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How to Face the Complexity of Plasmas?

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From Hamiltonian Chaos to Complex Systems

Part of the book series: Nonlinear Systems and Complexity ((NSCH,volume 5))

Abstract

This paper has two main parts. The first part is subjective and aims at favoring a brainstorming in the plasma community. It discusses the present theoretical description of plasmas, with a focus on hot weakly collisional plasmas. It comprises two subparts. The first one deals with the present status of this description. In particular, most models used in plasma physics are shown to have feet of clay, there is no strict hierarchy between them, and a principle of simplicity dominates the modeling activity. At any moment the description of plasma complexity is provisional and results from a collective and somewhat unconscious process. The second subpart considers possible methodological improvements, some of them specific to plasma physics and some others of possible interest for other fields of science. The proposals for improving the present situation go along the following lines: improving the way papers are structured and the way scientific quality is assessed in the referral process, developing new databases, stimulating the scientific discussion of published results, diversifying the way results are made available, assessing more quality than quantity, and making available an incompressible time for creative thinking and non-purpose-oriented research. Some possible improvements for teaching are also indicated. The suggested improvement of the structure of papers would be for each paper to have a “claim section” summarizing the main results and their most relevant connection to previous literature. One of the ideas put forward is that modern nonlinear dynamics and chaos might help revisiting and unifying the overall presentation of plasma physics. The second part of this chapter is devoted to one instance where this idea has been developed for three decades: the description of Langmuir wave–electron interaction in one-dimensional plasmas by a finite-dimensional Hamiltonian. This part is more specialized and is written like a classical scientific paper. This Hamiltonian approach enables recovering Vlasovian linear theory with a mechanical understanding. The quasilinear description of the weak warm beam is discussed, and it is shown that self-consistency vanishes when the plateau forms in the tail distribution function. This leads to consider the various diffusive regimes of the dynamics of particles in a frozen spectrum of waves with random phases. A recent numerical simulation showed that diffusion is quasilinear when the plateau sets in and that the variation of the phase of a given wave with time is almost non-fluctuating for random realizations of the initial wave phases. This led to new analytical calculations of the average behavior of the self-consistent dynamics when the initial wave phases are random. Using Picard iteration technique, they confirm numerical results and exhibit a spontaneous emission of spatial inhomogeneities.

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Notes

  1. 1.

    Which is made explicit by a series of examples in the following.

  2. 2.

    The TFTR tokamak was shut down in 1997. For years before, people no longer insisted into calling it Tokamak Fusion Test Reactor. When the ITER project officially started in 1992, “ITER” meant “International Tokamak Experimental Reactor.” Now “ITER” is the Latin noun meaning “the way” [125]. This is so true that the KTX machine, a large reversed-field pinch, is being funded in Hefei in the frame of the Chinese ITER domestic program.

  3. 3.

    Alfven’s Nobel citation reads: “(…) for fundamental work and discoveries in magnetohydrodynamics with fruitful applications in different parts of plasma physics”. Therefore his award was not meant as presented to a plasma physicist. In reality, Alfven was in an awkward position with respect to well identified fields of physics [124].

  4. 4.

    Increasing blowups are advised when using the Internet to watch a photograph of the painting.

  5. 5.

    An important exception is the ITER physics basis [79, 101], already mentioned in Sect. 4.1.1, that was written by a series of committees of experts. This trend toward a collective view about the tokamak has been present in the authorship of Wesson’s book [119] since its first edition in 1987. It is also worth noting that most modern plasma textbooks have at least two authors.

  6. 6.

    A very interesting example where this process was applied repeatedly is the theory of edge localized modes (ELMs) in tokamaks. First they were considered as current driven, then as pressure driven, then back as current driven, and now as both pressure and current driven (see Chap. 3, Sect. 2.6.3 of [101] and [112]).

  7. 7.

    As was told in Sect. 4.2.1.2, the applicability of MHD equations to fusion plasmas may be justified if perpendicular motion dominates, at least for the mass and momentum equations. Therefore adding more MHD equations to the latter does not mean necessarily a more accurate description. Furthermore the use of the mass conservation equation may bring an unphysical peaking of density on the axis of a pinch which is avoided by microscopic turbulence in an actual plasma (see Sect. 9.3.3 of [17]).

  8. 8.

    However plasma physicists are not desperate. To the contrary going ahead by using many approximate models is like going downhill rapidly on a scree made up with small stones: this is both fast and pleasant, though none of the stone be comfortable to stand quietly! However plasma physicists may sometimes be meditative: “(…) had the range of instabilities now known to beset tokamaks been discovered by theoreticians before the experimental program was undertaken, there might have been some hesitation” (p. 562 of [119]).

  9. 9.

    According to Popper’s falsifiability paradigm, this theory just survives the process of refutation, but it is not protected from refutation in the future [103].

  10. 10.

    This picture is quite flexible though and is justified for generic particle transport, provided there is enough randomness in the Hamiltonian describing the dynamics [57].

  11. 11.

    Some progress may be expected from a new technique tailored for periodically modulated experiments [58]. This technique avoids any a priori constraint on the profiles and computes them by simply inverting a 2D matrix. It also provides the uncertainty on the reconstruction. This is done by a controllable smoothing of the experimental data, instead of the ad hoc regularization of the profile of transport coefficients operated by transport codes (see Appendix 3 of Sect. 4.7).

  12. 12.

    The RFP is a magnetic configuration germane to the tokamak that produces most of its magnetic field by the currents flowing inside the plasma [53]. With respect to the tokamak and the stellarator, the RFP has a low imposed external field. It has a helical magnetic field like the stellarator, but it is more magnetically self-organized than a tokamak and much more than a stellarator.

  13. 13.

    “The history of science shows that the progress of science has constantly been hampered by the tyrannical influence of certain conceptions that finally come to be considered as dogma. For this reason, it is proper to submit periodically to a very searching examination principles that we have come to assume without discussion [33].”

  14. 14.

    The explanation goes along the following steps: (1) In the radial domain where the magnetic field is chaotic, transport is fast, and the equilibrium is almost force-free; therefore J = μ B where μ may be space-dependent. (2) Setting this in ∇  ⋅J = 0, implies B ⋅ ∇ μ = 0, which shows that μ must be constant along field lines; thus μ is constant in the chaotic radial domain. This straightforward derivation yields a result in full agreement with the fact that, in MHD simulations, μ is almost constant in most of the domain with a positive toroidal magnetic field, but not where it is reversed [22]. However this domain of almost constant value of μ was rather considered as a hint to the validity of TRT which predicts μ constant over the whole plasma radius instead [114].

  15. 15.

    However this description already reveals that the edge current does not matter to reach shallow reversal. This is important to guide the endeavor toward improving confinement of quasi-single-helicity states: one should enable the central part of the plasma to reach a genuine ohmic equilibrium. Indeed this should induce a low resistivity central part of the discharge diminishing the loop voltage and thus the ohmic power for the same plasma current.

  16. 16.

    It is hard for young physicists to imagine the age where numerical simulations were a tour de force with card punching, batch submission, and paper outputs. In 1976 the author published a one-dimensional Vlasovian simulation with 8,000 cells in phase space, while in present codes this number is larger by more than three orders of magnitude! However this lean code enabled to uncover the thermalization of a volume-created plasma due to the lack of static equilibrium [49].

  17. 17.

    It helped for the present chapter, even under the disguise of an extended summary! The reader is invited to write a claim section for his/her next paper, in order to ascertain the interest of the method, even for a private use. This section might be put as an appendix in the paper.

  18. 18.

    This is the motto of the Institute for Advanced Study in Princeton [126].

  19. 19.

    This limit is an edge density limit above which the discharge cannot be sustained.

  20. 20.

    From this point of view one might again take advantage of the analogy of the RFP with the tokamak, since in the RFP such barriers are related to shear reversal too [72].

  21. 21.

    This experiment studies dynamo action in the flow generated inside a cylinder filled with liquid sodium by the rotation of coaxial soft-iron impellers (von Karman geometry). It evidenced the self-generation of a stationary dynamo when the impellers do not rotate with the same angular velocity [70, 94]. The magnetic field averaged over a long enough time corresponds to a RFP magnetic state with a large m = 0 mode (see Fig. 7 of [94]).

  22. 22.

    In the fluid description of screw pinches, a classical model is provided by the combination of Faraday–Ohm’s law and of Navier–Stokes equation with Lorentz force. Then the Lundquist number is an obvious parameter. One may refine this description by adding a heat transport equation, which provides a self-consistent definition of the temperature profile and accordingly of the resistivity profile. Then the Lundquist number is no longer a parameter but an output of the model.

  23. 23.

    See for instance Sect. 14.2.1 of [52]. This sheds also a new light on the hydrodynamic or cold beam–plasma instability (Sect. 14.3.1 of [52]). Indeed a modulation with wavenumber k of the beam density generates a forcing of the plasma at pulsation ω = ku, where u is the beam velocity, which feeds back on the beam density modulation. The response of a harmonic oscillator scales like \({(\omega _{\mathrm{p}}^{2} {-\omega }^{2})}^{-1}\). For ω ≫ ω p the electrons react weakly due to their inertia, which rules out a positive feedback for such ω’s, and by continuity for ω > ω p. Then the plasma behaves like a classical dielectric, which screens the perturbing charge. As a result the unstable forcing must correspond to ω = ku ≤ ω p with a maximum for equality. This contrasts with the classical “negative energy” picture which rather suggests \(\omega = -\omega _{\mathrm{b}} + ku\), where ω b is the plasma frequency of the beam, and does not tell why the instability occurs rather for ku ≤ ω p and why it is the strongest for ku ≃ ω p. This forced harmonic oscillator picture works also for other reactive instabilities.

  24. 24.

    Moreover this calculation gives no clue to the plasma behavior in the actual nonlinear regime where damping is a manifestation of stability of an infinite-dimensional Hamiltonian system (see Sect. 4.3.1).

  25. 25.

    Using the blackboard is an efficient way to avoid “runaway lectures,” especially when calculations are presented.

  26. 26.

    Naturally this must be done without going up to a superficial presentation of the phenomena. Calculations are a way to anchor memory and to train students, especially at an undergraduate level.

  27. 27.

    In particular the development of new diagnostics to touch other parts of the “elephant.”

  28. 28.

    This is all the more justified, since plasma physicists contributed a lot to the development of these topics.

  29. 29.

    Unfortunately, the beauty and the flexibility of the derivation of the Hamiltonian description of magnetic field lines by a stationary action principle [28] have been largely overlooked. It was formulated in a simple way in [40, 102] showed a corresponding equivalence of canonical transformations and of changes of gauge.

  30. 30.

    For instance the existence of negative -specific heat in a magnetically self-confined plasma torus [82]. The saturation of the cold and water-bag beam–plasma instability can be computed analytically by using Hamiltonian Eq. (4.1) introduced in Sect. 4.3 with a single wave (cold: [68]; water bag: [8, 9]). The mean-field derivation of Vlasov equation was already mentioned in Sect. 4.2.1.2.

  31. 31.

    There are strong analogies with plasma turbulence, as exemplified by the Charney–Hasegawa–Mima model, but also strong differences since plasma turbulence is seldom fully developed. Furthermore the word “intermittency” is used with quite different meanings in the two fields.

  32. 32.

    The impact of instabilities was recently addressed in [4, 5] by taking into account the spontaneous emission of waves by particles which induces a corresponding drag on top of the “collisional” one.

  33. 33.

    This is a tremendous simplification with respect to the physics of many actual plasmas. In particular density fluctuations may bring dramatic changes in the dynamics of Langmuir waves by Anderson localization [36, 59], by a transfer of particle momentum over an increased range of velocities [48], and by nonlinear decay and scattering processes [121].

  34. 34.

    This relation makes sense, since we consider low-amplitude waves with phase velocities much above the thermal speed. If these conditions are not satisfied, the issue is a lot more involved [11–13].

  35. 35.

    By taking advantage of the intuition developed by this approach it is possible to derive a more pedestrian approach to wave–particle interaction [52].

  36. 36.

    In particular, though initially published with a caveat, the surfer model induces in the mind of the students the wrong feeling that trapping is involved in Landau effect.

  37. 37.

    This academic issue has a broader relevance since the QL approximation is used everywhere in plasma physics.

  38. 38.

    For a plateau with a finite width, the small remaining source brings a further evolution of the wave–particle system toward a Gibbsian state where the wave spectrum collapses toward small wavelengths together with the escape of initially resonant particles toward low bulk plasma thermal speeds [68]. This corresponds to a further step toward a new thermal equilibrium of the N-body system corresponding to the initial beam–plasma system. The description of the subsequent steps toward thermal equilibration requires to use a full N-body model.

  39. 39.

    This phenomenon, also called depression of nonlinearity, was introduced in fluid mechanics [84] and was identified as a result of the emergence of long-lived vortices where the enstrophy cascade is inhibited. It also exists in systems with quadratic nonlinearities [20, 84].

  40. 40.

    In the opposite limit when τ spread ∕ τ ac is small, the time evolution of the waves is slow with respect to the trapping motion in the instantaneous wave potential. Then chaotic dynamics may be described in an adiabatic way with the picture of a slowly pulsating separatrix [44, 45] (see also Sect. 5.5 of [46] and Sect. 14.5.2 of [52]). In this limit, for the case of the motion in two waves, the resonance overlap defined hereafter is large.

  41. 41.

    This criterion is a very useful rule of thumb which works, also experimentally [38], provided the two trapping domains are not too dissimilar. In particular, Δv 1 ∕ Δv 2 should not be too far from 1. Otherwise, one of the waves is a small perturbation for the other one, and the threshold of large scale chaos is a lot larger than 1 (see [50, 54] for more information). A more accurate way to understand the transition to large scale chaos is provided by a renormalization transformation [50, 54] (see also Sect. 5.4 of [46] and Sect. 14.5.4 of [52]). However Chirikov criterion can also be used to check whether high dimensional dynamics is chaotic enough. More specifically parameter s may be used as an observable whose Gibbsian estimate tells Gibbsian calculus makes sense when it is larger than 1 [60].

  42. 42.

    The necessity to go beyond τ spread to see the chaotic diffusion is a caveat for the numerical measurement of a chaotic diffusion coefficient. This minimum time comes from the locality in velocity of wave–particle interaction [10, 46]. Indeed it can be shown that at a given moment the waves making particle dynamics chaotic have a phase velocity within Δv ∼ 1 ∕ (kτ spread) from the particle velocity. Those out of this range act perturbatively. If waves have random phases, after visiting several “resonance boxes” of width Δv, a particle feels as having been acted upon by a series of independent chaotic dynamics, which triggers a diffusive behavior. This decorrelation makes it possible to numerically measure the diffusion coefficient by following the dynamics either of a single particle for a series of random outcomes of the wave phases or of many particles for a single typical outcome of the phases. By extension this enables to reconcile the uniqueness of each realization of an N-body system with models invoking a probabilistic average over independent realizations.

  43. 43.

    If the waves have random amplitudes A m and phases \(\varphi _{m}\) such that \(A_{m}\exp (\mathrm{i}\varphi _{m})\) is a gaussian variable, then the superquasilinear bump does not exist, and D ∕ D QL ≤ 1 for all values of s, but still goes to 1 when s becomes large [42].

  44. 44.

    This iteration technique is very convenient to alleviate the algebra of many perturbation calculations. This is the case for the perturbation calculation of the dynamics defined by Hamiltonian Eq. (4.5) in the wave amplitude A. In particular for a single wave, which provides Eq. (4.2).

  45. 45.

    “Qui autem omnia, quae ad cultum deorum pertinerent, diligenter retractarent et tamquam relegerent, sunt dicti religiosi ex relegendo, ut elegantes ex eligendo, ex diligendo diligentes, ex intellegendo intellegentes; his enim in verbis omnibus inest vis legendi eadem quae in religioso.” Cicero, De Natura Deorum, 2, 28. English translation [31]: “Those on the other hand who carefully reviewed and so to speak retraced all the lore of ritual were called ‘religious’ from relegere (to retrace or re-read), like ‘elegant’ from eligere (to select), ‘diligent’ from diligere (to care for), ‘intelligent’ from ‘intellegere’ (to understand) ; for all these words contain the same sense of “picking out” (legere) that is present in ‘religious’.”

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Acknowledgements

I am indebted to Y. Camenen, L. Couëdel, F. Doveil, and Y. Elskens, for a thorough reading of a first version of this paper and for providing me with an extensive feedback. My thanks also go to D. Bonfiglio, S. Cappello, and F. Sattin who did the same for a second version. Y. Elskens also helped me a lot in improving the English. F. Baldovin, M. Bécoulet, D. Bénisti, N. Bian, A. Boozer, P. Diamond, M.-C. Firpo, M. Henneaux, T. Mendonça, B. Momo, K. Razumova, S. Ruffo, M. Valisa, and F. Zolla are thanked for very useful comments and new references. I thank D. Guyomarc’h for drawing all the figures. My thanks go to M. Farge who pointed out to me reference [110]. The topic of my talk at Chaos, Complexity and Transport 2011 was about the description of self-consistent wave–particle interaction with a finite-dimensional Hamiltonian described in Sect. 4.3. However, two seminars I gave later on in the north and south campuses of Marseilles were the occasion to start developing the ideas of Sect. 4.2, in kind of an echo to Sect. 4.3. I thank the organizers of the conference for allowing me to extend the topic of my chapter beyond the original contents of my talk and to further develop my thoughts about plasma complexity and the way to tackle it.

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  1. UMR 7345 CNRS-Aix-Marseille-Université, case 321, Av. Normandie Niemen, FR-13397, Marseille CEDEX 20, France

    Dominique F. Escande

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  1. , Centre de Physique Théorique, Aix-Marseille Université, Luminy Case 907, Marseille Cedex 09, 13288, France

    Xavier Leoncini

  2. Scientifique, Institut de Recherche sur le Phénomènes, Centre National de la Recherche, rue F. Joliot-Curie 49, Marseille Cedex 13, 13384, France

    Marc Leonetti

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Escande, D.F. (2013). How to Face the Complexity of Plasmas?. In: Leoncini, X., Leonetti, M. (eds) From Hamiltonian Chaos to Complex Systems. Nonlinear Systems and Complexity, vol 5. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6962-9_4

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