Beltrami vector fields with an icosahedral symmetry

Alkauskas, Giedrius

doi:10.1016/j.geomphys.2020.103655

Cited by 3 publications

(2 citation statements)

References 39 publications

(66 reference statements)

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“…Indeed, using the separation of variable analysis, the eigenfunctions after the name of Chandrasekhar-Kendall (CK) modes can be obtained for the cases of periodic cylinder [17], sphere [19], specific types of spheromaks [27], finite cylinder [28], annular geometry [29], and periodic channel [30]. Specifically, Alkauskas [31] constructed two unique Beltrami vector fields with orientation-preserving icosahedral symmetry. For general domains, there are generally no analytical solutions for CK modes, and hence numerical solutions are necessary for finding the helical waves.…”

Section: Introductionmentioning

confidence: 99%

Numerical Computation of Helical Waves in a Finite Circular Cylinder Using Chebyshev Spectral Method

Xing-Liang Lyu,

Wei-Dong Su

2024

AAMM

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Helical waves are eigenfunctions of the curl operator and can be used to decompose an arbitrary three-dimensional vector field orthogonally. In turbulence study, high accuracy for helical waves especially of high wavenumber is required. Due to the difficulty in analytical formulation, the more feasible strategy to obtain helical waves is numerical computation. For circular cylinders of finite length, a semi-analytical method via infinite series to formulate the helical wave is known [E. C. Morse, J. Math. Phys., 46 (2005), 113511], where the eigenvalues are evaluated by iterating transcend equations. In this paper, the numerical computation for helical wave in a finite circular cylinder is implemented using a Chebyshev spectral method. The solving is transformed into a standard matrix eigenvalue problem. The large eigenvalues are computed with high precision, and the calculation cost to rule out spurious eigenvalues is significantly reduced with a new criterion suggested.

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Section: Introductionmentioning

confidence: 99%

Numerical Computation of Helical Waves in a Finite Circular Cylinder Using Chebyshev Spectral Method

Xing-Liang Lyu,

Wei-Dong Su

2024

AAMM

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“…The case where µ is not constant with respect to position is sometimes called a nonlinear force-free state [4], though, as long as µ is independent of B, the Beltrami equation is a linear elliptic PDE. Conversely, the case where µ is a spatial constant is sometimes called a linear force-free state and the Beltrami equation is also termed the Trkalian equation [5].…”

Section: Introductionmentioning

confidence: 99%

Nature of ideal MHD instabilities as described by multi-region relaxed MHD

Kumar¹,

Nührenberg²,

Qu³

et al. 2022

Plasma Phys. Control. Fusion

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In the following work, the Stepped Pressure Equilibrium Code (SPEC) [Hudson, Dewar\textit{ $\textit{et al.}$, Phys. Plasmas 19, 112502 (2012)}] which computes the equilibria of Multi-Region relaxed Magnetohydrodynamic energy principle (MRxMHD), has been upgraded to determine the MRxMHD stability in toroidal geometry. A theoretical formalism for SPEC is obtained by relating the second variation of the MRxMHD energy functional to the Hessian matrix, enabling the prediction of MHD linear instabilities. Negative eigenvalues of this matrix imply instability. Further, we demonstrate our method on simplified test scenarios in both tokamak and stellarator magnetic topologies, with a systematic comparison study between the marginal stability prediction of the SPEC with the ideal MHD stability code packages CAS3D and {MISHKA}-1.

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