Effects of ion motion on linear Landau damping

Transcription

1 Effcs of ion moion on linar Landau damping Hui Xu 1**, Zhng-Ming Shng 2,3,4, Xiang-Mu Kong 1, Fu-Fang Su 1 1 Shandong Provincial Ky Laboraory of Lasr Polarizaion and Informaion Tchnology, Dparmn of Physics, Qufu Normal Univrsiy, Qufu , China 2 SUPA, Dparmn of Physics, Univrsiy of Srahclyd, Glasgow G4 0NG, UK 3 Laboraory for Lasr Plasmas and Dparmn of Physics and Asronomy, Shanghai Jiao Tong Univrsiy, Shanghai , China 4 Collaboraiv Innovaion Cnr of IFSA, Shanghai Jiao Tong Univrsiy, Shanghai , China Absrac: Th ffcs of ion moion on Landau damping has bn sudid by us of on-dimnsional Vlasov-Poisson simulaion. I is shown ha h ion moion may significanly chang h dvlopmn of h linar Landau damping. Whn h ion mass is mulipl of proon mass, is moion will hal h linar Landau damping a som im du o h xciaion of ion acousic wavs. Th lar will domina h sysm voluion a h lar sag and hold a considrabl fracion of h oal nrgy in h sysm. Wih vry small ion mass, such as in lcron-posiron plasma, h ion moion can supprss h linar Landau damping vry quickly. Whn h iniial fild ampliud is rlaivly high such as wih h dnsiy prurbaion ampliud n/n 0 >0.1, h ffc of ion moion on Landau damping is found o b wak or vn ignorabl. Kywords:Linar Landau damping, Ion moion, Vlasov-Poisson simulaion PACS: Fp, Mw, Sb huixu3973@163.com

2 1. Inroducion Landau damping is a ky collciv bhavior of plasma discovrd 70 yars ago [1]. I is a kind of collisionlss damping rsuling from rsonan nrgy xchang bwn an lcrosaic wav and paricls. In h linar rgim whn an lcron disribuion funcion (EDF) in a spaially uniform plasma is prurbd by a small-ampliud lcrosaic wav, Landau s analysis prdics ha h im-asympoic voluion of h lcric fild xhibis xponnial damping (or growing). Th damping (or growing) ra k is proporional o h drivaiv of quilibrium EDF wih rspc o h vlociy v calculad a h phas vlociy v ph of h lcrosaic wav. ramn is rigorous whn h wav ampliud is infinisimal and h background ions ar assumd o b immobil. Th linar Landau damping has bn wll dmonsrad in Vlasov-Poisson simulaion by many rsarchrs in h pas dcads [2-12]. This collisionlss damping has bcom vry imporan in a wid conx (s rviws by Ryuov [13] and Ivanov [14] ) and som rcn works show ha progrss in undrsanding and modling his problm is sill bing mad [15-22]. In hs sudis, h ions ar usually assumd o b immobil as in h original hory [1]. Howvr, ion moion can chang h voluion of h plasma oscillaions a crain im scal. For xampl, i has bn shown ha boh ion moion and h rlaivisic ffc of lcron moion can lad o phas mixing phnomna if a plasma oscillaion has modra ampliud [23, 24]. In an unsabl nonlinar lcrosaic sysm, i is found ha ion moion can rduc h asympoic growh of wav ampliuds [25, 26]. In his papr, basd upon Vlasov-Poisson simulaions w show ha h ion moion can vn considrably chang h linar Landau damping. In paricular, i is found ha h linar Landau damping will disappar a a crain sag of h lcrosaic wav dcay du o h xciaion of ion acousic wavs. 2. Th modl and govrning quaions For h simpls cas, h lcrosaic oscillaions can b dscribd wih h on-dimnsional Vlasov-Poisson modl, which is govrn by h following quaions: Th f f f v E 0 x v, (1) fi fi m fi v i Z 0 x mi vi, (2) E Z f i dv i f dv x, (3) whr f (x, v, ) and f i (x, v i, ) ar h disribuion funcion of lcrons and ions normalizd by h quilibrium paricl dnsiy n 0, rspcivly, and E(x, ) is h lcric fild normalizd by

3 ( / m c) p 1. Hr im is normalizd o h invrs of h lcron plasma frquncy 1 p, spac coordina x o c / p, vlociy v and v i o h spd of ligh c, m i and m ar h rs mass of ions and lcrons, rspcivly, and Z is h charg numbr of h ions normalizd o h lcron charg. Th oscillaions ar xcid by iniializing a singl Fourir mod k m wih h following iniial lcron disribuion, 1 v f( x, v, 0) [1 cos( kmx)]xp( ), (4) 2 v 2v h whr is h iniial prurbaion ampliud of h lcron dnsiy, k 2 m / L is h wav numbr of h mod m, L is h whol lngh of h sysm which is 2 in all h following 2 2 h m simulaions, v K T m c 2 h B / is hrmal vlociy of lcrons wih h lcron mpraur T and K B is Bolzmann's consan. Ions ar iniially uniform in h x-spac and follow h Maxwllian disribuion in h vlociy spac f x v v v v, i(, i, 0) ( 2 hi) xp( i / 2 hi) whr v v T m / T m is h normalizd hrmal vlociy of ions wih h ion hi h i i mpraur T i. Whn solving h Vlasov quaion, w us h im spliing mhod inroducd by Chng and Knorr [27], which rss on spliing h Vlasov quaion ino sparad spaial and vlociy spac advcion quaions. For ach advcion quaion, w us h posiiv and flux consrvaiv (PFC) mhod [28]. Th fas Fourir ransform mhod is usd o solv h Poisson quaion wih h assumpion of a priodic boundary in h longiudinal spac. 3. Numrical simulaion rsuls 3.1 Disapparanc of linar Landau damping In h following simulaions, w kp a fixd wav numbr k m =1. Th lcron hrmal vlociy is v h =0.4, h ion mass is normalizd by lcron mass m by dfining m =1, m i0 =m i /m. In hs simulaions m i0 =1836=m p (hr m p is h mass of a proon), h ion mpraur is qual o h lcron mpraur (T i =T ) and h ion charg Z=1. Figur 1 shows how h fundamnal mod of h lcric fild volvs wih im undr diffrn iniial ampliuds of h wav. Whn h iniial prurbaion ampliud is high such as =0.1 shown in Fig. 1(a), h sysm is highly nonlinar. I indicas ha h sysms volv basically idnical ihr wih or wihou ion moion in his cas, i.., ion moion dos no lad o obvious chang of h fild voluion whn h iniial

4 ampliud is larg. Howvr, if h iniial ampliud is as small as nar h linar rgion, h ffc of ion moion bcoms obvious, as shown in Figs. 1(b)-1(d). I shows ha h linar Landau damping dvlops a h bginning. From abou =61, h fild voluion sars o dvia from h linar Landau damping. From h im =100, h Landau damping almos disappars. Insad, h fild voluion appars o dcay slowly wih a larg priodic srucur. Th priod of h fild voluion is as long as abou 340 1, which is abou 65 mulipls of h priod of h p lcron plasma wav govrnd by h disprsion rlaion 3 k v / 2. This larg p h srucur should b associad wih ion moion or h xciaion of ion acousic wavs. Th frquncy of h ion acousic wav is givn by k v Z k v (1 k ) ( T / T ). (5) i i hi hi D i 0.05 (a) ions immobil ions mobil (b) ions mobil ions immobil (c) ions mobil ions immobil (d) ions mobil 10-4 ions immobil Figur 1 Tim voluion of h lcric fild fundamnal mod undr diffrn iniial ampliuds: (a) 0.1, (b) 0.01, (c) 0.008, (d) 0.003, hr v h =0.4, m i0 =1836=m p, T =T i, and k m =1. For h givn paramrs in h simulaion and wih h adiabaic consans in h on-dimnsional siuaion 3, on finds h frquncis of lcron plasma wav and ion acousic wav i ar = p and i = p, rspcivly. Thrfor, h ion acousic cycl is abou 58 mulipls of h lcron plasma wav cycl, which is a lil shorr han h cycl of h

5 slowly dcaying srucur in h simulaion rsuls shown in Fig. 1. This diffrnc may b causd by h plasma wav phas vlociy shif whn h ion moion is includd, as discussd lar. Figur 2 Tim voluion of lcron dnsiy prurbaion n wih h ion moion includd during =[0,200] (a), [80,200] (b), and [200,400] (c), whr h ohr paramrs ar h sam as hos in Fig. 1(d). Figurs 2 and 3 illusra h im voluion of lcron dnsiy prurbaion ampliud n and ion dnsiy prurbaion ampliud ni, rspcivly, in diffrn im priods for Figur 2(a) shows ha h ampliud of h longiud lcron plasma wav is dampd o vry small lvl a abou =80. On h conrary, h ampliud of h ion acousic wav incrass wih h im and rachs h maximal valu a abou his momn =80 [s Fig. 3 (a)]. As shown in Figs. 3(c) and 3(d), h ion dnsiy prurbaion bcoms a h sam lvl as h lcron dnsiy prurbaion a h im =70. Afr his im, h lcron plasma wav undrgos furhr damping and h ion acousic wav sars o domina h voluion of h plasma sysm. Elcrons in h sysm mainly follow h ion moion in h lar priod as shown in Figs. 2(c) and 3(b). Thrfor, h slow lcric fild oscillaions in h lar sag corrspond o h ion acousic

6 wav frquncy as shown in Fig. 1(d). In passing, w poin ou ha ion dnsiy prurbaions can also dvlop whn h iniial lcron plasma wav is high (such as =0.1). Howvr, h lcron dnsiy prurbaions ar always much largr han h xcid ion dnsiy prurbaions. So h lcron dynamics rmains prdominan during h whol voluion of h sysm, i.., h ion mobil ffc on h fild voluion is vry small and can b nglcd whn h iniial prurb ampliud is larg. dnsiy (c) n i n x dnsiy (d) n i n x Figur 3 Tim voluion of ion dnsiy prurbaion ni during =[0,200] (a) and [200,400] (b), whr h ohr paramrs ar h sam as hos in Fig. 1(d). Th corrsponding snapshos of lcron and ion dnsiy disribuions in h x-spac a h im =70 and =250 ar also shown in (c) and (d), rspcivly. 3.2 Effc of ion mass To show mor xplicily ffc of ion moion, im voluion of h lcric fild fundamnal mod wih diffrn ion mass is shown in Fig. 4. Wih h incras of h ion mass, h influnc of ion moion on h linar Landau damping occurs lar wih largr oscillaion priods accordingly. Mos of h lcric fild nrgy has bn dampd o h paricls wih vlociy nar h phas vlociy. W also hav calculad im voluion of h lcric fild fundamnal mod

7 wih ohr diffrn ion charg and diffrn ion mass. Wih h incras of h ion charg, h im ha ion mobil bgins o influnc h linar Landau damping d is arly whn h ion mass is h sam. Th priod of simulad acousic wav i is almos h sam as long as h ion charg o mass raio is h sam, which is in accordanc wih horical scaling for ion acousic priod Z. For h havir ions (m i0 >200m p ), h im d is almos h sam whn h i m / i0 ion mass and h ion charg incras h sam mulipl (a) ions mobil ions immobil (b) m i =m m i =2m m i =10m (c) Figur 4 Tim voluion of h lcric fild fundamnal mod undr diffrn ion mass wih Z=1, (a) m i0 =10m p =18360, (b) m i0 =100m p, and (c) m i0 =1, 2, and 10. Th ohr paramrs ar sam as hos in Fig. 1(d).

8 Whn h ion mass is so small such as comparabl o h lcron mass, Fig. 4(c) shows im voluion of h lcric fild fundamnal mod wih diffrn ligh ion mass. Whn h normalizd ion mass m i0 =10, h linar Landau damping can sill b found clarly. Howvr, whn m i0 =1 or 2, h fild damping can only xis a h vry arly sag wih a rducd damping ra. From Fig. 5 i can b sn h ion dnsiy prurbaion can b xcid vry quickly (a abou =1.9 or abou half of an lcron oscillaion priod) whn h ion mass qual o lcron mass (i.., lcron-posiron plasma). Afrwards, h sysm voluion is mainly dominad by nrgy xchang bwn lcrons and posirons via lcric filds (c) (d) dnsiy dnsiy n n i n n i Figur 5 Tim voluion of lcron dnsiy prurbaion prurbaion n (a) and ion dnsiy ni (b) a h arly sag whn m i0 =1 (ha is for posiron), (c) and (d) show snapshos of h lcron and posiron dnsiy a =1.9 and 7.8, rspcivly. Hr m i0 =1, h ohr paramrs ar h sam as hos in Fig. 1(d). In ordr o xplain h fild voluion in lcron-posiron plasma, w calcula h avragd disribuion of h lcrons and ions in x-spac, ha is

9 L 1, i, i, i, i 0 f ( v ) L f ( x, v ) dx. (6) Wih h damping of h lcric fild, som paricls wih vlociy slighly lowr han h wav phas vlociy absorb nrgy from h wav and can b acclrad. As h diffrnial cofficin absolu valu of h lcron disribuion df / dv is small a h wav phas vlociy, h diffrnial cofficin can chang is sign vry asy o saisfy df / dv 0, which can b found in Fig. 6(a). Afr ha, hr ar mor paricls having vlociis slighly grar han h wav phas vlociy which will b dclrad and losing nrgy o h wav. Whn h ion moion is includd, h avragd disribuion in v-spac of lcron disribuion and ion disribuion ar always h sam in h lcron-posiron plasma du o hir sam mass and sam charg numbr [s Fig. 6(b)]. Figur 6(c) compars h avragd disribuion of lcrons ihr wih or wihou h posiron moion a =20. Th diffrnial cofficin of lcron disribuion wih h posiron moion volvs much slowr han hos wihou h posiron moion. This can xplain why h ion moion can dcras h damping ra of h lcric fild fundamnal mod. Compar Figs. 6(a) and 6(b), i can b sn ha h rsonanc domain in v-spac is movd o a largr vlociy dircion whn h ion moion is considrd. This implis h phas vlociy incras. According o h formula v / k, i can b dducd ha h lcron plasma wav ph frquncy incrass and oscillaion priod dcrass whn h posiron moion is considrd. From Fig. 6(d) i is shown ha h plasma wav frquncy incrass whn h posiron moion is considrd, which is in accordanc wih h hory analysis [29, 30]. And his is found o b ru for ions aking ohr mass raio alhough h plasma wav phas vlociy shif is no as obvious as in h posiron-lcron plasma h ion mass is larg. This can xplain why h raio of h ion acousic wav priod o h lcron plasma wav priod found from simulaion shown in Fig. 1 is a lil largr han h horical valu whr h ion moion is no considrd.

10 avrag(f ) (a) v =0.0 =155 =345 avrag(f,i ) (b) =0 av(f ) =100 av(f ) =100 av(f i ) v,i avrag(f ) =0 =20 ions mobil =20 ions immobil (c) v (d) ions mobil ions immobil Figur 6 Avragd disribuion of lcrons and posirons in x-spac, (a) v h =0.3, immobil posirons, (b) v h =0.3, posirons mobil, (c) v h =0.4, (d) voluion of h lcric fild fundamnal mod a h arly sag whn v h =0.6. Th ohr paramrs ar sam as hos in Fig. 6(d) wih m i0 =1, ha is lcron-posiron plasma. 4. Conclusions Using on-dimnsional Vlasov-Poisson simulaion, w hav sudid h ffc of ion moion on Landau damping. In h linar rgim of Landau damping wih a small iniial ampliud of lcron plasma wavs, i is shown ha h xcid ion acousic wav can significanly affc h sysm voluion. Whn h ion mass is mulipl of proon mass, is moion will sop h linar Landau damping a som im and h xcid ion acousic wav will domina h sysm voluion a h lar sag. As a rsul, h rsidual fild nrgy wih ion moion will b much largr (such as 100 ims) han ha wihou considring ion moion. Wih small ion mass, such as in lcron-posiron plasma, h ion moion significanly supprsss h linar Landau damping ra and h sysm voluion is mainly dominad by nrgy xchang bwn lcrons and posirons via lcric filds. Whn h iniial dnsiy prurbaion is larg in h quasi-nonlinar

11 rgim, h ffc of ion moion bcoms wak and h sysm is mainly drmind by h lcron dynamics. Acknowldgmn This work is suppord by h Naional Naural Scinc Foundaion of China (Gran No , , , , and ). ZMS acknowldgs h suppor of h suppor of a Lvrhulm Trus Rsarch Projc Gran a h Univrsiy of Srahclyd. REFERENCES [1] L. D. Landau, J. Phys. (Moscow) 10, 25 (1946). [2] J. Klimas, J. Compu. Phys. 68, 202 (1987). [3] A. Harn and S. Oshr, SIAM J. Numr. Anal. 24, 279 (1987). [4] A. Ghizzo, P. Brrand, M. Shoucri, T. W. Johnson, E. Filjakow and M. R. Fix, J. Compu. Phys. 90, 431 (1990). [5] J. Klimas and W. M. Farrll, J. Compu. Phys. 110, 150 (1994). [6] T. Usumi, T. Kunugi, and J. Koga, Comp. Phys. Commun. 108, 159 (1998). [7] E. Sonnndrückr, J. Roch, P. Brrand, and A. Ghizzo, J. Compu. Phys. 149, 201 (1998). [8] T. Nakamura and T. Yab, Compu. Phys. Commun. 120, 122 (1999). [9] E. Fijalkow, Compu. Phys. Commun. 116, 319 (1999). [10] E. Sonnndrückr, J. J. Barnard, A. Fridman, and D. P. Gro, and S. M. Lund, Nuclar Insrucion and Mhod in Physics Rsarch A 464, 653 (2001). [11] F. Filb and E. Sonnndruckr, Compu. Phys. Commun. 150, 247 (2003). [12] H. Xu and Z. Shng, Plasma Sci. Tch. 14, 181 (2012). [13] D. D. Ryuov, Plasma Phys. Conrolld Fusion 41, A1 (1999). [14] A. V. Ivanov, I. H. Cairns, and P. A. Robinson, Phys. Plasmas 11, 4649 (2004). [15] M. C. Firpo, F. Dovil, Y. Elskns, P. Brrand, M. Polni, and Guyomarc h, Phys. Rv. E 64, (2001); F. Dovil, M.-C. Firpo, Y. Elskns, D. Guyomarc h, M. Polni, and P. Brrand, Phys. L. A 284, 279 (2001). [16] Z. Sdláčk and L. Nocra, Phys. L. A 296, 117 (2002). [17] J. R. Danilson, F. Andrgg, and C. F. Driscoll, Phys. Rv. L. 92, (2004). [18] E. W. Laing and D. A. Divr, Phys. Plasmas 13, (2006). [19] V. Braanov, F. Jnko, D. Hach, and S. Brunnr, Phys. Plasmas 20, (2013).

12 [20] J. D. Calln, Phys. Plasmas 21, (2014). [21] J. Wsson, Phys. Plasmas 22, (2015). [22] B. Young, J. Mah. Phys. 57, (2016). [23] H. Xu, Z. M. Shng, and J. Zhang, Phys. Scr. 74, 673 (2006). [24] H. Xu, Z.M. Shng, and J. Zhang, Aca Phys. Sinica 56, 968 (2007) (in Chins). [25] J. D. Crawford, Phys. Plasmas 2, 97 (1995). [26] J. D. Crawford and A. Jayaraman, Phy. Rv. L. 77, 3549 (1996). [27] C. Z. Chng and G. Knorr, J. Compu. Phys. 22, 330 (1976). [28] F. Filb, E. Sonnndrückr, and P. Brrand, J. Compu. Phys. 172, 166 (2001). [29] N. Iwamoo, Phys. Rv. E 47, 604 (1993). [30] G. P. Zank and R. G. Gravs, Phys. Rv. E 51, 6079 (1995).