1. Topological interpretation of helicity

Ravello GNFM Summer School September 11-16, 2017

Course contents Renzo L. Ricca

Department of Mathematics & Applications, U. Milano-Bicocca renzo.ricca@unimib.it

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

magnetic fields in astrophysical flows and plasma physics

(TRACE mission 2002)

Vortex filaments in fluid mechanics (Kleckner & Irvine 2013)

vortex tangles in quantum systems (Villois et al. 2016)

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)

topological dynamics Knotted solutions

to Euler s equations

Energy relaxation methods Dynamical systems

and µ-preserving flows

Change of topology

- 3-D fluid topology - vortex solutions - fluid invariants

- topological stability - 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 fluid in : in

as

frozen field evolution:

:

~

topological equivalence class:

~

u = u X,t ( ) ^{∇ ⋅ u = 0}

u = 0 X → ∞

B

( X, t ) ^{= B}

( ^X

^{, 0} ) _∂X ^∂X

0 j

B X (

,0 ) B X,t ( )

3

3

B(X, t) ∈ ∂B

∂t = ∇ × u × B ( ) ∧ ∇ ⋅ B = 0; L

– norm

Re-arrangement of internal structure

Knotted pretzel

The concept of topological equivalence and invariants

~ ~ ~

Linked pretzel

Change of topology

reconnection via local surgery (dissipative effects)

Physical knots and links as tubular embeddings

C ⁱ Φ_i Let and :

Definition: A physical knot/link is a smooth immersion into of finitely many disjoint standard solid tori , such that

T

. T_i ⁼ S_i ^⊗C_i V_i = V

( )

in physical embedding:

by a standard foliation of the

B

such that on (material surface).

F

B ⋅ ˆ

ν

∂ T

volume and flux-preserving diffeomorphism:

V = V (B) _,

Φ

⁼ B ⋅ ˆ λ

{

}

A( )

∫

^d

^x

F

T ⁱ → Ki

K

^{:= supp B} ( )

supp B ( ) ^{: =} _∪

^K

^{→ L}

i = 1,..., n ( )

3

3

Analogies between Euler equations and magnetohydrodynamics

ω = ∇ × u

u

B = ∇ × A

∂B

∂t = ∇ × u × B ( )

H = u ⋅ ω ^d

^x

V_ω

∫ ^{H =} ∫

^{A ⋅ B d}

^x

not perfect

ω ↔ J

ω ↔ B

ω = ∇ × u

perfect (static)

J = ∇ × B J × B = ∇p u × ω = ∇h

h = p + 1

2 u

u

→ u

H = u ⋅ ω ^d

^x

V_ω

∫ ^{H =} ∫

^{J ⋅ B d}

^x

B^E

u^E

u₀ Β₀

∂ ω

∂t = ∇ × u × ( ω )

u

Helicity and linking numbers

Theorem (Moffatt 1969; Moffatt & Ricca 1992). Let be an essential physical link in an ideal fluid. Then, we have

L

. Helicity :

H (t)

where , with in .

B = ∇ × A ∇ ⋅ A = 0 H (t) = A ⋅ B

V (B)

∫ ^d

^X

Theorem (Woltjer 1958; Moreau 1961). Under ideal conditions helicity

is a conserved quantity (frozen in the flow), that is

= Lk

Φ

Φ

∑

⁺ ( ^{Wr +Tw} ) ^Φ

∑

H = A ⋅ B d

X

V (B)

∫ ^{= Lk}

^Φ

^Φ

∑

^{+ Lk}

^Φ

∑

.

Helicity in terms of Gauss linking number

Following Moffatt 1969 ( ) we have:

Field

B

N

No contribution to helicity from individual flux tubes

C₁

Proof. By Stokes’ theorem we have:

S

K

= 0

C

C

±Φ

C

C

For

Ν

C₂

H = A ⋅ B d

X

V (B)

∫ ^{= Lk}

^Φ

^Φ

∑

Lk

= 0

.

Written in integral form, we have

and summing over all tubes, we have an invariant integral over the whole

B

The

A

B

hence

that for discrete tubes becomes H

H

where

is Gauss linking number.

( )

+1 –1

+

+ c

= min(#)

Minimum number of crossings:

Number of components:

N

N = 4 c

= 0 Lk = 0 N = 2 c

= 2 Lk = +1

(Gauss) linking number between components:

ε

= ±1

First topological invariants

Lk = 1

2 ε

∑

Computations by hand: some examples

and introduce the concept of ribbon :

C

C R(C, C

)

C

→ C: x = x(s)

C

→ C

: x

= x(s) + ε ^N(s)

Suppose now that

Β

- internal winding

- tube axis knottedness

Suppose there is no other contribution to helicity.

Following Moffatt & Ricca 1992 ( ) we have:

Helicity in terms of Călugăreanu invariant

H = A ⋅ B d

X

V (B)

∫ ^{= Lk}

^Φ

∑

Lk

= 0

Călugăreanu-White invariant

Călugăreanu-White invariant:

writhing number:

total twist number:

(as )

ε → 0

and take the limit:

+

Călugăreanu 1961 White, 1969 limε→0 Lk₁₂(

ε

Lk(C, C

) = Wr(C) +Tw(C,C

)

T (C) = 1

2π τ^(s)

C

∫

Wr(C) = 1 4π

(x − x^*) ⋅ dx × dx^*

| x − x |^*3

C

∫

C

∫

Tw(C, C^*) ≡ 1

2

π θ

C

∫

N (C, C^*) = 1

2π ^{[Θ(x, x}

*)]R Let us re-consider the

Gauss linking formula:

Lk

= Lk(C

,C

) = 1 4 π

(x

− x

) ⋅ dx

× dx

x

− x

C

∫

C

∫

Properties of Lk, Wr, and Tw

Călugăreanu invariant:

Writhing number :

Total twist number :

- is a topological invariant of the ribbon;

- it is an integer;

- under cross-switching ( ):

- is a geometric measure of the curve C ; - it is a conformational invariant;

- under cross-switching ( ):

- 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

minimal projection generic projection

c

= 4

c = c

= 4 c = 10

Lk = +2 Lk = +2

Signed crossings and complexity measure

Wr = –3 Wr = +3 Wr = +1

ν χ_i

−1

−1

+1 ν

–1

(Fuller 1971)

Average crossing number:

C = 3 :

(Freedman & He 1991) Wr = Wr(C ) = n–(ν) – n₊(ν) = ε_r

r ∈

∑

C

ε

r ∈

∑

C

Writhing number in terms of signed crossings:

Tangle analysis by indented projections

χ_i Let

χ_i

Π_i = Π( ˆT_i) be the indented –projection of the oriented tangleTˆ_i component ; assign the value to each apparent crossing in .χ_{i Π}

i

writhing:

average crossing number:

linking:

Wr =Wr(T ^{) =} ε_r

r∈

∑

C_ij = C(χ_i^,χ_j) = ε_r

r∈χ_i∩χ_j

∑

, ;

;

.

−1

−1

+1

Π_i

Wr_i = Wr(χ_i) = ε_r

r ∈χ

∑

Lk_ij = Lk(χ_i, χ_j) = 1

2 ε_r

r∈χ_i∩χ_j

∑

r∈

∑

,

C = C_ij

r∈

∑

ε

= ±1

estimated values:

^{W r⊥ =} ε_r

r ∈

∑

⊥

, C_⊥ = ε_r

r∈

∑

⊥

T = χⁱ

∪

Tˆ_i

Energy-complexity relation: a 16 years-old test case

time

log H log C_⊥

log C log Wr_⊥

log L

mature tangle

O(164 s⁻¹)

O(82 s⁻¹)

log E

L = L / L

E = E / E

ABC-flow field super-imposed on an initial circular vortex ring

Energy-complexity relation (Barenghi et al. 2001)