Anomalous, quasilinear, and percolative regimes for magnetic-field-line transport in axially symmetric turbulence G. Zimbardo, P. Veltri, and P. Pommois Dipartimento di Fisica, Universita` della Calabria, I-87030 Arcavacata di Rende, Italy and Istituto Nazionale di Fisica della Materia, Unita` di Cosenza, I-87030 Arcavacata di Rende, Italy ~Received 9 September 1999! We studied a magnetic turbulence axisymmetric around the unperturbed magnetic field for cases having different ratios l i / l ’ . We find, in addition to the fact that a higher fluctuation level d B / B 0 makes the system more stochastic, that by increasing the ratio l i / l ’ at fixed d B / B 0 , the stochasticity increases. It appears that the different transport regimes can be organized in terms of the Kubo number R 5( d B / B 0 )( l i / l ’ ). The simulation results are compared with the two analytical limits, that is the percolative limit and the quasilinear limit. When R !1 weak chaos, closed magnetic surfaces, and anomalous transport regimes are found. When R ’1 the diffusion regime is Gaussian, and the quasilinear scaling of the diffusion coefficient D ’ ;( d B / B 0 ) 2 is recov- ered. Finally, for R @1 the percolation scaling of the diffusion coefficient D ’ ;( d B / B 0 ) 0.7 is obtained. PACS number~s!: 52.25.Fi, 02.50.Ey, 95.30.Qd, 05.45.2a I. INTRODUCTION The transport of heat and particles in magnetized plasmas depends on the electromagnetic turbulence in the plasma it- self, as the electromagnetic fluctuations induce ‘‘random’’ motions in the directions perpendicular to the average mag- netic field. For low frequency magnetic turbulence and strong background magnetic field B 0 , the particles approxi- mately follow the magnetic field lines. The quantitative de- scription of magnetic field line transport represents a long standing problem, since different transport regimes can be obtained, depending on the fluctuation level ~weak or strong!, on the anisotropy of magnetic turbulence, described by the values of the turbulence correlation lengths, on the Fourier spectral representation, and on the assumed dimen- sionality @i.e., two dimensions ~2D! or 3D# of the magnetic fluctuations @1–6#. We assume an unperturbed field B 0 5B 0 e ˆ z and magnetic fluctuations d B( r) depending on the three spatial coordinates, but frozen in time. The latter as- sumption corresponds to considering particle velocities larger than the typical magnetic wave velocity, e.g., Alfve ´ n velocity. Note that field line motion in such fields is formally equivalent to the problem of passive tracer transport in a two-dimensional, time dependent velocity field @7,8#, so that the main results obtained here can be applied to the problem of transport in fluid turbulence, too. In this paper, we would like to concentrate our attention on the effect of different correlation lengths l i and l ’ in the directions parallel and perpendicular to the mean magnetic field B 0 , respectively, that is, on the influence of anisotropy in turbulence with axial symmetry. Indeed, in many physical systems the magnetic turbulence is not spherically symmet- ric. Axially symmetric turbulence can develop in a plasma as a result of a background magnetic field @9–11#, or as a con- sequence of the geometrical features of a plasma device: in a toroidal configuration, for instance, the correlation length along the toroidal direction is usually much larger than those in the two other directions @12,13#. Also, magnetic turbu- lence with l i @l ’ is often assumed as an approximate model for the solar wind @9,6# as well as the interstellar @14# mag- netohydrodynamic turbulence. For axially symmetric turbulence, the Fourier spectral am- plitude d B ( k) can be represented as d B ~ k! } 1 ~ k ’ 2 l ’ 2 1k i 2 l i 2 ! g /411/2 , ~1! where k is the wave vector, k ’ ( k i ) is the projection of k in the plane perpendicular ~in the direction parallel! to B 0 , and g is the spectral index. Let us introduce a cut off of the spectrum at constant amplitude d B ( k). Then, for l i @l ’ the wavevectors are squeezed in the plane perpendicular to B 0 , forming a pancake ~or cre ˆ pe! in the k space. In such a case, the magnetic turbulence is termed quasi-2D @1,4#, the 2D case being obtained by taking the limit l i / l ’ →‘~and keep- ing only the Alfve ´ nic polarization for MHD turbulence, see later!. Conversely, for l i !l ’ the wave vectors are elongated along B 0 , forming a cigar ~or ‘‘spaghetto’’! in k space. In such a case, the domain of magnetic turbulence is quasi-1D, and this magnetic turbulence is termed slab model. Clearly, when l i 5l ’ the turbulence is spherically symmetric. We also note that if the turbulence is not axially symmetric, as it is in the cases considered here, it is necessary to use three corre- lation lengths, say l x , l y , and l z , in the expression of the Fourier amplitudes, Eq. ~1!. This case is of interest in many astrophysical plasmas, like the MHD turbulence in the solar wind and the magnetic fluctuations in the Earth’s magneto- pause @15–19#. In particular, transport of magnetic field lines in the case of anisotropy in the plane perpendicular to B 0 , that is when l x @l y , has been considered by Pommois et al. @11#. Several issues need to be taken into account when consid- ering magnetic field line transport in anisotropic turbulence. In the case of weak turbulence, that is when the level of fluctuation d B / B 0 is low, one has the quasilinear regime, in which the magnetic diffusion coefficient is D ;( d B / B 0 ) 2 l i @20–25#. It was shown by Kadomtsev and Pogutse @1# that the quasilinear regime is more properly obtained when the dimensionless parameter R 5( d B / B 0 )( l i / l ’ ), is very small, R !1. In the opposite limit, R @1, Kadomtsev and Pogutse PHYSICAL REVIEW E FEBRUARY 2000 VOLUME 61, NUMBER 2 PRE 61 1063-651X/2000/61~2!/1940~9!/$15.00 1940 ©2000 The American Physical Society