Deformation of the concentration profile of the weakly inhomogeneous plasma due to the modulated electron beam was studied via computer simulation using big-particles-in-cells method. At the first stages the electric field increase in the local plasma resonance region was observed. It resulted to the density cavity formation. For the isothermic plasma the density cavity transformed into the plasma concentration jump. For plasma with the hot electrons ion-acoustic turbulence was observed at the late time points

Anisimov, I.O.

Kelnyk, O.I.

Siversky, T.V.

Soroka, C.V.

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

DEFORMATION OF THE PLASMA CONCENTRATION PROFILE DUE FIELD OF THE MODULATED ELECTRON BEAM: NUMERICAL SIMU-LATIONI.O. Anisimov, O.I. Kelnyk, T.V. Siversky, C.V. SorokaTaras Shevchenko National University of Kyiv, Radio Physics Faculty, Kyiv, Ukraine, ioa@u-niv.kiev.uaDeformation of the concentration profile of the weakly inhomogeneous plasma due to the modulated elec-tron beam was studied via computer simulation using big-particles-in-cells method. At the first stages the electric field increase in the local plasma resonance region was observed. It resulted to the density cavity for-mation. For the isothermic plasma the density cavity transformed into the plasma concentration jump. For plasma with the hot electrons ion-acoustic turbulence was observed at the late time points.PACS: 52.35.Mw1. INTRODUCTIONDeformation of the plasma concentration profile due to the field of the incident electromagnetic wave was of-ten discussed (see, e.g., [1-2]). For the obliquely inci-dent p-polarized waves of the moderate power the con-centration profile deformation occurs mainly in the local plasma resonance region (LPRR). It has a form of the quasi-periodical generation of the density cavities [1] or formation of the sharp concentration jump [2]. Similar effects  were  predicted  in  [3]  for  the  field  excited  by modulated electron beam moving through the inhomo-geneous plasma. This problem was studied in [4-5] us-ing numerical  solution of  the non-linear  equations set for the electric field and plasma concentration. For this case problem was solved in the given beam current’ ap-proximation.Deformation of  the  inhomogeneous plasma profile due to the field of the modulated electron beam is treat-ed in this work. Numerical simulation was carried out using the big-particles-in-cells  method [6-7]  (i.e.  self-consistent approximation) for one-dimensional model.2. NUMERICAL SIMULATION METHOD AND PARAMETERS SELECTIONThe program used for simulation is the modification of the PDP1  code [4],  but it has some additional  re-source. It gives the possibility to introduce modulation of  the electron beam velocity or density and inhomo-geneity of the initial plasma concentration profile.In our simulation the beam density was modulated sinusoidally with the depth of 100%. The beam current and velocity v0 as well as the characteristic inhomogene-ity length L were selected so that the charge density pro-file in LPRR did not differ strongly from the sinusoidal shape (fig.1).After  the beam passage through LPRR significant deformation of the electron bunches’ profile did not oc-cur. This result justifies the validity of the given current approximation for this problem that was used in [4-5].Initial concentration profile was linear.  Concentra-tion varied from 0.5пс to 1.5пс (пс is the critical electron concentration for the beam modulation frequency). Thus LPRR was placed in the middle of the simulation re-gion. Mass of the plasma ions was selected so that char-acteristic time of the ion concentration profile deforma-tion did not exceed the time of simulation (a few hun-dreds periods of Langmuir oscillations). Ions  of hydro-gen satisfy this condition.abFig.1. Space distribution of the beam electrons’ con-centration in the steady-state regime for L=2.5m, ω/2π=1.3⋅108Hz, |v0|=7⋅107m/s, Те=25eV: a – v0>0; b – v0<0Both cases of the beam velocity sign (v0>0 – beam moving into the dense plasma and v0<0 – beam moving from the dense plasma) were studied. Simulation was carried out  for  isothermic  plasma  (Те=Ті) and  plasma with hot electrons (Те>>Ті).3. FIELD EXCITATION AT THE INITIAL TIME POINTSAt the initial time points after the injection start the space  periodicity  and  motion  direction  of  the  electric field wave are similar to the current wave of electron beam (Fig.2).However after some time periods of modulation the space period of electric field starts decreasing while its magnitude in the LPRR gradually increases. Field spa-tial distribution is symmetrical relatively to the LPRR for these time points. The field magnitude in LPRR in-creases during some units or some tens of the modula-tion periods.  The increase time is directly proportional to the plasma inhomogeneity length and inversely pro-portional to the thermal velocity of the plasma electrons. The field increase can be non-monotonic (in agreement with the calculation [8]).Later the spatial field distribution is modified. Peri-odic  oscillations  in  supercritical  plasma  disappear.  In the subcritical plasma Langmuir wave is formed movingFig.2. Time evolution of the electric field spatial distri-bution at the initial time points after the injection start  (v0>0, f⋅t1≈6; t1<t2<t3)out from the dense plasma. Oscillations in the LPRR have now the phase velocity directed antiparallelly to the beam velocity (Fig.3).Fig.3. Space-time course electric field in LPRR (v0>0, f⋅t1≈67; t1<t2<t3<t4)In this case condition of the weak plasma inhomogene-ity  k0L>>v03/cvTe2 (where  k0=ω/c,  v0 and vTe are  the beam velocity and thermal velocity of the plasma elec-trons respectively,  L is the inhomogeneity size) is not satisfied.  Therefore the simulation results  do not  de-pend sufficiently from the sign of the beam velocity [9-10].At the late time points deformation of the ion con-centration profile  predicted in  [3-5] was observed in the LPRR.4. DEFORMATION OF THE CONCENTRATION PROFILE OF ISOTHERMIC PLASMAFirstly the short-wave perturbation (in the scale of the LPRR width) appears. Instantaneous ion and elec-tron concentration profiles strongly differ on this stage (Fig.4a). Later the density cavity is formed in this re-gion (Fig.4b). Sometimes local maximum of the plas-ma concentration is observed in the center of the cavity (Fig.5).In the isothermic plasma (Те=Ті) at the late times points the density cavity is transformed. Its left border is smoothed (Fig.6a). At the same time its right border becomes like a step. It results to the cavity transforma-tion into the plasma concentration jump (Fig.6b). This result agrees qualitatively with the calculation [2]. This calculation was carried out for the plasma concentra-tion profile  deformation  due  to  the HF electric  field (stationary case).This concentration profile lasted out for the long pe-riod (the time of about some hundreds of the modulation periods).  It  indicates  the  balance  between the  plasma thermal pressure and the HF electric field excited in the LPRR by the modulated electron beam.After the interval of about a thousand of the modula-tion periods the concentration profile is smoothened due to the diffusion. It results to the decrease of the concen-tration jump.The case of the beam moving from the dense plasma was also examined. It does not differ significantly from the previous case (as well as the linear stage of the pro-cess). абFig.4: a – deformation of the ion concentration profile (f⋅t1≈65); b – formation of the density cavity (f⋅t1≈120). ne, ni is concentration of electron and ion respectively.  L=1m, Те=6eVFig.5. Local maximum in the center of the cavity (f⋅t1≈110)abFig.6. L=2.5m, Те=25eV: a – transformation of cavity (f⋅t1≈143); b – forming of concentration jump (f⋅t1≈195)abFig. 7 a – excitation of ion-acoustic wave (f⋅t1≈235), b – motion of ion-acoustic wave (t4>t3>t2>t1) Fig.8. Plasma concentration profile for the late time points (f⋅t1≈1650)5. DEFORMATION OF THE CONCENTRATION PROFILE OF PLASMA WITH HOT ELECTRONSWhen the electron plasma temperature is consider-ably  larger  relatively  to  the  ion  temperature  (plasma with hot electrons) the ion-acoustic type’ perturbations propagated from the cavity are observed (Fig.7).  This effect  was  predicted  in  [4]  for  the  weakly  non-linear regime.  Velocity  of  these  ion  perturbations  measured from the computer simulation was near to the ion-acous-tic wave’ velocity calculated for the simulation parame-ters. Electron concentration profile almost accurately re-peats the ion concentration profile (Fig.7).After the period of about two thousands of the mod-ulation  periods  the  ion  concentration  profile  becomes strongly indented in  the wide region around the local plasma resonance point (Fig.8). This result may be ex-plained by the ion-acoustic turbulence excitation.6. CONCLUSION1.  Interaction of the modulated electron beam with the inhomogeneous plasma has been studied by means of  computer  simulation  using  big-particles-in-cells method.2. The field in the local plasma resonance region ex-cited by the modulated electron beam results to the de-formation of the plasma concentration profile in this re-gion.  At  the  initial  time  points  the  density  cavity  is formed on the spot of LPRR.3.  Character of the plasma concentration profile on the spot of LPRR at the late time points depends strong-ly upon the relation of the temperatures of plasma elec-trons and ions. In the isothermic plasma (Те=Ті) the den-sity cavity is transformed into the plasma concentration jump. In the plasma with hot electrons the ion concen-tration profile becomes strongly indented in the wide re-gion around the spot of LPRR due to the ion-acoustic turbulence excitation.4. Quasi-periodical generation of the density cavities in the LPRR that was obtained in [2] due to homoge-neous external HF field was not observed in this simula-tion.REFERENCES1. L.M.Kovrizhnykh,  A.S.Sakharov.  Generation  of cavity  in  local  plasma resonance  region  //  Fizika plazmy. 1980, v. 6, №1, p. 150 - 158. (In Russian)2. V.B.Gildenburg,  G.M.Fraiman.  Deforma-tion  of plasma resonance region in strong electromagnetic field // ZhETF. 1975, v. 69, №5 (11), p.1601-1606. (In Russian).3. I.A.Anisimov,  D.G.Stefanovsky.  Electromagnetic fields excitation by the modulated electron stream in  the  weakly  inhomogeneous  isotropic  plasma // Ukr.  Fiz.  Zhurn.  1988,  v.  33,  №1,  p.38-40.  (In Russian).4. I.O.Anisimov,  O.A.Borisov.  Deformation  of  the concentration profile of the weakly inhomogeneous isotropic  plasma  due  to  the  modulated  electron beam //  VII Ukrainian Conf. on Controlled Fusion  and Plasma Physics. Abstracts. K. 1999, p.61. (In Ukrainian).5. I.O.Anisimov,  O.A.Borisov,  O.I.Kelnyk, S.V.Soroka.  Deformation  of  the  plasma concentration profile due to the modulated electron stream. // Int. Conf. and School on Plasma Physics  and  Controlled  Fusion.  Alushta,  Ukraine.  2002, Book of Absracts, p.153.6. Ch.K.Birdsall,  A.B.Langdon.  Plasma Physics,  via  Computer  Simulation.  McGraw-Hill  Book Company, 1985.7. I.O.Anisimov,  I.A.Blazhko,  T.V.Siversky.  Modi-fied PDP1 package for beam-plasma systems simu-lation. //  Proc. 2nd Int. Young Scientists Conf. on  Applied  Physics.  T.  Shevchenko  National  University of Kyiv, Faculty of Radiophysics. 2002, p.6-7.8. I.O.Anisimov, O.A.Borisov. Electrical field excita-tion in non-uniform plasma by the modulated elec-tron  beam.  // Physica  Scripta.  2000,  v.  62,  №5, p.375-380.9. N.S.Erokhin,  S.S.Moiseev.  Transformation  of waves  in  inhomogeneous  unstable  plasma.  // ZhETF.  1973,  v. 65,  №4,  p.1431-1447.  (In Russian).10. K.P.Shamrai.  Transformation and amplification of waves in a non-uniform plasma involving monoen-ergetic electron beam. //  J. Plasma Phys. 1984,  v. 31, №2, p.301-311.

Deformation of the plasma concentration profile due field of the modulated electron beam: numerical simulation

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Deformation of the plasma concentration profile due field of the modulated electron beam: numerical simulation

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