Grad-Shafranov Reconstruction: Overview and Improvement of the Numerical Solution Used in Space Physics

González, Arian Ojeda; Domingues, Margarete Oliveira; Mendes, Odim; Kaibara, M. K.; Prestes, A.

doi:10.1007/s13538-015-0342-y

Cited by 11 publications

(14 citation statements)

References 30 publications

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“…(21), com aplicação na Geofísica Espacial,é a solução de [1], sendo g(ξ) = f + 1 + f 2 e ιξ e Ψ = ln f cos X + 1 + f 2 cosh Y . Esta soluçãoé importante para comparar com a solução numérica [2,4].…”

Section: Solução Analíticaunclassified

“…Neste caso, utiliza-se a seguinte grade: 0 ≤ x ≤ 51 com ∆x = 51/101, e −5 ≤ y ≤ 5 com ∆y = 0.1 × ∆x [2,4]. A solução tem por objetivo estudar a maneira como o valor de A z (x, y 0 + ∆y) varia, quando ∆y = y − y 0 e uma quantidade muito pequena.…”

Section: Solução Numéricaunclassified

“…O primeiro termo da direita da equação de GS calcula-se a partir de um esquema de diferenças finitas de segunda ordem (ver Apendice em [2]),…”

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“…A maior contribuição de [2] foi a proposta de fazer a filtragem de B x (x, ±∆y) no final de cada passo da solução numérica. [4] e artigos posteriores não fizeram a filtragem.…”

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Revisão da solução analítica/numérica da equação de Grad-Shafranov

González¹

2018

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

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Resumo. Neste trabalho apresenta-se a solução analítica da equação de Grad-Shafranov (GS), importante ferramenta nos estudos de tubos de fluxo magnético no ambiente geoespacial. A equação de GS pode ser resolvida numericamente como um problema de Cauchy, considerando que a equação diferencial estará sujeita a condições iniciais. Se um satélite mede esses parâmetros iniciais, a equação de GS será resolvida, obtendo como resultado a reconstrução da estrutura de plasma na região entorno do satélite. Uma proposta para melhorar a solução numérica da equação de GS também será apresentada. Os tópicos abordados já foram publicadas pelo autor e colaboradores em Braz J Phys (2015) IntroduçãoA lei de Ampére-Maxwell ou Lei de Ampére Generalizadaé dada por:Considerando que não há dependência temporal nas grandezas físicas consideradas e substituindo a relação vetorial B = ∇ × A na Eq. (1), temos:Supondo que ∇ ∇ · A = 0, a Eq. (2) pode ser escrita como o seguinte sistema de três equações diferenciais não-homogêneas de segunda ordem,

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Section: Solução Analíticaunclassified

Section: Solução Numéricaunclassified

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Revisão da solução analítica/numérica da equação de Grad-Shafranov

González¹

2018

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

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“…It also has to treat the elliptic system as a Cauchy problem. However, it is mathematically ill posed or ill conditioned [ Li et al , ; González et al , ]. In the integration of the GS equation, singularities with large values of the magnetic vector potential arise after a certain number of integration steps away from the original data line.…”

Section: Introductionmentioning

confidence: 99%

Data completion with Hilbert transform over plane rectangle: Technique renovation for the Grad‐Shafranov reconstruction

Feng

et al. 2017

JGR Space Physics

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Hilbert transforms (HT) have first been used to build the essential technique of Grad‐Shafranov (GS) reconstruction by Li et al. (2013), where the problem of ill posedness in GS reconstruction has been thoroughly investigated. In this study, we present an extended Hilbert transform (EHT) over the plane rectangle. In contrast to previous one (HT over the unit circular region), corner singularities are introduced into these new formulae. It is confronted by problems like the integral with both endpoint singularities, and the semiinfinite integral with one endpoint singularity, as these EHT formulae are used to rebuild the essential technique of GS reconstruction. Two additional mathematic tools are adopted in this study. First, high‐accuracy quadrature schemes are constructed for those improper integrals based on the double exponential (DE) transformations. Benchmark testing with the analytic solutions on a rectangular boundary has shown the efficiency and robustness of the EHT formulae. Second, the data completion or the inverse boundary value problem is solved with the help of a truncated Chebyshev series, which approximates the unknown boundary gradients in very high efficiency under the only assumption that they are Lipschitz continuous on each side of the rectangle. Combining the introduced EHT formulae and the two needed mathematic tools, the essential technique of GS reconstruction is formulated into a linear system of Fredholm equations of the first kind. Then a three‐parameter Tikhonov regularization scheme is developed to deal with the ill‐posed linear operators appearing in the discretized linear system. This new approach for data completion over the plane rectangle is benchmarked with the analytic solutions. Numerical experiments highlight the efficiency and robustness of the proposed method.

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Grad-Shafranov equation: MHD simulation of the new solution obtained from the Fadeev and Naval models

González¹,

Santos²,

González-Avilés³

et al. 2020

Preprint

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The new analytical solution has magnetic field lines with neutral points and singular points.• This solution enter as initial condition in an MHD simulation by excluding the singular points.• The MHD simulation shows the fast evolution of magnetic islands into current sheets.• This is the first report shows the magnetic field polarity inversion related to analytical solution of the GS equation.• The importance of the direction and amplitude of the magnetic field near the singular points is explained.

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Grad-Shafranov Reconstruction: Overview and Improvement of the Numerical Solution Used in Space Physics

Abstract: How to citeComplete issue More information about this article Journal's homepage in redalyc.org Scientific Information System Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Non-profit academic project, developed under the open access initiative

Cited by 11 publications

References 30 publications

Revisão da solução analítica/numérica da equação de Grad-Shafranov

Revisão da solução analítica/numérica da equação de Grad-Shafranov

Data completion with Hilbert transform over plane rectangle: Technique renovation for the Grad‐Shafranov reconstruction

Grad-Shafranov equation: MHD simulation of the new solution obtained from the Fadeev and Naval models

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