1. Cascade of vortex loops intiated by single reconnection of quantum vortices Miron Kursa 1 Konrad Bajer 1 Tomasz Lipniacki 2 1 University of Warsaw 2 Polish Academy of Sciences, Institute of Fundamental Technological Research

2. 1.Self-similar solutions for LIA 2.Vortex rings cascades (BS, GP) 3.Energy dissipation in T→0 limit

3. 3 Motion of a vortex filament : non-dimensional friction parameter, vanishes at T=0

4. 4 Local Induction Approximation For T>0: >0 vortex ring shrinks

5. Self-similar and quasi-static solutions Lipniacki PoF 2003, JFM 2003 Quantum vortex shrinks: Frenet Seret equations

6. Shape-preserving (self-similar) solutions

7. The simplest shape-preserving solution (2003) In the case when transformation is a pure homothety we get analytic solution in implicit form: Self-crossings for Г<8º and sufficietly small α/β

8. Shape preserving solution: general case Logarithmic spirals on cones 4-parametric class

9. Wing tip vortices

10. 10 Buttke, 1988 THIS SOLUTION HAS CONSTANT CURVATURE ! Limit of shape preserving solution for α→0 ?

11. 11 When α→0 Shape preserving solutions „tend locally” to Buttke solution α=1, 0.1, 0.01, 0.001, Buttke YES

12. 12 Does LIA time-dependent dynamics tend to those similarity solutions ? Yes

13. 13 Does LIA time-dependent dynamics tend to those similarity solutions ? Yes

14. 14 LIA solutions for Г<8º have self-crossings DO THEY HAPPEN ALSO IN BIOT-SAVART DYNAMICS ?

15. 15 Biot-Savart Simulation

16. 16 Biot-Savart simulations

17. 17 Biot-Savart simulations

18. 18 Biot-Savart LIA Crossings happen below the respective lines

19. 19 Gross - Pitaevski equation vortex

20. 20 Gross - Pitaevski simulations Г=4º Dufort-Frankel scheme (Lai et al. 2004)

21. 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

22. Kerr, PRL 2011 Rings generation from reconnections of antiparallel vortices

23. Quasi-static solution, 2003 In the case when transformation is a pure translation we get analytic solution: where Self-crossings for α/β <0.45, Number of S-C tends to infinity as α/β tends to zero

24. Vortex loops cascades as a potential mechanism of energy dissipation?

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

26. 26 Diameters of subsequent rings form geometrical sequence Times of subsequent ring detachments form geometrical sequence „Lost” line length

27. 27 Average radius of curvature in the tangle ( Barenghi & Samuels 2004) Frequency of reconnections Total line length lost in single reconnection „transparent tangle”

28. 28 Mean free path of a ring of diameter in the tangle of line density „OPAQUE TANGLE” Total line length lost in single reconnection „opaque tangle”

29. 29 LINE LENGTH DECAY AT ZERO TEMPERATURE Transparent tangle Opaque tangle μ – Fraction of reconnections leading to cascades of rings

30. Waele, Aartz, 1994, μ=0 Uniform distribution of reconnection angles μ Thermally driven Mechanically driven Baggaley,Shervin,Barenghi,Sergeev 2012

31. 31 a Feynman's cascade, 1955 reconnections kelvons dissipation Line dissipation decreases like Loop cascade generation Line length dissipation decreases like Svistunov, 1995 … Efficient provided that μ is large enough