The influence of resonant electrons in phase space on the nonlinear dynamics of electron-cyclotron waves is investigated. It is shown that in the case when the frequency of externally applied electromagnetic waves is close to that of the electron-cyclotron waves, the nonlinear coupling between the electron-cyclotron waves and the resulting longitudinal perturbations is mainly caused by the pondermotive potential of the electrons. It is shown that in the competition of refraction caused by the non-resonant electrons and the relativistic nonlinearity, the influence of resonant electrons will increase strongly. The reaction of such longitudinal waves with electrons that are in resonance with high frequency electromagnetic field changes the character of the soliton propagation in an essential manner

Abbasi, H.

Hakimi Pajouh, H.

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

STATIONARY DYNAMICS OF NONLINEAR ELECTRON-CYCLOTRON WAVES AT THE PRESENCE OF RESONANT ELECTRONSH. Abbasi 1,2 and H. Hakimi Pajouh 11 Institute for Studies in Theoretical Physics and Mathematics,P. O. Box 19395-5531, Tehran, Iran2 Physics Department, Amir Kabir University, P. O. Box15875-4413, Tehran, IranThe  influence  of  resonant  electrons  in  phase  space  on  the  nonlinear  dynamics  of  electron-cyclotron  waves  is investigated. It is shown that in the case when the frequency of externally applied electromagnetic waves is close to that of  the  electron-cyclotron  waves,  the  nonlinear  coupling  between  the  electron-cyclotron  waves  and  the  resulting longitudinal perturbations is  mainly caused by the pondermotive potential  of the electrons.  It  is  shown that in the competition of refraction caused by the non-resonant electrons and the relativistic nonlinearity, the influence of resonant electrons will increase strongly. The reaction of such longitudinal waves with electrons that are in resonance with high frequency electromagnetic field changes the character of the soliton propagation in an essential manner.PACS: 52.35.-gINTRODUCTION       Circularly polarized electromagnetic waves are useful for  heating  of  fusion  plasma  [1],  modifications  of  the lower part of the earth's ionosphere [2]. Accordingly, it is essential to understand the linear as well as the nonlinear propagation characteristics of cyclotron waves. The linear propagation  of  electron  cyclotron  waves  is  fully understood  and  the  results  have  been  summarized  in  a monograph [3].  The nonlinear  propagation involves the study of  three wave interaction, wave modulation, self-filamentation, soliton formation, etc.       In the past, there have been substantial efforts [4] in the investigating of electron cyclotron waves. However, most  of  the  studies  were  concerned  with  the  study  of small  amplitude case.  Nevertheless,  the development of ultraintense  short  pulse  lasers  allows  exploration  of fundamentally new parameter regimes for nonlinear laser plasma interaction. In fact, a number of experiments have been carried out in which plasmas are irradiated by laser beams  with  intensities  up  to  219 /10 cmW .  At  such intensities,  the  electron  quiver  velocity  approaches  the speed  of  light  and  a  host  of  phenomena  have  been predicted  such  as  parametric  resonance  in  an  electron plasma  [5],  the  relativistic  self  focusing  [6]  and  the generation of large amplitude plasma waves (wake field) [7].Among the nonlinear phenomena, trapping of plasma particles  in  the  trough  of  low  frequency  longitudinal waves  of  plasma  exert  considerable  influence  on  the propagation characteristics of such waves. Determination  of  collisionless  distribution  function and its moments in nonstationary fields, which describes both  free  and  trapped  particles,  involves  difficult mathematics  demanding  simplifying  assumptions. Gurevich  [8],  assuming  slowly  varying   longitudinal fields and a collisionless trapping  mechanism, calculate the particle distribution function self-consistently. The validity of his solution is demonstrated by simulation and  experimental  results  of  Debye  shielding  and antishielding [9]. When applied field changes slowly, the distribution  function  is  a  continuous  function  of  total energy.  Then  the  distribution  of  untrapped  particle  is determined by that of particles coming from infinity with a  Maxwellian  distribution.  It  then  follows  from  the continuity  (continuity  of  particles  current  from  the trapped  to  untrapped  region)  conditions  that  trapped particle disteribution reduces to a constant. In this way, integrating over the free and trapped particle distributions results in the following  total density:  ,210 αααpiαTUTUerfenn effeffTUeff−+−−=>< −     (1) where  effU  describes  the  potential  well  and  αT is temperature of particles (α =species index; e,i). We will later explain the meaning of brackets.        In this paper,  we consider  a  class of  problems involving  the  interaction  of  relativistically  intense monochromatic radiation with a magnetized plasma when the effect of trapped electrons is taken into account. To our  knowledge,  the  present  study  represents  the  first investigation of the effects of the trapped electrons on the soliton formation.BASIC EQUATIONSConsider the propagation of a right-handed circularly polarized  electromagnetic  wave  along  the  external magnetic field zBB 00 =  in the form of,             ( ) ( ) ( ) .,.exp,21 cctitzEyixE +−+= ω            (2)where  ω  is  the  frequency  of  external  electromagnetic field and ..cc  stands for the complex conjugate.        The complete set of equations is:158                   Problems of Atomic Science and Technology. 2002. № 4. Series: Plasma Physics (7). P.  158-160                        JcEcB t pi41+∂=×∇ ,                       (3)                          Bc1E t∂−=×∇ ,                           (4)                                 pi ρ4=⋅∇ E,                              (5)                                ( )ei nne −=ρ ,                           (6)                              ( )eeii VnVneJ  −= ,                       (7)where  ee = .  In  order  to  close  the  above  systems  of equations,  ρ  and  J  should be properly defined. Since we are going to study the effect of trapped electrons we assume electron temperature, eT ,  is much larger that ions one. Therefore, electrons are treated kinetically [eq. (1)] while the fluid treatment is used for the ions dynamics.In our formalism there are two distinguishable time scales:  one  associated  with  the  fast  motion  with  the characteristic  time  of  the  order  of  the  high  frequency oscillation, ωpiτ 2= , and another with the slow motion resulting from the ions motion. Therefore, each quantity can be decomposed into the slowly and rapidly varying parts:                              ,~AAA +>= <                               (8) where  the  brackets  denote  averaging  over  the  time interval τ . We  consider  the  one  dimensional  case  in  which 0== ∂∂ yx .  Therefore,  the  nonlinear  interaction  of  a pump field with the background plasma that will generate a  slowly  varying  envelope  for  the  amplitude  of electromagnetic  wave  propagating  along  the  external magnetic  field  involving  the  weak  relativistic  quiver velocity is governed by [10]:EEcEiepzzte −Ω−−+ ∂∂ 2222 ωωω ωω                   (9) ( ) ,02 2222202=Ω−Ω−−Ω−− EcmEenneeeeepeωωωδωω ω                                                                                       where c  is light velocity, ep mnee 022 4piω = the electron plasma frequency, 0n  the unperturbed electron density, cmeB ee 0=Ω  the electron cyclotron frequency, em  the electron rest mass and 0nnn ee −〉〈=δ .       In order to define the slowly varying deviation of the density, enδ , taking into account the electron trapping we have  to  proceed  from eq.  (1).  In  eq.  (1)  we need  the definition of effU . For this purpose one should calculate the average of Vlasov equation over the time interval τ to find [11]:  ( ) ,0222=〉〈Ω−+−〉〈+〉〈∂∂−∂∂epeezezzetfmEeefvfzωωϕ     (10)                                                                                  where  ef  is the distribution of electrons. From here we can define the effective potential energy as follow:                ( ) .222eeeff mEeeUΩ−+−=ωωϕ               (11) We  assume  the  effective  potential  energy  has  the form  of  a  single   potential  well,  vanishing  at  infinity (soliton solution). Then, by substituting  eq. (11) in eq. (1) the  density  of  electrons  is  clearly  defined.  As  it  is mentioned,  the  ions  will  also  be  described  by  the hydrodynamic  equations  assuming  their  temperature  is negligibly  small.  Then  for  the  shallow  well  when eeff TU < <  ( 0nne < <δ )  and  for  the  case  when  quasi-neutrality condition is fulfilled ( ie nn δδ ≈ ), one can find for the enδ  the following expression:        ( )( ) ,2342232225222222220Ω−−+Ω−−=eeesseeesseTmEecucTmEecucnnωωpiωωδ      (12)where  u  is  the  velocity  of  stationary  envelope,  and ies mTc =2 ( im =mass of ion).  Substituting eq. (12) in eq. (9) and introducing the dimensionless variables as follows      ,8,,022ETnEzzcttepe →→→piωωωω                  (13)will leads to the following equation:      ( ),012322=−+−Ω−−∂+∂EEPEEREEEiepzzteωωω        (14)                                                                                  where,       ( ) ( ) ,222222222−−Ω−Ω−=ssTee cuccvR eωωωω         (15)159,34 2522225−Ω−=ssep cucPeωωωωpi                 (16)and  eeT mTv e = .       The  first  term  in  eq.  (15)  describes  the  weak relativistic effect and the second one is connected with the density  deviation  of  free  electrons.  It  is  essential  that these terms have different signs. The last term in eq. (14) describes the influence of  trapped electrons.  It  must  be mentioned that, as it is seen from eq. (12), in the case of eΩ>ω , the trapping of electrons is possible only in the supersonic regime when 22 scu > .SOLUTION AND RESULTS Now, we will consider the case when  eΩ>ω .  To solve the eq. (14) together with expressions (15) and (16) we use the standard method in which the amplitude of the high frequency field is expressed in the following form:                    )],,(exp[),(),( tzitzatzE ψ=              (17)where  a  and  ψ  are real functions. On substituting eq. (17) in eq. (14), one can find for the real and imaginary parts of eq. (14)  the following equations:             ( )( ) ,022 =∂+∂∂+∂ ψψ zzzzt aa                  (18)( ) ( ).0124322=−+−Ω−−∂−∂−∂paRaEaaepztzzeωωωψψ(19)                                                                                 In the stationary state when the amplitude and phase depend on the argument,                                 ,tcuz −=ξ                              (20)following the boundary conditions:                             ,0)(, →± ∞→ ξξ a                     (21)eqs. (18) and (19) are integrable and the equation for the amplitude can be presented in the following form:                      ( ) ( ) ,021 2=+∂ aVaξ                   (22)where,( ) ( ) ,5141121 542222PaRaacuaVepe−+−−Ω−−=ωωω(23)is  the effective potential  .  Equation (22) is  identical  in form  to  the  energy  equation  of  a  single  particle  with coordinate  a and  time  ξ .  The  relation  between  the maximum amplitude of soliton and its velocity is:     ( ) .52211 3max2max222PaRacuepe +−−Ω−=ωωω (24)        A glance at the effective potential )(aV  shows the effect of trapped electron ( )551 Pa− . As it is seen from eq.  (23)  to  have a bound solution for  the eq.  (22),  the equation ,                                    ,0)( =aV                                (25)should  at  least  have  two  roots.  In  the  competition  of refraction caused by the non-resonant electrons and the relativistic  nonlinearity,  R can  becomes  enough  small that  the  last  term  in  )(aV  (caused  by  the  trapped electrons)  dominates R and consequently it  can destroy the localized structure of the amplitude. REFERENCES[1] K. Matsuda, Phys. Fluids 29, 3490 (1986).[2] S. P. Kuo and M. C. Lee, Geophys. Res. Lett. 10, 979 (1983).[3] D. G. Lominadze, Cyclotron waves in plasmas (Pergamon, New York, 1981).[4] V. I.  Karpman and H. Washimi, J. Plasma Phys. 18, 173 (1977).[5] N. L. Tsintsadze, Sov. Phys. JETP 32, 684 (1971).[6] D. P. Garuchava, N. L. Tsintsadze, and D. D. Tskhakaya, Sov. Phys. JETP 71, 873 (1990).[7] T. Tajima and J. M. Dawson, Phys. Rev. Lett. 43, 267 (1979).[8] A. V. Gurevich, Sov. Phys. JETP 53, 953 (1967).[9] C. Hansen, A. B. Reimann, and J. Fajans, Phys. Plasmas 3, 1820 (1996).[10] N. N. Rao, P. K. Shukla and M. Y. Yu, Phys.       Fluids 27, 2664 (1984).[11] L. M. Kerashvili and N. L. Tsintsadze, Fiz. Plaszmy 9, 570 (1983). 160

Stationary dynamics of nonlinear electron-cyclotron waves at the presence of resonant electrons

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Stationary dynamics of nonlinear electron-cyclotron waves at the presence of resonant electrons

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Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)