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From Hamiltonian Chaos to Complex Systems pp 159–188Cite as

First Principle Transport Modeling in Fusion Plasmas: Critical Issues for ITER

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

Abstract

Tokamaks aim at confining hot plasmas by means of strong magnetic fields in view of reaching a net energy gain through fusion reactions. Plasma confinement turns out to be governed by small-scale instabilities which saturate nonlinearly and lead to turbulent fluctuations of a few percent. This paper recalls the basic equations for modeling such weakly collisional plasmas. It essentially relies on the kinetic, or more precisely the gyrokinetic, description, although some attempts are made to incorporate some of the kinetic properties, namely, wave-particle resonances, in fluid models by means of collisionless closures. Three main types of micro-instabilities are detailed and studied linearly, namely, drift waves, interchange, and bump-on-tail. Finally, some of the main critical issues in turbulence modeling are addressed: flux-driven versus gradient-driven models, the subsequent impact of mean profile relaxation on turbulent transport dynamics, and the role of large-scale flows, either at equilibrium or turbulence driven, on turbulence saturation and on the possible triggering of transport barriers. The significant progress in understanding and prediction of turbulent transport in tokamak plasmas thanks to first-principle simulations is highlighted.

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Notes

  1. 1.

    Notice that transverse drifts can also be derived within the fluid framework in the same adiabatic limit. At first order in the small ρ ∕ R parameter, with R the curvature—and or the gradient— length of B, they read: \(\mathbf{u}_{\perp }^{(1)} \equiv \mathbf{u}_{E} + \mathbf{u}_{s}^{{\ast}} = \frac{\mathbf{E}\times \mathbf{B}} {{B}^{2}} + \frac{\mathbf{B}\times \nabla p_{s}} {n_{s}e_{s}{B}^{2}}\). The first component, the electric drift u E , is also a particle drift. The latter one is not, since it depends on the pressure, which is a fluid quantity only. It is known as the diamagnetic drift u s  ∗ . It is the same order of magnitude for ions and electrons. Since it depends on the charge of the species, it carries transverse current. The second-order fluid drift is the so-called polarization drift. It is often approximated as follows: \(\mathbf{u}_{\perp }^{(2)} \equiv \mathbf{u}_{pol,\,s} = - \frac{m_{s}} {e_{s}{B}^{2}} \Big[\partial _{t} + (\mathbf{u}_{E} + \mathbf{u}_{s}^{{\ast}} + \mathbf{u}_{\parallel }).\nabla \Big]\nabla _{\perp }\phi\).

  2. 2.

    Notice that such a result intrinsically derives from the fast motion of the electrons in the parallel direction due to their small inertia. Therefore, only those modes which exhibit some structure in the parallel direction (i.e., such that k  ∥ ≠0) are subject to an adiabatic response of the electrons.

  3. 3.

    The ion density fluctuation δn i comes from the continuity equation, namely, \(\partial _{t}\delta n_{i} + u_{Er}\mathrm{d}n_{eq}/\mathrm{d}r = 0\), with \(u_{Er} = -\partial _{y}\phi /B\). For the considered plane wave, this reads as follows: \(-i\omega \delta n_{i} = i(k_{y}/B)(\mathrm{d}n_{eq}/\mathrm{d}r)\,\delta \phi\). The quasi-neutrality constraint \(\delta n_{i} =\delta n_{e}\) then leads to the result. See also Sect. 5.3.2

  4. 4.

    Indeed, it corresponds to 2 centered Maxwellians, for which Landau damping only is expected.

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Acknowledgment

It is my pleasure to acknowledge colleagues and friends who have most contributed to this paper through numerous enlightening discussions and common work on turbulence and transport for many years: X. Garbet and Ph. Ghendrih, P. Beyer, P.H. Diamond, G. Dif-Pradalier, and V. Grandgirard. Many thanks as well to the students J. Abiteboul, A. Strugarek, D. Zarzoso, and T. Cartier-Michaud. Last but not least, I wish to acknowledge C. Passeron for her precious support on numerical issues.

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  1. CEA, IRFM, F-13108, Saint-Paul-Lez-Durance, France

    Yanick Sarazin

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Correspondence to Yanick Sarazin .

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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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Sarazin, Y. (2013). First Principle Transport Modeling in Fusion Plasmas: Critical Issues for ITER. 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_5

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  • DOI: https://doi.org/10.1007/978-1-4614-6962-9_5

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