Skip to main content

Optimisation of confinement in a fusion reactor using a nonlinear turbulence model

  • E. G. Highcock (a1) (a2) (a3), N. R. Mandell (a4), M. Barnes (a2) and W. Dorland (a5)

The confinement of heat in the core of a magnetic fusion reactor is optimised using a multidimensional optimisation algorithm. For the first time in such a study, the loss of heat due to turbulence is modelled at every stage using first-principles nonlinear simulations which accurately capture the turbulent cascade and large-scale zonal flows. The simulations utilise a novel approach, with gyrofluid treatment of the small-scale drift waves and gyrokinetic treatment of the large-scale zonal flows. A simple near-circular equilibrium with standard parameters is chosen as the initial condition. The figure of merit, fusion power per unit volume, is calculated, and then two control parameters, the elongation and triangularity of the outer flux surface, are varied, with the algorithm seeking to optimise the chosen figure of merit. A twofold increase in the plasma power per unit volume is achieved by moving to higher elongation and strongly negative triangularity.

Corresponding author
Email address for correspondence:
Hide All
Abel, I. G. & Cowley, S. C. 2013 Multiscale gyrokinetics for rotating tokamak plasmas: II. Reduced models for electron dynamics. New J. Phys. 15, 023041.
Abel, I. G., Plunk, G. G., Wang, E., Barnes, M., Cowley, S. C., Dorland, W. & Schekochihin, A. A. 2013 Multiscale gyrokinetics for rotating tokamak plasmas: fluctuations, transport and energy flows. Rep. Progr. Phys. 76 (11), 116201.
Abramson, D., Peachey, T. & Lewis, A. 2006 Model optimization and parameter estimation with Nimrod / O. In Proceedings of the 6th Intl Conf. on Computational Science (ICCS’06), vol. 1, pp. 720727.
Barnes, M., Abel, I. G., Dorland, W., Görler, T., Hammett, G. W. & Jenko, F. 2010 Direct multiscale coupling of a transport code to gyrokinetic turbulence codes. Phys. Plasmas 17 (5), 056109.
Barnes, M., Parra, F. I., Highcock, E. G., Schekochihin, A., Cowley, S. C. & Roach, C. M. 2011a Turbulent transport in tokamak plasmas with rotational shear. Phys. Rev. Lett. 106 (17), 14.
Barnes, M., Parra, F. I. & Schekochihin, A. A. 2011b Critically balanced ion temperature gradient turbulence in fusion plasmas. Phys. Rev. Lett. 107 (11), 115003.
Beer, M. A., Cowley, S. C. & Hammett, G. W. 1995 Field-aligned coordinates for nonlinear simulations of tokamak turbulence. Phys. Plasmas 2 (7), 26872700.
Beer, M. A. & Hammett, G. W. 1996 Toroidal gyrofluid equations for simulations of tokamak turbulence. Phys. Plasmas 3 (11), 4046.
Beer, M. & Hammett, G. 1998 The dynamics of small-scale turbulence-driven flows. In Varenna Proc., pp. 111.
Bourdelle, C., Citrin, J., Baiocchi, B., Casati, A., Cottier, P., Garbet, X. & Imbeaux, F. 2016 Core turbulent transport in tokamak plasmas: bridging theory and experiment with QuaLiKiz. Plasma Phys. Control. Fusion 58 (1), 014036.
Budny, R. V. 2009 Comparisons of predicted plasma performance in ITER H-mode plasmas with various mixes of external heating. Nucl. Fusion 49 (8), 85008.
Burrell, K. H. 1997 Effects of ExB velocity shear and magnetic shear on turbulence and transport in magnetic confinement devices. Phys. Plasmas 4 (5), 14991518.
Chang, C. S. & Hinton, F. L. 1982 Effect of impurity particles on the finite-aspect ratio neoclassical ion thermal conductivity in a tokamak. Phys. Fluids 25 (1493), 3314.
Chen, C., Wicks, R., Horbury, T. & Schekochihin, A. 2010 Interpreting power anisotropy measurements in plasma turbulence. Astrophys. J. Lett. 711 (2), L79.
Citrin, J., Breton, S., Felici, F., Imbeaux, F., Aniel, T., Artaud, J. F., Baiocchi, B., Bourdelle, C., Camenen, Y. & Garcia, J. 2015a Real-time capable first principle based modelling of tokamak turbulent transport. Nucl. Fusion 55 (9), 92001.
Citrin, J., Garcia, J., Görler, T., Jenko, F., Mantica, P., Told, D., Bourdelle, C., Hatch, D. R., Hogeweij, G. M. D., Johnson, T. et al. 2015b Electromagnetic stabilization of tokamak microturbulence in a high- $\unicode[STIX]{x1D6FD}$ regime. Plasma Phys. Control. Fusion 57 (1), 014032.
Citrin, J., Jenko, F., Mantica, P., Told, D., Bourdelle, C., Dumont, R., Garcia, J., Haverkort, J., Hogeweij, G., Johnson, T. et al. 2014 Ion temperature profile stiffness: non-linear gyrokinetic simulations and comparison with experiment. Nucl. Fusion 54 (2), 023008.
Colyer, G. J., Schekochihin, A. A., Parra, F. I., Roach, C. M., Barnes, M. A., Ghim, Y. C. & Dorland, W. 2017 Collisionality scaling of the electron heat flux in ETG turbulence. Plasma Phys. Control. Fusion 59 (5), 055002.
Dimits, A. M., Bateman, G., Beer, M. A., Cohen, B. I., Dorland, W., Hammett, G. W., Kim, C., Kinsey, J. E., Kotschenreuther, M., Kritz, A. H. et al. 2000 Comparisons and physics basis of tokamak transport models and turbulence simulations. Phys. Plasmas 7 (3), 969.
Dorland, W. & Hammett, G. W. 1993 Gyrofluid turbulence models with kinetic effects. Phys. Fluids B 5, 812835.
Dorland, W., Jenko, F., Kotschenreuther, M. & Rogers, B. N. 2000 Electron temperature gradient turbulence. Phys. Rev. Lett. 85 (26 Pt 1), 55795582.
Doyle, E. J., Houlberg, W. A., Kamada, Y., Mukhovatov, V., Osborne, T. H., Polevoi, A., Bateman, G., Connor, J. W., Cordey, J. G., Fujita, T. et al. & ITPA Transport Physics Topical Group, ITPA Confinement Database, Modelling Topical Group & ITPA Pedestal Edge Topical Group, Physics 2007 Chapter 2: Plasma confinement and transport. Nucl. Fusion 47 (6), S18.
Esteco, M.2018 modeFrontier Home Page.
Federici, G., Kemp, R., Ward, D., Bachmann, C., Franke, T., Gonzalez, S., Lowry, C., Gadomska, M., Harman, J., Meszaros, B. et al. 2014 Overview of EU DEMO design and R&D activities. Fusion Engng Des. 89 (7–8), 882889.
Frieman, E. A. & Chen, L. 1982 Nonlinear gyrokinetic equations for low-frequency electromagnetic waves in general plasma equilibria. Phys. Fluids 25 (3), 502.
Galambos, J., Perkins, L., Haney, S. & Mandrekas, J. 1995 Commercial tokamak reactor potential with advanced tokamak operation. Nucl. Fusion 35 (5), 551573.
Glasstone, S. & Lovberg, R. H. 1960 Controlled Thermonuclear Reactions. D. Van Noatrand Company.
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 (25), 30193022.
Highcock, E. G., Barnes, M., Parra, F. I., Schekochihin, A. A., Roach, C. M. & Cowley, S. C. 2011 Transport bifurcation induced by sheared toroidal flow in tokamak plasmas. Phys. Plasmas 18 (10), 102304.
Highcock, E. G., Barnes, M., Schekochihin, A. A., Parra, F. I., Roach, C. M. & Cowley, S. C. 2010 Transport bifurcation in a rotating tokamak plasma. Phys. Rev. Lett. 105 (21), 215003.
Highcock, E. G., Schekochihin, A. A., Cowley, S. C., Barnes, M., Parra, F. I., Roach, C. M. & Dorland, W. 2012 Zero-turbulence manifold in a toroidal plasma. Phys. Rev. Lett. 109 (26), 265001.
Howes, G. G., TenBarge, J. M., Dorland, W., Quataert, E., Schekochihin, A. A., Numata, R. & Tatsuno, T. 2011 Gyrokinetic simulations of solar wind turbulence from ion to electron scales. Phys. Rev. Lett. 107 (3), 035004.
Jardin, S. C., Kessel, C. E., Mau, T. K., Miller, R. L., Najmabadi, F., Chan, V. S., Chu, M. S., Lahaye, R., Lao, L. L., Petrie, T. W. et al. 2006 Physics basis for the advanced tokamak fusion power plant, ARIES-AT. Fusion Engng Des. 80 (1–4), 2562.
Kawamori, E. 2013 Experimental verification of entropy cascade in two-dimensional electrostatic turbulence in magnetized plasma. Phys. Rev. Lett. 110 (9), 095001.
Kinsey, J. E., Staebler, G. M., Candy, J., Waltz, R. E. & Budny, R. V. 2011 ITER predictions using the GYRO verified and experimentally validated trapped gyro-Landau fluid transport model. Nucl. Fusion 51 (8), 083001.
Kotschenreuther, M., Rewoldt, G. & Tang, W. M. 1995 Comparison of initial value and eigenvalue codes for kinetic toroidal plasma instabilities. Comput. Phys. Commun. 88 (2–3), 128140.
Lewis, A.2004 Parallel optimisation algorithms for continuous, non-linear numerical simulations. PhD thesis, Griffith University, Brisbane, Australia.
Luce, T., Challis, C., Ide, S., Joffrin, E., Kamada, Y., Politzer, P., Schweinzer, J., Sips, A. C. C., Stober, J., Giruzzi, G. et al. 2014 Development of advanced inductive scenarios for ITER. Nucl. Fusion 54 (1), 013015.
Lütjens, H., Bondeson, A. & Sauter, O. 1996 The CHEASE code for toroidal MHD equilibria. Comput. Phys. Commun. 97, 219260.
Maggi, C. F., Groebner, R. J., Oyama, N., Sartori, R., Horton, L. D., Sips, A. C., Suttrop, W., Leonard, A., Luce, T. C., Wade, M. R. et al. 2007 Characteristics of the H-mode pedestal in improved confinement scenarios in ASDEX upgrade, DIII-D, JET and JT-60U. Nucl. Fusion 47 (7), 535551.
Mandell, N. & Dorland, W. 2014 Hybrid gyrokinetic/gyrofluid simulation of ITG turbulence. In Bull. Am. Phys. Soc., vol. 59. Abstract CP8.039.
Mandell, N., Dorland, W. & Landreman, M. 2018 Laguerre–Hermite pseudo-spectral velocity formulation of gyrokinetics. J. Plasma Phys. 84 (1), 905840108.
Manousopoulos, P. & Michalopoulos, M. 2009 Comparison of non-linear optimization algorithms for yield curve estimation. Eur. J. Oper. Res. 192 (2), 594602.
Marinoni, A., Brunner, S., Camenen, Y., Coda, S., Graves, J. P., Lapillonne, X., Pochelon, A., Sauter, O. & Villard, L. 2009 The effect of plasma triangularity on turbulent transport: modeling TCV experiments by linear and non-linear gyrokinetic simulations. Plasma Phys. Control. Fusion 51 (5), 055016.
Meneghini, O., Snyder, P. B., Smith, S. P., Candy, J., Staebler, G. M., Belli, E. A., Lao, L. L., Park, J. M., Green, D. L., Elwasif, W. et al. 2016 Integrated fusion simulation with self-consistent core-pedestal coupling. Phys. Plasmas 23 (4), 042507.
Merle, A., Sauter, O. & Yu Medvedev, S. 2017 Pedestal properties of H-modes with negative triangularity using the EPED-CH model. Plasma Phys. Control. Fusion 59 (10), 104001.
Mukhovatov, V., Shimomura, Y., Polevoi, A., Shimada, M., Sugihara, M., Bateman, G., Cordey, J. G., Kardaun, O., Pereverzev, G., Voitsekhovitch, I. et al. 2003 Comparison of ITER performance predicted by semi-empirical and theory-based transport models. Nucl. Fusion 43 (9), 942948.
Parail, V., Albanese, R., Ambrosino, R., Artaud, J. F., Besseghir, K., Cavinato, M., Corrigan, G., Garcia, J., Garzotti, L., Gribov, Y. et al. 2013 Self-consistent simulation of plasma scenarios for ITER using a combination of 1.5D transport codes and free-boundary equilibrium codes. Nucl. Fusion 53 (11), 113002.
Parker, J. T., Highcock, E. G., Schekochihin, A. A. & Dellar, P. J. 2016 Suppression of phase mixing in drift-kinetic plasma turbulence. Phys. Plasmas 23 (7), 070703.
Pochelon, A., Angelino, P., Behn, R., Brunner, S., Coda, S., Kirneva, N., Medvedev, S., Reimerdes, H., Rossel, J., Sauter, O. et al. 2012 Recent TCV results – Innovative plasma shaping to improve plasma properties and insight. Plasma Fusion Res. 7 (SPL.ISS.1), 18.
Pochelon, A., Goodman, T., Henderson, M., Angioni, C., Behn, R., Coda, S., Hofmann, F., Hogge, J.-P., Kirneva, N., Martynov, A. et al. 1999 Energy confinement and MHD activity in shaped TCV plasmas with localized electron cyclotron heating. Nucl. Fusion 39 (11Y), 18071818.
Roach, C., Walters, M., Budny, R., Imbeaux, F., Fredian, T., Greenwald, M., Stillerman, J., Alexander, D., Carlsson, J., Cary, J. et al. 2008 The 2008 public release of the international multi-tokamak confinement profile database. Nucl. Fusion 48 (12), 125001.
Rogers, B. N., Dorland, W. & Kotschenreuther, M. 2000 Generation and stability of zonal flows in ion-temperature-gradient mode turbulence. Phys. Rev. Lett. 85 (25), 53365339.
Rosenbluth, M. & Hinton, F. 1998 Poloidal flow driven by ion-temperature-gradient turbulence in tokamaks. Phys. Rev. Lett. 80 (4), 724.
Schekochihin, A. A., Cowley, S. C., Dorland, W., Hammett, G. W., Howes, G. G., Quataert, E. & Tatsuno, T. 2009 Astrophysical gyrokinetics: kinetic and fluid turbulent cascades in magnetized weakly collisional plasmas. Astrophys. J. Suppl. Ser. 182 (1), 310377.
Schekochihin, A. A., Parker, J. T., Highcock, E. G., Dellar, P. J., Dorland, W. & Hammett, G. W. 2016 Phase mixing versus nonlinear advection in drift-kinetic plasma turbulence. J. Plasma Phys. 82 (02), 905820212.
Simpson, T., Toropov, V., Balabanov, V. & Viana, F. 2008 Design and analysis of computer experiments in multidisciplinary design optimization: a review of how far we have come - or not. In 12th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference (September), pp. 122.
Snyder, P. & Hammett, G. 2001 A Landau fluid model for electromagnetic plasma microturbulence. Phys. Plasmas 8 (7), 31993216.
Sorbom, B. N., Ball, J., Palmer, T. R., Mangiarotti, F. J., Sierchio, J. M., Bonoli, P., Kasten, C., Sutherland, D. A., Barnard, H. S., Haakonsen, C. B. et al. 2015 ARC: a compact, high-field, fusion nuclear science facility and demonstration power plant with demountable magnets. Fusion Engng Des. 100, 378405.
Staebler, G. M. & John, H. E. S. 2006 Predicted toroidal rotation enhancement of fusion power production in ITER. Nucl. Fusion 46 (8), L6L8.
Staebler, G. M., Kinsey, J. E. & Waltz, R. E. 2007 A theory-based transport model with comprehensive physics. Phys. Plasmas 14 (5), 055909.
Stork, D., Agostini, P., Boutard, J. L., Buckthorpe, D., Diegele, E., Dudarev, S. L., English, C., Federici, G., Gilbert, M. R., Gonzalez, S. et al. 2014 Materials R&D for a timely DEMO: key findings and recommendations of the EU Roadmap Materials Assessment Group. Fusion Engng Des. 89 (7–8), 15861594.
Sugama, H. & Horton, W. 1998 Nonlinear electromagnetic gyrokinetic equation for plasmas with large mean flows. Phys. Plasmas 5 (7), 2560.
Synakowski, E. J. 1999 Formation and structure of internal and edge transport barriers. Plasma Phys. Control. Fusion 40 (5), 581596.
Tatsuno, T., Dorland, W., Schekochihin, A. A., Plunk, G. G., Barnes, M., Cowley, S. C. & Howes, G. G. 2009 Nonlinear phase mixing and phase-space cascade of entropy in gyrokinetic plasma turbulence. Phys. Rev. Lett. 103 (1), 25.
Tatsuno, T., Plunk, G., Barnes, M., Dorland, W., Howes, G. & Numata, R. 2012 Freely decaying turbulence in two-dimensional electrostatic gyrokinetics. Phys. Plasmas 19 (12), 122305.
Uckan, N. 1990 ITER Physics Design Guidelines: 1989, ITER Documentation Series, vol. 10. IAEA.
Wagner, F., Fussmann, G., Grave, T., Keilhacker, M., Kornherr, M., Lackner, K., McCormick, K., Müller, E. R., Stäbler, A., Becker, G. et al. 1984 Development of an Edge Transport Barrier at the H-Mode Transition of ASDEX. Phys. Rev. Lett. 53 (15), 14531456.
Weisen, H., Alberti, S., Berry, S., Behn, R., Blanchard, P., Bosshard, P., Bühlmann, F., Chavan, R., Coda, S., Deschenaux, C. et al. 1999 Effect of plasma shape on confinement and MHD behaviour in TCV. Plasma Phys. Control. Fusion 39 (12B), B135B144.
Wenninger, R., Arbeiter, F., Aubert, J., Aho-Mantila, L., Albanese, R., Ambrosino, R., Angioni, C., Artaud, J.-F., Bernert, M., Fable, E. et al. 2015 Advances in the physics basis for the European DEMO design. Nucl. Fusion 55, 063003.
White, A. E., Howard, N. T., Greenwald, M., Reinke, M. L., Sung, C., Baek, S., Barnes, M., Candy, J., Dominguez, A., Ernst, D. et al. 2013 Multi-channel transport experiments at Alcator C-Mod and comparison with gyrokinetic simulations. Phys. Plasmas 20 (5), 056106.
van Wyk, F., Highcock, E. G., Schekochihin, A. A., Roach, C. M., Field, A. R. & Dorland, W. 2016 Transition to subcritical turbulence in a tokamak plasma. J. Plasma Phys. 82 (6), 905820609.
Xanthopoulos, P., Mynick, H. E., Helander, P., Turkin, Y., Plunk, G. G., Jenko, F., Görler, T., Told, D., Bird, T. & Proll, J. H. 2014 Controlling turbulence in present and future stellarators. Phys. Rev. Lett. 113 (15), 14.
Xiao, Y., Catto, P. J. & Dorland, W. 2007 Effects of finite poloidal gyroradius, shaping, and collisions on the zonal flow residual. Phys. Plasmas 14 (5), 055910.
Zohm, H., Angioni, C., Fable, E., Federici, G., Gantenbein, G., Hartmann, T., Lackner, K., Poli, E., Porte, L., Sauter, O. et al. 2013 On the physics guidelines for a tokamak DEMO. Nucl. Fusion 53 (7), 073019.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Plasma Physics
  • ISSN: 0022-3778
  • EISSN: 1469-7807
  • URL: /core/journals/journal-of-plasma-physics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 48 *
Loading metrics...

Abstract views

Total abstract views: 125 *
Loading metrics...

* Views captured on Cambridge Core between 23rd April 2018 - 25th June 2018. This data will be updated every 24 hours.