Compound Perpendicular Diffusion of Cosmic Rays and Field Line Random Walk, with Drift

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1 Compound Perpendicular Diffusion of Cosmic Rays and Field Line Random Walk, with Drift G.M. Webb, J. A. le Roux, G. P. Zank, E. Kh. Kaghashvili and G. Li Institute of Geophysics and Planetary Physics, University of California Riverside, Riverside CA 9252, USA Space Science Center, University of New Hampshire, Morse Hall, Durham, NH 03824, USA Space Science Laboratory, University of California, Berkeley, CA Abstract. A Chapman-Kolmogorov equation describing compound transport of cosmic rays across the magnetic field, due to random walk of the field lines is investigated. The probability distribution (pdf for the particle transport across the field P, is given as a convolution of the pdf for random walk of the magnetic field, P FRW, with the pdf P p, for particle transport relative to the random walking field. The model generalizes the previous work of Webb et al. ], by including the effects of advection, drift and local perpendicular diffusion of the particles. At late times, it is found that the effective cross-field diffusion coefficient κ e f f = κ + κ F where κ is the local perpendicular diffusion coefficent, and κ F is the perpendicular diffusion coefficient due to field line random walk and due to advection and drift of the particles. At early times the particles undergo compound diffusion across the field. A telegrapher model for P p indicates that at the earliest times, the particles diffuse across the field due to field line random walk. Keywords: Cosmic Rays, Compound Diffusion, Advection, Diffusion, Field Line Random Walk PACS: sh, 05.0.Gg, a. INTRODUCTION Numerical simulations and analytical theory of compound and perpendicular diffusion of cosmic rays in turbulent and random magnetic fields is a subject of ongoing debate (e.g. Jokipii, 2], Jokipii and Parker 3], Kota and Jokipii, 4], Matthaeus et al. 5], Zank et al. 6], Giacalone and Jokipii 7], Shalchi 8], Webb et al. ]. In this paper, we extend the Chapman-Kolmogorov equation approach of Webb et al. ] to include local crossfield diffusion, advection and drift, and study their effects on the cross-field transport. 2. THE MODEL The basis of our analysis for the compound perpendicular transport of cosmic rays across the random walking magnetic field is the Chapman-Kolmogorov equation: P (x,y,t x 0,y 0,t 0 = dx dy dz P FRW (x,y z;x 0,y 0,z 0 P p (x,y,z,t x,y,z 0,t 0. ( In (, P is the probability that the particle moves a step x = (x x 0,y y 0,0 across the mean field, in a time t = t t 0 ; P FRW is the probability that the random magnetic

2 field consists of a step x = (x x 0,y y 0,0 across the mean field corresponding to a step z = z z 0 along the field; P p gives the probability distribution function (pdf, describing the particle transport relative to the random walking magnetic field. The pdf P FRW, for field line random walk in ( is assumed to obey Gaussian statistics, and to have the form: P FRW (x,y z = 4πD L z z 0 exp ( (x x (y y 0 2 4D L z z 0, (2 where D L = ( x 2 /(2 z is the field line random walk diffusion coefficient. The pdf P p for the particle propagation relative to the magnetic field is assumed to obey the advection-diffusion equation for particle transport derived by Parker 9], but in which, for analytical simplicity we neglect the energy changes of particles due to adiabatic deceleration (acceleration. Thus, f = P p is assumed to satisfy the advectiondiffusion and drift transport equation: f t + V f κ 2 f z 2 κ ( 2 f x f y 2 = 0, (3 where V = u + V D is the total advection speed due to both the plasma bulk velocity u and the drift velocity V D (e.g. Jokipii, Levy and Hubbard 0]. In general V D = (κ A e B, (4 is the drift velocity of the particles due to curvature and gradient drifts and drifts parallel to the magnetic field. Here κ A is the anti-symmetric component of the diffusion tensor associated with particle drifts. In the weak scattering limit, κ A vr g /3 where v is the particle speed and r g = pc/zeb is the particle gyro-radius, p is the particle momentum, Ze the particle charge and c is the speed of light. For the sake of analytical tractability, we assume that V, u, V D, κ and κ can be regarded as constants. This approximation can be justified in the present analysis, since the averaging implicit in the Chapman- Kolmogorov equation ( is over many correlation lengths of the field, which can still be over a limited spatial region, over which the above parameters are approximately constant (i.e. we are using a multiple scales approach. The particle propagator P p = f, is the solution of the advection-diffusion equation (3 with initial conditions: The required solution for P p is of the form: where P p (z,t z 0,t 0 = P p (x,y,t x,y,t 0 = f (x,y,z,t 0 = δ(x x δ(y y δ(z z 0. (5 P p = P p (z,t z 0,t 0 P p (x,y,t x,y,t 0, (6 ( exp (z z 0 V z t 2 (4πκ t /2 4κ t 4πκ t exp H( t, (7 ( (x x V x t 2 + (y y V y t 2 4κ t, (8

3 describe the particle transport parallel and perpendicular to the mean magnetic field, and t = t t FOURIER-LAPLACE TRANSFORMS AND ASYMPTOTICS To investigate the long-time, large space-scale behavior, and early time, short lengthscale behavior of P (x,y,t x 0,y 0,t 0, it is useful to determine the Fourier-Laplace transform of P defined as: P (k,k 2,s = d x dỹ d t exp s t + i(k x + k 2 ỹ]p (x,y,t x 0,y 0,t 0, (9 0 where x = x x 0 V x t, ỹ = y y 0 V y t, (0 are the transverse position coordinates of the particle in a frame moving with the net advection velocity V = (V x,v y,0 perpendicular to the field, and k = (k,k 2,0 is the transverse wavenumber. Using the Chapman-Kolmogorov equation expression ( for P in (9 note that P is a convolution with respect to x and y ], we obtain the equation: for P = /Φ(k,s where Φ(k,s = Φ(k,s P (k,s =, ( σ( σ + λ+ ( σ + λ, (2 σ + λ λ ± = k 2 D L κ ν cd, λ = k 2 D L κ, (3 σ = s + k 2 κ + ν cd, ν cd = V z 2. (4 4κ In (3 ν cd = /(4τ cd where τ cd = κ /Vz 2 is the advection diffusion time scale along the field, and l cd = κ /V z is the corresponding length scale. 3. Late time and long space scale expansion Lowest order balance of terms for small s and k 2, with s k2 in (2 gives Φ = s + k 2 κ + κ F k 2 κ2 F 4ν cd k 4 + O(s3, κ F = V z D L. (5 Here, κ F = V z D L is the perpendicular diffusion due to field line random walk, and the bulk advective transport of particles parallel to B.

4 Using the usual Laplace-Fourier transform associations: s t, k i, k 2 2, (6 (2 and (6 give the long wavelength, and long time scale evolution equation: t (κ + κ F 2 κ D 2 L 4 ] P ( x,t = δ( x δ(t, (7 for P ( x,t. Since x = x V t, (7 may be re-written in the form: t + V (κ + κ F 2 κ D 2 L 4 ] P ( x,t = δ( x δ(t. (8 Thus, at late times and long length scales, the cross-field diffusion coefficient κ e f f = κ + κ F consists of the kinetic, (microscopic perpendicular diffusion coefficient, plus the field line random walk diffusion coefficient κ F = V z D L due to bulk advection and drift parallel to B. The higher order, fourth derivative term κ D 2 L 4 represents the late time effects of compound diffusion. 3.2 Early time, short space-scale expansion An inspection of Φ(k,s reveals that at large s and k, balance of terms requires s k 4. Thus at large s and k we obtain: Φ(k,s = s + D L κ /2 s /2 k 2 + O(k 2 ]. (9 Inverting (9 back to ( x,t space, we obtain: t D L κ /2 ( t /2 2 ]P ( x,t = δ( x δ(t, (20 as the evolution equation for P ( x,t. The fractional Fokker Planck equation (20 can be written in the more revealing form: P t { t } D L κ t π (t t /2 2 P (x,t dt = δ(x δ(t, (2 0 which is the fractional diffusion equation obtained by Webb et al. ] to describe compound diffusion of cosmic rays in the case where the particles only diffuse parallel to the random walking magnetic field. Comment: The general solution for P can be expressed in the form: P = d z (4πκ t /2 4π(κ t + D L z ( exp ( z V z t 2 x 2 + ỹ 2. (22 4κ t 4(κ t + D L z

5 κ (t/κ 0.8 κ (t/κ V z 2 t/κ V z 2 t/κ FIGURE. Running diffusion coefficient κ ( t/κ versus t = V 2 z t/κ (left panel 0 < t <.6, right panel 0 < t < MOMENTS Of particular interest are the second order moments for x 2 and ỹ 2. We find that x 2 = ỹ 2 and ( x 2 = 2κ t + 2D L V z t erf V z t /2 κ D 2 L + 4 t /2 exp( V z 2 t. (23 π 4κ 2κ /2 The moment xỹ = 0. Using (23 we obtain x 2 4(κ D 2 L /π/2 ( t /2 at small t, which is characteristic of compound diffusion. At late times, we obtain: x 2 2(κ + V z D L t due to microscopic diffusion (κ and field line random walk (κ F = V z D L. The effective normalized, running diffusion coefficient from (23 is given by: κ ( t = x2 2 t = α + β erf ( t π t exp ( t 4 ]. (24 Here κ ( t = κ e f f /κ, t = Vz 2 t/κ and x = V z x/κ are the normalized perpendicular diffusion coefficient, time and x-position coordinate based on the advection-diffusion time and length scales. The parameters α and β are defined by the equations: α = κ /κ, β = κ F /κ, κ F = V z D L. (25 Figure shows a plot of κ ( t versus t for the case α = 0. and β = 0.2. The running diffusion coefficient κ ( t is a monotonic decreasing function of t, with κ ( t t /2 at small t, during the compound diffusion phase, and κ ( t α + β as t in the late time diffusion phase. The lower horizontal dashed line corresponds to the value of κ without field line random walk, whereas, the upper dashed horizontal line includes field line random walk, i.e., κ ( t = α + β. In compound diffusion the particle must

6 have undergone many scatters along the field line. This requires that t >> T = D 2 L /κ, or t >> β 2 (Chuvilgin and Ptuskin, ]. The vertical dashed line corresponds to the condition t = β 2. For t < β 2 other physical effects, not included in the model come into play (e.g. non-diffusive particle transport, due to ballistic or telegrapher equation effects, and field line decorrelation effects due to the magnetic field stochastic instability (e.g. Rechester and Rosenbluth 2]. 5. CONCLUSIONS In this paper we have investigated a Chapman-Kolmogorov equation model of compound and perpendicular diffusive transport in a model, in which the particles undergo diffusion and advective transport relative to the random walking magnetic field. The main result is that at late times, the effective cross-field diffusion coefficient κ e f f = κ + κ F where κ is the local cross field diffusion coefficient and κ F = V z D L is the perpendicular diffusion coefficient due to field line random walk, and advection and drifts (V z = (u + V D e B where e B = B/B is the unit vector along B. The above asymptotic value of κ e f f is achieved on a time scale of a few advection diffusion time scales T d = κ /Vz 2. Compound diffusion and perpendicular diffusion both occur at earlier times, with compound diffusion dominating at early times. ACKNOWLEDGMENTS GMW and GPZ were supported in part by NASA grants NN05GG83G and NSF grant nos. ATM and ATM EK was supported in part by NASA grants NNG05GM57G and NSF grants: ATM and ATM JAlR was supported in part by NASA grant NNX07AI64G. GL was supported in part by NASA grants NNG04GF83G and NNG05GH38G. REFERENCES. G. M. Webb, et al., Ap. J. 65, 2 ( J. R. Jokipii, Ap. J. 46, 480 ( J. R. Jokipii, and E. N. Parker, Ap. J. 55, 799 ( J. Kota, and J. R. Jokipii, Ap. J. 53, 067 ( W. H. Matthaeus, et al., Ap. J. Lett. 590, L3 ( G. P. Zank, et al., JGR 09, A0407 ( J. Giacalone, and J. R. Jokipii, Ap. J. 520, 524 ( A. Shalchi, JGR 0, A0903 ( E. N. Parker, Planet. Space Sci. 3, 9 ( J. R. Jokipii, E. Levy, and W. B. Hubbard, Ap. J. 23, 86 (977.. L. G. Chuvilgin, and V. S. Ptuskin, Astron. Astrophys. 279, 278 ( A. B. Rechester, and M. N. Rosenbluth, Phys. Rev. Lett. 40, 38 (978.