Nonlinear Dynamics of Single Bunch Instability in Accelerators *

Transcription

1 SLAC-PUB-7377 Deember 996 Nonlinear Dynami of Single Bunh Intability in Aelerator * G. V. Stupakov Stanford Linear Aelerator Center Stanford Univerity, Stanford, CA 9439 B.N. Breizman and M.S. Pekker Intitute for Fuion Studie, The Univerity of Texa, Autin TX, 787 Abtrat We develop a nonlinear theory of the weak ingle bunh intability in eletron and poitron irular aelerator and damping ring. A nonlinear equation i derived that govern the evolution of the amplitude of untable oillation with aount of quantum diffuion effet due to the ynhrotron radiation. Numerial olution to thi equation how a large variety of nonlinear regime depending on the growth rate of the intability and the diffuion oeffiient. Comparion with the obervation in the SLC Damping Ring at SLAC how qualitative agreement with the pattern oberved in the experiment. Submitted to Phyial Review E * Work upported by the Department of Energy ontrat DE-AC3-76SF55 and DE-FG3-96ER

2 . Introdution Mirowave ingle bunh intability in irular aelerator ha been known for many year. The intability uually arie when the number of partile in the bunh exeed ome ritial value, N, whih an vary depending on the parameter of the aelerating regime. Typially the intability lead to the growth of the bunh length ("turbulent bunh lengthening") and the inreaed energy pread of the beam []. The origin of the mirowave intability i uually aoiated with untable oillation of the bunh aued by high-frequeny part of the impedane of the vauum hamber. Reent obervation in the SLC damping ring at SLAC [] with a new lowimpedane vauum hamber revealed ome new intereting feature of the intability. It wa found that in ome ae, after initial exponential growth, the intability eventually aturated at a level that remained ontant through the aumulation yle. In other regime, a relaxation-type oillation were meaured in nonlinear phae of the intability. In many ae, the intability wa haraterized by a frequeny loe to the eond harmoni of the ynhrotron oillation. Similar effet have been oberved in LEP for the oillation of the bunh length [3]. A vat literature devoted to the mirowave intability motly foue on the linear theory. The main objetive of thi theory i to predit the frequeny, growth rate and the truture of the perturbation a a funtion of beam parameter. Epeially important for the experiment i determination of the threhold of the intability for a given wake in the aelerator. Mathematially, the linear problem redue to a et of integral equation whoe olution uually invoke elaborate numerial method [4-6]. A olution obtained in the linear theory, however, annot explain the time development of the intability above the threhold. Several attempt have been made to addre the nonlinear tage of the intability. Uing numerial imulation method D'yahkov and Baartman tudied a mehanim that generate awtooth oillation in a ingle bunh intability [7]. Simulation of the SLC damping ring intability that alo howed nonlinear oillation of the amplitude ha been performed in Ref. [8]. Reently Heifet propoed a theory of nonlinear oillation onidering nonlinear phae of the intability a a new equilibrium around a nonlinear reonane [9]. However, being baed on either omputer imulation or ome peifi aumption regarding the truture of the untable mode, thee work, in our view, do not give a onitent and univeral deription of the nonlinear tage of the intability. An attempt of a more general onideration of the problem baed on nonlinear Vlaov equation i arried out in thi paper. We adopt an approah reently developed in plama phyi for analyi of nonlinear behavior of weakly untable mode in dynami ytem [,]. Auming that the growth rate of the intability i muh maller than it frequeny, we find a time dependent olution to Vlaov equation and derive an equation for the omplex amplitude of the oillation valid in the nonlinear regime. Thi equation, after proper normalization, ontain only two dimenionle parameter, and an be eaily olved numerially. It turn out that even without detailed knowledge of the nature of the intability, we an qualitatively analyze and predit different pattern of the ignal that an be oberved in the experiment in a weakly nonlinear regime. The paper i organized a follow. In Setion we formulate the tability problem in term of Vlaov equation with a right hand ide due to the effet of ynhrotron radiation. In Setion 3, a brief review of the linear theory for a ingle bunh intability i given. Setion 4 ontain a general derivation of an equation for the evolution of the amplitude of weakly untable oillation near the threhold of the intability. A detailed alulation of nonlinear part of the equation i preented in etion 5. In Setion 6 we inlude ynhrotron radiation term into nonlinear equation and introdue dimenionle variable that minimize the number of free parameter in the equation. Analyi of the

3 olution and reult of numerial omputation are preented in Setion 7, and in Setion 8 we diu the main reult of the paper.. Bai Equation We tart from the equation of motion in longitudinal diretion (ee, e.g., Ref. []): z ηδ, δ K z, t, () where z i the longitudinal oordinate, δ i the relative energy deviation, η i the lip fator, the dot indiate differentiation with repet to time t, and = = ( ) ω Kzt z re (, )= dz n( z, t) w( z z) η Tγ. () z In Eq. () ω denote the unperturbed ynhrotron frequeny, T i the revolution period, r e i the laial eletron radiu, γ i the relativiti fator, nzt, ( ) = ( ) i the longitudinal beam denity, nztdz, N, where N i the number of partile in the beam, and wz () i the longitudinal wake funtion. The firt term in Eq. () orrepond to the potential of the aelerating voltage, and the eond term deribe the wakefield generated by the bunh. Equation of motion () an be obtained from the following Hamiltonian: ω Hz t z re (, δ, )= ηδ dz dz n( z, t) w( z z ) η Tγ, (3) z in whih z play a role of a oordinate, and δ i the onjugate momentum. We will ue a ditribution funtion ψ xpt,, integrating over δ give the partile denity ( )= ( ) z ( ) of the partile in the bunh uh that nzt, N ψ z, δ, tdδ. (4) Thi ditribution funtion atifie the Vlaov equation with a Fokker-Plank "olliion" term on the right hand ide, ψ { H, ψ}= R, (5) t where we have the Poion braket on the left hand ide, and R deribe the effet of the ynhrotron radiation (ee, e.g., [3]), R = D δ γ ψδ κ ψ δ. (6) In Eq. (6) γ D i the damping time for the amplitude of the ynhrotron oillation and κ i the diffuion oeffiient aoiated with the quantum nature of the radiation. 3

4 In the equilibrium tate the ditribution funtion ψ and the Hamiltonian H do not depend on time. The equilibrium olution of Eq. (6) wa given by Haiinki [4], ( ) H z, δ ψ( z, δ)= ont exp, (7) ησ E where σe = κ γ D i the rm energy pread of the beam in the abene of the wake, and H i the equilibrium Hamiltonian. It i onvenient to introdue dimenionle variable, z δ x =, p=, = tω, F = σzψ, (8) σz σe where σ z i the rm length of the beam without wake, σz = σeη ω. In thee variable the Hamiltonian (3) take the form Hxp (,, )= p Ux (, ), (9) where the "potential energy" U i with and U( x, )= x I dx S( x x) dpf( x, p, ), () x Nre I = T γω σ σ, () z E xσ S( x)= dzw() z. () Note that the funtion S i a dimenionle funtion of it argument. Let u now perform a anonial tranform from x and p to ation and angle variable, J and θ, of the equilibrium Hamiltonian H, and denote by Ṽ the deviation of the potential energy from the equilibrium in Eq. (9). Sine H depend on J only, the total Hamiltonian H θ, J, t ( ) take the form ( )= ( ) ( ) H θ, J, H J V θ, J,. (3) The Vlaov equation for F in term of ation angle variable i F ω F V F V F = R, (4) θ J θ θ J 4

5 where ω = ω( J) i the frequeny of ynhrotron oillation with the wake taken into aount, ω ( J)= dh dj. Suppoe that F( J) i the equilibrium ditribution funtion, and δfj (, θ, )= F F ( J) i it deviation from the equilibrium. Then δf atifie the following equation, where δf ω δ F δv F δv δf δv δf = R, (5) θ θ J θ J J θ δv θ, J, I djdθk J, J, θ, θ δf J, θ,, (6) and KJJ (,, θθ, )= SxJ ( (, θ) xj (, θ) ). We note that equation (5) and (6) are exat beaue we did not make any approximation in the derivation above. ( )= ( ) ( ) 3. Linear Theory In linear theory, the lat two term on the left hand ide in Eq. (5) mut be diarded. We aume that the perturbation of the ditribution funtion oillate with the frequeny ω, δf = f ( J θ) e iω,.., (7) where for the ake of brevity we ue the notation ".." to denote a omplex onjugate of the firt term. The perturbation of the potential Ṽ i Ṽ = Ve i ω... (8) Sine V i a periodi funtion of θ, we an expand it in Fourier erie, n n= = ( ) V v J e in θ. (9) For impliity we will neglet here the effet of the ynhrotron damping in the linear theory by dropping the R-term in Eq. (5). Thi greatly implifie the linear analyi and i uually aumed in the literature. However, a we will ee in Setion 7, the effet of the ynhrotron damping i ruial for the nonlinear tage of the intability and will later be inluded in the derivation of the nonlinear equation. Subtituting Eq. (7) and (9) into Eq. (5) give in linear approximation f iωf ω = n( ) θ where F = F J. A olution to Eq. () i inθ F inv J e, () n= 5

6 nvn J in f = F e θ. () n= ω n ω Now, ubtituting thi equation into Eq. (6) yield an infinite et of integral equation that determine eigenfrequenie and eigenfuntion for the olletive oillation of the bunh: with the kernel given by ( ) v J I dj F J K J, J n nm m= ( )= ( ) ( ) ππ ( ) ( ) mvm J ω mω J, () im n Knm( J, J)= d d e K J,, J, ( θ θ θθ ) ( θ θ). (3) π The integral on the right hand ide of Eq. () define an analytial funtion in the upper half plane of the omplex variable ω ; for Imω the integral mut be analytially ontinued into the lower half plane. For a real value of ω, the integration i performed along a ontour in the omplex plane whih bypae the ingular point of the integrand below the pole (ee, e.g., []). The reidue of the integral () are aoiated with the Landau damping effet. 4. Nonlinear Theory Let u aume that the intability ha a threhold orreponding to a ritial value of the parameter I = I with the frequeny at the threhold ω = ω (Imω = ). We will be intereted in the analyi of the nonlinear phae of the intability in the viinity of the threhold when the growth rate of the intability, Γ, i muh maller than ω, Γ<<ω. In other word, we aume that the intability i weak and develop on a time ale whih i muh larger then the period of the oillation. It turn out that in thi ae one an eparate a "low" time ale on whih the amplitude evolve from "fat" oillation with the frequeny ω and derive nonlinear equation for the evolution of the amplitude of the intability by averaging over ω. In thi etion we will give a general deription of the approah following a imilar analyi in the theory of nonlinear plama oillation [5]. Firt, we rewrite the reult of the previou etion in a onie form, Lˆ ω, I V ( ) =, (4) where the linear operator ˆL repreent a et of integral equation () and (3), ω Lˆ nv J inθ n ( ω, I) Vω e vn( J) I djdθkf ( J), (5) n= m= ω nω( J) and V ω i a Fourier harmoni of the funtion Ṽ orreponding to the frequeny ω, inθ V v J e. Note that at thi point we an alo inlude in ˆL a ontribution from the ω = ( ) n Fokker-Plank term R. A partiular form of the operator ˆL i not eential for the analyi in thi Setion. ( ) 6

7 The frequeny of the oillation ω at the threhold and the orreponding eigenfuntion V u are determined by the equation ω ( ) =. (6) Lˆ ω, I u We now need to define a alar produt of two funtion u and w of the phae pae variable J, θ. Let u denote thi produt by ( uw, ). Uually, alar multipliation in Hilbert pae i given in term of an integration of the produt uw * over J and θ with ome weight funtion. The exat hoie of the weight funtion i not important for what follow, and we do not peify it here. For a given alar produt, we an define an operator ˆL onjugate to ˆL atifying the following ondition for two arbitrary funtion u and w, ( ulw,ˆ )=( wlu,ˆ ). (7) We will aume that the operator ˆL i known and together with the olution of Eq. (6) the olution w of the onjugate problem Lˆ ( ω, I ) w = (8) i available. Note that olution of Eq. (8) repreent a linear problem and in eah partiular ae an be aomplihed by tandard method of numerial analyi. We now onider a ituation when I lightly exeed the threhold, I = I I, with I << I. Taking into aount nonlinear term in the Vlaov equation we will aume that they are muh maller than the linear one. That i to ay, we are expeting that the intability, after initial exponential growth, will eventually aturate at a level where the amplitude of the oillation i relatively mall. If thi i not a ae, and the intability evolve to a highly nonlinear regime, our theory will only be appliable for a relatively hort period of time following the linear growth. Fortunately, a we will ee in Setion 7, in many ae the damping aoiated with ynhrotron radiation indeed limit the growth of the intability, and the whole proe i deribed within a framework of a weakly nonlinear approximation. With nonlinear term, the equation for the V ω an now be written a Lˆ, I V Nˆ ( ) =, (9) ω ω ω where ˆN ω i a Fourier tranform of the nonlinear term negleted in the linear analyi. The operator ˆN ω depend on the parameter I, and at on the funtion V ω. Following a general preription of nonlinear theory of oillation [5], we will aume the following type of olution (in time repreentation) for Eq. (9) [ ] ( ) iω V = A( ) u e.. V J, θ,. (3) where Au >> V. The firt term in Eq. (3) deribe oillation with the eigenfuntion u, frequeny ω and varying amplitude A( ), and the eond term i a orretion due to the deviation of the exat eigenfuntion from u. It i important to emphaize here that 7

8 ( ) i uppoed to be a low funtion of time, ω A ln A <<. It alo mean that the petrum A ω of the funtion A( ) i repreented by a narrow peak (the width of the peak i muh maller than ω ) loalized near the zero frequeny. We now need to make a Fourier tranform of Eq. (3) and ubtitute it into Eq. (9). Sine we are intereted in the frequeny range loe to ω, an approximate relation hold: Ṽω Aω ω u V ω. (3) In Eq. (3) we negleted the term ontaining Aω ω whih i peaked around ω = ω. Eq. (9) now read Lˆ ( ω, I I A u V Nˆ )( ω ω ω)= ω. (3) Making a Taylor expanion of the linear part and negleting the produt I V ω one find ˆ Lˆ Lˆ L( ω, I V A u IA ˆ I u N ) ω ( ω ω) ω ω ω ω = ω, (33) ω where the derivative of the operator ˆL are evaluated at ω = ω, I = I. We an annihilate the firt term in Eq. (33) by a alar multipliation with w and uing Eq. (7) and (8). The reult i ( ω ω ) ω ω ω ω ω ω = A w L ˆ u IA w L ˆ,, ( ) I u w, Nˆ. (34) We now multiply Eq. (34) by e iω and make an invere Fourier tranform to time : where A i ω ω A ie i L = ( w N ), ˆ ˆ w u, ω, (35) ω = I w L ˆ I u w L ˆ,, u ω (36) i a linear frequeny hift due to the hange of I. Note that in Eq. (35), after the invere Fourier tranform, N repreent a funtion of time rather than ω. Without the right hand ide it follow from Eq. (35) that the amplitude A will vary with time a exp( i ω ) whih i a trivial onequene of the fat that in linear theory V exp( iω ) with ω = ω ω. In the next etion we will find the nonlinear term averaged over fat oillation whih add nonlinear dynami to Eq. (35). 5. Derivation of Nonlinear Equation 8

9 The nonlinear term in our problem arie from the lat term in kineti equation (5). We need to approximately olve thi equation and find N in Eq. (9). In order to implify the derivation, we firt onider the ae when R =. In the next etion a generalization for R will be given. Sine nonlinear term i aumed to be mall, it will be aurate enough to neglet V term in it evaluation. Hene, Ṽ A u e iω ( ).. where u i deompoed into Fourier erie over θ, u u J e = ( ) n= n in θ. (37) We will alo repreent the perturbation of the ditribution funtion δf a iω iω δf = [ f( J, θ, ) e.. ] f( J, θ, ) [ f( J, θ, ) e.. ], (38) where f, f and f are low funtion of time (a A( )) in the ene that t<< ω. Subtituting Eq. (38) into Eq. (5) we note that, a alulation how, the main ontribution ome from the reonant term in δf that are differentiated with repet to J. Thi allow u to neglet the lat term in Eq. (5) to obtain f ω f V * f V * f = θ θ J θ J, (39) f ω ω f V f i f =, (4) θ θ J f ω ω f V F V f V * f i f =, (4) θ θ J θ J θ J where the aterik indiate omplex onjugating. The lat two term in Eq. (4) imply that we an plit the funtion f into linear (L) and nonlinear (NL) part, where f L L NL f = f f, (4) atifie the equation of linear theory, L ω ω L f f V L F i f =, (43) θ θ J and f NL i the nonlinear orretion ariing from the higher order term in the kineti equation, * NL ω ω NL f NL f V f V f i f = θ θ J θ J In equation for f and f we an ubtitute f L for f.. (44) 9

10 Let u onider firt Eq. (43) for the linear part of the ditribution funtion. Thi i in fat the ame equation a Eq. (), however, we now want to find it olution in time domain rather than in frequeny domain. We expand f L in Fourier erie in θ, L in f = gn( J,) Fe θ, (45) n= and find from Eq. (43) an equation for g n, gn i ω nω g ina u Thi equation an be eaily olved, ( ) = ( ). (46) n n i( )( ω nω gn inun J A e ) d. (47) = ( ) ( ) We now onider equation (39) for f. The dominant term in thi equation will be thoe that do not depend on θ ; nonzero n term will aue only mall oillation in f at the frequeny nω, without ytemati hanging of it amplitude. Keeping only n = term we have f inau J g * F = n n... (48) n= J When differentiating with repet to J in Eq. (48), it i uffiient to differentiate the exponential term exp[ i( )( ω nω ( J) )] in the olution (47) only; all other term will be relatively mall beaue we aume that the time ale on whih the nonlinear effet beome eential i uh that ω >>, f 3 * * = in A( ) ununω F d A e n= Now we an integrate thi equation, yielding i ω nω ( ) ( ) ( ) ( ) i( ) ω nω n n ( ) ( ) n= 3 * * f = Re in ω Fu u d A( ) d A e In a imilar fahion the following equation an be obtained for f, f iωf with the olution ω f θ 3 n n=... (49) ( ). (5) inθ i ω nω in A u ω Fe da e ( )( ), (5) = ( ) ( ) 3 θ f = in ω Fu e d A( ) d A e n= n in i( ) ω nω ( ) ( ) ( ). (5)

11 We now have to ubtitute f and f into Eq. (44). A alulation how, the leading ontribution to f NL ome from f ; nonlinear term ariing from f turn out to be mall in parameter Γ ω. Keeping only f and performing differentiation with repet to J in the exponential term only we find NL f ω ω NL f V NL f i f = θ θ J * inθ * i ω nω = in ω u u F e Re d A d A e, n= with the olution ( ) ( ) 5 n n ( ) ( ) ( )( ) (53) NL 5 * f = in ω u u F d A n= * i 3 ω nω Re d A d A e. ( ) n n ( ) ( ) 3( 3) ( 3) ( )( ) Finally, ine time i uppoed to be muh larger than ω, one an ue the following mathematial identity when integrating over, (54) ixy dx dye = πδ( x) δ ( y), (55) whih in appliation to Eq. (54) after hanging the variable σ =, ζ =, yield NL 5 * f = πi n ω δ ω nω u u e n= n n ζ inθ F * da( ) da( ) A( ) J ζ ζ ζ σ ζ σ ζ σ. ( ) ( ) We have found a nonlinear part of the perturbation of the ditribution funtion f NL. A we ee, thi funtion i proportional to the third order of the amplitude A. On the linear tage of the intability, when A i mall, f NL an be negleted, however a A grow, the nonlinear term beome more important and eventually ompete with the linear part f L. Notie alo, that due to the preene of delta-funtion δ( ω nω ), the nonlinear term i peaked at the reonant value of the ation J n, uh that nω ( J )= ω. n (56) 6. Effet of Synhrotron Damping and Nonlinear Equation for the Amplitude In the previou etion we negleted the effet of the ynhrotron radiation in the Vlaov equation. To inlude the R-term we need to tranform it firt to J θ variable. In doing o we notie that, beaue of trong loalization near the reonant value J n of the perturbed ditribution funtion, the leading term in R will be the one ontaining the eond derivative with repet to J. In other word, the mot important effet of the ynhrotron

12 radiation will be the quantum diffuion of partile in the phae pae rather than energy lo. Keeping only the eond derivative in R give F R= D( J) J. (57) The diffuion oeffiient D wa found in Ref. [4] and equal ( )= ( ) DJ Jγ ω J. (58) D The derivation of f NL given in Setion 5 an now be repeated with the diffuion term R on the right hand ide of the Vlaov equation. For the ake of brevity we will omit thi derivation here referring the reader to Ref. [6] where a imilar problem wa worked out for nonlinear plama oillation problem. In our ae, the inluion of the diffuion redue formally to appearing of a exponential fator in the integrand of Eq. (56), NL 5 * f = πi n ω δ ω nω u u e n= n n ζ B n ζ σ ζ 3 F * da( ) da( ) A( ) e J ζ ζ ζ σ ζ σ ζ σ ( ) ( ) ( ) ( ) where Bn = n ω D Jn, and J n i the value of the ation at the n-th reonane, nω( Jn)= ω. We are now in poition to find the nonlinear term ˆN in Eq. (35). Sine it will be multiplied by exp( iω ), we need a omponent in ˆN that oillate a exp( iω ), o that the right hand ide in Eq. (35) would be a lowly varying funtion of time. From Eq. (6) and Eq. (38) we ee that uh a term in Ṽ i δv iω L NL Ie djdθk J, J, θ, θ f J, θ, f J, θ,, (6) whih give for ˆN = ( ) ( ) ( ) inθ ( ) = ( ) ( ) (59) iω NL N I e dj dθ K J, J, θ, θ f J, θ,. (6) With thi expreion, the right hand ide of Eq. (35) beome where 4 n( n) ω n= πi K J F n ζ Bζ n σ ζ 3 * da ζ ζ ζ da σ ζ σ A ζ σ e ( ) ( ) ( ), (6) K J w L in n( )= u d e w K unun ˆ *, θ θ (, ), (63) ω

13 ( ) in Eq. (63) i performed with repet to variable J and θ and the alar produt w, K in KJJ (,, θθ, ). To further implify the analyi we will aume here that only one term dominate in the um of Eq. (6). Thi aumption i orret if the variation of the frequeny ω ( J) within a ditribution funtion i not very large o that equation nω( Jn)= ω ha a olution only for one value of n. Omitting the um ign in Eq. (6) give the following nonlinear equation for the amplitude A, A i ω A π I K J Fn 4 = n( n) ω ζ B n ζ σ ζ 3 ( ) ( ) ( ) * da ζ ζ ζ da σ ζ σ A ζ σ e In thi form, Eq. (64) ontain two omplex and one real parameter. For numerial olution it i onvenient to redue the number of the parameter by hooing new variable. Firt, we denote the real part of the oherent frequeny hift by Ω, ω = Ωi Γ, and introdue the abolute value ρ and the phae φ of the omplex fator in front of the integral 4 i o that πik J Fn ω ρe φ ( ) =. With new variable n n. (64) equation (64) beome ρ iω Γ 3 a = A e, g =, ξ = Bn, (65) B 56 3 B n n ξ ξ ζ a ζ σ ζ iφ * 3 ga = e dζa( ξ ζ) ζ dσa( ξ ζ σ) a ( ξ ζ σ) e ξ. (66) The parameter g here play a role of dimenionle growth rate of the intability that i meaured in time unit related to the ynhrotron damping rate. Note that now Eq. (66) ontain only two real parameter, g and φ. 7. Analyi and Solution of Nonlinear Equation A omplete analyi of nonlinear dynami of the intability in any partiular ae require omputing of the oeffiient in Eq. (66) whih an only be done baed on the olution of the linear problem deribed in Setion 3. In the general ae, thi ontitute a major omputational tak, whih lie beyond the ope of the preent paper. Rather than trying to find a partiular olution to nonlinear problem for a given et of beam parameter we will outline here poible enario by numerially olving Eq. (66) for different value of g and φ. Firt, note that equation (66) admit an aymptoti olution in the form of a = ont exp( iλξ ) that orrepond to oillation with a ontant amplitude and a oherent frequeny hift λ. Thi olution i valid in the limit ξ and exit only if φ < π. It i given by the following formula that an be eaily verified by diret ubtitution into Eq. (66), 3

14 6 a = 8 g e i ξtan φ, (67) Γ3 oφ ( ) where Γ( 3) tand for the gamma funtion. Aording to thi olution, the teady tate amplitude a inreae in proportion to the quare root of the dimenionle growth rate, g. It turn out however, that thi olution i only table for relatively mall value of the parameter g []. We have olved numerially Eq. (66) for everal et of g and φ. The reult are preented in Fig. 3. In Fig. we how olution for φ = and variou value of g tarting with a uffiiently mall value of a o that initially the nonlinear term i unimportant. For mall value of g, g < 4., we ee that the olution, after initial exponential growth, reahe the equilibrium after everal oillation. With inreaing g, the oillation beome more pronouned, and finally at g = 48. a teady tate olution with periodi oillating amplitude et up. Further inreaing g beyond the value of 5. aue the period of thoe oillation to break up whih, after initial tranient period, reult in a relaxation-type behavior of the amplitude. For even larger g, g > 8., the nonlinear term annot tabilize the ytem any more and the amplitude tart to grow without limit. Fig. how olution for φ = π 4. In thi ae the amplitude oillation appear to be le table and runaway olution develop already for g = 5.. A wa mentioned above, a table aymptoti olution exit only if φ < π. Numerial olution indiate that for φ > π all the olution diverge with unlimited growth a ξ. An example of uh a olution i hown in Fig. 3. We ee that, in thi ae, the nonlinear term annot top the intability whoe amplitude ontinue to grow and eventually goe beyond the limit of appliability of the preent theory. 8. Conluion In thi paper we applied the theory of weakly nonlinear untable oillation to the ae of a ingle bunh intability in irular aelerator. We derived an equation whih deribe evolution of the amplitude of the intability and depend only on two dimenionle parameter a normalized linear growth rate of the intability g, and a phae of nonlinear term φ. We found that for mall value of φ the nonlinear term ha a tabilizing effet and, for not very large value of g, reult in the aturation of the intability at ome level. Larger value of g lead to relaxation type oillation of the amplitude. In the ae of φ > π, within the limit of the appliability of our theory, the nonlinear term doe not prevent the growth of the amplitude. A wa mentioned before, a omplete omparion of our theory with the experiment require olution of equation of the linear theory and determination of the parameter in the nonlinear equation. Due to omputational omplexity of thi problem we did not attempt to olve it in thi paper. However, even without knowing the exat parameter, we an try to ompare different pattern of the ignal that have been meaured in the experiment with olution obtained in the theory. In uh a omparion we only pay attention to qualitative behavior of the amplitude uh a growth, oillation and aturation at ome level. Even viual omparion of the intability ignal from Ref. [] how a lear reemblane to our urve. In one ae (Fig. 5 of Ref. []), after injetion in the ring, the amplitude of ignal from petrum analyzer tuned to a ideband frequeny began to grow monotonially and after ome time of the order of ynhrotron damping time aturated at 4

15 approximately ontant level. Thi ituation i very imilar to our Fig. a. In another ae (Fig. 4 of Ref. []), oillation with dereaing amplitude were oberved, whih an be identified with Fig. a or b. In later meaurement [7], amplitude oillation with approximately ontant modulation were meaured. Thi ituation remind our Fig. e. Unfortunately, at thi time we are not able to ompare with the experiment theoretial predition for the period of the nonlinear oillation, although preliminary rude etimate indiate they are about of the ame order. In onluion, our theory how qualitative agreement with the ignal oberved in the SLC Damping Ring ingle bunh intability. Further work i planned to make a more definite omparion of the theory and the experiment. 5

16 (a) (b) ().5 (d) (e).5.5 (f) (g) (h) Fig.. Plot of the abolute value of the amplitude, a, veru time ξ for φ =. a g =., b g = 3., g = 4., d g = 48., e g = 5., f g = 6., g g = 7., h g = 8.. 6

17 (a).6.4. (b) ().5.5 (d) (e) Fig.. Plot of the abolute value of the amplitude, a, veru time ξ for φ = π 4. a g =., b g =., g = 3., d g = 4., e g = 5.. 7

18 Fig. 3. Plot of the abolute value of the amplitude, a, veru time ξ for φ = π and g =.. 8

19 Referene. B. Zotter. Colletive Effet General Deription. Proeeding of CERN Aelerator Shool, General Aelerator Phyi. CERN 95-9 (Geneva, 985), p K. Bane et al. High-Intenity Single Bunh Intability Behavior In The New SLC Damping Ring Vauum Chamber. Proeeding of the 995 Partile Aelerator Conferene on High-Energy Aelerator, Dalla, Texa, May 995, (IEEE, 996), 5, D. Brandt, K. Corneli and A. Hofman. Experimental Obervation of Intabilitie in the Frequeny Domain at LEP. CERN SL/9-5, (99). 4. K. Oide. A Mehanim of Longitudinal Single-Bunh Intability in Storage Ring. KEK Preprint 94-38, M. D'yahkov and R. Baartman. Longitudinal Single Bunh Intability. Partile Aelerator, 5, 5, M. D'yahkov. Longitudinal Intabilitie of Bunhed Beam Caued by Short-Range Wake Field. Ph.D. Thei, The Univerity of Britih Columbia, R. Baartman and M. D'yahkov. Simulation of Sawtooth Intability. Proeeding of the 995 Partile Aelerator Conferene on High-Energy Aelerator, Dalla, Texa, May 995, (IEEE, 996), 5, K.L.F. Bane and K. Oide. Simulation of the Longitudinal Intability in the New SLC Damping Ring. Proeeding of the 995 Partile Aelerator Conferene on High- Energy Aelerator, Dalla, Texa, May 995, (IEEE, 996), 5, S.A. Heifet. Mirowave Intability Beyond Threhold. Phyial Review E, 54, p. 889, H.L. Berk, B.N. Breizman and M. Pekker. Nonlinear Dynami of a Driven Mode near Marginal Stability. Phyial Review Letter, 76, 56 (996).. B.N. Breizman, H.L.Berk, M.S. Pekker, F. Porelli, G.V. Stupakov, and K.L. Wong. Critial Nonlinear Phenomena for Kineti Intabilitie Near Threhold. Submitted to Phyi of Plama.. A. W. Chao. Phyi of Colletive Beam Intabilitie in High Energy Aelerator. Wiley, New York, A. Pivinky. Beam Loe and Lifetime. Proeeding of CERN Aelerator Shool, General Aelerator Phyi, CERN 95-9 (Geneva, 985), p J. Haiinki. Exat Longitudinal Equilibrium Ditribution of Stored Eletron in the Preene of Self-Field. Nuovo Cimento, 8, 7, A.A. Galeev and R.Z. Sagdeev. Review of Plama Phyi, edited by M.A. Leontovih (Conultant Bureau, New York, 979), 7, p.3. Tranlation of Voproy teorii plamy, Atomizdat, Moow, H.L. Berk, B.N. Breizman and M. Pekker, to be publihed. 7. B. Podobedov and R. Siemann, private ommuniation. 9