CHAPTER 6: Vlasov Equation Part 1 6.1 Introduction Distribution Function, Kinetic Equation, and Kinetic Theory : The distribution function ft (xv , , ) gives the particle density of a certitain speci es i n th th6die 6-dimensi onal ph ase space ofdtf xv and at time t. Thus, ftdxdv (xv , , )33 is the total number of particles in the differen- tial volume dxdv33 at point (xv , ) and time t . A kinetic equation describes the time evolution of f (xv , ,t ). The kinetic theories in Chs. 3-5 derive various forms of kinetic equations. In most cases,,,p however, the plasma behavior can be described by an approximate kinetic equation, called the Vlasov equation, which simply neglects the complications caused by collisions. By ignoring collisions , we may star t out without the knowledge of Chs. 3-5 and proceed directly to the derivation of the Vlasov equation

(also called the collisionless Boltzmann equation). 1

6.1 Introduction (continued) The Vlasov Equation : As shown in Sec. 2.9, a collision can result in an abrupt change of two colliding pa rticles' velocities and their instant escape from a small element in the xv - space, which contains the colliding particles before the collision. However, if collisions are neglected(validonatimescaled (valid on a time scale the collision time, see Sec. 1.6), particles in an element at position A in the xv- space will wander in continuous curves to v position B ( see fig fiure) . Th us, th e t ot al numbithber in the B element is conserved, and ft (xv , , ) obeys an equation A of continuity, which takes the form (see next page): x   t ft (xv , , )xv,  [ ft ( xv , , )( x , v )]  0 (1)     In (1), xv, [ ( , , , , , )] is a 6 - dimensional xyzvxyz  v  v

divergence operator and (x , v ) [ (xyzv  , , , x , v yz,)]v can be regarded as a 6-dimensional "velocity" vector in the xv - space. 2 6.1 Introduction (continued)   To show that t ft (xv , , )xv,  [ ft ( xv , , )( x , v )]  0 [(1)] implies conservation of particles, we integrate it over v ds6 an arbitrary volume VS66 enclosed by surface V6 S6 in the xv - space: x d 3 333 ftdx (xv , , ) dvxv, [(,,)(,)] fxv t x v dxdv  0 dt VV66  Nt() ft(,,)(,)xv x v d s6 s6 (by 6-dimensional divergence theorem) d  Nt ( ) f (x ,vx,td )( , vs )6 0, (2) dt s6 where Nt ( ) is the total number of particles in V6 , f (xv , , t )( x , v ) is the

6-dimensional "particle flux" in the xv - space, ds6 is a 6-dimensional

differential surface area of SS66 , with a direction normal to pointing outward. Thus, (2) states that the rate of increase (decrease) of the

total number of particles in V6 equals the partcle flux into (out of) V6 . 3

6.1 Introduction (continued)   Rewrite: t ft (xv , , )xv,  [ ft ( xv , , )( x , v )]  0 (1) In (1), xv,  [[(ft (xv,, )( x, v )]      (fx ) ( fy ) ( fz ) ( fv xyz )  ( fv  )  ( fv  ) xyzvxyz  v  v  ()ffxv    () (6.1) xv  xv ffffxxvvv+v  v+v 

where, because v is an independent variable, we have x v= 0 (3) q 1 In general, v vvEvB does not vanish. But for mc (   ),

v EE 0 [ does not depend on v ] we have  vvv()(vB B  v )( v B )000    q 1 which gives v v= m v (EvBc  ) 0 (4) q 1 Thus, xv, [(, [ffff (,xv ,)(,)], t)( x , v )]  v xff +mc ( E  v  B ) v  q 1 and (1) becomes t ffv+EvBxvmc (   )  f 0, (6.5) which is the Vlasov equation. 4 6.1 Introduction (continued) Physical Interpretation of the Vlasov Equation :  q 1 The Valsov equation t ffv+EvBxvmc (()0   )  f 0 (6.5) can be written q 1  vEvB mc ( ) dddft()(xv,,)0 fxv f f 0, (5) dtt dtxv dt df where is the total time derivative of f . The total time derivative dt (also called a convective derivative) fo llows the orbi t* of a particle in the xv - space. It evaluates the variation of f due to the change of the particle position in the xv - space as well as the explicit time variation of f . Thus , (6 .5) or (5) can be in terpreted as: Along a particle'sorbit s orbit in the xv - space, the particle density ft ( xv , , ) remains unchanged. *In the 6-dimensional xv - space, particles at a point ( xv , ) have the same velociy v. Hence, the orbit of a particle is also the orbitofant of an infinistesimal element containing the particle. In the 3-dimensional xx-space, by contrast, particles at a point have a range of velocities. 5

6.1 Introduction (continued) This interpretation is consistent with the fact that the sum of

(3) and (4) gives xv, (xvx, vxv= ))0  x x  v v=  0, (6) i.e. the 6-dimensional divergence of the 6-dimensional "velocity" (x , v ) vanishes. This implies that a collisionless v plliiasma is incompressibl iblHthle. Hence, the volume of B an elements (thus the particle density f ) will be A unchanged as it moves from AB to . x A specific example : Consider the simple case of a group of particles initially located in

a sqqp(,uare in the x -vAx space (area , lower figg)ure). If the particles are force free, those on the upper edge will move at the fastest (equal and

constant) speed, while those on the lower edge vx move at the slowest (equal and constant) speed. B Then, some time later, the square will become a A x parallelogram of the same area (area B , lower figure). 6 6.1 Introduction (continued) Effect of collisions : If there are collisions, they will cause a variation of f at the symbolic rate of ( f ) , which should be added to (6.5) to t coll give  q 1  t ffvEvBx mc (   ) v ff(),t coll  while the spp()pecfic form of ()t f coll depends on interparticle forces.  Throughout this course, the (t f )coll term will be neglected.

7

6.1 Introduction (continued) Complete Set of Equations : We now have the following set of self- consistent, coupled particle and field equations:   q 1 t ffvEvBmc(   ) v f   0 (6.5) B 0 (7)  For simplicity, we shall  E 4 (8)  henthforth denote x by .  1  EB  c t (9)  BEJ 1   4 (10)  cct ()(xxv,tftdt)(  q ftd (,, )3v (11)   where   3 Jx( ,tqftdv )   ( xv , , ) v (12)   Each particle species, denoted by the subscript " ", is governed by a separate Valsov equation, and q carries the sign of the charge. 8 6.2 Equilibrium Solutions General Form of Equilibrium Solutions : As shown in (5), the Vlasov equation can be written as a total time derivative: dddft (xv , , ) fxv f f 0, (5) dtt dt dt v where d follows the orbit of a particle whose position and velocity dt at time t is xv and , respectively. Thus, any fuction of constants of the motion along the orbit of the particle, CCii (xv , , t ), is a solution of d f dCi the Vlasov equation, i.e. fCC (12 , ...) 0, dti Ci dt because,,y by the definition of constant of the motion, dC i  CCddxv C 0. dtt ii dt dt v i The equilibrium solution ((yp)denoted by subscript "0") of interest to us is a steady-state solution formed of constants of the motion that do

not depend explicitly on tffCC , i.e. 0012 ( , ...) with CCii (xv , ). 9

6.2 Equilibrium Solutions (continued) Examples of Constants of the Motion :

1. If BE00 0, vvx , yz, and v are constants of the motion .

2. If EBe000 0, Bconstvvz .,  , z are constants of the motion. 3. The motion of a charged particle (mass mq and charge ) in EM field s ( represent ed by b pot enti al s A and ) idbL'is governed by Lagrange's equation: [Goldstein, Poole, & Safko, "Classical Mechanics," 3rd ed., p. 21 ] q d LL, i 1, 2, 3 (13)  mq d v EvB (13) dt qq dt c ii See Goldstein, Poole, & qi is a position coordinate. Safko, Sec. 1.5. where q Lmv1 2 vA qL [: Lagrangian; v : particle velocity]  2 c  In cylindrical coordinates, we have qrzqrzii( , , ), ( , , ), 22222 v r r  z, and vA  rArz  r A  zA . If the fields A and are ind epend ent of , (13) gi ives d LL  0 . H ence, dt   q L mrv rA  const [canonical angular momentum] (14)  c 10 6.2 Equilibrium Solutions (continued) Examples of Equilibrium Solutions: In contrast to the Boltzmann equation, which has only one equilibrium solution (the Maxwellian distribution), the Vlasov equation has an infinite number of possible equilibrium solutions. But they exist on a time scale short compared with the collision time. The choice of the equilibrium solution depends on how the plasma is formed. For example, if we inject two counter streaming

electrons of velilocity v0 ez and v0 ez into a neutralizi ng bkbackgroun d of cold ions, we have the following equilibrium solutions for the

electrons and ions (of equal density n0):  f1 nv( ) ( v )[ ( vv ) ( vv )]  e0 2 000xy  z  z  (15)  fnvvvi00 ()()()xyz which correctly represent the electron/ion distributions on a time scale short compared with the collision time. 11

6.2 Equilibrium Solutions (continued) Given sufficient time, collisions will first randomize electron velocities and eventually equalize elect ronandiontemperaturesron and ion temperatures. The fianl state will be an equilibrium solution in the form of the Maxwellian distribution for both the electrons and ions: 22 mmvv3/2 n0 fv00 ( ) n ( ) exp( )3/2 3 exp(2 ) (16) 22kT kT (2 ) vT 2vT where vkTmT  / is the thermal speed, T is the same for both species, mmei fthltfor the electrons and mm f fthior the ions. I(16)In (16),

fn0 has been normalized to give a uniform particle density of 0 3 in x-space [ fvdvn00 ( ) ].  2 135 (2n  1)  xdx2naxe   0 2nn1a a Useful formulae : (a  0) (17)   2  xdx21naxe  n! 0 2an1 22  2 eeaxdx11; x24 ax dx  ; x e  ax dx 3 (18) 0024aaa  08a2 a12 6.2 Equilibrium Solutions (continued)

Discussion : In constructing the equilibrium solution f0 , we must also consider the self -consistency of th e solution. For example, using

(11) and (12), we find that both (15) and (16) give 00 e  i0  0 and J=J000ei J 0. Hence, the plasma produces no self fields and the assumption of EB00 0 (which makes vvx , yz, and v constants of the motion) is valid. However, if fnvvvv1 ( ) ( )[ ( ) e0 2 00xy  z

(vvzxy000 )] in (15) is replace by fe0 nv ( ) ( v ) ( vvz ), then

Jvvvez  0 and there will be a self magnetic field, in which xyz , , are no longer constants of the motion. As a result, fvvve0 (xyz , , ) does not satisfy the Vlasov equation. In other words, the complete equilibrium solution includes not just f0 , but also the self-consistent fields. When the air in equilibrium [(16)] is disturbed, sound waves will be generated. When a plasma in equilibrium is disturbed, a great variety of waves may be generated. Some may even grow exponentially. These are subjects of primary interest in plasma studies. 13

6.3 Electrostatic Waves Rewrite the Vlasov-Maxwell equations:   q 1 t ffvEvBmc(   ) v f   0 (6.5) B 0 (7)   E  4 (8)  1  EB c t (9)  BEJ 1   4 (10)  cct (xxv ,tqftdv )  ( , , )3 (11)   where   3 Jx( ,tqftdv )   ( xv , , ) v (12)   Below, we present a kinetic treatme nt of the problem in Sec . 1 .4 . The electrons now have a velocity spread and, as a result, we will find that the electrostatic plasma oscillation becomes an electrostatic wave.14 6.3 Electrostatic Waves (continued)

At high frequencies (e.g.  ~pe ), the ions cannot respond fast enough to play a significant role. So we assume that the ions form a stationary background of uniform density n0 . Since we consider only electron dynamics, the species subscript " " in f will be dropped. Equilibrium (Zero - Order) Solution : Assume there is no field (including external field) at equilibrium. then, vvx ,y , and vz are constants of the motion and any fuction of v ,

ff00(()v ) f 0 (vvvxyz,, ), is an eq uilibri um sol ution for the electrons , i.e.  ff (vv ) ( v ) e ( E 1 vB  )  f ( v )  0 t 00  mce  000 v 00 0 0 provided f0 (v ), which represents a uniform distribution in real space, is normalized to the ion density n0 and it gives rise to zero current, 3 For clarity, all equilibrium quantities  f00()v dv n i.e.  3 are denoted by subscript "0". They (19)  fdv0 ()vv  0 are treated as zero-order quantities. so that there are no fields at equilibrium (EB00 0). 15

6.3 Electrostatic Waves (continued) First - Order Solution (Linear Theory) of Electrostatic Waves by the Normal - Mode Method : Consider small deviations from the equilibrium solution in (19)

[ff000 (vE ), B 0] and specialize to waves without a magnetic 1  fie ld (thus,  EBEE = c t BE  0  =   ). We may then writ e  ftfft(,,)xv () v (,,) xv For clarity, we denote all  01 small quantities by subscript  (xx ,tt ) ( , ) (20)  1 "1". They are treated as  Ex(,)tt E11 (,) x (,) x tfirst-order quantities. Sub. (20) into the Vlasov equation: 0  ffvEvB e ()0 1   f, (6. 5) t mce v we find the zero-order terms vanish. Equating the first-order terms, we obtain  ffv e  f (21) t 11me 10v 3 Sub. E111 and efdv 1 into the first-order field 23 equation, E11  4 , we obtain   1  4  efdv 1 (22)16 6.3 Electrostatic Waves (continued)   e  ff11v m  10v f (21) Rewrite: t e  23 114e f dv (22) Consider a normal mode (denoted by subscript "k " ) by letting  ikz z i t  ftfe1(xv,, ))() 11kk ( v ) [[():a f ( v ) : a small function of v] (2 3)  ikz z i t 1(x ,te ) 11kk [  : a small constant] (24) where it is understood that the LHS is given by the real part of the RHS. The normal-mode analysis is general because a complete solution can be expressed as a superposition of any number of normal modes.

e f0 ()v (21 ), (2 3), aad(nd (24) )gve( give ()()iikvfzz )11kk ()v m ik z e vz f ()v  f (v )e kz  0 (25) 11kkme kvzz vz f0 (v) 2 2 4ne0 1 vz 3 (22)-(25) give kkz 11kkm n z dv (26) e 0  kvzz 2 pe 17

6.3 Electrostatic Waves (continued)

f0()v 221 vz 3 Rewrit e (26) : kkdzpe  z v (26) 11kkn0  kvzz

f0()v 2  pe 1 vz 3 For   0, we must have 1 dv (()27) 1k k2 n0  vz z kz

The vvxy and integrations in (27) may be immediately carried out to result in a one-dimensional distribution function g0 ()()[vz [which, ,y by (19), is normalized to 1]: gv ( ) 1 f (v ) dvdv (28) 00zxyn0 

dg0()v 2  pe dvz dispersion Then, (27) becomes 1  dv  0 (29) k2  vz z relation z kz 

(29) has a singularity at vkzz  / , which will be addressed later.

For now, we circumvent this difficulty by assuming vkzz  / for the majority of electrons so that gv0 (zzz ) is negligibly small at v  / k . 18 6.3 Electrostatic Waves (continued)

Since g0 (vz  ) 0, integrating (29) by parts gives 22  peg00()vgvz  pe ()z 12  2 dvzz1 2 dv  0 (6.25) kz ()vz   (1kvzz )2 kz 

Note : The z -direction here is the x -direction in Nicholson. g0 ,,, kz , and vgkuz are, respectively, , , and in Nicholson. kvzz 2 Expanding (1  ) and keeping terms up to second order in 2 kv pe  kv kv 2 zz, we obtain 1gv ( )[1 2 zz 3( zz ) ]dv 0 (6.26)  2  0 zz A specific example : If the equilibrium soultion is Maxwellian: 222 n0 vvvxyz  f0 (v )3/2 3 exp(2 ) (6.23) (2 ) vTe 2vTe  2 then, using the formula in (18): eax dx  1  , we obtain 0 2 a 2 gv ( )1 f (v ) dvdv 1 exp( vz ) (6.24), (30) 00zxyn0  2 2v2 vTe Te 19

6.3 Electrostatic Waves (continued) 2 1 vz With gv0 (z ) exp( 2 ) [(30)] and the formulae: 2 vTe 2vTe 22 eeaxdx11 and x2 ax dx [(18)], we obtain 0024aaa   g ()vvdv1 ; v g () v dv0 ; 2g ()vdv v2 ( ()31) 00zz  z  z z   zzz 0 Te an odd function of vz 2 pe  kv kv 2 Sub. (31) into 1g () (v )[]1 2 zz3 (() zz )dv 0 [( 6.26 )], 2  0 z z 2222kv we obtain 1pe 3 zTe pe  0, (6.27) 24

By assump tion (30) ,   kzvTe. Th us, t o l owest ord er, (6 .27) gi ves

 pe and, to next order, we obtain the dispersion relation for the Langg(muir wave : (See Sec. 7.3 for a fluid treatment) 22 223 22 pe 3kv zTe [   pe (12 kv z Te )] ( 6.28) 20 6.4 Landau Contour We now address the singularity encountered in (29). First, a review of re levant d efi niti ons and th eorems i nvolilving compl ex vari iblables. (Reference: Mathews and Walker, "Math. Methods of Phys.," 2nd ed.) Laplace Transform : (M&W, Sec. 4.3)    pt  A tilde " " on top of a symbol Lt[()]0 ()te dt () p  initial value indicates a p -space quantity.   L[()] t pp () (t 0 ) (()32)  initial value initial value  Lt[()] pppt2  () ( 0)  ( t 0)  Im(p ) Inverse Laplace transform : pi   (t ) 1 0 ()pept dp poles (33) 2i p0 i of (p ) Re(p ) Note: p0 (0) ( 0) is sufficiently large so that all p path of  0 the poles of  (p ) lie to the left of the path of p-integration ptt-integration. Hence,  ( ) 0 if 0. (wy?h ) 21

6.4 Landau Contour (continued) Analytic Function : (M&W, Appendix A) A function f (zzzxi ) in the complex -plane (y rei ) is said to be analytic at a point z if it has a derivative there and the derivative fzh()() fz y fz() lim [[:h : a complex number] h z h0 r is independent of the path by which h approaches 0.  0 x The necessary and sufficient conditions for a function Wz ( ) Uxy ( , ) iVxy ( , ) UV  V  U to be analytic are: xy and  x  y (34) Examples : W z2 , W z , and W  ez are all analytic functions. Wz * ( z *: complex conjugate of z ) is not an analytic function.

22 6.4 Landau Contour (continued) Single - Valued Function : (M&W, Appendix A) A function Wz (() ) in the complex z- plane is single- valued if Wz ( rei ) Wz ( rein(2) ) [ n  1,2, ] y 2 z W  z isssgevued a single-valued fuuco.nction. r ElExamples :  1/2  Wz is not a single-valued function. 0 x For Wz 1/2 , we may draw a branch cut from y zz0t0 to i n th e -pl ane and df forbid z t o cross it. z r Then, Wz 1/2 is single-valued in the z -plane  0 x where  is restricted to the range 0 2 , i.e. branch cut the  value would have a 2 jump if z were to cross the branch cut. Regular Function : (M&W, Appendix A) A function is said to be regular in a region R if it is both analytic and single-valued in RWz . Thus,  2 is regular in the z -plane with an 1/2 arbitray  and Wz is regular in the z -plane with a branch cut. 23

6.4 Landau Contour (continued) Cauchy' Theorem : (M&W, Appendix A) If a function fz() ( ) is regular in a re gion R, then fzdz()  0 , c where CR is any closed path lying within . Hence, the line integral z2 y fzdz() is independent of the path of integration z2 z1 from zz to if the path lies within region R . z  12 1 x Theorem of Residues : (M&W, Sec. 3.3 & Appendix A) If fz (() ) is regular in a region R, except for a finite number of pol es, then, fzdz()  2iC residues inside , (35) c  where C is any closed path (in the counterclockwise direction) within R the residue of a pole 1 d n1 n and {}()[()()zz0 fz ]zz (36) of order nzz at  0 (1)!ndz 0 y gz() gz() is regular  Example:2(): dz  2() ig z0 z0 c zz 0 wihith no pol es. x b gx() xb0  gx() gx () Principal Value : P dx  lim dx dx (37)  24 a xx 0   axxx000  xx  6.4 Landau Contour (continued) Identity Theorem : (M&W, Appendix A) If two functions are each regular i naregionn a region R, and andhavingthe having the same values for all points within some subregion or for all points along an arc of some curve within R , then the two functions are identical everywhere in Rze. For example, i n the the -plane , z is the unique function in the zex -plane which equals x on the -axis.

Analytic Continuation : (M&W, Appendix A) R1

If fff12 (()z ) and f (()z ) are analyti l tic i n regi ions RR 12 and , f1()z respectively, and ff12 in a common region (or line), f12 f then fz212 ( ) is the analytic continuation of fz ( ) into R . f2()z

By identity theorem, it is the unique analytic continuation. R2 23 Example 1 : f1  1 z z  z  is analytic in the region z 1.

fz2 1/(1 ) is analytic everywhere except at t he pole z1. Since

ff12 in the common region z 1, f2 is the unique analytic continuation of fz1 into the  1 region (except for the pole at z 1). 25

6.4 Landau Contour (continued) Example 2 : Consider the following 2 analytic functions of p :  v -plane g()vz analtiilytic in th e upper z fp1( ) dvz ip / k (38)  vipkz  / z half plane, Re(p ) 0 z (kz is real and positive.) path of integration gv()z analytic in the fp2 ( ) dvz (39) L vipkz  / z entire p-plane vz -plane ip/ kz  LdLandau  gv()z dvz , Re(p ) 0 contour  vipkz  / z    gv() ip  Pdvz  ig(), Re()p  0 (()40)   z vipk / k  z z z definition  of Landau  gv()z ip dvz  2(), ig Re(p ) 0 Landau contour contour  vipkz  / z kz

Since fp12 ( ) fp ( ) in the upper half plane, fp 2 ( ) is the (unique) analytical continuation of fp ( ) into the lower half plane. 1 26 6.4 Landau Contour (continued) Electrostatic Waves by the Method of Laplace Transform : (Ref. Krall & Trivelpiece , Secs . 8 .3 3and and 84)8.4)  e  ff11v m  10v f (21) Return to (21) and (22): t e  23  11 4 e f dv (()22) Direct substitution of the nomal mode [(23), (24)] into (21) and (22) results in a singularity in (29). Landau resolved this problem by treating (21) and (22) as an initial value problem in t, while anal yzing a spatial Fourier component in zk (denoted by subscript " ").  ftfte(xv , , ) ( v , )ikz z  By assumption, the wave (41) Let  1 1k ikzz has no xy,-variation . 1(x,tte ) 1k ( ) (42) Sub. (41), (42) into (21), (22), we obtain   fikfikf ()(vv,t)(ik v f (),t)()e ik  ()t f0 ()v (43) t 11kkzzme z 1 kv  z kt2( ) 4 eftdv (v , )3 (44)  z 11kk 27

6.4 Landau Contour (continued)

  ftikvft(vv , ) ( , )e ikt ( )f0 ()v (43) t 11kkzzme z 1 kv Rewrite  z kt2( ) 4 eftdv (v , )3 (44)  z 11kk Perform a Laplace transform on (43) and (44) [see (32)], we obtain  pfff()(vv, p)(f (,t 0) ik ikfikv f (() v, p)() e ik  ()p f0 ()v (45)  11kkzz 1 kme z 1 kv   z kp2 ( ) 4 efpdv (v , )3 (46)  z 11kk   fp (,v ) f(v ,te ) pt dt (47)  11kk0 where    (ptedt ) ( ) pt (48)  11kk0 e f0()v  ft1k (,v  0)m ikz 1k ()p  e vz (45)fp1k (v , ) (49) pikv zz A note on notations : Subscripts "0" and "1" indicate, respe ctively, zero-order and first-order quantities. Subscript "k " indicates a Fourier component in zp . Symbols with a " " sign on top are -space quantities.28 6.4 Landau Contour (continued) Sub. (49) into (46), we obtain 4ne  gvt(,0)z   ft(,v  0) idv0 1k 1k 3 3  ip z 4e d v kz vz    pikv zz kz 1k ()p  (50) ikz fdgv00()v (z ) 2  n 2  kdv2[]1 pe 0 vz 3 1 pe dvz dv z 2  2  ip z kikkz pikzzv kz vz  kz  1 gv1k ( z ,0)tftdvdvn  1k (,0)v  xy Im(p ) where  0 (51) gv() 1 f ()v dvdv  00z n0  x y poles  Inverse Laplace transform : of 1k (p ) i Re(p ) 1 p0  pt By (33),  (tpedp )  ( ) , p (52) 11kk2i p i 0 path of 0 p-integration where pk0 ( 0) is a real number. In (50), z is real by assumption. If k  0, we see from (50) z Im(vz ) thatthepole(that the pole (/ip / kzz ))ofthe of the v -integrals integralslies lies ip/ kz  Re(vz ) above the Re(vptz ) axis. Hence, 11kk ( ) & ( ) path of v - integration are valid solutions without any singularity. z 29

6.4 Landau Contour (continued) 1 p0 i  pt Rewrite (52): 11kk (tpedp ) ( ) , (52) 2i p0 i 4ne  gvt(,0)z  idv0 1k Im(p ) 3  ip z kz vz   kz poles where 1k (p ) (50) dg0 () vz of  () (p ) 2  1k 1 pe dvz dv Re(p ) 2  ip z kz vz  p k 0 path of z p-integration The mathematical solution (52) is in a form in which the physics (e.g. normal modes and Im(vz ) dispersion relation, etc.) is not transparent. We ip/ k z Re(v ) need more work to obtain a physics solution . z path of v - integration This will require a detour of the path of the z pp-integration into the Re( ) 0 region, which implies that the pole at  ikip /illcrosstheRe()aisonhich()/ kzz will cross the Re(vp ) axis, on which 1k () is singu lar. Thu s, the  analytic region of 1k (pp ) is bounded by the Re( ) 0 line, and our first  30 step is to analytically continue 1k (pp ) from Re( ) 0 into Re( p ) 0. 6.4 Landau Contour (continued) Step 1: Analytic continuation : By the method in (38)-(40), we  may analytically continue 1k () (pp ) from Re( ) 0 into Re( p ) 0 by by changing the path of vz -integration in (50) from the straight line  dv to the Landau contour: dv . Im(v )  z L z z ip/ kz 4ne gvt(,0)z  idv0 1k Re(vz ) k3 L ip z z vz  original path of  kz Thus, 1k (p ) v z - inte g ration (53) dg0 ()vz 2 pe dvz 1 2 ip dvz kz L v  vz -plane z k z ip/ kz Note that this path change will not affect the same as p i 1 0  pt original path value of 11kk (tpedp ) ( ) in (52) 2i p0 i  because 1k () (p ) in (52) is evaluated along th e

Re(pp )0 ( 0) line, for which the Landau path Landau contour is the same as the original vz -path (see figures). 31

6.4 Landau Contour (continued)

 Spk(,z ) vz -plane Write (53) as 1k (p ) , (54) Dp(()p,kz ) ip/kz  4ne gvt(,0)z  Spk(, ) i0 1k dv  zz3 L ip kz vz   implying kz where  "source" dg0 () vz 2 Dpk( , ) 1pe dvz dv (55) zz2 L ip  kz vz   kz Landau contour 1 pi0   Spk()( , z ) pt then, (52) 1k (tedp ) (56) 2i pi0   Dpk(,z ) Im(p ) Since  (p ) is now regular (analytic and 1k poles single- valued) in the entire p-plane (except  of 1k (p ) at poles), we are free to deform the path of Re(p ) p p-integration in (56) (by Cauchy's theorem), 0 path of providdthided the new path thd does not cross any pole. p-integration

Note : We now have 2 complex planes: p -plane and vz -plane, and there are integrals along complex paths in both planes. 32 6.4 Landau Contour (continued) Step 2 : Deformation of the p - contour in (56) : Im(p )

pi0   Spk()( , ) 1 z ptt poles 1k (tedp ) (56) 2i pi0   Dpk(,z )  of 1k (p ) To bring out the physics in (56), we deform Re(p ) p the p-contour as shown to the right . Cauchy 's 0 original p -contour theorem requires the new path to encircle deformed (rather than cross) the poles it encounters. p-contour, Im(p ) pi0    radius   Athtllthlf()Assume that all the poles of 1k ()p [ Spk ( , ) / Dpk ( , )] are at the (1st-order) zz growing roots pjjz( 1,2, ) of Dpk( , ) 0. Then, mode Re(p ) Spk(,z ) p jt residues 1k ()tppe ()j damped integrand j Dpk(,z )pp at poles mode p tipt j  e r i  transient effects [] [integrations away from poles ] (57) pi  Path integrations away from the poles result in 0 pt 1k (tt ) 0 as  because e oscillates rapidly with ppi [ Im( )]. 33

6.4 Landau Contour (continued)

dg0 ()vz 2 Normal modes: Rewrite: 10 1 pe dvz dv  0 ( 55) 2 L ip z kz vz  kz it iikzt z Define  ip so that 1k ( t ) e and 1 (x , t ) e (58) dg0 ( vz ) 2 pe dvz dispersion then, (55) can be written 1 2  dvz  0 (59) kz L vz  relation kz  v (corresponding Landau contour: v z v z z )   0 i  0 i i  0 and (57) can be written as a sum of normal modes 1 Sk(, z ) i jt transient 1k (t) () j e (60) i D(, kz ) effects j  j  1 Sk(, z ) iikz jt z transient 1(x ,te ) () j  (61) i D(, kz ) effects j  j  where frequencies  jj (ip , j 1, 2, ) of the normal modes (in 34 gereral complex numbers) for a given kz can be found from (59). 6.4 Landau Contour (continued) Summary of techniques and theorems used : A Laplace transform bringsusintothecomplexvariabrings us into the complex variable territory. Cauchy's theorem is then used to deform the p -contour (left figure).  This requires the analytic continuation of 1k ( p ) to the entire p -plane, whichinturnleadstothewhich in turn leads to the Landau contour forthefor the vz -in tegration (right figure). With the deformed p -contour, we are able to apply the residue theorem to extract the essential physics by isolating the normal modfdes from th(the (non-essenti til)tital) transient effec ts. deformed v -plane p-contour, Im(p ) z ip/ k p0  i z radius  

growing mode Re(p ) damped integrand mode pt ipt  e r i

Landau contour 35 p0 i

6.4 Landau Contour (continued) A Recipe for Handling Singularities in Normal - Mode Method : Rewrite the solution [(29)] obtaine d by the normal-mode method in Sec. 6.3 and the new solution [(59) and (61)] obtained by the Laplace transform method in this section.

dg0 () vz 2  By comparison, we find that pe dvz 12  dvz 0 (29) kz  vz  the Laplace-transform method kz gives the additional information  dg0 () vz 2 ofdf mode amp lidilitude in terms of  pe dvz 12  dvz 0 the initial perturbation [see (61)]. (59)  kz L vz   kz  1 Sk()(, z ) iikz jt z tittransient 1(x ,te ) ()  j (61)  j i Dk(, z ) effects   j We find that (()29) and ()(59) have the same form excepppt for the path

of the vz -integration. This provides a simple recipe for removing the  singularity in (29): replacing dv with the Landau contour: dv .  z L z 36 6.4 Landau Contour (continued) The above recipe is of general applicability; namely, we may sol ve a va riety o f p rob le ms by t he (s imp lee)r) no rma l-mode met etod,hod, then remove similar singularities in the solutions by replacing the  dvdv contour with the Laudau contour .  zzL A question may arise as to whether the pole should remain above or below the Landau contour. As just shown, this depends on whether the original position of the pole ip is above or below the Re(v )-axis kz z  gvt(,0) 4ne0 1k z idv3 ip z kz  vz  kz in (50): 1k (p ) (50) dg0() vz 2  pe dvz 1 2 ip dvz kz  vz kz 1 p0 i  pt bfbefore we dfdeform t he p-contour in (52) : 11kk() (tpe ) ( ) dp 2i pi0  i.e. we determine the original position of ip by setting Re(pp ) 0. kz 0 37

6.4 Landau Contour (continued) p i  1 0  pt 11kk(tpedp ) ( ) , (52)  2i p0 i  4ne  gvt(,0)z   idv0 1k  k3  ip z Rewrite z vz    kz 1k ()p  (50 ) dg0 () vz  2  1 pe dvz dv  2  ip z kz vz   kz

where R(Re()pp )0 0W0. We see, if kz 0 , th e origi iinal posit ifthion of the pole ip lies above the path of v -integration, and should remain so kz z when the pv -contour in (52) is deformed and the z -contour in (50) is changed to the Landau contour. If kz  0, then the original position of the pole ip lies below, and should remain below, the v -contour. kz z If we convert the variable pip to a ne w variable   so that

the solution has the form of a normal mode: exp( it  ikzz ), the original position of the pole can be similarly determined as follows: 38 6.4 Landau Contour (continued) Recipe for Landau path in normal - mode analysis :

kvzz 0 (poles remain above -contour) :  v v  v z z z (62) itikz z    0   0 e dependence:  i i i  0 kz  0 (poles remain below vz -contour) :  v v  v z z z (63) i  0 i  0 i  0

kvzz 0 (poles remain below -contour) :  v v  v z z z (64) itikz     0   0 e z dependence:  i  0 i i kvzz 0 (poles remain above -contour) :  v v  v z z z (65)   0 i  0 i i  0 39

6.5 Landau Damping We have shown that a Laplace transform elegantly resolves the singularty problem in normal-mode analysis. The recipe is in the

form of the Landau contour for the vz -integration [(62)-(65)]. Waves considered so far are electrostatic in nature (i.e. without a B -field component). The Langmuir wave derived in Sec. 6.3 is only one example of such waves. In this section, we reconsider the Langgyppygmuir wave by properly accounting for the singgyularity throug h the use of the Landau contour. This leads to a very important new phenomenon known as Landau damping. We will then go beyond the scop e implied by the section title with an examination of two types of electostatic instabilities , the Landau growth and the two-stream instability, which occur in a plasma with a non -Maxwellian electron d istribution. We will also consider a different type of (low-frequency) electrostatic wave, the ion sound wave, which involves both the electrons and ions. 40 6.5 Landau Damping (continued)

Landau Damping in a Plasma with a Maxwellian g0 () vz : 2 pe 1 dg0() vz Rewrite (59): 1 dv 0, (59) k2 L vz dvz z z kz iikzt z which was derived under the e dependence. Thus, for kz  0, v v the vz -contour is [see (62)] vz z z   0   0 i i i  0 If i  0, we may take the path v z and write (see Nicholson, pp. 279-284)  11dg000() vzzz dg () v dg () v dv P dv i (66) L vvzzdvzzzzz dv dv kkzzvkzz/ gv() If  /kv , we may expand the principal 0 z z Te  1 dg0() vz k z value term to obtain  dv v L vz dvz z Te v kz z  dg kkvkvkvzzzzzzz[] 1   ( )23 ( )  dv  i 0 (67)  z dvz 41 vkzz/

6.5 Landau Damping (continued)

1 dg0()vz Rewrite (67):  dv L vz dvz z kz  dg dg kkvkvkvzzzzzzz00[] 1   ( )23 ( )  dv  i (68)  dvzzz dv vz  kz 2 gv( ) 1 exp( vz ) [see (30)]  0 z 2 v 2v2 For a Maxwellian g : Te Te 0 dg() v v 2 0 z  z exp(vz ) (69)  dv 3  2  z 2 vTe 2vTe 22 we use xdx24eeax1  and xdx ax 3 [(18)] to 004a aa8a2

1 dg00() vz kkzz242 dg obta in  dvvi() ( ) 3()3( )  (70) L vz dvzzz   Te dv k vz  z kz Sub. (70) into (59) and using (69), we obtain 2 2222 22 pekvz Te  pe 2kvz Te 12 (1 32 ) ie33  0 (71)   2 kzv Te 42 6.5 Landau Damping (continued) 2 2222 22 pekvz Te  pe 2kvz Te Rewrite (71): 1(13) 1 2 (1 32 )ie2 33  0 (71)   kzvTe 1    1 ii11 2 1 i Let r iii i and  1 22 2 (1  )  (1  2 ) rrr r r

Keeping terms up to first order, (71) gives 2 r 22222 22 pekvz Te pei  pe r 2kvz Te 122 (1 32 )  2ii 2 33 e  0 (72) rrr r kzvTe We may solve (72) by the method of iteration. To the lowest order,

(72) gives rpe . To first order, the real part of (72) gives 22 22 2 2 kkkzzvTTee3 k v r pe (1 322 )   r pe (1 ) (73) pe2 pe and the imaginary part of (72) gives 2  pe 3 4 22 2  pe 2kvz Te i  8 33 e [kz  0] (74) kzvTe 43

6.5 Landau Damping (continued)

The electron Debye length (De ) can be written 2 2 kTeeeT kT m v e De  222m (75) 44ne00e ne pe Thus, (73) and (74) can be written 22  kzv 22 (1 33Te ) (1k )  rpe222 pe z De  pe 2 (76)  pe 313   4 2222 2 2   pe22kkkz vTe pe kz  De i 8833ee  3 3  kkzzvTe De where r and i agree with (6.52) and (6.53) in Nicholson. Discussion: iikzt z (i) (76) is derived under the ek dependence with z  0. i t Hence,,g i is a negative number and   e , which implies that the wave is damped even though the plasma is assumed to be collisionless. This is known as Landau damping. 44 6.5 Landau Damping (continued) Rewrite (76): 22  kvz  (133Te )  (1k 22 )  rpe222 pe z De  pe 2 (76)  pe 313   4 2222 2 2   pe22kvz Te pe kz  De i   8833ee  3 3  kkzzvTe De (ii) In the limit of Te  0, we have rpe and i  0. Thus, (()76) reduce to the ( undamp p)ped) plasma oscillation of a cold plasma. discussed in Sec. 1.4. This shows that the plasma temperature is responsible for both the Landau damping and the change from an oscillation phenomenon to a wave phenome non. (iii) Mathematically, contribution to  comes i gv0 ()z from the residue of the pole at vkzz  / in the  kz vz -integration. Physically , this implies that v Te Landau damping is due to resonant electrons vz

moving at the phase velocity ( /kz ) of the wave. 45

6.5 Landau Damping (continued)

Question : How will the result be changed if kz  0, while still assuming the eiikzt z dependence? Answer : In this case, from (63), we use the contour v z instead of

v z . Thus, the only change is to replace "+i " in (66):  11dg000() vzzz dg () v dg () v vvdvdvzz P dv dv i dv L zzkkzzz vk/ zz zz with "i ". This will result in the same expression for r , but with a sign change in i i.e. gv0 ()z 2 pe 313   4 22kv2222 k 2 2  peeez Te pe z De kz i 8833 3 3 vTe kkzzvTe De vz Since kz  0, i is still a negative number and the wave will be damped at the same rate as is expected from symmetry considerations. 46 6.5 Landau Damping (continued) Landau Growth in a Plasma with a Bump - in - Tail Distribution : CidlConsider a plasma wh ose el ltectrons consist of 2 spati ally ll unif orm components with densities nn0a and 0b , and equilibrium distributions g0az (vv ) and g0bb ( z ) (see figure below). Assume that (1) nn0 a 0 ;

(2) g0az(vv ) has a z -spread of vTa centered at v z 0; and (3) g0b () v z has a vvzzz -spread of Tb centered at vvvv b Ta . Thus, g0 b ( ) looks like a small "bumppp" in the "tail" portion of g()0azv . To be self-consistent, we assume further that the ion density is equal to the total electron density, and the ions drift to the right with a current equal and opposite to the electr on current . Thus, there is no electric or magnetic field at equilibrium.

v g()0azv Tb g ()v 0b z vTa vz 0 v b 47

6.5 Landau Damping (continued) iikzt z Assuming ek dependence with z  0 and treating each component separately as before , we obtai n the dispersion relation: 2 2 pa11dg00a () vz pb dgb( vz ) 122dvz dvz  0 (77) kkzzL vvzzdvz L dvz kkzz  ()A ()B

For term (Akv ), we assume  /z  Ta (no resonant electrons). Thus, 22 g0az (vv ) ( z ). An integration by parts gives: Term (A) k z / (78)

Since g0b ()vvzaz g0 (), the real part of term (B ) is negligible compared with term (A ). However,  g ()v k we must keep the imaginary part of 0az z g()v v 0b z term (B ) because it determines i . Ta vz dg() v 0 v b Thus, Term (Bi )  0b z . (79) dvz vkzz / 2 2 pa pb dg0b() vz (77)-(79) give 12 i 2  0 (80)  kz dvz 48 vkzz / 6.5 Landau Damping (continued) 2 2 dg() v Rewrite (80): 1pai pb 0b z  0 (80) 2 2 dv  kz z vk /  zz Writing i  and assuming i  1, we obtain by r i r 11 i 2 1 i expansion: 22 2 (1  ii ) (1 2 ). Then (80) gives  r r r r

r  pa  : due to gg ;  : due to  rr0a 0b 2 3 (81)  pa pppb dg00bb() vzzpa n0b dg () v i  22 n  22kkzzdvzz0a dv  vkzrz// vk zrz

(81) shows that the sign of i depends on the sign of dg0b / dvz at vkzrz/ . Thus , i 0 (Landau growth) if rz/k falls on the positive slope of gk0b , and i  0 (Landau damping) if rz / falls on the negative slope of g0b . The  Landau g()v kz growth Landau growth is our first example 0az g()v 0b z of unstable equilibrium solutions. vTa vz 49 0 v b

6.5 Landau Damping (continued) A Qualitative Interpretation of Landau Damping and Landau Growth : v Assume that an electrostatic wave z vkphz / with phase velocity  /kz is present in the plasma. An electron moving with

velocity vz sees the wave at the Doppler-shifted frequency  :

  kvzz (82)

If  kvzz  0 (i.e. electron velocity  phase velocity), the electron experiences almost a DC electric field. In this field, it will gain or lose energy for an extended peri od of time (2/ ( 2 / ). This phenomenon is known as resonant interaction. Divide the electrons into  a slow electron  a fast electron slow electrons:    kvzz 0  vkph / z fast electrons:  kvzz  0 50 6.5 Landau Damping (continued) For both slow and fast electrons, some  a slow electron will lose energy to the wave and some wi ll  a ftltfast electron gain energy from the wave, depending on vkph / z the position of the electron relative to the phase of the wave.

If a slow electron loses energy in the resonant interaction, its vz decreases. Hence, its  (kvzz  0), which is a positive number 2 btAltthtifbecomes greater. As a result, the time for sustitained dit interacti on () ( ) becomes shorter. This will result in weaker resonance. On the other hand, if a slow electron gains energy in the resonant

itinteracti on, vz i ncreases and  b ecomes smaller. H ence, th e ti me for sustained interaction becomes longer (stronger resonance). This give the electrons in the energy-gaining phase the advantage and, on average, slow electrons gain energy from the wave. Similarly, fast electrons will, on average, lose energy to wave. 51

6.5 Landau Damping (continued) We have just concluded that, on average, slow electrons (relative to the phase velocity of the wave) gain energy from the wave and fast electrons lose energy to the wave. Thus, if the plasma contains more slow electrons than fast electrons (i.e. a negative slope of g0 at /lffi)hffi/kz , see left figure), the net effect is an energy transfffer from t he wave to the electrons (Landau damping). By similar argument, if the plasma contains more fast electrons than slow electrons (see right figure), there will be a net energy transfer from the electrons to the wave (Landau growth).

gv0 ()z   g()0az()v kz k g ()v z v 0b z v Ta Te v vz z 0 more slow electrons more fast electrons

than fast electrons than slow electrons 52 6.5 Landau Damping (continued) Kinetic Treatment vs Fluid Treatment : We have just considered a case in w hich details of the particle distribution function determine whether a wave grows or damps. On the other hand, fluid equations (derived in Sec. 1.4 of lecture notbtes by a si mpl e meth thd)fod) are formally lld deridfived from th thVle Vlasov equation (Sec. 7.2) by an integration procedure over the velocity space, in which details of the distribution function are lost. Hence, a fluid treatment w ill mi ss th e L and au d ampi/ing/growth h(S73) (Sec. 7.3) and other effects sensitive to the distribution function, but results of fluid equations are implicit in kinetic equations. Chapter 7 contains a fluid treatment of important plasma modes and instabilities. Here, as a supplement to Ch. 6, we will cover some of these topisc in the framework of the Vlasov equation. Our first study of fluid modes is on the two-stream instability. More will be considered in subsequent sections of this chapter. 53

6.5 Landau Damping (continued) Two - Stream Instability I: Consider aggpain the bump-in-tail model for Landau gg(pprowth (upper figure). The dispersion relation obtained by the kinetic approach is 2 2 pa 1 dg0a () vz pb 1 dg0b () vz 1 2  dvzz2  dv 0 (77) kz LLvz  dvz kz vz  dvz kz kz

Suppose the velocity spreads (vvTa , Tb ) of the 2 components vanish (lower figure). We then have a situation where one component streams thhhrough anoth er component. Integrat ing (77) by b parts andld letti ng g (vvv ), g ( ), we 0az0bb z v g0az()v Tb 2 2 pa pb g()v obtain 1  0 v 0 b z (83) 22()kv Ta v z b 0 v z This example shows that the b the kinetic result [(77)] can be  ()vz  ()vvz  b reduced to the fluid result [(83)] vz in the proper limit. 0 vb 54 6.5 Landau Damping (continued) 2 2 pa pb The dispersion relation: 122  0 [(83)] can be ()kvz b 2 kv 2 written (1pa )(1z b )2 pb  (84) 22      Daz ( ,k ) Dk(, ) Dka ()(, z ) b z 1 2 pa  pa Da (, kz )1  2 D (, kz ) where  b kv kz Dk(, ) (1z b )2  / v  b z  pa b (84) can be regarded as the coupling between the plasma mode

(0)dthbd(0)Thli(0)DDa  and the beam mode (0)b . The coupling itis strongest near the intersection of the two modes (see Figure). The intersecting pa point is at   pa and k z , which are solutions of vb Dkaz(, ) 0  55 Dkb (, z ) 0

6.5 Landau Damping (continued) 2 kv 2 Rewrite (84): (1pa )(1z b )2 pb (84) 22 To show that there is an instability, we will only look for the  value at kvzpa  / b , i.e. at the point of strongest interaction. Letting 11 1 1  pa  , we get (1 ); 22 (1 2 ) (85)  pa papa pa

Sub. (85) and kvzpa  / b into (84) and keeping terms up to first order in small quantities  and pb , we obta in 2 n 2 pb 0b 2 222 ( ) pa pa pa n0a 2 2in 33211nnbb0 33  pa()  1/3  pa ( )en ,  1,2,3 2 nna 2 0a 1 2  11n0b 3 3  1/3 pa ( )  i (86) 2 n0a 22  1 3 Unstable mode 22i 56 6.5 Landau Damping (continued) From (86), we find the frequency of the unstable mode: 2 11n0b 3 3  pa pa [1 1/3 ()( i )] 2 n0a 22  n 2 [11 (0b )3 ]  rpa 4/3 n  2 0a (87)  n 2  3 0b 3 ipa 4/3 ( ) [growth rate]  2 n0a Likeeecseo the case of Land dugow(uppau growth (uppeegue),eeeeegyr figure), the free energy available in a non-Maxwellian plasma drives a two-stream instability

(lower figure) . However, in the latter v g0az()v Tb case, there are no electrons at exactly g()()v v 0b z the phase velocity of the wave. Thus, Ta vz there is no singularity problem and a 0 v b rz/k fluid treatment will also be adequate .  ()vz rz/k (See Sec. 7.13 for a fluid treatment of  ()vvz  b a slightly different two-stream model.) vz57 0 vb

6.5 Landau Damping (continued) Comparing the Landau growth rate in (81) and the two-stream 3   pa n0b dg0b () vz i  2 n [Landau growth]  2 kz 0a dvz growth rate in (87): vkzrz /  2  3 n0b 3 i  4/3 pa () [Two - stream instability]  2 n0a we may show that ii [Landau growth] [Two-stream instability]. This is because, in the Landau growth (upper figure), electrons drifting slower than the wave absorb energy from the wave, whereas in the two-stream instability (lower v figure), all electrons deliver energy g0az()v Tb g()()v to the wave because they all drift v 0b z Ta faster than the wave. vz 0 v b rz/k As the g0b spread increases from 0  ()vz to a large value, the fluid instability rz/k  ()vvz  b will transition to a kinetic instability. vz58 0 vb 6.5 Landau Damping (continued) Two- Stream Instability II :  ()vv  ()vv Consider two cold electron beams z b z b of equal density nb streaming in an ion neutralizing background in opposite vz vb 0 vb directions (vvbb and  ). In (83), we 2 pb have already obtained the term for the forward stream: 2 . ()kvz b By symmetry, the backward beam will have a similar form, with vvbb replaced by  . Thus, following the same treatment leading to (83), we obtain the dispersion relation: 22 pb pb 122 0, (88) ()()kvzzbb kv which gives a quadratic equation in  2: 42222 22244  2(pb kvzzz b ) 2 pb kv b kvb  0 (89) 59

6.5 Landau Damping (continued) 422 22 2 22 44 Rewrite  2( pb kvzzz b ) 2 pb kv b kv b 0 (89) The solution for  2 as obtained from (89) is 1 2 222 2 222  pbkvzz b pb ( pb4 kv b ) (90) 222 It can be shown from (90) that for, kvz  2pb / b , all value of 222  are real (no instability). However, forkvz  2pb / b , two values of  will be a pair of complex conjugates and one of them gives rise to an instability. max We may find the wave number (kz ) for which the growth rate 222 maximizes by finding the value of kkzz for which dd / 0. The max 3pb result is kz  , which corresponds to a maximum growth 2vb rate of i  pb / 2. We will return to this problem again in Special

Topic II in connection with a new subject: the absolute instability. 60 6.5 Landau Damping (continued) Ion - Acoustic Waves : (See Sec. 7.3 for a fluid treatment) At low frequencies , the ion contrib ution to the dispersion relation cannot be ignored. For low frequency electrostatic waves, we simply add to (59) the ion term (the species subscript "ei " or " " is also added), 2 2 pe11dgei00()vz  pi dg ()vz 122dvzz dv 0 (91) kkzzLLvvzzdvzz dv kkzz Assume a Maxwellian distribution for both the electrons and ions:  v2 gv( ) 1 exp( z ) g () v (92)  e0 z 2v 2v2 i0 z Te Te gv()  2  / kz e0 z gv() 1 exp (() vz ) (93) i0 z 2v 2v2 v  Ti Ti z with vvTe Ti . Assume further v Te / kvz Ti (see figure) so that there is negligible electron Landau damp ing (because dgez0 /0) / dv  0) and negligible ion Landau damping (because  /kvz  Ti ). We will therefore neglect the imaginary part of both integrals in (91). 61

6.5 Landau Damping (continued)

dgee00()vdzzvz 1 g ()v For the electrons, sub.  g0 into  dvz dvzzv2 L vz  dv Te kz and neglect  ( v ) in the denominator, we obtain kz Te  (75) 1 dge0 () vz 1 11  dvzzz2222 g0 () v dv   (94) L vz  dvz vv pe kz Te Te De 1 dgi0 () vz  For the ion integral:  dvzz , we assume  v and L vz  dvzz k kz follow the same steps leading to (70). This gives the cold ion limit:

1 dgi0 () vz kz 2  dvz  ( ) (95) L vz  dvz  kz 2 1 pi Sub. (94) and (95) into (91), we obtain 122 2 0, (96) kz De  which is the most basic form of the disp ersion relation because ion thermal effects and electron Landau damp have all been neglected.

From (96), we find  pi , i.e. this is indeed a low-frequency wave. 62 6.5 Landau Damping (continued) 2 1 pi The dispersion relation 122 2 0 (94) kz De  k22 2 222 2 z De pi kkzzkTee4ne0 kT gives  22 222 mm  22 11kkzzDe De4ne0 ii 1  k z  De kC kT   z s with C  e (95) 22 s mi 1kz De Physically, when the ions are perturbed, electrons tend to follow the ions to shield their electric field, thereby reducing the restoring forces on the ions. So the wave has a frequency lower than pi ( pi would be the ion oscillation frequency if the electrons were immobile). However, the electrons cannot effectively shield the ion electric field if  (wavelength) De (see Sec. 1.2). Thus, when  De , or kz De 1,,( (94) shows that  will app roach  pi. Since the plasma motion is longitudinal and Cs is similar to the sound speed of a neutral gas, the wave is called an ion acoustic wave. 63 6.10 General Theory of Linear Vlasov Waves Part 2 We have so far treated only electrostatic waves in the absence of an external fi eld . I n thi s secti on, we l ay the ground work f or a general theory of linear waves, both electrostatic and electromagnetic, in an infinite and uniform plasma. We assume that the plasma is immersed iifin a uniform external magneti ifildlhc field along the zB-axibis: Be00 z , but there is no external electric field (E0  0). Equilibrium (Zero- Order) Solution :

An equilibrium solution f 0 (v ) must satisfy the zero-order Vlasov q equation:  ff(vv ) ( v )  ( E 1 v  Bf e )  ( v )  0 t00000   mc  z v  0 0 0

 (veBf00z )v  ( v ) 0 (101)

Thus, any function of the form fvv 0 ( ,z ) satisfies (101), provided that the total charge and current densit ies of all species vanish so that there is no net self field at equilibrium. This in turn makes vv and z constants of the motion in the only field present: B0ez . 1

6.10 General Theory of Linear Vlasov Waves (continued)

Examples of equilibrium solutions (normalized to n 0 ) are: 2  n0 v f0 3/2 3 exp(2 ) [Maxwellian] (102)  (2 ) vT 2vT   n v2 v2 f 0 exp( z ) [bi-Maxwellian]]( (103 )  0 (2 )3/2vv 2 22vv22  TTz TTz  n0 fvvv00( ) (z ) (104)  2v

In (103), th e parti cl es h ave two temperatures, vvTT and z . In

(104), all particles have the same vv(0 ) and vz ( 0). (104) is approximately self-consistent if the self magnetic field due to the gyrating particles is negligible. First- Order Equations : The linear ppproperties of a plasma is contained in the dispersion relation. To obtain the dispersion relation, we first linearize the set of Vlasov/Maxwell equations by writing 2 6.10 General Theory of Linear Vlasov Waves (continued)

 ftff(xv , , ) 0 ( v ) 1 ( xv , , t ) (105)  EEE()(x,tt)( E1 ()x, ) (106)  Bx ( ,tB ) e +B ( x , t ) As before, first-order (107)  0 z 1 quantities are denoted (()xx,tt)( (), ) by yp subscri p t "1". (()108)  1 Jx( ,tt ) J1 ( x , ) (109) Sub. (105)-(109) into the Vlasov/Maxwell equations. Zero-order terms give the equilibrium solution . Equating the first-order terms,   q ffvvemc()  Bf0 z v 1 t 11  q 1   mc(()EvB ) v f 0 (110)   11 B1 0 (111) we obtain E 4 (112)  11 1  E11c B (113)  t BEJ 1   4 (114)  111cct 3

6.10 General Theory of Linear Vlasov Waves (continued)

3 11(xxv ,tqftdv )   ( , , ) (115)   where  3 Jx11( ,tqftdv )   ( xvv , , ) (116)   Note : In Nicholson (6.145)-(6.153), zero-order fields depends on

x and tB . But from (6.154) on, E000 0 and B e z , as in our model. Particle dynamics : These first-order equations (110)-(116) are coupppyled. To examine the particle dynamics, we start from (()110): q  ffvve (  Bf )  t 11mc 0 z v 1 q  ()EvB 1   f (()110) mc 11v  0 The LHS is a total time derivative [d f ] along the zero-order dt 1 q orbit because the acceleration force is c ve B0 z . Thus, (110) can q be written d ff (EvB 1  )  (117) dt 111mc v  0 4 6.10 General Theory of Linear Vlasov Waves (continued)

Rewrite (117):  f 0 (,)vv z  q d f (xv , ,ttt ) [ E ( x , ) 1 v  B ( x , )]  f ( v ), (117) dt 111mc v  0 where d follows the zero-order orbit of a particle, which we denote dt bby xv (()tt ) an d  (() ). Un der the con diti()d()ions: xxvv()tt and () tt , y vtx () v cos[  ( t  t )] vt() vsin[ (() t  t )] v  y    vt() v   zz x we have  vv (118) xt() x sin[  ( t  t )]  sin    vv yt() y cos[  ( t  t )]  cos   zt() vz ( t  t ) z q B where  0 . (118) reduces to (1.25) if wes set  (polar angle of  mc

vx=xv=v ) / 2 and ttt 0. At  , (118) gives and as required.5

6.10 General Theory of Linear Vlasov Waves (continued) Change the variable tt in (117) to  and note that xx ( tt ) and vv(()tt ). A t - integration of (117) from   to t gives t d ftttdtftfttt [xv ( ),  ( ),  ]  ( xv , , ) [ xv  ( ),  ( ), ]t   dt 111 q t   dt{}EBE [()][()xv  t, t ]()[() 1  () tB [()]xv  t, t ] f ()(119)(  ) (119) mc  11v  0 We now consider a normal mode by assuming

Ex1(,)t E1k Bx(,)t B 1  1k eiti kx (120) Jx(,)t J 1 1k As before, subscript "k " ft1()(xv,,) f1k ()v denotes a normal mode . where EBJ111kkk , , are complex constants, and f 1 k ( v ) is a complex function of vJ . 11kk can be expressed in terms f ( v ) as 3 J1k   qf 1k (vv )dv (121)  Question: Why is 1 (x ,t ) not included in (120)? (see note below) 6 6.10 General Theory of Linear Vlasov Waves (continued) Sub. (120) into

fff11()[()()](xv,,t)[() f xttt, v (), ]t q t  dt{}Ex [  ( t ), t  ] 1 v  ( t )  Bx [  ( t ), t  ]  f ( v  ) (119) mc  11v  0 we obtain fe (vv )itikx f [ ( te )]  it  i kx 1k 1k t  it i() t  t ikx  i k ( x- x)  q t  dt [EvB 1 ( t )  ] eiti kx  f ( v  ) mc  11kkv  0 q t   edttiti  kx  [EvB1 ( ) ] f ( v ) mc  11kkv  0 eitti()kx-x) ( (122)

Note: In (122) , the opertaor operates o n f 0 (()v ) only .

7

6.10 General Theory of Linear Vlasov Waves (continued) We assume ( Im ) 0, then fte [v ( )]iti kx  0 and i 1k t (122) becomes iti kx fe1k (v ) q t  edttiti kx  [EvB 1 ( )  ]  f ( v ) mc  11kkv  0 eitti()kx-x) ( Factoring out the eiti  kx dependence, we obtain q t fdttf (vEvBv )  [ 1 ( )  ]  ( ) 111kkkmc  v  0 eiii()ttik)k (x-x) (123) Note : iti kx (i) 1 (x ,te ) is not in (120) because, with the assumed depen dence, 11kk is i mpli cit i n J th ough th e continuity equati on:  JkJkJ   0  ii    0   1  t 11kk 1 k 1 k 8 6.10 General Theory of Linear Vlasov Waves (continued)

(ii) Although (123) is derived under the assumption of i  0, the dispersion relation to be obtained from (123) can be analytically

continued to an arbitrary  by the use of Landau contour for the vz - integg[ral [see (135 )]. The ar gument follows the treatment of Landau damping. (iii) v as a vector is a function of t . However, it is understood

that fvv 0 (()v ) is a function of scalars  and z , both being constants

of the motion. Hence, in writing f 0 (v ) in (119), (122), and (123), v is not displayed as a function of t . (iv) The method we employed to obta in (123) is called "method of characteristics" or "integrating over unperturbed orbit".

9

6.10 General Theory of Linear Vlasov Waves (continued) Field equation : From the linearized Maxwell equations:  1  EB11 c t (113)  BEJ 1   4 (114)  111cct 2 we obtain  (() EBEJ )  11  4 (124) 1111c ttct22 c 2

Ex1(,)t E1k Bx(,)t B For a normal mode: 1  1k eiti kx (120) J ()(x,t) J 1 1k  ft1(,xv ,) f1k ()v (()113) and ()g(114) give kkE (() )2 E 4 i J (125) 11kkcc22 1 k A note on notations: Subscripts "0" and "1" indicate, respectively, zero and first order quantities. Subscript "k " indicates a normal mode. SbSubscr it""iditipt "" indicates parti tilcle speci es. (123) and (125) together with the orbit equations (118) form the basis for our treatment of linear plasma waves in Secs. 6.11 and 6.12. 10 6-11 Linear Vlasov Waves in Unmagnetized Plasma q t Rewrite (()123): fff ()vEvBv dt [ 1 ()]() t  ]  f ()( ) 111kkkmc  v  0 eitti()kx-x) ( (123)

Assume the absence of an external magnetic field ( 0). Then, vv()t  const (118) reduces to  , and (123) can be written xxv()ttt (    ) q t ffdte()vEvBv  (1 ) () itti()kv () tt (126) 111kkkmc v  0 

Again, assuming i  0, we obtain from (126) 1 q ()()EvB v f v f ()v   11kkc  0 (127) 1k m i()kv

Note : (127) can be readily derived by setting B0  0 and sub. the normal mode (120) into the linearized Vlasov equation (110): qq  ffvveEvB() B ff  ( 1  )  t 11mc0 z vv  1 m 1 c 1 0 but in the presence of B0ez , we must use (123) (see next section). 11

6-11 Linear Vlasov Waves in Unmagnetized Plasma (continued) 1 q ()()EvB v f v Rewrite f (v )  11kkc  0 (127) 1k m i()kv

Assume fffv000 (vv ) is an isotropic function, i.e. ( ) ( ).

Then, (v B1k ) v fv 0 ( ) 0. Since there is no external magnetic

field and fv 0 ( ) is isotropic, the plasma properties are also isotropic.

Without loss of generality, we assume ke kzz . Thus, (127) becomes

q E1k v fv 0() f1k (v )  (128) m i(kvz z ) 3 Sub. (128) into Jvv11kk  qf ( ) dv [(121)], we obtain  2 q E1kv fv 0() 3 J1k   m  v dv (129)   i()kvzz The field equation: kkE (() ) 2 E  4 i J [(125)] 11kkcc22 1 k 222 4 i may be written kEzzz111kzeEJ ( k 22 ) k  k (130) cc 12 6-11 Linear Vlasov Waves in Unmagnetized Plasma (continued) Electrostatic waves : 2  q E v f ()v 3 Jv  1k  0 d v ()(129)  1k  m  i()kv Rewrite   zz kE22eEJ( k 2 )4 i  0 (130)  zzz111kzcc22 k k

For electrostatic waves, Ek1k  ( kzz e ). So we set

Ee11kkz E z (131) (129) then gives f 0 ()v 2 q vz 3 Jeee1kkziEm 1  ( vxyzxyz  v  v ) dv (132)   kvzz 1 2222 In (132) , fff00 (()v )f [(vvvxyz ) ] is an even funct ion of v x and vxyy . Hence, the and components vanish upon, respectively, vvxy and integrations, and we have fv 0 () 2 vz q vz 3 J1kkziEm 1 ez  dv (133)   kvzz 13

6-11 Linear Vlasov Waves in Unmagnetized Plasma (continued)

f 0 ()v 2 vz q vz 3 Rewrite Je1kkz iEm 1 z  d v (133)   kvzz Defining g ( v ) 1 f ( v ) dv dv [as in (28)], we obtain 00zxyn 0 

dg 0 () vz 2  vz nq 0  dvz J1kkziEm 1 ez dvz    kzzv Writing vz 1 ( 1 ), we have kvzz k z kv zz 2  nq 0  1  dg 0 () vz J1kkziEm 1 ez (1 ) dvz ,   kkvz   zz dvz

The first term vanishes upon vgvzz -integration because  0 ( ) 0 at vz  . Thus,

dg 0 () vz 2  nq 0   dvz Je1kkz iEm 2 1 z  dvz , (134)   kz  vz  kz 14 6-11 Linear Vlasov Waves in Unmagnetized Plasma (continued)

dg 0 ()vz 2  nq 0   dvz Sub. EeJEeEiEzz and Je1  2 e  dvz 11kkzkm kz 1 kz vz   kz into the field equation: kE22eEJ ( k 2 ) 4 i (130) zzz111kzcc22 k k we obtain the dispersion relation for electrostatic waves:

dg 0 () vz 2 p dvz 1  2  dvz  0, (()135) kz L vz   kz  where, by the recipe in (62), we have replaced dv with dv .  zzL

In deriving (123) and (127 ), we have assumed i  0 . With the Landau contour,  in (135) can have any value provided the pole

/kz does not cross the Landau contour. (135) agrees with the electrostatic disperson relations in Sec. 6 .5 for the Langmuir wave, Landau damping/growth, two-stream insta- bilities, and ion acoustic waves. 15

6-11 Linear Vlasov Waves in Unmagnetized Plasma (continued) 1  4 Discussion : Rewrite BEJ111 cct  (114) For the normal mode in (120), the RHS of (114) gives 1 44i iti kx cccct EJ11 ( E 1kk  J 1 )e (136)

dg 0() vz 2  nq 0   dvz Inserting Je iE2  dvz z [(134)] and 1kkzm kz 1  vz  kz

Ee11kk E z z [(131)] into (136) , we find dg 0()vz 2  ii44  nq 0   dvz EJEiE ezz  2 e  dvz ccc11kk11 kz cm kz kz vz  kz dg 0() v 2 z i p dvz c Edv1kzezz{} 1 2   0 kz L vz  kz  0 by (135) This shows that, the displacement current and particle current exactly cancel out. Hence, we have an electrostatic wave. 16 6-11 Linear Vlasov Waves in Unmagnetized Plasma (continued) Electromagnetic waves : 2  q E v f ()v 3 Jv  1k  0 dv (129)  1k  m  i()kv Rewrite   zz kE22eEJ( k 2 )4 i  0 (130)  zzz111kzcc22 k k

For electromagnetic waves, Ek1k  (kzz e ). So, without loss of generality (because the plasma is isotropic), we set Ee11kky E y (137) Then, (129) and (130) give  fv 0 () 2  q vy 3 Jeee1kky iEm 1  ( vxyzxy  v  vz )dv (138)    kvzz (kE2 2 )eJ4 i 0 (139)  zycc2211ky k

fv 0 ( ) is an even function of vx . Hence, the x -component of (138)  vanishes upon vx -integration. fv 0 ( ) is an odd function of vy . vy

Hence, the zv -component of (138) vanishes upon y -integration. 17

6-11 Linear Vlasov Waves in Unmagnetized Plasma (continued) We are then left with only the y component of (138): fv 0 () 2 vy q vy 3 Je1kky iEm 1 y  dv (140)   kvzz

Integrating (140) by parts of over vy yields 2 q fv 0 () 3 Je1kky iEm 1 y  dv   kvzz Using the one-dimensional equilibrium distribution function: gv ( ) 1 fvdvdv ( ) , 00zxyn 0  2 nq00 g() vz we mayy() write (140) as J1kky iEm 1 e y  dvz (()141)   kvzz

Sub. (141) and Ee11kky E y into  kE22eEJ() k 2 4 i  0 (130) zzz111kkkkz cc22kk 222 gv 0 ()z we obtain kz 2  p  dvz  0 (142) c  kvzz 18 6-11 Linear Vlasov Waves in Unmagnetized Plasma (continued)

222 g 0 ()vz Rewrite kz 2  p  dvz  0 (142) c  kvzz EM waves in a plasma have a phase velocity  c [see (143) below].

Hence, we may assume  /kvzz and neglect the the kvzz term in the demonamitor of the integral in (142).  gv 0 ()z 11 dvzzgv 0 ( ) dvz  kvzz  This results in the dispersion rela tion:  pe k 2222 z kcz pe (143) where we have neglected the small ion contribution.

The  vs kz plot (see figure) is similar to that of the waveguide. There is a cutoff frequency  pe , below which EM waves can not pppgropagate. Short radio waves ( 10 MHz ) are hence reflected from the ionosphere. This has been exploited for long-range broadcasting. By comparison, the free space is non-dispersive with   kc . z 19

6-12 Linear Vlasov Waves in Magnetized Plasma (Ref.: Krall and Trivelpiece, Sec. 8.10) Dispersion Relation : We begin this section with a derivation of the general dispersion relation for waves in an infinite, uniform, and magnetized plasma on the basis of the following linearized equations derived in Sec. 6.10 for a normal mode with eiti kx dependence: t  q 1 fdttf111kkk()vEvBv [  ()  ] v  0 ( )  mc   itt(  )(ikx-x)  e (123)  2 4 i  kkE (11kk ) 22 E  J 1 k (125)  cc where we have assumed a uniform external magnetic field Be00 B z , and shown that the equilibrium distribution function in such a field is

ffvv 0 (()v )  0 ( , z ). So the plasma is isotropic in the xy, -dimensions, but it is 3-dimensional anisotropic. Thus, we expect the conductivity  to be in the form of a tensor : JE11kk . 20 6.12 Linear Vlasov Waves in Magnetized Plasma (continued) General form of the dispersion ralation :

xx xy xz E  1kx Write J11kk E yx yy yz E 1 ky (144)   zx zy zz E1kz Without loss of generality (for a plasma isotropic in xy , ), we let

kekk xzz e (145) Sub. (144) and (145) into the field equation: kkE ( )2 EJ4 i (125) 1k cc2211kk the xyz,, components are kc22 kkc2  (1z 44ii )EE   ( z  4  i  ) E  0  22xx11kx xy ky  xz 1 kz  22 444iii EEE (1kc  )   0 (146)   yx111kx2  yy ky yz kz  kkc2 kc22  (  z 44ii)(1)0EE 4  i  E  22zx11kx zy ky  zz 1 kz 21

6.12 Linear Vlasov Waves in Magnetized Plasma (continued) (146) can be written DDDE  xxxy xz 1kx DE11kkyDDDEyxyy yz   0 or (147) D DD zx zy zz E1kz kc22 kkc2 1z 44ii  z  4  i E 22xx xy  xz 1kx 22 4i  1 kc 44ii  E 0 (148)  yxyyyz2   1ky  kkc222 kc z 44ii 1 4  i  22zx zy  zz E 1kz For (147) or (148) to be solvable , the determinen toft of D must mustvanish: vanish: DDD  xxxy xz D DDDyxyy yyzz 0 (149)

DDDzx zy zz (149) is the most comprehensive form of the dispersion relation. 22 6.12 Linear Vlasov Waves in Magnetized Plasma (continued)  Particle dynamics : In (149), the conductivity tensor  is still unknown. To obtain the specific expressi on of the dispersion relation, we need to work on the equations for particle dynamics. Define tt and rewrite (123) and (118) in terms of q 0 fd (vEvBv )  [ 1  ( )  ]  f ( ) 111kkkmc  v  0  eiikx()-x[   ] (150) y vvx ()  cos( )  vv () sin( ) v  y    vv ()  x  zz  vv (151) xx() sin(  ) sin    vv yy()  cos(  ) cos      zvz()z 23

6.12 Linear Vlasov Waves in Magnetized Plasma (continued)

Using kekk xzz e [(145)] and the orbit equations in (151),

we may write kx[() [ ( ) x ] kx [[() ( ) x ] kzz [()[ ( ) z ]

kv  [sin(  )  sin  ] kvzz   kv ikvi( zz)  [sin( )  sin ]  eeiikx()-x[]   ixsin is Using the Bessel function identity: eJ  s ()xe , s kv  i sin( ) kv e   Je() is()   s  we obtain s  ikvsin   kv is e   Jes ()  s  kv ikvi( zz)[sin()sin]   eii kx()[   -x]  e 

kv kv  ikvs()zz  is() s  JJes ()()s (152) ss  24 6.12 Linear Vlasov Waves in Magnetized Plasma (continued)

Since ffvvvvvv00 (v ) ( ,zzz ) and ( ), ( ) are constants ofthf the moti on, we h ave v ff00() (v ) v (vv , z )

y v/ v e   ff001 f 0 v e v ee  z  v   vz  x ff00  222ve 2vzz (153) vv z  Th()()hus, Eveee1111kkxkykzv f 0 () 2()EEExyz  

f 0 f 0  ( 22ve  vzz) vv  z ff  2(Ev Ev )00  2 Ev (154) 11kxxy ky22 1 kz z vv z c From (113) and (120) , we obtain BkE11kk   . Then, cc vB1111kkkk v ( kE ) [( vEk  )  ( kvE ) ] (155) 25

6.12 Linear Vlasov Waves in Magnetized Plasma (continued) (153) and (155) give ff 12()v BEkkEB  f  [()()](v Ekk v E 00ve v ) c 111kkkv  0  22 zz vv z  f 2 [(vE )( kv ) ( kvv )( E )] 0   11kk2  v f 0   [(vE )kvzz  ( kv ) vE z ]  11k k vz2  2   [(vExyz111kx vE ky vE kz ) kv x  f  (kv kv )( vE  vE )]  0  xzzx11kx y ky 2 v f   [(vE vE vE ) kv  ( kv kv ) vE ]  0 xyzzzxzzz111kx ky kz 1 kz 2  vz  f 2 [( kvE kv E k v E) v  0  zx1kxz yyy11 ky x kz y 2 v f  (kvE kvE kvE ) v  0 ] (156) zx111kx zy ky x kz zv2 z 26 6.12 Linear Vlasov Waves in Magnetized Plasma (continued) Combining (154) and (156), we obtain vz 1  (EvB11kkc ) v fvXvYvZ 0 2xyz  2  2

ii()()    ve [ e ] X

ii()()   iv [ e e ] Y 2 vz Z (157)

 fff000vz XE()() kEz  kE   111kxvvv222 kx kz z    fff where YE000vz kE (  ) (158)  11kyvvv222 z ky  z  f 0 ZE 2  1kz vz

Note : (i) v , vz , and v are constants of the motion, but v ,vx and vy are time dependent.

(ii) XY , , and Z are functions of constants of the motion. 27

6.12 Linear Vlasov Waves in Magnetized Plasma (continued) Combining (152) and (157) gives 1  iikx[]()-x ()EvB1k c 1k  v fe 0 ii()()    ikvs()zz  is() s vXe [] e JJes s ss ii()()   ikvs()zz  is() s iv Y [ e e ]  JJes s ss ikvs()zz  is() s  2vZzsJJes (159) ss i() e ikvs()zz  is() s Write JJs  e s i() ss e ik[ vs( 1))] iss ( 1) e z z    JJs s  ss  ikvs[1)(1)zz (   )] iss  ns1  e  ns1 JJ n1 s ikv(()zz n  ) isn   e (160) ns JJ n1 s 28 6.12 Linear Vlasov Waves in Magnetized Plasma (continued) Sub. (160) into (159), we obtain 1  iikx()-x[]  ()EvB1k c 1k v f 0e

 [(vXJ J ) ivYJ (  J )2] vZJJzns ns nn11 nn  11  ik()() zzvn   i sn   ee (161) the only factor that depends on q 0 Su b. (161) into fd (()v )   [EBE 1 v (() )B ] 111kkkmc  iikx()-x[]  v fe 0 (v ) (150) and carrying out the  - integration, we obtain q vXJ()()2 J ivYJ  J vZJz n is() n f ()v    nn11 nn  11 Je  1k m ikvn() s ns   zz  (162) This is Eq. (8.10.8) in Krall & Trivelpiece. Note that all Bessel kv functions have the same argument: .  29

6.12 Linear Vlasov Waves in Magnetized Plasma (continued) The conductivity tensor : The perturbed current [(121)] can be written: 2 J q fd()vv3vqvd dddfv dd v f () vv 1k 11kk00  z  vvvcoseee sin  First consider the x -component of J1k : x yzz 2 Jqvdv ddv f (v )1 ve (ii e ) (163) 1kx   0 0  z 1k 2  

By (144), J1kx can be expressed in terms of the conductivity tensor as JEEE1kx  xx111 kx xy ky xz kz (164)

Then,  xx is the coefficient of the sum of all E1kx terms in (163) [which can be found from (162) and (158) ]: 2 q 2 1 ii xx m vdv d dv z 2 v  ( e e )   00 ffkkk v k v v [  0()1 zz zz  0 ]  2 2 v vz is() n   ()JJJenn11 s (165) ns ikvn()zz  30 6.12 Linear Vlasov Waves in Magnetized Plasma (continued) Using the Bessel function identities:  2n JxJxnn11   x Jxn    (166)  JxJxJxnn11 2 n 

kkkv k v 2n k v we may write JJnn11 ( ) ( ) Jn ( ) (167) kv  Sub. (167) into (165), we obtain q2 2 nv    vdv d dv   xx m  z  00 k ffkv kv 00()1zz zz vv22   z  is(1)(1) n is  n   JJn s ee , ns ikvn()zz    (168) where we see that only the sn  1 terms in the s sum will survive the -integration. 31

6.12 Linear Vlasov Waves in Magnetized Plasma (continued) Carrying out the  -integration in (168), we obtain 2q2 nv     vdv dv   xxm  z  n 0  k 2n ffkv kv  Jn 00(()1 zz) zz kv vv22    z  JJn ()nn11 J ikvn()zz   ff kv kv 00(1zz ) zz 2q2 n22 vv22    2vdv dv  J 2  z m 0  zn2  n k ikvn()zz  ffkkk v k v  00(1zz ) zz  2 22 p n22 vv  2vdv dv  J 2  z , 2 0  z 2 n  n k ikv()zz n   (169) 1 3 where ff00 n [hence fdv  0 1].  0  32 6.12 Linear Vlasov Waves in Magnetized Plasma (continued) The general dispersion relation : Rewrite the dispersion relation:  DDDxxxyxz D  DDDyx yy yz DDDzx zy zz kkk22cc 2 1zz44ii   4  i  22xx xy  xz 22c  4i  1  k  44ii  0 (149)  yx 2 yy yz kkc2 k22c  z 44ii 1  4  i  22zx zy  zz k22c Su b.  from (169) into D 1 z  4i  , we obtain xxxxxx2  22 k c 2  D12z 2  v dv dv xx2  p  z   n 0  fff ()(vv, )(kv kv f ()vv, ) 00zz(1zz ) zz n22 kv vv22   J 2 ( ) z (170) 2 n  33 k  kvzz n  

6.12 Linear Vlasov Waves in Magnetized Plasma (continued) By similar method, we obtain the other elements of the dispersion tThltlt(Ktensor. The complete results are (see Krall & T ri vel pi ece, pp. 405 -406) 22 22 kz c 2 2 n  2 kv DJxx 122  p n ( ) (171)   n k  nv 2i 2   kv dJn() k v / DJxy  p n ( ) (172)  n kdkv  ()/ 2 kkz ckv2 2 nv z 2 DJxz2  p n (() ) (173)   n k  DDyx xy (174) 222 ()kk  znc 2 2 2 dJ() kv / 2 Dvyy  12  p  [] (175)   n dkv()/

2i 2 kv dJn() k v / DvvJyz  p z n (() ) (176)  n  dk()v /

34 6.12 Linear Vlasov Waves in Magnetized Plasma (continued) 2 kkz c 2 2 n 2 kv DvJzxxn2  pz n ( ) (177)   n k 

2i 2 kv dJn (/) k v  DvvJzy  p z n ( ) (178)  n  dkv()/ 2 2 kc 2 2 22kv DvJzz 12  p z n ( )  (179)   n    Fv((), vz) F( v , vzz)2 ) 2 v dv dv (180)  0  kvzz n    fvv(,) fvv (,)  00zzkvzz kv zz where  22(1 ) (181)  vv   z  fvv0 (,) z n fvv00(,)z fvv (,z )  22  [  2 ] (182)  vvzzv Question: The plasma is isotropic in xD and y . Why are xx and Dyy unequal? 35

6.12 Linear Vlasov Waves in Magnetized Plasma (continued)

Waves Propagating Along B0ez (k = kzez): ex The dispersion relation for waves propagating B 0 e along B0 = B0ez may be obtained by letting z e k k  0 in (171)-(179). y FlltthBlftiFor a small argument, the Bessel functions Jn(x)d) and Jn(x)b) can be approximately written x  0 1 x n Jxn () ( )  n!2  all x x  0 (183) (1) n Jx() (1)n Jx ()  (x )n  n n n!2 For nn 0 and 1, we have in the limit x  0, Jx()1 , Jx ()xx , J () x   0122 1  (184) Jx()x , Jx()11 , J () x  0 2 1122 36 6.12 Linear Vlasov Waves in Magnetized Plasma (continued) kv In (171)-(180), the argument of all Bessel functions is  .  Us ing (183) an d (184), we fin d tha t in thliithe limit k  0 , DDDDxz zx zy zy 0 (185) and the other elements become 2 2 Dvv1kz c 2 221   2  (186)  xx 2   p 4 nn11   D  D (()187)  yy xx  DD  2i 221 vv 2  (188)  xy yx   p 4 nn11   D 1 2 22v  (189) zz   p z  n0   Thus, the dispersion relation (149) reduces to

DDxxxy0 22 DDyx yy 0DDD zzxxyyxyyxzzxxxy ( DD ) DD ( D ) 0 (190) 0 0Dzz 37

6.12 Linear Vlasov Waves in Magnetized Plasma (continued)

2 2 Rewrite the dispersion relation: DDzz ( xx D xy ) 0 (190) Several modes are contained in (190). To find these modes, we assume, for simplicity, that the plasma is isotropic in all 3 dimensions,

i.e. fvv 00(,) ( ,z ) fv ().( ). Electrostatic waves : One of the solutions of (190) is Dzz  0, which, by (189), (180), and (182), can be wr itten 2  vz f 0 2 2 vz2 1  p  2vdv dvz  0  0  kvzz d vgvzz 0() 1 2  dvz or 1 p  dvz 0, (191)   kvzz  where gv ( ) 2 vdvfvv ( , ).  00z 0   z

38 6.12 Linear Vlasov Waves in Magnetized Plasma (continued) The integral in (191) can be written dd vgvzz 00() [ ( kvgv zzz )] () dvzz1 dv dvzz dv kvzz k z  kv zz d  g 0()vz 11dvz d dvzzz dv g 0 ( v ) kkvkdvzzzzz       d 0 gv 0()z   dvz  2 dvz (192) kz  vz  kz d 2 gv 0()z p  dvz Sub. (192) into (191), we get 110 2  dvz   0,  kz  vz  kz which agrees with the electrostatic dispersion relation [(135)] for an unmagnetized plasma . This is because ele ctrostatic waves involve particle motion along B0 , and hence are unaffected by the magnetic field. The mode considered below will provide an opposite example. 39

6.12 Linear Vlasov Waves in Magnetized Plasma (continued) 22 Electromagnetic waves : Rewrite Dzz ( D xx D xy ) 0 (190) DiDxx xy  0 (193) (190) gives two other solutions:  DiDxx xy 0 (194)  We recall that the dispersion relation D  0 [(149)] is based on  the condition for the solvability of the field equations in DE1k 0

[(147)]. For kDDDDDD 0 , we have xzzxzyzyxxyy0 , , and DDyx xy [see (187)]. Then, for solutions (193) and (194), DDxxx yyyEE DDxx xy   (147) gi ives 11kx kx  0 (195) EE   DDyx yy11ky DD xy xx   ky D E   xy E DxxEDE11kx xy ky  0  11kxD ky  xx (196)   D DExy11kx DE xx ky 0 E  xx E  1kx Dxy 1ky  40 6.12 Linear Vlasov Waves in Magnetized Plasma (continued)  Dxy EE11kxD ky RitRewrite: xx (196)  D EE xx  11kxDxy ky The following information about the modes in (193) and (194) can be immediately learned from (196): 22 (1) The 2 equations in (196) are consistent only when DDxxxy , or when (190) is satisfied . This is a sp ecific example which shows the dispersion relation as the condition for solvability of field equations. We also find that (196) gives the relative amplitude (not the absolute values) of the field components , as is t ypical of linear solutions.

(2) The fields in (196) are in the xy- plane. With ke kzz and itikz z Ex11( ,te ) Ek [see (120)], we find itikz z Ex1 ( ,tik ) zz e  E1k e  0. Thus, the two solutions represent electromagnetic waves. 41

6.12 Linear Vlasov Waves in Magnetized Plasma (continued) D (3) Either of the equations in (196) gives EE xy . Hence, 11kxDxx ky iE1ky for Dxx iD xy 0 Ei[ee x y ] (198) E E 1k 11kxiEfor D iD 0 k E [ee i ] (199)  11kyxx xy k x y 2 21/2 where E1k (EE1kx 1ky ). Thus, both waves are circularly polarized*. Without y E loss of ggy(p),enerality (explained later), we assume positive 1 x  and Bz . At any fixed position , the field in (198) 0  itikz z Ei1kx () ee  y e rotates opposite to the gyration of electrons in B0ez . left circularly In another view , it rotates in the direc tion of left -hand polarized y fingers if the thumb points to the direction of B0ez , E1 hence the name "left circularly polarized wave". In x contrast, (199) gives a "right circularly pola rized itikz  z Ei1kx()ee ye wave" rotating in the same sense as the electrons. right circularly polarized *Note: The circular polarization is due to Dxx D yy . 42 6.12 Linear Vlasov Waves in Magnetized Plasma (continued)

DiDxx xy 0 (193) Turninggp to the dispersion relations:  DiDxx xy  0 (194) DDxx and xy are given by (186) and (188), respectively.  kz22c 2 221 2 Dxx 1 2  p v   v  (186)    4 n1 n1   2i 221  2 Dvvxyy  p 4      (188)   nn11 kz22c  22 2 DiDxx xy 1 2 2 p [ v  v    nn11  v2   v2  ]   n11  n 2c2  1kz  22v  0 (200) 2   p   n1 Similarly,  2c2 DiD 1kz  22 v  0 (201) xx xy 2   p   43   n1

6.12 Linear Vlasov Waves in Magnetized Plasma (continued) Using (181) and (182), (200) and (201) can be written  ff 00()1kvzz kv zz  vv22 222kc2 2 z vdvdv 3  0 (202)  zp   kv  z   zz   [left ci rcul arly l polidlarized wave]  ff  00()1kvzz kv zz  vv22 222 kc2 2 z vdvdv 3 0 (203)  zp   kv  z   zz   [right circularly polarized wave] These two disp ersion relations in their present forms allow an

anisotropic f 0 [e.g. (103) and (104)], which may lead to an instability (an example will be provided at the end of this section). 22 For an isotropic plasma, ff00 (()[evv  z ) [e. g. (102)], wehaee have ff00 22 . Then, (202) and (203) reduce to vv z 44 6.12 Linear Vlasov Waves in Magnetized Plasma (continued)

 f 0  v 222kc  2 vdvdv 2 0 (204)  zp   kv  z   zz   [left circularly polarized wave]   f 0  222 2v 2 kczp  vdvdv z 0 (205)   kvzz     [right circularly p olarized wave]

Integrating by parts with respect to v , we obtain  f 222kc20(206)20 2  0 vdvdv  (206)  zp   kv  z   zz   [left circularly polarized wave]   222 2 f 0  kczp2    vdvdv z  0 (207)   kvzz    [right circularly polarized wave] 45

6.12 Linear Vlasov Waves in Magnetized Plasma (continued) The basic properties of the waves can be most clearly seen in a 1 cold pl asma. S o, we l et f 0  ()() (vv ) (z ) (208) 2v 2 Note : f d3 v v dv d dv f 1 0000  z Then, (206) and (207) give 2  222 p  kcz  0[left circularly polarized] (209)     2 p 222kc 0[right circularly polarized] (210)  z      As an exercise in kinetic treatment of plasma waves, we have gone through great length to arrive at the above dispersion relations for a cold ppaslasma . In fact, (209) aadnd (210) can be readily deri ved from the fluid equations [see Nicholson, Sec. 7.10; Krall & Trivelpiece, Sec. 4.10]. 46 6.12 Linear Vlasov Waves in Magnetized Plasma (continued) Assume the plasma contains only one ion species of charge e and q B mass mq. IlltiIn all equations,  () (  0 ) carri ithes the sifign of and B . i  mc  0 eB eB 0 0 To be more explicit, we define the notations: e mc ; i mc e i Then, (209) and (210) can be writte n 2 2  222 pe pi kcz []  0 [left circularly polarized] (211)  e   i 2 2 222kc []pe  pi 0 [right circularly polarized] (212) z    e i Each equation can be put in the form of a 4th order polynomial in .

So, for a given kz , there are 4 solutions for  . However, with changed to  , one equation become the other equation. Thus, a negaitive- soltilution of one equati on i s id idtiltentical to a positive- sol uti on of th e oth er equation. So there must be 2 positive- and 2 negative- solutions for each equation. This results in a total of 4 independent solutions. 47

6.12 Linear Vlasov Waves in Magnetized Plasma (continued)

Furthermore, with a change of the sign of B0 , the two equations also reverse. So , without loss of generality [confirming the statement following (199)], we may restrict our consideration to positive-

solutions for a positive Bz00 (i.e. B is in the positive direction). The 4 independent , positive - soluti ons of (211) and (212) in a

positive B0 are shown in Fig. 1 or 2 in four branches, ranging from very low to very high frequencies. Various waves in these branches will b e cl assifi ed b el ow accordi ng to th eifir frequency range. (See Nicholson, Sec. 7.10 & 7.11; Krall & Trivelpiece, Sec. 4.10). Fig. 1 Fig. 2 2 1 2 e e (right circularly polarized)  (right circularly polarized) whistler 1 i i (left circularly polarized) (left circularly polarized) kz kz48 6.12 Linear Vlasov Waves in Magnetized Plasma (continued) A. High frequency electromagnetic waves - Faraday rotation

For high frequency waves () (  i ), the ion terms in (211) and (212) can be neglected. Thus,  2 222kc pe 0 [left circularlyyp polarized ] (213 )  z  e  2  222 pe  kcz 0 [right circularly polarized] (214)  e The high frequency branches are plotted in the top two curves in Figs. 1 and 2. Fig. 1 Fig. 2 2 1 2 e e (right circularly polarized)  (i(right hti circul arl y pol ari zed) whistler 1 i i (left circularly polarized) (left circularly polarized) kz kz49

6.12 Linear Vlasov Waves in Magnetized Plasma (continued)

Setting kz  0, we find the cut-off frequencies of the two branches: 2  2  1  ee (215)   pe 42 2

The 2 figures differ in plasma densities. When  pe 2 e , we

have 11epeee (Fig. 1). When   2  , we have  (Fig. 2).

As B0  0 (e 0), the 2 branches coalesce with the same cutoff 222 2 frequency pez and the same dispersion relation  kc  pe  0, consistent with (143).

Fig. 1 Fig. 2 2 1 2 e e ((gright circularl ypy polarized ) (right circularly polarized) whistler 1  i i (left circularly polarized) (left circularly polarized) kz kz50 6.12 Linear Vlasov Waves in Magnetized Plasma (continued) As shown in Figs. 1 and 2, at a given frequency, the right circularly polarized wave has a greater phase veloc ity than the left ciucularly polarized wave. Hence, if a linearly polarized wave is injected into the plasma, it may be regarded as the superposition of a right circularly polarized wave and a left circularly pol arized wave of equal amplitude , each traveling at a different phase velocity. The combined wave is still linearly polarized but its E field (i.e. its polarization) will rotate as the wave propagates. Thi s i s call ed th e F aradttidilitdday rotation and is exploited for plasma density measurement because the degree of polarization rotaion depends on the plasma density. In an unmagnetized plasma, there is no electromagnetic wave below the cutoff frequency  pe . A magnetized plasma, however, can support other branches of electromagnetic waves at frequencies below the cutoff frequencyies of the top two branches, as discussed below. 51

6.12 Linear Vlasov Waves in Magnetized Plasma (continued) B. Intermediate frequency electromagnetic waves - whistler wave and electron cyclotron wave

In the intermediate frequency range, we still have   i , hence (213) and (214) still apply. (213) has no other solution in this range. (214) has a solution marked as "whistler" & "electron cyclotron wave " in Figs. 1 and 2. The electron cyclotron wave can be exploited for electron cyclotron resonance heating since it has the same frequency as the el ect ron cycl ot ron f requency and it rot at es i n th e same sense as the electrons. Fig. 1 Fig. 2 2 1 2 e e ((gright circularl ypy polarized ) (right circularly polarized) whistler 1  i i (left circularly polarized) (left circularly polarized) kz kz52 6.12 Linear Vlasov Waves in Magnetized Plasma (continued)

For the whistler wave, the group velocity vddkg (  /z ) increases as  increases . When a lightning stroke o n earth generates a pluse of EM waves containing many frequencies, the pulse may reach the ionosphere and propagate along the earth magnetic field as a whistler wave. Some the wave will eventually leav e the ionosphere to impinge on the earth, where it can be received by a radio and generate a sound like that of a whistle, hence the name whistler wave. The radio signal hdtilhas a duration longer th ththiilan the original pulblse because component tts at

different frequencies travel at differnet vg in the ionosphere. Fig. 1 Fig. 2 2 1 2 e e ((gright circularl ypy polarized ) (right circularly polarized) whistler 1  i i (left circularly polarized) (left circularly polarized) kz kz53

6.12 Linear Vlasov Waves in Magnetized Plasma (continued) C. Low frequency electromagnetic waves - Alfven wave and ion cyclotron wave As the frequency gets lower, the ions participate more and more.

At frequencies near or below i , the ions play a major role and we must use (211) and (212). In the vicinity of i , (211) gives the low- frequency end of the whistler wave (slightly modified by the ions), and (212) gives a new wave called the ion cyclotron wave.

Fig. 1 Fig. 2 2 1 2 e e (right circularly polarized)  (right circularly polarized) whistler 1  i i (left circularly polarized) (left circularly polarized) kz kz 54 6.12 Linear Vlasov Waves in Magnetized Plasma (continued)

When   i , we have 11 (1  )1 1 (1 ) (216)      ei,, ei ei , ei , ei , 11 (1 )1 1 (1 ) (217)      e,,iii e i e ,i e ,i e ,i Sub. (216) and (217) into either (211) or (212), we get the same results (i.e the two low-frequency branches merge into one): 2 2 222kc 2 (pe  pi ) 0, (218) z 22 e i 2 22 Since pppe  pi  pi , we neglect the electron term to get 22 e e i i 2 4ne222 mc 2222kcpi  2222 kc ii0  0 (()219) zz222m eB i i  0 4 2 nmcii0 B2 55 0

6.12 Linear Vlasov Waves in Magnetized Plasma (continued)

Defining a speed VBA (called the Alfven speed) in terms of 0

and the ion mass density ii nmi0 :

B0 VA  , (220) 4i we obtain from (219) the diepersion relation of the Alfven wave: 22 22 kV 222kc  c  0 or 2 z A (221) z V 2 22 A 1/VA c

Fig. 1 Fig. 2 2 1 2 e e (right circularly polarized)  ((gright circularly yp polarized) whistl er 1  i i (left circularly polarized) (left circularly polarized) k kz z56 6.12 Linear Vlasov Waves in Magnetized Plasma (continued) Physics of the Alfven wave : We first develop the useful concept of magnetic pressure and magnetic tension. Using 4 the static law: BJ  c (approximately applicable at very low frequencies), we may express the magnetic force density f (for ce per unit volume) entirely in terms of the B -field: a solenoid B-field fJB11 ( BB ) lines  c 4 a uniform ()()(ab  a  b+b  ) a+a  ( b )+( b  a ) electron beam 2 B 1 (BB  ) (222) 84    magnetic pressure magnetic tension force density, force density as if a curved B -field line B-field tended to become a straight line . B lines In regions where Jf 0, we have 0, i.e. the pressure and tension force densities cancel out. uniform current 57

6.12 Linear Vlasov Waves in Magnetized Plasma (continued) Return to the Alfven wave. Since the two lower branches merge

at   i , the left and right circularly polar ized waves have the same phase velocity. So a linearly polarized wave will remain linearly polarized (no Faraday rotation). The figure below shows a linearly

polarized wave with BE11 in the yx-direction a nd in the -direction . Since   ie , , the electron and ion behavior can be described

by their EB1  0 drift motion (same speed and same direction). The wave elect ri c fi eld E1xex cause b oth th e el ectditdiftitrons and ions to drift in the  y -direction, while the wave magnetic z total B -field

field B1ye y bends the external B0 in the direction of the plasma drift (see figure). A quantitative analysis (Nicholson, y pp). 163) shows that the field lines and Linearly polarized EBE =eEB; B  e the plasma move together as if the field 11x x 11y y lines were "frozen" to the plasma (or the plasma frozen to field lines). 58 6.12 Linear Vlasov Waves in Magnetized Plasma (continued) On the other hand, when the magnetic field lines are bent, there is a "tension force density" on the plas mawhichactsasarestoringma, which acts as a restoring force to drive the plasma back so that the field lines (which are frozen to the plasma) become straight. As the field lines are straightened, the momentumoftheplasmacarriesthefieldmomentum of the plasma carries the field lines further back , thus bending the field lines again, in the opposite direction. The tension force then acts again to start another oscillation cycle.

NtNote thtthat we h ave assumed k  kzze ; h ence, all quantiti es vary only with the z -variable. This implies z total B -field that, at a given time, the E1xeBx  0 drift in the y -direction has the same speed at all points along y . Thus, y the drift motion will not compress/ Linearly polarized EBE =eEB; B  e decompress the plasma to produce 11x x 11y y a density variation. The plasma remains uniform in the processes. 59

6.12 Linear Vlasov Waves in Magnetized Plasma (continued) Alternative derivation of the Alfven wave dispersion relation :

The EB1  d rift s cause th e pl asma el ect ron andid ions t o move i n the same direction with the same speed, hence generating no current.

However, there is another drift motion due to the time variation of E1 , which results in a polarization drift current given by (2.43) of Sec. 2.5: 22 mmcicE1  it ikz z JEp 221ke BB00t 2 icm or JEpk 2 1 k , (223) B0 where m  nmii00 nmee is the plasma mass density . Note that the polarization drift speed is much greater for the ions than electrons.

J pk plays a critical role in the Alfven wave. It generates the wave magnetic fi eld B1k and h ence th e magneti c t ensifion force d ditIensity. In fact, we may derive the dispersion relation based on (223). 60 6.12 Linear Vlasov Waves in Magnetized Plasma (continued) 2 icm Sub. JJ1111kpk2 E k [(223)], E k Ek kx ex , and k zz e B0 into the field equation derived earlier: kkE ( ) 2 E 4 i J (125) 11kkcc22 1 k we obtain 2 2 2  4m  (kEzx22 )1kxe  0, cB0 which gives the same dispersion relation as (221):

22222c  kcz 2  0 VA

61

6.12 Linear Vlasov Waves in Magnetized Plasma (continued)

Waves propagating perpendicular to Bk0 ezz (0) : 22 Assume isotropic distribution, ff00 (vv  z ), we have ff00 22 . Then, with kz  0, we obtain from (173)-(178) vvz  2 f 0 vJz n n  v2 Dvdv2 2  2 dv z (224) xz   p 0  z  n k n  dJ f vvJ n  0 z  n 2 2i 2  dkv(/) vz Dvdvdvyz  p 2 z (225)  n 0  n 

2 f 0 vJz n n  v2 D2 2  2 v dv dv  (226) zx  p 0  z  n k n  dJn f 0 vvJz n  dkv(/) v2 Dvdvdv2i 2 2   (227) zy  p 0  z  n n  62 6.12 Linear Vlasov Waves in Magnetized Plasma (continued)

Factoring out the vz -integrals from (224) and (225), we have

 f 0 DDyz , xz dvv z z  vz2  f  11dv  0 f ()v 0 22 zzvz  0  Factoring out the vz -integrals from (226) and (227), we have  f DD ,  dvv 0 0 zx zy z z 2 v

because fv 0 is an even function of z . Thus,,() DyzDDD xz zx zy and (147) reduces to  DxxD xy 0 E1kx  DyxDE yy 001ky  ( (8)228)  00D zz E1kz 63

6.12 Linear Vlasov Waves in Magnetized Plasma (continued) where 2 p n22 f D1122 2 2 v dv dv J 2  0 (229) xx  n  p 0  z k2 n 2  n   v 2  2i p nv  DDxy yx    2 vdvdv z  n n  0  k dJn f 0  Jn (230) dkv( /) 2   v 22 2  kc 2 p 2 Dyy122   vdv dvv z    n n  0  dJ f  [n ]2  0 (231) dkv(/) 2  v 2 k2c2 p  f D 1  2 2vdv dvvJ2 2  0 (232) zz 2   n 0  zz n 2  n  vz In (229)- (232), the argument of al l Bessel functions is kv  /  .

In the limit of a cold plasma (vv ,z  0), we only need to keep the lowest-order, non-vanishing terms in the sum over n . 64 6.12 Linear Vlasov Waves in Magnetized Plasma (continued) (1) n Using (183): limJx ( )1 (xx )nn ; lim J ( x ) ( ) , xx00nnnn!2 ! 2 we find that the lowest-order, non-vanishing terms for Dxx are the nnD1 terms. Thus, the sum over in xx is 2 p n22 f 2vdvdvJ 2  0  n 0    z k2 n 2 n   v 222 kv22f []2pp  vdv dv    0   0  z 222  k 4 v 2  f  p dv22 dv v  0 22 0  z 2  v 2 2  integration by parts over v p 2  22dv dvz f 0   0  1  fldfor a cold pll()()asma: f 0  ()()vv z 2 2v  p (233) 22) 2( 65

6.12 Linear Vlasov Waves in Magnetized Plasma (continued) Sub. (233) into (229), we obtain 2 p Dxx  1 22 (234)    Similarly, the lowest-order, non-vanishing terms for Dxy , D yx , and Dyy are also the n  1 terms, and we obtain 2 p DxyDi yx  22 (()235)  ()  22 2 kc p Dyy  1222  (236)    

The lowest-order, non-vanishing term for Dnzz is the  0 term, 2 2 2 k c p which gives Dzz 1 22  (237)   

66 6.12 Linear Vlasov Waves in Magnetized Plasma (continued)

 DDxx xy E ((8)gves228) gives 1kx  0 a adnd D E =0 ( 238) E zz 1kz DDxy yy 1ky Using (234)-(237), we find from (238) the dispersion relations: 2 2 p p 1 22 i 22 DDxx xy   ()     0 (239) 2222 DDxy yy pp kc i22 1 2 22 () 22 2 kc p and Dzz  122   0 (240) 

Note : We have assumed ke k xzz k e ex [(145)]. Thus, with kz  0, (239) and (240) k apply to waves with ke k x . This explains B0 ez why DDxx and yy are unequal, although the e y system is isotropic in xy and 67

6.12 Linear Vlasov Waves in Magnetized Plasma (continued) Ordinary mode : (see Nicholson, Sec. 7.9 for a fluid treatment) 22 2 kc p Rewrite Dzz  122   0 (240)   Since pe  pi , we may neglect

the ion contribution and get  p e 2222 k kc pe 0 (241) This i s th e di spersi on rel ati on f or the or dinary mod e. Th e ordi nary mode is a pure electromagnetic mode, which propagates in a direction

perpendicular to BB00eezz , with the electric field parallel to . The dispersion relation (241) has the same form as that of electromagnetic

waves in an unmagnetized plasma [see (143)] ex because the electron motion is along B0ez and k B0 hence is unaffected by the external magnetic ez E1 field. e y 68 6.12 Linear Vlasov Waves in Magnetized Plasma (continued) General properties of modes in D D D2 0: xx yy xy  DDxx xy E Rewrite the field equations: 1kx  0 (238) E DDxy yy 1ky and the dispersion relation: DD D2  0 (242) xx yy xy (238) gives the following information about the modes in (242):

(1) Eee1 [=EE11kxxy + ky ] of these modes lies on the xy - plane. Dxy (2) Under (242), either equation in (238) gives EE11kxD ky e xx From (239), we see that Dxx is real and x k DEExy is imaginary. Thus, 11kx and ky differ by a factor "iEE ", implying and 11xy B0 E1 e are 900 out of ppghase while having z unequal amplitudes ( E  E ), 1x 1y e y An ellipse on the xy - plane traced by the E vector i.e. E1 is elliptically polarized. 1 69

6.12 Linear Vlasov Waves in Magnetized Plasma (continued) (3) As shown in the figure, we have ex Ee1=EE11kkkx x + ky ey  k ke k x Thus, in general, these modes are E B0 1 ez neihither el ectrostati c ( kE1 0) nor electromagnetic (kE1 0), except e y An ellipse on the xy - plane at particular freqq(uencies (such as traced by the E1 vector

 ) or wave numbers (such as k  ). Consider, for example, the relative amplitude of EE11kx and ky in the relation: D EE  xy 11kxDxx ky If, for some  or kD , we have xx 0. Then, Ee1 E1kx x and the mode becomes electrostatic . If , Dkxy  0 at some  or  , then Ee1  E1ky y and the mode becomes electromagnetic. In either case, 70 E1 also becomes linearly polarized. 6.12 Linear Vlasov Waves in Magnetized Plasma (continued) Extraordinary mode: (see Nicholson, Sec. 7.9 for a fluid treatment) At hig h freque nc ies, we may neg lect tteohe ion cotbuto.contribution. The e,n, 2 2  1 pe i pe e 22( 2 2) (239) can be written e e  0, (243) 22 22 pe ekc pe -1i 22  2 22 ()ee    eB where, as before,  0 . (243) gives e mce 2 22 242 pe kc pe pe e ()() 122 1 2 22 222 0 (244) eee ()  After some algebra, (()244) can be written 22222 2 22 22 22 ()pe ee ()  pe  identical to kc  22 2 = 22 ,  (245) pe e  UH (7.178) 2 where UH , called the upper hybrid frequenc y, is defined as 222 UH pe e (246) 71

6.12 Linear Vlasov Waves in Magnetized Plasma (continued)

2 22 22 22 ()pe  e Rewrite kc  22 (245)  UH This is the dispersion relation for the extraordinary mode. It has two branches with the following limiting frequencies:

eeeB0 /( mc ) 2 1 2 ee UH kk0;  pe       42 kc 2    Thus, as shown in the figure, the frequency of the lower branch  ordinary mode  2 goes from 1 to UH and the UH pe frequency of the upper branch goes 1 extraordinary mode from  to infinity. Note that at 2 k UH , we have DExx 0 . Hence , E11xex [see (238)] and the wave is electrostatic (called upper hybrid resonance). As   ,

we have DExy  0E11  yey and the wave is electromagnetic. 72 6.12 Linear Vlasov Waves in Magnetized Plasma (continued) Magnetosonic Wave : (see Nicholson, Sec. 7.12 for a fluid treatment) 22 pp   122 i 22 DDxx xy ()  Rewrite  0 (239) 2 22 2 DDyyyyx yy p kc p i22 1 2 22 ()  22 At very low frequencies (  i ), ions play a major role and we 22 must ret ai n th e i on t erms i n (239) . U nd er the con dition:   i 222 2 pppep  i Dxx  122  1 2  1 2  2     e i 22 2 222 22 2 kc p kkccpp i Dyy  1222   1 221 22   i 222 2 p ppep i   i DDixy yx  22  i = []   0 ()  e i 2 = pi/  i 73

6.12 Linear Vlasov Waves in Magnetized Plasma (continued)

Thus, (239) gives DDxx yy 0. Because D xx 0, we have 22 2 kc pi Dyy  122   0 (247)  i

(247) has a form identical to (219) if kz in (219) is replaced with k .

Thus, the solution is simply (221) with kkz changed to  : 22 2 kV A   22 (248) 1/VcA where the Alfven speed VA is defined in (220) as B0 VnmA  [][ iii0 ] (220) 4i (248) gives the dispersion relation for the ex magg,netosonic wave. Because D  0, it has an k B yy 0 e electric field EeE . Hence, kE 0 and B z 11y 1 e 1 y E the wave is electromagnetic with Be11 B z . 1 74 6.12 Linear Vlasov Waves in Magnetized Plasma (continued) A physical picture : Like the low-frequency Alfven wave, particle dynamics can be described by 2 types of drift motion. The EB1eeyz 0 ik x drifts move the plasma in the  xE -direction. Since 1 ( e ) also varies with x , the drift motion will compress/decompress the plasma, resulting in a density variation along x. On the other hand, the wave magnetic field BB1eezz , when superposed with 0 will cause a similar density variation of the magnetic field lines (see figure). The field lines are again frozen to the plasma, si milar to the Alfven wave. However, the restoring force (thus the oscillation mechanism) is now provided by the magnetic pressure force density: B2 / 8 [(222)]. As in the Alfven wave, there is ex also a polarization drift current in the k y-direction due to the time variation B1 of E e . This current generates the 1 y B0 ez wave magnetic field B e , hence E1 1 z B1 the magnetic pressure. e y 75

6.12 Linear Vlasov Waves in Magnetized Plasma (continued) Alternative derivation of the magnetosonic wave dispersion relation : The pol ari zati on d rift current i s gibiven by 22E mmcic1  itikx  JEp 221ke BB00t 2 icm or J pk  2 E1k , (249) B0 2 icm Sub. JJ1111kpk2 EE kkky, Ek ey , and k  ex into B0 the field equation: kkE ( ) 2 E 4 i J (125) 11kkcc22 1 k 2 2 2  4m  we obtain (kE 22 )1kye y  0 cB0 22222c or  kc 2  0, VA which gives the same dispersion relation as (248):

76 6.12 Linear Vlasov Waves in Magnetized Plasma (continued) Asymptotic behavior of the magnetosonic wave dispersion relation : The dispersion relation for the mag netosonic wave: 222 22  kV AA /(1 V / c ) (245) 22 is valid under the condition   i . It breaks down as k . To 22 22 find the behavior at k  , we assume   i and  e . 2 2 2 ppep   i Then, Dxx  1 22  1 2  2 and from   e 22 pp 122 i 22 DxxxyD ()   0 (239) 2 22 2 DDyx yy p kc p i22 1 2 22 ()    we find that, as k  but  remains finite, we must have 2 2 2 p pe pi Dxx  1 22  1 220, (247)    e  77 which implies E11 E xex [see (238)]. Hence, the wave is electrostatic.

6.12 Linear Vlasov Waves in Magnetized Plasma (continued) 2 2 pe pi Rewrite 122 0 (247) e  (247) gives 22 if  pe  e 22 22 22ppiiee 22 2 e i LH , [for k  ] (248) e pe pe

where LH e i (()249) is called the lower hybrid frequency. extraordinary mode This justifies the assumption: ordinary mode 22 22   i and  e 2 UH we made in obtaining (248). pe  Finallyypp, all the perpendicular 1 magnetosonic wave LH modes (kB 0 ) discussed so far are summarized in the figure. k  78 6.12 Linear Vlasov Waves in Magnetized Plasma (continued) Discussion : (()i) We have covered a number of the most familiar modes in a uniform plasma in the framework of the kinetic theory. These modes are treated in Ch. 7 of Nicholson by the fluid theory. However, some other familiar uniform - plasma modes have been left out , for example , the Bernstein modes (Krall & Trivelpiece, Sec. 8.12.3). (ii) We have only considered waves either along or perpendicular to BB0eezz. In practice, waves can exist at any angle to 0 (Nicholson , p. 165), with complicated expressions and mixed properties. The general dispersion relation (149) can be the basis for a detailed study of such uncovered uniform- plasma modes. (iii) There are also modes which are not contained in (149). For example, an inhomogeneous (equilibrium) distribution in density or temperatitdture introduces new mod es, such hdif as drift(Nihlt waves (Nicholson, Sec. 7.14; Krall & Trivelpiece, Secs. 8-15 and 8.16). Plasmas in 79

6.12 Linear Vlasov Waves in Magnetized Plasma (continued) some devices (e.g. tokamaks) are futher complicated by a complex magnetic field configuration . Such plasm as are usually the subjects of research papers. (iv) Modes considered in Sec. 6.12 are for a cold plasma. Basic properties and th e und erl yi ng ph ysi cs are more clearl y exhibit ed i n this limit. However, cold modes are stable because there is no free energy to drive an instability. In the next topic, we will demonstrate shhhow how a mod e can b ecome unstabl blie in an aniilisotropic plasma. (v) The relativistic Valsov equation can be derived by the same steps as in the derivation of the Vlasov equation in Sec. 6.1. For the case we considered (EBe000 0,B z ), the relativistic factor  is a constant of the motion in zero-order orbit equations. Hence, the derivation of the relativistic dispersio n relation takes exactly the same steps which lead to (171)-(182). In Special Topis I, we will derive the relativistic Vlasov equation and consider a relativistic instability. 80 6.12 Linear Vlasov Waves in Magnetized Plasma (continued) A Slow - Wave Instability on the Electron Cyclotron Wave : The dispersion relation for "right circu larly polarized waves" is ff 001kvzz kv zz 22() 222 2vv z 3 kcz 2  p vdvdvz  0 (203)   kvzz  At high frequencies, we may neglect the ions. Then, for a cold 2 eB 222 pe 0 plasma, (203) reduces to  kcz 0, [] [e  mc ] ( 214) e e which gives the 2 "right circularly polarized" branches in the figure. Below, we will show that the "electron cyclotron wave" portion of the lower branch can be 2 destabilized by an anisotropy in e  (right circularly polarized) velocity distribution, resulting 1 i in a velocity-space instability. (left circularly polarized) 81 kz

6.12 Linear Vlasov Waves in Magnetized Plasma (continued) For an anisotropic plasma, we cannot use (214), but must go back to (203). Again, assume high frequency a nd neglect the ions . (203) ffee00(1kvzz ) kv zz vv22 gives 222kc 2 2  z v3 dv dv  0 (250) z pe kv z  zze  The integral I can be written I  f e0 1kvzz kz  f e0  ()  11v 23vz I v dv dvzz v  dv dv 22kvzzee  kv zz 2  integration by parts ff()1kvzzk z ee00vdvdv 1 vdvdv3 z 2 2 z  kvzz e  (kvz z e) 1 Now, assume f e0  ()(),vv 0 vz [see (104)] (251) 2v which represents a uniform distribution of electrons in random-phase gyrational motion, with vv0 and vz 0 for all electrons. 82 6.12 Linear Vlasov Waves in Magnetized Plasma (continued) kv22 111 z 0 Then, I []  2 (252) 22e ()e Sub. (252) into (250), we obtain the dispersion relation: kv22 222 2  1 z 0  kcz  pe[]  2  0,,() (253) e 2 ()e which reduces to (214) as v0  0. frequency The two right circularly polarized branches are plotted for  pe 10 e and vc0  0.2 . The top figure plots r (wave frequency) vs kz . The upper branch is a fast wave (()r / kcz  ), while the lower branch is a slow wave (r / kz growth rate  c). The bottom figure plots i (growth rate) vs kz . We see that the slow wave is destab ilized by the gyrational particles, which feed energy to the wave through cyclotron resonances. 83