Waves and Oscillations in Plasmas

cooperative phenomena and shock waves in collisionless plasmas and surface flute waves in plasmas theory and applications, propagation of electromagnetic waves in plasmas
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Dr.LeonBurns,New Zealand,Researcher
Published Date:21-07-2017
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Waves  in  plasmas T.  Johnson 2/8/16 Dispersive  Media,  Lecture  5  -­Thomas  Johnson 1Outline • Introduction  to  plasma  physics – Magneto-­Hydro  Dynamics,  MHD • Plasmas  without  magnetic  fields – Cold  plasmas • Transverse  waves  –plasma  modified  light  waves • Longitudinal  plasma  oscillations – Warm  plasmas  –longitudinal  waves • Debye  shielding  – incomplete  screening  at  finite  temperature • Langmuir  waves • Ion-­acoustic  (ion-­sound)  waves • Alfvénwaves  -­low  frequency   waves  in  magnetised  plasmas – Shear  and  fast  Alfvén waves • Magnetoionic waves – Wave  resonances  &  cut-­offs – CMA  diagram 2/8/16 Dispersive  Media,  Lecture  5  -­Thomas  Johnson 2Introduction  of  plasma • Plasmas  are  ionised  gases • High  temperature  and  low  concentration;;  use  Newtonian  mechanics 𝑚𝐯̇ =𝑞 𝐄+𝐯×𝐁 • High  conductivity – difficult  to  produce  charge  separation  – plasmas  are  quasineutral;; – except  at  high  frequency,  𝜔𝜔 ,  electrons  too  heavy  to  react , - • Magnetic  fields:  cause  particles  to  follow  gyro  orbits – 𝐯×𝐁force  cause  particles  to  gyrate  around   the  magnetic  field  lines. – Gyro  frequency:  Ω= 𝑞𝐵/𝑚 – Gyro  radius:  ρ=𝑣/Ω=𝑚𝑣/𝑞𝐵 – Plasmas  can  be  very  anisotropic,   in  fusion  plasmas:   𝜎 ∥ 9 10 𝜎 5 i.e.  it  is  almost  impossible  to  conduct   currentsperpendicular  to  𝐵 2/8/16 Dispersive  Media,  Lecture  5  -­Thomas  Johnson 3Plasma  models • There  are  many  mathematical  plasma  models • We  have  the  cold  plasma models,  i.e.  without  thermal  motion – Magneto-­ionic  theory  (inc. electron  response,  while  ions  are  static) – Cold  plasma  –inc. both  electron  and  ion  responses • And  the  general  warm  plasma model  from  lecture  5. – This  one  can  also  be  generalised  to  include  magnetised  plasmas. • Many  of  these  models  are  too  complicated  to  analyse  and  solve – Almost  all  practical  solutions  involve  further  approximations – Typically:  expand  for  a  specific  range  of  frequencies,  wave  lengths,  or   phase  velocity – Example:   𝜔≪𝜔 ,  𝜔 ≪Ω ,  𝑉 𝑘≪ Ω … , ; ; = ; These  are  the  main  conditions  for  the  so  called   magneto-­hydro  dynamics model 2/8/16 Dispersive  Media,  Lecture  5  -­Thomas  Johnson 4The  MHD  model  for  a  plasma • Magneto-­Hydro  Dynamics  (MHD),  the  most  famous  plasma  model • It  assumes  low  frequencies  compared  to  the  ion  cyclotron   frequency,  𝜔 ≪Ω and  the  plasma  frequency,  𝜔 ≪ 𝜔 . ; , - • Assumption:  electrons  have  an  infinite  small  mass: 𝐄+𝐯×𝐁= 𝜂𝐉 This  equation  is  referred  to  as  Ohms  law and  𝜂 is  the  resistivity. – In  the  rest  frame,  𝐯=0,  we  have  the  conventional  Ohms  law:  𝐄= 𝜂𝐉 – In  many  plasma  application  resistivity  can  be  neglected 𝐄+𝐯×𝐁=0 • Thus,  there  is  no  parallel  electric  field,  𝐄 = 0 ∥ • When  in  the  rest  frame  without  resistivity,  𝐄 =0 • The  mass  continuity  equation for  the  mass  density  𝜌 : C 𝜕𝜌 C +𝛻 G(𝐯𝜌 ) =0 C 𝜕𝑡 • …but  there  are  more  equations 2/8/16 Dispersive  Media,  Lecture  5  -­Thomas  Johnson 5Maxwell’s  equations  for  MHD • The  plasma  moves  with  velocity  𝐯,  momentum  conservation: 𝜕𝐯 𝜌 + 𝐯G𝛻𝐯 = 𝐉×𝐁−𝛻𝑝 C 𝜕𝑡 – Note:  the  term  𝐯G𝛻𝐯is  non-­linear  in  𝐯,  thus  it’s  negligible  for  small   amplitude  waves MN • Adiabatic  pressure:    𝑝𝑛 =const ,  where  𝛾 =  adiabatic  index – Apply  gradient:      𝑛𝛻𝑝 = 𝛾𝑝𝛻𝑛 – Linearize  for  homogeneous  plasmas:    𝑛 𝛻𝑝 =𝛾𝑝 𝛻𝑛 U V U V – Defined  equilibrium  temperature,  𝑇 ,  such  that  𝑝 =𝑛 𝑇 : U U U U 𝛻𝑝 = 𝛾𝑇 𝛻𝑛 V U U • The  low  frequency  assumption  in  MHD  simplify  Maxwell’s  equations – Charge  separation  is  impossible 𝛻G𝐉= 0  &  𝛻G𝐄=0 – Slow  events  means  that  the  phase  velocity  is  smaller  than  𝑐: 𝛻×𝐁=𝜇 𝐉 U 2/8/16 Dispersive  Media,  Lecture  5  -­Thomas  Johnson 6Maxwell’s  equations  for  MHD MHD  equations 𝜕𝜌 C 𝛻×𝐁= 𝜇 𝐉 𝛻G𝐁=0 +𝛻G(𝐯𝜌 ) =0 U C 𝜕𝑡 𝜕𝐯 𝜕𝐁 𝛻G𝐄= 0 𝜌 + 𝐯G𝛻𝐯 =𝐉×𝐁−𝛻𝑝 𝛻×𝐄= − C 𝜕𝑡 𝜕𝑡 MN 𝑝𝑛 =const 𝐄+𝐯×𝐁= 𝜂𝐉 𝛻 G𝐉 =0 • The  MHD  equations  then  includes  4  vector  equations  and  5  scalar   equations;;  altogether  17  coupled  differential  equations • Non-­linear  equations  – linearisation often  required • MHD  equations  are  simplified  by  elimination  of  variable   – E.g.  eliminate  𝐉using  Ampere’s  law  and  𝐄 from  Ohm’s  law • Common  simplification,  resistivity  𝜂 =0 ,  known  as  ideal  MHD \ • The  plasma  drift  perpendicular  to  the  field  lines:  𝐯 = 𝐄×𝐁/𝐵 5 2/8/16 Dispersive  Media,  Lecture  5  -­Thomas  Johnson 7Transverse  waves  -­ Modified  light  waves • The  waves  equation  in  an  unmagnetised plasmas,  when  𝐤= 𝑘𝐞 _ \ 𝜔 , 𝐾 = 𝐾 𝜔 𝛿 = 1− 𝛿 ;a ;a ;a \ 𝜔 \ 𝐾 𝜔 −𝑛 0 0 𝐸 0 V \ \ 𝐸 𝑛 𝜅 𝜅 −𝛿 +𝐾 𝐸 = = 0 0 𝐾 𝜔 −𝑛 0 \ ; a ;a ;a a 𝐸 0 0 0 𝐾 𝜔 e \ \ • Dispersion  equation:   𝐾 𝜔 −𝑛 𝐾 𝜔 = 0 \ \ \ 𝜔 𝑐 𝑘 , – Transverse  waves \ \ \ \ 𝜔 −𝜔 −𝑐 𝑘 =0 1− − =0 , \ \ 𝜔 𝜔 \ \ \ \ • Dispersion  equation:    𝜔 =𝑐 𝑘 +𝜔 , • These  waves  are  very  weakly  damped;;   – Phase  velocity: thus  no resonant  particles  and  thus  no  Landau  damping – damping  can  be  obtained  from  collisions;; for  “collision  frequency”  =  ν the  energy  decay  rate  is: e 2/8/16 Dispersive  Media,  Lecture  5  -­Thomas  Johnson 8Plasma  oscillations • Plasma  oscillations:  “the  linear  reaction  of  cold  and   unmagnetised electrons  to  electrostatic  perturbations” – “Cold  electrons”  =  the  temperature  is  negligible. \ \ • Dispersion  equation  (previous  page):       𝐾 𝜔 −𝑛 𝐾 𝜔 =0 • Longitudinal part  of  dispersion  eq.:    𝐾 𝜔 = 0 \ 𝜔 , \ \ 1− =0 →𝜔 =𝜔 , \ 𝜔 • Note: Thus,  plasma  oscillation  is  not  a  wave since  no  information  is   propagated  by  the  oscillation • However,  if  we  let  the  electrons  have  a  finite  temperature  the   plasma  oscillations  are  turned  into  Langmuir  waves 2/8/16 Dispersive  Media,  Lecture  5  -­Thomas  Johnson 9Physics  of  plasma  oscillations • Model  equations: – Electrostatic  perturbations  follow  Poisson’s  equation where                                                    is  the  charge  density. – Electron  response – Ion  response;;  ions  are  heavy  and  do  not  have  time  to  move: – Charge  continuity 2/8/16 Dispersive  Media,  Lecture  5  -­Thomas  Johnson 10Plasma  oscillations • Consider  small  oscillations  near  a  static  equilibrium: Non-­linear (small  term) – where  all  the  small  quantities  have  sub-­index  1. • Next  Fourier  transform  in  time  and  space ω is  the  plasma  frequency   pe (see  previous  lecture) • Dispersion  relation:   2/8/16 Dispersive  Media,  Lecture  5  -­Thomas  Johnson 11Debye  screening • Debye  screening  is  a  static  electron  response  to  electrostatic   perturbations • Principle:   – Electrons  tries  to  screen  electrons  field – But  due  to  thermal  motion,  fields  that  are  static  in  rest  frame  are  not   static  in  the  frame  of  a  moving  electron – A  moving  electron  reacts  only  slow  changes,  ω 𝜔 , - – A  static  perturbation  will  in  a  moving  frame  appear  as:  𝜔 =𝑘𝑣 – Electrons  moving  with  the  thermal  speed  𝑉 will  only  screen  𝑘 𝜔 /𝑉 = , - = 𝑒𝜙 V Force  balance: 0= −𝑞 𝑛 𝛻𝜙 −𝑇𝛻𝑛 →𝑛 =𝑛 - -U V -,V -V -U 𝑇 𝑒 𝑒 𝑒 →𝜙 = 𝑛 = 𝑛 𝜙 Poisson’s  eq.:    𝜀 𝛻G𝛻𝜙 = −𝑒𝑛 V -,V -U V U V -,V \ \ 𝜀 𝑘 𝜀 𝑘 𝑇 U U \ \ \ 𝑛 𝑒 𝜔 𝜔 -U , - , - \ →𝑘 = = = \ 𝜀 𝑇 𝑚 𝑇 𝑉 U - = \ Define  the  Debye  length:    𝜆 ≔1/𝑘=𝜔 /𝑉 =𝑛 𝑒 /𝜀 𝑇 j , - = -U U 2/8/16 Dispersive  Media,  Lecture  5  -­Thomas  Johnson 12Langmuir  waves • At  finite  temperature  plasma  oscillations  turn  into  waves • Assume  the  plasma  is  isothermal,  i.e.  the  temperature  constant 𝛻𝑝 𝑡,𝑥 = 𝛾𝑇 𝛻𝑛 𝑡,𝑥 - E.g.  in  collisionless plasmas. • The  equation  of  motion,  of  momentum  continuity  eq.,  then  reads 𝜕𝐯 - 𝑛 𝑚 = −𝑞 𝑛 𝛻𝜙−𝛾𝑇 𝛻𝑛 - - - -U - - 𝜕𝑡 • Linearise: 𝜕𝐯 -V 𝑛 𝑚 = −𝑞 𝑛 𝛻𝜙 −𝛾𝑇 𝛻𝑛 -U - - -U V - -V 𝜕𝑡 • Divergence: 𝜕 𝑚 𝑛 𝛁G𝐯 =−𝑞 𝑛 𝛁G𝛻𝜙 −𝛾𝑇 𝛁G𝛻𝑛 - -U -V - -U V - -V 𝜕𝑡 Charge  continuity Poisson’s  equation 𝜕 𝜕𝑛 𝑞 -V - 𝑚 =−𝑞 𝑛 𝑛 −𝛾𝑇 𝛁G𝛻𝑛 - - -U -V - -V 𝜕𝑡 𝜕𝑡 𝜀 U \ \ \ \ \ 𝑉 =𝑇 /𝑚 is   Dispersion  equation:   𝜔 −𝜔 +𝛾𝑉 𝑘 = 0 - - = , - = the  thermal  speed \ \ \ \ Dispersion  relation    :      𝜔 =𝜔 −𝛾𝑉 𝑘 , - = 2/8/16 Dispersive  Media,  Lecture  5  -­Thomas  Johnson 13Langmuir  waves • Derivation  of  Langmuir  waves  from  warm  plasma  tensor,  𝐤 =𝑘𝐞 _ 𝜆 =𝑣 /𝜔 j; =,; , ; 𝜔 𝑦 = ; 2𝑘𝑣 =,; 𝑣 = 𝑇/𝑚 =,; ; ; • Langmuir  wave,  the  longitudinal  solution: • Neglect  ions  response  and  expand  in  small  thermal  electron   velocity  (almost  cold  electrons);;  use  expansion  in  Eq.  (10.30) \ v \ v 𝑘 𝑣 3𝑘 𝑣 1 3 =,- =,- 𝜙 𝑦 =1+ + +⋯ = 1+ + +⋯ \ v \ v 2𝑦 4𝑦 𝜔 𝜔 Letting  v =0 give   the plasma  oscillations 2/8/16 Dispersive  Media,  Lecture  5  -­Thomas  Johnson 14Polarization  and  damping  of  Langmuir  waves • Polarization  vector  e can  be  obtained  from  wave  equation  when   i inserting  the  dispersion  relation • Thus,  the  wave  damping  can  be  written  as where 2/8/16 Dispersive  Media,  Lecture  5  -­Thomas  Johnson 15Absorption  of  Langmuir  waves • Inserting  the  dispersion  relation  and  the  expression  for  K gives   L the  energy  dissipation  rate – Damping  (dissipation)  is  due  to  Landau  damping,  i.e.  for  electrons   with  velocities  vsuch  that   – Here  N is  proportional  to  the  number  of  Landau  resonant  electrons res • Damping  is  small  for  small  &  large  thermal  velocities – Maximum  in  damping  is  when   2/8/16 Dispersive  Media,  Lecture  5  -­Thomas  Johnson 16Ion  acoustic  waves • In  addition  to  the  Langmuir  waves  there  is  another  important   longitudinal  plasma  wave  (i.e.  K =0)  called  the  ion  acoustic  wave. L • This  mode  require  motion  of  both  ions  and  electrons.  Assume: – Fast  electrons:  v ω/k,  expansions  (10.29)   the – Slow  ions:  v ω/k,  expansions  (10.30) thi electron ion – Here  v is  the  sounds  speed:    𝑣 = 𝑇/𝑚 s x ; – Again,  N is  proportional  to  the  number  of  Landau  resonant  electrons res • Ion  acoustic  waves  reduces  to  normal  sounds  waves  for  small  kλ De 2/8/16 Dispersive  Media,  Lecture  5  -­Thomas  Johnson 17Physics  of  ion-­acoustic  waves • Assume  a  hydrogen  plasma:  𝑞 =−𝑞 =𝑒 ; - • Electrostatic  wave;;  we  need  Poisson’s  eq.:    −𝜀 𝛻G𝛻𝜙 =𝑒𝑛 −𝑒𝑛 U ; - • For  electrons  ion-­acoustic  waves  have  low  frequency,  𝜔≪ 𝜔 ;;   , - neglect  electron  mass.   𝑒𝜙 V 0=−𝑞 𝑛 𝛻𝜙−𝑇𝛻𝑛 → 𝑛 = 𝑛 - -U - -V -U 𝑇 \ MV • Poission eq.:       −𝜀 𝛻G𝛻𝜙 +𝑒 𝑛 𝑇 𝜙 =𝑒𝑛 U V -U V ;,V 𝑒𝑛 ;,V \ M\ 𝑘 +𝜆 𝜙 = V j 𝜀 U • For  ions,  ion-­acoustic  waves  have  low  frequency, 𝜔 ≪𝜔 ;; , ; ̇ 𝑚 𝑛 𝐯 =−𝑒𝛻𝜙−𝛾 𝑇𝛻𝑛 Not  included  on  previous  slide ; ; ; ; ; • Divergence  of  ion  eq.  of  motion;;  ion  mass  continuity  &  Poission eq. 𝜕 Ion-­acoustic  waves 𝑚 𝑛 𝛻G𝐯 = −𝑒𝑛 𝛻G𝛻𝜙 −𝑇𝛻G𝛻𝑛 ; ;,U ;,V ;,U V ;,V 𝜕𝑡 \ \ \ \ 𝑉 𝑘 𝑒 𝑛 𝑘 𝑛 x ;,U ;,V \ \ \ \ \ 𝜔 = +𝛾 𝑉 𝑘 −𝜔 𝑚 𝑛 = − −𝛾𝑇𝑘 𝑛 ; x ; ;,V ;,V \ \ M\ \ 𝜆 𝑘 +1 𝜀 𝑘 +𝜆 U j j 2/9/16 Dispersive  Media,  Lecture  5  -­Thomas  Johnson 18Outline • Introduction  to  plasma  physics – Magneto-­Hydro  Dynamics,  MHD • Plasmas  without  magnetic  fields – Cold  plasmas • Transverse  waves  –plasma  modified  light  waves • Longitudinal  plasma  oscillations – Warm  plasmas  –longitudinal  waves • Debye  shilding – incomplete  screening  at  finite  temperature • Langmuir  waves • Ion-­acoustic  (ion-­sound)  waves • Alfvénwaves  -­low  frequency   waves  in  magnetised  plasmas – Shear  and  fast  Alfvén waves • Magnetoionic waves – Wave  resonances  &  cut-­offs – CMA  diagram 2/8/16 Dispersive  Media,  Lecture  5  -­Thomas  Johnson 19Alfven  waves  (1) • Next:  Low  frequency  waves  in  a  cold  magnetised  plasma  including   both  ions  and  electrons • These  waves  were  first  studied  by  Hannes  Alfvén,  here  at  KTH  in   1940.  The  wave  he  discovered  is  now  called  the  Alfvén  wave. • To  study  these  waves  we  choose: B e and  k = ( k , 0 , k ) z x • The  dielectric  tensor  for  these  waves  were  derived  in  the  previous   lecture  assuming                                             V =  ”Alfvén  speed” A 2/8/16 Dispersive  Media,  Lecture  5  -­Thomas  Johnson 20

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