Vortex knots dynamics and momenta of a tangle:

Transcription

1 Lecture 2 Vortex knots dynamics and momenta of a tangle: - Localized Induction Approximation (LIA) and Non-Linear Schrödinger (NLS) equation - Integrable vortex dynamics and LIA hierarchy - Torus knot solutions to LIA - Linear and angular momentum in terms of signed area information Selected references Maggioni, F, Alamri, SZ, Barenghi, CF & Ricca, RL 2010 Velocity, energy and helicity of vortex knots and unknots. Phys Rev E 82, Ricca, RL 1993 Torus knots and polynomial invariants for a class of soliton equations. Chaos 3, 83 [Erratum. Chaos 5, 346]. Ricca, RL 2008 Momenta of a vortex tangle by structural complexity analysis. Physica D 237, 2223.

2 Localized induction approximation (LIA) homogeneous incompressible inviscid Space curve C t : C t u = ux,t ω = u u =0 u =0 fluid in R 3 : in as, given by: X(s,t):= X t (s) C ν s [0,L] R 3 Intrinsic reference on (Frenet frame) t ˆ := X (s,t) = X s Vortex line on C t : ω = ω 0ˆ t ω 0 C t, = constant circulation = constant κ asymptotic theory:, given by: t ˆ, n ˆ, b ˆ { } R/a>>1 ˆ t ˆ n R 3 X ˆ b C t X(s,t) = X t = X X = c ˆb = u LIA (Da Rios 1906; Hama & Arms 1961)

3 Intrinsic equations under LIA and NonLinear Schrödinger (NLS) equation u =(u t,u n,u b ) Intrinsic description:, curvature, torsion; u LIA =cˆ b cs,t u t = u n = 0 u b = c under LIA:,,. τ( s,t) C t := ϕ t (C) c = (cτ ) c τ τ = c cτ 2 c + c c Da Rios 1906 Betchov 1965 NLS eq. via Madelung transform: dξ ψ( s,t) = cs,t e i τξ,t 0 s τ R 2 c ψ Im C Re NLS eq.: i ψ + ψ + 1 ψ 2 ψ = 0 2 (Hasimoto 1972) Intrinsic equations by log-derivative: let = s 0 τ ξ,t Θ s,t dξ and consider ψ = c e iθ Da Rios- Betchov eqs.

4 Integrable vortex dynamics and LIA hierarchy Integrable vortex dynamics: u = ( α + βs)cˆ b + ( γ + δs)ˆ β,δ : inhomogeneities (Lakshmanan & Ganesan 1985) LIA hierarchy: γ t + µ 1 c 2ˆ t + c n ˆ + cτˆ b 2 : non-linear stretching (Onuki 1985) µ : axial flow and vorticity (Fukumoto & Miyazaki 1991) u ( 0) = u LIA ( u j+1 ) = ˆ t ˆt + cs,t j u b ( j ) +τun ds (Langer & Perline 1991) = ˆt j+1 F ν ( u 0 ) ν=1 s ν = R u ( 0 ) integrable class: ψ = P ψ ( u n e iθ )+ Q ψ u b e iθ (Nakayama et al. 1992) Hirota class NLSE mkdv sine-g

5 Stationary solutions: Hasimoto soliton τ = const. cs ()= 2 sech s () (Hasimoto 1972)

6 Steady solutions to LIA: torus knots and unknots Look for stationary solutions to LIA in the shape of torus knots and unknots: (Da Rios 1933) Theorem (Ricca 1993). LIA admits torus knot solutions in closed T p,q analytic form, given by: Theorem (Kida 1981). There exists a class of steady torus knot solutions T p,q ( p > 1, q >1 co-prime integers) to LIA in terms of incomplete elliptic integrals, given by r 2 = F r ( J ), α = F α ( E), z = F z Π. r = r 0 +ε sin( wξ ) α = s + ε cos( wξ ) r 0 wr 0 z = t +ε 1 1 r 0 w 2 1/2 cos( wξ ).

7 Proof. First let s write LIA in terms of cartesian coordinates (x, y, z) and cylindrical polar coordinates (r, α, z). From and x = r cos α, y = r sin α, z u LIA = X(s,t) = X X ( ) ( ), by taking time-derivative, we have and by the last equality of ( ), we have We now take the arc-length derivative of to get LIA in cylindrical polar coordinates:. ( ) and substitute in the above,. ( )

8 Now let s consider small-amplitude perturbations. For the vortex ring we have r = R, and α = s/r, so that previous equations reduce to Small-amplitude perturbations are given by taking.. By substituting these into ( ) and taking first-order terms, we have By looking for a traveling-wave solution ξ = s κt as torus knot, we have r = r 0 +ε sin wξ cos( wξ ) α = s + ε r 0 wr 0 z = t +ε 1 1 r 0 w 2 1/2 cos( wξ ).

9 Small-amplitude torus knots and unknots under LIA torus knots torus unknots

10 Linear stability of LIA torus knots T p,q Theorem (Ricca 1995). For any given w = q/ p, is linearly stable iff w >1. T 2,3 ( w >1) T 3,2 ( w <1) (Ricca et al 1999)

11 U 1,m T p,q U m,1 LIA knots and unknots, (Maggioni et al. 2010)

12 Creation and dynamics of trefoil vortex knot in water (Klecker & Irvine 2013)

13 Linear and angular momentum by signed area information χ ω = ω 0 ˆt ω 0 = cst. Let denote the centreline of a vortex filament with. Theorem (Ricca 2008). The components of linear and angular momentum of a vortex filament of circulation Γ, given respectively by P = (P x, P y, P z ) and M = (M x,m y,m z ), can be expressed in terms of signed areas of the projected graph regions:,.

14 Proof. χ ω = ω 0 ˆt ω 0 = cst. Let denote the vortex axis:, and ; - the linear momentum is given by - the angular momentum is given by since X = ˆt = dx, we have X ds = dx and ds How to compute? For individual components, we have where M = 1 3 P = 1 2 X ω d 3 X = 1 2 Γ X X ds V( ω) Λ = ˆν(χ): 3 2 = Γ d 2 X L χ A χ X ( X ω)d 3 X = 1 3 Γ X ( X X )ds V( ω) P x : Λ x P y : Λ y P z : Λ z and and and A x = A Λ x A y = A Λ y A z = A Λ z denotes the standard graph projected along the x-axis and the signed area of. Similarly for the y- and z-axis and M x, M y, M z. = 2 L χ 3 Γ d 2 X A χ,.

15 Computation of signed area of a planar graph To each sub-region of Λ we assign an index that weights the relative contribution from the circulation of neighbouring strands: R j χ 3 where according to the r.h. sign convention of the reference. ˆt ˆ ρ R The signed area of Λ is thus given by where of. is the standard area

16 Thus, by the standard definitions of momenta and the observations above we have: P = 1 2 Γ X X ds = Γ = Γ d 2 X L χ A( χ ) ds = 2 L χ 3 Γ X d 2 X A( χ ) M = 1 3 Γ X X X = 2 3 Γ where, and the signed areas of the projected graphs. Computation of signed area of interacting vortices ( Γ=1 ) Corollary. The components of linear and angular momentum of a vortex tangle T can be computed in terms of signed areas of the projected graph regions.

17 Head-on collision of vortex rings a b (Lim & Nickels 1992)