Kinetic processes in plasmas

1. Kinetic processes in plasmas A. Mangeney Observatoire de Paris Basic processes of turbulent plasmas, 2003

2. A composite fluctuation spectrum in the Solar wind Basic processes of turbulent plasmas, 2003

3. From a system of N particles to "the typical" particle N particles of mass At the finest level: moving according to Newton's law under the action of forces, both external and internal with interaction forces Basic processes of turbulent plasmas, 2003

4. From a system of N particles to "the typical" particle : dimensionnal phase space, {X } Among functions defined on phase space: Klimontovich distribution: "counts" the number of particles in a volume dxdv of Basic processes of turbulent plasmas, 2003

5. Instead of all the details of the distribution of particles in consider only a small number of velocity moments: From a system of N particles to "the typical" particle How to loose information? Density: Momentum density: Kinetic energy density: Kinetic energy flux etc… Basic processes of turbulent plasmas, 2003

6. : random because • complicated motion of the particles because of mutual • pair wise interactions, even chaotic… • -unknown initial conditions, • -etc…. From a system of N particles to "the typical" particle How to loose more information? Basic processes of turbulent plasmas, 2003

7. A typical particle At the kinetic level: The identity of the particles has been lost; a small number of smooth functions, describing the statistics of the fine grained distribution What happens to the interaction forces? Mean field, resulting from the linear superposition of the fields of all particles; the discrete character of the particles has been lost Basic processes of turbulent plasmas, 2003

8. Collisionnal/collisionless But the interaction force is also a random quantity, with fluctuations determined by f2 (Correlation function, retaining part of the memory of particle discreteness) A) Mean field description - Vlasov equation if otherwise B) Collisional fluid - Boltzmann ( Fokker-Planck) equation Basic processes of turbulent plasmas, 2003

9. What makes a plasma to differ from another fluid? • For most neutral fluids interparticle forces are short range! • In a plasma, or a gravitating stellar system, interparticle forces are long range! 1D, electrostatic: Particles are actually charged sheets, with charge ±e. A particle at xi is the source of a piece-wise constant electric field: E x Basic processes of turbulent plasmas, 2003

10. Thus, total electric field E(x) at a given x depend only on the total number of particles of both signs at left and at right of x, but not on their precise location; if these numbers are large, E(x) vary slowly with x, with little jumps each time a particle is crossed (discreteness effects) x 1/n <E>varying on scales greater than particle separation on scales comparable to particle separation Screening effects have to be taken into account Basic processes of turbulent plasmas, 2003

11. Poisson equation: Charge neutrality What is really the scale of variation of the average field? Debye-Huckel (1923) Electrons move fast to cancel any notable average charge separation Debye length and electron plasma frequency Basic processes of turbulent plasmas, 2003

12. l Collisionless plasma 1d: N particles of both signs ~ nl Charge density fluctuation Potential fluctuation in 3d: Basic processes of turbulent plasmas, 2003

13. Vlasov (Mean field): Charged particles move in a self consistent mean electric field Vlasov Poisson distribution functions remain constant along a particle trajectory: if these trajectories are complicated, the distribution function may become also very complex (see later) Stationary states : Infinite number of invariants Basic processes of turbulent plasmas, 2003

14. Collisionnal case In that case, one has to include the fluctuating electric field due to discreteness: When averaging over the fluctuations, one obtains a Fokker planck type of equation Particle recoil for sponatneous emission Random walk in the fluctuating potential Basic processes of turbulent plasmas, 2003

15. Lennard Balescu equation: Not too far from equilibrium i.e. fluctuation spectrum ~ what is expected from free streaming particles Still extremely complex due to dielectric effects, screening, etc… However, in the absence of external forces, only one stationary solution, the maxwellian distribution, at temperature T: Basic processes of turbulent plasmas, 2003

16. From a typical particle to fluid-like quantities How to loose STILL more information? Moments: From an infinite number of fields to 3 hydrodynamic fields! Basic processes of turbulent plasmas, 2003

17. Infinite hierarchy of equations! etc… (for each particle species) Basic processes of turbulent plasmas, 2003

18. Closure: A) Collisions -Local maxwellian: gaussian random variables in v for all (x,t): Ideal Euler equations -ETL: Transport processes, Navier Stokes equations B) No a priori valid closures for the collisionless case Several "nested" closure: - correlations - moments of f1 Importance of boundary conditions! Basic processes of turbulent plasmas, 2003

19. "Thermal" noise in the Solar Wind Here only quietest solar wind state, far from Shocks, etc… Basic processes of turbulent plasmas, 2003

20. Collisionless evolution Phase mixing, Landau damping Violent relaxation: virialisation- attempt to reach mechanical equilibrium holes in phase space, observed almost everywhere in space as soon as time resolution sufficient Development of microscopic instabilities Basic processes of turbulent plasmas, 2003

21. Suprathermal electrons with energies above about 80 eV at 1 AU continually stream out along magnetic field lines with a velocity distributions, f(v) usually consisting of • a dominant field-aligned component directed outward from the Sun, the strahl (found in high speed solar wind) • a weaker and • more isotropic halo component; Wind observations • significant variability of the strahl and/or halo, • other types of distributions, such as counterstreaming strahls, angular depletions and enhancements, and sunward streaming conics Basic processes of turbulent plasmas, 2003

22. Electric fluctuations at lower frequency • Quasi thermal noise • (Issautier et al.,1999) • with Gaussian statistics (B: bandwidth, t integration time ) • Intermittent non thermal emission Basic processes of turbulent plasmas, 2003

23. Histogram of electric fluctuations at two frequencies At f = 4.27 kHz, non thermal emission is observed above 5 10-13V2/Hz, with a power law distribution. Above 7 kHz, these nonthermal Emissions disappear. Basic processes of turbulent plasmas, 2003

24. At high time resolution: Langmuir waves « Ion acoustic waves » In the «quiet » Solar wind, all events recorded by the Time Domain Sampler (above a threshold of ~ 50mV/m) are coherent waveforms ( Mangeney et al., 1999) • Langmuir waves Basic processes of turbulent plasmas, 2003

25. Weak Double Layers (WDL) About 30% of these CEW are Isolated Electrostatic Waveforms with a measurable net potential jump: The corresponding electric field is almost always directed towards the Earth Basic processes of turbulent plasmas, 2003

26. Phase mixing Basic processes of turbulent plasmas, 2003

27. Phase mixing All moment perturbations decrease because of velocity integration which washes out fine structures developping in the velocity dependance. One may even prepare the system to obtain a wave propagating at an arbitrary velocity by ajusting the initial distribution Damping rate is diminished Basic processes of turbulent plasmas, 2003

28. Phase relationships between moments Suggests closure (non local) : depending on k, may be imaginary Basic processes of turbulent plasmas, 2003

29. Phase relationships between moments Suggests closure (non local) : depending on k, may be imaginary Basic processes of turbulent plasmas, 2003

30. Landau damping and phase mixing In the free streaming case no restoring force and no wave modes. If one retains the electric field, there is now a restoring force and wave modes; however the same phenomenon occurs:there are a continuum of wave modes in phase space, while velocity averages decrease, now only exponentially (in a stable plasma), due to a subtler phase mixing (Landau damping). Landau closures: compare a linearized fluid theory, with ad-hoc transport coefficient and the "exact" Vlasov linear theory, and try to fit one theory with the other; leads to non local transport coefficient Basic processes of turbulent plasmas, 2003

31. Example : Heat transport in fluids and collisionless plasmas Fluids: small deviations from ETL Collisionless plasmas: apparition of strong electric fields Some particles travel almost freely: ballistic mixing while others are strongly affected Landau closures: attempt to mimic collisionnal theory with Landau damping Basic processes of turbulent plasmas, 2003

32. Nonmaxwellian plasma Stationary fluid equilibrium Two maxwellian electron distribution: cold and hot Cold, at rest: Hot, speed uh Basic processes of turbulent plasmas, 2003

33. Fluid like equilibrium, not Vlasov equilibrium! 1d, open boundary Vlasov simulation (x,v), electrons and ions, to test Landau closures (for this summer school) Basic processes of turbulent plasmas, 2003

34. t=0 Basic processes of turbulent plasmas, 2003

35. "Ballistic evolution" Basic processes of turbulent plasmas, 2003

36. Ballistic evolution, electric pulse formation and proton acceleration Basic processes of turbulent plasmas, 2003

37. Evolution of the electric potential Basic processes of turbulent plasmas, 2003

38. Evolution of electron temperature Does not seem compatible with a fluid like closure ! Basic processes of turbulent plasmas, 2003

39. Random forcing (mimic discreteness effects) A)Full N-body calculation - Heavy!!! B)Random forcing: B1) « self consistent » gaussian force leading to the Landau equation (Qiang et al, 2000, for example) B2) Constant temperature molecular dynamics method: a random force is introduced to allow the system to sample a canonical or microcanonical ensemble B3) Dirty way : artificial random forcing Here, B3! (Collaboration F. Califano) Basic processes of turbulent plasmas, 2003

40. (1) External force acting only on the protons, deriving from potential Y(x,t) (2) Random « external » electric potential: F(x,t) acting on electrons and protons h=0 : forcing only on protons = 1: forcing both on protons and electrons Basic processes of turbulent plasmas, 2003

41. Random forcing: I-transient compressions or expansions c(x) : spatial profile (compression/expansion) q(t) : time profile l t • (xj, tj): independant random points and times • (sj , lj ,tj): randomly distributed around typical values s*, l*,t* Basic processes of turbulent plasmas, 2003

42. II- random charge fluctuations When the forcing concerns both electrons and protons (h=1), it is equivalent to the introduction of external charges t x Space - time distribution of random charges Spatial profile « Discreteness » introduced by random external charges Basic processes of turbulent plasmas, 2003

43. However, « thermal »charge fluctuations related to particle discreteness have a spectral density while the random charges used here have very different space time properties, and smaller level! Basic processes of turbulent plasmas, 2003

44. Two sets of 1Druns: (I) Nx=512, Nv=401, L=1000 lDe RUN A: h = 0, F = 0, Y ≠ 0, l*=10 RUN B: h = 1, F ≠ 0,Y= 0, l*=10 (II) Nx=2048, Nv=501, L=5000 lDe RUN C: h = 1, F ≠ 0,Y ≠ 0, l*=100 RUN D: h = 1, F ≠ 0,Y = 0, l*=100 Quasineutralityrandom forcing only on the protons Basic processes of turbulent plasmas, 2003

45. Two runs with same amplitude of forcing: • (A) forcing only on the protons, h=0 • (B) forcing on electrons and protons, h=1 dq2 (A) dnp2 (B) dne2 A B A) electric neutrality maintained at all times B) smaller density fluctuations but much larger charge fluctuations at forcing times! Basic processes of turbulent plasmas, 2003

46. Forcing on protons (Y≠0) leads to formation of long lived, small scale, stuctures Life time ≥ 2000 tpe>> t*~20 ; spatial scale ~ 50 lDe<< l evolving time scale comparable with proton phase mixing time (If forcing sufficiently strong: formation of electron holes with their associated bipolar electric field signature; not considered here) Basic processes of turbulent plasmas, 2003

47. When the forcing is only on protons • Heating of protons • Generation of electron plasma waves and electron heating • but no halo formation • If some external charge fluctuations are added • Heating of protons • Generation of plasma waves • Formation of a stationary halo for large t Basic processes of turbulent plasmas, 2003

48. Proton density variation Langmuir wave power density Long lived coherent density cavities generated by LF proton forcing trap Langmuir waves Basic processes of turbulent plasmas, 2003

49. Spectral electric density integrated in a band around the electron plasma frequency Run C LF proton forcing produces a broader k-spectrum of Langmuir wave Run D Basic processes of turbulent plasmas, 2003

50. Electron distribution function: Phase space modulation for v>0 and v<0 Proton distribution function • slow phase mixing on the proton distribution function • no significant proton heating • strong interaction of tail electrons with Langmuir waves Basic processes of turbulent plasmas, 2003