Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-29T04:55:01.746Z Has data issue: false hasContentIssue false
  • English
  • Français

Plasma stability theory including the resistive wall effects

Part of: Focus on Fusion

Published online by Cambridge University Press:  09 December 2015

V. D. Pustovitov
V. D. Pustovitov*
Affiliation:
National Research Centre ‘Kurchatov Institute’, Kurchatov Sq., 1, Moscow 123182, Russian Federation National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Moscow 115409, Russian Federation
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Plasma stabilization due to a nearby conducting wall can provide access to better performance in some scenarios in tokamaks. This was proved by experiments with an essential gain in ${\it\beta}$ and demonstrated as a long-lasting effect at sufficiently fast plasma rotation in the DIII-D tokamak (see, for example, Strait et al., Nucl. Fusion, vol. 43, 2003, pp. 430–440). The rotational stabilization is the central topic of this review, though eventually the mode rotation gains significance. The analysis is based on the first-principle equations describing the energy balance with dissipation in the resistive wall. The method emphasizes derivation of the dispersion relations for the modes which are faster than the conventional resistive wall modes, but slower than the ideal magnetohydrodynamics modes. Both the standard thin wall and ideal-wall approximations are not valid in this range. Here, these are replaced by an approach incorporating the skin effect in the wall. This new element in the stability theory makes the energy sink a nonlinear function of the complex growth rate. An important consequence is that a mode rotating above a critical level can provide a damping effect sufficient for instability suppression. Estimates are given and applications are discussed.

Type
Research Article
Information
Journal of Plasma Physics , Volume 81 , Issue 6 , December 2015 , 905810609
Copyright
© Cambridge University Press 2015 

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

ITER Physics Expert Group on Disruptions Plasma Control and MHD and ITER Physics Basis Editors 1999 Chapter 3: MHD stability, operational limits and disruptions. Nucl. Fusion 39, 2251–2389.Google Scholar
Hender, T. C., Wesley, J. C, Bialek, J., Bondeson, A., Boozer, A. H., Buttery, R. J., Garofalo, A., Goodman, T. P., Granetz, R. S., Gribov, Y. et al. 2007 Chapter 3: MHD stability, operational limits and disruptions. Nucl. Fusion 47, S128–S202.CrossRefGoogle Scholar
Bernstein, I. B., Frieman, E. A., Kruskal, M. D. & Kulsrud, R. M. 1958 An energy principle for hydromagnetic stability problems. Proc. R. Soc. Lond. A 244, 17–40.Google Scholar
Kadomtsev, B. B. 1966 Hydromagnetic stability of a plasma. In Reviews of Plasma Physics (ed. Leontovich, M. A.), vol. 2, p. 153. Consultants Bureau.Google Scholar
Freidberg, J. P. 1987 Ideal Magnetohydrodynamics. Plenum.CrossRefGoogle Scholar
Miyamoto, K. 1997 Fundamentals of Plasma Physics and Controlled Fusion. Iwanami.Google Scholar
Wesson, J. A. 2004 Tokamaks, 3rd edn. Clarendon.Google Scholar
Goedbloed, J. P. & Poedts, S. 2004 Principles of Magnetohydrodynamics. Cambridge University Press.Google Scholar
Mikhailovskii, A. B. 1998 Instabilities in a Confined Plasma. Institute of Physics Publishing.Google Scholar
Chu, M. S. & Okabayashi, M. 2010 Stabilization of the external kink and the resistive wall mode. Plasma Phys. Control. Fusion 52, 123001.Google Scholar
Garofalo, A. M., Jensen, T. H., Johnson, L. C., La Haye, R. J., Navratil, G. A., Okabayashi, M., Scoville, J. T., Strait, E. J., Baker, D. R., Bialek, J. et al. 2002 Sustained rotational stabilization of DIII-D plasmas above the no-wall beta limit. Phys. Plasmas 9, 1997–2005.CrossRefGoogle Scholar
Strait, E. J., Bialek, J., Bogatu, N., Chance, M., Chu, M. S., Edgell, D., Garofalo, A. M., Jackson, G. L., Jensen, T. H., Johnson, L. C. et al. 2003 Resistive wall stabilization of high-beta plasmas in DIII-D. Nucl. Fusion 43, 430–440.CrossRefGoogle Scholar
Garofalo, A. M., Jensen, T. H. & Strait, E. J. 2003 Analysis of stable resistive wall modes in a rotating plasma. Phys. Plasmas 10, 4776–4783.CrossRefGoogle Scholar
Chu, M. S., Bondeson, A., Chance, M., Liu, Y. Q., Garofalo, A. M., Glasser, A. L., La Haye, R. J., Lao, L. L., Navratil, G. A. et al. 2004 Modeling of feedback and rotation stabilization of the resistive wall mode in tokamaks. Phys. Plasmas 11, 2497–2504.Google Scholar
Strait, E. J., Bialek, J. M., Bogatu, I. N., Chance, M. S., Chu, M. S., Edgell, D. H., Garofalo, A. M., Jackson, G. L., Jayakumar, R. J., Jensen, T. H. et al. 2004 Resistive wall mode stabilization with internal feedback coils in DIII-D. Phys. Plasmas 11, 2505–2513.CrossRefGoogle Scholar
La Haye, R. J., Bondeson, A., Chu, M. S., Garofalo, A. M., Liu, Y. Q., Navratil, G. A., Okabayashi, M., Reimerdes, H. & Strait, E. J. 2004 Scaling of the critical plasma rotation for stabilization of the $n=1$ resistive wall mode (ideal kink) in the DIII-D tokamak. Nucl. Fusion 44, 1197–1203.Google Scholar
Okabayashi, M., Bialek, J., Bondeson, A., Chance, M. S., Chu, M. S., Garofalo, A. M., Hatcher, R., In, Y., Jackson, G. L., Jayakumar, R. J. et al. 2005 Control of the resistive wall mode with internal coils in the DIII-D tokamak. Nucl. Fusion 45, 1715–1731.CrossRefGoogle Scholar
Strait, E. J. 2005 Stability limits of high-beta plasmas in DIII-D. Fusion Sci. Technol. 48, 864–874.CrossRefGoogle Scholar
Turnbull, A. D., Brennan, D. P., Chu, M. S., Lao, L. L. & Snyder, P. B. 2005 Theory and simulation basis for magnetohydrodynamic stability in DIII-D. Fusion Sci. Technol. 48, 875–905.CrossRefGoogle Scholar
Garofalo, A. M. 2005 Resistive wall mode stabilization studies at DIII-D. Fusion Sci. Technol. 48, 918–930.CrossRefGoogle Scholar
Reimerdes, H., Hender, T. C., Sabbagh, S. A., Bialek, J. M., Chu, M. S., Garofalo, A. M., Gryaznevich, M. P., Howell, D. F., Jackson, G. L., La Haye, R. J. et al. 2006 Cross-machine comparison of resonant field amplification and resistive wall mode stabilization by plasma rotation. Phys. Plasmas 13, 056107.CrossRefGoogle Scholar
Reimerdes, H., Garofalo, A. M., Jackson, G. L., Okabayashi, M., Strait, E. J., Chu, M. S., In, Y., La Haye, R. J., Lanctot, M. J., Liu, Y. Q. et al. 2007 Reduced critical rotation for resistive-wall mode stabilization in a near-axisymmetric configuration. Phys. Rev. Lett. 98, 055001.Google Scholar
Strait, E. J., Garofalo, A. M., Jackson, G. L., Okabayashi, M., Reimerdes, H., Chu, M. S., Fitzpatrick, R., Groebner, R. J., In, Y., La Haye, R. J. et al. 2007 Resistive wall mode stabilization by slow plasma rotation in DIII-D tokamak discharges with balanced neutral beam injection. Phys. Plasmas 14, 056101.CrossRefGoogle Scholar
Okabayashi, M., Bogatu, I. N., Chance, M. S., Chu, M. S., Garofalo, A. M., In, Y., Jackson, G. L., La Haye, R. J., Lanctot, M. J., Manickam, J. et al. 2009 Comprehensive control of resistive wall modes in DIII-D advanced tokamak plasmas. Nucl. Fusion 49, 125003.Google Scholar
Okabayashi, M., Matsunaga, G., Degrassie, J. S., Heidbrink, W. W., In, Y., Liu, Y. Q., Reimerdes, H., Solomon, W. M., Strait, E. J., Takechi, M. et al. 2011 Off-axis fishbone-like instability and excitation of resistive wall modes in JT-60U and DIII-D. Phys. Plasmas 18, 056112.CrossRefGoogle Scholar
Kamada, Y., Barabaschi, P., Ishida, S., Ide, S., Lackner, K., Fujita, T., Bolzonella, T., Suzuki, T., Matsunaga, G., Yoshida, M. et al. 2011 Plasma regimes and research goals of JT-60SA towards ITER and DEMO. Nucl. Fusion 51, 073011.Google Scholar
Kamada, Y., Barabaschi, P. & Ishida, S. 2013 Progress of the JT-60SA project. Nucl. Fusion 53, 104010.Google Scholar
Sabbagh, S. A., Sontag, A. C., Bialek, J. M., Gates, D. A., Glasser, A. H., Menard, J. E., Zhu, W., Bell, M. G., Bell, R. E., Bondeson, A. et al. 2006 Resistive wall stabilized operation in rotating high beta NSTX plasmas. Nucl. Fusion 46, 635–644.Google Scholar
Sontag, A. C., Sabbagh, S. A., Zhu, W., Menard, J. E., Bell, R. E., Bialek, J. M., Bell, M. G., Gates, D. A., Glasser, A. H., Leblanc, B. P. et al. 2007 Investigation of resistive wall mode stabilization physics in high-beta plasmas using applied non-axisymmetric fields in NSTX. Nucl. Fusion 47, 1005–1011.Google Scholar
Takechi, M., Matsunaga, G., Aiba, N., Fujita, T., Ozeki, T., Koide, Y., Sakamoto, Y., Kurita, G., Isayama, A., Kamada, Y.& The JT-60 Team 2007 Identification of a low plasma-rotation threshold for stabilization of the resistive-wall mode. Phys. Rev. Lett. 98, 055002.Google Scholar
Zakharov, L. & Putvinskii, S. V. 1987 Effect of plasma rotation in a tokamak on the stabilizing effect of a conducting wall. Sov. J. Plasma Phys. 13, 68.Google Scholar
Bondeson, A. & Ward, D. J. 1994 Stabilization of external modes in tokamaks by resistive walls and plasma rotation. Phys. Rev. Lett. 72, 2709–2712.Google Scholar
Gimblett, C. G. & Hastie, R. J. 1999 The effect of rotation on the stability of Resistive Wall Modes in tokamaks. In Theory of Fusion Plasmas, pp. 319–332. Editrice Compositori.Google Scholar
Gimblett, C. G. & Hastie, R. J. 2004 The interaction of error fields and resistive wall modes. Phys. Plasmas 11, 1019–1027.CrossRefGoogle Scholar
Haney, S. W. & Freidberg, J. P. 1989 Variational methods for studying tokamak stability in the presence of a thin resistive wall. Phys. Fluids B 1, 1637–1645.Google Scholar
Betti, R. & Freidberg, J. P. 1995 Stability analysis of resistive wall kink modes in rotating plasmas. Phys. Rev. Lett. 74, 2949–2952.CrossRefGoogle ScholarPubMed
Taylor, J. B., Connor, J. W., Gimblett, C. G., Wilson, H. R. & Hastie, R. J. 2001 Resistive wall modes and nonuniform wall rotation. Phys. Plasmas 8, 4062–4072.CrossRefGoogle Scholar
Chu, M. S., Chan, V. S., Chance, M. S., Edgell, D. H., Garofalo, A. M., Glasser, A. H., Guo, S. C., Humphreys, D. A., Jensen, T. H., Kim, J. S. et al. 2003 Modelling of feedback and rotation stabilization of the resistive wall mode in tokamaks. Nucl. Fusion 43, 196–201.CrossRefGoogle Scholar
Frassinetti, L., Brunsell, P. R. & Drake, J. R. 2009 Experiments and modelling of active quasi-single helicity regime generation in a reversed field pinch. Nucl. Fusion 49, 075019.Google Scholar
Igochine, V. 2015 Chapter 6 resistive wall mode (RWM). In Active Control of Magneto-Hydrodynamic Instabilities in Hot Plasmas (ed. Igochine, V.), Springer Series on Atomic, Optical, and Plasma Physics, vol. 83, pp. 183–226. Springer.Google Scholar
Bialek, J., Boozer, A. H., Mauel, M. E. & Navratil, G. A. 2001 Modeling of active control of external magnetohydrodynamic instabilities. Phys. Plasmas 8, 2170–2180.Google Scholar
Bondeson, A., Liu, Y. Q., Gregoratto, D., Fransson, C. M., Lennartson, B., Breitholtz, C., Gribov, Y. & Pustovitov, V. D. 2002 Physics and control of resistive wall modes. Phys. Plasmas 9, 2044–2050.Google Scholar
Chu, M. S., Chance, M. S., Glasser, A. H. & Okabayashi, M. 2003 Normal mode approach to modelling of feedback stabilization of the resistive wall mode. Nucl. Fusion 43, 441–454.CrossRefGoogle Scholar
Medvedev, S. Yu. & Pustovitov, V. D. 2004 Modeling of the feedback stabilization of the resistive wall modes in a tokamak. Plasma Phys. Rep. 30, 895–906.Google Scholar
Mauel, M. E., Bialek, J., Boozer, A. H., Cates, C., James, R., Katsuro-Hopkins, O., Klein, A., Liu, Y., Maurer, D. A., Maslovsky, D. et al. 2005 Dynamics and control of resistive wall modes with magnetic feedback control coils: experiment and theory. Nucl. Fusion 45, 285–293.CrossRefGoogle Scholar
Chu, M. S. & Ichiguchi, K. 2005 Effect of the resistive wall on the growth rate of weakly unstable external kink mode in general 3D configurations. Nucl. Fusion 45, 804–813.Google Scholar
Lanctot, M. J., Reimerdes, H., Garofalo, A. M., Chu, M. S., Liu, Y. Q., Strait, E. J., Jackson, G. L., La Haye, R. J., Okabayashi, M., Osborne, T. H. et al. 2010 Validation of the linear ideal magnetohydrodynamic model of three-dimensional tokamak equilibria. Phys. Plasmas 17, 030701.Google Scholar
Lanctot, M. J., Reimerdes, H., Garofalo, A. M., Chu, M. S., Hanson, J. M., Liu, Y. Q., Navratil, G. A., Bogatu, I. N., In, Y. G., Jackson, G. L. et al. 2011 Measurement and modeling of three-dimensional equilibria in DIII-D. Phys. Plasmas 18, 056121.Google Scholar
Shiraishi, J., Aiba, N., Miyato, N. & Yagi, M. 2014 Effects of centrifugal modification of magnetohydrodynamic equilibrium on resistive wall mode stability. Nucl. Fusion 54, 083008.Google Scholar
Chu, M. S., Greene, J. M., Jensen, T. H., Miller, R. L., Bondeson, A., Johnson, R. W. & Mauel, M. E. 1995 Effect of toroidal plasma flow and flow shear on global magnetohydrodynamic MHD modes. Phys. Plasmas 2, 2236–2241.CrossRefGoogle Scholar
Taylor, T. S., Strait, E. J., Lao, L. L., Mauel, M., Turnbull, A. D., Burrell, K. H., Chu, M. S., Ferron, J. R., Groebner, R. J., La Haye, R. J. et al. 1995 Wall stabilization of high beta plasmas in DIII-D. Phys. Plasmas 2, 2390–2396.CrossRefGoogle Scholar
Hu, B., Betti, R. & Manickam, J. 2005 Application of the low-frequency energy principle to wall modes. Phys. Plasmas 12, 057301.CrossRefGoogle Scholar
Ham, C. J., Gimblett, C. G. & Hastie, R. J. 2009 Rotational stabilization of the resistive wall mode by coupling to a dissipative rational surface. Plasma Phys. Control. Fusion 51, 115010.Google Scholar
Ham, C. J., Gimblett, C. G. & Hastie, R. J. 2011 Analytical modelling of resistive wall mode stabilization by rotation in toroidal tokamak plasmas. Plasma Phys. Control. Fusion 53, 025001.Google Scholar
Liu, Y. Q., Chu, M. S., Chapman, I. T. & Hender, T. C. 2008 Toroidal self-consistent modeling of drift kinetic effects on the resistive wall mode. Phys. Plasmas 15, 112503.CrossRefGoogle Scholar
Liu, Y. Q., Chu, M. S., Chapman, I. T. & Hender, T. C. 2009 Modelling resistive wall modes in ITER with self-consistent inclusion of drift kinetic resonances. Nucl. Fusion 49, 035004.Google Scholar
Berkery, J. W., Sabbagh, S. A., Reimerdes, H., Betti, R., Hu, B., Bell, R. E., Gerhardt, S. P., Manickam, J. & Podestà, M. 2010 The role of kinetic effects, including plasma rotation and energetic particles, in resistive wall mode stability. Phys. Plasmas 17, 082504.Google Scholar
Hao, G. Z., Liu, Y. Q., Wang, A. K., Jiang, H. B., Lu, G., He, H. D. & Qiu, X. M. 2011 Stabilization of the resistive wall mode instability by trapped energetic particles. Phys. Plasmas 18, 032513.Google Scholar
Reimerdes, H., Berkery, J. W., Lanctot, M. J., Garofalo, A. M., Hanson, J. M., In, Y., Okabayashi, M., Sabbagh, S. A. & Strait, E. J. 2011 Evidence for the importance of trapped particle resonances for resistive wall mode stability in high beta tokamak plasmas. Phys. Rev. Lett. 106, 215002.Google Scholar
Hao, G. Z., Wang, A. K., Liu, Y. Q. & Qiu, X. M. 2011 Effect of trapped energetic particles on the resistive wall mode. Phys. Rev. Lett. 107, 015001.CrossRefGoogle ScholarPubMed
Berkery, J. W., Betti, R. & Sabbagh, S. A. 2011 Investigation of multiple roots of the resistive wall mode dispersion relation, including kinetic effects. Phys. Plasmas 18, 072501.Google Scholar
Yadykin, D., Liu, Y. Q. & Paccagnella, R. 2011 Effects of kinetic resonances on the stability of resistive wall modes in reversed field pinch. Plasma Phys. Control. Fusion 53, 085024.Google Scholar
Hao, G. Z., Liu, Y. Q., Wang, A. K. & Qiu, X. M. 2012 Kinetic effects of trapped energetic particles on stability of external kink modes with a resistive wall. Phys. Plasmas 19, 032507.Google Scholar
Wang, Z. R., Guo, S. C. & Liu, Y. Q. 2012 Drift kinetic effects on the resistive wall mode stability – comparison between reversed field pinches and tokamaks. Phys. Plasmas 19, 072518.CrossRefGoogle Scholar
Pustovitov, V. D. 2012 Thick-wall effects in the theory of resistive wall modes. Phys. Plasmas 19, 062503.Google Scholar
Pustovitov, V. D. 2013 Thick-wall effects on the rotational stabilization of resistive wall modes in tokamaks. Nucl. Fusion 53, 033001.Google Scholar
Pustovitov, V. D. 2013 Energy approach to stability analysis of the locked and rotating resistive wall modes in tokamaks. Plasma Phys. Rep. 39, 199–208.Google Scholar
Strumberger, E., Merkel, P., Sempf, M. & Günter, S. 2008 On fully three-dimensional resistive wall mode and feedback stabilization computations. Phys. Plasmas 15, 056110.Google Scholar
Albanese, R., Liu, Y. Q., Portone, A., Rubinacci, G. & Villone, F. 2008 Coupling between a 3-D integral eddy current formulation and a linearized MHD model for the analysis of resistive wall modes. IEEE Trans. Magn. 44, 1654–1657.CrossRefGoogle Scholar
Portone, A., Villone, F., Liu, Y., Albanese, R. & Rubinacci, G. 2008 Linearly perturbed MHD equilibria and 3D eddy current coupling via the control surface method. Plasma Phys. Control. Fusion 50, 085004.Google Scholar
Villone, F., Liu, Y. Q., Paccagnella, R., Bolzonella, T. & Rubinacci, G. 2008 Effects of three-dimensional electromagnetic structures on resistive-wall-mode stability of reversed field pinches. Phys. Rev. Lett. 100, 255005.Google Scholar
Gribov, Y. & Pustovitov, V. D. 2002 Analytical study of RWM feedback stabilisation with application to ITER. In Proceedings of the 19th IAEA Fusion Energy Conference (Lyon, 2002), IAEA. IAEA-CN-94/CT/P-12; http://www.iaea.org/programmes/ripc/physics/fec2002/html/fec2002.htm.Google Scholar
Chance, M. S., Chu, M. S., Okabayashi, M. & Turnbull, A. D. 2002 Theoretical modelling of the feedback stabilization of external MHD modes in toroidal geometry. Nucl. Fusion 42, 295–300.CrossRefGoogle Scholar
Pfirsch, D. & Tasso, H. 1971 A theorem on MHD-instability of plasmas with resistive walls. Nucl. Fusion 11, 259–260.Google Scholar
Reimerdes, H., Bialek, J., Chance, M. S., Chu, M. S., Garofalo, A. M., Gohil, P., In, Y., Jackson, G. L., Jayakumar, R. J., Jensen, T. H. et al. 2005 Measurement of resistive wall mode stability in rotating high- ${\it\beta}$ DIII-D plasmas. Nucl. Fusion 45, 368–376.Google Scholar
Bondeson, A., Liu, Y. Q., Gregoratto, D., Gribov, Y. & Pustovitov, V. D. 2002 Active control of resistive wall modes in the large-aspect-ratio tokamak. Nucl. Fusion 42, 768–779.CrossRefGoogle Scholar
Pustovitov, V. D. 2004 Error field amplification near the stability boundary of the modes interacting with a conducting wall. Plasma Phys. Rep. 30, 187–195.Google Scholar
Strumberger, E., Günter, S., Merkel, P., Riondato, S., Schwarz, E., Tichmann, C. & Zehrfeld, H. P. 2005 Numerical MHD stability studies: toroidal rotation, viscosity, resistive walls and current holes. Nucl. Fusion 45, 1156–1167.Google Scholar
Pustovitov, V. D. 2008 General formulation of the resistive wall mode coupling equations. Phys. Plasmas 15, 072501.Google Scholar
Pustovitov, V. D. 2012 Energy approach to the problem of plasma stability in tokamaks with a resistive wall. Phys. Lett. A 376, 2001–2003.Google Scholar
Pustovitov, V. D. 2012 A unified approach to description of the fast and slow resistive wall modes in tokamaks. Plasma Phys. Rep. 38, 697–707.Google Scholar
Katsuro-Hopkins, O., Sabbagh, S. A., Bialek, J. M., Park, H. K., Bak, J. G., Chung, J., Hahn, S. H., Kim, J. Y., Kwon, M., Lee, S. G. et al. 2010 Equilibrium and global MHD stability study of KSTAR high beta plasmas under passive and active mode control. Nucl. Fusion 50, 025019.CrossRefGoogle Scholar
Manickam, J., Chance, M. S., Jardin, S. C., Kessel, C., Monticello, D., Pomphrey, N., Reiman, A., Wang, C. & Zakharov, L. E. 1994 The prospects for magnetohydrodynamic stability in advanced tokamak regimes. Phys. Plasmas 1, 1601–1605.CrossRefGoogle Scholar
Pustovitov, V. D. 2009 Resistive ferromagnetic wall modes in theory and experiment. Phys. Plasmas 16, 052503.Google Scholar
Pustovitov, V. D. & Villone, F. 2010 Effect of ferromagnetic structures on RWM growth rates: a cylindrical model and a verification on JET. Plasma Phys. Control. Fusion 52, 065010.Google Scholar
Pustovitov, V. D. & Yanovskiy, V. V. 2015 Rotational stabilization of the resistive wall modes in tokamaks with a ferritic wall. Phys. Plasmas 22, 032509.Google Scholar
Levesque, J. P., Hughes, P. E., Bialek, J., Byrne, P. J., Mauel, M. E., Navratil, G. A., Peng, Q., Rhodes, D. J. & Stoafer, C. C. 2015 Active and passive kink mode studies in a tokamak with a movable ferromagnetic wall. Phys. Plasmas 22, 056102.Google Scholar
Zheng, L.-J. & Kotschenreuther, M. T. 2005 Wall thickness effect on the resistive wall mode stability in toroidal plasmas. Phys. Plasmas 12, 072504.CrossRefGoogle Scholar
Pustovitov, V. D. & Yanovskiy, V. V.2007 Dependence of the resistive wall mode growth rate on the wall thickness. In Proceedings of the 34th EPS Conference on Plasma Phys (Warsaw, Poland, 2007) vol. 31F (EPS, Mulhouse 2007) Paper P4.115.Google Scholar
Pustovitov, V. D. & Mayorova, M. S. 2006 Feedback stabilization of RWM with in-vessel coils. Plasma Phys. Control. Fusion 48, 51–59.Google Scholar
Pustovitov, V. D. 2009 The rotating wall effect on the error field induced torque and the error field shielding in tokamaks. Nucl. Fusion 49, 045003.Google Scholar
Pustovitov, V. D. 2011 Integral torque balance in tokamaks. Nucl. Fusion 51, 013006.Google Scholar
Pustovitov, V. D. 2007 Dynamic diagnostics of the error fields in tokamaks. Nucl. Fusion 47, 563–571.Google Scholar
Pustovitov, V. D. 2008 Decoupling in the problem of tokamak plasma response to asymmetric magnetic perturbations. Plasma Phys. Control. Fusion 50, 105001.Google Scholar
Pustovitov, V. D. 2011 The linear model and experimentally observed resonant field amplification in tokamaks and reversed field pinches. Plasma Phys. Rep. 37, 35–42.Google Scholar
Pustovitov, V. D. 2007 Rotational stabilization in the Boozer model. Phys. Plasmas 14, 022501.Google Scholar
Boozer, A. H. 1999 Feedback equations for the wall modes of a rotating plasma. Phys. Plasmas 6, 3180–3187.Google Scholar
Boozer, A. H. 2004 Robust feedback systems for resistive wall modes. Phys. Plasmas 11, 110–114.Google Scholar
Fitzpatrick, R. & Aydemir, A. Y. 1996 Stabilization of the resistive shell mode in tokamaks. Nucl. Fusion 36, 11–38.Google Scholar
Pustovitov, V. D. 2007 Rotation and plasma stability in the Fitzpatrick–Aydemir model. Phys. Plasmas 14, 082506.Google Scholar
Liu, Y. Q., Kirk, A. & Nardon, E. 2010 Full toroidal plasma response to externally applied nonaxisymmetric magnetic fields. Phys. Plasmas 17, 122502.Google Scholar
Turnbull, A. D. 2012 Plasma response models for non-axisymmetric perturbations. Nucl. Fusion 52, 054016.Google Scholar
Wade, M. R., Nazikian, R., Battaglia, D., Buttery, R. J., Degrassie, J. S., Evans, T. E., Fenstermacher, M. E., Ferraro, N. M., Grierson, B. A., Hansen, J. M. et al. 2012 Advances in the physics understanding of ELM suppression using resonant magnetic perturbations in DIII-D. In Proceedings of the 24th IAEA Fusion Energy Conference (San Diego, USA, 2012), IAEA. Paper EX/3-1.Google Scholar
Reiman, A., Ferraro, N. M., Turnbull, A., Park, J. K., Cerfon, A., Evans, T. E., Lanctot, M. J., Lazarus, E. A., Liu, Y., McFadden, G. et al. 2015 Tokamak plasma high field side response to an $n=3$ magnetic perturbation: a comparison of 3D equilibrium solutions from seven different codes. Nucl. Fusion 55, 063026.Google Scholar
Okabayashi, M., Bialek, J., Chance, M., Chu, M. S., Fredrickson, E. D., Garofalo, A. M., Gryaznevich, M., Hatcher, R. E., Jensen, T. H., Johnson, L. C. et al. 2001 Active feedback stabilization of the resistive wall mode on the DIII-D device. Phys. Plasmas 8, 2071–2082.Google Scholar
Baruzzo, M., Bolzonella, T., Liu, Y. Q., Manduchi, G., Marchiori, G., Soppelsa, A., Takechi, M. & Villone, F. 2012 RWM control studies on RFX-mod with a limited set of active coils. Nucl. Fusion 52, 103001.Google Scholar
Chapman, I. T., Hender, T. C., Howell, D. F., Erents, S. K., Gryaznevich, M. P., Shibaev, S., Stamp, M. F., De La Luna, E., Savchkov, A., Scannell, R.& The MAST Team & JET EFDA Contributors 2007 Perturbation of tokamak magnetic surfaces by applied toroidally asymmetric magnetic fields. Nucl. Fusion 47, L36–L40.Google Scholar
Liu, Y. Q., Chu, M. S., Guo, W. F., Villone, F., Albanese, R., Ambrosino, G., Baruzzo, M., Bolzonella, T., Chapman, I. T., Garofalo, A. M. et al. 2010 Resistive wall mode control code maturity: progress and specific examples. Plasma Phys. Control. Fusion 52, 104002.Google Scholar
Shafranov, V. D. & Zakharov, L. E. 1972 Use of the virtual-casing principle in calculating the containing magnetic field in toroidal plasma systems. Nucl. Fusion 12, 599–601.Google Scholar
Hirshman, S. P., Lazarus, E. A., Hanson, J. D., Knowlton, S. F. & Lao, L. L. 2004 Magnetic diagnostic responses for compact stellarators. Phys. Plasmas 11, 595–603.Google Scholar
Merkel, P. 1986 An integral equation technique for the exterior and interior Neumann problem in toroidal regions. J. Comput. Phys. 66, 83–98.Google Scholar
Gryaznevich, M. P., Liu, Y. Q., Hender, T. C., Howell, D. F., Beurskens, M., Chapman, I. T., Challis, C. D., Joffrin, E., Koslowski, H. R., Buratti, P., Solano, E.& JET-EFDA Contributors 2012 Determination of plasma stability using resonant field amplification in JET. Nucl. Fusion 52, 083018.Google Scholar
Schaffer, M. J., Snipes, J. A., Gohil, P., De Vries, P., Evans, T. E., Fenstermacher, M. E., Gao, X., Garofalo, A. M., Gates, D. A., Greenfield, C. M. et al. 2011 ITER test blanket module error field simulation experiments at DIII-D. Nucl. Fusion 51, 103028.Google Scholar
Pustovitov, V. D. & Yanovskiy, V. V. 2013 Analysis of the dispersion relation for resistive wall modes in tokamaks with account of the skin effect. Plasma Phys. Rep. 39, 345–353.Google Scholar
Pustovitov, V. D. & Yanovskiy, V. V. 2013 Modeling of the rotational stabilization of tokamak plasmas with account of skin effect in the resistive wall. Plasma Phys. Rep. 39, 779–786.Google Scholar
Villone, F. & Pustovitov, V. D. 2013 Skin effect modifications of the resistive wall mode dynamics in tokamaks. Phys. Lett. A 377, 2780–2784.Google Scholar
Sakasai, A., Masaki, K., Shibama, Y. K., Sakurai, S., Hayashi, T., Nakamura, S., Ozaki, H., Yokoyama, K., Seki, Y., Shibanuma, K.& The JT-60SA Team 2012 Manufacturing and development of JT-60SA vacuum vessel and divertor. In Proceedings of the 24th IAEA Fusion Energy Conference (San Diego, USA, 2012), IAEA. Paper FTP/P7-20; www-naweb.iaea.org/napc/physics/FEC/FEC2012/html/proceedings.pdf.Google Scholar
Liu, Y. Q., Chapman, I. T., Chu, M. S., Reimerdes, H., Villone, F., Albanese, R., Ambrosino, G., Garofalo, A. M., Gimblett, C. G., Hastie, R. J. et al. 2009 Progress in physics and control of the resistive wall mode in advanced tokamaks. Phys. Plasmas 16, 056113.Google Scholar
Strauss, H. R., Paccagnella, R. & Breslau, J. 2010 Wall forces produced during ITER disruptions. Phys. Plasmas 17, 082505.Google Scholar
Baruzzo, M., Bolzonella, T., Guo, S. C., Liu, Y. Q., Marchiori, G., Paccagnella, R., Soppelsa, A., Villone, F. & Wang, Z. R. 2011 3D effects on RWM physics in RFX-mod. Nucl. Fusion 51, 083037.Google Scholar
Villone, F., Chiariello, A. G., Mastrostefano, S., Pironti, A. & Ventre, S. 2012 GPU-accelerated analysis of vertical instabilities in ITER including three-dimensional volumetric conducting structures. Plasma Phys. Control. Fusion 54, 085003.Google Scholar
Jin, W., Ding, Y. H., Rao, B., Hu, Q. M., Jin, X. S., Wang, N. C., Zhang, X. Q., Wang, Z. J., Chen, Z. Y., Zhuang, G.& The J-Text Team 2013 Dependence of plasma responses to an externally applied perturbation field on MHD oscillation frequency on the J-TEXT tokamak. Plasma Phys. Control. Fusion 55, 035010.Google Scholar
Tsuzuki, K., Kamiya, K., Shinohara, K., Bakhtiari, M., Ogawa, H., Kurita, G., Takechi, M., Kasai, S., Sato, M., Kawashima, H. et al. 2006 Compatibility of reduced activation ferritic steel wall with high performance plasma on JFT-2M. Nucl. Fusion 46, 966–971.Google Scholar
The JET Team, The JET Project – Design Proposal, Rep. EURJET-R5, CEC, Brussels (1975).Google Scholar
Brunsell, P. R., Kuldkepp, M., Menmuir, S., Cecconello, M., Hedqvist, A., Yadikin, D., Drake, J. R. & Rachlew, E. 2006 Reversed field pinch operation with intelligent shell feedback control in EXTRAP T2R. Nucl. Fusion 46, 904–913.Google Scholar
Cecconello, M., Menmuir, S., Brunsell, P. R. & Kuldkepp, M. 2006 Rotation in a reversed field pinch with active feedback stabilization of resistive wall modes. Plasma Phys. Control. Fusion 48, 1311–1331.Google Scholar
Pustovitov, V. D. 2001 Feedback stabilization of resistive wall modes in a tokamak with a double resistive wall. Plasma Phys. Rep. 27, 195–204.Google Scholar
Hutchinson, I. H. 2001 Electromagnetic wall torques from magnetically confined plasmas. Plasma Phys. Control. Fusion 43, 145–153.Google Scholar
Degtyarev, L., Martynov, A., Medvedev, S., Troyon, E., Villard, L. & Gruber, R. 1997 The KINX ideal MHD stability code for axisymmetric plasmas with separatrix. Comput. Phys. Commun. 103, 10–27.Google Scholar
Pustovitov, V. D. & Yanovskiy, V. V. 2014 Resistive wall stabilization of rotating edge modes in tokamaks. Plasma Phys. Control. Fusion 56, 035003.Google Scholar
Paz-Soldan, C., Brookhart, M. I., Hegna, C. C. & Forest, C. B. 2012 Wall-locking of kink modes in a line-tied screw pinch with a rotating wall. Phys. Plasmas 19, 056104.Google Scholar
Zheng, L. J., Kotschenreuther, M. T. & Valanju, P. 2014 Diamagnetic drift effects on the low- $n$ magnetohydrodynamic modes at the high mode pedestal with plasma rotation. Phys. Plasmas 21, 062502.Google Scholar
Menard, J. E., Wang, Z., Liu, Y., Bell, R. E., Kaye, S. M., Park, J.-K. & Tritz, K. 2014 Rotation and kinetic modifications of the tokamak ideal-wall pressure limit. Phys. Rev. Lett. 113, 255002.Google Scholar
deGrassie, J. S. 2009 Tokamak rotation sources, transport and sinks. Plasma Phys. Control. Fusion 51, 124047.Google Scholar
Callen, J. D. 2011 Effects of 3D magnetic perturbations on toroidal plasmas. Nucl. Fusion 51, 094026.Google Scholar
Chapman, I. T., Liu, Y. Q., Asunta, O., Graves, J. P., Johnson, T. & Jucker, M. 2012 Kinetic damping of resistive wall modes in ITER. Phys. Plasmas 19, 052502.Google Scholar
Wang, Z. R., Guo, S. C., Liu, Y. Q. & Chu, M. S. 2012 Kinetic damping of resistive wall mode in reversed field pinch. Nucl. Fusion 52, 063001.Google Scholar
Hao, G. Z., Liu, Y. Q., Wang, A. K., Matsunaga, G., Okabayashi, M., Mou, Z. Z. & Qiu, X. M. 2013 Destabilization of low- $n$ peeling modes by trapped energetic particles. Phys. Plasmas 20, 062502.Google Scholar
Cai, H., Cao, J. & Li, D. 2014 The influence of energetic ions on resistive wall mode in reversed field pinch. Nucl. Fusion 54, 032001.Google Scholar
Berkery, J. W., Liu, Y. Q., Wang, Z. R., Sabbagh, S. A., Logan, N. C., Park, J.-K., Manickam, J. & Betti, R. 2014 Benchmarking kinetic calculations of resistive wall mode stability. Phys. Plasmas 21, 052505.CrossRefGoogle Scholar
Liu, Y. Q., Chapman, I. T., Graves, J. P., Hao, G. Z., Wang, Z. R., Menard, J. E., Okabayashi, M., Strait, E. J. & Turnbull, A. 2014 Non-perturbative modelling of energetic particle effects on resistive wall mode: anisotropy and finite orbit width. Phys. Plasmas 21, 056105.Google Scholar
He, Y., Liu, Y. Q., Liu, Y., Hao, G. & Wang, A. 2014 Plasma-resistivity-induced strong damping of the kinetic resistive wall mode. Phys. Rev. Lett. 113, 175001.Google Scholar
Berkery, J. W., Betti, R., Sabbagh, S. A., Guazzotto, L. & Manickam, J. 2014 The effect of an anisotropic pressure of thermal particles on resistive wall mode stability. Phys. Plasmas 21, 112505.CrossRefGoogle Scholar
Garofalo, A. M., Jackson, G. L., La Haye, R. J., Okabayashi, M., Reimerdes, H., Strait, E. J., Ferron, J. R., Groebner, R. J., In, Y., Lanctot, M. J. et al. 2007 Stability and control of resistive wall modes in high beta, low rotation DIII-D plasmas. Nucl. Fusion 47, 1121–1130.Google Scholar
Mikhailovskii, A. B., Lominadze, J. G., Churikov, A. P. & Pustovitov, V. D. 2009 Progress in theory of instabilities in a rotating plasma. Plasma Phys. Rep. 35, 273–314.Google Scholar
Lepikhin, N. D. & Pustovitov, V. D. 2014 Dispersion relations for slow and fast resistive wall modes within the Haney–Freidberg model. Phys. Plasmas 21, 042504.Google Scholar
Pustovitov, V. D. 2015 Energy principle for the modes interacting with a resistive wall in toroidal systems. Nucl. Fusion 55, 033008.Google Scholar
Whyte, D. G., Hubbard, A. E., Hughes, J. W., Lipschultz, B., Rice, J. E., Marmar, E. S., Greenwald, M., Cziegler, I., Dominguez, A., Golfinopoulos, T. et al. 2010 I-mode: an H-mode energy confinement regime with L-mode particle transport in Alcator C-Mod. Nucl. Fusion 50, 105005.Google Scholar
García-Muñoz, M., Martin, P., Fahrbach, H.-U., Gobbin, M., Günter, S., Maraschek, M., Marrelli, L., Zohm, H.& The ASDEX Upgrade Team 2007 NTM induced fast ion losses in ASDEX upgrade. Nucl. Fusion 47, L10–L15.Google Scholar
Isayama, A., Oyama, N., Urano, H., Suzuki, T., Takechi, M., Hayashi, N., Nagasaki, K., Kamada, Y., Ide, S., Ozeki, T.& The JT-60 Team 2007 Stabilization of neoclassical tearing modes by electron cyclotron current drive in JT-60U. Nucl. Fusion 47, 773–782.Google Scholar
Van Zeeland, M. A., Yu, J. H., Chu, M. S., Burrell, K. H., La Haye, R. J., Luce, T. C., Nazikian, R., Solomon, W. M. & West, W. P. 2008 Tearing mode structure in the DIII-D tokamak through spectrally filtered fast visible bremsstrahlung imaging. Nucl. Fusion 48, 092002.Google Scholar
Menard, J. E., Bell, R. E., Gates, D. A., Gerhardt, S. P., Park, J.-K., Sabbagh, S. A., Berkery, J. W., Egan, A., Kallman, J., Kaye, S. M. et al. 2010 Progress in understanding error-field physics in NSTX spherical torus plasmas. Nucl. Fusion 50, 045008.Google Scholar
Takahashi, H., Fredrickson, E. D. & Chance, M. S. 2002 Unusual low frequency magnetic perturbations in the TFTR tokamak. Nucl. Fusion 42, 448–485.Google Scholar
Burrell, K. H., Austin, M. E., Brennan, D. P., Deboo, J. C., Doyle, E. J., Fenzi, C., Fuchs, C., Gohil, P., Greenfield, C. M., Groebner, R. J. et al. 2001 Quiescent double barrier high-confinement mode plasmas in the DIII-D tokamak. Phys. Plasmas 8, 2153–2162.Google Scholar
Suttrop, W., Hynönen, V., Kurki-suonio, T., Lang, P. T., Maraschek, M., Neu, R., Stäbler, A., Conway, G. D., Hacquin, S., Kempenaars, M. et al. 2005 Studies of the ‘Quiescent H-mode’ regime in ASDEX upgrade and JET. Nucl. Fusion 45, 721–730.Google Scholar
Oyama, N., Gohil, P., Horton, L. D., Hubbard, A. E., Hughes, J. W., Kamada, Y., Kamiya, K., Leonard, A. W., Loarte, A., Maingi, R. et al. 2006 Pedestal conditions for small ELM regimes in tokamaks. Plasma Phys. Control. Fusion 48, A171–A181.Google Scholar
Snyder, P. B., Burrell, K. H., Wilson, H. R., Chu, M. S., Fenstermacher, M. E., Leonard, A. W., Moyer, R. A., Osborne, T. H., Umansky, M., West, W. P. et al. 2007 Stability and dynamics of the edge pedestal in the low collisionality regime: physics mechanisms for steady-state ELM-free operation. Nucl. Fusion 47, 961–968.Google Scholar
Oyama, N. 2008 Progress and issues in understanding the physics of ELM dynamics, ELM mitigation, and ELM control. J. Phys.: Conf. Ser. 123, 012002.Google Scholar
Osborne, T. H., Snyder, P. B., Burrell, K. H., Evans, T. E., Fenstermacher, M. E., Leonard, A. W., Moyer, R. A., Schaffer, M. J. & West, W. P. 2008 Edge stability of stationary ELM-suppressed regimes on DIII-D. J. Phys.: Conf. Ser. 123, 012014.Google Scholar
Garofalo, A. M., Solomon, W. M., Park, J.-K., Burrell, K. H., Deboo, J. C., Lanctot, M. J., Mckee, G. R., Reimerdes, H., Schmitz, L., Schaffer, M. J. et al. 2011 Advances towards QH-mode viability for ELM-stable operation in ITER. Nucl. Fusion 51, 083018.Google Scholar
Sontag, A. C., Canik, J. M., Maingi, R., Manickam, J., Snyder, P. B., Bell, R. E., Gerhardt, S. P., Kubota, S., Leblanc, B. P., Mueller, D. et al. 2011 Pedestal characterization and stability of small-ELM regimes in NSTX. Nucl. Fusion 51, 103022.Google Scholar
Yan, Z., McKee, G. R., Groebner, R. J., Snyder, P. B., Osborne, T. H. & Burrell, K. H. 2011 High-frequency coherent edge fluctuations in a high-pedestal-pressure quiescent H-mode plasma. Phys. Rev. Lett. 107, 055004.Google Scholar
Snyder, P. B., Osborne, T. H., Burrell, K. H., Groebner, R. J., Leonard, A. W., Nazikian, R., Orlov, D. M., Schmitz, O., Wade, M. R. & Wilson, H. R. 2012 The EPED pedestal model and edge localized mode-suppressed regimes: studies of quiescent H-mode and development of a model for edge localized mode suppression via resonant magnetic perturbations. Phys. Plasmas 19, 056115.Google Scholar
Burrell, K. H., Garofalo, A. M., Solomon, W. M., Fenstermacher, M. E., Orlov, D. M., Osborne, T. H., Park, J.-K. & Snyder, P. B. 2013 Quiescent H-mode operation using torque from non-axisymmetric, non-resonant magnetic fields. Nucl. Fusion 53, 073038.Google Scholar
Takemura, Y., Sakakibara, S., Narushima, Y., Okamoto, M., Watanabe, K. Y., Suzuki, Y., Ohdachi, S., Ida, K., Yoshinuma, M., Tanaka, K. et al. 2012 Mode locking phenomena observed near the stability boundary of the ideal interchange mode of LHD. Nucl. Fusion 52, 102001.Google Scholar
Neuhauser, J., Bobkov, V., Conway, G. D., Dux, R., Eich, T., Garcia-Munoz, M., Herrmann, A., Horton, L. D., Kallenbach, A., Kalvin, S. et al. 2008 Structure and dynamics of spontaneous and induced ELMs on ASDEX upgrade. Nucl. Fusion 48, 045005.Google Scholar
Soldatov, S., Krämer-flecken, A., Kantor, M., Unterberg, B., Sun, Y., van Oost, G., Reiter, D.& The TEXTOR Team 2010 Turbulence, flows and edge localized mode (ELM) dynamics in limiter H-mode plasmas in TEXTOR. Plasma Phys. Control. Fusion 52, 085001.Google Scholar
Oyama, N., Hayashi, N., Aiba, N., Isayama, A., Urano, H., Sakamoto, Y., Kamada, Y., Takizuka, T.& The JT-60 Team 2011 Characteristics and control of the type I edge localized mode in JT-60U. Nucl. Fusion 51, 033009.Google Scholar
Sommer, F., Günter, S., Kallenbach, A., Maraschek, M., Boom, J., Fischer, R., Hicks, N., Luhmann, N. C. Jr, Park, H. K., Reiter, B. et al. & The ASDEX Upgrade Team 2011 Characterization and interpretation of the edge snake in between type-I edge localized modes at ASDEX upgrade. Plasma Phys. Control. Fusion 53, 085012.Google Scholar
Jiang, M., Xu, G. S., Xiao, C., Guo, H. Y., Wan, B. N., Wang, H. Q., Wang, L., Zhang, L., Naulin, V., Gan, K. F. et al. 2012 Characteristics of edge-localized modes in the experimental advanced superconducting tokamak (EAST). Plasma Phys. Control. Fusion 54, 095003.Google Scholar
Wenninger, R. P., Zohm, H., Boom, J. E., Burckhart, A., Dunne, M. G., Dux, R., Eich, T., Fischer, R., Fuchs, C., Garcia-Munoz, M. et al. 2012 Solitary magnetic perturbations at the ELM onset. Nucl. Fusion 52, 114025.Google Scholar
Sechrest, Y., Munsat, T., Battaglia, D. J. & Zweben, S. J. 2012 Two-dimensional characterization of ELM precursors in NSTX. Nucl. Fusion 52, 123009.Google Scholar
Bak, J. G., Ahn, J.-W., Kim, H.-S., Lee, J. W., Yoon, S. W., Maingi, R., Han, H. S., Jeon, Y. M., England, A. C., Park, Y. S. et al. 2012 ELM and pedestal structure studies in KSTAR H-mode plasmas. In Proceedings of the 24th IAEA Fusion Energy Conference (San Diego, USA, 2012), IAEA. Paper EX/P4-12.Google Scholar
Huysmans, G. T. A., Pamela, S., van der Plas, E. & Ramet, P. 2009 Non-linear MHD simulations of edge localized modes (ELMs). Plasma Phys. Control. Fusion 51, 124012.Google Scholar
Fenstermacher, M. E., Becoulet, M., Cahyna, P., Canik, J., Chang, C. S., Evans, T. E., Gohil, P., Kaye, S., Kirk, A., Liang, Y. et al. 2010 ELM control by resonant magnetic perturbations: overview of research by the PEP ITPA group. In Proceedings of the 23th IAEA Fusion Energy Conference (Daejon, Korea, 2010), IAEA. Paper ITR/P1-30.Google Scholar
Callen, J. D., Hegna, C. C., Rice, B. W., Strait, E. J. & Turnbull, A. D. 1999 Growth of ideal magnetohydrodynamic modes driven slowly through their instability threshold: application to disruption precursors. Phys. Plasmas 6, 2963–2967.Google Scholar
Takechi, M., Fujita, T., Ishii, Y., Ozeki, T., Suzuki, T., Isayama, A.& The JT-60 Team 2005 MHD instabilities leading to disruptions in low beta JT-60U reversed shear plasmas. Nucl. Fusion 45, 1694–1699.Google Scholar
Papp, G., Pokol, G. I., Por, G., Magyarkuti, A., Lazányi, N., Horváth, L., Igochine, V., Maraschek, M.& ASDEX Upgrade Team 2011 Low frequency sawtooth precursor activity in ASDEX upgrade. Plasma Phys. Control. Fusion 53, 065007.Google Scholar