A plasmastatic model of the galathea-belt magnetic trap

K. V. Brushlinskii   ; P. A. Ignatov

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 50 (2010) no. 12, pp. 2184-2194

Résumé

The mathematical apparatus of plasmastatics, which includes the MHD equilibrium equation and steady-state Maxwell equations, is reduced, in two-dimensional problems arising due to symmetry, to a single scalar second-order elliptic equation with a nonlinear right-hand side known as the Grad-Shafranov equation. In this paper, we numerically solve a series of boundary value problems for this equation that model equilibrium plasma configurations in the magnetic field of the belt-like galathea trap in a cylinder with two plasma embedded conductors. The mathematical model is outlined, the results of calculations of the magnetic field and plasma pressure in the cylinder depending on the parameters of the problem are presented, and the main integral characteristics of the trap are calculated. The existence and uniqueness of the solution is discussed; the limiting values of the maximal pressure at which there exists a solution of the equilibrium problem are found.

Bibliographie

[1] Brushlinskii K. V., Matematicheskie i vychislitelnye zadachi magnitnoi gazodinamiki, BINOM. Laboratoriya znanii, M., 2009

[2] Morozov A. I., “O galateyakh — plazmennykh lovushkakh s omyvaemymi plazmoi provodnikami”, Fiz. plazmy, 18:3 (1992), 305–316

[3] Morozov A. I., Frank A. G., “Toroidalnaya magnitnaya lovushka-galateya s azimutalnym tokom”, Fiz. plazmy, 20:11 (1994), 982–989

[4] Morozov A. I., Murzina M. V., “Prosteishie ravnovesnye konfiguratsii galatei tipa “Poyas””, Fiz. plazmy, 22:6 (1996), 551–563

[5] Morozov A. I., Savelev V. V., “Figury ravnovesiya idealnoi plazmy s $\beta=1$ v magnitnom pole”, Fiz. plazmy, 19:8 (1993), 977–989

[6] Savelev V. V., “Figury ravnovesiya idealnoi plazmy s tokom v magnitnom pole”, Fiz. plazmy, 21:3 (1995), 216–220

[7] Brushlinskii K. V., Gorshenin K. P., “Ploskaya MGD model obrazovaniya plazmennoi konfiguratsii s pogruzhennymi v nee provodnikami”, Matem. modelirovanie, 9:5 (1997), 28–36 | Zbl

[8] Dudnikova G. I., Morozov A. I., Fedoruk M. P., “Chislennoe modelirovanie pryamykh plazmennykh konfiguratsii — galatei tipa “Poyas””, Fiz. plazmy, 23:5 (1997), 387–396

[9] Morozov A. I., Savelev V. V., “O galateyakh-lovushkakh s pogruzhennymi v plazmu provodnikami”, Uspekhi fiz. nauk, 168:11 (1998), 1153–1194

[10] Brushlinskii K. V., Savelev V. V., “Magnitnye lovushki dlya uderzhaniya plazmy”, Matem. modelirovanie, 11:5 (1999), 3–36 | MR

[11] Shafranov V. D., “O ravnovesnykh magnitogidrodinamicheskikh konfiguratsiyakh”, Zh. eksperim. i teor. fiz., 33:3(9) (1957), 710–722 | Zbl

[12] Dnestrovskii Yu. N., Kostomarov D. P., Matematicheskoe modelirovanie plazmy, Nauka, M., 1982

[13] Brushlinskii K. V., Chmykhova N. A., “O ravnovesii plazmy v magnitnom pole lovushek galatei”, Matem. modelirovanie, 22:6 (2010), 3–14 | MR

[14] Brushlinskii K. V., Zueva N. M., Mikhailova M. S. i dr., “Chislennoe modelirovanie pryamykh vintovykh shnurov s provodnikami, pogruzhennymi v plazmu”, Fiz. plazmy, 20:3 (1994), 284–292

[15] Arsenin V. Ya., Metody matematicheskoi fiziki i spetsialnye funktsii, Nauka, M., 1974 | MR

[16] Brushlinskii K. V., “Dva podkhoda k zadache ob ustoichivosti ravnovesiya plazmy v tsilindre”, Prikl. matem. i mekhan., 65:2 (2001), 235–243 | MR