Cascade of vortex loops intiated by single
reconnection of quantum vortices
Miron Kursa1
Konrad Bajer1
Tomasz Lipniacki2
1University
2Polish
of Warsaw
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
Motion of a vortex filament
: non-dimensional friction parameter, vanishes at T=0
3
Local Induction Approximation
For T>0:
>0
vortex ring shrinks
4
Self-similar and quasi-static solutions
Lipniacki PoF 2003, JFM 2003
s(, t )  s  s  s  c(b  n)
Quantum vortex shrinks:

    c 2 d 
0

c
   2c  c    c  c 2  c 3    c  c 2 d
t
0




 c  c

c 
2c  c  

2
2


   
    

2

c



c
d


t
c
2
c




0


2
2
Frenet Seret equations
t  cn, n  ct  b, b  n
Shape-preserving (self-similar) solutions
t  2t ,
   ,
1  
c
K  ,
t  t
1  
  T  ,
t  t
l

t
.
  
s(, t )  t (t ) S 
 t
K  lK 
  2 K T  KT    K   KT  K  K  K dl 
2
0

2
3

l
2
l


 K   KT 2 K 2 
 2 K T  KT 
T  lT 
2
2
   
 

  2TK   T  K dl 
K
2 
K
2




0
The simplest shape-preserving solution (2003)
In the case when transformation is a pure homothety we get analytic
solution in implicit form:
 
l(K )  

2
2 K
K
K0
dK
ln(K0 / K )  p ( K  K )
2
0
2
,
K 
T
K
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
Buttke, 1988
THIS SOLUTION HAS CONSTANT CURVATURE !
Limit of shape preserving solution for α→0 ?
2   2
dK
l(K )  
,

2
2
 K K ln(K0 / K )  p ( K0  K )
K
0
K 
T
K
10
YES
When α→0
Shape preserving solutions
„tend locally” to
Buttke solution
α=1, 0.1, 0.01, 0.001, Buttke
11
Does LIA time-dependent dynamics tend to those similarity solutions ?
Yes
12
Does LIA time-dependent dynamics tend to those similarity solutions ?
Yes
13
LIA solutions for Г<8º have
self-crossings
DO THEY HAPPEN ALSO IN
BIOT-SAVART DYNAMICS ?
14
Biot-Savart simulations
16
Biot-Savart simulations
17
Biot-Savart
LIA
Crossings happen below the respective lines
18
Gross - Pitaevski equation
vortex
19
Gross - Pitaevski simulations
Г=4º
Dufort-Frankel scheme (Lai et al. 2004)
20
Kursa, M.; Bajer, K. & Lipniacki, T.
Cascade of vortex loops initiated by a single reconnection of
quantum vortices Phys. Rev. B, 2011, 83, 014515
21
Rings generation from reconnections of antiparallel vortices
Kerr, PRL 2011
Quasi-static solution, 2003
s(, t )  W(t )  (t ) s(,0)
In the case when transformation is a pure translation we get analytic
solution:
c  c0sech(A ),
A
 c0
 
2
2
,
  Btanh(A )
B
 c0
2  2



2
2
s( , t )    Rcos(q ) d ,  R sin (q ) d , tc0     ln(cosh(A )) / A 

 

where R  sech( A ),
q  c02  A2
Self-crossings for α/β <0.45,
Number of S-C tends to infinity as α/β tends to zero
Vortex loops cascades
as a potential mechanism of
energy dissipation?
Evaporation of a packet of quantized vorticity, Barenghi, Samuels, 2002
Diameters of subsequent
rings form
geometrical sequence
Times of subsequent ring
detachments form
geometrical sequence
„Lost” line length
26
Total line length lost in single
reconnection
„transparent tangle”
Average radius of curvature
in the tangle
Frequency of reconnections
(Barenghi
& Samuels 2004)
27
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”
28
LINE LENGTH DECAY AT ZERO TEMPERATURE
Transparent tangle
Opaque tangle
μ – Fraction of reconnections leading to cascades of rings
29
μ
1  cos  BS
Uniform distribution of reconnection angles  
 10 3
2
Waele, Aartz, 1994, μ=0
Thermally driven
Mechanically driven
Baggaley,Shervin,Barenghi,Sergeev
2012
Feynman's cascade, 1955
a
Svistunov, 1995 …
reconnections
kelvons
dissipation
Line dissipation decreases like
Loop cascade generation
Line length dissipation decreases like
Efficient provided that μ is large enough
31