The Tale of Two Tangles: Dynamics of "Kolmogorov" and "Vinen"

turbulences in 4He near T=0

Paul Walmsley, Steve May, Alexander Levchenko, Andrei Golov

(thanks: Henry Hall, JOE VINEN)

Euromech 491, Exeter 2007

1. Different types of tangles and their dissipation at T=0 2. Production of random and structured tangles3. Detection of turbulence by ballistic vortex rings4. Results for both types of tangles5. Conclusions

In classical turbulence dissipation is via vorticity and viscosity :

In superfluid, turbulence is made of quantized vortices and their tangles of density L

Superfluid 4He has zero molecular viscosity, = 0

Conversion of flow energy into heat is mediated by quantized vortices:(’ – “effective kinematic viscosity”)

dE/dt = -’(L)2

dE/dt = - 2

Introduction

Circulation quantum, = h/m = 10-3 cm2s-1

Core a0 ~ 0.1 nm

L = 10 – 105 cm-2

l = L-1/2 = 0.03 – 3 mm

Random (“Vinen”) vs. structured (“Kolmogorov”) state

No correlations between vortices, hence only one length scale, l = L-1/2

All energy in vortex line tension

Vinen’s equation: dL/dt = - L2

Free decay: E(t) ~ t -1

L(t) = 1.2 ’-1t -1

Expectations: ’ ~

Eddies of different sizes >> l

Most of energy in the largest eddy

If the largest eddy saturates at d and decays within turnover time:

Free decay: E(t) ~ t -2

L(t) = (1.5/)d ’-1/2 t -3/2

Expectations: T > 1K, ’ ~ T = 0, ’ - ?

Dissipation: -dE/dt = ’(L)2

Simulations by Tsubota, Araki, Nemirovskii (PRB 2000)

T = 1.6 K T = 0

dphononemission

kl = L-1/2Quasi-classical Quantum

Kolmogorov Kelvin waves(Svistunov PRB 1995)

0.03 mm - 3 mm4.5 cm ~ 40 nm

T = 1.6 K

Bottleneck? (L’vov, Nazarenko, Rudenko PRB 2007)

T = 0

Possible scenarios in 4He at T=0

The nature of the transfer of energy from Kolmogorov to Kelvin cascade is debated:

- Accumulation of energy/vorticity at scale ~ l (L’vov, Nazarenko, Rudenko, PRB2007): ’(Vinen) / ’(Kolmogorov) ~ (ln(l/a))5 ~ 106

- Reconnections should ease the problem (Kozik-Svistunov, cond-mat 2007):’(Vinen) / ’(Kolmogorov) ~ ln(l/a) ~ 15

Kolmogorov cascade Kelvin-wave cascade

’(Vinen) ~ ’(Kolmogorov) - ?

From Kolmogorov to Kelvin-wave cascade (Kozik & Svistunov, 2007)

SII ~ vv crossover to QT

reconnections of vortex bundles

reconnections between neighbors

in the bundle

self – reconnections

(vortex ring generation)

purely non-linear cascade of Kelvin waves

(no reconnections)

length scale

phonon radiation

Random (“Vinen”) vs. structured (“Kolmogorov”) state

No correlations between vortices, hence only one length scale, l = L-1/2

All energy in vortex line tension

Vinen’s equation: dL/dt = - L2

Free decay: E(t) ~ t -1

L(t) = 1.2 ’-1t -1

Expectations: ’ ~

Eddies of different sizes >> l

Most of energy in the largest eddy

If the largest eddy saturates at d and decays within turnover time:

Free decay: E(t) ~ t -2

L(t) = (1.5/)d ’-1/2 t -3/2

Expectations: T > 1K, ’ ~ T = 0, ’ - ?

Dissipation: -dE/dt = ’(L)2

Available information

Towed grid in 4He(Oregon):Vibrating grid in 3He-B

(Lancaster): 0.0 0.5 1.0 1.5 2.00.01

0.1

1

n/

nn/

Towed Grid(Stalp et al. 2002)

(t-3/2)Vinen & Niemela

(t-1)

Tsubota et al. (t-1)(2000-2003)

' /

T (K)

Schwarz (counterflow turb.)

3He-B vibrating grid(Bradley et al., 2006):

?

Experimental challenges:

- How to produce turbulence at T < 1K?

- How to detect it?

Vortex rings are nucleated by such ions at T < 1 K; electron stays trapped by vortex (binding energy ~ 50 K)

Ring dynamics: E ~ R , v ~ 1/R

Rings as injected: E0 = 30 eV, R0 = 0.8 m, v = 11 cm/s

In liquid helium, an injected electron creates a bubble of radius ~ 20 A

Charged rings have large capture diameter ~ 1m (c.f. typical inter-vortex distance of ~ 30 - 3000 m)

E

Ions in helium

Turbulence detection: We developed techniques to measure L by scattering a beam of probe particles:

1. Free ions (T > 0.8 K), trapping diameter ~ 0.1 m

2. Charged quantized vortex rings (T < 0.8 K), trapping diameter ~1 m

Rotating cryostat was used to calibrate trapping diameter vs. electric field and temperature

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

Am

plit

ude

of P

ulse

(p

A)

(rad/s)

T=140 mK, P=0Pulse length=0.1 sDiverging field:

10 V/cm 20 V/cmI=I

0exp(-2D/)

Cross-sections :10 V/cm: 2.14 ± 0.17 m20 V/cm: 3.45 ± 0.21 m

Turbulence production:

We developed techniques to produce either structured or random tangles:

1. Impulsive spin-down to rest (works at any temperatures)Energy injected at the largest scale (structured tangle)

2. Jet of free ions in stationary helium (T > 0.8 K)Energy injected at the largest scale (structured tangle)

3. Beam of small vortex rings in stationary helium (T< 0.8 K)Energy mainly injected on scale << l (random tangle)

d phononsk

l = L-1/2Quasi-classical Quantum

Kolmogorov Kelvin waves0.03 - 3 mm4.5 cm ~ 40 nm

4.5 cm

Experimental Cell

We can inject rings from the side

We can also inject rings from the bottom

We can create an array of vortices by rotating the

cryostat

The experiment is a cube with sides of length 4.5 cm containing 4He (P = 0.1 bar).

1. Random Tangles Produced by Charged Vortex Rings

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.2

0.4

0.6

0.8

1.0

Cu

rre

nt (

pA

)

Time (s)

Pulse Spacing (s): 5 10 16 25 50 150 500

4.5 cm

1 10 100 1000100

101

102

103

V

ortex

Lin

e D

ensi

ty, L (cm

-2)

Time between pulses, t (s)

Vertical Pulses:20 V/cm field

0.1 s, -160 V 0.3 s, -160 V 0.3 s, -125 V

10 V/cm field 0.3 s, -160 V

L = 2(t)-1

1 10 100 1000100

101

102

103

Horizontal Pulses:20 V/cm field

0.1 s, -400 V 0.3 s, -400 V

V

ortex

Lin

e D

ensi

ty, L (cm

-2)

Time between pulses, t (s)

Vertical Pulses:20 V/cm field

0.1 s, -160 V 0.3 s, -160 V 0.3 s, -125 V

10 V/cm field 0.3 s, -160 V

’ver = 0.17

’hor = 0.13

0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

Cur

rent

(pA

)

Time after stopping main injection, t (s)

Tangle decay

We probe the decay after a long injection by sending a short pulse a time, t, after stopping injection.

Signal applied to injector:

50 s initial injection t

Probe pulse

1 10 100 1000101

102

103

Vor

tex

Line

Den

sity

(cm

-2)

t (s)

Decay of tangle generated by short pulses

Decay of tangle generated by long pulse

t-1

Tangle decay

Tangle Growth & Decay in Centre of Cell

1 10102

103

Vor

tex

Line

Den

sity

, L

(cm

-2)

Time after injection from left, t (s)

L=(0.08 t)-1

We can probe the growth of the tangle by first sending a pulse from the left tip and then use a pulse from the bottom tip to probe the vortex line density in the centre of the cell.

The tangle grows and fills the whole cell. L~1/t, agrees well with our other measurements.

1 sinjection

0.3 s probe pulse a time,t, after injection from left

10 V/cmfield

Maximum line density occurs at about 4 seconds

Tangle decay: varying temperature

1 10 100 1000101

102

103

Vortex

Lin

e D

ensi

ty (cm

-2)

Time between pulses (s)

T (K) 0.61 0.50 0.30 0.15

Bottom tip to top collector

For T = 0.08 K – 0.5 K, ’ = (0.15 ± 0.03)

Stopping rotation2. Structured tangles

Impulsive stopping rotation:

(from a vortex array to L=0 through 3D turbulence)

~ 1 rad/s = 0

Horizontal vs. vertical direction

100 101 102 103 104101

102

103

104 = 1.5 rad/s0.5 rad/s

0.15 rad/s

Lt,

cm-2

t, s

t -3/2

0.05 rad/s

T = 0.15 K

100 101 102 103 104101

102

103

104

1.5 rad/s

= 0.5 rad/s

0.15 rad/s

La,

cm

-2

t, s

t -3/2

0.05 rad/sT = 0.15 K

Horizontal Vertical

Scaling with Angular Velocity

10-1 100 101 102 103 104

101

102

103

104

105

axial

horizontal

L-3

/2, c

m-2s3

/2

t

0.05 rad/s 0.15 rad/s 0.5 rad/s 1.5 rad/s1.5 rad/s1.5 rad/s 1.5 rad/s

5x106(t)-3/2

one initial revolution

Low vs. High Temperature: horizontal

10-1 100 101 102 103 104

101

102

103

104

105

5x105(t)-3/2

L-3

/2, c

m-2s3

/2

t

T = 0.15 K: 0.05 rad/s 0.15 rad/s 0.5 rad/s 1.5 rad/s1.5 rad/s1.5 rad/s 1.5 rad/s

T = 1.6 K: 0.5 rad/s 1.5 rad/s

5x106(t)-3/2

Horizontal measurement

High Temperatures: spin-down vs. ion-injection

1 10 100102

103

104

105

T = 1.60 KHorizontal

1.5 rad/s 0.5 rad/s Ion-induced, 20 V/cm towed grid

Vortex

Lin

e D

ensi

ty (cm

-2)

Time (s)

Low vs. High Temperatures

1 10 100 1000 10000101

102

103

104

105

Ions and Spin-down, 1.60 K' = 0.2

Spin-down, 0.15 K' = 0.003

Vo

rte

x L

ine

De

nsi

ty (

cm-2)

Time (s)

Towed grid (Oregon), 1.6K

Ions, 0.15 K' = 0.13

Spin-down from 1.5 rad/svs. Ion-induced tangles

Low vs. High Temperature

“Kolomogorov”(structured)tangle

“Vinen”(random)tangle

0.0 0.5 1.0 1.5 2.010-3

10-2

10-1

100 '

/

T (K)

Transverse Axial

Towed grid Theory

n

Summary• We have used charged vortex rings to probe turbulence in superfluid 4He in the T=0 limit.

• The decay of a tangle produced by either injected current or impulsive spin-down have been studied.

• Random tangles decay as L = t-1. This is consistent with Vinen’s equation with the effective kinematic viscosity of 0.15 .

• Structured tangles decay as L ~ t-3/2 which is consistent with a developed Kolmogorov cascade saturated at cell size. The effective kinematic viscosity is 0.003 .

• ‘(random) / ’(Kolmogorov) ~ 50. Bottleneck between the two cascades? However, not as huge an effect as if reconnections were suppressed.

• Techniques of great potential. More detailed studies to follow.