Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-23T16:02:43.753Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalized fluid theory including non-Maxwellian kinetic effects

Part of: Solved and Unsolved problems in Plasma Physics

Published online by Cambridge University Press:  29 March 2017

Olivier Izacard Open the ORCID record for Olivier Izacard [Opens in a new window]
Olivier Izacard*
Affiliation:
Lawrence Livermore National Laboratory, 7000 East Avenue, L-637, Livermore, CA 94550, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The results obtained by the plasma physics community for the validation and the prediction of turbulence and transport in magnetized plasmas come mainly from the use of very central processing unit (CPU)-consuming particle-in-cell or (gyro)kinetic codes which naturally include non-Maxwellian kinetic effects. To date, fluid codes are not considered to be relevant for the description of these kinetic effects. Here, after revisiting the limitations of the current fluid theory developed in the 19th century, we generalize the fluid theory including kinetic effects such as non-Maxwellian super-thermal tails with as few fluid equations as possible. The collisionless and collisional fluid closures from the nonlinear Landau Fokker–Planck collision operator are shown for an arbitrary collisionality. Indeed, the first fluid models associated with two examples of collisionless fluid closures are obtained by assuming an analytic non-Maxwellian distribution function (e.g. the INMDF (Izacard, O. 2016b Kinetic corrections from analytic non-Maxwellian distribution functions in magnetized plasmas. Phys. Plasmas 23, 082504) that stands for interpreted non-Maxwellian distribution function). One of the main differences with the literature is our analytic representation of the distribution function in the velocity phase space with as few hidden variables as possible thanks to the use of non-orthogonal basis sets. These new non-Maxwellian fluid equations could initiate the next generation of fluid codes including kinetic effects and can be expanded to other scientific disciplines such as astrophysics, condensed matter or hydrodynamics. As a validation test, we perform a numerical simulation based on a minimal reduced INMDF fluid model. The result of this test is the discovery of the origin of particle and heat diffusion. The diffusion is due to the competition between a growing INMDF on short time scales due to spatial gradients and the thermalization on longer time scales. The results shown here could provide the insights to break some of the unsolved puzzles of turbulence.

Keywords

fusion plasmamagnetized plasmasplasma confinement
Type
Research Article
Information
Journal of Plasma Physics , Volume 83 , Issue 2 , April 2017 , 595830201
Check for updates
Copyright
© Cambridge University Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bacus, G. E. 1960 Linearized plasma oscillations in arbitrary electron velocity distributions. J. Math. Phys. 1, 178–191.Google Scholar
Bartiromo, R., Hesse, M., Söldner, F. X., Burhenn, R., Fussmann, G., Leuterer, F., Murmann, H., Eckhartt, D., Eberhagen, A., Giuliana, A. et al. 1986 Experimental study of non-thermal electron generation by lower hybrid waves in the ASDEX tokamak. Nucl. Fusion 26, 1106–1111.CrossRefGoogle Scholar
Beausang, K. V., Prunty, S. L., Scannell, R., Beurskens, M. N., Walsh, M. J., De La Luna, E.& JET EFDA Contributors 2011 Detecting non-Maxwellian electron velocity distributions at JET by high resolution Thomson scattering. Rev. Sci. Instrum. 82, 033514.Google Scholar
Bernstein, I. B., Trehan, S. K. & Weenink, M. P. H. 1964 Plasma oscillations: II. kinetic theory of waves in plasmas. Nucl. Fusion 4, 61–104.Google Scholar
Birdsall, C. K. & Langdon, A. B. 1991 Plasma Physics via Computer Simulation. Institute of Physics.Google Scholar
Bitter, M., Gu, M. F., Vainshtein, L. A., Beiersdorfer, P., Bertschinger, G., Marchuk, O., Bell, R., LeBlanc, B., Hill, K. W., Johnson, D. et al. 2003 New benchmarks from tokamak experiments for theoretical calculations of the dielectronic satellite spectra of Helium like ions. Phys. Rev. Lett. 91, 265001.Google Scholar
Boltzmann, L. E. 1877 Kinetic Theory. Oxford Press; translated by S. G. Brush (1966).Google Scholar
Braginskii, S. I. 1965 Transport processes in a plasma. Rev. Plasma Phys. 1, 205–311; adsabs.harvard.edu/abs/1965RvPP....1..205B.Google Scholar
Brush, S. G. 2003 The Kinetic Theory of Gases – An Anthology of Classic Papers with Historical Commentary. Pergamon.CrossRefGoogle Scholar
Canik, J. M., Maingi, R., Soukhanovskii, V. A., Bell, R. E., Kugel, H. W., Leblanc, B. P. & Osborne, T. H. 2011 Measurements and 2-D modeling of recycling and edge transport in discharges with lithium-coated PFCs in NSTX. J. Nucl. Mater. 415, S409–S412.Google Scholar
Case, K. M. 1959 Plasma oscillations. Ann. Phys. 2, 349–364.Google Scholar
Chang, Z. & Callen, J. D. 1992 Unified fluid/kinetic description of plasma microinstabilities. Part I: basic equations in a sheared slab geometry. Phys. Fluids B 4, 1167–1181.CrossRefGoogle Scholar
Chankin, A. V. et al. 2007a Discrepancy between modelled and measured radial electric fields in the scrape-off layer of divertor tokamaks: a challenge for 2D fluid codes? Nucl. Fusion 47, 479–489.CrossRefGoogle Scholar
Chankin, A. V et al. 2007b A possible role of radial electric field in driving parallel ion flow in scrape-off layer of divertor tokamaks. Nucl. Fusion 47, 762–772.Google Scholar
Chew, G. F. M., Goldberger, L. & Low, F. E. 1956 The Boltzmann equation and the one-fluid hydromagnetic equations in the absence of particle collisions. Proc. R. Soc. Lond. A 236, 112–116.Google Scholar
Dawson, J. M. 1983 Particle simulation of plasmas. Rev. Mod. Phys. 55, 403–447.Google Scholar
De Guillebon, L. & Chandre, C. 2012 Hamiltonian structure of reduced fluid models for plasmas obtained from a kinetic description. Phys. Lett. A 376, 3172–3176.Google Scholar
De La Luna, E., Krivenski, V., Giruzzi, G., Gowers, C., Prentice, R., Travere, J. M. & Zerbini, M. 2003 Impact of bulk non-Maxwellian electrons on electron temperature measurements (invited). Rev. Sci. Instrum. 74, 1414–1420.Google Scholar
Dimits, A. M., Joseph, I. & Umansky, M. V. 2014 A fast non-Fourier method for Landau-fluid operators (invited paper). Phys. Plasmas 21, 055907.CrossRefGoogle Scholar
Dorf, M. A., Cohen, R. H., Dorr, M., Rognlien, T., Hittinger, J., Compton, J., Colella, P., Martin, D. & McCorquodale, P. 2013 Simulation of neoclassical transport with the continuum gyrokinetic code COGENT. Phys. Plasmas 20, 012513.CrossRefGoogle Scholar
Dorf, M. A., Dorr, M. R., Hittinger, J. A., Cohen, R. H. & Rognlien, T. D. 2016 Continuum kinetic modeling of the tokamak plasma edge. Phys. Plasmas 23, 056102.Google Scholar
Dorland, W., Jenko, F., Kotschenreuther, M. & Rogers, B. N. 2000 Electron temperature gradient turbulence. Phys. Rev. Lett. 85, 5579–5582.Google Scholar
Fidone, I., Giruzzi, G. & Taylor, G. 1996 Plasma diagnostics in the Tokamak Fusion Test Reactor using emission of electron cyclotron radiation at arbitrary frequencies. Phys. Plasmas 3, 2331–2336.CrossRefGoogle Scholar
Fonseca, R. A., Silva, L. O., Tsung, F. S., Decyk, V. K., Lu, W., Ren, C., Mori, W. B., Deng, S., Lee, S., Katsouleas, T. et al. 2002 OSIRIS: A Three-Dimensional, Fully Relativistic Particle in Cell Code for Modeling Plasma Based Accelerators, Computational Science — ICCS 2002: International Conference Amsterdam, The Netherlands, April 21–24, 2002 Proceedings, Part III, pp. 342–351. Springer.Google Scholar
Gaffey, J. D. Jr. 1976 Energetic ion distribution resulting from neutral beam injection in tokamaks. J. Plasma Phys. 16, 149–169.Google Scholar
Grad, H. 1949 Note on N-dimensional Hermite polynamials. Commun. Pure Appl. Maths 2, 325–330; 1949 On the kinetic theory of rarefied gases. Commun. Pure Appl. Maths 2, 331–407.Google Scholar
Grad, H. 1963 Asymptotic theory of the Boltzmann equation. Phys. Fluids 6, 147–181.CrossRefGoogle Scholar
Gradshteyn, I. S. & Ryzhik, I. M. 2007 Table of Integrals, Series, and Products, 7th edn. Academic.Google Scholar
Grandgirard, V., Sarazin, Y., Garbet, X., Dif-Pradalier, G., Ghendrih, P., Crouseilles, N., Latu, G., Sonnendrücker, E., Besse, N. & Bertrand, P. 2006 GYSELA, a full-f global gyrokinetic Semi-Lagrangian code for ITG turbulence simulations. In AIP Conference Proceedings, vol. 871, p. 100.Google Scholar
Grasso, D. & Tassi, E. 2015 Hamiltonian magnetic reconnection with parallel electron heat flux dynamics. J. Plasma Phys. 81, 495810501.CrossRefGoogle Scholar
Gross, E. P. & Krook, M. 1956 Model for collision processes in gases: small-amplitude oscillations of charged two-component systems. Phys. Rev. 102, 593–604.Google Scholar
Groth, M., Brezinsek, S., Belo, P., Beurskens, M. N. A., Brix, M., Clever, M., Coenen, J. W., Corrigan, C., Eich, T., Flanagan, J. et al. 2013 Impact of carbon and tungsten as divertor materials on the scrape-off layer conditions in JET. Nucl. Fusion 53, 093016.CrossRefGoogle Scholar
Hammett, G. W. & Perkins, F. W. 1990 Fluid moment models for Landau damping with application to the ion-temperature-gradient instability. Phys. Rev. Lett. 64, 3019–3022.Google Scholar
Heikkinen, J. A., Janhunen, S. J., Kiviniemi, T. P. & Ogando, F. 2008 Full of gyrokinetic method for particle simulation of tokamak transport. J. Comput. Phys. 227, 5582–5609.Google Scholar
Hirvijoki, E., Candy, J., Belli, E. & Embréus, O. 2015 The Gaussian radial basis function method for plasma kinetic theory. Phys. Lett. A 379, 2735–2739.Google Scholar
Idomura, Y., Tokuda, S. & Kishimoto, Y. 2003 Global gyrokinetic simulation of ion temperature gradient driven turbulence in plasmas using a canonical Maxwellian distribution. Nucl. Fusion 43, 234–243.Google Scholar
Izacard, O. 2013a Seminar: Foundations of the Hamiltonian Maxwell-Fluid Theory and of the INMDF. General Atomics, CA.Google Scholar
Izacard, O. 2016a On the generalization of fluid theory including kinetic effects. In Nathaniel J. Fisch Symposium ‘Solved and Unsolved Problems in Plasma Physics’, Princeton, NJ, LLNL-ABS-681092, LLNL-POST-686288.Google Scholar
Izacard, O. 2016b Kinetic corrections from analytic non-Maxwellian distribution functions in magnetized plasmas. Phys. Plasmas 23, 082504.Google Scholar
Izacard, O., Scotti, F., Soukhanovskii, V. A., Rensink, M. E., Rognlien, T. D. & Umansky, M. V. 2016c UEDGE modeling of snowflake divertors in NSTX-U. In American Physical Society – Division of Plasma Physics Conference, NP10.00031, LLNL-ABS-697742, LLNL-POST-707405.Google Scholar
Jenko, F., Dorland, W., Kotschenreuther, M. & Rogers, B. N. 2000 Electron temperature gradient driven turbulence. Phys. Plasmas 7, 1904–1910.Google Scholar
Ji, J. Y. & Held, E. D. 2013 Exact linearized Coulomb collision operator in the moment expansion. Phys. Plasmas 13, 102103.Google Scholar
Jolliet, S., Bottino, A., Angelino, P., Hatzky, R., Tran, T. M., Mc Millan, B. F., Sauter, O., Appert, K., Idomura, Y. & Villard, L. 2007 A global collisionless PIC code in magnetic coordinates. Comput. Phys. Commun. 177, 409–425.Google Scholar
Kotschenreuther, M., Rewoldt, G. & Tang, W. M. 1995 Comparison of initial value and eigenvalue codes for kinetic toroidal plasma instabilities. Comput. Phys. Commun. 88, 128–140.Google Scholar
Landau, L. D. 1946 J. Phys. USSR 10, 25–34; English transl. 1965 JETP 16, 574–586. Reproduced in Collected papers of L. D. Landau, Ed. ter Haar D., Pergamon (1965).Google Scholar
Lee, W. W. & Kolesnikov, R. A. 2009a On higher order corrections to gyrokinetic Vlasov–Poisson equations in the long wavelength limit. Phys. Plasmas 16, 044506.Google Scholar
Lee, W. W. & Kolesnikov, R. A. 2009b Phys. Plasmas 16, 124702; response to comment on ‘On higher-order corrections to gyrokinetic Vlasov–Poisson equations in the long wavelength limit’. Phys. Plasmas (2009) 16, 124701.Google Scholar
Maxwell, J. C. 1872 Theory of Heat. Greenwood Press.Google Scholar
Mazon, D., Fenzi, C. & Sabot, R. 2016 As hot as it gets. Nat. Phys. 12, 14–17.Google Scholar
Meier, E. T., Soukhanovskii, V. A., Bell, R. E., Diallo, A., Kaita, R., Leblanc, B. P., Mclean, A. G., Podesta, M., Rognlien, T. D. & Scotti, F. 2014 Modeling NSTX snowflake divertor physics. In 21st International Conference on Plasma Surface Interaction P2-022 Kanazawa Ishikawa, Japan.Google Scholar
Meneghini, O. & Lao, L. 2013 Integrated modeling of tokamak experiments with OMFIT. Plasma Fusion Res. 8, 2403009.Google Scholar
Meneghini, O., Smith, S. P., Lao, L. L., Izacard, O., Ren, Q., Park, J. M., Candy, J., Wang, Z., Luna, C. J., Izzo, V. A. et al. 2015 Integrated modeling applications for tokamak experiments with OMFIT. Nucl. Fusion 55, 083008.CrossRefGoogle Scholar
Ong, R. S. B. 1963 An integral equation occurring in plasma oscillations. Q. Appl. Maths 21, 162–166; jstor.org/stable/43635294.Google Scholar
Parra, F. I. & Catto, P. J. 2008 Limitations of gyrokinetics on transport time scales. Plasma Phys. Control. Fusion 50, 065014.Google Scholar
Parra, F. I. & Catto, P. J. 2009 Comment on ‘On higher order corrections to gyrokinetic Vlasov–Poisson equations in the long wavelength limit’. Phys. Plasmas 16, 124701; Phys. Plasmas (2009) 16, 044506.Google Scholar
Parker, S., Kim, C. & Chen, Y. 1999 Large-scale gyrokinetic turbulence simulations: effects of profile variation. Phys. Plasmas 6, 1709–1716.Google Scholar
Plunk, G. G., Cowley, S. C., Schekochihin, A. A. & Tstsuno, T. 2010 Two-dimensional gyrokinetic turbulence. J. Fluid Mech. 664, 407–435.Google Scholar
Pomeau, Y. 2016 The long and winding road. Nat. Phys. 12, 198–199.Google Scholar
Rognlien, T. D. 2014 A convective divertor utilizing a 2nd-order magnetic field null. In 56th Annual Meeting of APS-DPP 59, GI1.00004 (invited talk), meetings.aps.org/Meeting/DPP14/Session/GI1.4.Google Scholar
Rosenbluth, M. N., Macdonald, W. M. & Judd, D. L. 1957 Fokker–Planck equation for an inverse-square force. Phys. Rev. 107, 1–6.Google Scholar
Scott, B. 2010 Derivation via free energy conservation constraints of gyrofluid equations with finite-gyroradius electromagnetic nonlinearities. Phys. Plasmas 17, 102306.Google Scholar
Sedlàček, Z. 1967 Landau damping and the moments method. Nucl. Fusion 7, 161–168.Google Scholar
Snyder, P. B., Hammett, G. W. & Dorland, W. 1997 Landau fluid models of collisionless magnetohydrodynamics. Phys. Plasmas 4, 3974–3985.Google Scholar
Staebler, G. M., Kinsey, J. E. & Waltz, R. E. 2007 A theory-based transport model with comprehensive physics. Phys. Plasmas 14, 055909.Google Scholar
Sulem, P. L. & Passot, T. 2014 Landau fluid closures with nonlinear large-scale finite Larmor radius corrections for collisionless plasmas. J. Plasma Phys. 81, 325810103.Google Scholar
Taylor, G., LeBlanc, B., Murakami, M., Phillips, C. K., Wilson, J. R., Bell, M. G., Budny, R. V., Bush, C. E., Chang, Z., Darrow, D. et al. 1996 ICRF heating of TFTR plasmas fuelled by deuterium - tritium neutral beam injection. Plasma Phys. Control. Fusion 38, 723–750.Google Scholar
Van Kampen, N. G. 1955 On the theory of stationary waves in plasmas. Physica 21, 949–963.Google Scholar
Vlasov, A. A. 1938 Zh. Eksp. Teor. Fiz. 8, 291–318; English transl. 1968 The vibrational properties of an electron gas. Sov. Phys. Uspekhi 10, 721–733 (Translated by D. ter Haar).Google Scholar
Waltz, R. E., Candy, J. & Rosenbluth, M. N. 2002 Gyrokinetic turbulence simulation of profile shear stabilization and broken gyroBohm scaling. Phys. Plasmas 9, 1938–1946.Google Scholar
Wang, W. X., Tang, W. M., Hinton, F. L., Zakharov, L. E., White, R. B. & Manickam, J. 2004 Global of particle simulation of neoclassical transport and ambipolar electric field in general geometry. Comput. Phys. Commun. 164, 178–182.Google Scholar
Wang, W. X., Lin, Z. W., Tang, M., Lee, W. W., Ethier, S., Lewandowski, J. L. V., Rewoldt, G., Hahm, T. S. & Manickam, J. 2006 Gyro-kinetic simulation of global turbulent transport properties in tokamak experiments. Phys. Plasmas 13, 092505.Google Scholar
Watanabe, T. H. & Sugama, H. 2006 Velocityspace structures of distribution function in toroidal ion temperature gradient turbulence. Nucl. Fusion 46, 24–32.CrossRefGoogle Scholar