Enhancing Learning of the Grad-Shafranov Equation through Scientific Literature: Part 1 of a Physics Education Series

Ojeda-González, A.; Santos, L. Nunes dos; De La Luz, V.; Cristaldo-Oliveira, M.; Sousa, A.N. Laurindo; Prestes, A.; Klausner, V.; Pilling, S.

doi:10.1590/1806-9126-RBEF-2023-0104

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Articles • Rev. Bras. Ensino Fís. 45 • 2023 • https://doi.org/10.1590/1806-9126-RBEF-2023-0104 copy

Enhancing Learning of the Grad-Shafranov Equation through Scientific Literature: Part 1 of a Physics Education Series

Authorship SCIMAGO INSTITUTIONS RANKINGS

This article provides a comprehensive review of relevant studies in the fields of plasma physics, electromagnetism, and space physics. The aim is to demonstrate how the study of the scientific literature can be used to enhance problem-solving abilities and develop innovative solutions in physics. In this paper, we focus on the study of solutions of the specific Grad-Shafranov equation. Two of the new solutions proposed by Yoon and Lui (2005) are used as a basis for the development of a new solution. The new solution presented has singular points similar to the Yoon-Lui-2 solution, but with an inverted configuration, and also presents less rounded double islands compared to the Yoon-Lui-2 solution. Additionally, the new solution does not exhibit the formation of a current ring, a characteristic of the Yoon-Lui-1 solution, and varying its parameters may lead to higher plasma confinement efficiency. In summary, we illustrate how a thorough analysis of literature can serve as a powerful means for generating innovative approaches to resolving theoretical issues in physics.

Keywords:
Grad-Shafranov equation; Magnetic flux-ropes; Plasma confinement; Singularity analysis

Solution:	$g (ζ)$ =	$Ψ$ =
1- Harris^a a [24]	$e^{ι ζ}$	$\ln (\cosh Z)$
2- Fadeev^b b [25]	$f_{p} + \sqrt{1 + f_{p}^{2}} e^{ι ζ}$	$\ln (f_{p} \cos X + \sqrt{1 + f_{p}^{2}} \cosh Z)$
3- Kan³ c [8]	$e^{ι ζ - \frac{ι b}{ξ}}$	$\ln \frac{\cosh [Z (1 + b / R^{2})]}{\sqrt{{(1 + b / R^{2})}^{2} - 4 b Z^{2} / R^{4}}}$ ^g g R 2 = X 2 + Z 2
4- Manankova^d d [26, 28, 29]	$f_{p} + \sqrt{1 + f_{p}^{2}} e^{(i ζ - \frac{i b}{ζ - a})}$	$\ln \frac{f_{p} \cos (X - \frac{b (X - a)}{R_{0}^{2}}) + \sqrt{1 + f_{p}^{2}} \cosh (Z (1 + \frac{b}{R_{0}^{2}}))}{{\sqrt{{(1 + \frac{b}{R_{0}^{2}})}^{2} - \frac{4 b Z^{2}}{R_{0}^{4}}}}^{h}}$ ^h h R 0 2 = ( X − a ) 2 + Z 2
5- H-F-K-M⁵ e [1, sections 3.5 (H-F-K-M), 3.7 (1), 3.8 (2), 3.9 (3)]	$f_{p} + \sqrt{1 + f_{p}^{2}} e^{(i ζ - \frac{i b}{{(ζ - a)}^{k}})}$	$\ln \frac{f_{p} \cos (a + R_{0} \cos θ - \frac{b \cos k θ}{R_{0}^{k}}) + \sqrt{1 + f_{p}^{2}} \cosh (R_{0} \sin θ + \frac{b \sin k θ}{R_{0}^{k}})}{\sqrt{{(1 + \frac{k b}{R_{0}^{k + 1}})}^{2} - \frac{4 k b}{R_{0}^{k + 1}} \sin^{2} \frac{(k + 1) θ}{2}}}$ ⁱ i θ = tan − 1 Z X − a
6- B-W ^f f [27]	$e^{[- β ρ (ζ)]}$ ^j j ζ = α cosh ρ , ρ ( ζ ) = cosh − 1 ( ζ α ) = ln ζ ± ζ 2 − α 2 α	$\ln \frac{\sqrt{r_{+} r_{-} \cosh (β ξ)}}{β}$ ^k k r ± = ( X ± α ) 2 + Z 2 , ξ = ln ( τ + τ 2 − 1 ) $ a n d $ τ = r + + r − 2 α
7- Yoon-Lui-1^e e [1, sections 3.5 (H-F-K-M), 3.7 (1), 3.8 (2), 3.9 (3)]	$ζ^{ν}$	$\ln \frac{R (R^{ν} + R^{- ν})}{2 ν}$
8- Yoon-Lui-2^e e [1, sections 3.5 (H-F-K-M), 3.7 (1), 3.8 (2), 3.9 (3)]	$ζ - \frac{a}{ζ}$	$\ln \frac{{(R^{2} + a)}^{2} + R^{2} - 4 a X^{2}}{2 {[{(R^{2} + a)}^{2} - 4 a Z^{2}]}^{1 / 2}}$
9- Yoon-Lui-3^e e [1, sections 3.5 (H-F-K-M), 3.7 (1), 3.8 (2), 3.9 (3)]	$\frac{ζ}{(1 - a^{2} ζ^{2})}$	$\frac{1}{2} \ln (\frac{S {(S + R^{2})}^{2}}{2 T})$ ^l l S = ( 1 − a 2 R 2 ) 2 + ( 2 a Z ) 2 , T = ( 1 − a 4 R 4 ) 2 + ( 4 a 2 X Z ) 2

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[1] * Correspondence email address: ojeda.gonzalez.a@gmail.com