Makoto Tsubota,Tsunehiko Araki and Akira Mitani (Osaka), Sarah Hulton (Stirling), David Samuels (Virginia Tech) INSTABILITY OF VORTEX ARRAY AND POLARIZATION OF SUPERFLUID TURBULENCE Carlo F. Barenghi

ordered vortex array L=2Ω/Γ 2. Counterflow : 1. Rotation: disordered vortex tangle L=γ² V² where V=V n -V s L = vortex line density

Rotating counterflow

Experiment by Swanson, Barenghi and Donnelly, Phys. Rev. Lett. 50, 90, (1983) -For V<V c1 : vortex array -What are the critical velocities V c1 and V c2 ? -What is the state V c1 <V<V c2 ? -What is the state V>V c2 ?

Vortex dynamics Velocity at point S(ξ,t): Self-induced velocity:

What is the first critical velocity V c1 ? Small amplitude helix of wavenumber k=2π/λ: kA<<1, ψ=kz-ωt, ξ≈z Growth rate: σ=α(kV n -βk²), Max σ at k=V n /2β, Frequency: ω=(1+α’)V n ²/4β -It is an instability of Kelvin waves Assume Then

T=1.6K, Ω=4.98 x 10ˉ ² s ˉ¹ V ns =0.08 cm/sec What happens for V>V c1 ?

t=0 t=12 s t=28 st=160 s Numerical simulation of rotating vortex array in the presence of an axial counterflow velocity

Vortex line density L vs time Ω=9.97 x 10ˉ³ sˉ¹ Ω=4.98 x 10ˉ² sˉ¹ Ω=0 After an initial transient, L saturates to a statistical steady state

Polarization vs time Ω=9.97 x 10ˉ³ sˉ¹ Ω=0 Ω=4.98 x 10ˉ² sˉ¹ Thus for V>Vc1 we have “polarized turbulence”

Analogy with paramagnetism L*=[(L H +L R )-L]/(bL H ) vs Ω*=a L R /L H, with a=11 and b=0.23 The observed L is always less than the expected L H +L R, where Vortices are aligned by the applied rotation Ω and randomised by the counterflow Vns L R =2 Ω/Г L H =γ²V ns ²

What is the second critical velocity V c2 ? T1= characteristic time of growing Kelvin waves ≈ 1/σ max =4β/(α V ns ²) T2= characteristic friction lifetime of vortex loops created by reconnections ≈ 2ρ s πR²/(γβ) where 2R≈δ, δ≈1/√L and γ=friction coeff If T2>T1 vortex loops have no time to shrink before more loops are created → randomness Thus polarized tangle is unstable if L<C V ns ² with C≈50000 which has the same order of magnitude of the finding of the experiment of Swanson et al Conclusion: probably for V>V c2 the tangle is random

Classical turbulence Fourier transform the velocity: Energy spectrum: Kolmogorov -5/3 law: Dissipation: The energy sink is viscosity, acting only for k>1/η η=small scale (Kolmogorov length) D=large scale

Turbulence in He II Experiments show similarities between classical turbulence and superfluid turbulence, for example the same Kolmogorov spectrum indipendently of temperarature..\Application Data\SSH\temp\poster Maurer and Tabeling, Europhysics Lett 43, 29 (1998) (a) T=2.3K (b) T=2.08K (c) T=1.4K

Thus BOTH normal fluid and superfluid have independent reasons to obey the classical Kolmogorov law. Can the mutual friction provide a small degree of polarization to keep the two fluids in sync on scales larger than δ (k<1/ δ) ? Yes The superfluid alone (T=0) obeys the Kolmogorov law for k<1/δ, where δ=1/ √ L is the average intervortex spacing; the sink of kinetic energy here sound rather than viscosity Araki et al, Phys Rev Lett 89, (2002)

A straight vortex (red segment in figure), initially in the plane θ=π/2, in the presence of a normal fluid eddy V n =(0,0,Ωr sinθ), moves according to dr/dt=0, dφ/dt=α’ and dθ/dt=-αΩsin(θ) Hence θ(t)=2 arctan(exp(-αΩt))→0 for t→∞ However the lifetime of the eddy is only τ ≈ 1/Ω so the segment can only turn to the angle θ(τ)= π/2-α A SIMPLE MODEL OF POLARIZATION

The normal fluid spectrum in the inertial range 1/D<k<1/η is: (D=large scale, η=Kolmogorov scale) In time 1/ω k re-ordering of existing vortices creates a net superfluid vorticity ω s ≈αLГ/3 in the direction of the vorticity ω k of the normal fluid eddy of wavenumber k. Since ω k ≈√(k³E k ), matching of ω s and ω k gives But ω k is concentrated at smallest scale (k≈1/η) so a vortex tangle of given L and intervortex spacing δ ≈1/√L can satisfy that relation only up to a certain k. Since ε¼=ν n ¾/η we have Conclusion: matching of ω k and ω s (hence coupling normal fluid and superfluid patterns) is possible for the entire inertial range !

Consider the evolution of few seeding vortex rings in the presence of an ABC (Arnold, Beltrami, Childress) normal flow of the form: Vorticity regions of driving ABC flow Resulting polarized tangle MORE NUMERICAL EVIDENCE OF POLARISATION

Results: = at various A,α

Scaled results: /α vs t/τ where τ=1/ω n and ω n is the normal fluid vorticity No matter whether the tangle grows or decays, the same polarization takes place for t/τ≤1

CONCLUSIONS -Provided that enough vortex lines are present, vorticity matching ω s ≈ω n can take place over the inertial range up to k≈1/δ, consistently with experiments -Instability of vortex lattice and new state of polarized turbulence References: Phys Rev Letters 89, 27530, (2002), Phys Rev Letters 90, 20530, (2003) Phys Rev B, submitted