Marcowith, A

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29th International Cosmic Ray Conference Pune (2005) 3, 221224On cosmic-ray self-generated turbulence of supernova remnant shocksA. Marcowith , G. Pelletierand M. Lemoine(a) Centre d’Etude Spatiale des Rayonnements, 9 av du colonel Roche, 31028 Toulouse, France(b) Laboratoire d’Astrophysique de Grenoble, rue de la piscine, 38000 Grenoble, France(c) Institut d’Astrophysique de Paris, Bvd Arago, 75014 Paris, FrancePresenter: A. Marcowith (Alexandre.Marcowith@cesr.fr), fra-marcowith-A-abs1-og14-oralIt has been claimed that young Supernovae remnants (SNr) could be at the origin of cosmic-rays up to energiesof the order of 	 because of the strong amplification of the magnetic field by the stream of the accel-erated particles in the precursor of the shock. Moreover the turbulence spectrum so generated would allow aBohm regime for cosmic ray transport, which would provide the maximum efficiency of the acceleration. Inthis work, we have investigated these issues through a spectral analysis of the magnetic turbulence excited up-stream the forward SNR shock. Both resonant and non-resonant regimes of the streaming instability are treatedconcurrently. The turbulence spectra in a mean magnetic field as well as the turbulence level at the shock arederived for a quasi-parallel shock configuration accounting for different non-linear transfer processes. Wefinally discuss the consequences in the downstream flow of the stationary upstream turbulence.1. IntroductionThe Fermi acceleration of cosmic rays in astrophysical shock fronts depends in a crucial way on their transportproperties in the turbulent magnetic field on both sides of the shock. Often the turbulent field spectrum andintensity are arbitrarily prescribed, assuming that it has been built by the ambient medium independently ofthe shock acceleration process. However, the anisotropy of the cosmic ray distribution function triggers aninstability upstream of the shock ([1], [2]). Recently, [3] have shown that the amplification of the turbulentmagnetic energy could be quite significant and could produce a magnetic field intensity suitable for pushing thehigh energy cut-off of the proton distribution up to the “knee” of the cosmic ray spectrum (  	 eV).This theory has then been developed further by [4] analysing the competition between turbulence generationand advection towards the shock front leading to the saturation of the spectrum. These authors have also carriedout a preliminary examination of the role of the Kolmogorov cascade for the energy transfer among the excitedwaves. More recently [5] described a non-resonant regime of the streaming instability and has shown that itsgrowth rate should be dominant in the high wavenumber range. In this work, we analyse the excitation ofAlfve´n waves as a function of the location in the upstream flow and of the wavenumber taking into accountthe two instability regimes (Section 2). In Section 3, we calculate the saturation mechanism of the instabilityconsidering the advection effect as a function of the wavenumber and the location in the upstream flow. Wecalculate the contribution due to two nonlinear effects: the transverse nonlinear transfer among turbulent Alfve´nwaves, and the non-linear backscattering of Alfve´n waves off slow magneto-sonic waves. These two nonlinearprocesses are shown to be relevant and essential to the determination of the anisotropic turbulence spectra. Wefinally derive these spectra mandatory for calculating the transport coefficients of the cosmic rays. In section4, we examine the astrophysical consequences of our work with some emphasizes on the phases of dominanceof the two regimes of instability and on the properties the turbulence downstream the shock.2. The two regimes of the streaming instabilityThe instability triggered by the super-Alfve´nic flow of cosmic rays upstream of a shock has been analysed intwo ways: one is related to the resonant interaction of the cosmic rays with the Alfve´n waves (see [2]) and222 A.Marcowith, G. Pelletier and M. Lemoineis essentially described by the kinetic theory. The other one has recently been proposed by [5], it emphasisesthe importance of non-resonant interactions in which the DC-electric current ﬁﬀ of cosmic-rays generates aLorentz force responsible for the amplification of the MHD perturbations. In fact, this is the return current ﬂﬃin the background plasma which generates the perturbations. The plasma remains locally neutral  ﬁﬀ"!#$ﬂﬃ&%with' ﬁﬀ !' ﬂﬃ %' . Hereafter, (*) and "+) denote the mean magnetic field and the Alfve´n velocity inthe interstellar medium; i.e. far from the forward SN shock. We assumed that the magnetic field obliquityallows a de Hofmann-Teller transformation, such that the cosmic rays motion, at velocity'", with respect tothe upstream medium, is along the mean field.The non-resonant streaming instability:Following the analysis of the Alfve´n wave equation in [5] one can obtain for &- )/.  ,10  , an unstablebranch of the dispersion relation when 243#%65783- ):9<;=>;?5;3=A@CB . The wavenumber; reads as;EDF ﬂ>ﬃ (G)F. IHJ3- )%F Kﬂ 5KMLFN(*) .9POﬂ  - )@Q, . - ) where IHR%OﬂGS H ,KﬂNS H andKMLS H are the densityfraction in proton and electron cosmic-ray respectively. Once the instability grows, the magnetic field and theAlfve´n speed in; have to include the perturbed magnetic field component and are written as barred quantities.For instance, we have T"+C%UQ+A) .9V5XW@	YZ3 . The quantity WD\[(R3 .9[(J3]!^(R3)@determines the strengthof turbulence; W`_  corresponds to [( . (8)a_b!8c . The unstable mode is a purely growing mode ofright or left circular polarization depending on the main composition of the CR-fluid and the orientation ofthe magnetic field. For the likely case of a proton dominated CR-fluid, the mode is left-handed with respectto'() ; in other words, it rotates in the same sense as the protons. This instability does apply downto a scale[H%\d.2eﬂAf beyond which the MHD approximation is no longer valid; i.e. for;Z[H%F KﬂJ5KLF,A.gTQ- . Asstated by [5] at short waves  .hi]0;=0; , the non-resonant growth rate is jJkl"ﬀL,9<;=@nmT-9<;;Q=@	YZ3 .The Larmor radius h i corresponding to the minimum CR distribution momentum is an increasing quantitywith the distance to the shock. Note that this non-resonant instability is not operative for a wavenumber;e=at distances o0qpr9hs%t.;=@since the correspondinghi9o@0.;=. The exact spatial dependence ofthe growth rate will be specified further on. For long waves,;0.Nhi the CR-response dominates, i.e. thenon-resonant instability is inactive, the resonant instability takes over.The resonant streaming instability:Both electrons and ions, moving forward or backward, can resonate either with forward modes or backwardmodes. The small instability growth rate is the same, within an angular factor of order unity, and is given byjﬀL,9<;=uo@%j*H9<;=uo@	v9o.pAr9<;=@w@, wherepAr9x;Q=@should be understood aspr9hs%.;=@; the exact spa-tial dependence ofvis given by the convection-diffusion equation; in the case of uniform diffusion coefficienty,v%{z|~}95?o.NpAr@. The growth rate j8H (from [6]) isj*H9<;=uo@%H9@[HSiSH{I~ŁFA~ŁF",45!-M9<;=hi9o@	@&uthe coefficientH9@%39"!@9!@F gF	&9e53@according to [4]. Only modes propagating forwardare destabilized when  , is sufficiently larger than the Alfve´n velocity. Note that the backward waves aredamped at the same rate than the forward waves are amplified. The growth rate j H9o@depends on x due to theamplification of the magnetic field by the non-resonant instability. To derive the previous relation, we had toassume a CR power-law distribution 9@nlQl between"+ andQfk . The parameteris either positiveor negative depending on the back-reaction effects of the CR and turbulence generation on the shock structure.On cosmic-ray self-generated turbulence of supernova remnant shocks 2233. Stationary spectra and saturation mechanismsUpstream turbulence:It is convenient to describe the profiles of the wave spectra with the help of a dimensionless variable  definedbyo% p r9h s% .;Q=uo@ , then the functionv%6l" . The quantity h i9o@is defined by ¡%¢ forhs % h i , Since pr9hs %£ .;Q=Auo@%9 . @9d . ",@;l=9<;=.; f¤k@ﬁ¥l¦Wl9o@hence 1%¨§;Q=h i©3l¥ ª&«. Thenon-resonant dominated regime is to be taken in the interval BB+ %9<;$fk hNi@3l¥«(far from theshock front) while the resonant regime is to be taken in the interval BB (close to the shock front). Theparameters ¬"­¯® and ¬Q® stem for the non-resonant and resonant spectral indices respectively. The spectrumevolution equation normalised the timescale to the advection time °+9x;Q=@% pAr9x;Q=@. ", can be constructedwith the dimentionless variable y. The perpendicular k spectrum is fixed by the non-linear transfer producedin Alfve´nic turbulence (see [7], [8]) while the parallel spectrum is fixed by the dominant streaming instabilityregime in the shock precursor. For instance, using the Goldreich-Sridhar scaling for strong turbulence thesolutions have the form ±e² r9x;Q=uZ;I³@´;l¥=;l µ³with !·¶ ¬%`¸¹$º»%¡¼G5^ . Hence for ¬1% we get%{½ .  corresponding to a Kolmogorov-type spectrum.At high y or at large distances from the shock front, the non-resonant instability prevails and if one comparesthe instability growth to the advection, the evolution equation leads to a hard spectrum ( ¬ ­¯® %` ) and a veryhigh turbulence level proportional to hs+>I.hsfk:%Q+>.Qfk , which can be larger than I¾ . In fact, thenon-resonant growth should saturate much earlier, either through a quenching at;hNi%¿ or through non-linear transfer, leading to a 1D spectrum ±9<;"=@G;l=, i.e. ¬"­¯®%À . The saturation level is;"=±9x;Q=Au@Gm9ÂÁAÃ®.Ä@²,.9ÅÅ3+)d@whereÁÃ®Æ%ÇÃ®.HÈI3,is the ratio of CR pressure to the kinetic energy at the shockfront (see [5]) andÄmÉS9Ê+>.f¤k@Åm . Closer to the shock, the resonant streaming instability takesover, the spectrum is built from the solution at y = 1. Balancing the advection and the growth rate leads tosaturation level;"=±9x;Q=@Gm9PÁAÃ®.$Ä@9Q,."+)@and a 1D spectrum ±9<;Q=@V;l=, i.e. ¬Q®{%Ë . However, inthe resonant regime, for the CR to be scattered back and forth the shock, backward Alfve´n modes have to begenerated efficiently upstream the shock. The Alfve`n turbulence transfers the energy only in the perpendiculardirection to the magnetic field and one have to invoke another non-linear transfer mechanism to transfer energyfrom the forward Alfve´n waves destabilised by the CR towards the backward Alfve´n waves. This transfer ispossible in magnetic field dominated plasma with ¬ ﬂ %{ÇÍÌÂÎ.ÇÐÏ^Ñ¡ through the interplay of sonic waves (see[1]). This is the only way to produce similar spectrum for both forward and backward Alfve´n waves.Downstream turbulence:The turbulence properties downstream (level of turbulence, spectral index) can be constrained using the obser-vation of the X-ray filaments in young SNr (see [9]). It is shown that the relativistic electrons with energiesabout a few tens of TeV producing the observed synchrotron radiation have a diffusion coefficient close to theBohm value. However the previous analysis has been made assuming an isotropic turblence spectrum, whichis not correct at least for two reasons: the turbulence is already anisotropic upstream and the magnetic fieldamplified upstream is compressed in the direction parallel to the shock front.The way the turbulence acts on a relativistic particle can be characterized by competing the non-linear Alfve´ntransfer time Ò	kﬃ9<;=@%9<;=TQ-9x;Q=@w@lm9É=.gT-@9É=>;=@¦ÓÊ¥l²ZÔYZ3±l	YZ3Huwith the downstream return timescaleÒﬀLÌE%ÖÕ9d.Q,@3¶.Ò	,uwith Ò	,]%9¶@l¥Wl9É=.d@9hs×.É=@3l¥. The prefactor ÕB accounts for the short-ening of the return timescale in compressed turbulence. The condition ÒZﬀLÌBÒkﬃ (using ¬`%t ) means theparticle does explore distances smaller than the relaxation length of the turbulence downstream and particlesexperience compressed rather than relaxed turbulence during their journey downstream. Using typical valuesof the magnetic field, mean density and shock velocity in our problem (see next section) and acknowledging224 A.Marcowith, G. Pelletier and M. Lemoinea saturation level of Ø +Á Ã®g.Ä , we find the previous condition is fulfilled unless the shock velocity is lowerthan  l 3 c.4. Astrophysical consequencesBoth regimes of streaming instability occur in SNr but the dominance of one regime over the other dependsstrongly on the Alfvenic Mach number of the shock Ù + %¢ ,. Q- ) . This number depends as much ason the evolution phase of the SN remnant than on the physical properties of the external interstellar medium(ISM). Let us consider the two first phases of SN evolution: the very early free expansion phase where theshock velocity reaches values as high as  ,mÚ¤d , the late free expansion phase (or early Sedov self-similarphase) where the shock velocity drops to  ,ml3 d . In the latter phase the remnant may expand eitherin a hot rarefied ( ÛmI¾ÅÜuºml ²Yl3Ýl ² ) interior of a massive star wind bubble or in a warmpartially ionised typical ISM ( ÛmÜuºml	YHÝl ² ). The mean ISM magnetic field in both cases isconservatively taken as ×Þ¹$ßw. In the very early phase of the SNr evolution, the medium is probably muchdenser with ºm85à$AÝl ² . The ratio ±e­¯® . ±&®mÙâáQ, . dm½$$*(l²wãä¯åS Å3, l, where ", lis theshock velocity in units of 0.1c. The non-resonant regime appears to dominate for the very early free-expansionphase as already pointed out by [5], while the resonant regime dominates in the late free-expansion phase. Theresonant instability is reinforced in low density and highly magnetised media.5. ConclusionsUpstream of an astrophysical shock, the cosmic ray streaming triggers an instability occuring under eitherresonant condition and dominates the larger wave-lengths of the Alfve´n spectrum off resonance and dominatesat shorter wavelengths. The non-resonant instability prevails at large distance to the shock front and cansaturates either through a balance of non-linear transfer cascade with the growth rate or by a quasi-linearquenching. The stationary 1D spectrum solution scales as;l=. The main saturation mechanism of the resonantinstability stems from the fact that the shock front catches up with the growing waves over a diffusion length.This mechanism determines the spectrum, namely scaling also in;l=. The;³dependence of the spectrumis remodeled by the non-linear cascade of Alfve´n waves, that essentially works transversally, the transfer timeis short enough compared to the advection time. The resonant regime is found to always dominate in the latefree-expansion phase once the shock velocity has enough decreased. The compression effects downstreamhave to be taken into account in the transport of the particles during a Fermi cycle.References[1] Skilling J., 1975, MNRAS, 172, 557; 173, 245.[2] McKenzie, J.F., Vo¨lk H.J., 1982, Astron. & Astrophys., 116, 191.[3] Bell A.R., Lucek S.G., 2001, MNRAS, 321, 433.[4] Ptuskin V.S., Zirakashvili V.N., 2003, Astron. & Astrophys., 429, 755.[5] Bell A.R., 2004, MNRAS, 353, 550.[6] Melrose, D.B., 1980, Plasma Astrophysics, Dover.[7] Galtier S., Nazarenko S.V., Newell A.C., Pouquet A.,2000, J. Plasma Phys., 63, 447.[8] Goldreich P., Shridar S., 1995, ApJ 438, 763.[9] Parizot, E., Marcowith, A., Ballet, J. & Gallant, Y.A., Astron. & Astrophys., 2005, submitted.

On cosmic-ray self-generated turbulence of supernova remnant shocks

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