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Probing diffusion parameters of suprathermal ions near heliospheric shocks
Advances in Space Research 34 (2004) 157–160 www.elsevier.com/locate/asr
Probing diffusion parameters of suprathermal ions near heliospheric shocks R. Kallenbach
a,*
, K. Bamert b, M. Hilchenbach c, B. Klecker
d
a
International Space Science Institute, Hallerstrasse 6, CH-3012 Bern, Switzerland Physikalisches Institut, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland c Max-Planck-Institut f€ ur Aeronomie, Postfach 2, D-37189 Katlenburg-Lindau, Germany Max-Planck-Institut f€ ur Extraterrestrische Physik, Postfach 1603, D-85740 Garching, Germany b
d
Received 2 December 2002; received in revised form 27 January 2003; accepted 3 March 2003
Abstract The distribution functions of suprathermal ions in the energy range 35–2000 keV/amu associated with the interplanetary coronal mass ejection of 14–16 July 2000 are analyzed. The ion spectra agree with a theoretical model on momentum diffusion in twodimensional magnetohydrodynamic turbulence. Ó 2004 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Heliospheric shocks; Suprathermal ions; Diffusion parameters; Response: coronal mass ejection
1. Introduction It has been shown for the Bastille Day event (CME of 14–16 July 2000) that the evolution of the spectra of suprathermal solar wind ions follows a model scheme, which is based on spatial and momentum diffusion coefficients that scale with particle speed and mass-percharge ratio A=Q, with Q denoting the mean solar wind charge state for each ion species (Bamert et al., 2003; Kallenbach et al., 2003). In this paper, the plasma parameters related to momentum diffusion are evaluated quantitatively and compared to a recent theoretical model by le Roux et al. (2002).
2. Observations The spectra of suprathermal ions in the energy range 35–2000 keV/amu associated with the Bastille Day event have been derived from the Highly Suprathermal Time of Flight (HSTOF) sensor on board the Solar and
*
Corresponding author. Tel.: +41-31-631-4897; fax: +41-31-6314891. E-mail address: [email protected] (R. Kallenbach).
Heliospheric Observatory (SOHO) at 1 AU (Bamert et al., 2003). These spectra have been fitted by formulae that describe two ion populations propagating away from the shock and its downstream turbulence region: 1. One population is generated by first-order Fermi acceleration at the shock. This population has power-law spectra near the shock and diffuses into the upstream and downstream regions. Although ions from first-order Fermi acceleration usually represent the dominant energetic particle population, they are not dominant in the CME event studied here – at least in the energy range, where HSTOF is sensitive. 2. The second suprathermal ion population is generated in the strong magnetic turbulence downstream of the shock. The data in Fig. 1 give evidence that the suprathermal ion flux in the turbulence region of length L is larger than at the shock. The distribution function fS (E; x) fitted to the data is E A 1 Q c exp ½E1 þ E2 ðxÞ ; fS ðE; xÞ ¼ fS;0 E exp E0 Q E A ð1Þ where fS;0 is a normalizing constant. The variable E denotes the ion energy-per-mass ratio in MeV/amu, A and Q are the ion mass and charge, respectively, in
0273-1177/$30 Ó 2004 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2003.03.067
158
R. Kallenbach et al. / Advances in Space Research 34 (2004) 157–160
Fig. 1. Overview of data from the Bastille Day coronal mass ejection event, adapted from Bamert et al. (2003). The three panels show from top to bottom: the He intensities derived from HSTOF rates data; the proton density from proton monitor (PM) data of the CELIAS experiment (Hovestadt et al., 1995) on board SOHO (http://umtof.umd.edu/pm/crn/); the magnetic field strength from Level 2 data of the MAG experiment on board ACE (http://www.srl.caltech.edu/ACE/ASC/level2/lvl2DATA-MAG.html).
atomic units, and E2 ðxÞ describes spatial diffusion and is related to the distance x from the turbulence region. The parameters E0 , E1 , and c can be identified in the distribution functions derived from a simple model on stochastic acceleration described below.
3. Theoretical considerations The evolution of suprathermal ion populations near a parallel shock between upstream and downstream plasma with bulk speeds jVj V1 and jVj V2 , respectively, is described by the transport equation of p of þ V rf ¼ r ðjrf Þ þ I S þ rV ot 3 op 1 o of þ 2 p2 Dpp ; p op op
ð2Þ
where p ¼ v=v0 , with v0 the speed of 1 MeV protons. The left hand side denotes the explicit time dependence and convection of suprathermal ions, whereas the terms on the right hand side describe spatial diffusion, sources (I), sinks (S), adiabatic deceleration, and momentum diffusion. We start our considerations with the most standard theory on diffusion coefficients in Alfvenic turbulence by Hasselmann and Wibberenz (1968). For symmetrically counter-propagating Alfven waves and isotropic distribution functions f , the spatial and momentum diffusion parameters j and Dpp are related by Dpp j ¼ VA2 p2 =9, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where the Alfven speed VA ¼ B0 = l0 mp np depends on the proton density np , the ambient magnetic field B0 , the proton mass mp in case of a proton-dominated plasma, and on the vacuum magnetic permeability l0 ¼ 1:257 106 H/m. The momentum diffusion parameter
Dpp is related to the energy diffusion parameter DEE by Dpp ¼ p2 DEE . For DEE we use the expression derived from the hard-sphere approximation, given in M€ obius et al. (1982) as a function of the amplitude dBres of the magnetic field irregularities with power spectral density P ¼ P0 f a resonating with ions of atomic mass A and charge Q: 2a 4 VA2 Q Eðaþ1Þ=2 ; DEE;res ¼ 9 j0 A B0 j0 ¼ 6 105 2 T m2 s1 ; ð3Þ dBres where the subscript ‘‘res’’ denotes gyro-resonant interaction between Alfven waves and ions. This yields 2a 4 VA2 Q pa1 and Dpp;res ¼ 9 j0 A a2 1 Q jres ¼ j0 p3a : ð4Þ 4 A The data from the Bastille Day event on suprathermal ions have shown that the spatial evolution of the spectra can be fitted by distribution functions which describe stationary spatial and momentum (energy) diffusion, with the momentum diffusion essentially restricted to the turbulence region downstream of the main shock. We neglect the explicite time derivative, the adiabatic deceleration term, and the convection term by choosing the downstream plasma frame as reference frame. It is further assumed that sources and sinks of the populations in the suprathermal energy range studied here can be neglected, i.e., injection into the process of spatial and energy diffusion occurs at lower energies, the solar wind bulk energies. Finally, it is assumed that one spatial coordinate x is sufficient to describe the situation of a parallel shock. We obtain a partial differential
R. Kallenbach et al. / Advances in Space Research 34 (2004) 157–160
equation solvable by separating the variables i.e., f ðx; pÞ ¼ gðxÞhðpÞ: o of 1 o of 2 j p Dpp;res þ 2 ¼ 0: ð5Þ ox ox p op op All Q=A- and p-dependent terms are incorporated into the momentum part of the equation, whereas all x-dependent parts are absorbed in the spatial differential equation. Both parts must equal a constant C because x and p are independent variables. Therefore, 42a o2 h a þ 1 oh A C þ p42a h ¼ 0: ð6Þ 2 op p op Q This is solved by 3 hðpÞ ¼ h0 p2c expðp2F gp2F Þ; c ¼ ; 4 pffiffiffiffi 2a C A 3a 9 6a ; g¼ ¼ ; ; F ¼ 2F Q 2 32F 2
ð7Þ
if we neglect terms of order p2F 2 and p4F 2 . The latter terms fall off more rapidly than others for a < 3 and for the parameters that apply in our study (this needs to be checked if applied to another problem). A similar solution for a similar problem can be found in the article by M€ obius et al. (1982). The spatial differential equation is evaluated by substituting x with a dimensionless variable n: 9j0 o j0 ogðxÞ ¼ CgðxÞ; 4VA2 ox 4 ox Z x 4VA 0 o2 gðnÞ 1 oVA ogðnÞ þ CgðnÞ ¼ 0: n :¼ dx ) þ VA on on on2 0 3j0 ð8Þ The constant C is related to the scale length L (in meters) of the turbulence region, centered at x ¼ 0, by C ¼ fnum
9j200 ; 2 16L2 VA;0
j00 ¼ j0 ðx ¼ 0Þ;
ð9Þ
VA;0 ¼ VA ðx ¼ 0Þ: The numerical parameter fnum depends on the exact spatial evolution of j0 ðxÞ and VA ðxÞ. For instance, with 1 1 VA ¼ VA;0 ¼ const, pffiffiffiffi and j0 ¼ j00 2expðjxj=LÞ, we get gðnÞ / expð C nÞ, and fnum ¼ ln ½gð0Þ=gðn1 Þ, where gðn1 Þ is the distribution function far away from the turbulence region, e.g., jxj > 5L. From the first panel in Fig. 1 we see that fnum 15 does not strongly depend on ion energy, which justifies our approach.
4. Comparison of observational results with theory The data of the Bastille Day event matched the above model if the spectral index a ¼ 1 was assumed.
159
However, the absolute quantity of the momentum diffusion parameter Dpp;res is somewhat difficult to reconcile assuming realistic magnetic field turbulence levels. Mathematically, the parameter C relates the plasma parameters of the turbulence region to the energy scale factors g and E, where the latter is A=ðQEo Þ for a spectral index a ¼ 1. The parameter ¼: p02 characterizes the momentum p0 , where the distribution function hðpÞ rolls over; high wave power gives a high roll-over momentum, and low wave power yields a roll-over at low momentum (or energy). In our case, the ambient magnetic field B0 in the turbulence region is 20 nT, the length of the turbulence region is L 5 109 m, the density is np 5 106 m3 , and the Alfven speed is VA 200 km s1 (Smith et al., 2001). Therefore, the parameters j00 , C, , and g are j00 3 1013 B20 =dB2res m2 s1 ; C 0:01B20 =dB2res ; 0:05AB0 =ðQdBres Þ; g 2QdBres =ðAB0 Þ: ð10Þ The Bastille Day spectra give values of order 2A=Q and g 0:25Q=A (Bamert et al., 2003), requiring dBres =B0 0:1. As this ratio is rather large, we also compare our results to the diffusion parameter Dpp;2D ¼ 2p3 Xl2
pv 2 hdB2 i 2 A ? x v B20
100xqDpp;res ;
q¼
hdB2? i hdB2? i ; B20 dB2res
ð11Þ
derived by le Roux et al. (2002, see also this issue) from quasi-linear kinetic theory for transport of superAlfvenic ions in two-dimensional turbulence. In the above formula, l is the cosine of the pitch-angle, and w ¼ rA r2c ð1 þ rA Þ2 =4, with rc the cross-helicity of two-dimensional turbulence (Matthaeus et al., 1991) and rA the ratio of the spectral energy density in the velocity fluctuations to the spectral energy density in magnetic field fluctuations. Here, we restrict ourselves to a rough comparison between Dpp;res and Dpp;2D and do not discuss the other coefficients Dlp;2D and Dll;2D . Assuming that the parameter w is of order unity, the value of Dpp;2D is larger than that of Dpp;res , if q is larger than 0.01. This may well be the case at relative turbulence levels of 1% because hdB2? i is derived from both resonant and nonresonant contributions of the turbulent magnetohydrodynamic field, while dB2res only represents the first-order resonant interaction of the turbulent magnetic field with the ions. Note, that the suprathermal solar wind particles observed in the April 11, 2001 CME (see the accompanying paper by Bamert et al.) show a similar
160
R. Kallenbach et al. / Advances in Space Research 34 (2004) 157–160
behavior as the ions observed during the Bastille Day event, but the dominance of stochastic acceleration is less pronounced.
Acknowledgements This work is supported by the Swiss National Science Foundation and by INTAS Grant WP 270.
5. Conclusions References The efficiency of stochastic acceleration of suprathermal solar wind ions in the energy range 35–2000 keV/amu in CME events can be larger than that of first-order Fermi acceleration. The momentum diffusion parameter derived from quasi-linear kinetic theory for charged-particle transport in two-dimensional turbulence (le Roux et al., 2002) can reconcile this high efficiency for stochastic ion acceleration observed for the Bastille Day CME and for other events. Standard quasi-linear theory for one-dimensional Alfvenic turbulence underestimates the momentum diffusion at relative turbulence levels above about 1%. Our study suggests, that the efficiency for momentum diffusion depends on the total turbulent wave power and simply on particle rigidity. This is consistent with the suggestion of Dr€ oge (2000), based on investigations of particles at higher rigidities from 0.1 to a few GV. Studies on the diffusion parameters of suprathermal ions in plasma turbulence near interplanetary shocks will promote understanding of the processes that preaccelerate suprathermal ions near the termination shock. The latter ions are the source population of the anomalous component of the Cosmic Rays and of energetic neutral atoms from the heliospheric interface region.
Bamert, K., Wimmer-Schweingruber, R.F., Kallenbach, R., Hilchenbach, M., Klecker, B. Charge-to-mass fractionation during injection and acceleration of suprathermal particles associated with the Bastille Day event: SOHO/CELIAS/HSTOF data, in: SOLAR WIND TEN: Proceedings of the 10th International Solar Wind Conference, AIP Conference Proceedings, vol. 679, pp. 668–671, 2003. Dr€ oge, W. The rigidity dependence of solar particle scattering mean free paths. Astrophys. J. 537, 1073–1079, 2000. Hasselmann, K., Wibberenz, G. Scattering of charged particles by random electromagnetic fields. Zeitschrift f€ ur Physik 34, 353, 1968. Hovestadt, D. et al. CELIAS – Charge, element and isotope analysis system for SOHO. Solid Phys. 162, 441–481, 1995. Kallenbach, R., Bamert, K., Wimmer-Schweingruber, R.F. Charge-tomass fractionation of suprathermal ions associated with interplanetary CMEs, in: SOLAR WIND 10: Proceedings of the 10th International Solar Wind Conference, AIP Conference Proceedings, vol. 679, pp. 672–675, 2003. le Roux, J.A., Zank, G.P., Milano, L.J., Matthaeus, W.H. A quasilinear kinetic theory for charged-particle transport in two-dimensional turbulence. Astrophys. J. 567, L155–L158, 2002. Matthaeus, W.H., Klein, L.W., Ghosh, S., Brown, M.R. Nearly incompressible magnetohydrodynamics, pseudosound, and solar wind fluctuations. J. Geophys. Res. 96, 5421–5435, 1991. M€ obius, E., Scholer, M., Hovestadt, D., Klecker, B., Gloeckler, G. Comparison of helium and heavy ion spectra in He-3-rich solar flares with model calculations based on stochastic fermi acceleration in Alfven turbulence. Astrophys. J. 259, 397–410, 1982. Smith, C.W. et al. ACE Observations of the Bastille day 2000 interplanetary disturbances. Solid Phys. 204, 229–254, 2001.
Probing diffusion parameters of suprathermal ions near heliospheric shocks R. Kallenbach
a,*
, K. Bamert b, M. Hilchenbach c, B. Klecker
d
a
International Space Science Institute, Hallerstrasse 6, CH-3012 Bern, Switzerland Physikalisches Institut, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland c Max-Planck-Institut f€ ur Aeronomie, Postfach 2, D-37189 Katlenburg-Lindau, Germany Max-Planck-Institut f€ ur Extraterrestrische Physik, Postfach 1603, D-85740 Garching, Germany b
d
Received 2 December 2002; received in revised form 27 January 2003; accepted 3 March 2003
Abstract The distribution functions of suprathermal ions in the energy range 35–2000 keV/amu associated with the interplanetary coronal mass ejection of 14–16 July 2000 are analyzed. The ion spectra agree with a theoretical model on momentum diffusion in twodimensional magnetohydrodynamic turbulence. Ó 2004 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Heliospheric shocks; Suprathermal ions; Diffusion parameters; Response: coronal mass ejection
1. Introduction It has been shown for the Bastille Day event (CME of 14–16 July 2000) that the evolution of the spectra of suprathermal solar wind ions follows a model scheme, which is based on spatial and momentum diffusion coefficients that scale with particle speed and mass-percharge ratio A=Q, with Q denoting the mean solar wind charge state for each ion species (Bamert et al., 2003; Kallenbach et al., 2003). In this paper, the plasma parameters related to momentum diffusion are evaluated quantitatively and compared to a recent theoretical model by le Roux et al. (2002).
2. Observations The spectra of suprathermal ions in the energy range 35–2000 keV/amu associated with the Bastille Day event have been derived from the Highly Suprathermal Time of Flight (HSTOF) sensor on board the Solar and
*
Corresponding author. Tel.: +41-31-631-4897; fax: +41-31-6314891. E-mail address: [email protected] (R. Kallenbach).
Heliospheric Observatory (SOHO) at 1 AU (Bamert et al., 2003). These spectra have been fitted by formulae that describe two ion populations propagating away from the shock and its downstream turbulence region: 1. One population is generated by first-order Fermi acceleration at the shock. This population has power-law spectra near the shock and diffuses into the upstream and downstream regions. Although ions from first-order Fermi acceleration usually represent the dominant energetic particle population, they are not dominant in the CME event studied here – at least in the energy range, where HSTOF is sensitive. 2. The second suprathermal ion population is generated in the strong magnetic turbulence downstream of the shock. The data in Fig. 1 give evidence that the suprathermal ion flux in the turbulence region of length L is larger than at the shock. The distribution function fS (E; x) fitted to the data is E A 1 Q c exp ½E1 þ E2 ðxÞ ; fS ðE; xÞ ¼ fS;0 E exp E0 Q E A ð1Þ where fS;0 is a normalizing constant. The variable E denotes the ion energy-per-mass ratio in MeV/amu, A and Q are the ion mass and charge, respectively, in
0273-1177/$30 Ó 2004 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2003.03.067
158
R. Kallenbach et al. / Advances in Space Research 34 (2004) 157–160
Fig. 1. Overview of data from the Bastille Day coronal mass ejection event, adapted from Bamert et al. (2003). The three panels show from top to bottom: the He intensities derived from HSTOF rates data; the proton density from proton monitor (PM) data of the CELIAS experiment (Hovestadt et al., 1995) on board SOHO (http://umtof.umd.edu/pm/crn/); the magnetic field strength from Level 2 data of the MAG experiment on board ACE (http://www.srl.caltech.edu/ACE/ASC/level2/lvl2DATA-MAG.html).
atomic units, and E2 ðxÞ describes spatial diffusion and is related to the distance x from the turbulence region. The parameters E0 , E1 , and c can be identified in the distribution functions derived from a simple model on stochastic acceleration described below.
3. Theoretical considerations The evolution of suprathermal ion populations near a parallel shock between upstream and downstream plasma with bulk speeds jVj V1 and jVj V2 , respectively, is described by the transport equation of p of þ V rf ¼ r ðjrf Þ þ I S þ rV ot 3 op 1 o of þ 2 p2 Dpp ; p op op
ð2Þ
where p ¼ v=v0 , with v0 the speed of 1 MeV protons. The left hand side denotes the explicit time dependence and convection of suprathermal ions, whereas the terms on the right hand side describe spatial diffusion, sources (I), sinks (S), adiabatic deceleration, and momentum diffusion. We start our considerations with the most standard theory on diffusion coefficients in Alfvenic turbulence by Hasselmann and Wibberenz (1968). For symmetrically counter-propagating Alfven waves and isotropic distribution functions f , the spatial and momentum diffusion parameters j and Dpp are related by Dpp j ¼ VA2 p2 =9, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where the Alfven speed VA ¼ B0 = l0 mp np depends on the proton density np , the ambient magnetic field B0 , the proton mass mp in case of a proton-dominated plasma, and on the vacuum magnetic permeability l0 ¼ 1:257 106 H/m. The momentum diffusion parameter
Dpp is related to the energy diffusion parameter DEE by Dpp ¼ p2 DEE . For DEE we use the expression derived from the hard-sphere approximation, given in M€ obius et al. (1982) as a function of the amplitude dBres of the magnetic field irregularities with power spectral density P ¼ P0 f a resonating with ions of atomic mass A and charge Q: 2a 4 VA2 Q Eðaþ1Þ=2 ; DEE;res ¼ 9 j0 A B0 j0 ¼ 6 105 2 T m2 s1 ; ð3Þ dBres where the subscript ‘‘res’’ denotes gyro-resonant interaction between Alfven waves and ions. This yields 2a 4 VA2 Q pa1 and Dpp;res ¼ 9 j0 A a2 1 Q jres ¼ j0 p3a : ð4Þ 4 A The data from the Bastille Day event on suprathermal ions have shown that the spatial evolution of the spectra can be fitted by distribution functions which describe stationary spatial and momentum (energy) diffusion, with the momentum diffusion essentially restricted to the turbulence region downstream of the main shock. We neglect the explicite time derivative, the adiabatic deceleration term, and the convection term by choosing the downstream plasma frame as reference frame. It is further assumed that sources and sinks of the populations in the suprathermal energy range studied here can be neglected, i.e., injection into the process of spatial and energy diffusion occurs at lower energies, the solar wind bulk energies. Finally, it is assumed that one spatial coordinate x is sufficient to describe the situation of a parallel shock. We obtain a partial differential
R. Kallenbach et al. / Advances in Space Research 34 (2004) 157–160
equation solvable by separating the variables i.e., f ðx; pÞ ¼ gðxÞhðpÞ: o of 1 o of 2 j p Dpp;res þ 2 ¼ 0: ð5Þ ox ox p op op All Q=A- and p-dependent terms are incorporated into the momentum part of the equation, whereas all x-dependent parts are absorbed in the spatial differential equation. Both parts must equal a constant C because x and p are independent variables. Therefore, 42a o2 h a þ 1 oh A C þ p42a h ¼ 0: ð6Þ 2 op p op Q This is solved by 3 hðpÞ ¼ h0 p2c expðp2F gp2F Þ; c ¼ ; 4 pffiffiffiffi 2a C A 3a 9 6a ; g¼ ¼ ; ; F ¼ 2F Q 2 32F 2
ð7Þ
if we neglect terms of order p2F 2 and p4F 2 . The latter terms fall off more rapidly than others for a < 3 and for the parameters that apply in our study (this needs to be checked if applied to another problem). A similar solution for a similar problem can be found in the article by M€ obius et al. (1982). The spatial differential equation is evaluated by substituting x with a dimensionless variable n: 9j0 o j0 ogðxÞ ¼ CgðxÞ; 4VA2 ox 4 ox Z x 4VA 0 o2 gðnÞ 1 oVA ogðnÞ þ CgðnÞ ¼ 0: n :¼ dx ) þ VA on on on2 0 3j0 ð8Þ The constant C is related to the scale length L (in meters) of the turbulence region, centered at x ¼ 0, by C ¼ fnum
9j200 ; 2 16L2 VA;0
j00 ¼ j0 ðx ¼ 0Þ;
ð9Þ
VA;0 ¼ VA ðx ¼ 0Þ: The numerical parameter fnum depends on the exact spatial evolution of j0 ðxÞ and VA ðxÞ. For instance, with 1 1 VA ¼ VA;0 ¼ const, pffiffiffiffi and j0 ¼ j00 2expðjxj=LÞ, we get gðnÞ / expð C nÞ, and fnum ¼ ln ½gð0Þ=gðn1 Þ, where gðn1 Þ is the distribution function far away from the turbulence region, e.g., jxj > 5L. From the first panel in Fig. 1 we see that fnum 15 does not strongly depend on ion energy, which justifies our approach.
4. Comparison of observational results with theory The data of the Bastille Day event matched the above model if the spectral index a ¼ 1 was assumed.
159
However, the absolute quantity of the momentum diffusion parameter Dpp;res is somewhat difficult to reconcile assuming realistic magnetic field turbulence levels. Mathematically, the parameter C relates the plasma parameters of the turbulence region to the energy scale factors g and E, where the latter is A=ðQEo Þ for a spectral index a ¼ 1. The parameter ¼: p02 characterizes the momentum p0 , where the distribution function hðpÞ rolls over; high wave power gives a high roll-over momentum, and low wave power yields a roll-over at low momentum (or energy). In our case, the ambient magnetic field B0 in the turbulence region is 20 nT, the length of the turbulence region is L 5 109 m, the density is np 5 106 m3 , and the Alfven speed is VA 200 km s1 (Smith et al., 2001). Therefore, the parameters j00 , C, , and g are j00 3 1013 B20 =dB2res m2 s1 ; C 0:01B20 =dB2res ; 0:05AB0 =ðQdBres Þ; g 2QdBres =ðAB0 Þ: ð10Þ The Bastille Day spectra give values of order 2A=Q and g 0:25Q=A (Bamert et al., 2003), requiring dBres =B0 0:1. As this ratio is rather large, we also compare our results to the diffusion parameter Dpp;2D ¼ 2p3 Xl2
pv 2 hdB2 i 2 A ? x v B20
100xqDpp;res ;
q¼
hdB2? i hdB2? i ; B20 dB2res
ð11Þ
derived by le Roux et al. (2002, see also this issue) from quasi-linear kinetic theory for transport of superAlfvenic ions in two-dimensional turbulence. In the above formula, l is the cosine of the pitch-angle, and w ¼ rA r2c ð1 þ rA Þ2 =4, with rc the cross-helicity of two-dimensional turbulence (Matthaeus et al., 1991) and rA the ratio of the spectral energy density in the velocity fluctuations to the spectral energy density in magnetic field fluctuations. Here, we restrict ourselves to a rough comparison between Dpp;res and Dpp;2D and do not discuss the other coefficients Dlp;2D and Dll;2D . Assuming that the parameter w is of order unity, the value of Dpp;2D is larger than that of Dpp;res , if q is larger than 0.01. This may well be the case at relative turbulence levels of 1% because hdB2? i is derived from both resonant and nonresonant contributions of the turbulent magnetohydrodynamic field, while dB2res only represents the first-order resonant interaction of the turbulent magnetic field with the ions. Note, that the suprathermal solar wind particles observed in the April 11, 2001 CME (see the accompanying paper by Bamert et al.) show a similar
160
R. Kallenbach et al. / Advances in Space Research 34 (2004) 157–160
behavior as the ions observed during the Bastille Day event, but the dominance of stochastic acceleration is less pronounced.
Acknowledgements This work is supported by the Swiss National Science Foundation and by INTAS Grant WP 270.
5. Conclusions References The efficiency of stochastic acceleration of suprathermal solar wind ions in the energy range 35–2000 keV/amu in CME events can be larger than that of first-order Fermi acceleration. The momentum diffusion parameter derived from quasi-linear kinetic theory for charged-particle transport in two-dimensional turbulence (le Roux et al., 2002) can reconcile this high efficiency for stochastic ion acceleration observed for the Bastille Day CME and for other events. Standard quasi-linear theory for one-dimensional Alfvenic turbulence underestimates the momentum diffusion at relative turbulence levels above about 1%. Our study suggests, that the efficiency for momentum diffusion depends on the total turbulent wave power and simply on particle rigidity. This is consistent with the suggestion of Dr€ oge (2000), based on investigations of particles at higher rigidities from 0.1 to a few GV. Studies on the diffusion parameters of suprathermal ions in plasma turbulence near interplanetary shocks will promote understanding of the processes that preaccelerate suprathermal ions near the termination shock. The latter ions are the source population of the anomalous component of the Cosmic Rays and of energetic neutral atoms from the heliospheric interface region.
Bamert, K., Wimmer-Schweingruber, R.F., Kallenbach, R., Hilchenbach, M., Klecker, B. Charge-to-mass fractionation during injection and acceleration of suprathermal particles associated with the Bastille Day event: SOHO/CELIAS/HSTOF data, in: SOLAR WIND TEN: Proceedings of the 10th International Solar Wind Conference, AIP Conference Proceedings, vol. 679, pp. 668–671, 2003. Dr€ oge, W. The rigidity dependence of solar particle scattering mean free paths. Astrophys. J. 537, 1073–1079, 2000. Hasselmann, K., Wibberenz, G. Scattering of charged particles by random electromagnetic fields. Zeitschrift f€ ur Physik 34, 353, 1968. Hovestadt, D. et al. CELIAS – Charge, element and isotope analysis system for SOHO. Solid Phys. 162, 441–481, 1995. Kallenbach, R., Bamert, K., Wimmer-Schweingruber, R.F. Charge-tomass fractionation of suprathermal ions associated with interplanetary CMEs, in: SOLAR WIND 10: Proceedings of the 10th International Solar Wind Conference, AIP Conference Proceedings, vol. 679, pp. 672–675, 2003. le Roux, J.A., Zank, G.P., Milano, L.J., Matthaeus, W.H. A quasilinear kinetic theory for charged-particle transport in two-dimensional turbulence. Astrophys. J. 567, L155–L158, 2002. Matthaeus, W.H., Klein, L.W., Ghosh, S., Brown, M.R. Nearly incompressible magnetohydrodynamics, pseudosound, and solar wind fluctuations. J. Geophys. Res. 96, 5421–5435, 1991. M€ obius, E., Scholer, M., Hovestadt, D., Klecker, B., Gloeckler, G. Comparison of helium and heavy ion spectra in He-3-rich solar flares with model calculations based on stochastic fermi acceleration in Alfven turbulence. Astrophys. J. 259, 397–410, 1982. Smith, C.W. et al. ACE Observations of the Bastille day 2000 interplanetary disturbances. Solid Phys. 204, 229–254, 2001.