PoS(ICRC2021)172 International th 12–23 July 2021 July 12–23 37 SICS CONFERENCE SICS CONFERENCE Berlin | Germany Berlin Berlin | Germany Cosmic Ray Conference Ray Cosmic ICRC 2021 THE ASTROPARTICLE PHY ICRC 2021 THEASTROPARTICLE PHY https://pos.sissa.it/ ONLINE 0 and Yutaka Ohira ∗ 0, , [email protected] [email protected] International Cosmic Ray Conference (ICRC 2021) Presenter ∗ th Department of Earth and Planetary Science,7-3-1 The Hongo, University Bunkyo-ku, of Tokyo, Tokyo, 113-0033 Japan E-mail: Our previous work revealed thatdiffusive the shock energy acceleration (DSA) spectrum can of be modified cosmicinhomogeneity by of rays downstream the produced sound waves shock by originated upstream the from medium.by standard linear However, the analytical background solutions plasma and waswork, nonlinear described evolution we of investigate sound how waves nonlinear wasaffect effects not particle such acceleration included. by as using In steepening test-particle this steepened simulations. and from dissipation First, sound we of imitate waves weak sound by shock analytical waves efficiently waves sawtooth accelerate waves and particles show earlier thatpropagation large-scale than into waves they can an dissipate. inhomogeneouswaves. medium and Our Next, results discuss we imply particle numerically additionalthey acceleration acceleration solve evolve nonlinearly. by by shock downstream downstream sound waves works even when Copyright owned by the author(s) under the terms of the Creative Commons 0 © Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). Shota L. Yokoyama 37 July 12th – 23rd, 2021 Online – Berlin, Germany Particle acceleration by sound waves generatedshock in downstream the region PoS(ICRC2021)172 B − ? , and 1 ∝ are the D ? 1 = d : / ) = d G,C and ( 1 D , Xd Shota L. Yokoyama )] C 1 D is used for the upstream − 1 G ( 1 : [ sin 1 Xd + 1 d = ) 2 G,C ( d in which this mechanism works is estimated in [14]. 2nd , acc C mean the density, shock normal component of velocity measured % for strong shocks. , and 2 D , = d B is used for downstream ones. Thanks to this treatment, we were able to relate the 2 , where 1 % = ) Diffusive shock acceleration (DSA) in supernova remnants (SNRs) is believed to be one of the In this study, we examine the effect of the nonlinear evolution of sound waves and its impact on In [14], we treated the background plasma which determines the shock structure as a fluid However, due to the linear approximation, sound waves generated in the shock downstream Although in the conventional theory of DSA, the upstream medium was assumed to be uniform, G,C ( % most plausible mechanisms by which energizeproblems galactic cosmic are rays pointed (CRs) out [e.g.2]. for However,maximum this some energy picture achievable by and the it standard DSA needshighest in some energy typical of supernova modification remnants galactic [5,6]. CRs does (that not For is, reachCRs the knee example, observed energy). on the the DSA Earth also and cannot the explainSNRs, diversity the since of spectral the spectral index standard indices of DSA of uniquely radio predicts synchrotron a emissions power-law from momentum distribution Sound wave acceleration in the shock downstream 1. Introduction in the shock rest frame, and pressure of the background plasma, respectively. with spectral index particle acceleration by solving boththat, test-particle we imitate motion the and weak shock fluid waves produce equationswaves by numerically. and steepening of investigate Before sound particle waves by acceleration analyticalworks sawtooth in and clarify such the problems. waves. We briefly In introducein the Section3. Section2, numerical methods The we used results review in our of our simulations simulations previous are shown in Section4 and discussed in2.5. Section Review and the upstream fluctuation asfollowing conditions a for linear the shock monochromatic upstream: entropy mode. That is, we assumed the region extended infinitely far without anywaves, they damping. steepen into Considering weak the shock waves nonlinear and behavior dissipate of in sound some finite time scale. This time-scale it is natural to consider inhomogeneous medium andfluctuations indeed the is existence confirmed of various by scales observations of1,4]. [e.g. density inhomogeneity We on have particle investigated acceleration the using influence test-particle simulations of and medium generated revealed that from sound waves the interaction betweenadditional shock particle waves acceleration and in upstream the density shockfluctuations downstream fluctuations in region can the [13, background14 ]. cause plasma However, in aremedium those treated inhomogeneity works, as affects linear particle perturbations. acceleration It in is more still realistic not situations. unclear how amplitude and wavenumber of thequantities, upstream while entropy mode. Subscript downstream quantities to the upstreamshowed ones that sound analytically, waves following originated the from upstream linear mediumin analysis inhomogeneity the can of shock accelerate downstream [7]. particles region. Acceleration We mechanismas by second-order downstream acceleration sound by waves is large-scale identified compressible turbulence,Particles which are is stochastically discussed accelerated by in local e.g. velocitywaves. differences [9]. produced The by acceleration downstream time-scale sound PoS(ICRC2021)172 2 (2) (3) (1) XD are _ 4 / c ) , 2 → ) A = of each time 2 C : ) Δ 0 . ? G ( Shota L. Yokoyama sc g are normalized by the 1 − sinusoidal wave sawtooth wave dis  C 1 1 d Xd . )]( ) . In each time step, we follow 5 . 0 0   0 ? ? ) ( 0 + 2 ? sc 2 ( g XD of particles with injection momentum 1 XD _ ) − mfp _ ?  _ ( 1 )/  1 C sc d 1 g s2 Xd − E  and it is concluded that sound waves dissipate   − sh 2 ) G D 3 0 2nd  , 2 ? (( ( ) XD acc 2 C _ − and the dissipation time-scale 1 mfp 2 floor _ 2 " and sound Mach number, respectively. In the second equal  XD − " 1 + 1 ) 2 _ d 2 − 1 /  XD 2 " − )]( _ and scattering time sh d 2 C 2 A D ) )/ ( s2 +  C ? E ( 1 s2 6 ( . − E A 5 mfp G − ( _ ' = 2 G 2 : 2 [( [ XD 2 XD _ sin XD 2 = . And in the last similarity, numerical values for strong shock limit 6 2 is the floor function which returns the greatest integer less than XD √ dis ) C XD ( G ( = ) are the compression ratio are the wavenumber and sound velocity of the downstream sound waves, respectively. The floor G,C " ( s2 When we investigate particle acceleration under sawtooth waves generated from steepening of To model the particle motion in a turbulent magnetic field, we treat particles as test particles For simulations that numerically solve both particle motion and fluid equations, we adopt a E D can be evaluated as follows10]. [ . In [14], this time-scale is compared to and 0 dis ? sound waves, we use the following analytical fluid fields. the following operations; (i) translationthe of new particle particle positions, positions, (iii) (ii) Lorentz obtainingdiffusion, transformation (v) the to inverse the plasma Lorentz local velocity transformation plasma at to rest the frame, original (iv) isotropic frame. and their motion as isotropiccontinuously diffusion injected in the at local the rest main frame shock of with background initial plasma. momentum Particles are earlier than the second-order process works.generated On by the steepening other hand, is accelerationevolution not by of weak discussed sound shock quantitatively waves waves impedes there. orthe promotes one-dimensional Therefore, downstream acceleration monochromatic whether is case. the stillsimulations. nonlinear unknown even In for this work, we tackle this problem by test-particle 3. Test-particle simulations sign, we used the linear estimationsound by [7] waves for the amplitude ofused. velocity The fluctuations wavelength by of downstream sound waves scattering mean free path where step is calculated by the minimumtime. of The that velocity determined field by calculated the in CFL this condition manner and is particle referred scattering to in step (ii) of the particle part. numerical factors appeared in the above equationthe are sawtooth introduced waves to is guarantee equal that to energyplotted. density that of of the sinusoidal waves. In1, Figure these analytical fieldsGodunov-type are scheme. Riemann problems are approximately solvedand by the fifth-order HLLC accuracy method in [e.g. space12] is achieved by the MP5 scheme [11]. Time width Sound wave acceleration in the shock downstream and C A PoS(ICRC2021)172 s2 E . 1 . The 1 d d 5 . 5 0 . 0 = = 1 1 Xd Xd Shota L. Yokoyama and and sound velocity 2 and 2 1 . 2 0 XD 1 . = 0 sh = D sh D and vary the wavelength of waves to 0 = G 4 . ) 0 ? ( Analytical velocity fields. mfp _ . The amplitude of sound waves 2 ) 0 10 ? as they propagate. The left panel of3 Figure is the velocity field ( is determined to assure that linear estimate of the wavelength of Figure 1: ) G 1 mfp 1 : _ : 4 ( 10 sin , 1 Xd and , + 3 and clearly shows steepening and dissipation of sound waves. In this simulation, 1 ) 10 d 0 , 2 ? = ( ) 10 sc , g G,C 5 ( 10 d 10 = In order to evaluate particle acceleration under weak shock-train produced from steepen- The resulted momentum spectra are shown in3. Figure For short wavelengths, sinusoidal Because, by the previous simulations, efficient acceleration by weak shock waves is confirmed = 2 C XD are calculated so as to coincide with downstream quantities when ing of soundin waves,1. Figure we use We the continuously analytical inject fluid particles field at introduced in Equation (3) and shown 4. Results 4.1 Particle acceleration by sawtooth shock waves sound waves and sawtooth shockthat waves particle act acceleration in by almost sawtoothfor these waves the wavelengths. works same On in way the a other forof similar hand, particles. second-order when way the acceleration as wavelengths This becomes of the lower waves implies diffusion second-order get since becomes larger, process the the smaller. efficiency velocity Even difference foraccelerators. these felt wavelengths, We by speculate however, sawtooth particles that waves this during remain efficient efficient many acceleration times is achieved in by the crossing same weak shock way waves as DSA. 4.2 Particle acceleration by nonlinear sound waves at least for someshock cases, wave we and investigate downstream particlethe waves. evolution acceleration of in Applying fluid the the fieldsequations system numerical is are including numerically methods solved solved in both explained as the main mode in well upstream as Section3, rest diffusive frame motion and of shock particles. wave interacts Fluid with upstream entropy Sound wave acceleration in the shock downstream at shock velocity and amplitude of entropy mode are set to be wavelength of entropy mode downstream sound waves becomes _ PoS(ICRC2021)172 . ) 0 ? ( sc g 5 10 × . 2 ) 0 = ? C ( g Shota L. Yokoyama 5 10 = C . Because the number of injected ) . 0 ) ? 0 ( ? ( sc g 5 mfp _ 10 4 10 = 5 C and , 3 10 , 2 10 , 10 = 2 XD _ Velocity field (left) and momentum spectrum (right) at Figure 3: Momentum spectra of particles accelerated by sinusoidal or sawtooth waves at The right-hand panel shows the resulted momentum spectrum (orange) and compares that in particles is the same forcase of both inhomogeneous cases, upstream. this But indicates we cannot that see high-energy any particles change in are the increased maximum energy. in the the case of uniform shock upstream (blue) at The wavelengths are set to be Figure 2: Sound wave acceleration in the shock downstream PoS(ICRC2021)172 = 2 10. XD _ 10.1088/ Shota L. Yokoyama in the case for dis C 10.1088/1361-6587/ab49eb 10.1093/mnras/182.2.147 is comparable to or less than this. Because even 6 dis C , we expect that spectral modification from the spectrum of DSA ) 0 and dissipation time ? ( ) 0 ? mfp ( _ 4 mfp 10 _ 2 10 and , 10.1007/s11214-013-9988-3 = ) 0 2 ? XD ( _ mfp 2 Figure implies that if there are sound waves whose wavelengths are much larger than par- Although our works so far are limited for one-dimensional monochromatic density fluctuations, Numerical computations were carried out on the Cray XC50 at the Center for Computational _ 1086/175515 0954-3899/27/7/316 201, doi: 3 ticle mean free paths,waves, waves while can the efficiently acceleration acceleratecause is particles this inefficient acceleration by if time-scale steepening they is into would shorter weak remain than shock purely the sinusoidal dissipation waves. time Be- particle acceleration by turbulenceConsidering is the expected oblique incidence to of a be shockcan much wave, incompressible also more turbulence accelerate is complex generated particles and in this [e.g.8].make realistic situations situations. more Multi-dimensional complicated. fluctuations and Includingturbulence cascades the or back-reaction of magnetic of turbulence field accelerated might particles,our be results upstream amplified show that [e.g.3] medium and inhomogeneity affect doesthe particle modify simplest the case, acceleration. energy shock spectrum Because acceleration of inin particles an more even inhomogeneous realistic in medium situations and should compared be with investigated further observations. Acknowledgement Astrophysics, National Astronomical ObservatoryKAKENHI of grant Japan. numbers JP21J20737Excellent Young This Researchers (S.Y.), MEXT, work Japan. JP19H01893 is (Y.O.), supported and by Leading JSPS Initiative for References [1] Armstrong, J. W., Rickett, B. J., & Spangler, S. R. 1995, Astrophys. J., 443, 209, doi: Sound wave acceleration in the shock downstream 5. Discussion in this case, we findhappens the in the increase case of of larger high-energy wavelengthslonger. or particles, slower However, shock we we velocity, where guess met dissipation numerical that timeto difficulty becomes significant improve in modification our doing methods. simulations This for will those be parameters done and in need our future work(s). certainly occurs for these wavelengths. Onwhere the other hand, the spectrum in3 Figure is for the case [2] Bell, A. R. 1978, Mon. Not. R. Astron. Soc., 182, 147, doi: [4] Ferrière, K. 2020, Plasma Phys. Control. Fusion, 62, 1, doi: 10 [3] Bykov, A. M., Brandenburg, A., Malkov, M. A., & Osipov, S. M. 2013, Space Sci. Rev., 178, [5] Kirk, J. G., & Dendy, R. O. 2001, J. Phys. G Nucl. Part. Phys., 27, 1589, doi: PoS(ICRC2021)172 10.2514/6. Shota L. Yokoyama 10.1088/0034-4885/ 10.1063/1.1691825 10.1086/151757 10.3847/1538-4357/ab93c3 10.22323/1.358.0162 https://ui.adsabs.harvard.edu/abs/ 10.1007/s00193-019-00912-4 10.1088/2041-8205/767/1/L16 7 1988SvAL...14..255P/abstract 64/4/201 1997-2037 [8] Ohira, Y. 2013, Astrophys. J. Lett., 767, 5,[9] doi: Ptuskin, V. S. 1988, Sov. Astron. Lett., 14, 255. [10] Stein, R. F., & Schwartz, R. A. 1972,[11] Astrophys. J.,Suresh, 177, A., 807, & doi: Huynh, H. T. 1997, 13th Comput. Fluid Dyn. Conf., 99, 965, doi: [13] Yokoyama, S., & Ohira, Y. 2019, 36th ICRC, 358,[14] doi: S.Yokoyama, L., & Ohira, Y.2020, Astrophys. J., 897, 50, doi: [7] Mckenzie, J. F., & Westphal, K. O. 1968, Phys. Fluids, 11, 2350, doi: [12] Toro, E. F. 2019, Shock Waves, 29, 1065, doi: Sound wave acceleration in the shock downstream [6] Malkov, M. A., & Drury, L. O. 2001, Reports Prog. Phys., 64, 429, doi: