Moduli spaces of vortex knots for an exact fluid flow
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Abstract
The moduli spaces S(D) of non-isotopic vortex knots are introduced for the ideal fluid flows in invariant domains D. The analogous moduli spaces of the magnetic fields B knots are defined. We derive and investigate new exact fluid flows (and analogous plasma equilibria) satisfying the Beltrami equation which have nested invariant balls Bk3 with radii Rk ≈ (k + 1) π, k⟶∞. The first flow is z-axisymmetric; the other ones do not possess any rotational symmetries. The axisymmetric flow has an invariant plane z = 0. Due to an involutive symmetry of the flow, its vortex knots in the invariant half-spaces z > 0 and z < 0 are equivalent. It is demonstrated that the moduli space 𝒮(ℝ3) for the derived fluid flow in ℝ3 is naturally isomorphic to the set of all rational numbers p/q in the interval J1:0.25<q<M̃1≈0.5847, where q is the safety factor. For the fluid flow in the first invariant ball B13, it is shown that all values of the safety factor q belong to a small interval of length ℓ ≈ 0.1261. It is established that only torus knots Kp,q with 0.25 < p/q < 0.5847 are realized as vortex knots for the constructed flow in ℝ3. Each torus knot Kp,q with 0.25 < p/q < 0.5 is realized on countably many invariant tori Tk2 located between the invariant spheres Sk2 and Sk+12, while torus knots with 0.5<p/q<M̃1 are realized only on finitely many invariant tori. The moduli spaces Sm(Ba3)(m=1,2,…) of vortex knots are constructed for some axisymmetric steady fluid flows that are solutions to the boundary eigenvalue problem for the curl operator on a ball Ba3.
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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