Ravello GNFM Summer School September 11-16, 2017 Renzo L. Ricca Department of Mathematics & Applications, U. Milano-Bicocca renzo.ricca@unimib.it 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
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.
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)
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
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) ~
The concept of topological equivalence and invariants Re-arrangement of internal structure ~ ~ ~ Linked pretzel Knotted pretzel
Change of topology reconnection via local surgery (dissipative effects)
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.
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 + 1 2 u2 Β 0 B E u E u 0 B 0 B E u 0 u E
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.
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
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.
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,,,, +
Computations by hand: some examples
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
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
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:
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
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
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
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
Energy-complexity relation (Barenghi et al. 2001)