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Reduced Vlasov-Poisson model and it's instabilities

Reduced Vlasov-Poisson model and it's instabilities. Denis Silantyev 1 Harvey Rose 2 Pavel Lushnikov 1 Mathematics & Statistics Department, University of New Mexico Los Alamos National Laboratory. 8 Jun 2012. Vlasov-Poisson collisionless plasma. - electron density distribution function.

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Reduced Vlasov-Poisson model and it's instabilities

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  1. Reduced Vlasov-Poisson model and it's instabilities Denis Silantyev1Harvey Rose2 Pavel Lushnikov1 Mathematics & Statistics Department, University of New Mexico Los Alamos National Laboratory 8 Jun 2012

  2. Vlasov-Poisson collisionless plasma - electron density distribution function (1) (2) Units: electron charge and mass =1; length, λD; time, 1/ωpe; electrostatic potential, Te/e, with Tethe initial electron temperature.

  3. Framework • One preferred direction - laser propagation direction • Laser intensity is high enough (near the instability threshold) → sparse array of laser intensity speckles • LW energy density is small compared to thermal → the most probable transverse electron speed is thermal Effects to observe: • electron trapping in LW electrostatic potential wells parallel to the laser beam • LW self-focusing ( sparse array of intense laser intensity speckles)

  4. Reduced Vlasov-Poisson Model (Vlasov Multi-Dimensional Model) (0') (1') (2')

  5. Isotropy in transverse direction 2D case. 2 streams 3D case. N streams (uniformly distributed over angle φ) . φ θ

  6. z k θ x - plasma dispersion function - phase velocity Dispersion relation (2D) 2D Vlasov model. 2 streams. u1=u2=u=1 where 2D VMD model. 2 streams. u1=u2=u=1 • well-known cold plasma • two stream dispersion relation При

  7. Unstable Stable 2D case. 2 transverse streams

  8. 2D case. N transverse streams such that

  9. 2D case. N→ transverse streams Vlasov: VMD:

  10. where θ φ Dispersion relation (3D) 3D Vlasov model. N streams. k z 3D VMD model. N streams. y k┴ x When

  11. Stable Unstable

  12. Stable Unstable

  13. Measure of anisotropy

  14. Convergence of anisotropy in 3D as N→ θ1 and θ2match up to 16 digits for N≥24 Llinear fit:

  15. Envelope curve of different cross-sections w.r.t. φ N=4,6,8,10,12

  16. k=0.3 θ→0 Anisotropy (inφ) of Langmuir branch withθ→0

  17. Isotropy (inφ) of Langmuir branch withθ=0 k=0.3 θ→0

  18. Langmuir branchwith θ=0. N=2,4,6,8

  19. Anisotropy (inφ) of Langmuir branch withθ=0.3

  20. Langmuir branchwithθ=0.3 N=2

  21. Langmuir branchwithθ=0.3 N=4

  22. Langmuir branchwithθ=0.3 N=6

  23. Langmuir branchwithθ=0.3 N=8

  24. Anisotropy (inφ) of Langmuir branch withθ=0.5

  25. Langmuir branchwithθ=0.5 N=2

  26. Langmuir branchwithθ=0.5 N=4

  27. Langmuir branchwithθ=0.5 N=6 ~1%

  28. Langmuir branchwithθ=0.5 N=8

  29. Anisotropy (inφ) of Langmuir branch withθ=1

  30. Langmuir branchwith θ=1 N=2

  31. Langmuir branchwith θ=1 N=4

  32. Langmuir branchwith θ=1 N=6

  33. Langmuir branchwith θ=1 N=8

  34. Conclusions • VMD model can be used for 3D simulations in regimes when plasma waves are confined to a narrow (θ<0.6) conewith quite good precision with only 6 or 8 transverse streams • Using VMD model we can drastically reduce necessary computing power since we only have to compute 6 or 8 equations that are effectively 4D instead of one 6D equation.

  35. Thank you! Denis Silantyev1 Harvey Rose2 Pavel Lushnikov1 Mathematics & Statistics Department, University of New Mexico Los Alamos National Laboratory 8 Jun 2012

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