Ravello GNFM Summer School September 11-16, Renzo L. Ricca. Department of Mathematics & Applications, U. Milano-Bicocca

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1 Ravello GNFM Summer School September 11-16, 2017 Renzo L. Ricca Department of Mathematics & Applications, U. Milano-Bicocca Course contents 1. Topological interpretation of helicity 2. Vortex knots dynamics and momenta of a tangle 3. Magnetic knots and groundstate energy spectrum 4. Topological transition of soap films 5. Helicity change under reconnection: the GPE case 6. Topological decay measured by knot polynomials

2 Lecture 1 Topological interpretation of helicity: - Coherent structures and topological fluid mechanics - Diffeomorphisms and topological equivalence - Kinetic and magnetic helicity of flux tubes - Gauss linking number - Călugăreanu invariant and geometric decomposition Selected references Barenghi, CF, Ricca, RL & Samuels, DC 2001 How tangled is a tangle? Physica D 157, 197. Moffatt, HK 1969 The degree of knottedness of tangled vortex lines. J Fluid Mech 35, 117. Moffatt, HK & Ricca, RL 1992 Helicity and the Călugăreanu invariant. Proc Roy Soc A 439, 411. Ricca, RL & Berger, MA 1996 Topological ideas and fluid mechanics. Phys Today 49, 24.

3 Coherent structures Vortex filaments in fluid mechanics (Kleckner & Irvine 2013) vortex tangles in quantum systems (Villois et al. 2016) magnetic fields in astrophysical flows and plasma physics (TRACE mission 2002)

4 150 years of topological dynamics Linking number formula (Gauss 1833) Knot tabulation (Tait 1877) Applications to magnetic fields (Maxwell 1867) Applications to vortices (Kelvin 1867) Knotted solutions to Euler s equations topological dynamics - 3-D fluid topology - vortex solutions - fluid invariants - topological stability Energy relaxation methods Dynamical systems and µ-preserving flows Change of topology - magnetic knots - charged knots - groundstate energy - Theorems for vector fields - closed and cahotic orbits - Hamiltonian structures - reconnection mechanisms - singularity formation

5 Diffeomorphisms of frozen fields ideal, incompressible perfectly conducting u =0 in 3 fluid in 3 : u = ux,t ( ) as u =0 X frozen field evolution: B(X, t) topological equivalence class: B i B t = ( u B) B = 0; L 2 norm ( X, t) = B ( j X 0, 0) X i : ~ X 0j BX 0,0 ( ) B( X,t) ~

6 The concept of topological equivalence and invariants Re-arrangement of internal structure ~ ~ ~ Linked pretzel Knotted pretzel

7 Change of topology reconnection via local surgery (dissipative effects)

8 Physical knots and links as tubular embeddings Let T i = S i C i and V i = V T i : T i K i in 3 ( ) F { pi,q i } physical embedding: by a standard foliation of the B-lines, B ˆν = 0 T i ( ) K i := supp B F pi,q i such that on (material surface). C i { } ˆλ Φ i Definition: A physical knot/link is a smooth immersion into 3 of finitely many disjoint standard solid tori, such that supp B ( ) := K i T i L n i = 1,..., n i ( ) volume and flux-preserving diffeomorphism:. V = V (B), Φ i = B ˆλ d 2 x A( S i ) ; signature {V, Φ i } constant.

9 Analogies between Euler equations and magnetohydrodynamics ω t H = ω = u u ( ) = u ω not perfect ω B B = A B t = ( u B) u ω d 3 x H = A B d 3 x V ω VB u ω = u J = B H = u ω = h perfect (static) J B = p ω J H = J B d 3 x u ω d 3 x V ω V J h = p u2 Β 0 B E u E u 0 B 0 B E u 0 u E

10 Helicity and linking numbers Helicity H(t) : H(t) = B = A A = 0 A B d 3 X V (B) 3 where, with in. Theorem (Woltjer 1958; Moreau 1961). Under ideal conditions helicity is a conserved quantity (frozen in the flow), that is Theorem (Moffatt 1969; Moffatt & Ricca 1992). Let L n be an essential physical link in an ideal fluid. Then, we have. H = A B d 3 X = Lk ij Φ i Φ j + Lk i Φ 2 i V (B) i j = Lk ij Φ i Φ j + ( 2 Wr +Tw)Φ i i j i i.

11 Helicity in terms of Gauss linking number Field B entirely localized in N unlinked, unknotted flux tubes No contribution to helicity from individual flux tubes Lk i = 0 H = A B d 3 X = Lk ij Φ i Φ j Following Moffatt 1969 ( ) we have: V (B) Proof. By Stokes theorem we have: i j. K 1 = 0 if C 1 and C 2 are not linked; ±Φ 2 if C 1 and C 2 are singly linked. For Ν tubes multiply linked with each other, we have C 1 S 1 C 2

12 Written in integral form, we have and summing over all tubes, we have an invariant integral over the whole B-field: H The A field is given by the Biot-Savart law due to B-field, i.e. ( ) hence H that for discrete tubes becomes where is Gauss linking number.

13 First topological invariants Number of components: N Minimum number of crossings: c min = min(#) (Gauss) linking number between components: Lk = 1 2 ε r r +1 1 ε r = ±1 + N = 4 c min = 0 Lk = 0 N = 2 c min = 2 Lk = +1,,,, +

14 Computations by hand: some examples

15 Helicity in terms of Călugăreanu invariant Suppose now that Β-field lines inside individual flux tubes contribute to helicity: - internal winding - tube axis knottedness Suppose there is no other contribution to helicity. Following Moffatt & Ricca 1992 ( ) we have: H = A B d 3 X = Lk i Φ 2 i V (B) i Lk ij = 0 R(C,C * ) and introduce the concept of ribbon : C 1 C: x = x(s) C 2 C * : x * = x(s)+εn(s) C * C

16 Călugăreanu-White invariant Let us re-consider the Gauss linking formula: Lk 12 = Lk(C 1,C 2 ) = 1 4π C 1 C 2 (x 1 x 2 ) dx 1 dx 2 x 1 x 2 3 and take the limit: lim ε 0 Lk 12 (ε) = Lk(C,C * ) Călugăreanu-White invariant: Lk(C,C * ) = Wr(C) +Tw(C,C * ) (as ε 0 ) Călugăreanu 1961 White, 1969 writhing number: Wr(C) = 1 4π C C (x x * ) dx dx * x x *3 total twist number: Tw(C,C * ) 1 2π θ(x, x* ) ds = C T (C) = + N (C,C * ) = 1 τ(s) 2π ds C 1 2π [Θ(x, x* )] R

17 Properties of Lk, Wr, and Tw Călugăreanu invariant: - is a topological invariant of the ribbon; - it is an integer; - under cross-switching ( ): Writhing number : - is a geometric measure of the curve C ; - it is a conformational invariant; - under cross-switching ( ): Total twist number : - is a geometric measure of the ribbon ; - it is a conformational invariant; - it is additive:

18 Topological crossing number versus average crossing number 2-component oriented link with topological crossing number c min = 4 minimal projection generic projection c = c min = 4 c =10 Lk =+2 Lk =+2

19 Signed crossings and complexity measure Writhing number in terms of signed crossings: χ i 1 +1 (Fuller 1971) ν ν C Wr = Wr(C ) = n (ν) n + (ν) = ε r r C C 1 +1 Average crossing number: 1 C = C (C ) = r C C ε r (Freedman & He 1991) C = 3: Wr = 3 Wr =+3 Wr =+1

20 Tangle analysis by indented projections Let Π i = Π( ˆT i ) be the indented ˆT i projection of the oriented tangle χ i component ; assign the value to each apparent crossing in. writhing: Wr i = Wr(χ i ) = r χ i ε r ε r =±1, Wr =Wr(T ) = ; r T ε r T = i χ i Π i linking: Lk ij = Lk(χ i, χ j ) = 1 2 ε r, Lk tot = Lk ij r χ i χ j r T ; χ i i average crossing number: ˆT i C ij = C(χ i, χ j ) = ε r, C = C ij ; r χ i χ j r T 1 +1 estimated values: W r = ε r, r T C = ε r r T. Π i 1

21 Energy-complexity relation: a 16 years-old test case ABC-flow field super-imposed on an initial circular vortex ring log H log C O(164 s 1 ) log C log Wr O(82 s 1 ) L = L / L0 log E log L E = E / E0 mature tangle time

22 Energy-complexity relation (Barenghi et al. 2001)