A form of weakly relativistic dielectric tensor valid for oblique propagation has been developed in previous works. Nevertheless this form was somehow cumbersome to deal with. In present work, authors start from those results to show that the weakly relativistic dielectric tensor can be written in a compact form that is consistent with the more elegant form of the non-relativistic dielectric tensor

Castejon, F.

Pavlov, S.S.

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

COMPACT FORM OF WEAKLY RELATIVISTIC DIELECTRIC TENSOR FOR UNIFORM MAXWELLIAN PLASMAF.Castejon* and S.S.Pavlov*** Laboratorio Nacional de Fusión, Asociación EURATOM-CIEMAT, 28040 Madrid, Spain;**Institute of Plasma Physics of NSC KhIPT, 61108, Kharkov, UkraineA form of weakly relativistic dielectric tensor valid for oblique propagation has been developed in previous works. Nevertheless this form was somehow cumbersome to deal with. In present work, authors start from those results to show that the weakly relativistic dielectric tensor can be written in a compact form that is consistent with the more elegant form of the non-relativistic dielectric tensor.PACS: 52.25.-b1. INTRODUCTIONIt is known that theoretical investigation of electron cyclotron wave propagation and absorption in magneto-active  plasma  requires  a  relativistic  treatment  for  the temperatures  that  are  usually  achieved  in  the  magnetic confinement devices [1]. This treatment is essential in the case  of  quasi-perpendicular  wave  propagation,  when relativistic effects are dominant. In this regime the non-relativistic  treatment  can  even  give  wrong  results, producing zero absorption in some cases in which it  is non-zero.There  exists  a  vast  literature  on  this  topic.  In  the original  paper  of  Trubnikov  [2]  there  was  introduced relativistic treatment but derived form of dielectric tensor was  too  hard  tractable.  Dnestrovskii,  Kostomarov  and Skrydlov  [3]  derived  the  tractable  form  of  tensor  for perpendicular  waves  propagation  on  the  base  weakly relativistic treatment,  in terms of so called Dnestrovsky dispersion  functions.  Shkarofsky  [4]  generalized  the weakly  relativistic  treatment  in  the  case  of  quasi-perpendicular  propagation  and  introduced  more  general Shkarofsky dispersion functions,  that  nevertheless,  were difficult  to  estimate.  The  method to  derive  the  weakly relativistic  tensor  for  arbitrary  waves  propagation  in numerically tractable form was suggested by Airroldi and Orefice  [5].  This  method  allows  one,  in  principle,  to obtain dielectric tensor as an expansion in the relativistic parameter 1/µ (µ=m0c2/Te,  m0 is the electron rest mass) and the finite Larmor radius parameter  λ=(k⊥ρe)2  (k⊥ is the  perpendicular  wave  number  and  ρe is  the  electron Larmor  radius).  However,  this  tensor  is  somewhat cumbersome to deal with. Krvenski and Orefice [6], using this  method,  received  dielectric  tensor  in  the  lowest significant  approximation  in  the  parameter   1/µ,  as  an expansion  in  the  parameter   λ.  This  form of  dielectric tensor  is  far  from  having  the  compact  form that  non-relativistic  tensor  expansions have,  because  authors did not pay attention in obtaining a final expressions as close as possible to the non-relativistic ones.In the present work authors, starting from results of the  work  [6],  show  that  the  lowest  significant approximation  in  the  parameter  1/µ of  the  weakly relativistic  dielectric  tensor  may  be  presented  in  the compact form that is consistent with one of the possible forms  for  non-relativistic  one.  On  this  way  there  were received two next approximations in the parameter  1/µ.2. MUTUALLY CONSISTENT FORM OF NON-RELATIVISTIC AND WEAKLY RELATIVISTIC DIELECTRIC TENSOR It  is  known  that  using  analytical  properties  of functions  that  appear  in  the  non-relativistic  dielectric tensor  (modified  Bessel  functions  and  non-relativistic plasma dispersion function)  it  is  possible  to  write  it  in different  forms.  The  choice  of  any  concrete  form  is defined  either  by  tendency  to  the  extremely  possible simplicity of final expression for tensor [7] or by some other considerations [8]. It is easy to verify that this tensor can be presented also in the following form:( ) ( ) ( )[ ]∑+ ∞− ∞=′′+′++=nnZZnSnZZnRnZZnPpZ ˆˆˆ201ˆˆωωε          (1)where  ω  is incident wave frequency and  pω  is plasma frequency, )( nZZ  is the plasma dispersion function of the argument  )//2/()( tVkcnZ ωω −= ,  cω  and  tV  are electron cyclotron frequency and thermal velocity and′−′′−=nAnAnAnnAinAinnAnnP0002/20/2ˆ λλλ′′−=02/2/2/002/00ˆλλλλnAinnAnAinnAnR=2/00000000nAnS)()exp( λλ nInA −= , 2)( ρλ ⊥= k , //k  and ⊥k  are parallel and perpendicular wave numbers.The main peculiarity of such presentation is total separation  of  perpendicular  and longitudinal  dispersion. The perpendicular dispersion is described by tensors  nPˆ , nRˆ ,  nSˆ  and the longitudinal one. Is described by plasma dispersion  function,  Z ,  and  its  first  and  second derivatives in the parameter nZ . It is easy to see that this 52                              Problems of Atomic Science and Technology. 2002. № 5. Series: Plasma Physics (8). P. 52-54additional  condition  totally  defines  single  form  of dielectric tensor from all possible ones. Now, let us show that this form of non-relativistic tensor obeys to a deeper sense,  from the  point  of  view of  consistency  with  the weakly  relativistic  tensor.  In  the  work  [6]  the  weakly relativistic  tensor  was  derived  in  the  lowest  significant approximation in the parameter 1/µ in the form:Dp ˆ21ˆˆ −=ωωµε                                                            (2)( ) ( ) ( )0,1,2!0 0211 =+−+++∞=∞== ∑ ∑ hksQksksakskss ksD λ ,( ) ( ) ( ) ( )0,1,2!,2112 =−−++++−=−= ∑ hksQksksakskskskssiDD λ ,( ) ( ) ( )0,1,2!,22 =+−+++= ∑ hksQksksbksksksD λ ,( ) ( ) ( )1,1,2!,3113 =−−+++== ∑ hksQksksakskskssDD λµ λ ,( ) ( ) ( ) ( )1,1,2!,3223 =+−++++=−= ∑ hksQksksaksksksksiDD λµ λ( ) ( ) ( )2,1,,2!33 =+−+++= ∑ hksQksksksaksksD λµ λ .Above were used next denotings( ) ( ) ( )[ ]( )( )[ ] ( ) !!22!221!21,kkskskskskksa++++−= , (here ( ) 0!1 =− n )( ) ( )2,1, −= kaksb ,   (for s=0)( ) ( ) ( ) ( )−+−+−−++−= 1,122,1,141, ksaksksksaksaksb ,   (for s≠0)( ) µωλ /20/ Ω⊥= N ,  cmeB 0/00 =Ω ,( ) ( ) ( )hksQhksQhksQ ,,, −±=± ,     (s≠0)( )hkQkQ ,0,0 =+ ,                        (s=0).   The tensor  Dˆ  is development in the parameter  λ . In this  development  Shkarofsky  functions  of  half  integer index appear, ),( azqF :( ) ( ) ( ) −−∞−−=== ∫ itatiztqitdtiazqFhksQ 12exp01),(0,  ,where  ( )ωωµ /01 csz −= ,  2/2//Na µ= ,  2/3++= ksq  (k=0,1,…). ( ) { }1//1, +−== qFqFNhksQ ,( ) [ ]{ }122//112, +−++++−== qFqFqFNqFhksQ µµ .Now,  let  us  change  argument  z in  Shkarofsky functions  into  ( )aznZ 2/= .  It  is  homotheticaly  correct, since Shkarofsky functions depend mono-valuably on this parameter for any finite value of longitudinal refractive index //N  and have derivatives for any real value of this parameter.  We  have  in  mind,  of  course,  only  those functions  ),( azqF  that appear in dielectric tensor, i.e. we consider  2/5≥q .  Obviously  )//2/()( tVkcnZ ωω −= .  The advantage  of  such  a  change  consists  of  the  fact  that Shkarofsky functions depend now on the same argument that the non-relativistic plasma dispersion function. It is not difficult to verify term by term that weakly relativistic tensor (2) can be written in the form:( ) ( ) ( )[ ]∑+ ∞− ∞=′′∗+′∗+∗−=nnZFnSnZFnRnZFnPp  ˆˆˆ21ˆˆωωµε        (3)where  ( )nZF  is  an  infinite  vector  containing  all Shkarofsky  functions  with  gradually  increasing  index starting from 2/5=q :( )nZF = ( ) ( ) ( )[ ],...2/9,2/7,2/5 nZFnZFnZF ,the sign ∗  denotes the scalar product of the infinite vector ( )nZF  with every left-side matrix element considered as a infinite vector resulting from finite Larmor radius (FLR) – expansion beginning from the term of zeroth order. So, for instance,( )[ ] ( )nZFnAnnZFnP  ∗=∗ λ/211ˆ ,where = ,.../2,/1,/0/ λλλλ nAnAnAnA. Here the superscript in λ/knA  denotes the order in FLR – expansion of λ/nA . The order in FLR-expansion in tensor  nRˆ ,  that is connected with elements of tensor  13ε and 23ε , must be considered after dividing these elements by λ .Thus  one  can  see  that  the  form  (1)  of  non-relativistic tensor is very close to the form (3) of weakly relativistic tensor. The perpendicular dispersions defined by tensors nPˆ , nRˆ , nSˆ  are the same in both cases and the longitudinal dispersions defined by analytical  properties of  non-relativistic  plasma  dispersion  function  and  the whole  family  of  Shkarofsky  functions  are  very  close, cause analytical properties of plasma dispersion function and every Shkarofsky function are very close, especially for oblique propagation.The  comparison  of  nonrelativistic  (1)  and  weakly relativistic  forms (3)  of  dielectric  tensor  shows directly that it is enough to take into account the dependence of relativistic  electron  mass  on  speed  to  obtain  a  split  of plasma dispersion function into the family of Shkarofsky functions  of  half-integer  order  2/5≥q .  In  the  weakly relativistic case the role of plasma dispersion function is played by the whole family of Shkarofsky functions and a single Shkarofsky function is directly connected with only one  corresponding  order  in  FLR-expansion  of perpendicular dispersion for the tensor elements 11ε , 12ε , 22ε ,  with  the  corresponding  and  previous  one  for  the elements 13ε , 23ε  trough its first derivative, and with the corresponding  and two previous  ones  for  elements  33ε  trough its second derivative.3. WEAKLY RELATIVISTIC DIELECTRIC TENSOR IN FURTHER ORDER APPROXIMATIONS IN PARAMETER µ .Since the weakly relativistic tensor can be presented in  a  very  close  form to  the  non-relativistic  one  in  the lowest significant order of the expansion in the parameter 53µ , one would expect that the next approximations can be written in a more or less compact form using the same argument  for  Shkarofsky  functions  and  tending  to  the form  of  non-relativistic  tensor  (1).  These  next approximations are necessary for numerics of EC waves in  burning  plasmas,  when  the  condition  µ >> λ  is fulfilled.Really,  using the method suggested by Airoldi and Orefice [5] and described in detail in a recent book by Brambila  [7],  the  next  to  lowest  approximation  in  the parameter µ  can be presented in the form:( ) ( ) ( ) ( )[ ]∑+ ∞− ∞=′′∗+′∗+∗−−=nnZFnSnZFnRnZFnPpKe1ˆ1ˆ1ˆ2221ˆωωµµµε  (4),where ( )µ2K is McDonald function and subscript 1 stands for  first  order  in  µ -approximation  of  dielectric  tensor. The  vector  1F is  easily  obtained  from  the  vector  of Shkarofsky  functions  F,  used  in  Equation  (3),  by multiplication by matrix Bˆ , that has comparatively simple structure:=...............00...0..................0200......010ˆnbbbB , ( ) 2283 ++= nnnb ,1F= Bˆ F.Coefficients nb  are defined by the expansion coefficients appearing  in  Airoldi  and  Orifice’s  method.  In  order  to derive  this  approximation,  we  have  neglected  the  even order derivatives of Shkarofsky functions, ( )kqF 2  (k≥1), that are small in comparison with the function itself, in the  tensor  elements  33,22,12,11 εεεε  and  the  odd  order derivatives,  ( )12 +kqF  (k≥1),  in  comparison  with  first derivatives,  ( )1qF ,  in  the  elements  13ε  and  23ε .  These approximations  are  right  for  any  value  of  longitudinal refractive  index  //N  in  ECR frequency range  and  it  is easily verified numerically. It is worthwhile to note that coefficients 1b  and 2b , corresponding  to first and second harmonics,  differ  of  ones  in  Brambilla’s  book  by  the constant  83 , which appears after taking into account all terms of order µ1 .In  this  way,  one  can  see  that  every  next approximation differs from previous one by the structure of the matrix Bˆ . In the lowest significant approximation, matrix  Bˆ  has units  in main diagonal,  i.e.  it  is  the unit matrix. In the next one it has also single nonzero terms diagonal, which is next right diagonal to the to the main one. In each more deep approximation matrix Bˆ  has also single  nonzero  diagonal,  again,  which  is  next  right diagonal to the previous one. The coefficients nb  for 2−µ- approximation are:( )++−++−+−= 32321211721542153223 nnnnnnnb( ) ( )22212++ nnnand for 3−µ -approximation( ) ( ) ( )+++−+−−= 102105821056221054242 nnnnnnnb( ) ( ) ( )+−−+−−+72154211532313211nnnnn( ) ( )++−++3232222212nnnn( ) ( ) ( )323213+++ nnnnObviously the coefficients  nb can be written for any  µ - approximation and with every step they become more  and  more  cumbersome,  but  the  coefficients presented  above  are  enough  for  laboratory  and  reactor plasmas.REFERENCES1. Fidone I.,  Granata G.,  and  Meyer R.L. Physics of Fluids 25 (1982) 2249.2. Trubnikov  B.A.  1959  Plasma  Physics  and  the Problem of Controlled Thermonuclear Reactions, (ed. M.A. Leontovich), vol.3, Pergamon.3. Dnestrovskii  Ya.N.,  Kostomarov  D.P.,  Skrydlov N.V., 1964, Soviet Phys. Tech. Phys., 8, 691.4. Shkarofsky I.P., 1966, Phys. Fluids, 9, 561.5. Airoldi A.C., Orefice A., 1982, J. Plasma Phys., 27, 515.6. Krivenski V. & Orefice A.,  1983, J.  Plasma Phys., 30, 125.7. Akhiezer A.I,  Akhiezer I.A., Polovin A.V.,  Sitenko A.G.,  Stepanov  K.N.  Plasma  Electrodynamics, Pergamon Press, Oxford, 1975.8. Brambilla  M.,  Kinetic  theory  of  plasma  waves, homogeneous  plasmas,  Oxford  University  Press, Oxford, 1998.54

Compact form of weakly relativistic dielectric tensor for uniform Maxwellian plasma

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Compact form of weakly relativistic dielectric tensor for uniform Maxwellian plasma

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