Abstract It has been proved theoretically that particle acceleration along and across a magnetic field and electric field across a magnetic field can be induced by nonlinear Landau damping of almost perpendicularly propagating electrostatic waves in a relativistic magnetized plasma

M Sugawa

R Sugaya

T Maehara

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Cross-Field Plasma Acceleration and Potential Formation Induced by 
Electromagnetic Waves in a Relativistic Magnetized Plasma 
 
                   R. Sugaya, T. Maehara and M. Sugawa 
 
             Department of Physics, Faculty of Science, Ehime University 
                   Bunkyo-cho 2-5, Matsuyama 790-8577, Japan 
 
Abstract 
  It has been proved theoretically that particle acceleration along and across a magnetic field 
and electric field across a magnetic field can be induced by nonlinear Landau damping of 
almost perpendicularly propagating electrostatic waves in a relativistic magnetized plasma. 
 
1.  Introduction 
  Particle acceleration along and across a magnetic field and generation of electric field 
transverse to the magnetic field, both induced by nonlinear Landau damping (nonlinear 
wave-particle scattering) of almost perpendicularly propagating electromagnetic waves in a 
relativistic magnetized plasma have been investigated theoretically on the basis of the 
relativistic transport equations derived from relativistic Vlasov-Maxwell equations1-6.  Two 
electromagnetic waves interact nonlinearly with particles, satisfying the resonance condition  
of  ( ) ( ) 2/k k d cs dk k v k k v mω ω ω γ′ ⊥ ⊥′ ′− − − − − =  , where v  and 0 0/dv cE B=  are the 
parallel and perpendicular velocities of particles, respectively, ( ) 1/ 22 21 /d dv cγ
−
= − and  
0 /cs s s se B m cω γ=   ( ( )
1/ 22 2 21 /s sp m cγ = + )  is the relativistic cyclotron frequency.     The  
relativistic transport equations show that the electromagnetic waves accelerate plasma 
particles in the ′′k  direction ( ′′ ′= −k k k ).  As a result, the strong plasma acceleration or 
transport across a magnetic field appears.  Simultaneously the intense cross-field electric 
field E0=B0×vd/c is generated via the dynamo effect of perpendicular particle drift to satisfy 
the generalized Ohm’s law, that is, the electromagnetic waves can produce the cross-field 
particle drift that is identical to E×B drift.  Moreover, the relativistic transport equations for 
the relativistic cross-field particle acceleration were derived by means of Lorentz 
transformation of the relativistic momentum-space diffusion equation in the frame moving 
with the cross-field drift velocity3,5.  They can be applied usefully to the theoretical study of 
the highly relativistic electron beam acceleration in a magnetized plasma5,7,8 as well as the 
relativistic cross-field particle acceleration which may occur possibly in space plasmas. 
 
2.  Relativistic Transport Equations 
  The relativistic kinetic wave equations and the relativistic transport equations for nonlinear 
Landau and cyclotron damping of two electromagnetic waves propagating almost 
perpendicularly in a homogeneous relativistic plasma immersed in uniform magnetic and  
electric fields  ( )0 0 ,0,0B=B   and  ( )0 00, ,0E=E   can be derived from the relativistic  
 
                                    - 1 - 
33rd EPS Conference on Plasma Phys. Rome, 19 - 23 June 2006 ECA Vol.30I, P-1.055  (2006)
Vlasov-Maxwell equations1-6.   At first those for the nonrelativistic cross-field particle 
acceleration ( / 1dv c ) are given as follows: 
 
       
* *
, ,
, , ,
* *
, ,
, , ,
( ) ( ) ( )
, ,
, , ,
2 , (1)
4
2 , (2)
4
2 2
4
jl l l j l l l
j l l l
jl l l j l l l
j l l l
s s s jl l ls
j l l l
U
U A E E E E
t
U
U A E E E E
t
U
U U A E
t
ω
γ
π
ω
γ
π
ω
γ γ
π
′′ ′ ′ ′′
′′ ′ ′ ′
′ ′′
′′ ′ ′ ′′′ ′
′ ′ ′′ ′ ′ ′
′ ′′
′′ ′′′
′ ′ ′′ ′
′ ′′
∂
= +
∂
∂
= −
∂
∂
= − − −
∂
∑
∑
∑
k k
k k k k k k k k k
k k
k k k k k k k k k
k
k k k k k k k
* *
( ) ( )
( ) * *
, ,
, , ,
, (3)
2 2
, (4)
4
j l l l
s s
s jl l l j l l ls
j l l l
E E E
U U A E E E E
t
γ γ
ω ω π
′ ′′
′ ′
′′ ′ ′ ′′′
′ ′′ ′ ′ ′
′ ′′′
′ ′′∂
= − − −
∂ ∑
k k k k
k k
k k k k k k k k k
k k
P k k k
 
 
where ( ), , ( , , , , , )x y zE E E j l l l x y z′ ′′= =k k k kE  is the wave electric field, *− =k kE E ,  
*ω ω− = −k k , ω ω ω′′ ′= −k k k , ( ),0,k k⊥=k , ( ),0,k k⊥′ ′ ′=k  and the wave energy density Uk , 
the energy and momentum densities of particles of species s, sU  and sP  are given by 
( ) ( )( )*1/ 8U π ω⎡ ⎤′= ∂ −⎣ ⎦k k k k k kE Ei iε N , ( )( )2 2 2/jl jl j lN c k k kω δ= −k ds s sU w g= ∫ p , s =P
( )d ,0,s s s sg P P⊥=∫ pp , s s dsP n p⊥ = , ds s s dp m vγ= , s s s s sP n m vγ= , 2s s s sw n m cγ= , s sn=p p  
and ( ){ }1/ 22 20 , ,s s x ds yg g p p p p t⎡ ⎤= − +⎣ ⎦  ( 0 0 ( , , )s sg g p p t⊥= ) is the background relativistic 
momentum distribution function of particles of species s containing the fluctuation-induced 
cross-field drift velocity dd x sv v g= ∫ p ( ( ) 20 0 0,0,0 /d dv c B= = ×v E B , / 1dv c ).  The 
linear damping rates Imγ ω=k k  and 
( )sγ k  are shown by 
( )s
s
γ γ=∑k k  and 
( )( )( ) * ( )/ 8s sUγ ω π ′′= −k k k k k kE Ei iε , the dielectric tensor ( )s
s
=∑k kε ε  is described in Ref. 6, 
and the relativistic nonlinear wave-particle coupling coefficients , ,
jl l lA ′′ ′′′ ′k k k  are represented by 
 
      ( )( ) ( ) ( ) ( ), , , , , , , , , ,,jl l l s jl l l s jl l l s jl l l s jl l lAH
s
A A A P C D′′ ′ ′′ ′ ′′ ′ ′′ ′ ′′ ′′′ ′ ′′ ′ ′′ ′ ′′ ′ ′′ ′= = − +∑k k k k k k k k k k k k k k k     ,             (5)  
      
2 2
( )
, , , , 02 d ( )
ps ss jl l l jl ll
AH m s
e
P C g
πω
δ ξ
ω ω ω
′′ ′ ′′ ′
′′ ′ ′ ′
′ ′−
= Ω Λ∫k k k k k k k
k k k
p    ,                      (6) 
      ( )
2
( )
, , , , , , 0d ( ) ( )
ps s ss jl l l j ll j l l jl j
AH m m s
j
e m
P D X g
πω
β β δ ξ
ω ω ω
′′ ′ ′ ′ ′ ′ ′′ ′
′′ ′ ′ ′ ′ ′′− −
′′ ′′
= + Ω∑ ∫k k k k k k k k k k
k k k
p p   ,      (7) 
  ( )( )
2
*
, , , 0d ( ) ( ) ( )
j
ps sj ll j l l ll
m ds m s
s
e
J g
πω
β β δ ξ µ Λ
ω ω ω
′
′ ′ ′ ′ ′
′ ′ ′′ ′′ ′′ ′− −
′′ ′−
⎡ ⎤+ = +⎣ ⎦∑ ∫ ik k k k k k k k k
k k k
p u p pκ   ,  (8) 
( ) ( )
2
*
2
0
1d ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) , (9)
s jl s jl
AH
ps j
m m j ds m
s
l
m l ds m m s
P
u J s p J
u J s p J U g
ε ε
πω
δ ξ µ µ
ω γ
µ µ
′′ ′′
′′ ′′ ′′
′′
′′ ′′ ′′
′′ =
⎡ ⎤= − +⎣ ⎦
′′⎡ ⎤× +⎣ ⎦
∫
k k
k k k
k
k k k
p p
p k
 
 
                                    - 2 - 
33rd EPS 2006; R.Sugaya et al. : Cross-Field Plasma Acceleration and Potential Formation Induced by Electrom...2 of 4
  
( )
(
)
( )
*
, , ,2
*
*
1 , 1 , 2 2
1( ) ( ) ( )
( ) ( )
( ) ( ) , (
jl j l
j l m q q q m q m
q d cs
j l
q q mj l
q q q m q m
d cs cs
J a J a J
k v k v q
b J b J
a J a J
k v k v q
Ω η η µ µ µ
ω ω
µ µ
µ µ
ω ω ω
∞
′′ ′′
′ ′′ ′′ ′ ′− −
=−∞ ⊥
′′
′ ′−′′
′ ′− −
⊥
⎧⎪ ⎡ ⎤ ⎡ ⎤= + ⎨ ⎣ ⎦ ⎣ ⎦
+ − +⎪⎩
⎫⎡ ⎤ ⎡ ⎤ ⎪⎣ ⎦ ⎣ ⎦⎡ ⎤ ⎡ ⎤− + ⎬⎣ ⎦ ⎣ ⎦
+ − + − ⎪⎭
∑k k k k k k k
k
k k k k
k k k k
k
10)
 
   
( )
,
, , 2
, ,
1
( )
, (11)
2 ( )
ll
ll l
l l m m
rs cs s d cs
ll ll
s cs d cs d cs
Vk J
k v k v r
V V
k v k v r k v k v rς
Λ ηη µ Ψ
γ ω γ ω ω
ς
γ ω ω ς ω ω ω
′∞
′′ ⊥
′ ′ ′′ ′′
=−∞ ⊥
′ ′
′ ′⊥
=± ⊥ ⊥
⎧′′ ⎪= + ⎨
+ − +⎪⎩
⎫⎪+ + ⎬+ − + − + − + ⎪⎭
∑
∑
k k
k k k k
k
k k k k
k k
 
 
where    m ds s s s s csk p k p m m mξ γ ω γ ω′′⊥′′ ′′= + − +k  ,   
†
AHP′′ ′′ ′′ ′′ ′′′′= − =k k k k ki iκ χ χ ε χ  , 
( ) 1N −′′ ′′ ′′= −k k kχ ε  , and † ′′kχ  is the complex conjugate transpose matrix of ′′kχ .  The 
differential operators are defined as 
 
  
( ) ( )
* *
, , , , ,
* *
, , , ,
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ,
ll l l l l
r r r m r m r m r
l l l l
cs r s s r m cs r m s s r
V k a J k a J
k v r m m k v r m
µ Χ µ Χ
ω Χ γ Χ ω Χ γ Χ
′ ′ ′
′ ′ ′ ′− − −
′ ′
′ ′− −
′ ⎡ ⎤ ⎡ ⎤= − +⎣ ⎦ ⎣ ⎦
′ ⎡ ⎤ ⎡ ⎤+ + − − +⎣ ⎦ ⎣ ⎦
k k k k k k k k
k k k k
p p
p p p p
 
* *
, , ,( ) ( ) ( ) ( ) ,
ll l l l l l l
r r m r m r m r rV k b J J k b J Jς ςµ ξ µ µ Ψ µ ξ µ µ Ψ
′ ′ ′ ′
′ ′ ′ ′ ′ ′ ′⊥ ⊥ − − − ⊥ − −′ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤= − +⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦k k k k k k k k k k k k k k  
*
, , , , ,( ) ( )
ll l l l l l l
s r r m r r m r m rV m J Jµ ξ µ µ ξ µ Ψ Ψ Ψ Ψ
′ ′ ′ ′
′ ′ ′ ′ ′ ′− − −⎡ ⎤ ⎡ ⎤ ⎡ ⎤= −⎣ ⎦ ⎣ ⎦ ⎣ ⎦k k k k k k k k k k k k    ,    
, ,( ) ( )
l l l
r r rJΧ µ ξ µ Ψ⎡ ⎤= ⎣ ⎦k k k k kp   ,  ( ),
x
r cs d cs
k k v
r k v r
k p k v p
ω
Ψ ω ω⊥
⊥ ⊥ ⊥ ⊥
−∂ ∂
= + +
∂ ∂
k
k     , 
,
dy
r cs
k k v k v
k p k v p
ω
Ψ ω ⊥
⊥ ⊥ ⊥ ⊥
⎛ ⎞− −∂ ∂
= +⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠
k
k   , ( ),z r d cs cs
p
k v r r
p p p
Ψ ω ω ω⊥
⊥ ⊥
∂ ∂
= − − +
∂ ∂k k
 , 
     ( ),x q cs d
k
a q k v
k
ω ⊥
⊥
= +k ,  ,
d
d
y
qa ik v µ⊥
=k
k
 ,  ,
z
q d csa k v qω ω⊥= − −k k   , 
  xb k vω= −k k  ,   
d
d
y
csb iω µµ
=k k
k
 ,  xb k v⊥=k    , 
 
  1 , / /
x
q cs da q ck k v v cω ω ⊥= +k k ,  ( )1 , /y q da i k v v cω ⊥ ⊥= −k k  , ( )1 , /z q da k v v cω ⊥= −k k  , 
AHP  indicates the anti-Hermitian part of the tensor, and ( )rU k , ,
j
qak , 
jbk , 
jξk , µk , js , jη   
and psω  are defined in Ref. 6. 
It is found easily that Eqs. (1)-(4) lead to the conservation laws for the total energy and 
momentum densities of two electromagnetic waves and plasma particles,  
 
                                    - 3 - 
33rd EPS 2006; R.Sugaya et al. : Cross-Field Plasma Acceleration and Potential Formation Induced by Electrom...3 of 4
          0s
s
U U U
t ′
∂ ⎡ ⎤+ + =⎢ ⎥∂ ⎣ ⎦
∑ k k    ,  0s
s
U U
t ω ω ′′
⎡ ⎤′∂
+ + =⎢ ⎥∂ ⎣ ⎦
∑ k k
k k
k kP   .    (12) 
The transport equations and the conservation laws (3), (4), (12) predict obviously that the 
electromagnetic waves can generate strong particle acceleration or transport along and across 
the magnetic field via nonlinear Landau and cyclotron damping of electromagnetic waves. 
  Next the relativistic transport equations for the relativistic cross-field particle acceleration3,5  
( / 1dv c < ) are provided by the same way as the previous work described in Ref. 6.  The 
relativistic transport equations in the laboratory (stationary) frame of reference with an 
electric field and ×E B  drift can be derived by means of Lorents transformation for the 
momentum-space integration of the relativistic momentum-space diffusion equation in the 
frame of reference moving with  ×E B  drift velocity dv .   The details are described in 
Ref. 6. 
 
3.  Conclusion 
  It was verified theoretically that relativistic particle acceleration along and across the 
magnetic field and electric field transverse to the magnetic field can be induced by nonlinear 
Landau damping of almost perpendicularly propagating electromagnetic waves in a 
relativistic magnetized plasma.  The relativistic transport equations show that the 
electromagnetic waves produce strong particle acceleration.  Simultaneously the intense 
cross-field electric field is created via the dynamo effect of the cross-field particle drift.  The 
obtained transport equations can be available for the theoretical investigation of the 
relativistic and nonrelativistic cross-field particle acceleration and transport which may occur 
possibly in space and fusion plasmas.  It can be available usefully to the acceleration of the 
highly relativistic electron beam5,7,8.   
  The authors wish to thank Professor Y. Kitagawa for valuable discussions.  This research 
was performed partially under the collaborating research programs at the Institute of Laser 
Engineering, Osaka University, and was supported partially by the Grant-in-Aid for Scientific 
Research from the Japanese Ministry of Education, Culture, Sports, Science and Technology. 
 
 
 
References 
[1] R. Sugaya, J. Phys. Soc. Jpn. 59, 3227 (1990); 60, 518 (1991).  
[2] R. Sugaya, Phys. Plasmas 1, 2768 (1996); 3, 3485 (1996).  
[3] R. Sugaya, J. Plasma Phys. 56, 193 (1996); 64, 109 (2000); 66, 143 (2001).  
[4] R. Sugaya, Phys. Plasmas 6, 4333 (1999); J. Plasma Phys. 70, 331 (2004). 
[5] R. Sugaya, Phys. Plasmas 10, 3939 (2003); 5634 (2004).  
[6] R. Sugaya, Proceedings of 11th International Congress on Plasma Physics, Sydney, 
Australia, 2002, Vol. 669, p. 816 (2002).    
[7] Y. Kitagawa, Y. Sentoku, S. Akamatsu et al., Phys. Rev. Lett. 92, 205002 (2004).  
[8] M. J. Hogan, C. D. Barnes, C. E. Clayton et al., Phys. Rev. Lett. 95, 054802 (2005). 
 
                                    - 4 - 
33rd EPS 2006; R.Sugaya et al. : Cross-Field Plasma Acceleration and Potential Formation Induced by Electrom...4 of 4


Cross-Field Plasma Acceleration and Potential Formation Induced by Electromagnetic Waves in a Relativistic Magnetized Plasma

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Cross-Field Plasma Acceleration and Potential Formation Induced by Electromagnetic Waves in a Relativistic Magnetized Plasma

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