Existence of Vortex Rings in Beltrami Flows

Abe, Ken

doi:10.1007/s00220-022-04331-y

Cited by 9 publications

(11 citation statements)

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“…In particular, we shall discuss a class of spherical vortices with swirl parameter κ, normalising the radius of the sphere to 1 for convenience. For an in-depth review of such vortices, we direct the interested reader to [68].…”

Section: Higher Symplectic Reductionsmentioning

confidence: 99%

Monge-Ampere Geometry and Vortices

Napper¹,

Roulstone²,

Rubtsov³

et al. 2023

Preprint

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We introduce a new approach to Monge-Ampère geometry based on techniques from higher symplectic geometry. Our work is motivated by the application of Monge-Ampère geometry to the Poisson equation for the pressure that arises for incompressible Navier-Stokes flows. Whilst this equation constitutes an elliptic problem for the pressure, it can also be viewed as a non-linear partial differential equation connecting the pressure, the vorticity, and the rate-of-strain. As such, it is a key diagnostic relation in the quest to understand the formation of vortices in turbulent flows. We study this equation via the (higher) Lagrangian submanifold it defines in the cotangent bundle to the configuration space of the fluid. Using our definition of a (higher) Monge-Ampère structure, we study an associated metric on the cotangent bundle together with its pull-back to the (higher) Lagrangian submanifold. The signatures of these metrics are dictated by the relationship between vorticity and rate-of-strain, and their scalar curvatures can be interpreted in a physical context in terms of the accumulation of vorticity, strain, and their gradients. We show explicity, in the case of two-dimensional flows, how topological information can be derived from the Monge-Ampère geometry of the Lagrangian submanifold. We also demonstrate how certain solutions to the three-dimensional incompressible Navier-Stokes equations, such as Hill's spherical vortex and an integrable case of Arnol'd-Beltrami-Childress flow, have symmetries that facilitate a formulation of these solutions from the perspective of (higher) symplectic reduction.

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Section: Higher Symplectic Reductionsmentioning

confidence: 99%

Monge-Ampere Geometry and Vortices

Napper¹,

Roulstone²,

Rubtsov³

et al. 2023

Preprint

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show abstract

“…The existence and the stability of axisymmetric nonlinear force-free fields with continuous factors f ∈ C(R 3 ) may be studied by replacing g(s) = 2s + of generalized magnetic helicity in (6.5) with sufficiently regular g(s), e.g., g(s) = 2s α + , α > 1, cf. [Abe22].…”

Section: Appendix a Existence Of Axisymmetric Leray-hopf Solutionsmentioning

confidence: 99%

“…The nonlinear system (1.3) is an overdetermined problem in general [EPS16], [CK20], cf. [EPS12], [EPS15], and existence results are available only under symmetry, e.g., [Cha56], [Tur89], [Abe22]. In the axisymmetric setting, both the system (1.3) and the steady Euler flow can be reduced to the Grad-Shafranov equation [GR58], [Sha58]; see [Gav19], [CLV19], and [DVEPS21] for the existence of compactly supported axisymmetric steady Euler flows.…”

Section: Introductionmentioning

confidence: 99%

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Stability of Chandrasekhar's nonlinear force-free fields

Abe¹

2022

Preprint

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Chandrasekhar's nonlinear force-free field is an axisymmetric magnetic field with a swirl whose current field is supported on a ball in a uniform magnetic field at infinity. We prove its orbital stability in ideal MHD for axisymmetric disturbances in terms of the existence of weak ideal limits of Leray-Hopf solutions introduced by Faraco and Lindberg. The proof is based on a derivation of Taylor's relaxation theory for nonlinear forcefree fields under axisymmetry: Woltjer's principle and Taylor's conjecture with generalized magnetic helicity. The main result demonstrates the stability of axisymmetric magnetic fields with swirls in ideal MHD even though blow-up occurs.

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“…Benjamin [12, Section I] in 1976 suggested variational principles for a broad class of steady vortex rings and inferred their stability up to a translation. Saffman also suggested in his textbook [96, footnote in p. 25] that one can employ conservation of mass and momentum to produce nonlinear stability in an 𝐿 1 and 𝐿 2 norm. However, to the best of our knowledge, there is still no rigorous proof for such stability.…”

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confidence: 99%

Stability of Hill's spherical vortex

Choi

2023

Comm Pure Appl Math

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We study stability of a spherical vortex introduced by M. Hill in 1894, which is an explicit solution of the three‐dimensional incompressible Euler equations. The flow is axi‐symmetric with no swirl, the vortex core is simply a ball sliding on the axis of symmetry with a constant speed, and the vorticity in the core is proportional to the distance from the symmetry axis. We use the variational setting introduced by A. Friedman and B. Turkington (Trans. Amer. Math. Soc., 1981), which produced a maximizer of the kinetic energy under constraints on vortex strength, impulse, and circulation. We match the set of maximizers with the Hill's vortex via the uniqueness result of C. Amick and L. Fraenkel (Arch. Rational Mech. Anal., 1986). The matching process is done by an approximation near exceptional points (so‐called metrical boundary points) of the vortex core. As a consequence, the stability up to a translation is obtained by using a concentrated compactness method.

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Existence of Vortex Rings in Beltrami Flows

Cited by 9 publications

References 100 publications

Monge-Ampere Geometry and Vortices

Monge-Ampere Geometry and Vortices

Stability of Chandrasekhar's nonlinear force-free fields

Stability of Hill's spherical vortex

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