The Poisson equation in the context of electrostatic shocks can be written:
d 2j/ d x2 =
4 p e n o (1/ Ö (1+ 2e j/me v eo 2 – 1/ Ö 1 – 2 e j)/ mi vio 2 )
= ¶ v (j) / ¶ j
Where v (j) is a pseudo-potential or the Sagdeev potential. On integrating:
v (j) =
- 4 p n o { me v eo 2 (Ö (1+ 2e j/me v eo 2 ) + mi vio 2 Ö 1 – 2 e j)/ mi vio 2 )
And: me v eo 2 = mi vio 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/ Te
And: x = x/ le 2 where le = v eo / we 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:
No comments:
Post a Comment