IRREVERSIBLE DYNAMICS OF SUPERFLUID VORTEX RECONNECTIONS DAVIDE - PowerPoint PPT Presentation

manuscript in preparation IRREVERSIBLE DYNAMICS OF SUPERFLUID VORTEX RECONNECTIONS DAVIDE PROMENT, UNIVERSITY OF EAST ANGLIA (UK) Joint work with: Alberto Villois and Giorgio Krstulovic

RECONNECTIONS IN SUPERFLUIDS δ − ( t ) [Paoletti et al., PNAS 2008] φ Vortex reconnections in superfluid liquid helium (top) and BEC of cold atoms (bottom) δ + ( t ) t [Serafini et al., PRL 2015]

THE GROSS-PITAEVSKII MODEL [Pitaevskii & Stringari, 2003] ∂ t + ℏ 2 ı ℏ∂ ψ 2 m ∇ 2 ψ − g | ψ | 2 ψ = 0 Madelung transformation u = ℏ / m ∇ ϕ , ρ = m | ψ | 2 ψ = ρ exp( ı ϕ ) ∂ ρ ∂ t + ∇ ⋅ ( ρ u ) = 0 given bulk density ρ 0 ∇ 2 ρ mV + ℏ 2 ∂ u ∂ t + ( u ⋅ ∇ ) u = ∇ − g m ρ + 1 ℏ 2 /(2 mg ρ 0 ) ξ = 2 m 2 ρ c = g ρ 0 / m ‣ inviscid, compressible, and irrotational fluid ‣ vortices are topological defects ‣ circulation quantised κ = h / m

TWO ENERGY FAMILIES IN THE GP MODEL [Nore et al., PoF 1997] ‣ total energy is a constant of motion E = ∫ | ∇ ψ | 2 dV + 1 2 2 ∫ ( | ψ | 2 − ρ 0 ) dV E kin = 1 2 E q = 1 2 E int = 1 4 ∫ ( 4 ∫ ( 2 ∇ 2 ∫ ( ρ − ρ 0 ) 2 dV ρ v ) ρ ) dV , dV , ρ v = ( + ( i c ρ v ) ρ v ) Using Helmoltz’s decomposition 2 4 ∫ [ ( ] c kin = 1 ρ v ) density perturbations: E c dV 2 4 ∫ [ ( ] i kin = 1 ρ v ) quantised vortices: E i dV ‣ energy transfers between vortices and sound families

OUR NUMERICAL EXPERIMENTS ON RECONNECTIONS ‣ decay of two linked rings ‣ vary the offset , spanning over 49 different d configurations ‣ track vortex filaments and measure sound

<latexit sha1_base64="NAuFnSTeLNicUZ4QmEFrCuwL6QY=">ACBHicdVDLSgMxFM3UV62vqks3wSK4cjUPndFNy4r2Ae0Y8mkaRubmQxJRihDt67d6je4E7f+h5/gX5hpK2jRAxcO59zLvfd4IWdKI/RhpVZW19Y30puZre2d3b3s/kFTiUgS2iC9n2sKcBbShmea0HUqKfY/Tlje+TPzWPZWKieBGT0Lq+ngYsAEjWBup2Q1H7Pasl80hG5WcaqUEkV0o56uFoiFOvlhCDo2miEHFqj3sp/dviCRTwNOFaq46BQuzGWmhFOp5lupGiIyRgPacfQAPtUufHs2ik8MUofDoQ0FWg4U39OxNhXauJ7ptPHeqSWvUT80sULQRXSwfoQcWNWRBGmgZkvn8QcagFTBKBfSYp0XxiCaSmRcgGWGJiTa5ZUw23wHA/0kzbzvndv6kKtdLFJKgyNwDE6BA8qgBq5AHTQAXfgETyBZ+vBerFerbd5a8pazByCX7DevwBQJpkJ</latexit> ABOUT RECONNECTION: LINEAR THEORY APPROXIMATION [Nazarenko & West, JLTP 2003] ∂ t + ℏ 2 ı ℏ∂ ψ δ ± ( t ) ≤ ξ 2 m ∇ 2 ψ − g | ψ | 2 ψ = 0 ⟹ δ ± ( t ) = A ± κ | t − t r | ϕ − = 2 arctan( A + / A − ) ‣ same scaling before δ ∝ t 1/2 φ − and after, only the pre- A ± factors change ‣ filaments follow locally the branches of an hyperbola

ASYMMETRY IN RATES OF APPROACH AND SEPARATION 2.5 δ ± ( t ) = A ± κ | t − t r | 2 1.5 1 0.5 0 0 0.5 1 1.5 2 2.5 Red circles correspond to data of the present work, all other symbols are from [Villois et al., PRFluids 2018] HOW TO EXPLAIN THIS ASYMMETRY?

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

COMPRESSIBLE KINETIC ENERGY GROWTH 10 -4 Example of energy growth during a reconnection kin ∆ E C 10 -5 ∆ E C kin 10 10 -6 10 0 10 1 A + /A − Growth of the compressible HOW TO PREDICT THIS kinetic energy during the BEHAVIOUR? reconnection vs. the ratio A + / A − for the 49 different realisations

A PHENOMENOLOGICAL MATCHING THEORY ‣ when nonlinear theory using vortex filament model δ ( t ) ≥ δ lin or local induction approximation (LIA) ‣ when linear theory as described before δ ( t ) ≤ δ lin ‣ matching of the two theories at δ ( t ) = δ lin

A PHENOMENOLOGICAL MATCHING THEORY from nonlinear theory fil = κ 2 ∮ R ± × d R ± momentum: P ± Δ P fil = P + fil − P − fil ⟹ Δ E fil = E + fil − E − ∮ | d R ± | energy: E ± fil ∝ | κ | 2 fil [Pismen, 1999]

CONVERSION OF FILAMENT’S MOMENTUM INTO SOUND Δ P wav = − Δ P fil ∝ ( 0, 0, 1 + A + / A − A + / A − ) ⟹ Δ P wav,z > 0 Example of sound pulse emission propagating orthogonally to the reconnection plane ‣ propagation at almost speed of sound (dashed green lines) 1.01 ‣ some dispersion 1 ‣ reduction in the sound minimum 0.99 0.98 1 ∝ ( t − t r − z / c ) 2 0.97 -30 -20 -10 0 10 20 30

CONVERSION OF FILAMENT’S ENERGY INTO SOUND (using LIA) Δ E wav = − Δ E fil ∝ Δℒ / ℒ ‣ 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

CONCLUSIONS ‣ phenomenological matching between linear and nonlinear theory ‣ linear momentum always lost in the negative direction (orthogonal to the reconnection plane), sound pulse has positive momentum ‣ because it is A + ≥ A − energetically favourable ‣ energy radiated depends on the reconnecting angle ϕ − = 2 arctan( A + / A − )

manuscript in preparation THANKS FOR YOUR ATTENTION! Joint work with: Alberto Villois and Giorgio Krstulovic 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.