More Plasma Physics: Looking At Nonlinear Electrostatic (BGK) Waves (Part II)

 We combine equations (a), (b) and (c) from end of previous post to obtain an equation for  n_s  in terms of:  v_so ,     n _so   and  q _s j .

Then:

n _s  =  n _so  v_so / v_s

And obtain   v_s  to substitute into the preceding:

v_s  =  Ö (v_so ²   -    2q _s j  /m _s )

Þ

n _s  =  n _so  v_so /Ö (v_so ²   -    2q _s j  /m _s ) 

Yielding same result as previous method.

Now, assume:   n _so  =  n _o

Then: 

d ²j/ d x²    =    

4 p  n _o e (n _e – n _i) [(1/ Ö 1 + 2e j /m _e v_eo²    -   (1/ Ö 1 + 2e j /m _i v_io² )

 =    -   ¶ V( j) / ¶ j

Where V( j)  is the pseudo potential or Sagdeev potential,

 On integrating:

V( j)  = 

- 4 p n _o { m_e v _eo ² (Ö (1+ 2e j/m_e v _eo ² )  +  m_i v_io ² Ö 1 – 2 e j)/ m_i v_io ² )

And:  m_e v _eo ² =   m_i v_io ²   = const.

The wave speed is fixed with respect to the relative motion of the electrons, ions.  This allows a determination of the wave speed, i.e. we look for a class of nonlinear waves which satisfy this equality. (This leads to the "BGK" waves.)

In pursuing shock solutions-solitons it is customary to make the above non-dimensional to simplify it. To that end, one can use the non-dimensional constant:

 F = e j/ T_e

 And:   x =  x/ l_e ²    where    l_e  = v _eo  / w_e Ö(2)

Then:   d ² F/ d x ²  =   exp (F) -  1Ö(1 – (2 e F)/ (M _s )²  =    - ¶  y/ ¶ j 

Which is the dimensionless Poisson equation

Where:   Y  =  exp (F) - 1/ Ö(1 – (2 e F)/ M _s ²)  +  C

 One requires for the solution:

1)  y _I  (0)  =  0       Þ    C 1 =  1 +  n _ot  M _s ² +  2/3   F _max n _or 

2) y _I  (F _max ) =  y _II (F _max  )  Þ    C 1 =  C2

 3) y _I " (0)  <  0   Þ     M _s ²   > n _ot  / 1 +  n _or /2F _max  

_{4) y (F max ) =    0}  

Þ 

 1 - exp (F) +   n _ot  M _s ² (1 -  Ö(1 – (2 e F)/ M _s ²)  +  2/3   F _max n _or = 0

(Also:   n _ot    +   n _or   =   1)

5) y _II (F)   >   0

I.e. One solves for y _II (F)  =   0  over:

F _min   <  F  <   F _max      And seeks a monotonic function

For reference, a sketch of the graph of the relevant potentials is given below which in terms of the dimensionless Poisson solutions can be regarded as the shock profile:

The maximum  V( j) is limited by k, the input constant related to the streaming speed v_s .  This type of potential is able to support a wave train solution only. If the potential is symmetric the wave form will be completely symmetric i.e. like a sine wave of j( x)  vs. x.

We next revisit the equation:

d ²j/ d x²    =    

4 p  n _o e (n _e – n _i) [(1/ Ö 1 + 2e j /m _e v_eo²    -   (1/ Ö 1 + 2e j /m _i v_io² )

And take:

ò   (n) x   2d j/ d x  Þ 

d/dx (d j/ d x ) ²    =  

4 p  n _o 2k  d/dx [Ö 1 + 2e j /k    -   (1/ Ö 1 -  2e j /k )]

ò   d j/ [Ö 1 + 2e j /k    -   (1/ Ö 1 -  2e j /k )]  =   x + 

And  x =   f (k, j )

The preceding is a sketch of one particular type of BGK wave with the wave train form of solution that can only support a cold streaming plasma.

Next:  Case 2 - Application to ion-acoustic solitons and shocks