Espedal, Magne S.

English

University of Bergen

byIUNIVERSITY OF BERGENBergen, NorwayDepartmentofAPPLIED MATHEMATICSThe effects of trapped and untrappedparticles on an electrostatic wave packetMagne S. Espedalleport No. 46 December 1973ISSN 0084-778xThe effects of trapped and untrappedparticles on an electrostatic wave packetbyMagne S. EspedalReport No. 46 Decemher 1973Ahstract.The propagation of an electrostativ wavepacket in acollissionless plasma is studied. ¥e get a change in amplitudecaused hy interaction hetween the packet and particles propagating with velocities near to the group velocity. Also, weget modulation of the plasma in the front of the plasma causedhy trapping effects.1r2)Introduction.In this paper we shall study the interaction betweenparticles and an electrostatic wave packet. The evolutionof a large amplitude wave packet, has heen studied earlier hynumerical simulation (j. Denavit and R.N. Sudan 1972), but amore complete theory has not been given. As in nonlinear opticsand water-wave theory, we shall try to find a waveequation.In collisionless plasmas, the nonlinearity often comes from thetrapping of particles in the potential troughs of the waves.Therefore we have to find a procedure which takes care of thiseffect.I. The wave equation.The equations to govern the onedimensional motion ofcollisionless plasmas are the Vlasov - Poisson equations:(i.O(1.The suffix J denotes the species of plasma particles.representing -- A , /vvij = f*y\ for electrons, and Xy =SL ,Wj = M for the ions.øxj C x>Ki) -+ t UA) £*» Kt )3* Ei*, i) = V/7 Jijj f} (.x. v, i) d-V 2problem.(1.5)(1.4)(1.5)(1.6)(1.7)(1.8)(1.9)¥e shall solve eqs. (1.1 - 2) as an initialvalueE tx, o)are given consistently.In order to solve eqs. (1.1 - 4), we assume thatFurther we define:7-C* ,-t)Now, eqs. (1.1 - 8) gives :where :>) ' /) a t X (*. o)p < *,*. o) - jt iv) + p (x.y.o)j._ i % (*< °)L Lx, o) x(.X, i/,i) * C-f_ cXu,i)b. (.*,+ ) - t (X,UJ i X , l )9* % (y,i) -(o-t -* ) + u,v>i)6 (“- uv) -fi tx, t ''•* )  310)11)12)13)\(fi + L k ) E cx t i) (1Integrating eq. (1.9) along the characteristics we get:dr * /(1«it * htq * rr -/'"'j t- -2-(i¥e shall solve eq. (1.11) with the following conditions:X(1s/v// JEL Jjj u, ki) dv% - % E- VfXj ( r=vs C r = i)which gives:i, JL *T r T 0.14)£ J )Ar _ ij(w-XKi)<<S QA \-a 4 *415)16)Now„ eqs. (1.10) and (1.14) combine to.(i¥e assume that 0(x, i/, 4 ) exists in the sense that theright hand side of eq. (1.1 5) is finite.Integrating the last integral in eq. (1.15) by parts.(1 iOp _ ,( . i «%)cLt >ff +_ y *V'T Ij'1 ff i f C /fr \ A _ LJ~ (J dl/ jB) g X t <d- *TJ0we get: éQ £ .. r- f A i(t )ci 'T+Lk & r 1/77 fij j , LT*o) Jl o £ V -f-r *,/_i L( E rl[l \ Jl d7l * ' (0T t (_L__ a f_L_ ifX0,Lu f_L_ é [J— å f-£- &)))•+ L j r [d7 l w dT V U>- Vt W,. Qy. yj SL J.i>5it in a(1.17)(1.18)(1.19)(1.20)(1.21)fEq. (1.16) is rather complicated, so we want to writemore attractive form.¥e define:£C^,k)/ 9£0K / 3\9é. f. x, -t)(H ) ''q, (x, i)T* ( Uj . K ) £ f U>,K) -29* Ok ) 4_ i[U il-K~ ]),jU -zÉv))]^il-u. d/HnWV<*V IW J Ia;-KV (aj-ku dl/VuT^-KV// J_ uffA * f* * £ Vi Jnr \ 'f2 ITSi(.x,i ] - 2 i116(1.22)(1.25)(1.24)(1.25)(1.26)(1.27)(1.28)v 4 3,Jwhere J-t = øi+[/;andUsing eqs. (1.17 “ 22), eq. (1.16) reduces to:. tfd fVT 2 r wr 9 p].£f 121 2 c.TZ 3 cb[/D z\2t ty T 3X ( Qw'3K9tC cSK* S* 1-,%( T = 'i) — (p L* < -i) -fIt + Vc\ ff ~ Lvo £ (uj - k ) 4V __vD <L -1 c CX, I ) -ts( £ t x, i)C " /where:£-r c f / I (u> - K ]/y ) fltjgI, = - W JLJ fj (T* o) J.Ti-r  c- un-»' frlii 1 '/(U.-KI/Ji r-i, > i f(T i ytr. [ <• { .x j J i " ' f7- - S 'Æ*s( f—léf £ v ‘ ' j—3 ' j L V ~ K /J7- 0 Æl/ é-J 01*1 flJ—i Ui(V)]x 4 (tu '****-S 0<j- '“varl w - Kt jvJ r= o au' i TT - is: Vfj;/ ,JÅ7 U/J LIJsLf iJé-])) •<’!*"'*" <j- ati Lo-KV-jj/jl * J07The left hand side of eq. (1.25) is a nonlinear Schroedingertype of wave equation. But the term on the right handside contains £(X,i ), so we still have a integrodifferentialequation to solve.il. The lowest order solution.In order to solve eq. (1.25), we shall introduce thecharacteristic time and space scales connected to the problem.The frequency and wavenumher of the waves making up the packetare given hy LO(/ f i) and |< ( i), and define the fast time-and space-scales.If X is the characteristic length for the variation ofthe amplitude of the wave packet, we may define:£* vxand we shall assume thate <.c / (2.1)Therefore we may define the slow space- and timescales hy:(2.2)i, * e*Now, our hasic assumtions are that the amplitude of thewave packet, the frequency and the wavenumher vary only on theslow time- and space-scales.x, = £ X8E(x,t) = E(xv t 1 )(2.3). There are two other characteristic timescales which enterinto the problem:(2.4)Tp is the typical time which a particle with the velocityCO jf uses to get through the wave packet. We may note that ifwe have a very long wave packet, or a finite amplitude wave,Å should be taken as the damping or growth scale of the amplitude/ tv- is the oscillation time for the trapped particles, andit depends on the particle mass(2.5)This means that we have to distinguish between electronwaves and ion-waves.¥e assume thatrO > rp (2.6)for electron-waves, andjk(x,t) = K(x 1 ,t 1 )2.LTp - /s -*/Zi -ir,r CI^Ekiy*V * (—) r/,A1C1 £ j9rJl ? r„ (2.7)for ion-waves.Eq. (2,5-6) means that the trapped electrons make lessthan one oscillation in the potential well, while the ions feelno trapping effects.Eqs. (2.5) and (2.7) means that the ions make less thanone oscillation in the potential well, in which the electronsmay oscillate several times, However, in many cases the electronshehave as an ideal fluid and the electron trapping effects may heneglected.wave packet, are given hy;(2.8)and trapping effects will he important in the same range of thevelocity-space.Us ing eq. (2.3), we notice that S2( x j j xij is the onlyterm on the left hand side of eq. (1.23) which depends on thefast time and space scales.In order to eliminate this dependence, we integrate over thefast variahles in the following way;where (p is defined hy eq. (1.22)Eqs . (?. f7 -8), (1,11) and (1 .22) givesU)The phase velocities, , of the waves making up theK* £ t < K,2.77HK.O * *TT , *, 0 i,) A f (2. 9 )10(2.10)Writing eqs. (1.26-28) in a more explicit form, we have:(2.11)r(T, s 5T j) )*“ J ~i,f r U , / E \1f) =  'J- "F5j [f j^ v [oicri^-* ° -ft {å f ztU rj i'"-**. d*j(u> - 0 J•LT , A \_ / y f r */ j /-rk A- [. 1 Af E ]\l i ds~J~l/(*,jt,j f) - y- [ co-Kw-y cUli U " kl j' dtli £ * I/ ' J y~ o-t(M. i /--L- ))*£}[J I°°~ * j' d<Sy lU3 - K (LO - K /*nj L~J -f-+ UJ-L-<L /_£_ jtf & \*icj (2>12)J Ts f ,/•L/%,/_§_ i f_i— A flL V'i /ia rfl .-3/^—.)isJ lu> UJ - ' *-j j yf x J w fy. ( J 7Zj- tj jg, o J11—(. f ~ J — — / 4- —/ 1 - /£_M: \l j Clrfl U>~kVj}J  tu-kkyl f‘f[å 7 /‘ '*j ( “'* l ' i)dS I di i £ %$. \\ e.y rJ dlA < Ldin[*-**sr*A *= ‘ 6 ’£ 7hi*>-* v s Jtr(‘^“‘'j'r J cL 7 -f. ; £ l 0 i< P (dr [Jj_i cL. / e w. _"  J ' d£7 ito-KIJ )-., E +t)i,<L( / rå t±_ cL f £-55-i f_L cL i WlVi rlMj.f /— fL / e \\ «v p nT At7\ ~crrztjj ft.rjl +-i c,‘ 3f fe ir n .. I /_/_ i! g £) 3 )J UO- K Uj-jy/V u / Ji T.. å f E %é- Wii rlMfli £ Tjd / £ * Ui ~ uo - K V/j- >//( /*'»y W j -+«iTi<f[. .uk”-**)** Ui i A( E \\ a f£jé]lf*i'c]l-f JL * idVj UJ-uujJJ L JJ '012U)15)16)17)18)Assuraing thatlAJ (2.Lf*all the coefficients of i. „ n = 1. 2. 5., are slowlyic* 0varylng functions compared to X ' . One should note that thecoefficients depend on (f> through which is periodicin . Therefore,, taking the mean value of eq. (1.23) andeqs. (2.11-13)* we get to 0(£?):(2.V£ I.t Sl, 2' fit(2.(2.o(e)t(vt, + j -1* £< *>'*)L- i/ 0 n- • w eVl /21Z 1 9 c 21L 2r rfl[ft, to 1 2>i, L '1w5k 2a, t) 2a, ( 0 w2>« C -) 1 9 *•> 3*;L = Iwhere“ ,/2 2j } __ 2 2H) +o l ,'S lk Jf I d (_L Vj .XL ~u i -L j2> uo 7)t t >K / ' Jlf{ Jli(2. L, r !,1 z z -Mf*13(2.19)In eqs.(2.l6) and (2.20), we have neglected the self action term(Dysthe whlch gives an amplitude dependent frequency shift.In order to solve eq.(2.8), we shall make the followlngassumptions:R ( w, K ) (2.21)(2.21a)¥ith the condition (2.21), eq. (2.15) is a hyperholic typeof equation.Eq. (2.21a) means that the suhcharacteristics (J.D. Cole 1968)given hyn * I(2.22)d*.'ki' 0Furthermore, we divide the ( ; {, ) space into two partsaccording to:-*~2 £ JJyjL dyj^so7 . i £*? i. (j — i fÉJL Vi} (2.20)_l v  C J J i dz ct£? i  J'Jrv ti A fr o?0iajia) / {A.VO 0 7 TIuUJare timelike, and the initial value prohlem may he solved.14(2.23)(2.24)out erin(3.1)(3.2)(3.3)(3-UThe solution in the region (2.12), we shall call thesolution., and the other one the inner solution.III. The outer solution.In this region it is natural to search for a solutionthe formwhereiL * J,l t 3jEqs. (2.15) and (3.1-2) glve:whereu i -i ( t-m-MLo K,i, c - /2 2. u;+ Kj **, i t1 2 ai+ v % f< e-<*»»or—= 1 1 ck)/ L = 0"oSi.fr*  - £' ; "i -&’*>)15(3.5)(3.6)(3.7)(3.8)The solution of eq. (5-3) is given by:r 0. /. 2,( ids n I* É (-M * x i-fy 2 i fro&.OL* lr>l* i,) *h, (-7L, K + 71, 1])e^L&.i.) - i i i\ + + K* ++ y 0 I si = 2,3, VflÉ-L t*,' 1 ,) = JOy> [L I \\s D ( ?(“'*)  + tSX t )di')E L (*,C3^o) y o) foL , s~h i* ; i 2 |V 0 ( £(**,*) +i J2, )d\a i . / s))0 1 7d/,ds ' V 016(5.9)}(3.10),a) +(3.11)11 )in{TVvi-Because fc t as a s l° w ly varying quantlty.V,,(f(w,*0 must be zero to lowest order:Expanding £(^ f K ) in a Taylor-series around =U 0CO = U> 0 , we get :Because 1/ d (5(w,k ) =/9{£w), ’/ a ri, =d(£ w), which means thatwe may meglect this term to lowest order.This gives :To get the explicit expressions of eqs. (3.7) and (3.we have to solve eqs. (1.11) and (3.8), which will he donesectlon V.XC oV 0 tO*M)-- AW- *OM,C^ u f)® UJ* W, LO-LOc \ swhere Aw = w-uj 0 3 = k- K„We may note that /V-o - AK r 0 ( & 60 } £/-i- f f-5'O0 * W-w* '4 j J ((k-., u>„)ZiK) * j(Axfjdsji/*/*},*)0 i17IV. The inner solution.In the inner region, we define: 3*; ' ' 5x, 4£ W. 4 t% Zti i  ' '(4.1)Eq. (4.1) is consistent with the assumption (2.24).Furthermore, we have from Eqs. (1.7-8):(4.2)(4.3)(4.4)Now, eqs. (2.15), (4.1-3) give to £) (£ 3 ) :— 2 c 1 , r ? jLwhere i  = ti,X L = £ V ; / = /. «, 3 , - • •Qu) 2K2*, 4 Si, = 0 * orw 'l* / r 1 \ Al S' sx ' a*,  ( c ? *> f k 3 +  J 00 - (/£ 2t ; _)¥e define:»EUv-v r' — * , — Uv-l-j r <L £„A7 - oJ v-> -j % V' J = 0 j i* 1.1.3."-18Using eqs. (1.18), (4.1-2), we have:which gives : -£( ttJ| * ) _ qo X i (4.7)(4.8)i h j "h. ' hAs in the out er region, <£ {to, k) = d9(£a;). Otherwise£(X,i) should have a variation on the fast time scale.Now, eq. (4.5) reduces to:(4.9)ctav - i(4.10)(k + £<-.«) - n )C -i ; (*? g. C +(4.5)7 1 *) i !>/ ''a* 'V J h, l v^)\ai, f„ = £ Jl/ o -£ t ('//ij-j£ k*  % ! nr }( £ f* £ * |£} + • ( 4 • 1 6 :)V % ' V % (J X 3 J1c- Er -i i & i-;, fr - £ r1 fc- 0cW r ' / V° J = x a19We define :(4.11)(4.12)JUs ing eqs. (4.Y-8), we may fouriertransform eq. (4.12)(4.13)i r j fJ 1'i 1 ; v J 1 | *- )Al’ («0 (4.14)(4.15)6Ur~ ft ° = b o (-vnoting that t = t ,*) - Eqs . (4.9), (4.11) give :(jL ‘ "i ' 3d-C- U ET- I Li. x C ** o—-«o £E? 1 *,*)* (vryiJfEjCLL-o ) f— ou- -=» & / i 20(4.16)i.<,eClNoting that X* is the fouriertransform ofj ( '-) =( W L )«* SL ' lw: , so, using the convolution theoremfor f ouriertransf oms* we have that;Eq. (4.16) descrihes a diffusion in X, -space, during theevolution.J7-* 0 7 0Ti 1 r d-tfvuV^-(J.D. Cole 1968)."D .'>? - Corvt^A*-L x oOe 0 (* *) »&w o<2- i iL*-y) l (- / j$cU) \ JL 0 ) -i,fi < f f f . Ja* <**) 1 t i"f Jds ((J K &y J £ ti (y, * ) A-yj) \Introducing an intermediate scaleLV T' ? tn Li , — 1 * 00 ->Choosing = £3 , it is easy to show thatCr^Jr— 9 h0 £>21As in the outer solution we have to solve eqs. (1,11),(4.10) to get an explicit expression in eq. (4.16).V. The uniform solution. To solve eqs. (1.11), (1.1j5)* there are two differenteffects which. have to be taken into account.Particles with velocities:(5.1)(2.8)(5.2)are trapped by the waves making up the wave packet., whileparticles with velocities outside this region., are untrapped.Furthermore, the wave packet will behave similar to anelectrostatic pulse. This means that particles with velocities:(5.3)c / Aj. r~where (5U)are accellerated by the packet, while particles with velocities(5.5)K 'J < i“ +S K< * < v;V-(£y Ei-s> < i <“) <22are retarded (M.S, Espedal 1971).Particles outside the regions (5.1), (5.7) and (5.5)are passing the wave packet, and get no final change invelocity. They may get a change in phase if the packet isunsymmetrical.Because of eqs. (2.6-7), the regions (5.1) and (5.5),(5.5) are seperated.Using eqs. ( 1 .11 ), (1.13) and (2.10), we get;ii r , v /Å LZ T) ( *?C ('?) + (X1 "i ) (5.6)rX (7) -(co.ccr) - U (£.r) l/j (i)) d.7 (5.7)iEqs. (5.6-7)j we shallsolve approximately, dividingthe velocity space into the following regions:(5.8)(5.9)(5.10)(5.11)hl 9 h C 1 / (5.12)V s 3V °i v %-f <C V £ x~ S K< V M * S K. : anolgoi gnlvjollol sricl- odnl sosqa23Further, we shall approximate E(£T) to lowest orderby:(5.13)In the regions (5.8), (5.10) and (5.12), the particlesare passing the packet. Therefore we take V toJlowest order, which gives:nj luj lit)- *iit) v] £7tNow, in the region (5.9)j we define:(5.13)(5.14)(5.15)= t a (£x (-r), o)rKj ('f) -w + j£ ( t(x-n^(7-é)) l bj (-%[ f U.iijdf (i 1 - 14 )-fcwhere' K, CLT)X(t)J(lo - k ) jjr d y (7)x c '% l (7 )0'T Ty'{i) - j Si7) dr * x(t) -x- J K dr l - 124Eqs. (5.6-7) and give:(5.16)The lowest order solution of eq. (5.16) is :(5.17)We should note that particles which have velocities! / = ± oi , |oc| £ , before the interaction with the packet,get a velocity V = ih- + after the interaction.Similarly, in the region of trapped particles, (5.10),we define:+ (JL(7)v s tT) (5.18)J (5.19)Xfr) kV C7) X,'frJ (5.20)(5.21)d-It-4 j = £.(e*£r>, 4 ) -£ # ?x'n) N iS J0 (T) = +åj E 0 U *n>,o) + f}lx'a)j' c O '(£ T ) -f -Sj 6 1 'i )r t*/n )-x - J Ar = j ta t) drBo,, eq. (5*6) reduces to;= F 0 (£X(7),») w (%l +f) - e Ti 7 (%)25which gives to lowest order:(5.22)If T = P is the time when the interaction between thepartlcle and the packet start, and t- the Interaction time,we may write eqs. (5.14), (5.1?) and (5.25):1/V- (T) rJft'- ' (5.25)p t Tun TT° solve eqs. (5.8) and (4,10) to lowest order, we haveto approxlmate \/ c . This may he done, us ing the fact thatfc W) k ) * 0(t K) (5.24)Therefore eqs. (5.11). (4.16) and (5.25) may be usedas a first step in a successive approximation procedure.In this section, we shall study the podulation of a electronplasma wace packet propagating through a collisionless electronplasma in a background of fixed ions.¥e assume that:(6.1)M 7 -> * - ((?- l/>) 1’ xYjliV --r < pf <r r < p +Vjt Ct= p * r^j.)i = 1 > 2, 3VI. The electron plasma wave packet.1 ( £<,)*)ct-sswo 89V.926(6.2)(6.3)(6.4)(6.5)(6.6)(6.7)the(5.12),To calculate , i = 1, 2, 3, 4, we have to estimateinteraction time, TWa . In the regions (5.8), (5.10),XXi is the passing time;27(6.8)In the regions (5.9). C^l is twice the time neededto accelerate a particle from 1/This is approximately:s| <s rKiMlfiiy (6.9)The interaction time in the region (5.10), is also thepassing time, which is approximately:(6.10)Å rough figure of the phase-plane is;~ |1/ - != l/j to (/ = l/j + 5at /, to-Ki/ \Tua * /i/-iy l KS 0 )C -(— F -M *i- D K /2811 )With these assumptions , the main contribution from theintegrals (2,18-20), is:(6.Jdt__ /Jlzlli'C - K l / -fwhere the plus-sign should be used for X, - 4, > 0 andthe minus-sign for X, - IX, -£ ; < 0 •The plus-sign should be used for V - > 0 and the minus-sign for \/ ~ < OKr. <* i (i _4 lE (''--M £ 'f“ 0 'J V ~ 1 j2_* C - ; + 0(t i)f fr,, r (&%,(> Mit]'X (x 1 14 J + (6-12)(I - ))'+ £ )S j29Eqs. (3.11), (4.16), (6.12-13) give :J - Ij, l‘ ('--'• ; 3) give that the effect of trapped particles, I trJL } P ro P a g a tes as a free streaming effect. Becausek “ : tl > ' 9 faster than the packet, and shouldhe observed as a modulation in the front of the packet.(J.N. Denavit and R.N. Sudan, 1972).ihe term in eq. (6.1 3) takes care of the "pulseeffect (M.S. Espedal, 1971). This interaction effect propagateswith the velocity Vp , and modulates the packet itself.VII. Conclusion.ihe interaction between particles and an electrostaticwave packet results mainly in two different effects. We get amodulation of the packet caused by particles propagating withvelocities near to v . The evolution of these effects isrepresented by eq. (4.16).Particles with velocities near to — get a net change invelocity during the interaction. The evolution of these effectsis given by eq. (3.11).*-») =  + R< j T i) d 5 +4030¥e may note that* taking into account wave-wave interaction effects, the average equatlon is no longer linear. Inmodels where these effects appear., we may get similar equationas those obtained by Y.H.Ichikawa and T.Taniutl.References.J.Denavit and R.N.Sudan. Phys. Rev. Letters. Vol.28 (1972)pp. 404-407.K.B.Dysthe. To appear in J. ofM.S.Espedal. J. Plasma PhysicsPlasma Physics (1974)Vol. 5 (1971) pp. 343-355.Y.H.Ichikawa and T.Taniuti. J. of Physical Society of Japan.Vol. 34 (1973) PP. 513-521.al .TnenJtl n tc 'cu 1no ' r i . o-xr •f ' 3 , ii : j nno'tf iXJoe r; nl.ro:  , m ov  o1llfl 1H MB1 ni<r ‘i !it !; ii», 1 1ki ] Ji i 1i {i j:IIiH

The effects of trapped and untrapped particles on an electrostatic wave packet

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The effects of trapped and untrapped particles on an electrostatic wave packet

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