IRREVERSIBLE DYNAMICS OF SUPERFLUID VORTEX RECONNECTIONS DAVIDE - - PowerPoint PPT Presentation

IRREVERSIBLE DYNAMICS OF SUPERFLUID VORTEX RECONNECTIONS

DAVIDE PROMENT, UNIVERSITY OF EAST ANGLIA (UK)

Joint work with: Alberto Villois and Giorgio Krstulovic

manuscript in preparation

RECONNECTIONS IN SUPERFLUIDS

[Serafini et al., PRL 2015] [Paoletti et al., PNAS 2008]

δ−(t) δ+(t)

φ t

Vortex reconnections in superfluid liquid helium (top) and BEC of cold atoms (bottom)

THE GROSS-PITAEVSKII MODEL Madelung transformation

∂ρ ∂t + ∇ ⋅ (ρu) = 0 ∂u ∂t + (u ⋅ ∇)u = ∇ − g m ρ + 1 mV + ℏ2 2m2 ∇2 ρ ρ

ψ = ρ exp(ıϕ)

u = ℏ/m∇ϕ , ρ = m|ψ|2

• inviscid, compressible, and irrotational fluid
• vortices are topological defects
• circulation quantised

ıℏ∂ψ ∂t + ℏ2 2m ∇2ψ − g|ψ|2ψ = 0

given bulk density ρ0 ξ = ℏ2/(2mgρ0) c = gρ0/m

κ = h/m

[Pitaevskii & Stringari, 2003]

TWO ENERGY FAMILIES IN THE GP MODEL

Ekin = 1 4 ∫ ( ρ v)

2

dV , Eq = 1 4 ∫ (2∇ ρ)

2

dV , Eint = 1 2 ∫ (ρ − ρ0)

2 dV

E = ∫ |∇ψ|2dV + 1 2 ∫ (|ψ|2 − ρ0)

2

dV

[Nore et al., PoF 1997]

ρ v = ( ρ v)

i

+ ( ρ v)

c

density perturbations: Ec

kin = 1

4 ∫ [( ρ v)

c

]

2

dV quantised vortices: Ei

kin = 1

4 ∫ [( ρ v)

i

]

2

dV

Using Helmoltz’s decomposition

• total energy is a constant of motion
• energy transfers between vortices and sound families

OUR NUMERICAL EXPERIMENTS ON RECONNECTIONS

• decay of two linked rings
• vary the offset , spanning over 49 different

configurations

• track vortex filaments and measure sound

d

ABOUT RECONNECTION: LINEAR THEORY APPROXIMATION δ±(t) = A± κ|t − tr| ϕ− = 2 arctan(A+/A−)

[Nazarenko & West, JLTP 2003]

φ−

δ±(t) ≤ ξ ⟹ ıℏ∂ψ ∂t + ℏ2 2m ∇2ψ − g|ψ|2ψ = 0

• same scaling before

and after, only the pre- factors change

• filaments follow locally the

branches of an hyperbola δ ∝ t1/2 A±

ASYMMETRY IN RATES OF APPROACH AND SEPARATION

Red circles correspond to data of the present work, all other symbols are from [Villois et al., PRFluids 2018]

0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5

HOW TO EXPLAIN THIS ASYMMETRY?

δ±(t) = A± κ|t − tr|

A TYPICAL EVOLUTION Evolution of various measurable quantities during the reconnection, including energy components

COMPRESSIBLE KINETIC ENERGY GROWTH

∆EC

kin

10 A+/A−

100 101

∆EC

kin

10-6 10-5 10-4

Growth of the compressible kinetic energy during the reconnection vs. the ratio for the 49 different realisations A+/A−

Example of energy growth during a reconnection

HOW TO PREDICT THIS BEHAVIOUR?

A PHENOMENOLOGICAL MATCHING THEORY

• when nonlinear theory using vortex filament model
• r local induction approximation (LIA)
• when linear theory as described before
• matching of the two theories at

δ(t) ≥ δlin δ(t) ≤ δlin δ(t) = δlin

A PHENOMENOLOGICAL MATCHING THEORY momentum: P±

fil = κ

2 ∮ R± × dR± energy: E±

fil ∝ |κ|2

∮ |dR±| ⟹ ΔPfil = P+

fil − P− fil

ΔEfil = E+

fil − E− fil

[Pismen, 1999]

from nonlinear theory

CONVERSION OF FILAMENT’S MOMENTUM INTO SOUND ΔPwav = − ΔPfil ∝ (0, 0, 1 + A+/A− A+/A− ) ⟹ ΔPwav,z > 0

• 30
• 20
• 10

10 20 30 0.97 0.98 0.99 1 1.01

Example of sound pulse emission propagating orthogonally to the reconnection plane

• propagation at almost

speed of sound (dashed green lines)

• some dispersion
• reduction in the

sound minimum

∝ 1 (t − tr − z/c)2

CONVERSION OF FILAMENT’S ENERGY INTO SOUND

• a range of values are allowed, when considering that reconnecting

filaments do not lie exactly on a plane

• is a measure of the concavity in the z direction

γ ΔEwav = − ΔEfil ∝ Δℒ/ℒ (using LIA)

CONCLUSIONS

• linear momentum always lost

in the negative direction (orthogonal to the reconnection plane), sound pulse has positive momentum

• because it is

energetically favourable

• energy radiated depends on

the reconnecting angle

• phenomenological matching

between linear and nonlinear theory A+ ≥ A− ϕ− = 2 arctan(A+/A−)

THANKS FOR YOUR ATTENTION!

Acknowledgments G.K., D.P . and A.V. were supported by the cost-share Royal Society International Exchanges Scheme (IE150527) in conjunction with CNRS. A.V. and D.P . were supported by the EPSRC First Grant scheme (EP/P023770/1). D.P . acknowledges the Fédération Doeblin for the support while visiting Nice in November 2017.

Joint work with: Alberto Villois and Giorgio Krstulovic

manuscript in preparation