Transient Evolution of Nonlinear Localized Coherent Structures of Kinetic Alfvén Waves

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1 Solar Phys (2007) 243: DOI /s Transient Evolution of Nonlinear Localized Coherent Structures of Kinetic Alfvén Waves H.D. Singh R.P. Sharma Received: 13 February 2007 / Accepted: 20 July 2007 / Published online: 6 September 2007 Springer Science+Business Media B.V Abstract We present numerical simulations of kinetic Alfvén waves (KAWs) and inertial Alfvén waves (IAWs) applicable to the solar wind, the solar corona, and the auroral regions, respectively, leading to the formation of coherent magnetic structures when the nonlinearity arises from ponderomotive effects and Joule heating. The nonlinear dynamical equation satisfies the modified nonlinear Schrödinger equation. The effect of nonlinear coupling between the main KAW/IAW and the perturbation, producing filamentary structures of the magnetic field, has been studied. Scalings in the spectral index of the power spectrum at different times have been calculated. These filamentary structures can act as a source for particle acceleration by wave particle interaction because the KAWs/IAWs are mixed modes and Landau damping is possible. 1. Introduction It has long been known that Alfvén waves are commonly observed in space plasmas, especially in the solar wind (Coleman, 1966; Unti and Neugebauer, 1968; Belcher and Davis, 1971), the magnetosheath (Sahraoui et al., 2003), and the auroral regions (Norqvist et al., 1996; Stasiewicz et al., 2000). At the equatorial magnetosphere, where the magnetospheric plasma is hot and the electron thermal speed exceeds the Alfvén speed, the kinetic Alfvén wave (KAW) is the appropriate limit. For cold plasma at low altitudes up to 3 4 R E in geocentric distance, where the electron thermal speed becomes much smaller than the Alfvén speed, the inertial Alfvén wave (IAW) is the appropriate wave mode. Clearly, IAWs arise in alow-β plasma with β = 8πn 0 T/B 2 0 <m e/m i,wheren 0 is the unperturbed plasma number density, T(= T e T i ) is the plasma temperature, B 0 is the strength of the ambient magnetic field, and m e and m i are the masses of the electron and the ion, respectively, whereas KAWs appear in an intermediate-β plasmas with m e /m i <β<1. The KAW/IAW is obtained when the magnetohydrodynamic (MHD) Alfvén wave develops a large wavenumber, k, transverse to the ambient magnetic field B 0.TheKAW/IAW H.D. Singh ( ) R.P. Sharma Centre for Energy Studies, Indian Institute of Technology, Delhi , India hdsingh_iit@yahoo.co.in

2 220 H.D. Singh, R.P. Sharma carries nonzero parallel electric and magnetic field perturbations, which play an important role in accelerating and heating plasma particles. Electric field fluctuations parallel to B 0 develop, leading to Landau damping (Stefant, 1970; Lysak and Lotko, 1996) and plasma heating. Thus the KAWs/IAWs can energize plasma from longer spatial scales to shorter spatial scales (Tsiklauri, Sakai, and Saito, 2005). One such means for transferring energy from the large scale to small scale is the transverse collapse, which leads to the formation of strong magnetic filaments parallel to the ambient field, as asymptotically predicted by the nonlinear Schrödinger equation (NLSE) for the wave envelope (Champeaux, Passot, and Sulem, 1997, 1998; Laveder, Passot, and Sulem, 2001, 2002). Hollweg and Isenberg (2002) proposed resonant interaction with cyclotron waves as the possible mechanism responsible for heating and accelerating the coronal hole ions to generate the fast solar wind. Vinas, Wong, and Klimas (2000) showed that high-frequency electron plasma oscillations excited by low-frequency obliquely propagating electromagnetic waves can heat the corona. Vointenko and Goossens (2004) have studied possible heating produced by oblique KAWs and found that the perpendicular heating of the heavy ions observed in the corona may be explained by the superadiabatic acceleration of the ions across the magnetic field induced by the KAWs. Some other alternative mechanisms for solar wind heating such as heating from shocks and rotational discontinuities have also been investigated (Lee and Wu, 2000; Vasquez and Hollweg, 2001). Tsurutani et al. (2005) argued that the ion acceleration is associated with the dissipation of phase-steepened Alfvén waves. Although a number of mechanisms for the coronal heating and acceleration of the solar wind have been investigated and can explain many observations, problems still remain. The filamentation process (hot spot formation) may provide a clue to the dissipation problem, as it is a fast (catastrophic) way to transport energy at small scales. Therefore, the filamentation of Alfvén waves is important in the context of the solar wind, the solar corona, and the auroral regions. Results from the Freja spacecraft show that there are several regions of intense fieldaligned currents having several scale sizes, ranging from large scale of 500-km width to narrow subkilometer scales (Lanchester et al., 2001). Simulation results of Lanchester et al. (2001) showed that the formation and evolution of the small-scale filamentary currents are determined by the propagation and reflection of Alfvén waves. As the Alfvén waves propagate toward the ionosphere, they generate filamentary structures extending along the magnetic field lines that connect spatial gradients in the magnetosphere and the ionosphere, providing an efficient magnetosphere ionosphere coupling. It is believed that auroral forms and vortex structures with a thickness of km are related to the nonlinear IAW phenomena. In fact, filamentary structures are frequently observed in the solar atmosphere (van Ballegooijen, 2004), near the Earth s bow shock (Alexandrova et al., 2004), in the auroral regions (Lühr et al., 1994), as well as in laboratory plasmas. Simulations and analytical studies show that the formation and evolution of small-scale filamentary structures are determined by the collapse of an Alfvén wave or coupling of an Alfvén wave with a small perturbation (Shukla and Stenflo, 1989; Champeaux, Passot, and Sulem, 1997; Laveder, Passot, and Sulem, 2002; Laveder et al., 2003; Shukla and Sharma, 2001; Shukla, Sharma, and Malik, 2004). The high-intensity filaments can induce other decaying waves as well as a source of further collapse of the KAW/IAW, changing the spectrum of the KAW/IAW turbulence, and hence the spectral index is expected to be changed. Goldstein et al. (1999) solvedmhd equations in spherical geometry to study the temporal evolution of Alfvénic solar wind turbulence. They computed the power spectra for various time series and found that, for later times and larger distances, the spectral index approached a value of 5/3.

3 Transient Evolution of Nonlinear Localized Coherent Structures 221 The present paper focuses on the transient filamentation process arising from the coupling between the main KAW/IAW (having a uniform intensity distribution in a plane transverse to the direction of propagation) and the transverse perturbation by taking the adiabatic response for the background density, in the presence of nonlinear electron heating and the ponderomotive force. To develop a fully numerical solution of Alfvén wave filamentation, the envelope nonlinear dynamical equation must satisfy the modified nonlinear Schrödinger equation (MNLSE) in the presence of significant dispersion. The motivation of this paper is to study this MNLSE numerically and examine the transient evolutionary nature of the filaments. Relevance of these studies to power spectral index and particle acceleration in many space regions will also be pointed out. The formation of transient filaments has also been studied semianalytically, by using the model equation as studied here, by Shukla and Sharma (2001) when the KAW has a Gaussian intensity distribution along its wavefront. However, these studies are limited by the paraxial ray approximation. Transient filamentation was also studied by Laveder, Passot, and Sulem (2001, 2002) but these studies were limited within the Hall MHD case and considered pure transverse electromagnetic waves. Therefore, their studies are not applicable when the pump Alfvén wave is a mixed mode (partly electrostatic and partly electromagnetic) as is the case for KAWs/IAWs. The organization of this paper is as follows: The model equations appropriate for KAWs in the intermediate-β regime (m e /m i β 1) and IAWs in the low-β regime (1 β m e /m i ) are presented in Section 2. The simulation results and their applications to the solar wind and the auroral regions are given in Section 3. Finally, the last section provides conclusions. 2. Model Equations The dynamical equation governing the propagation (in the x z plane) of low-frequency, long-wavelength, and finite-amplitude KAW/IAW (magnetized by a uniform ambient magnetic field B 0 along the z direction) can be obtained by using the standard method (Bellan and Stasiewicz, 1998; Shukla and Stenflo, 1999, 2000; Shukla, Stenflo, and Bingham, 1999; Shukla and Sharma, 2002) and can be written as 2 B y t 2 = Ɣ 1 λ 2 4 B y e x 2 t Ɣ 2v 2 4 B y 2 te λ2 e x 2 z + 2 v2 A ( 1 δn ) s 2 B y n 0 z, (1) 2 where Ɣ 1 = 1andƔ 2 = 0forlow-β(β m e /m i ) plasmas, and Ɣ 1 = 0andƔ 2 = 1for intermediate-β (m e /m i β 1) plasmas, λ e (= c 2 m e /4πn 2 0 ) is the collisionless electron skin depth, v te (= T e /m e ) is the electron thermal speed, and v A (= B0 2/4πn 0m i ) is the Alfvén speed, δn s = n e n 0 is the number density change, with n e the modified electron density. Consider a plane wave solution of Equation (1): B y = B y (x,z,t) e i(k 0x x+k 0z z ωt). (2) Using Equation (2) inequation(1), one gets the dynamical equation

4 222 H.D. Singh, R.P. Sharma 2i ω ( ) 1 + Ɣ1 k 2 B y va 2 0x λ2 e t ω (Ɣ ik 0x λ 2 e v 2 A B y 2ik 0z z + 1 vte 2 Ɣ 2 k 2 va 2 0z v 2 A ) By x δn s k2 0z B y = 0, n 0 ( Ɣ1 λ 2 e ω2 Ɣ 2 v 2 te λ2 e k2 0z) 2 B y x 2 (3) where z B y k 0z B y,k 0x (k 0z ) is the component of the wave vector perpendicular (parallel) to ẑb 0,andω is the Alfvén wave frequency. Also δn s [ ξ 1 B y 2], (4) n 0 for low β (Shukla and Stenflo, 1999), and δn s n 0 [ ξ 2 B y 2], (5) for intermediate β (Shukla and Stenflo, 2000), where ξ 1 = (1+8k 2 0x λ2 e )/48πn 0T e,ξ 2 ={[1 Δ(1+δ)v 2 A k 0z]/16πn 0 T e ω 2 },Δ= ω 2 /ω 2 ci,δ= m ek 2 0x /m ik 2 0z,andω ci (= cb 0 /m i c) is the ion gyrofrequency. It should be mentioned that the changes in the density considered here are due to both Joule heating and the ponderomotive force in IAWs and are only due to the ponderomotive force in KAWs. Using Equations (4) and(5), we rewrite Equation (3) in a dimensionless form as i B y t i B y z ± 2iƔ B y x ± 2 B y x 2 ± B y 2 B y = 0, (6) where the upper sign in the two sign conventions (and hereafter also unless specified) correspondsto low-β plasmas and the lower sign to intermediate-β plasmas and Ɣ is a parameter characterizing the normalized perpendicular wave number in terms of electron s collisionless skin depth, given by ( 1 + k 2 Ɣ = 0x ρs 2 ) 1/2 k 0xλ 1 + k0x 2 e, (7) λ2 e for low β, and Ɣ = (v te /v A ) k 0x λ e, (8) for intermediate β, whereρ s (= c s /ω ci ) is the ion gyroradius at the electron temperature and c s (= T e /m i ) is the ion sound speed. The normalizing values are t n = (2ω/vA 2 k2 0z )(1+k2 0x λ2 e ), z n = 2/k 0z,x n ={(1+k0x 2 c2 s /ω2 ci )/(1+k2 0x λ2 e )}1/2 λ e,andb n ={(1+ 8k0x 2 λ2 e )/48πn 0T e } 1/2 for low-β plasmas and t n = (2ω/vA 2 k2 0z )(1 + k2 0x λ2 e ), z n = 2/k 0z,x n = v te λ e /v A,andB n =[{1 Δ(1 + δ)}va 2 k2 0z /16πn 0T e ω 2 ] 1/2 for intermediate-β plasmas. In the next section we will carry out the numerical solution of Equation (6) with approximate initial conditions, which is a plane wave at which a nonuniform perturbation is superimposed. 3. Numerical Simulations We have looked for a numerical solution to Equation (6) by using a 2D pseudospectral method in a (2π/α x ) (2π/α z ) periodic spatial domain with α x,α z = 1and(128) 2 grid

5 Transient Evolution of Nonlinear Localized Coherent Structures 223 Figure 1 The magnetic field intensity profiles of KAW. (a) t = 1.5, (b) t = 2, (c) t = 2.5, and (d) t = 4.5. points. The initial condition of the simulation was B y (x,z,0) = B y0 ( cos(αx x) )( cos(α z z) ), (9) where B y0 = 1 is the amplitude of the homogenous pump KAW/IAW. A finite-difference method with a predictor corrector scheme was used for the integration in the time domain with a time step of dt = Before solving Equation (6) numerically, we wrote the algorithm for the well-known 2D cubic NLSE. Accordingly, it was compared with the well-known results and then modified for our case of Equation (6). The results of filament formation of KAWs/IAWs at various times by keeping Ɣ(=0.01) fixed are presented in the following. First we present the simulation results for KAWs that are applicable to the solar wind. The time evolution of the intensity of the transverse magnetic field is exemplified in Figures 1(a d) by means of snapshots at four instants of time (t = 1.5, 2, 2.5, and 4.5). The filamentary structures having one or two filaments that were more intense at early times (Figures 1(b) and (c)) end up with several filaments at later times. It is evident from the figure that at early times more intense filaments are formed and, with the advancement of time, these become chaotic structures. Once the chaotic structure is formed, the filaments become more or less chaotic at later times. Figures 2(a d) depict the contour plots of spectra of B y in the (k x,k z ) Fourier space at the same instants of time as in Figures 1(a d). The most intense localized structure is found in Figure 2(b). But in Figure 2(d), when t = 4.5, a fully chaotic state is found. It is evident from the figures that as time advances the location of the localized structures vary. With the advancement of time, the spectra become more chaotic and energies are distributed from a large scale to smaller scales available for dissipation.

6 224 H.D. Singh, R.P. Sharma Figure 2 Contours of B yk 2 against Fourier modes of KAW. (a) t = 1.5, (b) t = 2, (c) t = 2.5, and (d) t = 4.5. Next, we studied the power spectral index at different times. Figures 3(a d) show the variation of B yk 2 against k z for a fixed value of k x = 0 at different times. It is evident from the wavenumber spectrum that the spectral index is steeper than the Kolmogorov k 5/3 scaling at early times. The KAW spectra show k 15,k 5,k 3,andk 5/3 scalings at t = 1.5, 2, 2.5, and 4.5, respectively. However, as time advances, it approaches the Kolmogorov k 5/3 scaling. At later times also the spectrum is nearly the Kolmogorov k 5/3 type of scaling. This is in agreement with the power spectra for various time series found by Goldstein et al. (1999) for later times and larger distances. The wavenumber spectrum with its typically k 5/3 Kolmogorov form indicates quasisteady spectral transfer and strong nonlinear couplings. So the inferred power spectrum of magnetic field fluctuations indicates that nonlinear interactions may be distributing energy among large and intermediate wavenumbers with only a small amount of energy flowing into the dissipation range. Many authors (Coleman, 1968; Denskat, Beinroth, and Neubauer, 1983; Leamon et al., 1998) found a solar wind power spectrum with a much steeper branch ( k α 1,α 1 3 4) on the high-frequency end. In this paper we have discussed the power spectrum of the magnetic fluctuations propagating parallel to the magnetic field at fixed transverse fluctuations. But numerical simulations and analytical studies indicate that the spectral cascade to high wavenumber occurs most strongly for transverse (k ) fluctuations (see, e.g., Montgomery, 1982; Shebalin, Matthaeus, and Montgomery, 1983; Matthaeus et al., 1996; Goldreich and Sridhar, 1997; Medvedev, 2000). Observations of in situ solar wind fluctuations support the view that high-k modes dominate (Matthaeus, Goldstein, and Roberts, 1990; Bieber, Wanner, and Matthaeus, 1996), but there does seem to be evidence for high-k modes that dissipate via ion gyroresonance (e.g., Leamon et al., 1998).

7 Transient Evolution of Nonlinear Localized Coherent Structures 225 Figure 3 Variation of B yk 2 against k z at k x = 0forKAW.(a)t = 1.5, (b) t = 2, (c) t = 2.5, and (d) t = 4.5. The thick line indicates the spectral index scaling. Figure 4 Time evolution of B yk of three Fourier modes for KAW. To understand the time evolution of the y-component of the magnetic field, we analyze in the following the evolution of their major Fourier modes. Figure 4 shows the time evolution of the amplitudes of Fourier modes of the magnetic field at a fixed k x = 0 and at different k z = 1, 2, and 3. From the figure it is evident that the active participants in the energysharing process are mostly confined to the low-wavenumber modes. It can be seen that at t = 0 all the energies of mode A are there and it decays with time. At t 12.5 the nonlinear growth of mode A occurs. Since the k 5/3 scaling has already formed at t = 4.5, the growth of mode A at t = 12.5 occurred under the turbulent state of waves. Modes B and C grow first and decay, showing an oscillatory evolution. For application purposes, we discuss the relevance of our work to solar wind plasma parameters. The typical values of several solar wind parameters as measured by He-

8 226 H.D. Singh, R.P. Sharma Figure 5 The magnetic field intensity profiles of IAW. (a) t = 2, (b) t = 2.5, (c) t = 3, and (d) t = 3.5. lios 2 at 1 AU are B 0 6nT,n 0 3cm 3,T 10 ev; then β 0.335,v A cm s 1,v te cm s 1, ω ci 0.6 Hz, and ρ s cm. For these typical parameters at ω/ω ci = 0.02 and k x ρ s = 0.01, the magnitudes of the fluctuations are δb y0 /B ,Ɣ 0.01, and the k 5/3 scaling starts forming when the actual time scale of the temporal evolution is 12.5 minutes. In the solar wind region and over a wide range of heliospheric distances, the quantity δb y0 /B 0 tends to be small ( 5%) (Matthaeus and Goldstein, 1982; Roberts et al., 1987). The observed filamentary structures of Figures 1(a d) have a characteristic transverse scale size of the order of the ion gyroradius ρ s (at e 1 of the intensity peak). It is worth pointing out that our results can be quite useful for understanding nonlinear wave filamentation in the cusp region of the magnetopause where the solar wind can directly access the ionosphere. The typical plasma parameters in this region are B nt, n 0 5cm 3,andT 100 ev; then β 0.02, v A cm s 1, v te cm s 1, ω ci 9.58 Hz, and ρ s 10 6 cm. For these typical parameters at ω/ω ci = 0.02 and k x ρ s = 0.01, we get Ɣ 0.01 and the k 5/3 scaling starts forming when the actual time scale of the temporal evolution is 47 seconds. The Cluster multispacecraft mission carried out in situ simultaneous multipoint measurements in the high-altitude cusp region (Sundkvist et al., 2005) and revealed the presence of coupled drift and KAWs as well as electromagnetic ion-cyclotron waves whose characteristic perpendicular scale lengths are of (2 4)ρ s. Chmyrev et al. (1988) found the perpendicular scale size of the filaments of Alfvén vortex tubes propagating transverse to the background magnetic field of the order of ρ s when β>m e /m i. Next, we studied the transient filament formation and spectral index for IAWs found in the solar corona and auroral regions. We present here the magnetic field intensity profiles of IAWs in Figures 5(a d) for different times (t = 2, 2.5, 3, and 3.5). In Figures 6(a d) we illustrate the effect of filament formation on the wavenumber spectrum

9 Transient Evolution of Nonlinear Localized Coherent Structures 227 Figure 6 Variation of B yk 2 against k z for IAW. (a) t = 2, (b) t = 2.5, (c) t = 3, and (d) t = 3.5. The thick line indicates the spectral index scaling. ( B yk 2 against k z for a fixed k x = 0). The IAW spectra show k 7,k 5,k 3,andk 5/3 scalings at t = 2, 2.5, 3, and 3.5, respectively. Chandran (2005) has also calculated the power spectra ( k 3 and k 7/2 ) of weakly turbulent Alfvén waves and fast magnetosonic waves in low-β plasmas applicable to the solar corona. It is evident from Figures 5 and 6 that the pattern of filament formation for IAWs is almost the same as that of KAWs except that the spectra reach a k 5/3 scaling at earlier times than for KAWs. Moreover, like KAWs, here also the IAW spectra are almost k 5/3 at later times. For application purposes, we discuss the relevance of our work to auroral region plasma parameters. The typical plasma parameters at an auroral altitudes of 1700 km are B T, n cm 3,andT 1eV;thenβ , v A cm s 1, v te cm s 1, ω ci Hz, λ e cm, and ρ s cm. For these typical parameters at ω/ω ci = 0.02 and k x ρ s = 0.001, the magnitudes of the fluctuations are δb y0 /B 0 = and Ɣ = Morales et al. (1999) studied density and temperature filaments in the laboratory at fluctuation levels of δb /B , under conditions of relevance to the auroral ionosphere. The observed filamentary structures of Figure 5 have the characteristic transverse scale size of the order of λ e (at e 1 of the intensity peak). Morales et al. (1999) analyzed filamentary structures in laboratory nonthermal plasmas with transverse scale size of the order of λ e, spontaneously generating Alfvénic turbulence spatially localized in the filament region. 4. Conclusions We have investigated the nonlinear temporal evolution of large-amplitude KAWs/IAWs propagating at an angle to the background magnetic field. These filaments may act as the source for parametric instabilities of the Alfvén waves and may act as a source of decay waves as well as a source of further collapse. The formation of filaments is generally attributed to the cascade of energy among nonlinearly interacting modes. The transverse collapse (filamentation) of a KAW/IAW produces small-scale Alfvénic structures that allow

10 228 H.D. Singh, R.P. Sharma dissipation processes such as ion-cyclotron resonance or Landau damping to act, leading to the heating of the plasma. These studies involve mainly the nonlinear evolution of a coherent pump Alfvén wave for filamentation process. However, the interaction between two Alfvén waves may be another mechanism for generating the filamentation structures as studied by Wang and Lin (2006). Besides this, Shukla and Sharma (2001) and, recently, Sharma and Malik (2006) have also studied the mutual nonlinear interaction between two kinetic Alfvén beams and the effect on the filament formation process. When two KAWs are present, the background density modification by ponderomotive nonlinearities depends on the intensity of both the KAWs (in contrast to the single coherent beam case) and the filamentation threshold is considerably reduced by increasing the power of the second beam. This filamentation process (hot spot formation) may provide a clue to the dissipation problem, as it is a fast way to transport energy at small scales. The present mechanism may have some interesting relevance to the initiation of turbulence in the solar wind and heating of the solar wind ambient plasma. Selfconsistent models can be developed based on kinetic theory to estimate plasma heating. The wavenumber spectrum can be used to estimate the additional heating by calculating the velocity space diffusion coefficient in the Fokker Planck equation. Acknowledgements H.D. Singh is grateful to UGC, India, for providing the financial assistance for the present work. This work is also partially supported by DST (India). References Alexandrova, O., Mangeney, A., Maksimovic, M., et al.: 2004, J. Geophys. Res. 109, A Belcher, J.W., Davis, L., Jr.: 1971, J. Geophys. Res. 76, Bellan, P.M., Stasiewicz, K.: 1998, Phys.Rev.Lett.80, Bieber, J.W., Wanner, W., Matthaeus, W.H.: 1996, J. Geophys. Res. 101, Champeaux, S., Passot, T., Sulem, P.L.: 1997, J. Plasma Phys. 58, 665. Champeaux, S., Passot, T., Sulem, P.L.: 1998, Phys. Plasmas 5, 100. Chandran, B.D.G.: 2005, Phys. Rev. Lett. 95, Chmyrev, V.M., Bilichenko, S.V., Phokhotelov, O.A., et al.: 1988, Phys. Scr. 38, 841. Coleman, P.J., Jr.: 1966, Phys.Rev.Lett.17, 207. Coleman, P.J., Jr.: 1968, Astrophys. J. 153, 371. Denskat, K.U., Beinroth, H.J., Neubauer, F.M.: 1983, J. Geophys. Res. 54, 60. Goldreich, P., Sridhar, S.: 1997, Astrophys. J. 485, 680. Goldstein, M.L., Roberts, D.A., Deane, A.E., Ghosh, S., Wong, H.K.: 1999, J. Geophys. Res. 104(A7), Hollweg, J.V., Isenberg, P.A.: 2002, J. Geophys. Res. 107(7), SSH Lanchester, B.S., Rees, M.H., Lummerzheim, D., et al.: 2001, J. Geophys. Res. 106, Laveder, D., Passot, T., Sulem, P.L.: 2001, Physica D , 694. Laveder, D., Passot, T., Sulem, P.L.: 2002, Phys. Plasmas 9, 293. Laveder, D., Passot, T., Sulem, C., Sulem, P.L., Wang, D.S., Wang, X.P.: 2003, Physica D 184, 237. Leamon, R.J., Smith, C.W., Ness, N.F., Matthaeus, W.H., Wong, H.K.: 1998, J. Geophys. Res. 103, Lee, L.C., Wu, B.H.: 2000, Astrophys. J. 535, Lühr, H., Warnecke, J., Zanetti, L.J., Lindqvist, P.A., Hughes, T.J.: 1994, Geophys. Res. Lett. 21, Lysak, R.L., Lotko, W.: 1996, J. Geophys. Res. 101, Matthaeus, W.H., Goldstein, M.L.: 1982, J. Geophys. Res. 87, Matthaeus, W.H., Goldstein, M.L., Roberts, D.A.: 1990, J. Geophys. Res. 95, Matthaeus, W.H., Gosh, S., Oughton, S., Roberts, D.A.: 1996, J. Geophys. Res. 101, Medvedev, M.V.: 2000, Astrophys. J. 541, 811. Montgomery, D.: 1982, Phys. Scr. T2/1, 83. Morales, G.J., Maggs, J.E., Burke, A.T., Peñano, J.R.: 1999, Plasma Phys. Control. Fusion 41, A519. Norqvist, P., Andre, M., Eliason, L., et al.: 1996, J. Geophys. Res. 101, Roberts, D.A., Klein, L.W., Goldstein, M.L., Matthaeus, W.H.: 1987, J. Geophys. Res. 92, Sahraoui, F., Pincon, J.L., Belmont, G., et al.: 2003, J. Geophys. Res. 108, Sharma, R.P., Malik, M.: 2006, Astron. Astrophys. 457, 675.

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