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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. Self-similar solutions for LIA

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cascade of vortex loops intiated by single reconnection of quantum vortices
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

slide2

Self-similar solutions for LIA

Vortex rings cascades (BS, GP)

Energy dissipation in T→0 limit

motion of a vortex filament
Motion of a vortex filament

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

slide4

Local Induction Approximation

For T>0: >0 vortex ring shrinks

slide5

Self-similar and quasi-static solutions

Lipniacki PoF 2003, JFM 2003

Quantum vortex shrinks:

Frenet Seret equations

slide7

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

slide8

Shape preserving solution: general case

Logarithmic spirals on cones

4-parametric class

slide10

Buttke, 1988

THIS SOLUTION HAS CONSTANT CURVATURE !

Limit of shape preserving solution for α→0 ?

slide11

YES

When α→0

Shape preserving solutions

„tend locally” to

Buttke solution

α=1, 0.1, 0.01, 0.001, Buttke

slide14

LIA solutions for Г<8º have self-crossings

DO THEY HAPPEN ALSO IN

BIOT-SAVART DYNAMICS ?

slide17

Biot-Savart

LIA

Crossings happen below the respective lines

slide19

Gross - Pitaevski simulations

Г=4º

Dufort-Frankel scheme (Lai et al. 2004)

slide20

Kursa, M.; Bajer, K. & Lipniacki, T.

Cascade of vortex loops initiated by a single reconnection of quantum vortices Phys. Rev. B, 2011, 83, 014515

slide22

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

slide25

Diameters of subsequent rings form

geometrical sequence

Times of subsequent ring detachments form

geometrical sequence

„Lost” line length

slide26

Total line length lost in single reconnection

„transparent tangle”

Average radius of curvature

in the tangle

Frequency of reconnections

(Barenghi & Samuels 2004)

slide27

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”

slide28

LINE LENGTH DECAY AT ZERO TEMPERATURE

Transparent tangle

Opaque tangle

μ – Fraction of reconnections leading to cascades of rings

slide29

μ

Uniform distribution of reconnection angles

Thermally driven

Mechanically driven

Baggaley,Shervin,Barenghi,Sergeev 2012

Waele, Aartz, 1994, μ=0

slide30

a

Feynman's cascade, 1955

Svistunov, 1995 …

reconnections

kelvons

dissipation

Line dissipation decreases like

Loop cascade generation

Line length dissipation decreases like

Efficient provided that μ is large enough