Summary
Asymptotic solutions of the magnetohydrostatic equations ∇×(J×B)=0, ∇·B=0,J=∇×B (with single-valued ∫dx·J×B) have been recently obtained to all orders in a torus with arbitrary section for a quasi-axisymmetric, quasi-azimuthal field configuration and quasivacuumB. We now consider a much weaker asymptotic ansatz, namely i) the spatial domain where the solution is sought merely belongs to the torus homology class, ii) a vacuum, closed-lineB dominates together with a parallelJ; and prove that the corresponding problem of constructing an asymptotic solution can be reduced to solving a weakly singular Hammerstein integral equation to the lowest significant order and a weakly singular nonhomogeneous Fredholm integral equation of the 2nd kind for each of the subsequent orders.
Riassunto
Soluzioni asintotiche delle equazioni magnetoidrostatiche ∇×(J×B)=0, ∇·B=0,J=∇×B (con ∫dx·J×B ad un sol valore) sono state recentemente ottenute a tutti gli ordini in un toro di sezione meridiana arbitraria, per una configurazione di campo quasi assisimmetrica e quasi azimutale, e conB quasi di vuoto. Un ansatz asintotico molto più debole è considerato nel presente lavoro, e cioè 1) il dominio spaziale dove si ricerca la soluzione è omotopico al toro, 2) unB di vuoto a linee chiuse ed unaJ parallela dominano. Si dimostra che il corrispondente problema di costruire soluzioni asintotiche può essere ridotto alla soluzione di un'equazione integrale debolmente singolare di Hammerstein al più basso ordine significativo e di un'equazione integrale debolmente singolare non omogenea di Fredholm di seconda specie per ciascuno degli ordini successivi.
Резюме
Недавно были получены во всех порядках асимптотические решения магнитногидростатических уравнений ∇×(J×B)=0, ∇·B=0,J=∇×B (с одно-значной величиной ∫dx·J×B) в торе с произвольным сечением для квази-осесим-метричной, квази-азимутальной конфигурации поля и квази-вакуумаB. В этой работе мы рассматриваем более слабый асимптотический подход. Доказывается, что соответствующая проблема конструирования асимптотического решения может быть сведена к решению слабо сингулярного интегрального уравнения Хаммер-стейна и слабо сингулярного неоднородного интегрального уравнения Фредгольма второго рода для каждого последующего порядка.
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Lo Surdo, C. Hydromagnetic asymptotic equilibria with vacuum, closed-line dominantB and parallel dominantJ . Nuovo Cim B 32, 1–39 (1976). https://doi.org/10.1007/BF02726742
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DOI: https://doi.org/10.1007/BF02726742