/S S min0 min

Lecture 5

Helicity change under reconnection; the GPE case:

- Reconnection and change of helicity - Application to solar cornal loops

- Writhe conservation under reconnection

- Cascade of quantum vortex links under Gross-Pitaevskii equation - Iso-phase surfaces as Seifert surfaces

- Evolution of iso-phase surfaces of minimal area

Selected references

Laing, CE, Ricca, RL & Sumners, DeW L 2015 Conservation of writhe helicity under anti-parallel reconnection. Nature Sci Rep 5, 9224.

Zuccher, S & Ricca, RL 2015 Helicity conservation under quantum reconnection of vortex rings. Phys Rev E 92, 061001.

Zuccher, S & Ricca, RL 2017 Relaxation of twist helicity in the cascade process of linked quantum vortices. Phys Rev E 95, 053109.

Example of high-energy fields relaxation

(Battye & Sutcliffe 1998)

Simulation of magnetic flux tubes reconnection Cross-switching at reconnection sites

Isosurfaces ( ) of force-free magnetic field with uniform twist.

Gold-Hoyle model ( ).

(Dahlburg & Norton 1995)

1

2 B _max

η = 0.005

−1

(a) (b) (c)

(d) (e) (f)

Simulation of vortex tubes reconnection

DNS of vorticity iso-surfaces at 40% of maximum initial vorticity (Hussain & Duraisamy 2011)

Helicity-complexity change in a dissipative fluid

In presence of dissipation: , . Let .

H = H (t)

Theorem (Ricca 2008). Let be a zero-framed, essential physical

link, embedded in an incompressible, dissipative fluid. Then, we have

L

C(t)

H (t) ≤ Φ

C(t) ^{d H (t)}

dt ≤ sign(H ) sign(Ω) Φ

dC(t)

I) ; II) .

dt

Proof.

Since and , we have

Ω = d ω

∑

H (t) = Φ² d

ω

∫

∑

^{C(t) =} ∑

∫

^{d ω}

H (t) = Φ

d ω

∫

∑

^C(t) _d

ω

∫

∑

≤ Φ

∑

∫

d ω

^C(t)

d ω

∫

∑

^{= Φ}

2

C(t)

Thus I) is proved.

Consider then the elementary change in helicity

d H (t) = d sign(t) H (t) = sign(H ) dH (t) = sign(H ) Φ

d ω

∫

∑

and in average crossing number

dC(t) = d ω

∑

We have:

d H (t) = sign(H) Φ

d ω

∑

d ω

∑

dC(t) ≤ sign(H) Φ

d ω

∑

d ω

∑

^dC(t)

= sign(H ) sign(Ω) Φ

dC(t)

In the limit, by taking the time derivative of the above, we have (II).

.

.

Applications to solar coronal loops

solar coronal braid; minimal number of strings (loops)

In presence of random photospheric motion, we have (Berger 1993):

B

c

∝ N

b( B ^{) = N}

E

∝ c

Φ

N

L E

∝ Φ

L N

and in our case, by using Ohyama (1993) inequality, we have:

M

∝ Φ

L (N −1)

In presence of dissipation:

H (t) ≤ Φ

C(t) ^{d H (t)}

dt ≤ Φ

dC(t)

and

dt

and ,

M

≥ 16 π

1/ 3

b ( ) B ^{− 1}

( )

Wr ≈ +1:

(Scheeler et al. 2014) Writhe helicity under reconnection

anti-parallel

Wr ≈ −1

Anti-parallel reconnection near a crossing does not change writhe:

Writhe under anti-parallel reconnection

Wr ≈ −1

– 1

(Zabusky & Melander 1989)

– 1 – 1 – 1

Oriented polygons:

A = {a

,...,a

} B = {b

,...,b

}

Writhe of by oriented edges:

Wr(A) = 1

4 π

^w(a

^,a

⁾

n i =1

∑

n

∑

Lk(A, B) = 1

4 π

^w(a

^,b

⁾

m i =1

∑

n

∑

Linking number of disjoint oriented polygons , by oriented edges:

A

B

w(e

,e

) = w(e

,e

)

Signed area contribution on the unit sphere covered by direction vectors pointing from

oriented edge to oriented edge :

e

^e

e

e

PL- curves

A

B

.

A B

A a

b

Proof.

Writhe of disjoint union

Wr(A∪ B)= 1

4 π

^w(a

^,a

⁾

n i =1

∑

n

∑ ^{+ w(a}

^,b

⁾

j =1 m i =1

∑

n

∑ ^{+ w(b}

^,a

⁾

j =1 m i =1

∑

n

∑ ^{+ w(b}

^,b

⁾

j =1 m i =1

∑

m

∑

A∪ B

Writhe is conserved by anti-parallel reconnection

Theorem (Laing et al. 2015). Writhe of and is conserved under anti-parallel reconnection of and , that is

A B

A B

Wr(A∪ B) = Wr(A# B)

A∪ B (A# B) * A# B

= Wr(A) + 2Lk(A, B) +Wr(B)

Anti-parallel reconnection: from disjoint union through the theta-curve intermediate to the reconnected curve :

A∪ B

A# B

(A# B) *

Wr(A∪ B) = Wr(A) + 2Lk(A, B) +Wr(B) = Wr[(A# B)*]

(ii) oppositely oriented edges cancel out, hence

Wr[(A# B)*] = Wr(A# B)

thus

Wr(A∪ B) = Wr(A# B)

In the continuum limit from PL-curves , to smooth curves , , we have

;

; .

Wr( χ

∪ χ

^{) = Wr(} χ

) +Wr( χ

) + 2Lk( χ

^, χ

^{) = Wr(} χ

^# χ

⁾

A B χ

χ

. Helicity of flux tubes

α

β

χ

χ

H ( α ∪ β ^{) = Wr(} χ

∪ χ

) +Tw( α ∪ β ^{) = Wr(} χ

^# χ

) +Tw( α ∪ β ⁾

Φ = 1

From rigid translation and internal cancellation of common edge:

(i) translation preserves , , ; hence

Wr(A) Wr(B) Lk(A, B)

Helicity and energy changes due to intrinsic twist change

“Empty” tube

γ

Tw ( ) γ ^{= T} ( ) ^{γ + N} ( ) ^γ

=0 =0

∆H = H ( α ∪ β ) ^{− H} ( ^{α # β} ) ^{= ∆Tw}

hence, change in helicity due to change in torsion:

H ( α ∪ β ) ^{= Wr} ( ^χ

^{# χ}

) ^{+ Tw ( α # β ) + ∆Tw}

= H ( α ^# β ) ^{+ ∆Tw}

or

.

;

In time: change in twist implies change in energy ( ):

δ ^{H (t) =} δ Tw(t) ≤ 4E δ ^E(t)

Φ = 1

Since change in twist , we have:

Tw ( α ∪ β ) ^{= Tw} ( ^{α # β} ) ^{+ ∆Tw}

∆H = ∆T ∆H = 0

Quantum vortex knots under the Gross-Pitaevskii equation GPE Vortex lines as phase defects of :

ψ

→ 1 X → ∞

ψ

ρ

2

u = ∇ θ

Gross-Pitaevskii equation:

with as . Fluid dynamical interpretation of GPE:

GPE:

H = K + I = constant

Conserved hamiltonian:

K = 1

2 ∫ ∇ ψ ⋅ ^{∇ ψ}

^d

^X I = 1

4 ^∫ ( 1− ψ

)

^d

^X

Cascade process of a Hopf link of quantum vortices ( )Γ = 1

Total writhe and twist contributions to helicity ( )

Wr

H = Wr

+ Tw

Tw

Γ = 1

Wr

= Wr

+Wr

+ 2Lk

Twist analysis by ribbon construction on isophase surface

Individual writhe and twist contributions

(Zuccher & Ricca 2017)

Evolution of iso-phase (Seifert) surface of minimal area

(Zuccher & Ricca 2017 – in preparation)

Iso-phase (Seifert) surface of minimal area versus time

(Zuccher & Ricca 2017 – in preparation)

t

S

t

Phase

θ

θ

t

u = ∇ θ

0 5 10 15 20 25 30 0.5

0.6 0.7 0.8 0.9 1

t Smin/Smin0

T₂₃, R = 6, r = 2, L = 40

Evolution of iso-phase surface of minimal area of trefoil knot

t

S = S_min S_{min 0}

Evolution of iso-phase surface of minimal area of knot types

(Zuccher & Ricca 2017 – in preparation)

t

T

T

T

T

T

T

T

T

S = S_min S_{min 0}