Two scale modeling of superfluid turbulence Tomasz Lipniacki

He II: -- normal component,-- superfluid component,-- superfluid vortices.

Relative proportion of normal fluid and superfluid as a function of temperature.

vortex line core diameter ~ 1Å

• Solutions of single vortex motion in LIA

• Fenomenological models of anisotropic turbulence

Localized Induction Approximation

nbssss cct ),(

binormal - ,normal-

,tangent-

, and where

bssns

ts

ssss

cc

dd

dtd

c - curvature- torsion

sss ),( tIdeal vortex

)()()(),( utttWt ss

Quasi static solutions:

Totally integrable, equivalent to non-linear Schrödinger equation.Countable (infinite) family of invariants:

)(),(),(),( uttutctc

dcccdcdcddL 222

222 '

4,,,,

Figure 1

:

Hasimoto soliton on ideal vortex (1972)

sss ),( t

20

20

0

000

4 ,2

,2/)(sech

cu

utccc

Kida 1981

sss ),( tGeneral solutions of in terms of elliptic integrals including shapes such as

circular vortex ringhelicoidal and toroidal filamentplane sinusoidal filamentEuler’s elasticaHasimoto soliton

Quasi static solutions for quantum vortices ?

Quantum vortex shrinks:

0

2dc

)(),( nbssss ct

Quasi-static solutionsLipniacki JFM2003

)0,()()(),( sWs ttt

)(),( ),(),( tctc

)0,()(0

2

sωVtnb dcc

0

2222

0

232

222

0

20

dccc

cccccc

dccccccc

nbbtnnt , , ccFrenet Seret equations

In the case of pure translation we get analytic solution

220

220

0

,

)tanh( ),sech(

cBcA

ABAcc

AAtcdRdRt /))ln(cosh(,)(qsin,)cos(q),( 220

s

where 220q ),sech( AcAR

Vtnb

0

2)( dcc

0

22

22

dccc

cc

2/

For 254.2/ Solution has no self-crossings

Quasi-static solution: pure rotation

Vortex asymptotically wraps over coneswith opening angle

0

3/13/2

00

3/1 )cot()(,23)(,||)(

KzK

Kr

for in cylindrical coordinates ),,( zr

/)cot(,2

where

00

303 K

)0('),0(),0('),0( 0000 cccc

Quasi-static solution: general case

For vortex wraps over paraboloids

0

3/203/2

00

3/1 3)(,23)(,||)(

KTzK

Kr

Results in general case

I. 4-parametric class of solutions, determined by initial condition

)0( ),0( ),0( ),0( cc

II. Each solution corresponds to a specific isometric transformation (t)related analytically to initial condition.

III. The asymptotic for

is related analytically via transformation (t) to initial condition.

Shape-preserving solutionsLipniacki, Phys. Fluids. 2003

ssss ),( t

, ,2 tt

Equation

is invariant under the transformation

If the initial condition is scale-free, than the solution is self-similar

t

ttt Ss )(),(

In general scale-free curve is a sum of two logarithmic spirals on coaxial cones; there is four parametric class of such curves.

Shape-preserving solutions

t

ttt Ss )(),(

. ,1 ,1t

lt

Ttt

Kt

c

solutions withdecreasing scale

222

2

22

0

2222

0

232

TlTldKTTKK

TKTKKK

KTK

KlKldKKKKTKTKTK

l

l

ssss ),( t

Analytic result: shape-preserving solution

In the case when transformation is a pure homothety we get analytic solution in implicit form:

KK

KKpKKKKdKl

K

K

T ,

)()/ln()(

0

2200

22

scale growing 1p 0T scale, decreasing 1 p

Shape preserving solution: general case

scale growing 1p scale decreasing 1pLogarithmic spirals on cones

Results in general case

I. 4-parametric class of solutions, determined by initial condition

)0( ),0( ),0(K ),0( TTK

II. Each solution corresponds to a specific similarity transformation

III. The asymptotics for

is given the initial condition of the original problem (with time).

l

Self-similar solution for vortex in Navier-Stokes?

ωωvω 2)( v

t

N-S is invariant to the same transformation

,/, ,2 rVVrrtt

The friction coefficient corresponds to

b

n

b

n

VVv

VV

Wing tip vortices

-- Euler equation for superfluid component

-- Navier-Stokes equation for normal component

Macroscopic description

Coupled by mutual friction force LVF nsns ~

where snns VVV and L is superfluid vortex line density.

Microscopic description of tangle superfluid vortices

Assuming that quantum tangle is “close” to statistical equilibriumderive equations for quantum line length density L and anisotropy parameters of the tangle.

Aim

Two cases

I – rapidly changing counterflow, but uniform in space

II – slowly changing counterflow, not uniform in space

Statistical equilibrium?

Model of quantum tangle“Particles” = segments of vortex line of length equal to characteristicradius of curvature. Particles are characterized by their tangent t, normal n, and binormal b vectors. Velocity of each particle is proportional to its binormal (+collective motion).Interactions of particles = reconnections.

STAdtdQVn /

Reconnections

1. Lines lost their identity: two line segments are replaced by two new line segments

2. Introduce new curvature to the system

3. Remove line-length

Motion of vortex line in the presence of counterflow

Evolution of its line-length

snns VVV

dl is

Evolution of line length density

where

I

average curvature

average curvature squared

Average binormal vector (normalized)

dL 1

Rapidly changing counterflow

Generation term: polarization of a tangle by counterflow

Degradation term: relaxation due to reconnections

Generation term: polarization of a tangle by counterflow

evolution of vortex ringof radius R and orientation in uniform counterflow

unit binormal

Statistical equilibrium assumption: distribution of f(b) is the most probable distribution giving right I i.e.

I

bIexp)( Cbf

Degradation term: relaxation due to reconnections

Reconnections produce curvature but not net binormal

Total curvature of the tangle

Reconnection frequency 2/2/51 Lcfr

Relaxation time: LfCT

rr

2

Finally

Lipniacki PRB 2001

Model prediction: there exists a critical frequency of counterflow above which tangle may not be sustained

max78.0 L max94.0 L

Quantum turbulence with high net macroscopic vorticity

nsns I VVI 0We assume now

Let denote anisotropy of the tangent

Vinen type equation for turbulence with net macroscopic vorticity

Vinen-Schwarz equation

Drift velocity LV

where

Mutual friction force

drag force

nsF

where

finally

Helium II dynamical equations

Lipniacki EJM 2006

Specific cases: stationary rotating turbulence

Pure heat driven turbulence

Pure rotation

„Sum”

Fast rotation Slow rotation

Plane Couette flow

V- normal velocityU-superfluid velocity

q- anisotropyl-line length density

“Hypothetical” vortex configuration

Summary and Conclusions

• Self-similar solutions of quantum vortex motion in LIA

• Dynamic of a quantum tangle in rapidly varying counterflow

• Dynamic of He II in quasi laminar case.

In small scale quantum turbulence, the anisotropy of the tangle in addition to its density L is key to analyzedynamic of both components.

bEVmt

i *0

22

2

Non-linear Schrodinger equation (Gross, Pitaevskii)

*2

)/(

mmA

mV

s

s

Madelung Transformation

superfluid velocity

superfluid density

)exp( iAFor

k

jk

jjss

ss xx

PVVt

V

2

20

2mVP s

kj

ssjk xxm

)ln(

4

2

2

2

Modified Euler equation

With

pressure

Quantum stress

Mixed turbulence

Kolgomorov spectrum for each fluid

3/5

3/2

)(k

CkE

- energy flux per unit mass

Two scale modeling of superfluid turbulence Tomasz Lipniacki

He II: -- normal component,-- superfluid component,-- superfluid vortices.

Relative proportion of normal fluid and superfluid as a function of temperature.

vortex line