terms a Xp a Yp as well as the spatial diffusion coefficients κ zx κ zy athe cosmic ray charge-sign dependence. For a specific symmetric in μ) choice of the pitch-angle Fokker Planck coefficients we show that κ zx = κ zy =0.Theratioof the remaining perpendicular to parallel spatial diffusion coefficients apart from a constant is determined by the spatial first derivatives of the non-constant cosmic ray Larmor radius in the non-uniform magnetic field.

11 Cosmic ray transport in non-uniform magnetic fields 327 Acknowledgements R. Schlickeiser gratefully acknowledges partial support by the Deutsche Forschungsgemeinschaft through the grant Schl 201/19-1. References Adriani, O., arbarino, G. C., azilevskaya, G. A., ellotti, R., oezio, M., ogomolov, E. A., onechi, L., ongi, M., onvicini, V., ottai, S. et al An anomalous positron abundance in cosmic rays with energies GeV. Nature 458, 607. Antonsen, T. M. Lane, Kinetic equations for low frequency instabilities in inhomogeneous plasmas. Phys. Fluids 23, ieber, J. W., Dröge, W., Evenson, P. A., Pyle, R., Ruffalo, D., Pinsook, U., Tooprakai, P., Rujiwarodom, M., Khumlumlert, T. Krucker, S Energetic Particle Observations during the 2000 July 14 Solar Event. ApJ 567, 622. oyd, T. J. M. Serson, J. J Plasma Dynamics. London: Thomas Nelson Sons. Cap, F Einführung in die Plasmaphysik I. erlin: Akademie-Verlag, erlin. Catto, P. J., Tang, W. M. aldwin, D. E Generalized gyrokinetics. Plasma Phys. 23, 639. Compton, A. H. Getting, I. A An apparent effect of galactic rotation on the intensity of cosmic rays. Phys. Rev. 47, 817. Earl, J. A The diffusive idealization of charged-particle transport in rom magnetic fields. ApJ 193, 231. Hasselmann, K. Wibberenz, G Scattering of charged particles by rom electromagnetic fields. Z. Geophysics 34, 353. Hauff, T., Jenko, F., Shalchi, A. Schlickeiser, R Scaling theory for cross-field transport of cosmic rays in turbulent fields. ApJ submitted). Jaekel, U. Schlickeiser, R Cosmic ray transport I. The Fokker-Planck coefficients in rom electromagnetic fields. J. Phys. G 18, Jokipii, J. R Cosmic-Ray Propagation. I. Charged Particles in a Rom Magnetic Field. ApJ 146, 480. Kunstmann, J A new transport mode for energetic charged particles in magnetic fluctuations superposed on a diverging mean field. ApJ 229, 812. Roelof, E. C Propagation of solar cosmic rays in the interplanetary magnetic field. In: Lectures in High Energy Astrophysics eds. H. Ögelman J. R. Wayl), NASA SP Scientific Technical Information Division, Office of Technology Utilization, NASA, Washington, DC, p Schlickeiser, R Cosmic ray transport acceleration I. Derivation of the kinetic equation application to cosmic rays in static cold media. ApJ 336, 243. Schlickeiser, R Cosmic Ray Astrophysics. Heidelberg, Germany: Springer. Schlickeiser, R First-order distributed Fermi acceleration of cosmic ray hadrons in nonuniform magnetic fields. Modern Phys. Lett. A 24, Schlickeiser, R., Dohle, U., Tautz, R. C. Shalchi, A A new type of cosmic ray anisotropy from perpendicular diffusion I. Modification of the spatial diffusion tensor the diffusion-convection cosmic ray transport equation. ApJ 661, 185 paper I). Schlickeiser, R. Shalchi, A Cosmic ray diffusion approximation with weak adiabatic focusing. ApJ 686, 292. Shalchi, A Nonlinear Cosmic Ray Diffusion Theories, Astrophysics Space Science Library, Vol erlin: Springer. Skilling, J Cosmic ray streaming. I - Effect of Alfven waves on particles MNRAS 172, 557. Spangler, S. R. asart, J. P A model for energetic electron transport in extragalactic radio sources. ApJ 243, 1103.