The self-consistent formation, observed in experiments, of the solitary barrier for plasma electrons and ions has been analytically described

Barchuk, S.V.

Lapshin, V.I.

Maslov, V.I.

Onishchenko, I.N.

Tretyakov, V.N.

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

ELECTRIC BARRIER FORMATION IN KIND OF CONNECTED DIP AND HUMP OF ELECTRIC POTENTIAL NEAR ECR POINTS.V.Barchuk*, V.I.Lapshin, V.I.Maslov, I.N.Onishchenko, V.N.Tretyakov*NSC Kharkov Institute of Physics and Technology, Kharkov 61108, Ukraine,E-mail: vmaslov@kipt.kharkov.ua;*Karazin Kharkov National University, Kharkov, 61108, UkraineThe self-consistent formation, observed in experiments, of the solitary barrier for plasma electrons and ions has been analytically described.PACS: 52.25.-bTHERMAL BARRIER FORMATION FOR PLASMA ELECTRONS IN ECR POINTIn [1] the formation of the thermal barrier for plasma particles was observed near the point of electron cyclotron resonance (ECR) in inhomogeneous magnetic field. In this paper two mechanisms of similar thermal electric barriers formation for electrons and ions of the plasma and plasma flow in ECR points on ends of the magnetized cylindrical trap are described (see fig. 1). Fig. 1. Scheme of the thermal electrical barriers formation for plasma electrons and ions in ECR points on ends of the magnetized cylindrical trapWe consider the case of inhomogeneous magnetic field. Namely the magnetic field is minimum in the center of the cylindrical trap and it grows to the ends of the cylindrical trap. Near ECR point the transversal electron velocity V⊥o is increased. At electron motion from the system they are reflected  from  the  magnetic  barrier  back.  Further  the electrons move inside the system. In the inhomogeneous magnetic  field  the  electron  transversal  velocity  V⊥(z) decreases  V⊥(z)=V⊥o(H(z)/Ho)1/2,  but  the  electron longitudinal  velocity  V||(z)  increases  V||(z)=V⊥o(1-H(z)/Ho)1/2 . It results in average electron velocity. Growth of electron velocity leads to that electrons in the area, in which they are penetrated, form noncompensated negative charge  δne.  According  to  Poisson  equation  it  is  the perturbation  of  the  dip  of  the  electric  potential,  from which the plasma electrons are reflected.  Growth of the electron longitudinal velocity with respect to the ions near ECR point provides nonequilibrium state. The reflection of the electrons with nonequilibrium distribution function from the dip of the electric potential leads to growth of the dip’s amplitude. So these current-carrying electrons excite the  electric  potential  dip  with  amplitude  ϕo on  an  ion mode with velocity Vc,  close to zero,  and are  reflected from  it.  From  Vlasov  equation  for  electrons  and hydrodynamic equations for ions one can derive evolution equation,  describing  this  dip.  Really  we  use  slow evolution  of  the  dip  for  its  description.  In  zero approximation,  taking  into  account  that  the  resonant electrons are reflected from the dip, one can derive from Vlasov equation the expression for  electron distribution function fe=foe[-(V2-2e(ϕ±∆ϕ)/me)1/2±V||], V><A(ϕ)sign(z)  (1)A(ϕ)=[2e(ϕo+ϕ)/me]1/2.We use the normalized values: φ≡eϕ/Te, N-≡no-/no+, Ne≡noe/no+,  Q±=q±/e,  Vs±=(Te/M±)1/2.  We  normalize  x  on Debye radius of electrons rde, V|| on Vthe, time t on plasma frequency  of  positive  ions  ωp+-1,  velocity  of  solitary perturbation Vc on ion-acoustic velocity (Te/M+)1/2. Te, is the  temperature  of  electrons,  no-,  no+ are  unperturbed densities of negative and positive ions, q± is the charge of positive and negative ions.Integrating (1), one can derive the electron density in first approximation on Vо ne≈noeexp(φ)[1-(2∆φ/√pi)∫oβdx exp(-x2)--2V||(2/pi)1/2∫oβdx (x2-φ)1/2exp(-x2)]  (2)Far from the dip the plasma is quasineutral ne(z)z→∞ =ne(z)z→-∞=1. From here one can derive,  using (2),  the expression for potential jump near the dip ∆φ=V||(2/pi)1/2(1-exp(-φo))/[1-(2/√pi)∫o√φo dx exp(-x2)] (3)From hydrodynamic equations for ions one can obtain for perturbations of densities of positive and negative ions ni±=n±NL+n±τ , n±NL=no±/[1-(±q±)2ϕ/M±Vc2]1/2,  (4)∂n±τ/∂z=±2(∂ϕ/∂t)(no±q±/M±Vc3)××[1-(±q±)ϕ/M±Vc2]/[1-(±q±)2ϕ/M±Vc2]3/2Substituting  (2),  (4)  in  Poisson  equation  one  can derive nonlinear evolution equation∂3zφ+{Q+2V2s+(1-2φQ+V2s+/Vc2)-3/2(1-φQ+V2s+/Vc2)+98                     Problems of Atomic Science and Technology. 2002. № 5. Series: Plasma Physics (8). P. 98-100   +Q-2N-V2s-(1+2φQ-V2s-/Vc2)-3/2(1+φQ-V2s-/Vc2)}2∂tφ/Vc3+(∂zφ/Vc2){Q+2V2s+(1-2φQ+V2s+/Vc2)-3/2++Q-2N-V2s-(1+2φQ-V2s-/Vc2)-3/2}-  (5)-{exp(φ)-sign(z)V||(2/pi)1/2{(φo/(φo+φ))1/2exp(-φo)--∫√-φ√φo dy(1-2y2)exp(-y2)/(y2+φ)1/2++(1-exp(-φo))[1-(2/√pi)∫o√φo dxexp(-x2)]-1××[exp(-φo)/(φo+φ)1/2+2(φo+φ)exp(-φo)++4∫√-φ√φo dy y(y2+φ)1/2exp(-y2)]/√pi}}∂zφ=0From nonlinear equation (5) one can show that the dip propagates with the slow velocity Vc≈0. From (5) one can get also the growth rate of the dip small amplitude γnl≈ωp+(V||/Vthe)3/2(q+/e)(n+/ne)1/2{1++[1/3-(ne/n+)(e/q+)](eϕo/Te)(pi/2)1/2(Vthe/2V||)} (6)One can see that  the dip is  formed at  ratio  of  electron current  to thermal velocity V||/Vthe larger than threshold. The threshold decreases at decreasing of ratio of electron and  positive  ion  densities  ne/n+ and  equal  zero  at ne/n+<q+/3e. The threshold is maximum at ne/n+=1. BARRIER FORMATION FOR PLASMA IONS IN KIND OF ELECTRIC POTENTIAL HUMP NEAR ECR POINTAs the  electrons  are  reflected  from the  dip  and  the flow  ions  pass  with  Vo+ through  the  dip  freely, noncompensated volume charge of ions is formed after the dip,  in  which  field  the  ions  are  slowing  down  and reflected.  This volume charge forms perturbation of  the electric potential hump. The ion flow enhances this hump of the electric potential. We describe the quasistationary properties  of  the  hump,  neglecting  the  nonequilibrium condition.  Taking  into  account  the  nonequilibrium condition leads to the hump’s excitation, in other words to growth of the hump’s amplitude.  Further we will show, that  the  electric  potential  hump is  almost  nonmobile  in space. In linear approximation the perturbation excitation by plasma  flow,  propagating  relative  to  negative  and motionless  positive  ions,  is  described  by  the  following dispersion ratio:1+1/(krde)2-ω2p+/(ω-kVo+)2-ω2p-/ω2-ω2pq/ω2=0. (7)Here  ω,  k  are  the  frequency  and  wavevector  of  the perturbation;  ωp± are  the  plasma  frequencies  of  the negative  and  flow’s  positive  ions;  ωpq is  the  plasma frequency of the positive motionless ions; rde is the Debye radius of electrons; Vo+ is the velocity of the positive ion flow.From (7)  one  can  see  that  one  can  select  the  flow velocity suchVph=ω/k≈(Vo+/24/3)[(n-m+q-2/ n+m-q+2)++(n+qq+q2/ n+q+2)]1/3<<Vs+,λ=2pi/k=2pirde/(Vs+2n+q+2/Vo+2nee2-1)1/2>>rde, (8)that the  perturbation  is  almost  motionless,  that  is Vph<<Vs+. Vs+=(T/m+)1/2 is the ion-acoustic velocity of the flow positive ions.  Here  n-,  m-,  q- (n+,  m+,  q+ )  are  the density, mass and charge of the negative (positive) ions.From  (7)  one  can  derive  the  growth  rate  of  the perturbation excitation: γ=(1.5)1/2(Vo+/rde)[(n-m+q-2/n+m-q+2)++(n+qq+q2/n+q+2)]1/3(Vs+2q+/Vo+2e-1)1/2. (9)On non-linear stage of the instability development the electric  potential  perturbation  ϕ represents  the  solitary hump of finite amplitude ϕo. The distribution function of nontrapped electrons, that is arranged outside the separatrix, looks like:fe(v)=[noe/Vte(2pi)1/2]exp(eϕ/Te-mev2/2Te).  (10)For  trapped  electrons,  that  is  for  electrons  located inside  the  separatrix,  the  distribution  function  does  not depend  on  energy  because  of  an  adiabaticity  of  the evolution.Integrating  the  electron  distribution  function  on velocity,  we  receive  following  expression  for  electron density:ne=(no/(2pi)1/2)(2/T)3/2∫∞odε(ε+eϕ)1/2exp(-ε/T).  (11)From hydrodynamic equations  for  positive ions it  is possible to receive following expression for their density:n+=no+/[1-2q+ϕ/m+(Vo+-Vh)2]1/2. (12)Here Vh is the velocity of the solitary perturbation. As a result from (11),  (12) and  Poisson equation  we have  equation for  the spatial  distribution of the electric potential of the perturbation of any amplitude:φ’’=(2/√pi)∫ ∞odae-a(a+φ)1/2-1/(1-2Qφ/voh2)1/2.  (13)Q=q+/e, φ=eϕ/T, «’»=∂/∂x, x=z/rde, voh=(Vo+-Vh)/Vs+. From the condition  φ’|φ=φo=0 and (13)  we obtain  the hump velocity, voh:voh2/Q=(A-2)2/2(A-2-φo),A=(8/3√pi)∫ ∞odae-a(a+φ)3/2. (14)In the approximation of small amplitudes from (13), (14) we receive:voh2≈ Q,  L≈[15√pi/4(1-1/√2)]1/2φo-1/4.  (15)If  Vo+ close  to  (q+/e)1/2Vs+,  then  the  perturbation  is approximately motionless. Taking into account the small densities negative and motionless  positive  ions,  we  obtain  from  the  Poisson equation the evolution equation:2ω2p+∂3ϕ/∂t3/(Vo+-Vh)3=-(ω2p-+ω2pq)∂3ϕ/∂z3. (16)99From  (16)  the  growth  rate  of  the  non-linear perturbation amplitude is followed:γNL≈ωp+(eϕo/T)1/2[(no-m+q2-/no+m+q2+)++(noqq2+q/no+q2+)]1/3 .  (17)ELECTRON MECHANISM OF BARRIER FORMATION FOR PLASMA IONS NEAR ECR POINTLet us consider the mechanism of the electric potential hump formation by plasma electrons near the dip of the electric potential.The  potential  jump accelerates ions.  Hence on first front of the electric potential dip the ion density becomes smaller. If into this region the electron flow penetrate with electron velocity only a little smaller than electron thermal velocity, hence on the first front of the potential dip the electron drift velocity becomes more than electron thermal velocity due  to  flow continuity law.  Due  to  Bunemann mechanism interaction of electron flow with this region an electric potential hump is excited.Let  us  describe  a  solitary  perturbation  in  type  of electric potential hump. We will show that it represents a nonlinear perturbation on a slow electron-sound mode. As it  is  slow,  resonant  electrons  can  be  trapped  by  such perturbation. From Vlasov equation the expression for perturbation of  electron  distribution  function  follows.  Integrating which on  velocities  in  case  of  small  amplitudes  of  the solitary perturbation ϕo one can derive the expression for perturbation of electron density δn'=∂tϕ[y+(1-2y2)(1-R(y))/y]++ϕ'R(y)+ϕϕ'[1-y2+(3/2-y2)(R(y)-1)]  (18)R(y)=1+(y/√pi)∫-∞∞dtexp(-t2)/(t-y), y=Vh/Vth√2Here  point  means  derivation  on  time,  and  prime  is  a spatial  derivation.  Vh,  ϕ are  velocity  and  potential  of soliton.  φ=eϕ/Te.  Substituting (18)  in  Poisson  equation, one can derive an equation, describing spatial distribution of potential:(φ')2 =φ2R(y)-[1+(2y2-3)R(y)]φ3/6 (19)Let us determine approximately the soliton width from (19):  ∆x=(48Te/ϕo)1/2.  The  soliton  width  decreases  with amplitude growth. In  case  of  large  amplitudes,  eϕo/Te>1,  from Vlasov equation we have the expression for electron distribution function  f=fo[(u2-2eϕ/m)1/2+Vhsign(u)]  for  u=V-Vh>(2eϕ/m)1/2. Here fo is Maxwell distribution function. Thus we obtain the equation for the soliton shape(φ')2==-φ+(2/√pi)1/2∫-∞∞dt(t-y)2exp(-t2){[1+φ/(y-t)2]1/2-1} (20)From (20) ∆x=[2eϕo/Te(√2-1)]1/2 (21)we  conclude  that  the  soliton  width  grows  with  ϕo. Therefore, it is necessary to take into account electrons, trapped by the soliton field. Assuming distribution of their density as ntr(x)=n2exp[eϕ(x)/Ttr],  we derive similarly to (21),  that  width  and  velocity  of  the  soliton  grow with amplitude  growth  (in  difference  from  case  of  small amplitudes of the solitary perturbation).Thus,  at  a  neglect  of  ion  mobility  this  solitary perturbation is  stationary and electron one.  However at taking  into  account  of  ion  mobility  it  is  necessary  to expect  occurrence  of  slow growth of  the  perturbation’s amplitude,  as  a  result  of  Bunemann  instability development.  In  the  following  order  of  the  theory  of disturbances from (18) one can derive the correction of the next order to a spatial derivative from electron density ne1'=∂tϕ[y+(1-2y2)(1-R(y))/y] (22)This expression, as follows from a spatial derivative from Poisson equation, should be equal to a spatial derivative from ion density perturbation ni'. ni' is possible to find in linear approximation from ion hydrodynamic equations ∂2tni=(me/mi)φ'' (23)Equating the second time derivative from (10) and first spatial derivative from (23), we obtain ∂3tφ=(6me/mi)φ''' (24)The solution of (24) we search asφ(x,t)=φo(t)η[x-∫-∞t dt1δvo(φo(t1))] , (25)η(x)  is  quasistationary  shape  of  the  perturbation, assuming, that ∂tφo(t)=γφo(t). In (25) the change of soliton velocity  with  change  of  its  amplitude  is  taken  into account.Substituting ∂tφ through γφ-δvhφ’, we obtain from (24)γ≈(me/mi)1/3φo1/2 REFERENCES1. T.Kaneko, R.Hatakeyama, N.Sato. //  IEEE Trans. on  Plasma Science. 2000. v. 28, p. 1747.100

Electric barrier formation in kind of connected dip and hump of electric potential near ECR point

http://dspace.nbuv.gov.ua/bitstream/handle/123456789/78923/31-Barchuk.pdf?sequence=1

Electric barrier formation in kind of connected dip and hump of electric potential near ECR point

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