Time-dependent perpendicular transport of fast charged particles in a turbulent magnetic field
TTime-dependent perpendicular transport of fast charged particlesin a turbulent magnetic field
F. Fraschetti , and J.R. Jokipii Departments of Planetary Sciences and Astronomy, University of Arizona, Tucson, AZ,85721, USA LUTh, Observatoire de Paris, CNRS-UMR8102 and Universit´e Paris VII, 5 Place JulesJanssen, F-92195 Meudon C´edex, France.Received ; accepted a r X i v : . [ a s t r o - ph . H E ] A p r ABSTRACT
We present an analytic derivation of the temporal dependence of the perpen-dicular transport coefficient of charged particles in magnetostatic turbulence, fortimes smaller than the time needed to charged particles to travel the turbulencecorrelation length. This time window is left unexplored in most transport mod-els. In our analysis all magnetic scales are taken to be much larger than theparticle gyroradius, so that perpendicular transport is assumed to be dominatedby the guiding center motion. Particle drift from the local magnetic field linesand magnetic field lines random walk are evaluated separately for slab and 3Disotropic turbulence. Contributions of wavelength scales shorter and longer thanthe turbulence coherence length are compared. In contrast to slab case, particlesin 3D isotropic turbulence unexpectedly diffuse from local magnetic field lines;this result questions the common assumption that particle magnetization is inde-pendent on turbulence geometry. Extensions of this model will allow for a studyof solar wind anisotropies.
Subject headings:
1. Introduction
The behaviour of individual fast charged particles in magnetic turbulence is relevant toa number of problems in plasma astrophysics, from the solar wind (e.g. Bruno & Carbone2005) to interstellar medium (e.g. Elmegreen & Scalo 2004) and cosmic rays at highestenergy (e.g. Fraschetti 2008). However, in contrast to cosmic rays with energies beyond theGeV scale, a thorough understanding of the particle transport properties can be attainedonly in interplanetary space, where in situ measurements of both magnetic turbulenceenergy spectrum and particles energy are possible. Diffusion theory (e.g. Jokipii 1966), as amain tool to study charged particle propagation in magnetic turbulence, yields a statisticaldescription of a population of particles and relies on the approximation (Jokipii 1972) thata characteristic time T exists much larger than the correlation time t c of the magneticfield fluctuations (as seen by the particle) but also much smaller than the time-scale ofboth the variation of these fluctuations and of the average distribution function. TheVlasov-Boltzmann equation for the charged particles phase-space distribution function canbe therefore considerably simplified to terms of the second order moments of the magneticfield fluctuations. In this scenario higher-order moments are not necessary to determinethe particles motion as the process is markovian; diffusion is governed by the central limittheorem (Chandrasekhar 1943).Observational constraints posed by heliospheric environment on perpendicular diffusionacross the average magnetic field involving, e.g. jovian electrons (Chenette et al. 1977),have not yet been included in a first-principles unified theoretical picture. Perpendiculardiffusion occurring in the ecliptic plane is invoked as a plausible explanation of the timedelays in solar energetic particle events detected by Helios (Wibberenz & Cane 2006). Amore remarkable longitudinal separation in the combined electron observations by
Stereo
A/B and
SOHO from the January 17, 2010 event suggests a strong diffusion perpendicularto the mean magnetic field. The access of high energy particles to high-latitude heliosphericregions observed by Ulysses (Malandraki et al. 2009) is dominated by particle propagationalong the mean magnetic field lines, although cross-field diffusion cannot be excluded.Perpendicular transport in a magnetic field depending on two or fewer space coordinatesoriginates only from the meandering of the magnetic field lines (Jokipii et al. 1993) whereasin an arbitrary three-dimensional turbulence also emerges as a general property of theparticle motion (Giacalone & Jokipii 1994). Kinetic approach has been applied toperpendicular scattering of strongly magnetized charged particles by using a model forthe collision integral (Chuvilgin & Ptuskin 1993). Recent numerical simulations (Minnieet al. 2009) investigated the common assumption that the charged-particle gyrocenterfollows the magnetic field lines: the approaching of the guiding center cross-field motionto the transverse field line random walk for various parallel mean free paths is studiedfor a solar-wind like turbulence. However we notice that the assumption that guidingcenter follows the magnetic field lines does not result directly from the equation of motion;therefore it may be realized only for particular turbulence models.An approximate diffusive perpendicular transport model based on the guiding centermotion (Non-Linear Guiding Center, NLGC) has been put forward in Matthaeus et al.(2003), which provided a method to compute magnetic fluctuations along perturbed particletrajectory. However, the NLGC’s assumption that the probability density of perpendiculardisplacement is diffusive at all times is a limitation of this model. Subdiffusive nature ofperpendicular transport in a slab turbulence was not recovered, contrary to the expectationfrom the conservation of canonical momentum of ignorable coordinate (Jokipii et al.1993) and to the findings from test particle numerical simulations in turbulence having a Available at URL http : .physik.uni − kiel.de/stereo/downloads/sept electron events.pdf
2. Transverse guiding center drift
We consider a process of propagation of a charged particle in magnetic turbulencewhich is statistically homogeneous in time, i.e., the velocity correlation depends only ontime difference along the orbit. The instantaneous mean square displacement along thespace coordinate x after a time ∆ t = t − t (cid:48) for a particle propagating in an arbitrarymedium can be defined as (Taylor 1922; Green 1951; Kubo 1957) (cid:104) (∆ x ) (cid:105) = 2 (cid:90) ∆ t dξ (∆ t − ξ ) (cid:104) v x ( t (cid:48) + ξ ) v x ( t (cid:48) ) (cid:105) , (1)where t (cid:48) is an arbitrary initial time, ξ the time lag and the ensemble average (cid:104) .. (cid:105) is meantto be the average over a population of particles and over an ensemble of turbulencerealizations. We notice that Eq.(1) applies for any value of ∆ t . We define d xx ( t ) ≡ ddt (cid:104) (∆ x ) (cid:105) = (cid:90) t dξ (cid:104) v x ( t (cid:48) + ξ ) v x ( t (cid:48) ) (cid:105) . (2)The standard perpendicular coefficient of diffusion can then be defined as κ xx = lim t →∞ d xx ( t ) . (3)We here investigate the transverse motion of a low-rigidity particle for a time smaller than t c in general 3D magnetostatic turbulence. The diffusion approximation therefore may notbe valid.We consider a spatially homogeneous, fluctuating, time-independent magnetic field.The amplitude of the fluctuation ( δB ) is assumed to be much smaller than the averagefield magnitude ( B ). We represent such a magnetic field as B ( x ) = B + δ B ( x ), with anaverage component B = B e z and (cid:104) δ B ( x ) (cid:105) = 0 and δB ( x ) /B (cid:28)
1. This approximationis known to be valid in several turbulent media, as the solar wind, where the propagationof the magnetic fluctuation is much smaller than the velocity of the bulk ionized fluid. Wewill make use of the first-order orbit theory (Rossi & Olbert 1970): the particle gyroradius 7 – r g is much smaller than the length-scale of any magnetic field variation: r g (cid:28) min i,j =1 , (cid:12)(cid:12)(cid:12)(cid:12) B i ∂ j B i (cid:12)(cid:12)(cid:12)(cid:12) , (4)where B i is the i -th component of the perturbed field B ( x ). No further assumption ismade on the spatial dependence of δ B or geometry. In this approximation, we considerthe guiding center motion. In a spatially varying magnetic field, the guiding center maysignificantly drift from the average field direction due to the action of the field gradienton the particle magnetic moment. We therefore consider non-zero gradient and curvaturedrifts. We estimate that drift and resulting displacement after a time shorter than thecorrelation time. The guiding position X ( t ) = ( X, Y, Z ) for a particle of mass m , charge Ze and momentum p having coordinate x ( t ) = ( x, y, z ) is described, in c.g.s. units, by X ( t ) = x ( t ) − cZe B × p ( t ) B . (5)If the scales of magnetic fluctuation are much larger than the gyroradius r g , the guidingcenter motion defined in Eq. (5) has the role of effective gyroperiod-averaged motion.Therefore, Eq. (5) can well describe the motion perpendicular to the local magnetic field.In the case of “finite Larmor radius”, the gyroradius only represents the typical scale ofparticle motion and Eq. (5) provides the instantaneous guiding center position whereasgyroperiod-average becomes meaningless. In the magnetostatic field described above, theguiding center velocity transverse to the field B ( x ) is given at the first order in δB ( x ) /B by the gyroperiod average (Rossi & Olbert 1970) V G ⊥ ( t ) = vpcZeB (cid:20) µ B × ∇ B + µ B ( ∇ × B ) ⊥ (cid:21) (6)where α is the particle pitch angle and µ = cos α . Eq. (6) gives the first order most generalexpression of the guiding center velocity orthogonal to the local magnetic field direction(Balescu 1988). Here the variation of α is assumed to be negligible over a gyroperiod.Being V G ⊥ ( t ) a gyroperiod average, magnetic field can be computed at the guiding center 8 –position during that gyroperiod. In contrast to Matthaeus et al. (2003), the transversemotion of the guiding center from the field line is not parametrized in the present paperthrough some constants to be inferred from numerical simulations, but described directlyfrom the equation of motion of the guiding center. The finite-time average square transversedisplacement of the particle from the direction of local B due to drift d D ( t ) can then bewritten in this approximation using the Eq.(2): d D ii ( t ) = (cid:90) t dξ (cid:104) V G ⊥ ,i ( t (cid:48) ) V G ⊥ ,i ( t (cid:48) + ξ ) (cid:105) (7)where i stands for any transverse coordinate, X or Y . The average square displacement iscomputed from the following expression, to the lowest order in δB/B , d D ii ( t ) (cid:39) (cid:18) vpcZeB (cid:19) × (cid:90) t dξ (cid:104) (cid:20) − µ ∂ j δB + µ ∂ δB j (cid:21) [ x ( t (cid:48) )] × (cid:20) − µ ∂ j δB + µ ∂ δB j (cid:21) [ x ( t (cid:48) + ξ )] (cid:105) , (8)where ( i, j ) = (1 ,
2) or (2 ,
1) and the fields are evaluated at the perturbed particle position x ( t ). The average square of the displacement d D ii ( t ) does not depend on the sign ofthe electric charge Ze , at variance from the drift velocity in Eq. (6). We notice thatin the first-order orbit approximation the particular case of 2D turbulence defined by δB ( x, y ) = ( δB x , δB y ,
0) provides a zero transverse velocity drift; this is because, on theright hand side of Eq. (8), this form of turbulence has δB = 0 and also δB j does notdepend on the z coordinate. Thus, this analytic method cannot be applied to the compositeslab/2D solar wind model of Bieber et al. (1996), a very useful but empirical descriptionof the MHD-scale turbulence in slow solar wind. Moreover, the slab/2D model couldbe incomplete as the non-wave (“2D”) turbulence might have an additional componentalong B . This implies that, due to the sub-diffusive nature of the perpendicular particletransport in slab turbulence, drifts from local field-line found in numerical simulations for 9 –composite model (Minnie et al. 2009) are second-order contributions. Different anisotropiesmay be compatible with large-scales solar wind observations; in this paper we indicate apossible alternative method.We may simplify the derivation by using the Fourier representation of δ B ( x ): δ B ( x ) = (cid:60) (cid:90) ∞−∞ d k δ B ( k ) e i k · x ( t ) (9)where (cid:60) ( · ) stands for the real part and x ( t ) is the particle position at time t . Therefore theaverage displacement in Eq.(8) contains terms of type ∂ l δB j ( x ) = (cid:60) (cid:90) ∞−∞ d k δB j ( k )( ik l ) e i k · x ( t ) (10)with l, j = 1 , ,
3. We compute the particle position in Eq.(10) along the localmagnetic field: x ( t ) = x ( t ) + x MF L ( z ( t )), where the unperturbed particle orbit is x ( t ) = ( v sin φ (cid:112) − µ / Ω , − v cos φ (cid:112) − µ / Ω , v (cid:107) t ); here v (cid:107) is the unperturbed particlevelocity along z , φ is the particle azimuth angle in the plane orthogonal to B andΩ = ZeB / ( mγc ) the particle gyrofrequency in the background field containing the Lorentzfactor γ ; x MF L ( z ( t )) = ( x MF L , y
MF L , z ( t )) is the offset in the plane orthogonal to B dueto the magnetic field line random walk (MFLRW) at z = z ( t ). The assumption of ballisticmotion along B , i.e., z = v (cid:107) t , relies on the choice δB (cid:28) B ; at times smaller than thecorrelation time of the perpendicular fluctuation, a fortiori we cannot assume paralleldiffusion. At the small length-scales considered here, parallel and perpendicular motions canbe disentangled and any non-markovian parallel motion, e.g., memory effect of a particletracing back its trajectory, is not expected to interfere with the perpendicular transport, incontrast to the case of compound diffusion. We can write e i k · x ( t ) (cid:39) e i k · x ( t ) e i k · x MFL ( z ( t )) . (11)The magnetic field lines (MFL) are defined by d x MF L × B = 0. This implies thata finite distance ∆ x MF L in the ballistic approximation is a first order term in δ B : 10 –∆ x MF L (cid:39) ( δ B /B ) v (cid:107) t . Therefore, the exponential e i k · x MFL ( z ( t )) contribute only at zeroorder in Eq. (11) and the fuctuation in Eq. (10) can be computed along the unperturbedtrajectory: e i k · x ( t ) (cid:39) e i k · x ( t ) , which is equivalent to the quasi-linear approximation. To firstorder in Eq.(10) we can replace the exponential as e i k · x ( t ) (cid:39) e i ( W sin( ψ − φ )+ k (cid:107) v (cid:107) t ) , (12)where ψ = tg − ( k y /k x ), k (cid:107) = k z , k ⊥ = (cid:112) k x + k y and W = k ⊥ v (cid:112) − µ / Ω = k ⊥ r g .By using the Bessel function identities (see Abramowitz & Stegun (1964), Eq. (9.1.41)) e iz sin φ = ∞ (cid:88) n = −∞ J n ( z ) e inφ , (13)Eq.(11) is rewritten as (see also Schlickeiser (2002), Sect. 12.2.1): e i k · x ( t ) = ∞ (cid:88) n = −∞ J n ( W ) e ik (cid:107) v (cid:107) t + in ( ψ − φ +Ω t ) . (14)The magnetic fluctuation space derivative in Eq.(10) can be then written in the followingway ∂ l δB j ( x ) = (cid:60) ∞ (cid:88) n = −∞ (cid:90) ∞−∞ d k δB j ( k )( ik l ) J n ( W ) × e ik (cid:107) v (cid:107) t + in ( ψ − φ +Ω t ) (15)with l, j = 1 , ,
3. The typical term in Eq.(8) is of type (cid:60) (cid:18) vpcZeB (cid:19) F ( µ ) (cid:90) t dξ (cid:104) ∂ l δB r [ x ( t (cid:48) )] · ∂ p δB ∗ q [ x ( t (cid:48) + ξ )] (cid:105) , (16)here F ( µ ) represents various µ factors resulting from the expansion of Eq.(8). UsingEq.(15), we obtain for Eq.(16) (cid:60) (cid:18) vpcZeB (cid:19) F ( µ ) × (cid:90) t dξ (cid:104) ∞ (cid:88) n = −∞ ∞ (cid:88) m = −∞ (cid:90) ∞−∞ d k (cid:90) ∞−∞ d k (cid:48) δB r ( k ) × J n ( W )( ik l ) δB ∗ q ( k (cid:48) ) J ∗ m ( W )( − ik (cid:48) p ) × e [ i ( k (cid:107) − k (cid:48)(cid:107) ) v (cid:107) t (cid:48) + i ( n − m )( ψ − φ +Ω t (cid:48) ) − i ( k (cid:48)(cid:107) v (cid:107) + m Ω) ξ ] (cid:105) . (17) 11 –We assume the standard inertial range magnetic turbulence power spectrum which isuncorrelated at different wavenumber vectors: (cid:104) δB r ( k ) δB ∗ q ( k (cid:48) ) (cid:105) = δ ( k − k (cid:48) ) P rq ( k ) . (18)Thus Eq.(17) reduces to (cid:60) (cid:18) vpcZeB (cid:19) F ( µ ) ∞ (cid:88) n = −∞ (cid:90) ∞−∞ d kR ( k , t ) P rq ( k ) k l k p J n ( W ) (19)whose time-dependence is entirely contained in R ( k , t ) ≡ (cid:90) t dξe − i ( k (cid:107) v (cid:107) + n Ω) ξ = e − i ( k (cid:107) v (cid:107) + n Ω) t − − i ( k (cid:107) v (cid:107) + n Ω) . (20)Since (cid:61) J n ( W ) = 0, where (cid:61) ( · ) stands for imaginary part, we may consider (cid:60) R ( k , t ): (cid:60) R ( k , t ) = sin[( k (cid:107) v (cid:107) + n Ω) t ] k (cid:107) v (cid:107) + n Ω . (21)The orthogonal scale 1 /k ⊥ can be estimated as | B i /∂ j B i | , thus Eq.(4) states W ∼ k ⊥ v/ Ω (cid:28) . (22)For W (cid:28)
1, it holds J ( W ) (cid:29) J n ( W ) for n ≥
1; moreover, (cid:60) R ( k , t ) ∼ /n for large n . Wemay therefore approximate the sum in Eq.(19) as its term with n = 0. Therefore Eq.(19)yields, using Eq.(21), four terms of type: (cid:18) vpcZeB (cid:19) F ( µ ) (cid:90) ∞−∞ d kP rq ( k ) k l k p sin[ k (cid:107) v (cid:107) t ] k (cid:107) v (cid:107) , (23)with indexes ( r, q, l, p ) = (3 , , , , , , , , , , , ,
3) for d D XX and( r, q, l, p ) = (3 , , , , , , , , , , , ,
3) for d D Y Y . Eq. (23) represents thegeneral term contributing to the first-order transverse drift coefficient of a particle in astatic first-order perturbed magnetic field. 12 –
3. Magnetic-field-line random walk
In the present section we compute the contribution to the time-dependent particletransverse transport due to MFLRW. If the correlation function of the magnetic fluctuationis homogeneous in space, the mean square displacement of the MFL orthogonal to the z-axiscan be defined, in analogy to Eq.(2), as d MF L ( z ) ≡ ddz (cid:104) (∆ x MF L ) (cid:105) ( z )= 1 B (cid:90) z dz (cid:48) (cid:104) δB x [ x ( z (cid:48) )] δB x [ x (0)] (cid:105) . (24)We compute the magnetic turbulence δ B ( x ) in Eq.(24) along the unperturbed trajectory ofa particle travelling with zero pitch-angle, as in Eqs.(9, 12). The motion along the averagefield is then ballistic, i.e. z = v (cid:107) t . In these approximations the transverse displacement of aMFL corresponding to a distance v (cid:107) t along B travelled by a small rigidity particle can bewritten as d MF L ( t ) = 1 B (cid:90) ∞−∞ d kP rq ( k ) sin[ k (cid:107) v (cid:107) t ] k (cid:107) . (25)Equation (25) is in agreement with Eq. (17) of Shalchi (2005) derived for pure slabturbulence. The MFL coefficient diffusion describing the random walk of the field lines canthen be defined as κ MF L = lim t →∞ d MF L ( t ) . (26)In our approach, MFL diffusion cannot be assumed because travelled distance z smallerthan parallel correlation lengths is considered; nevertheless, a simple ballistic motion alongthe z-axis allows to recover the standard result of MFL perpendicular diffusion in QLT. Thediscussion of the previous two section implies that the instantaneous coefficient diffusionperpendicular to the average magnetic field B is given, in the presence of weak turbulenceand neglecting parallel scattering, by two contributions: the random walk of the field line 13 –and the guiding center drift from the field line: d ( t ) = d D ( t ) + v (cid:107) d MF L ( t ) ,κ = κ D + v (cid:107) κ MF L = lim t →∞ [ d D ( t ) + v (cid:107) d MF L ( t )] . (27)In the next section the previous results are applied to the slab and 3D isotropic turbulences.
4. The turbulence power spectrum
In this section we apply the approach developed in previous sections to derive theinstantaneous transverse particle transport coefficients both of guiding center drifting fromlocal MFL (see Eq.(23)) and of the MFL from the average field direction in Eq.(25) by usingthe coherence length of the turbulence to disentangle small from large scale contributions toperpendicular diffusion. We will consider two cases: 1) slab turbulence, introduced (Jokipii1966) to represent the static limit of the solar wind magnetic fluctuations and extensivelystudied with Monte Carlo numerical simulations; we will compare the result with the QLTlimit; 2) 3D isotropic turbulence, idealized case likely to provide an unperturbed model foranisotropies observed in the solar wind.
We consider the slab turbulence, static limit of transverse and longitudinal-propagating Alfven waves: δ B = δ B ( z ) and δ B ( x ) · e z = 0. In this case, from Eq.(8), d sD XX ( t ) = d sD Y Y ( t ) = d sD ( t ). The turbulence wave number is aligned to the average magneticfield, thus we adopt the following form of the power spectrum: P rq ( k ) = G ( k (cid:107) )( δ ( k ⊥ ) /k ⊥ ) δ rq with r, q = 1 ,
2, and P i ( k ) = 0 with i = 1 , ,
3. The 1D spectrum is assumed to be ofKolmogorov type. Observations of electron-density fluctuations inferred from scintillationmeasurements exhibit a Kolmogorov power law, with index approximately equal to 5/3,over 5 orders of magnitude (Armstrong et al. 1995). Several other observations of magneticturbulent media, from earth’s magnetosphere to galaxy clusters, validate the Kolmogorov 14 –power spectrum up to a range of 12 orders of magnitude. We mention that solar windobservations show that at scales smaller than the ion thermal gyroradius ( ∼ cm aroundthe earth), much smaller than the scales considered in this paper, the magnetic turbulencespectrum deviates from the Kolmogorov, having an index of − .
12 (Bale et al. 2005).At length-scales larger than the coherence length the measured interplanetary magneticturbulence is well described by a flattening power spectrum (Hedgecock 1975; Bieber et al.1994). On the other hand, a consistent comparison with the quasi-linear limit requires thepower spectrum to be defined at scales larger than coherence length, i.e. for k (cid:107) < k min (cid:107) , upto the physical scale of the system 2 π/k (cid:107) ; we will adopt here a simplified form: G ( k (cid:107) ) = G (cid:107) k − q (cid:107) if k min (cid:107) < k (cid:107) < k max (cid:107) G (cid:107) ( k min (cid:107) ) − q if k (cid:107) < k (cid:107) < k min (cid:107) , (28)where k max (cid:107) corresponds to the scale where the dissipation rate of the turbulence overcomesthe energy cascade rate. The choice of a constant power spectrum at large scales insteadof a function smoothly connected to the inertial range already used in the literature ismerely dictated by easier mathematical tractability. Here q = 5 / G (cid:107) isdetermined from the normalization( δB ) = (cid:90) ∞−∞ d k ( P + P + P ) (29)implying, using cylindrical coordinate ( d k = dk (cid:107) k ⊥ dk ⊥ dψ ), G (cid:107) = ( δB ) ( q − πq ( k min (cid:107) ) − q , (30)with the assumption k (cid:107) (cid:28) k min (cid:107) (cid:28) k max (cid:107) .We consider first the transverse drift in Eq.(23). We average F ( µ ) over an isotropicpitch angle distribution. Using cylindrical coordinate ( d k = dk (cid:107) k ⊥ dk ⊥ dψ ) we have d sD ( t ) = (cid:18) vpcZeB (cid:19) π (cid:90) k max (cid:107) k (cid:107) dk (cid:107) k (cid:107) G ( k (cid:107) ) sin[ k (cid:107) v (cid:107) t ] k (cid:107) v (cid:107) (31) 15 –In units of the Bohm coefficient diffusion ( κ B = (1 / r g v ) and approximating r g /v (cid:107) (cid:39) Ω − ,we obtain d sD ( t ) κ B = 320 (cid:18) δBB (cid:19) q − q F ( y m (cid:107) , y (cid:107) , q ) (32)where we defined F ( y m (cid:107) , y (cid:107) , q ) = k min (cid:107) r g (cid:34) I (2 − q, y m (cid:107) )( y m (cid:107) ) − q + sin y (cid:107) − y (cid:107) cos y (cid:107) y (cid:107) (cid:12)(cid:12)(cid:12) y m (cid:107) y (cid:107) (cid:35) (33)where the time-dependence is contained in the new variable y m (cid:107) = k min (cid:107) v (cid:107) t (cid:39) k min (cid:107) r g Ω t (and y (cid:107) = k (cid:107) v (cid:107) t (cid:39) k (cid:107) r g Ω t ) and we used I ( a, u ) = (cid:90) ∞ u y a − sin y dy = i/ e − i π a Γ( a, iu ) − e i π a Γ( a, − iu )] (34)where Γ( a, z ) is the incomplete gamma function (see Gradshteyn & Ryzhik (1973),Eq.(3.761.2)). The time evolution of the drift coefficient d sD is depicted in Fig.1. Thefirst term in Eq. (33), corresponding to scales smaller than coherence scale 2 π/k min (cid:107) ( k > k min (cid:107) ), dominates over the second term, corresponding to scales larger than 2 π/k min (cid:107) ( k < k min (cid:107) ). The diffusive behaviour can be found by using the approximation of Γ( a, z )for | z | = y m (cid:107) = k min (cid:107) v (cid:107) t (cid:28) y m (cid:107) (cid:28)
1; since we assume a weak magnetic fluctuation, it is reasonable to assume that theperpendicular diffusion time-scale is shorter than the parallel scattering time-scale, i.e.,1 /k min (cid:107) v (cid:107) , or in other terms the diffusion limit is the dominant term in Eq. (33) for large t and t < /k min (cid:107) v (cid:107) : Γ( a, z ) ∼ Γ( a ) − z a /a ; therefore I (2 − q, y (cid:107) ) ∼ sin( qπ/ − q ) (dashedline in Fig.1). In diffusive regime, the second term in Eq.(33), representing the large scales( l > π/k min (cid:107) or k < k min (cid:107) ), does not significantly contribute to the particle drift, as it ismanifest in Fig. 1. We find that for slab turbulence, transverse particle drift coefficient 16 –from local MFL is given by d sD ( t ) κ B → (cid:18) δBB (cid:19) q − q sin( qπ/ − q )( k min (cid:107) r g ) − q (Ω t ) − q , (35)thus subdiffusive with behaviour κ sD ( t ) ∼ t − (2 − q ) (depicted as the dashed line in Fig. 1).Transverse subdiffusion has also been found by considering time-scales longer than theparallel scattering time and therefore allowing parallel scattering in K´ota & Jokipii (2000).However, in that case particles are assumed to propagate back and forth along the MFLand to be tied to the MFL. We notice that the drift-coefficient time evolution in Eq. (35)confirms that charged-particles in a turbulence depending on less than 3 space coordinatesremain confined within a gyroradius from the local field line (Jokipii et al. 1993; Jones etal. 1998).The time-integration of Eq. (35) up to t = L (cid:107) /v ∼ π/ ( k min (cid:107) r g Ω), gives in case ofweak turbulence ( δB (cid:28) B ) the condition (cid:104) ∆ x (cid:105) (cid:28) r g . We notice that the time-integral of κ sD ∼ t q − , which provides (cid:104) (∆ x ) (cid:105) ∼ t q − , is an increasing function of time for any observedphysical value of q ; however, as shown above, this result does not contradict the theoremof reduced dimensionality. In summary, the present result has been obtained under threeassumptions: 1) ballistic motion in the z coordinate ( z = v (cid:107) t ); 2) average displacementtransverse to the local field B due to first-order drift; 3) Kolmogorov power spectrum formagnetic fluctuations. Equations (32, 33) represent the average transverse displacementcomputed in the first-order orbit approximation at any time smaller than the parallelscattering time-scale, so that the approximation of ballistic motion parallel to the meanmagnetic field holds.From Eq. (25), the MFLRW in units of magnetic coherence length L (cid:107) = 2 π/k min (cid:107) isgiven by d sMF L ( t ) k min (cid:107) = (cid:18) δBB (cid:19) q − q H ( y (cid:107) , q ) (36)where we defined H ( y (cid:107) , q ) = ( k min (cid:107) r g Ω t ) q I ( − q, y (cid:107) ) + Si( y min (cid:107) ) , (37) 17 –where Si( x ) is the Sine integral function. The first term in Eq. (37), corresponding toscales smaller than coherence scale 2 π/k min (cid:107) ( k > k min (cid:107) ), is dominated by the second term,corresponding to scales larger than 2 π/k min (cid:107) ( k < k min (cid:107) ). From Eq.s (36, 37), the MFLRWdiffusion coefficient is given by κ sMF L = ( δB/B ) π ( q − / (4 qk min (cid:107) ). In Fig. 2, the κ sMF L isshown to recover the quasi-linear limit and is dominated by large wavelengths, given by theSi( x ) term in Eq. (37): D MF L = π G ( k (cid:107) = 0) /B = κ sMF L , where the quasi-linear limit isexpressed, as known, as power spectrum at zero parallel wavenumber.Equations (36, 37) provide the MFLRW for distances ∆ z along B shorter than L (cid:107) .The guiding center perpendicular scattering in a slab turbulence is described as a series ofbumps in the transverse drift which are asymptotically suppressed confining the transversemotion to follow the MFL meandering, as also found in low-rigidity test particle numericalsimulations (Qin et al. 2002). Therefore, we confirm that slab transverse transport is dueto the meandering of MFLs but we also model the transport across the MFL for first-ordermagnetic fluctuations not considered in previous treatments (K´ota & Jokipii 2000). Forthe slab turbulence, the transverse particle diffusion can be disentangled in two energeticcontributions: drift coefficient is dominated by length scales smaller than coherence lengthwhereas the MFLRW is dominated by length scales larger than coherence length. We consider 3D isotropic turbulence, in which the turbulence δB depends on all threespace coordinates. We adopt the following power spectrum (Batchelor 1970): P rq ( k ) = G ( k )8 πk (cid:20) δ lm − k l k m k (cid:21) (38)with G ( k ) = G k − q if k min < k < k max G k − qmin if k < k < k min , (39) 18 –where k max corresponds to the scale where the dissipation rates of the turbulence overcomesthe energy cascade rate, the coherence length is given by L = 2 π/k min , and the physicalscale of the system by 2 π/k (about the spectrum at large scales, see the discussion inSect.4.1). The constant G is fixed by normalization: G = ( δB ) ( q − q ( k min ) − q , (40)assuming k (cid:28) k min (cid:28) k max . We use spherical coordinate for the wavenumber k = k (sin θ cos ψ, sin θ sin ψ, cos θ ) and 3D-turbulence: δ B ( x ) = ( δB x , δB y , δB z ). Wecompute first d iD XX ( t ). In reference to Eq. (23), the non-zero terms are ( r, q, l, p ) = (3 , , , r, q, l, p ) = (2 , , ,
3) and ( r, q, l, p ) = (2 , , , d iD ( t ) κ B = 34 (cid:18) δBB (cid:19) q − q ( k min r g ) × (cid:20) F i ( y m , q ) y − qm + F i ( y, q ) | y m y y m (cid:21) (41)with y m = k min v (cid:107) t (cid:39) k min r g Ω t (and y (cid:39) k r g Ω t ); here, the term integrated over scalessmaller than coherence scale 2 π/k min (or k > k min ) is recast as F i ( y, q ) = 215 − y − q − q (cos y + y Si( y )) + 65 y − q sin yq − y − − q sin y q + 2 3 q − q + 1815 q (2 − q ) ¯ I (1 − q, y )+ 145 1 + q q ¯ I ( − − q, y ) (42)and the term integrated over scales larger than coherence scale 2 π/k min (or k < k min ) isrecast as F i ( y, q ) = y
15 (cos y + y Si( y )) −
15 Si( y )+ sin y − y y −
75 cos yy (43) 19 –where we approximated r g /v (cid:107) (cid:39) Ω − as in Eq. (32) and we used¯ I ( a, u ) = (cid:90) ∞ u y a − cos y dy = 1 / e − i π a Γ( a, iu ) + e i π a Γ( a, − iu )] , (44)from Gradshteyn & Ryzhik (1973), Eq.(3.761.7). The instantaneous transverse coefficientdiffusion in Eq.(41) is represented in Fig.3. The dominant term in the diffusive limit, withthe condition y m = k min v (cid:107) t <
1, is given by d iD ( t ) κ B → π (cid:18) δBB (cid:19) q − q − k min r g ) = κ iD κ B (45)and represented in Fig.3. In contrast to the slab, the particle drift from the MFL doesnot depend only on the power spectrum at length-scales smaller than L (cid:107) . As for the slab,transverse diffusion is axisymmetric: d iD XX ( t ) = d iD Y Y ( t ) = d iD ( t ). Statistically, chargedparticle motion is not tied to local MFL in 3D isotropic turbulence. Theorem on reduceddimensionality turbulence in Jokipii et al. (1993) and Jones et al. (1998) allows chargedparticle to be magnetized to local MFL within gyroradius scale only in turbulence dependingon a reduced number of space coordinates. Any three-dimensional extension could dependon specific geometry but would not be justified in general, as our result shows.Comparison with previous numerical simulations (see, e.g., Giacalone & Jokipii (1999))requires the evaluation of the MFLRW, for isotropic turbulence. Using Eq. (25), we find forthe average square displacement of MFL (same contribution along x - and y -axes) d iMF L ( t ) k min = 14 (cid:18) δBB (cid:19) q − q (cid:2) H i ( y m , q ) + H i ( y m , y , q ) (cid:3) ; (46)here the term integrated over scales smaller than coherence scale 2 π/k min (or k > k min ) isrecast as H i ( y, q ) = 1 qy (cos y + y Si( y )) + y − q sin y − q + 2 q + 1 q (2 + q ) y q I ( − − q, y ) (47) 20 –and the term integrated over scales larger than coherence scale 2 π/k min (or k < k min ) isrecast as H i ( y m , y , q ) = (cid:90) y m y dy Si( y ) y + (cid:18) Si( y )2 − sin y y + cos y y (cid:19) (cid:12)(cid:12)(cid:12) y m y . (48)Figure 4 shows the diffusive behaviour of MFL for a magnetic turbulence with isotropicwave-number spectrum. As for the slab case, scales larger than coherence length ( k < k min )dominate over the turbulent contribution (see Fig. 4). The leading terms in the diffusivelimit of H i ( y, q ) are found in the integral and in the first term in parenthesis of Eq. (48). Byintegrating by parts and using Gradshteyn & Ryzhik (1973), Eq.(4.421.1), it can be found d iMF L ( t ) k min → (cid:18) δBB (cid:19) q − q (cid:20) Si( y )log( y ) | y m y + π (cid:18) C + 12 (cid:19)(cid:21) = κ iMF L k min , (49)where C = 0 . y m = k min v (cid:107) t (cid:39) k min r g Ω t ( y (cid:39) k r g Ω t ).We find that the MFL of a 3D isotropic weakly turbulent magnetic field are superdiffusiveaccording to Eq. (49). This result does not imply the superdiffusion of the particles whichpropagate diffusively in a 3D isotropic turbulence (Giacalone & Jokipii 1999). Before beingtransported superdiffusively along a field line, a particle will undergo parallel scattering,not taken into account in this paper, and eventually decorrelate to another field line. Theseresults might be qualitatively extended to MHD-turbulence with a small k-anisotropy sothat results in diffusion regime found here still apply. In this case, particle drift retains itsdiffusive character shown in Fig. 3, even if only scales smaller than the coherence length( k > k min , in dotted-dashed) are taken into account. Therefore, at small length-scales, theperturbative approach based on δB (cid:28) B could be applied to solar wind turbulence. Atlength-scales much larger than the coherence scale ( k (cid:28) k min ), where the approximation δB (cid:28) B breaks down in the solar wind, these conclusions cannot be extended. 21 –
5. Discussion and conclusion
We have described analytically the time evolution of individual charged particles driftand MFLRW across a static magnetic field to first-order in the magnetic fluctuations. Weconsider the case where the motion perpendicular to the average magnetic field is dominatedby guiding center motion which includes the meandering of the MFL and the drift from thefirst-order orbit theory, in the approximation that the particle gyroradius is much smallerthan the length scale of magnetic field variations. In contrast to previous models for theperpendicular transport, we do not assume diffusive scattering at all times ; this allows usto treat consistently the slab turbulence perpendicular diffusion. Drift and MFL transversetransport are explicitly computed for both the slab and 3D isotropic cases. In the slabcase, the time-evolution of the drift displacement shows how the particle diffusion from theMFL is suppressed. The instantaneous slab drift coefficient diffusion transverse to the localfield depends on the turbulence power-law spectral index; for a Kolmogorov turbulence isfound to decrease as t − / , slower than compound diffusion displacement ( t − / ), which ishowever computed transversally to the average field and not to the local magnetic fieldas in this paper. The recovery of the MFL coefficient diffusion of QLT shows that thisresult does not depend on assuming MFL parallel diffusion. Secondly, we provide analyticaltime-dependence of drift and MFL coefficients of diffusion for a 3D isotropic turbulence.We found that for a 3D isotropic turbulence the particle drift from the local field line isdiffusive, whereas the field line itself is superdiffusive. Previous numerical simulations arenot contradicted by our result which is obtained neglecting scattering parallel to the meanfield. For the slab, we find that MFLRW is dominated by length-scales larger than thecoherence length whereas particle drift from the local field line is dominated by length-scalessmaller than the coherence length. This disentaglement does not hold for 3D isotropicturbulence: MFLRW is still dominated by large length-scales whereas as far as drift isconcerned turbulent energy contribute at all scales, both below and above the coherence 22 –length. The study carried out here provides a framework for particle transport in the solarwind and supernova remnant blast wave turbulence and questions the common assumptionthat cosmic-rays trajectory follow the magnetic field line.It is a pleasure to acknowledge the fruitful discussions with J. Giacalone, J. K´ota and D.Ruffolo. We thank the anonymous referee for useful suggestions and comments. The workof FF was supported by NSF grant ATM0447354 and by NASA grants NNX07AH19G andNNX10AF24G; the work of JRJ was partially supported by NASA grant NNX08AH55G. 23 – REFERENCES
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1. The horizontal line representsthe quasi-linear limit. The field-line wandering, by computing the magnetic fluctuation alongthe unperturbed trajectory from times smaller than the time needed to particles to travelthe turbulence correlation length, reaches asymptotically the QLT limit. 28 –Fig. 3.— Total average transverse drift is compared with the distinct contributions at length-scales smaller and larger than the coherence length in the diffusive regime in units of κ B as afunction of Ω t for k max r g = 10 − , k max /k min = 10 , k min /k = 10 , q = 11 / δB/B = 0 . κ iD /κ B , is shown. In contrast to the slab, length-scaleslarger than the coherence length give a non-negligible contribution to diffusion. 29 –Fig. 4.— Total average magnetic field line transverse displacement for a 3D isotropic tur-bulence is compared with the distinct contributions at length-scales smaller and largerthan the coherence length in the diffusive regime as a function of Ω t for k max r g = 10 − , k max /k min = 10 , k min /k = 10 , q = 11 / δB/B = 0 .
1. The diffusive limit, κ iMF L k minmin