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

The putative basis for BGK (Bernstein- Greene-Kruskal mode) waves in the wave frame is the 1D Vlasov equation, e.g.

v ¶ f _s / ¶ x  - q _s /m _s [¶ je/ ¶ x  · ¶ f _s/ ¶ v]  = 0

Where f _s   is any function of the constant of the motion for particles species s.  And q _s ,  m _s  refer respectively to the charges and masses for plasma species s.

We expect the solution to be a function of velocity and potential, i.e. f _s (v, j)

With general solution:

f _s  =  f _s  (v ²   +    2q _s j  /m _s -  v _s²  )

 With no magnetic field present, the only Maxwell equation we need is Poisson's  e.g.

 ¶ ²j/ ¶ x²    =   - 4 p e (n _e – n _i) =     - 4 p r

For the Poisson equation one obtains the graph below as one "double layer"  solution to the potential equation:

        Solution to the potential equation for cold streaming plasma: electric potential vs distance  (x)

The applicable constant of the motion (total energy) is:

¶ e/ ¶ m _s   =  v ²   +    2q _s j  /m _s -  v _s² 

Where v _s  is the streaming velocity.

Cases to consider: 

1)  Cold, electron-ion streaming plasma,

2) Ion-acoustic solitons and shocks,

 We examine case (1) first for which:

f _s  =  A n _so d  (v ²   +    2q _s j  /m _s -  v_s²  )

Where A is a normalization constant,

n _so  is n _s  the particle species number density at locations where the potential:

j  = 0.

 v _s  is the streaming velocity at j  = 0.

Where:  <  v_s >] _f=0 = v _s

   To get A we first integrate:

  n _s  =   ò -_¥ ¥  f _s  dv      

Þ

 A n _so / 2v   

Evaluated at: v =  Ö v_s ²   -    2q _s j  /m _s

Whence:   n _s  =    A n _so / 2Ö v_s ²   -    2q _s j  /m _s

_{Expression for number density at any potential} j .

Then:    n _s (j  = 0)   =  n _so   =   A n _so / 2  v_so 

Þ

A  =  2  v_so 

Then the distribution function becomes:

f _s  =  2 v _so n _so d  (v ²   +    2q _s j  /m _s -  v_so²  )

And:

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

n _so  / Ö (1 - q _s j  /½m _s v_so²  )

By the continuity equation:

a) n _s  v_s =  const.  =   n _so  v_so 

And the momentum equation:

b) m _s  n_s    dv_s  / dt  =   q _s  n_s E  =   -q _s n_s ¶ j/ ¶ x

Also:

c)  ½m _s v_s²   + q _s j  =  const.  = ½m _s v_so² 

Problem:

Combine equations (a), (b) and (c) above to obtain an equation for  n_s  in terms of:  v_so ,     n _so   and  q _s j .

To be continued...