Some mathematics for quasi-symmetry

Burby, J. W.; Kallinikos, Nikos; MacKay, Robert S.

doi:10.48550/arxiv.1912.06468

Cited by 9 publications

(23 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As noted in Ref. 26, the quasisymmetric GS equation does not possess a natural variational principle unless the infinitesimal generator of quasisymmetry u satisfies (∇ × u) × u + ∇(u • u) = 0. It is therefore curious that the GGS equation always possesses a variational principle.…”

Section: Comparison With Previous Workmentioning

confidence: 98%

See 1 more Smart Citation

Generalized Grad-Shafranov equation for non-axisymmetric MHD equilibria

Burby,

Kallinikos,

MacKay

2020

Preprint

View full text Add to dashboard Cite

The structure of static MHD equilibria that admit continuous families of Euclidean symmetries is well understood. Such field configurations are governed by the classical Grad-Shafranov equation, which is a single elliptic PDE in two space dimensions. By revealing a hidden symmetry, we show that in fact all nondegenerate solutions of the equilibrium equations satisfy a generalization of the Grad-Shafranov equation. In contrast to solutions of the classical Grad-Shafranov equation, solutions of the generalized equation are not automatically equilibria, but instead only satisfy force balance averaged over the one-parameter hidden symmetry. We then explain how the generalized Grad-Shafranov equation can be used to reformulate the problem of finding exact three-dimensional smooth solutions of the equilibrium equations as finding an optimal volume-preserving symmetry.

show abstract

Section: Comparison With Previous Workmentioning

confidence: 98%

“…In previous work 26 we identified a quasisymmetric variant of the Grad-Shafranov equation. (See Theorem 10.5 in Ref.…”

Section: Comparison With Previous Workmentioning

confidence: 99%

Generalized Grad-Shafranov equation for non-axisymmetric MHD equilibria

Burby,

Kallinikos,

MacKay

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…For a more detailed analysis of continuous symmetries of Hamiltonian systems and, in particular, the guiding-center Hamiltonian, see Refs. [28,15], respectively.…”

Section: Quasisymmetry Vectormentioning

confidence: 98%

“…( 44) with B we find that B • ∇Γ = 0, which allows us to set Γ = Γ(ψ) (for more details see Ref. [15]). Using Eqs.…”

Section: Quasisymmetry Vectormentioning

confidence: 99%

Construction of Quasisymmetric Stellarators Using a Direct Coordinate Approach

Jorge,

Sengupta,

Landreman

2020

Preprint

View full text Add to dashboard Cite

Optimized stellarator configurations and their analytical properties are obtained using a near-axis expansion approach. Such configurations are associated with good confinement as the guiding center particle trajectories and neoclassical transport are isomorphic to those in a tokamak. This makes them appealing as fusion reactor candidates. Using a direct coordinate approach, where the magnetic field and flux surface functions are found explicitly in terms of the position vector at successive orders in the distance to the axis, the set of ordinary differential equations for first and second order quasisymmetry is derived. Examples of quasi-axisymmetric shapes are constructed using a pseudospectral numerical method. Finally, the direct coordinate approach is used to independently verify two hypotheses commonly associated with quasisymmetric magnetic fields, namely that the number of equations exceeds the number of parameters at third order in the expansion and that the near-axis expansion does not prohibit exact quasisymmetry from being achieved on a single flux surface.

show abstract

“…In this formulation, B naturally appears as an input to the problem. Cast in this form, the formulation generalizes the classical form 1,10 and is less constraining than its strong form 6,16,17 . It is also convenient to write the magnetic field in its covariant form,…”

Section: A Magnetic Equationsmentioning

confidence: 99%

Weakly Quasisymmetric Near-Axis Solutions to all Orders

Rodríguez,

Sengupta,

Bhattacharjee

2021

Preprint

View full text Add to dashboard Cite

Some mathematics for quasi-symmetry

Cited by 9 publications

References 2 publications

Generalized Grad-Shafranov equation for non-axisymmetric MHD equilibria

Generalized Grad-Shafranov equation for non-axisymmetric MHD equilibria

Construction of Quasisymmetric Stellarators Using a Direct Coordinate Approach

Weakly Quasisymmetric Near-Axis Solutions to all Orders

Contact Info

Product

Resources

About