Cascade of vortex loops intiated by single reconnection of quantum vortices
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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
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1. Self-similar solutions for LIA
2. Vortex rings cascades (BS, GP)
3. Energy dissipation in T→0 limit
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Motion of a vortex filament
Biot-Savartlaw
: non-dimensional friction parameter, vanishes at T=0
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Local Induction Approximation
For T>0: >0 vortex ring shrinks
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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
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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
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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
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Shape preserving solution: general case
Logarithmic spirals on cones
4-parametric class
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Wing tip vortices
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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 ?
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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
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Does LIA time-dependent dynamics tend to those similarity solutions ?
Yes
® = 0; ¡ = 57±
® = 0; ¡ = 57± :
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Does LIA time-dependent dynamics tend to those similarity solutions ?
Yes
® = 0:001; ¡ = 1:4± :
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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 ?
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Biot-Savart simulations
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Biot-Savart simulations
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Biot-Savart
LIA
Crossings happen below the respective lines
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Gross - Pitaevski equation
ª 0 = f(r)ei© : f(r)¡¡¡!r! 1
1 :.
vortex
ª
· =Hdl¢v = 2¼~=m :
.
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Gross - Pitaevski simulations
¡ = 4± :
Г=4º Dufort-Frankel scheme (Lai et al. 2004)
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Kursa, M.; Bajer, K. & Lipniacki, T. Cascade of vortex loops initiated by a single reconnection of quantum vortices Phys. Rev. B, 2011, 83, 014515
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Kerr, PRL 2011
Rings generation from reconnections of antiparallel vortices
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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
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Vortex loops cascades as a potential mechanism of
energy dissipation?
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Evaporation of a packet of quantized vorticity, Barenghi, Samuels, 2002
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Diameters of subsequent rings form
geometrical sequence
Times of subsequent ring detachments form
geometrical sequence
„Lost” line length
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°B S
Average radius of curvature in the tangle
(Barenghi & Samuels 2004)
Frequency of reconnections
Total line length lost in single reconnection
„transparent tangle”
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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”
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LINE LENGTH DECAY AT ZERO TEMPERATURE
Transparent tangle
Opaque tangle
μ – Fraction of reconnections leading to cascades of rings®
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Waele, Aartz, 1994, μ=0
Uniform distribution of reconnection angles
μ310
2
cos1
BS
Thermally driven Mechanically driven
Baggaley,Shervin,Barenghi,Sergeev 2012
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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