Cascade of vortex loops intiated by single reconnection of quantum vortices

Miron Kursa1

Konrad Bajer1

Tomasz Lipniacki2

1University of Warsaw

2Polish Academy of Sciences, Institute of Fundamental Technological Research

1. Self-similar solutions for LIA

2. Vortex rings cascades (BS, GP)

3. Energy dissipation in T→0 limit

3

Motion of a vortex filament

Biot-Savartlaw

: non-dimensional friction parameter, vanishes at T=0

4

Local Induction Approximation

For T>0: >0 vortex ring shrinks

Self-similar and quasi-static solutions Lipniacki PoF 2003, JFM 2003

Quantum vortex shrinks:

0

2dc

)(),( nbssss ct

0

2222

0

232

22

2

2

dccc

ccc

c

cc

t

dccccccct

c

nbbtnnt , , ccFrenet Seret equations

Shape-preserving (self-similar) solutions

t

ttt Ss )(),(

. ,1

,1

, ,2

tl

tTtt

Kt

c

tt

22

2

2

22

0

2222

0

232

TlTldKTTK

K

TKTKK

K

KTK

KlKldKKKKTKTKTK

l

l

The simplest shape-preserving solution (2003) In the case when transformation is a pure homothety we get analytic solution in implicit form:

K

K

KKpKKK

KdKl

K

K

T ,

)()/ln()(

0

2200

22

Self-crossings for Г<8º

and sufficietly small α/β

Shape preserving solution: general case

Logarithmic spirals on cones

4-parametric class

Wing tip vortices

10

c(»;t)=c0pt

¿(»;t)=»2t

Buttke, 1988

THIS SOLUTION HAS CONSTANT CURVATURE !

K

K

KKpKKK

KdKl

K

K

T ,

)()/ln()(

0

2200

22

Limit of shape preserving solution for α→0 ?

11

RootM ean SquareDistance(RM SD)asa function of®

W hen ® ¡! 0

When α→0

Shape preserving solutions

„tend locally” to

Buttke solution

α=1, 0.1, 0.01, 0.001, Buttke

YES

12

Does LIA time-dependent dynamics tend to those similarity solutions ?

Yes

® = 0; ¡ = 57±

® = 0; ¡ = 57± :

13

Does LIA time-dependent dynamics tend to those similarity solutions ?

Yes

® = 0:001; ¡ = 1:4± :

14

Buttke,JCP,1988

·0 = 1:3

¡ ¼ 5±

® = 0; ®0= 0

LIA solutions for Г<8º have self-crossings

DO THEY HAPPEN ALSO INBIOT-SAVART DYNAMICS ?

16

Biot-Savart simulations

17

Biot-Savart simulations

18

Biot-Savart

LIA

Crossings happen below the respective lines

19

Gross - Pitaevski equation

ª 0 = f(r)ei© : f(r)¡¡¡!r! 1

1 :.

vortex

ª

· =Hdl¢v = 2¼~=m :

.

20

Gross - Pitaevski simulations

¡ = 4± :

Г=4º Dufort-Frankel scheme (Lai et al. 2004)

21

Kursa, M.; Bajer, K. & Lipniacki, T. Cascade of vortex loops initiated by a single reconnection of quantum vortices Phys. Rev. B, 2011, 83, 014515

Kerr, PRL 2011

Rings generation from reconnections of antiparallel vortices

Quasi-static solution, 2003 In the case when transformation is a pure translation we get analyticsolution:

22

0

22

0

0

,

)tanh( ),sech(

cB

cA

ABAcc

AAtcdRdRt /))ln(cosh(,)(qsin,)cos(q),( 220

s

where 220q ),sech( AcAR

Self-crossings for α/β <0.45, Number of S-C tends to infinity as α/β tends to zero

)0,()()(),( sWs ttt

Vortex loops cascades as a potential mechanism of

energy dissipation?

Evaporation of a packet of quantized vorticity, Barenghi, Samuels, 2002

26

Diameters of subsequent rings form

geometrical sequence

Times of subsequent ring detachments form

geometrical sequence

„Lost” line length

27

°B S

Average radius of curvature in the tangle

(Barenghi & Samuels 2004)

Frequency of reconnections

Total line length lost in single reconnection

„transparent tangle”

28

Mean free path of a ring of diameter in the tangle of line density

„OPAQUE TANGLE”

d

L

Total line length lost in single reconnection

„opaque tangle”

29

LINE LENGTH DECAY AT ZERO TEMPERATURE

Transparent tangle

Opaque tangle

μ – Fraction of reconnections leading to cascades of rings®

Waele, Aartz, 1994, μ=0

Uniform distribution of reconnection angles

μ310

2

cos1

BS

Thermally driven Mechanically driven

Baggaley,Shervin,Barenghi,Sergeev 2012

31

a

Feynman's cascade, 1955

reconnections kelvons dissipation

Line dissipation decreases like

Loop cascade generation Line length dissipation decreases like

t¡ 2

Svistunov, 1995 …

Efficient provided that μ is large enough