electron transport in coronal loops: the influence of the ... - IOPscience

The Astrophysical Journal, 719:1912–1917, 2010 August 20 C 2010.

doi:10.1088/0004-637X/719/2/1912

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ELECTRON TRANSPORT IN CORONAL LOOPS: THE INFLUENCE OF THE EXPONENTIAL SEPARATION OF MAGNETIC FIELD LINES R. Bitane1 , G. Zimbardo2 , and P. Veltri2 1

Université de Nice-Sophia Antipolis, CNRS, Observatoire de la Côte d’Azur, Laboratoire Cassiopée, Blvd. De l’Observatoire, 06300 Nice, France 2 Dipartimento di Fisica, Universit´ a della Calabria, Ponte P. Bucci, Cubo 31C, I-87036 Arcavacata di Rende, Italy; [email protected] Received 2010 May 26; accepted 2010 June 30; published 2010 August 4

ABSTRACT Observations by the TRACE spacecraft have shown that coronal emission in the extreme ultraviolet is characterized by filamentary structures within coronal loops, with transverse sizes close to the instrumental resolution. Starting from the observed filament widths and using the concepts of braided magnetic fields, an estimate of the turbulence level in the coronal loops can be obtained. Magnetic turbulence in the presence of a background magnetic field can be strongly anisotropic, and such anisotropy influences the separation of magnetic field lines, as well as the magnetic field line diffusion coefficient. Careful computations of the magnetic field line diffusion coefficient Dm and of the rate of exponential separation of magnetic field lines h, also allowing for the possibility of anisotropic magnetic turbulence, enable computation of the effective perpendicular diffusion coefficient for electrons. When compared with observations this yields magnetic turbulence levels on the order of δB/B0 = 0.05–0.7, which are larger than previous estimates. These values of the magnetic fluctuation level support the idea that magnetic turbulence can contribute to coronal heating by means of MHD turbulence dissipation. It is also found that field line transport is not governed by the quasilinear regime, but by a nonlinear regime which includes an intermediate and the percolation regimes. Key words: chaos – diffusion – plasmas – Sun: corona – turbulence sizes close to the instrumental resolution (Aschwanden & Nightingale 2005). The distribution of loop widths is found to be in the range w = 1420 ± 340 km, the loop lengths in the range L = 32 ± 17 Mm (neglecting a high-end tail with lengths larger than 70 Mm), and temperatures T = (1.05 ± 0.12) × 106 K (with secondary peaks at 1.4 and 2.1 × 106 K; Aschwanden & Nightingale 2005). Aschwanden & Nightingale have also shown that the loop strands have an isothermal cross section over the width w 1400 km, which implies that cross field heat conduction in coronal loops is rather efficient, unless the heating itself occurs uniformly over the loop cross section. The isothermal cross section of the strands and the fact that the emitting filament cross section is nearly constant when moving along the loop suggest that the energy is released at small spatial scales and is transported diffusively: heat is quickly transported along the magnetic field lines and then propagates transversely to the magnetic field at a length-independent pace, so that the flaring of large-scale magnetic field lines in the coronal loop structure is not so evident. Assuming that emission is due to hot electrons, using the concepts of tangled (i.e., braided) magnetic fields, and considering a transport regime in which both magnetic turbulence and collisions play a role, known as Rechester and Rosenbluth diffusion (Rechester & Rosenbluth 1978), Galloway et al. (2006) derived estimates of the magnetic turbulence level, obtaining δB/B0 = 0.025–0.075. These estimates are much smaller than the turbulence levels quoted above, envisaged by Nigro et al. (2005), Buchlin & Velli (2007), and Grappin et al. (2008). On this basis, Galloway et al. (2006) questioned the possibility to heat coronal loops by dissipation of magnetic turbulence, since not much magnetic energy would be available in the fluctuations. Here, we revisit this idea, using up-to-date results on the transport of magnetic field lines in turbulent magnetic fields (Zimbardo et al. 2000; Pommois et al. 2001b) and on the evaluation of the rate of separation of close magnetic field lines, also called Kolmogorov entropy (Zimbardo et al. 2009). In fact,

1. INTRODUCTION It was suggested by Parker (1988) that coronal heating could be due to the release of magnetic energy in small time and spatial scale events called microflares and nanoflares. Such events are thought to convert magnetic energy into heat and particle kinetic energy by means of magnetic reconnection. Einaudi & Velli (1999) proposed that coronal dissipation occurs in bursts at very small spatial scales, which form the building blocks of coronal activity. Berger & Asgari-Targhi (2009) considered the braiding of magnetic field lines and its relation to reconnection in coronal loops. In order to assess the contribution of magnetic energy dissipation to the heating of coronal loops, it is fundamental to determine the level of magnetic fluctuations. Magnetohydrodynamic (MHD) simulations of nonlinear wave cascade to small scales and subsequent dissipation in coronal loops have shown that energy can indeed be released at short, intermittent time scales (Nigro et al. 2005; Buchlin & Velli 2007), and that turbulence dissipation can give a substantial contribution to heating coronal loops. Those numerical studies indicated a level of magnetic fluctuations δB/B0 0.2, which is in agreement with the estimate δB/B0 0.15 based on the propagation of MHD disturbances along coronal loops by Grappin et al. (2008). Here, δB represents the rms value of the fluctuating field and B0 the background average field of the loop. On the other hand, the presence of magnetic fluctuations influences particle perpendicular transport. This issue has been long considered in cosmic-ray transport (Jokipii 1966; Matthaeus et al. 2003; Ruffolo et al. 2004; Shalchi & Dosch 2008; Hauff et al. 2010) for the transport of solar energetic particles in the solar wind (Jokipii & Parker 1969; Reames 1999; Pommois et al. 2001a; Ippolito et al. 2005) and for particle transport in coronal loops (Galloway et al. 2006; Gkioulidou et al. 2007). Recent observations by the TRACE spacecraft have shown that coronal emission in the extreme ultraviolet is characterized by filamentary structures within coronal loops, with transverse 1912

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ELECTRON TRANSPORT IN CORONAL LOOPS

Galloway et al. (2006) assume a simple quasilinear form of the magnetic field line diffusion coefficient Dm and a simple estimate of the Kolmogorov entropy h. The quantities Dm and h enter the electron diffusion coefficient DRR obtained by Rechester and Rosenbluth (see below), so that the thickness of an emitting strand in coronal loops can be estimated as Δx 2 = 2DRR Δt. If the thickness and the propagation time (or the lifetime) are known, one can invert this expression to estimate DRR , from which the magnetic turbulence level is obtained. However, if the expression of DRR as a function of δB/B0 is overestimated, upon inverting, the inferred value of δB/B0 will be underestimated. In addition, an important property which needs to be taken into account is that magnetic turbulence in the presence of a background magnetic field B0 is likely to be highly anisotropic (Nigro et al. 2005; Buchlin & Velli 2007; Cranmer et al. 2007), since the MHD nonlinear cascade is faster for the wavevectors perpendicular to the magnetic field. In such a case, a spectrum with k⊥ k forms, which is referred to as two-dimensional (2D, e.g., Matthaeus et al. 1995) or quasi2D (Zimbardo et al. 2000). As a consequence, the turbulence correlation length parallel to the magnetic field l is much larger than the perpendicular one l⊥ , l l⊥ . This should be taken into account, as the turbulence anisotropy influences both the values of Dm and h; for instance, the value of the transverse gradients of B, which has a major influence on the separation of field lines (see, e.g., Matthaeus et al. 1995, 2003; Zimbardo et al. 2000, 2004; Pommois et al. 2007), strongly depends on the turbulence anisotropy. Further, the very nature of the transport regime depends on the turbulence anisotropy. Indeed, for large values of the Kubo number, defined as R = (δB/B0 )(l / l⊥ ), transport regimes substantially different from the quasilinear one are obtained (Isichenko 1991a, 1991b, 1992; Zimbardo et al. 2000; Zimbardo 2005). Therefore, in this paper, we extend the work of Galloway et al. (2006) by considering the results of numerical simulations of the field line transport and exponential separation, which lead to improved estimates of Dm and h. When comparing with the observations, magnetic turbulence levels on the order of δB/B0 = 0.05–0.7 are obtained, which are substantially larger than the values proposed by Galloway et al. (2006). In Section 2, we discuss the electron transport regimes for different Kubo numbers; in Section 3, we present the simulation results for the exponential separation rate of magnetic field lines, and in Section 4, we apply the results to TRACE observations of coronal loops, from which the values of δB/B0 are obtained. 2. ELECTRON TRANSPORT REGIMES In a magnetized plasma, transport perpendicular to the magnetic field would be limited to the particle gyroradius, unless either collisions or electromagnetic fluctuations allow for perpendicular displacement. Here, we recall the main properties of particle transport in the presence of magnetic turbulence, as this seems to be the governing effect for hot electrons in coronal loops (Galloway et al. 2006). In the presence of a background magnetic field B 0 = B0 eˆz and of collisions only, particle transport is characterized by a diffusive law with (Δx)2 = 2D⊥ t

(1)

(Δz)2 = 2D t,

(2)

and

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where the collisional diffusion coefficients can be estimated as D⊥ ρ 2 /τc ,

D λ2 /τc .

(3)

Here, (Δx)2 ((Δz)2 ) represents the mean square deviation perpendicular (parallel) to B 0 , ρ is the particle gyroradius and λ is the collisional mean free path along the average magnetic field, and τc is the collision time. For typical coronal conditions, λ ρ. On the other hand, the presence of magnetic turbulence causes a field line random walk, due to the deflection of magnetic field lines, which can be quantified as (Δx)2 = 2Dm Δz,

(4)

where Dm is the diffusion coefficient for magnetic field lines. The random walk of magnetic field lines influences particle propagation; in the small gyroradius limit, as appropriate for electrons, particles follow the magnetic field lines and in the case of free motion along the magnetic field, i.e., in the absence of collisions or pitch angle diffusion, particles move as Δz = vt, with v the particle speed, and the particle perpendicular transport would be given by (Δx)2 = 2Dm vt (e.g., Jokipii 1966; Rechester & Rosenbluth 1978). However, in the presence of collisions, plasma particles would diffuse along the magnetic field according to Equation (2), and the perpendicular displacement would be given by (Δx)2 = 2Dm Δz = 2Dm 2D t. (5) This expression shows that a subdiffusive regime can be obtained; this is called compound diffusion and is sometimes envisioned for cosmic rays (Kóta & Jokipii 2000; Matthaeus et al. 2003). However, the separation of magnetic field lines also matters (e.g., Jokipii 1973), and it was shown by Rechester & Rosenbluth (1978) that a diffusive (i.e., faster) transport regime can be recovered in the case when magnetic field lines are subject to stochastic instability, that is, if close magnetic field lines separate exponentially. Indeed, the exponential separation of field lines prevents the particles from retracing their path after a collision, otherwise a subdiffusive regime would be found. In this connection, it is important to understand what degree of field line exponentiation is necessary to recover the diffusive behavior (Qin et al. 2002; Matthaeus et al. 2003; Ruffolo et al. 2004; Zimbardo et al. 2006). The exponential separation of magnetic field lines is quantified by a parameter h, called Kolmogorov entropy, which is defined as (Zaslavsky & Chirikov 1972; Rechester et al. 1979; Zimbardo et al. 1984, 1995; Bickerton 1997) 1 d(s) h = lim lim ln , (6) s→∞ d0 →0 s d0 where d0 is the initial distance between two close field lines and d(s) is the corresponding distance at the field line length s. This quantity has received considerable attention in the theory of chaos, and in practice the value of h is obtained by studying the exponentiation of field lines in the presence of magnetic turbulence. Introducing the Kolmogorov length as LK = 1/ h, Rechester & Rosenbluth (1978) have shown that in the presence of both collisions and magnetic turbulence the perpendicular diffusion coefficient can be expressed as DRR =

Dm D l⊥ D 1/2 LK ln LK D ⊥

(7)

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BITANE, ZIMBARDO, & VELTRI

(for a recent derivation of this expression, see Galloway et al. 2006). It can be seen that several parameters can influence the electron diffusion coefficient. On the other hand, the transport regimes for magnetic field lines are conveniently classified in terms of the Kubo number R (Kadomtsev & Pogutse 1979; Isichenko 1991a, 1991b; Zimbardo et al. 2000, 2004; Pommois et al. 2001b), given by R=

δB l . B0 l⊥

(8)

In the quasilinear regime of magnetic field line transport, obtained for R 1, the magnetic field line diffusion coefficient scales as 2 δB l2 QL D⊥ ∼ βQL l = βQL R 2 ⊥ , (9) B0 l where βQL is a numerical prefactor. It was shown by Kadomtsev & Pogutse (1979) that for large Kubo numbers, R 1, a percolation regime is found; in such a case, the diffusion coefficient scales as (Gruzinov et al. 1990; Isichenko 1991b) 0.7 1.3 δB l⊥ l2 P Dm ∼ βP = βP R 0.7 ⊥ , (10) 0.3 B0 l l where βP is a prefactor. Further, in the transition from the quasilinear to the percolation regime, an intermediate regime is found for 1 < R < 30, for which the diffusion coefficient of magnetic field lines has been computed by Pommois et al. (2001b), and can be expressed as int Dm 0.0358 × R 1.2

l⊥2 . l

(11)

Considering the separation of magnetic field lines, an estimate of h for the case of uniform background magnetic field without shear and three-dimensional (3D) turbulence yields (Kadomtsev & Pogutse 1979; Zimbardo et al. 1984, 1995; Isichenko 1991a) 2 l δB R2 hQL ∼ = . (12) 2 B0 l l⊥ In the percolation regime, the Kolmogorov entropy is expected to scale as (Gruzinov et al. 1990; Isichenko 1991b, 1992) hP ∼ R

1/2

ln R/ l .

(13)

P The above percolation scaling of the diffusion coefficient Dm has been verified, with some approximation, by several authors (Ottaviani 1992; Vlad et al. 1998; Zimbardo et al. 2000; Pommois et al. 2001b; Hauff et al. 2010). On the other hand, the proposed scaling of hP in the percolation regime has never been studied until Zimbardo et al. (2009), with the only exception of a study for a six-wave perturbation model by Isichenko et al. (1992). In summary, we note that the dependence of DRR on the magnetic turbulence level and on the Kubo number dramatically changes with the transport regime. If we overlook the weak dependence contained in the logarithmic factor of the Rechester and Rosenbluth diffusion coefficient, Equation (7), in the quasilinear regime we have QL DRR ∼ (δB/B0 )4 ,

(14)

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while, assuming for Dm and h the scalings proposed by Gruzinov et al. (1990) and Isichenko (1991b), in the percolation regime we get P DRR ∼ (δB/B0 )1.2 ln(δB/B0 ), (15) a much slower scaling. As we shall see, the actual scaling of h with the turbulence level is even slower than that proposed by Gruzinov et al. (1990) and Isichenko (1991b), Equation (13), so that an even slower dependence of the electron diffusion coefficient on the turbulence level is obtained. Clearly, such a slow dependence of h on δB/B0 influences the final estimate of the fluctuation level in coronal loops, as shown in Section 4. 3. COMPUTATION OF THE EXPONENTIAL SEPARATION RATE OF MAGNETIC FIELD LINES A number of studies on the separation of magnetic field lines have been done. In particular, Jokipii (1973) considered the diffusive growth of the distance between field lines in the case when the distance traveled along the background magnetic field is much larger than the parallel correlation length l and the field lines are no longer very close; Rechester et al. (1979) considered the exponential separation of close field lines when the background magnetic field is characterized by magnetic shear, that is, for configurations relevant to fusion plasmas; Zimbardo et al. (1984) considered the exponential separation for magnetic configurations both with and without shear, but only in the quasilinear regime; Ruffolo et al. (2004) discussed several different regimes of field line separation as a function of the distance traveled along B0 . The exponential separation rate of magnetic field lines, or Kolmogorov entropy h, is defined by Equation (6), and it corresponds to the sum of all positive Lyapunov exponents associated with the magnetic field line equations (Benettin et al. 1976). Also, the Lyapunov exponents are the exponential separation (or convergence) rates along independent directions in space. In order to evaluate the Kolmogorov entropy, we used a 3D numerical model in which, following Zimbardo et al. (2009), we integrate the field line equations in a magnetic field given by the sum of a constant homogeneous background field B0 and static magnetic perturbations δB(r). These equations can be written as B0 + δB(r) dr = . (16) ds |B0 + δB(r)| The perturbed part of the magnetic field is defined by its Fourier series expansion as ) (σ ) δB(r) = (17) δBk e(σ k exp i k · r + φk , k,σ ) (σ ) where e(σ k are two polarization unit vectors, σ = 1, 2 and φk are random phases. The polarization vectors are given by

e(1) k =i

k × B0 , |k × B0 |

e(2) k =i

k × e(1) k . |k|

(18)

) The orthogonality of e(σ k with respect to k ensures that ∇ ·B = 0. We note that the wavevectors k are distributed on a 3D grid in phase space, so that the fluctuations are fully 3D with δBx , δBy , δBz = 0. This is true even for the strongly anisotropic cases considered below, so that we call those cases either quasi-2D or quasi-slab. The magnetostatic assumption is done in this paper. This is a good enough approximation for hot

No. 2, 2010


electrons (Te ∼ 106 K), whose thermal speed is on the order of 6000 km s−1 , while the Alfvén speed in coronal loops is on the order of 1000 km s−1 . The spectral amplitudes δBk , characterized by transversal and longitudinal correlation lengths l and l⊥ , are given by

5

(19)

where α = 5/3 is the typical Kolmogorov spectral index for fluids and plasmas, whereas C is a normalization constant. The turbulence anisotropy can be changed by varying l⊥ and l in order to investigate from the quasi-2D case (l⊥ l ) to the quasi-slab case (l⊥ l ). Accordingly, the constant amplitude surfaces in k space change from an oblate to a prolate ellipsoid. The spectrum presents two cutoffs for short and long wavelengths both in the transversal and in the longitudinal direction (band spectrum). In this way, we avoid the spurious periodicity effects due to the discretization of the k space (Pommois et al. 1998). We considered several values of δB/B0 between 0.01 and 1 and also several values of (l / l⊥ ) between 1 and 100 in order to obtain a Kubo number varying in a large range from 0.01 to 90, that is going from the quasilinear to the percolative regime. Compared to Zimbardo et al. (2009), we extended the upper limit of the Kubo number from 60 to 90. Cross sections of the magnetic flux tube structures for various anisotropy ratios are given by Zimbardo et al. (2004); those cross sections can give an intuitive idea of field line separation and tangling. According to the case, we used from 40 to 100 different initial conditions of the magnetic field lines to evaluate numerically the Lyapunov exponents, from which the Kolmogorov entropy is obtained. Zimbardo et al. (2009) have shown that even for different anisotropy ratios l / l⊥ , the values of h depend only on R, and not separately on δB/B0 and on l / l⊥ . Therefore, here we report only the results for l / l⊥ = 100. Figure 1 shows the simulation results, and we can see that the Kolmogorov entropy grows up to ∼4 for R ∼ 10 and then decidedly more slowly. This shows that in the large Kubo number regime the quasilinear estimate of h is not appropriate. We can fit our numerical results for the Kolmogorov entropy for 2 < R < 50 with the following empirical expression: h = (0.9957 × ln R + 1.5695)/ l⊥ .

6

(20)

The comparison between the numerical results and the fitting expression is shown in Figure 1. As can be seen, for values of the Kubo number below ∼50 the behavior of h is approximated by a logarithmic dependence on R, whereas for larger values h grows more slowly. It is interesting to note that the growth of h with R is even slower than that predicted for the percolation regime by Gruzinov et al. (1990) and Isichenko (1991b), h ∝ R 1/2 ln (R). This slow growth for large R has been investigated theoretically by Milovanov et al. (2009). Here, we make the fit for 2 < R < 50 because this range is appropriate for the application to coronal loops. However, even using different fitting ranges, it is seen that for large R the growth of h with R is even slower than logarithmic. 4. ELECTRON TRANSPORT IN CORONAL LOOPS We apply the above results for h to the TRACE observations of emitting filaments, considering a simple scenario for the propagation of hot electrons in coronal loops, and that electrons

4

h*l⊥

C δBk =

α/4+1/2 , k ⊥ 2 l⊥ 2 + k 2 l 2

1915

3

2

1

0

fit with (0.9957 ln(R)+1.5695)

0

10

20

30

40

50

60

70

80

90

R Figure 1. Numerical results on the Kolmogorov entropy (big dots with error bars) as a function of the Kubo number in the parameter range 1 < R < 90. The anisotropy level is (l⊥ / l ) = 100, so that the variation of R is due to the variation of δB/B0 from 0.01 to 0.9. The solid line represents a best logarithmic fit, and the error bars correspond to the statistical uncertainty on the values of h.

are heated at localized reconnection events. The observations suggest that electron transport along the magnetic field is much faster than electron transport perpendicular to it, in agreement with expectations. In particular, we assume that (1) the hot electrons are injected in a region of limited size, which here we assume to be pointlike and (2) the parallel propagation of electrons is due to collisional diffusion. Clearly, such assumptions represent a simplification of the physical situation, since we neglect all of the physics of electron acceleration and subsequent thermalization. However, a study of those effects is beyond the scope of the present work. We will compare our estimates of δB/B0 with those of Galloway et al. (2006). In this regard, we improve their estimates in two respects. First, they assumed for the magnetic field line diffusion coefficient Dm = (δB/B0 )2 l , that is, a quasilinear scaling where even the prefactor (that is, βQL in Equation (9)) is set to one, while numerically the prefactor is always found to be less than 0.1 (Gray et al. 1996; Zimbardo et al. 2000; Pommois et al. 2001b). Second, they assume that the Kolmogorov length LK = 1/ h equals the turbulence parallel correlation length l ; this, however, is certainly an oversimplification, given that LK is defined as the scale for exponential separation, not as a correlation length. The physical features of coronal loops, such as length and thickness, have been described by Aschwanden et al. (2000) for several cases; here, we consider the same set of six strands studied by Galloway et al. (2006; a subset of the catalog by Aschwanden et al. 2000), and whose dimensions are given in Table 1. We can compare an “observed” quantity such as the ratio of width w over the length L with the same ratio that we can obtain from electron transport. We assume that the transverse propagation of electrons is given by the Rechester and Rosenbluth diffusion coefficient, Equation (7), while the parallel propagation of electrons is due to collisional diffusion, with D λ2 /τc , see Equations (2) and (3). Here, we neglect the possible pitch angle diffusion due to the electron interaction

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Table 1 Loop Parameters Observed by TRACE and Obtained Magnetic Fluctuation Levels l / l⊥ = 30 Loop Width (km) 2300 1800 2100 2900 2400 3100

± ± ± ± ± ±

Loop Length (km)

100 300 300 300 200 100

20000 34000 64000 74000 92000 142000

± ± ± ± ± ±

4000 2000 4000 4000 12000 8000

w L

=

2DRR t 2D t

=

Dm h 1/2 . ln l⊥ h(D /D⊥ )

(21)

From this expression, we can obtain an estimate of the magnetic turbulence level in coronal loops, once the expressions of Dm and h are given. Considering the previous estimate of δB/B0 = 0.15–0.2 (Nigro et al. 2005; Grappin et al. 2008) and realistic values of l / l⊥ ∼ 30, we expect Kubo numbers of order R ∼ 5 or more. Therefore, we expect to be either in the intermediate or in the percolation regime, rather than in the quasilinear regime, see Section 2. The diffusion coefficient of magnetic field lines Dm in the intermediate regime 1 < R < 30 is given by Equation (11), while for the Kolmogorov entropy h we use Equation (20). By using the expression already given for D⊥ and D , the above ratio only depends on the level of turbulence δB/B0 and on the anisotropy ratio l / l⊥ : w L

=

0.0358×(δBl /B0 l⊥ )1.2 (l⊥ / l )(0.99×ln(δBl /B0 l⊥ )+1.57) . ln[(0.99×ln(δBl /B0 l⊥ )+1.57)(λe /ρe )]

δB/B0

R

± ± ± ± ± ±

24 8.0 4.2 5.3 3.1 2.5

0.794 0.268 0.140 0.178 0.104 0.083

with higher frequency waves (Veltri & Zimbardo 1993a, 1993b). Then we have

(22)

For electron temperatures of 106 K and electron number densities of 109 cm−3 , typical of the coronal loops (Aschwanden & Nightingale 2005), and assuming a magnetic field B0 ∼ 100 G, we obtain for the electron mean free path typical values of λe = 92.8 km and for the electron gyroradius of ρe = 0.21 cm (note that the collision time cancels in the ratio D /D⊥ in the argument of the logarithm at the denominator of the above equation). We note that Aschwanden et al. (2000) give electron temperatures ranging from 0.8 × 106 to 1.2 × 106 K for coronal loops. The varying value of Te would influence the electron mean free path λe . However, we neglect this variation for the present analysis, since this variation is not too strong and it is contained in the logarithm of Equation (21), so that its influence on the values of DRR is very small. In order to extract the turbulence level from the above equation, we also need to know the anisotropy ratio l / l⊥ . Clearly, we do not have direct access to such a ratio for coronal loops. On the other hand, the evolution of magnetic turbulence in the presence of a background magnetic field shows that an anisotropy with l / l⊥ 1 develops (Dobrowolny et al. 1980; Matthaeus et al. 1998), corresponding to the socalled quasi-2D anisotropy. Therefore, in order to take into account the possibility of such a strong anisotropy, we consider l / l⊥ 30 to be a reasonable assumption, and we assume a further value of l / l⊥ = 100 for comparison. Numerical inversion of Equation (22) then yields for δB/B0 and for R the values reported in Table 1 for six different loops. These values of δB/B0 0.05–0.79 are larger by almost an order of magnitude than those reported by Galloway et al. (2006). Also, we can see

l / l⊥ = 100

0.369 0.092 0.042 0.041 0.035 0.010

δB/B0

R

± ± ± ± ± ±

57 19 9.5 12 6.9 5.5

0.569 0.186 0.095 0.122 0.069 0.055

0.271 0.067 0.030 0.029 0.024 0.007

that the larger the anisotropy ratio l / l⊥ , the smaller the obtained turbulence levels. However, l / l⊥ = 100 should be considered as an upper limit, due to the curvature of loops which would lead to a defocusing of wavevectors. Conversely, smaller values of l / l⊥ would lead to larger values of the turbulence level. For the first loop in the table, a rather large value of δB/B0 = 0.79 is obtained: this is the same loop for which Galloway et al. obtained δB/B0 = 0.075, that is their largest value, and we consider that this is due to the fact that this loop has a typical thickness but it is rather short, compared to the average loops. The observed thickness may indicate that very strong magnetic turbulence was present in this loop, and indication of a very dynamical situation. On the other hand, we note that some of the loops considered in Table 1 are very elongated, especially the fifth and sixth, which correspond to low values of the ratio w/L and consequently to low values of δB/B0 . Conversely, if we consider the average width of 1420 km and average length of 32,000 km, as given by Aschwanden & Nightingale (2005), i.e., values close to the second loop in Table 1, we can see that a turbulence level δB/B0 0.2–0.25 can be assumed as typical for coronal loops. We may wonder whether such turbulence levels can influence the visible loop structure; indeed, two out of three of the TRACE extreme ultraviolet images reported in Figure 1 of Galloway et al. (2006) show considerable tangling and distortion of the loops, consistently with a relevant level of magnetic fluctuations. On the other hand, magnetic fluctuations can also be found at small scales, that is at scales smaller than the instrumental resolution of about 1000 km. Indeed, the nonlinear cascade typical of turbulence causes the energy to be distributed over a long spectrum which extends down to the dissipation scale (see, e.g., Nigro et al. 2005), so that only a fraction of the turbulence energy is found at scales above the instrumental resolution. Therefore, a loop observed in the extreme ultraviolet can appear to be relatively smooth and undistorted even with a fluctuation level δB/B0 0.2. The fact that we can estimate a level of magnetic fluctuations on the order of 0.2 or more is in agreement with previous MHD numerical simulations of turbulence evolution and dissipation in coronal loops (Nigro et al. 2005; Buchlin & Velli 2007; Grappin et al. 2008), and this suggests that a sufficiently strong level of magnetic turbulence is present in coronal loops to give an important contribution to coronal heating. Also, the corresponding values of the Kubo number are given in Table 1, R ∼ 2.5–57. The fact that R > 1 shows that transport is definitely not into the quasilinear regime, but rather either in the intermediate or in the percolation regime. Therefore, the frequently assumed quasilinear approximation is not appropriate for astrophysical plasmas in the presence of a strong background magnetic field, which often leads to a quasi-2D anisotropy. This result alone emphasizes the importance of understanding turbulent transport outside of the quasilinear regime.

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5. CONCLUSIONS

REFERENCES

In this paper, we have considered the perpendicular transport of electrons in solar coronal loops, taking into account the presence of anisotropic magnetic turbulence with a parallel correlation length l much larger than the perpendicular correlation length l⊥ . We have tried to point out the importance of turbulence anisotropy, which determines whether the magnetic field line transport belongs to either the quasilinear, small Kubo number regime, or the intermediate, or the percolation, large Kubo number regime. We have revisited the work of Galloway et al. (2006), which shows that electron transport in coronal loops is described by the Rechester and Rosenbluth regime, in order to use improved, more quantitative evaluations of the magnetic field line diffusion coefficient Dm and of the rate of exponential separation of magnetic field lines h. In particular, we have carried out a computation of h extending into the high Kubo number regime. This has allowed us to estimate the electron perpendicular diffusion coefficient also in the case of strong quasi-2D anisotropy, l / l⊥ 1, as it is appropriate to coronal loops. By making a comparison of the diffusion coefficient with TRACE UV observations of coronal loop widths and lengths, we can deduce typical magnetic turbulence levels in the range δB/B0 0.05–0.7. These values are about an order of magnitude larger than those obtained by Galloway et al. (2006) using the same approach, but with a less quantitative estimate of Dm and h. We point out that the Rechester and Rosenbluth diffusion coefficient assumes that particles closely follow magnetic field lines. We consider that this assumption is well satisfied by electrons in coronal loops, due to their very small gyroradius. The situation may be different for ions, since a decorrelation of the particle trajectory from the magnetic field lines can indeed occur, as shown by particle simulations in the presence of magnetic turbulence (Pommois et al. 2007; Minnie et al. 2009). We also note that, for particles in the absence of collisions, superdiffusive transport regimes are obtained both from numerical studies (Zimbardo et al. 2006; Pommois et al. 2007; Shalchi & Kourakis 2007) and from data analysis in the solar wind (Perri & Zimbardo 2007, 2008, 2009). In those cases, a generalized compound diffusion scenario can be envisaged (e.g., Zimbardo 2005), and it would be interesting to single out the influence of the exponential separation of field lines in those cases. Our results show that enough magnetic turbulence is present in coronal loops to give a substantial contribution to the loop heating by means of localized, fragmented dissipation events, as considered by Einaudi & Velli (1999), Nigro et al. (2005), and Buchlin & Velli (2007). We also find that field line transport happens in the large Kubo number regime and not in the quasilinear regime. This result has implications for magnetic field line transport in the solar wind, too, where strong turbulence levels, δB/B0 0.5–1, and quasi-2D anisotropy are frequently envisaged. Finally, this study is an interesting example of the application of concepts originally developed for fusion plasmas, like the exponential separation of field lines and the existence of a percolation transport regime, to astrophysical plasmas.

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We are grateful to Pierre Pommois for help in the numerical study. This work was supported in part by the Italian Space Agency, contract ASI no. I/015/07/0 “Esplorazione del Sistema Solare.”