Makoto Tsubota,Tsunehiko Araki and Akira Mitani Osaka,

1
INSTABILITY OF VORTEX ARRAY AND POLARIZATION
OFSUPERFLUID TURBULENCE

Carlo F. Barenghi

• Makoto Tsubota,Tsunehiko Araki and Akira Mitani
  (Osaka),
• Sarah Hulton (Stirling),
• David Samuels (Virginia Tech)

2
1. Rotation
2. Counterflow
disordered vortex tangle L?² V²

• ordered vortex
• array L2O/G

where VVn-Vs
L vortex line density
3
Rotating counterflow
4
Experiment by Swanson, Barenghi and
Donnelly,Phys. Rev. Lett. 50, 90, (1983)
-For VltVc1 vortex array -What are the critical
velocities Vc1 and Vc2 ? -What is the state
Vc1ltVltVc2 ? -What is the state VgtVc2 ?
5
Vortex dynamics
Velocity at point S(?,t)
Self-induced velocity
6
What is the first critical velocity Vc1 ?
-It is an instability of Kelvin waves

• Small amplitude helix of
• wavenumber k2p/?
• kAltlt1, ?kz-?t, ?z

Assume
Then
Growth rate sa(kVn-ßk²), Max s at kVn/2ß,
Frequency ?(1a)Vn²/4ß
7
T1.6K, O4.98 x 10² s ¹ Vns0.08 cm/sec
What happens for VgtVc1 ?
8
Numerical simulation of rotating vortex array in
the presence of an axial counterflow velocity
t12 s
t0
t28 s
t160 s
9
Vortex line density L vs time
O9.97 x 10³ s¹
O4.98 x 10² s¹
O0
After an initial transient, L saturates to a
statistical steady state
10
Polarization ltszgt vs time
O9.97 x 10³ s¹
O4.98 x 10² s¹
O0
Thus for VgtVc1 we have polarized turbulence
11
Analogy with paramagnetism
Vortices are aligned by the applied rotation O
and randomised by the counterflow Vns
The observed L is always less than the
expected LHLR, where
LR2O/?
LH?²Vns²
L(LHLR)-L/(bLH) vs Oa LR/LH, with a11 and
b0.23
12
What is the second critical velocity Vc2 ?

• T1 characteristic time of growing Kelvin waves
• 1/smax4ß/(a Vns²)
• T2 characteristic friction lifetime of vortex
  loops created by
• reconnections
• 2?spR²/(?ß) where 2Rd, d1/vL and
  ?friction coeff
• If T2gtT1 vortex loops have no time to shrink
  before more
• loops are created ? randomness
• Thus polarized tangle is unstable if LltC Vns²
  with C50000
• which has the same order of magnitude of the
  finding of
• the experiment of Swanson et al

Conclusion probably for VgtVc2 the tangle is
random
13
Classical turbulence
Fourier transform the velocity
Energy spectrum
Dissipation
Kolmogorov -5/3 law
The energy sink is viscosity, acting only for
kgt1/? ?small scale (Kolmogorov length) Dlarge
scale
14
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) T2.3K (b) T2.08K (c) T1.4K
..\Application Data\SSH\temp\poster
15
The superfluid alone (T0) obeys the Kolmogorov
law for klt1/d, where d1/vL 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 d (klt1/ d) ?
• Yes

16
A straight vortex (red segment in figure),
initially in the plane ?p/2, in the presence of
a normal fluid eddyVn(0,0,Or sin?), moves
according todr/dt0, df/dta and
d?/dt-aOsin(?)Hence ?(t)2 arctan(exp(-aOt))?0
for t?8
A SIMPLE MODEL OF POLARIZATION
However the lifetime of the eddy is only t 1/O
so the segment can only turn to the angle ?(t)
p/2-a
17
The normal fluid spectrum in the inertial range
1/Dltklt1/? is(Dlarge scale, ?Kolmogorov scale)
In time 1/?k re-ordering of existing vortices
creates a net superfluid vorticity ?saL?/3 in
the direction of the vorticity ?k of the normal
fluid eddy of wavenumber k. Since ?kv(k³Ek),
matching of ?s and ?k gives
But ?k is concentrated at smallest scale (k1/?)
so a vortex tangle of given L and intervortex
spacing d 1/vL can satisfy that relation only
up to a certain k. Since e¼?n¾/? we have
Conclusion matching of ?k and ?s (hence coupling
normal fluid and superfluid patterns) is possible
for the entire inertial range !
18
Consider the evolution of few seeding vortex
rings in the presence of an ABC (Arnold,
Beltrami, Childress) normal flow of the form
MORE NUMERICAL EVIDENCE OF POLARISATION
Vorticity regions of driving ABC flow
Resulting polarized tangle
19
Results ltcos(?)gtltszgt at various A,a
20
Scaled results ltcos(?)gt/a vs t/twhere t1/?n
and ?n is the normal fluid vorticity
No matter whether the tangle grows or decays, the
same polarization takes place for t/t1
21
CONCLUSIONS

• Provided that enough vortex lines are present,
  vorticity matching ?s?n can take place over the
  inertial range up to k1/d, 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