Abstract
Galathea traps for confining plasma in a magnetic field created by current-carrying conductors immersed in the plasma volume constitute a promising class of objects for development in the field of controlled thermonuclear fusion. After describing the main properties and quantitative characteristics of the traps, the central problem is to study the stability of equilibrium magnetoplasma configurations. Mathematical modeling and calculations performed in terms of the differential equations of magnetohydrodynamics play an essential role here. These equations are described using the example of a toroidal “Galathea belt” trap straightened into a cylinder. The article presents its plasmastatic model and several approaches to studying the stability of configurations: the convergence of iterative methods for establishing equilibrium in two-dimensional models, a brief survey of the research into the magnetohydrodynamic stability of one-dimensional configurations surrounding a straight current-carrying conductor, and a rigorous study of the stability of configurations with respect to two-dimensional perturbations in the linear approximation. Stability criteria and their relationship to each other are obtained numerically in various approximations. Ways of generalizing the results to three-dimensional perturbations are indicated.
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REFERENCES
Morozov, A.I., Galathea—plasma confinement systems in which the conductors are immersed in the plasma, Sov. J. Plasma Phys., 1992, vol. 18, no. 3, pp. 159–165.
Morozov, A.I. and Pustovitov, V.D., Stellarator with levitating windings, Sov. J. Plasma Phys., 1991, vol. 17, p. 740.
Morozov, A.I. and Frank, A.G., Galateya toroidal multipole trap with azimuthal current, Plasma Phys. Rep., 1994, vol. 20, pp. 879–886.
Morozov, A.I., Bugrova, A.I., Bishaev, A.M., Lipatov, A.S., and Kozintseva, M.V., Plasma parameters in the upgraded Trimyx-M Galathea, Tech. Phys., 2007, vol. 52, no. 12, pp. 1546–1551.
Brushlinskii, K.V., Zueva, N.M., Mikhailova, M.S., Morozov, A.I., Pustovitov, V.D., and Tuzova, N.B., Numerical simulation for straight helical sheaths with conductors immersed in plasma, Plasma Phys. Rep., 1994, vol. 20, no. 3, pp. 257–264.
Brushlinskii, K.V. and Ignatov, P.A., A plasmastatic model of the Galathea-belt magnetic trap, Comput. Math. Math. Phys., 2010, vol. 50, no. 12, pp. 2071–2081.
Brushlinskii, K.V. and Kondrat’ev, I.A., Comparative analysis of plasma equilibrium computations in toroidal and cylindrical magnetic traps, Math. Model. Comput. Simul., 2019, vol. 11, no. 1, pp. 122–132.
Tao, B., Jin, X., Li, Z., and Tong, W., Equilibrium configuration reconstruction of multipole Galatea magnetic trap based on magnetic measurement, IEEE Trans. Plasma Sci., 2019, vol. 47, no. 7, pp. 3114–3123.
Morozov, A.I. and Savel’ev, V.V., On Galateas—magnetic traps with plasma-embedded conductors, Phys.-Usp., 1998, vol. 41, no. 11, pp. 1049–1089.
Brushlinskii, K.V., Matematicheskie i vychislitel’nye zadachi magnitnoi gazodinamiki (Mathematical and Computational Problem of Magnetohydrodynamics), Moscow: BINOM. Lab. Znanii, 2009.
Brushlinskii, K.V., Matematicheskie osnovy vychislitel’noi mekhaniki zhidkosti, gaza i plazmy (Mathematical Foundations of Computational Mechanics of Fluid, Gas, and Plasma), Dolgoprudnyi: Intellekt, 2017.
Shafranov, V.D., On magnetohydrodynamic equilibrium configurations, Sov. Phys. JETP, 1958, vol. 6, pp. 545–554.
Grad, H. and Rubin, H., Hydrodynamic equilibria and force-free fields, Proc. 2nd U. N. Int. Conf. Peaceful Uses At. Energy. Geneva, 1958, vol. 31, pp. 190–197.
Shafranov, V.D., Plasma equilibrium in a magnetic field, in Rev. Plasma Phys., Leontovich, M.A., Ed., New York, 1966, no. 2, pp. 103–152.
Kadomtsev, B.B., Hydromagnetic stability of a plasma, in Rev. Plasma Phys., Leontovich, M.A., Ed., New York, 1966, no. 2, pp. 153–206.
Solov’ev, L.S., Hydromagnetic stability of closed plasma configurations, in Voprosy teorii plazmy. Vyp. 6 (Plasma Theory Issues. Issue 2), Leontovich, M.A., Ed., Moscow: Atomizdat, 1972, pp. 210–290.
Bateman, G., MHD Instability, Cambridge, MA–London: The MIT Press, 1979. Translated under the title: MGD Neustoichivosti, Moscow: Atomizdat, 1982.
Medvedev, S.Yu., Martynov, A.A., Drozdov, V.V., Ivanov, A.A., Poshekhonov, Yu.Yu., Konovalov, S.V., and Villard, L., MHD stability and energy principle for two-dimensional equilibria without assumption of nested magnetic surfaces, Plasma Phys. Rep., 2019, vol. 45, no. 2, pp. 108–120.
Brushlinskii, K.V., Two approaches to the stability problem for plasma equilibrium in a cylinder, J. Appl. Math. Mech., 2001, vol. 65, no. 2, pp. 229–236.
Brushlinskii, K.V., Krivtsov, S.A., and Stepin, E.V., On the stability of plasma equilibrium in the neighborhood of a straight current conductor, Comput. Math. Math. Phys., 2020, vol. 60, no. 4, pp. 686–696.
Brushlinskii, K.V. and Stepin, E.V., Mathematical models of equilibrium configurations of plasma surrounding current-carrying conductors, Differ. Equations, 2020, vol. 56, no. 7, pp. 872–881.
Brushlinskii, K.V. and Stepin, E.V., Mathematical model and stability investigation of plasma equilibrium around a current-carrying conductor, J. Phys.: Conf. Ser., 2020, vol. 1686, p. 012030.
Brushlinskii, K.V. and Stepin, E.V., Plasma equilibrium and stability in a current-carrying conductor vicinity, J. Phys.: Conf. Ser., 2020, vol. 1640, p. 012018.
Peaceman, D.W. and Rachford, H.H., The numerical solution of parabolic and elliptic differential equations, J. Soc. Ind. Appl. Math., 1955, vol. 3, no. 1, pp. 28–42.
Douglas, J., On the numerical integration of \(\partial ^2 u /\partial x^2+\partial ^2 u /\partial y^2=\partial u /\partial t \) by implicit method, J. Soc. Ind. Appl. Math., 1955, vol. 3, no. 1, pp. 42–65.
Arsenin, V.Ya., Metody matematicheskoi fiziki i spetsial’nye funktsii (Methods of Mathematical Physics and Special Functions), Moscow: Nauka, 1984.
Vedenov, A.A., Velikhov, E.P., and Sagdeev, R.Z., Stability of plasma, Sov. Phys. Usp., 1961, vol. 4, no. 2, pp. 332–369.
Bernstein, I.B., Frieman, E.A., Kruskal, M.D., and Kulsrud, R.M., Energy principle for the hydromagnetic stability problem, Proc. R. Soc., 1958, vol. 244, pp. 17–50.
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This work was supported by the Moscow Center for Fundamental and Applied Mathematics, Agreement with the Ministry of Science and Higher Education of the Russian Federation no. 075-15-2019-1623.
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Translated by V. Potapchouck
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Brushlinskii, K.V., Stepin, E.V. Stability Issues in Two-Dimensional Mathematical Models of Plasma Equilibrium in Magnetic Galathea Traps. Diff Equat 57, 835–847 (2021). https://doi.org/10.1134/S0012266121070016
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DOI: https://doi.org/10.1134/S0012266121070016