Existence of ideal magnetofluid equilibria without continuous Euclidean symmetries

Sato, Naoki

doi:10.1088/1361-6587/ab5001

Cited by 3 publications

(3 citation statements)

References 33 publications

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“…The condition ∂g ij /∂x 3 = 0 restricts the class of admissible symmetries to continuous Euclidean isometries (i.e. transformation that preserve the distance between points in three dimensional Euclidean space, see [23,24,3]). These transformations are translations, rotations, or a combination of them.…”

Section: Symmetry Of Solutionsmentioning

confidence: 99%

“…At present, a rigorous mathematical treatment of system (1) with boundary conditions (2) is not available. This difficulty stems from the mixed nature of these equations: they define a nonlinear twice hyperbolic twice elliptic first order system of PDEs for the variables B x , B y , B z , and P (see [2,3] on this point). For this reason, the existence of solutions with appropriate regularity and symmetry properties represents an open mathematical problem with both practical and theoretical implications [4].…”

Section: Introductionmentioning

confidence: 99%

“…If one is willing to allow discontinuities, class p solutions of (1), ( 2) can be constructed in asymmetric tori [27], and class p asymmetric solutions of the boundary value problem (1), (2) can be obtained in arbitrary domains [3]. Nonetheless, the existence of class p regular solutions in asymmetric bounded domains, and the existence of class p regular asymmetric solutions in bounded domains remain open issues.…”

Section: Introductionmentioning

confidence: 99%

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Symmetric Ideal Magnetofluidostatic Equilibria with Non-Vanishing Pressure Gradients in Asymmetric Confinement Vessels

Sato

2020

Preprint

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We study the possibility of constructing steady magnetic fields satisfying the force balance equation of ideal magnetohydrodynamics with tangential boundary conditions in asymmetric confinement vessels, i.e. bounded regions that are not invariant under continuous Euclidean isometries (translations, rotations, or their combination). This problem is often encountered in the design of next-generation fusion reactors. We show that such configurations are possible if one relaxes the standard assumption that the vessel boundary corresponds to a pressure isosurface. We exhibit a smooth solution that possesses an Euclidean symmetry and yet solves the boundary value problem in an asymmetric ellipsoidal domain while sustaining a nonvanishing pressure gradient. This result provides a definitive answer to the problem of existence of regular ideal magnetofluidostatic equilibria in asymmetric bounded domains. The question remains open whether regular asymmetric solutions of the boundary value problem exist.

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Section: Symmetry Of Solutionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Symmetric Ideal Magnetofluidostatic Equilibria with Non-Vanishing Pressure Gradients in Asymmetric Confinement Vessels

Sato

2020

Preprint

Self Cite

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Symmetric ideal magnetofluidostatic equilibria with nonvanishing pressure gradients in asymmetric confinement vessels

Sato

2020

Physics of Plasmas

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We study the possibility of constructing steady magnetic fields satisfying the force balance equation of ideal magnetohydrodynamics with tangential boundary conditions in asymmetric confinement vessels, i.e., bounded regions that are not invariant under continuous Euclidean isometries (translations, rotations, or their combination). This problem is often encountered in the design of next-generation fusion reactors. We show that such configurations are possible if one relaxes the standard assumption that the vessel boundary corresponds to a pressure isosurface. We exhibit a smooth solution that possesses a Euclidean symmetry and yet solves the boundary value problem in an asymmetric ellipsoidal domain while sustaining a nonvanishing pressure gradient. This result provides a definitive answer to the problem of existence of regular ideal magnetofluidostatic equilibria in asymmetric bounded domains. The question remains open whether regular asymmetric solutions of the boundary value problem exist.

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Quasisymmetric magnetic fields in asymmetric toroidal domains

Sato

Pfefferlé

et al. 2021

Physics of Plasmas

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We explore the existence of quasisymmetric magnetic fields in asymmetric toroidal domains. These vector fields can be identified with a class of magnetohydrodynamic equilibria in the presence of pressure anisotropy. First, using Clebsch potentials, we derive a system of two coupled nonlinear first order partial differential equations expressing a family of quasisymmetric magnetic fields in bounded domains. In regions where flux surfaces and surfaces of constant field strength are not tangential, this system can be further reduced to a single degenerate nonlinear second order partial differential equation with externally assigned initial data. Subclasses of solutions are then constructed by specifying as input the form the flux function, which enforces boundary shape and nested flux surfaces. In particular, we exhibit smooth quasisymmetric vector fields, which correspond to local solutions of anisotropic magnetohydrodynamics in asymmetric toroidal domains such that tangential boundary conditions are fulfilled on a portion of the bounding surface. These solutions are local because they lack periodicity in the toroidal angle. The problems of boundary shape and locality are also discussed. We find that magnetic fields with Euclidean isometries can be fitted into asymmetric domains and that the mathematical difficulty encountered in the derivation of global quasisymmetric magnetic fields lies in the topological obstruction toward global extension affecting local solutions of the governing nonlinear first order partial differential equations.

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Existence of ideal magnetofluid equilibria without continuous Euclidean symmetries

Cited by 3 publications

References 33 publications

Symmetric Ideal Magnetofluidostatic Equilibria with Non-Vanishing Pressure Gradients in Asymmetric Confinement Vessels

Symmetric Ideal Magnetofluidostatic Equilibria with Non-Vanishing Pressure Gradients in Asymmetric Confinement Vessels

Symmetric ideal magnetofluidostatic equilibria with nonvanishing pressure gradients in asymmetric confinement vessels

Quasisymmetric magnetic fields in asymmetric toroidal domains

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