The magnetic fluctuations due to, e.g., magnetohydrodynamic turbulence cause a magnetic-field line random walk that influences many cosmic plasma phenomena. The results of a three-dimensional numerical simulation of a turbulent magnetic field in plane geometry are presented here. Magnetic percolation, L´evy flights, and non-Gaussian random walk of the magnetic-field lines are found for moderate perturbation levels. In such a case plasma transport can be anomalous, i.e., either superdiffusive

or subdiffusive. Increasing the perturbation level a Gaussian diffusion regime is attained. The implications on the structure of the electron foreshock and of planetary

magnetopauses are discussed

Veltri, P.

Zimbardo, G.

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IL NUOVO CIMENTO VOL. 20 C, N. 6 Novembre-Dicembre 1997Plasma transport in the interplanetary space: Percolationand anomalous diffusion of magnetic-field lines()G. ZIMBARDO and P. VELTRIDipartimento di Fisica, Universita` della Calabria - I-87030 Arcavacata di Rende (CS), Italy(ricevuto il 5 Dicembre 1996; approvato il 18 Marzo 1997)Summary. — The magnetic fluctuations due to, e.g., magnetohydrodynamic turbulencecause a magnetic-field line random walk that influences many cosmic plasma phenom-ena. The results of a three-dimensional numerical simulation of a turbulent magneticfield in plane geometry are presented here. Magnetic percolation, Le´vy flights, andnon-Gaussian random walk of the magnetic-field lines are found for moderate pertur-bation levels. In such a case plasma transport can be anomalous, i.e., either superdif-fusive or subdiffusive. Increasing the perturbation level a Gaussian diffusion regime isattained. The implications on the structure of the electron foreshock and of planetarymagnetopauses are discussed.PACS 52.25.Fi – Transport properties.PACS 95.30.Qd – Magnetohydrodynamics and plasmas.PACS 01.30.Cc – Conference proceedings.1. – IntroductionThe plasma universe is shaped by the magnetic field, and, as a consequence, the mag-netic turbulence influences plasma transport in many cosmic systems: in practice, a parti-cle gyrating in the magnetic field follows a field line in its wandering due to the magneticfluctuations. The concept of field line random walk was introduced by Jokipii [1] in connec-tion with the confinement of cosmic rays in the galactic disc a long time ago, and contem-porarily to the earliest studies of stochastic transport in plasma fusion devices [2]. Amongthe astrophysical problems to which magnetic-field line diffusion is relevant, we can men-tion the effect of magnetic fluctuations in the solar wind on the propagation of solar flareenergetic particles [3], and the flaring of extragalactic radio jets, due to stochastic diffu-sion of the relativistic electrons [4]. In the space plasmas, the entry of solar-wind particlesinto the planetary magnetospheres is achieved by a sort of magnetic percolation [5].() Paper presented at the VII Cosmic Physics National Conference, Rimini, October 26-28, 1994.G Societa` Italiana di Fisica 897898 G. ZIMBARDO and P. VELTRICommon features of particle transport in astrophysical systems are: i) in the great ma-jority of cases the particle mean free path is very large, so that collisions can be neglected;ii) in many cases the relevant particles are superalfvenic, so that the magnetostatic caseis appropriate. Under these conditions, the particle diffusion coefficient perpendicular tothe mean magnetic field is [6-8]D?= vkDm;(1)where Dmis the magnetic-field line diffusion coefficient and vkthe velocity along themagnetic field. On the other hand, the relevant perturbation amplitudes are large,B=B  0:1–1, so that a quasi-linear treatment is not appropriate to obtaining Dm.2. – Numerical simulation of magnetic-field line transportThe magnetic-field line equations can be written asdrds=B(r)jB(r)j;(2)whereB(r) is the magnetic field at a generic point r, and s is the field line length. Equation(2) constitutes a set of three non-linear equations which, in the strong turbulence limit,has to be solved by numerical integration. In particular, we consider a magnetic fieldB = B0e^z+ B, with three-dimensional magnetic fluctuations given byB(r) =Xk;B(k)e^(k) exp[i(k  r+ 'k)];(3)where B(k) is the Fourier amplitude of the mode with wave vector k and polarization ,e^(k) are the polarization unit vectors, and'kare random phases. The field periodicity isthat of a cube of side L, so that k = 2(nx; ny; nz)=L. The Fourier amplitudes are givenby [9]jB(k)j = (2=L)3=2[C=(k22+ 1)=4+1=2];(4)where C is a normalization constant, and  is the spectral index. The spectrum is trun-cated at kmax= 2N=L, where N is the ratio between L and the smallest turbulencewavelength present. For the present isotropic calculations,  is set equal to L, while thespectral index is fixed as  = 3=2. This spectrum can be representative of the solar-windmagnetic turbulence. The fluctuation amplitude is defined as A =phB2i=B0, wherehB2i is obtained from eq. (4) summed over the relevant wave vectors. More details onthe numerical code are found in ref. [10].3. – Le´vy flights and anomalous diffusionExtensive numerical simulations have shown that, grossly speaking, two regimes oftransport can be identified, the first for A  1 and the second for A  1. For A  1,i.e. in the weak-turbulence regime where good, nested magnetic surfaces exist, magneticislands are found, and these are separated from each other by stochastic (percolating)layers. Increasing A the field line motion is more and more chaotic, until the last closedPLASMA TRANSPORT IN THE INTERPLANETARY SPACE: ETC. 899Fig. 1. – The projection on the xy plane of a field line with the same initial conditions is plotted for zvarying from zero to 1000 (all lengths normalized to L) for different values of the fluctuation level A.a) For A = 0:05 good nested magnetic surfaces exist, and the field line evolves on the surface of aflux tube (a KAM torus). b) For A = 0:15 the magnetic surface opens, becoming a cantorus [14],and the field line trapped in a magnetic flux tube passes from one trap to the next. c) For A = 0:35the random walk of the field line is composed of temporary trapping and of Le´vy-like flights, as, inparticular, the trajectory from about y =  4 to about y =  8. d) For A = 0:70 the random walk ismore isotropic, tends to fill the plane like a Gaussian random walk, and both field line trapping andshort flights happen frequently.magnetic surface is destroyed when A  1. For low A field lines can be trapped for sometime in a region close to a good magnetic surface, a so-called cantorus, and then travel longpaths, which are called “Le´vy flights” [11-14]. These are legs of almost ballistic motionwhich occur with small but nonvanishing probability in the percolation layer [15]. This isshown in fig. 1, where a sample field line is plotted projected in the xy plane for N = 12and A = 0:05, 0.15, 0.35, and 0.70. The trapping and subsequent Le´vy flight of field linesis apparent in fig. 1c).900 G. ZIMBARDO and P. VELTRIFig. 2. – The anomalous diffusion exponents x, y, and zvs. the fluctuation amplitude A. Circles:N = 3; crosses: N = 6; squares: N = 9; triangles: N = 12.The Le´vy random walk corresponds to a non-Gaussian probability distribution p(l) ofmaking a jump of length l in a given time interval of the form p(l)  l (+1) for l ! 1[13]. The cases with  > 2 correspond to a normal random walk, whereas  < 2 gives riseto a random walk law of the formhx2i = 2Ds; s!1;(5)with  = 2= [13]. Here, the exponent  characterizes the anomalous diffusion:  = 1in the diffusion regime,  = 2 in the ballistic regime, 1 <  < 2 in the case of Le´vyrandom walk [12,13] and  < 1 in the case of trapping (subdiffusive regime). On the otherhand, when  < 1 the random walk is time intermittent and the microscopic dynamics isgoverned by long waiting times  distributed according to q()   (+1) for  !1. For < 1 subdiffusion results with  = . More generally, the anomalous diffusion exponentis given by  = 2= [10].Figure 1 shows that, because of the increase of stochasticity of field lines with A, theprobability of going from a “trapped” to a “percolating” path or vice versa increases withthe amplitude of the fluctuating field, so that eventually the field line motion approachesa Gaussian random walk with a finite scale length. At last, for large values of A, theprobability of long flights is very small, and the normal random walk is recovered (seebelow). We point out that the non-Gaussian nature of microscopic motion is particularlyimportant in media which exhibit continuum percolation [16]. Indeed, many successivemoves are in the same direction in the case of percolation, and this corresponds to a powerlaw probability distribution p(l).PLASMA TRANSPORT IN THE INTERPLANETARY SPACE: ETC. 901Anomalous diffusion is studied by computing hx2i, hy2i, and hz2i as a function ofs, rather than z, since in the case of strong turbulence the field line length s can departconsiderably from its projection on the direction of the average field. The above meansquare deviations are computed over an ensemble of 2000 lines of force integrated untilan asymptotic regime is attained [10]. Here we give the results of the fit of eq. (5) bythe numerical simulations, while further data are to be found in ref. [10]. The values forx; y; and zas a function of A are reported in fig. 2 for N = 3; 6; 9, and 12. Bothsuperdiffusive and subdiffusive behaviour is found for the motion in the xy plane for lowA, and a normal diffusion regime is obtained for A ' 1. In addition, the spreading of zaround its mean value goes from a ballistic regime (z' 2) for lowA to a diffusion regimefor larger values of A. In particular, note that the diffusion regime is reached the sooner,the higher N . This matches with the fact that global stochasticity is attained earlier forhigher N , an indication that is obtained from the Poincare´ sections, too (not shown).4. – Applications to space plasmas and conclusionsAt the time of writing, two different astrophysical applications are under way, the firstto the electron foreshock and the second to the Earth’s magnetopause.The upstream boundary of the electron foreshock is defined as the path of the fastelectrons originating from the tangent point between the Interplanetary Magnetic Field(IMF) and the bow shock. When we consider that the magnetic turbulence in the solarwind can be strong, A  1, we realize that the foreshock upstream boundary is not a verysmooth surface. Rather, because of the magnetic fluctuations, this boundary will graduallybecome distorted and entangled, and will develop a very fine structure. Magnetic-field linediffusion implies that the upstream boundary is scattered in space, with a typical widthgiven by hx2i = 2Ds, where s is the distance from the tangent point measured alongthe magnetic-field lines. Therefore, when analysing the experimental data collected bythe spacecrafts crossing the foreshock, care must be used in determining the position ofthe bow shock: the broadening of the boundary tends to anticipate the detection of thetangent field line, for a spacecraft coming from upstream of the foreshock boundary [17].The Earth’s magnetopause, like the other planetary magnetopauses, is the boundarylayer between the shocked solar wind and the actual magnetosphere. What amount ofplasma can cross the magnetopause because of motion along magnetic-field lines? In thecase that the IMF is northward, i.e., parallel to the Earth’s magnetic field at the sub-solar point, magnetic reconnection (in the sense of a tearing instability) is not favouredat the low-latitude dayside magnetopause. Therefore, the magnetic connection betweenthe magnetosheath and the magnetosphere could be due to the effect of magnetic fluc-tuations, which have been both observed [18, 19] and simulated numerically [20, 21], andwhich destroy the magnetic surfaces. On the other hand, when the IMF is southward,magnetic reconnection is strongly favoured at the subsolar magnetopause, and plasmatransport along the reconnected magnetic-field line certainly plays an important role inthe creation of the low-latitude boundary layer [5]. A numerical simulation dedicated tothese configurations will be carried out in a future work.The results of our numerical simulation of magnetic-field line transport in the presenceof 3D fluctuations with cubic periodicity can be summarized as follows: for low fluctuationlevels magnetic-field line transport is anomalous and the random walk is composed oftrapping in the cantori and of long ballistic flights (Le´vy flights) in the percolation layer.The transport law is hx2i / 2Ds with  > 1 and  < 1. On the other hand, for higherfluctuation levels a Gaussian diffusion regime is attained; the fluctuation threshold for this902 G. ZIMBARDO and P. VELTRIregime is lowered as the number of modes, dependent on N , is increased. This suggeststhat many cosmic plasmas, where N is very large, are mostly diffusive.  This work was supported in part by a contract of the Italian Space Agency (ASI 94-RS-164), and by contracts of the Italian Ministero dell’Universita` e della Ricerca Scientificae Tecnologica (MURST), and of the Italian Consiglio Nazionale delle Ricerche (CNR No.93.01383.CT02).REFERENCES[1] JOKIPII J. R., Astrophys. J., 146 (1966) 480.[2] ROSENBLUTH M. N., SAGDEEV R. Z., TAYLOR G. B. and ZASLAVSKY G. M., Nucl. Fusion,6 (1966) 297.[3] JOKIPII J. R. and PARKER E. N., Astrophys. J., 155 (1969) 777.[4] CORBELLI E. and VELTRI P., Astrophys. J., 340 (1989) 679.[5] MILOVANOV A. V. and ZELENYI L. M., in Physics of the Magnetopause, Geophys.Monograph, Vol. 90, edited by B. U. O. SONNERUP, P. SONG and M. F. THOMSEN (AGU,Washington, D.C.) 1995, p. 357.[6] LAVAL G., Phys. Fluids B, 5 (1993) 711.[7] RECHESTER A. B. and ROSENBLUTH M. N., Phys. Rev. Lett., 40 (1978) 38.[8] ISICHENKO M. B., Plasma Phys. Control. Fusion, 33 (1991) 809.[9] CARBONE V. and VELTRI P., Astron. Astrophys., 188 (1987) 239.[10] ZIMBARDO G., VELTRI P., BASILE G. and PRINCIPATO S., Phys. Plasmas, 2 (1995) 2653.[11] SHLESINGER M. F., WEST B. J. and KLAFTER J., Phys. Rev. Lett., 58 (1987) 1100.[12] ZASLAVSKY G. M., SAGDEEV R. Z., CHAIKOVSKIJ D. K. and CHERNIKOV A. A., Sov. Phys.JETP, 68 (1989) 995.[13] BOUCHAUD, J.-P. and GEORGES A., Phys. Rep., 195 (1990) 127.[14] SHLESINGER M. F., ZASLAVSKY G. M. and KLAFTER J., Nature, 363 (1993) 31.[15] ZIMBARDO G. and VELTRI P., Phys. Rev. E, 51 (1995) 1412.[16] ISICHENKO M. B., Rev. Mod. Phys., 64 (1992) 961.[17] ZIMBARDO G. and VELTRI P., in Magnetic Turbulence and Transport, edited by P.HENNEQUIN and M. A. DUBOIS (Editions de Physique, Orsay) 1993, pp. 257–260.[18] SONG P., ELPHIC R. C., RUSSELL T. C., GOSLING J. T. and CATTELL C. A., J. Geophys.Res., 95 (1990) 6375.[19] SONG P., RUSSELL R. C., STRANGEWAY R. J., WYGANT J. R., CATTELL C. A.,FITZENREITER R. J. and ANDERSON R. R., J. Geophys. Res., 98 (1993) 187.[20] BERCHEM J. and OKUDA H., J. Geophys. Res., 95 (1990) 8133.[21] OKUDA H., J. Geophys. Res., 98 (1993) 3953.

Plasma transport in the interplanetary space: Percolation

and anomalous diffusion of magnetic-field lines

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Plasma transport in the interplanetary space: Percolation and anomalous diffusion of magnetic-field lines

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