INSTABILITY OF VORTEX ARRAY AND POLARIZATION OF SUPERFLUID

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

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

Rotating counterflow

Experiment by Swanson, Barenghi and Donnelly, Phys. Rev. Lett. 50, 90, (1983) -For V<Vc 1: vortex array -What are the critical velocities Vc 1 and Vc 2 ? -What is the state Vc 1<V<Vc 2 ? -What is the state V>Vc 2 ?

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

What is the first critical velocity Vc 1 ? -It is an instability of Kelvin waves Small amplitude helix of wavenumber k=2π/λ: k. A<<1, ψ=kz-ωt, ξ≈z Assume Then Growth rate: σ=α(k. Vn-βk²), Max σ at k=Vn/2β, Frequency: ω=(1+α’)Vn²/4β

What happens for V>Vc 1 ? T=1. 6 K, Ω=4. 98 x 10ˉ² s ˉ¹ Vns=0. 08 cm/sec

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

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 <s’z> vs time Ω=9. 97 x 10ˉ³ sˉ¹ Ω=4. 98 x 10ˉ² sˉ¹ Ω=0 Thus for V>Vc 1 we have “polarized turbulence”

Analogy with paramagnetism Vortices are aligned by the applied rotation Ω and randomised by the counterflow Vns The observed L is always less than the expected LH+LR, where LR=2Ω/Г LH=γ²Vns² L*=[(LH+LR)-L]/(b. LH) vs Ω*=a LR/LH, with a=11 and b=0. 23

What is the second critical velocity Vc 2 ? T 1= characteristic time of growing Kelvin waves ≈ 1/σmax=4β/(α Vns²) T 2= characteristic friction lifetime of vortex loops created by reconnections ≈ 2ρsπR²/(γβ) where 2 R≈δ, δ≈1/√L and γ=friction coeff If T 2>T 1 vortex loops have no time to shrink before more loops are created → randomness Thus polarized tangle is unstable if L<C Vns² with C≈50000 which has the same order of magnitude of the finding of the experiment of Swanson et al Conclusion: probably for V>Vc 2 the tangle is random

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

Turbulence in He II Experiments show similarities between classical turbulence and superfluid turbulence, for example the same Kolmogorov spectrum indipendently of temperarature Maurer and Tabeling, Europhysics Lett 43, 29 (1998) (a) T=2. 3 K (b) T=2. 08 K (c) T=1. 4 K . .

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, 145301 (2002) 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

A SIMPLE MODEL OF POLARIZATION A straight vortex (red segment in figure), initially in the plane θ=π/2, in the presence of a normal fluid eddy Vn=(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

The normal fluid spectrum in the inertial range 1/D<k<1/η is: (D=large scale, η=Kolmogorov scale) In time 1/ωk re-ordering of existing vortices creates a net superfluid vorticity ωs≈αLГ/3 in the direction of the vorticity ωk of the normal fluid eddy of wavenumber k. Since ωk≈√(k³Ek), matching of ωs and ωk gives But ωk 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=νn¾/η we have Conclusion: matching of ωk and ωs (hence coupling normal fluid and superfluid patterns) is possible for the entire inertial range !

MORE NUMERICAL EVIDENCE OF POLARISATION 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

Results: <cos(θ)>=<s’z> at various A, α

Scaled results: <cos(θ)>/α vs t/τ where τ=1/ωn and ωn 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 ωs≈ωn 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