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P. M. Walmsley, A. A. Levchenko, S. May, S. L. Chan, B. White, A. I. Golov

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Ion source

R ~ 1 mm

X=Y=Z = 4.5 cm

Vortex dynamics at steady rotation and upon spin-up and spin-down in 4He at T = 0

P. M. Walmsley, A. A. Levchenko, S. May,

S. L. Chan, B. White, A. I. Golov

- 0. Introduction, ion technique
- 1. Vortex dynamics after starting and stopping rotation
- 2. Vortex dynamics at steady rotation
- 3. Dynamics of discharging of rectilinear vortices
- 4. Turbulence in rotation

T ~ 0.1 K

4He, p = 0

Vortex dynamics at T=0 – why and how?

Two limiting cases: regular vortex array or irregular vortex tangle

Vortex line excitations – Kelvin waves

With normal component, vortex dynamics is governed by their interaction with it

(via mutual friction and viscosity of normal component)

Dynamics can be overdamped (3He at T > 0.5 Tc) or underdamped (4He, 3He at T < 0.5 Tc)

- What if there is NO normal component:
- - Dynamics of rectilinear vortices in an array?
- New mechanisms of turbulence evolution/decay? (reconnections-driven, Kelvin waves)
- New statistical properties of turbulence? (non-Kolmogorov spectrum)
- Tractable discrete turbulence? (discrete sibling of continuous classical turbulence)

Vortex dynamics at T=0 – brief history

Guenin and Hess (1972): vortex tangle decay in 4He at 0.4 KMcClintock et al. (late 70s): turbulence injected by ions decayingScwartz (1985): mechanisms of relaxation in T=0 limit? Friction at walls?Svistunov (1995), Vinen (~2000): Kelvin wave cascade and reconnectionsTsubota, Araki, Nemirovskii (2000): numerical simulations of vortex tangle at T=0New experimental era:McClintock et al. (2000): vibrating grid in 4He, turbulence probed by free ionsBradley et al. (2000) vibrating wire shedding off vortices in 3He-B, then vibrating grid …

S.I.Davis, P.C.Hendry, P.V.E.McClintock, H.Nichol, in “Quantized Vortex Dynamics and Superfluid Turbulence”,

ed. C.F.Barenghi, R.J.Donnelly and W.F.Vinen, Springer (2001).

Ions in liquid helium

An excess electron creates a bubble of radius ~ 20 AIons are attracted to vortex lines (binding energy ~ 50 K) However, capture diameter very small (<100 A at T<1K)

Ostermeier and Glaberson (1974);

Vortex rings are nucleated by such ions at T < 1 K Initial rings: KE0 = 34 eV,R0 = 0.8 mm, v = 10.6 cm/s After gaining another 90eV, KEf = 124 eV, Rf= 2.7 mmTypically: <R> = 2 mm, <v> = 5 cm/s

Such rings have capture diameter s = 1.6 R = 2 - 3 mm(Schwarz and Donnelly, 1966)c.f. typical inter-vortex spacing of ~ 100 - 1000 mm

E

Scwartz and Donnelly on vortex rings:

Back in 1966, Schwarz and Donnelly wrote: “quantized vortex rings are very sensitive vortex-line detectors, making them suitable probes for a number of problems in quantum hydrodynamics.”

We have measured the attenuation of pulses of charged vortex rings due to their interaction with rectilinear array of vortices produced by rotating at an angular velocity, .

Trapped charge moves along vortices and leaves the cell

Diverging field helps remove trapped ions

Pulse amplitude as a function of . The measured trapping diameter is comparable to the diameter of vortex rings.

vortex lines

Starting rotation: spin-up

(from Landau state to vortex array)

Stopping rotation: spin-down

(from vortex array to Landau state)

Experiments at T > 1K: Hall-Vinen, Reppy-Lane, Campbell-Krasnov, Tsakadze:- satisfactory description in terms of mutual friction and surface friction - role of turbulence acknowledged but not quantifiedExpectations for T = 0: relaxation should: - slow down as mutual friction decreases - accelerate if turbulent burst occurs

- Tractable cases:
- Regular (overdamped) vortex dynamics (hardly in 4He): t ~ a-1
- - No mutual friction but no seed vortices either (vc too high): Landau state forever

- With turbulence at T=0, we might expect three processes to matter:
- Vortex multiplication / turbalisation
- Turbulent front moves in
- Turbulence decays
- - 4He (lots of pinned remnant vortices)
- in an awkward cell (lots of surface friction and stirring up)
- at T < 0.3 K (no mutual friction at all).
- Hence we might expect fast multiplication of vorticity followed by a decay specific to T = 0.

Vortex relaxation from HVBK (T>1 K)(conservative estimate)

0.01 t0 = 500 s

Idowu, Henderson, Samuels (2000)

1. Speculations on what follows after turbalisation

Kinetic energy of solid body rotation: KE = (1/4)rVR2W2

Vortex energy per unit lengthg = (rk2/4p) ln(l/a), whereln(l/a) ~ 12

Hence,in Vinen state, LVinen= KE / (gpR2) = pR2W2 / (12 k2)

After comparing to Leq = 2W/k : LVinen/Leq = pR2W / 24k ~ 3000.

However, if there is a Kolmogorov cascade E(k) = C e2/3k-5/3

between k1 = 2p/2R and k2 = 2p/2l,

its energy ~ 0.15k2L4/3(2R)2/3 (using C=1.62, e =nw2=nk2L2, n=k/10).

ThenLKolm/Leq~ 70 .

at W = 1 rad/s

1. Developed tangle decays fast!

Walmsley et al. (talk on Thursday)

[4He, T ~ 0.1 K]

D.I.Bradley et al., PRL 96, 035301 (2006)

[3He-B, T ~ 0.15 Tc]

1. Compare with BEC: Spin Up & Down

Formation and Decay of Vortex Lattices in Bose-Einstein Condensates at Finite Temperatures

Abo-Shaeer, Raman, Ketterle, PRL 88, 070409 (2002)

The dynamics of vortex lattices in stirred Bose-Einstein condensates have been studied at finite temperatures.

The decay of the vortex lattice was observed nondestructively by monitoring the centrifugal distortions of the rotating condensate. The formation of the vortex lattice could be deduced from the increasing contrast of the vortex cores observed in ballistic expansion. In contrast to the decay, the

formation of the vortex lattice is insensitive to temperature change.

Decay (“Spin-down”)

Formation (“Spin-up”)

150 ms

300 ms

50 ms

2. Vortex Diffusion at T=0, W = const

Questions and expectations:It is known that vortices migrate and thus move trapped charge with them (Packard et al., mid-70s) One known reason - mechanical vibrations. Can we stabilize the equilibrium (vortex array) state? Or study the dynamics of rectilinear vortices?We can exert controllable force on individual vortices (not the usual Magnus force!)

2. Vortex (charge) Diffusion, W = const

(1) charging

(2) waiting

(3) collecting

Re-orient the field to the vertical direction and collect the surviving charge at the top collector.

Stop the injection of charge and then wait a particular time t in a horizontal field Eh.

Prepare a density of charge while preventing it escaping in the vertical direction.

2. Vortex Diffusion, W = const

In the high-Eh regime, the diffusion time is proportional to

Q ~ (t / )-1.2

Benefit of having large (4.5 cm) cell: a truly macroscopic system.

at Ω = 3 rad/s, total 105 lines;

at Ω = 0.075 rad/s, total 3x103 lines.

2. Why do vortices migrate at all?

The equilibrium (a regular lattice of straight lines) might be hard to achieve due to:

- long relaxation time (2-4 hours: Awschalom & Schwatz 1983, DeConde et al. 1974) ;

- agitation (E-field, vibrations, current injection);

- surface pinning.

Hence, probably a disordered bunch of kinky vortex lines

Remnant turbulence (more in section 4):

High Ω = 3 rad/s - “weakly upset vortex array”

Low Ω = 0.075 rad/s - “weakly polarised vortex tangle”

3. Ion motion along vortex lines – expectations:

Trapped ions can slide along the vortex lines, the mobility is reduced due to the interaction with Kelvin waves.

Donnelly, Glaberson, Parks (1967), Ostermeier and Glaberson (1976)

At T < 0.5 K, limiting velocity ~ 10 m/s; ions travel 5 cm in ~ 5 ms.

The transients of the charge arriving at the collector after the vertical field is turned on

Two arrival times: one shorter than 0.1 s and another around 0.3 s.

Trapped ions can slide along the vortex lines, the mobility is reduced due to the interaction with Kelvin waves.

At T < 0.5 K, limiting velocity ~ 10 m/s; ions travel 5 cm in ~ 5 ms.

(E = 20 V/cm, R = 2.25 cm)

Ions on vortex rings:

- Minimal time for smallest rings: 4peER2/rk3(h-1)2 = 0.3 s

- Typical time for developed rings: 8peR/rk3(h-1)2 = 1.3 s

(of energy e = 300 eV)

3. Tail = turbulent tangle (W = 0)

The zoo

- Fast train: absent at slow rotation, dominates at high W,
- correlated with redirecting E-field, very fast (v > 40 cm/s).
- A: Ions sliding along vortices.
- 2. Slow train: needs rotation but vanishes at highest W,
- correlated with redirecting E-field, arrives after 0.30 s (v = 7 cm/s).
- A: Freshly made singly-charged rings (Hashimoto solitons)?
- Q-A: Mechanism: reconnection of a Kelvin wave after a shake?
- Q: Why suppressed by remnant turbulence
- (at low W and waiting times < 10 s)?
- 3. Tail: exists even at W = 0, correlated with turning source off.
- A: Remnant turbulence.

4. Tangle + rotation = polarised turbulence?

Previous research at T > 1K:

Rotating 4He in co-axial counterflow; two critical velocities

(Swanson, Barenghi and Donnelly,1983):

First critical velocity - “Glaberson instability” = Kelvin wave excitation

Second threshold – Kelvin waves overlap and entangle = polarised tangle?

(Tsubota, Araki, Barenghi, 2003)

4. Tangle + rotation = polarised turbulence?

Low Ω - “weakly polarised vortex tangle”?

High Ω - “weakly upset vortex array”?

1 s

20 s

1 s

60 s

- We have used negatively charged vortex rings to detect and manipulate vortices in superfluid 4He in the T = 0 limit.
- Success – one can detect vortices by ring-ions down to 30 mK
- The dynamics of vortex motion upon starting and stopping rotation have been probed. The characteristic timescale for flow relaxation was ~ 200 - 400 s.
- The drift of vortices under normal force at steady rotation has a threshold. Above the threshold, charge decays as Q ~ ( / t)1.2
- Along rectilinear vortices, trapped charge arrives by two distinct trains
- Turbulence can nicely co-exist with rotation

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