Andrew Lucas - Stanford Universityajlucas/LowT2dSFTurbulenceTalk.pdf · Andrew Lucas Harvard...

Vortex dynamics in low temperature two-dimensionalsuperfluid turbulence

Andrew Lucas

Harvard Physics

Leiden, Lorentz Seminar

September 8, 2014

Collaborators 2

Paul CheslerHarvard Physics

Piotr SurowkaHarvard Physics

[Lucas, Surowka, arXiv:1408.5913]

[Chesler, Lucas, in preparation]

Two-Dimensional Superfluids 3

Superfluid Vortices

Gross-Pitaevskii action: low energy EFT of a superfluid.

L = i~ψ∂tψ −~2|∇ψ|2

2m+ µ|ψ|2 − λ

2|ψ|4.

I µ > 0: |ψ|2 > 0. U(1) symmetry ψ → eiθψ spontaneouslybroken.

I vortices: topological point defects near which ψ ∼ eiWθ

I SF velocity field: v =~m∇θ ≈ W~

m

φ

r


Superfluid Vortices


L = i~ψ∂tψ −~2|∇ψ|2

2m+ µ|ψ|2 − λ

2|ψ|4.




m

φ

r


Superfluid Vortices


L = i~ψ∂tψ −~2|∇ψ|2

2m+ µ|ψ|2 − λ

2|ψ|4.




m

φ

r


Superfluid Vortices


L = i~ψ∂tψ −~2|∇ψ|2

2m+ µ|ψ|2 − λ

2|ψ|4.




m

φ

r

Classical Turbulence 4

d = 3: Direct Cascade

I what is turbulence?

I strongly nonlinear, chaotic motion ofvortices (incompressible approximation)

I ≈ scale invariant in inertial rangeI d = 3 spatial dimensions: (approximate)

flux of energy (ε) from IR to UVdissipative scale: direct cascade

I Kolmogorov’s 5/3 law (dimensional analysis): energystored at wave vector k (in inertial range)

E(k) ∼ ε2/3k−5/3

[Kolmogorov, Proceedings of the USSR Academy of Sciences 30 299 (1941)]



I what is turbulence?I strongly nonlinear, chaotic motion of

vortices (incompressible approximation)

I ≈ scale invariant in inertial rangeI d = 3 spatial dimensions: (approximate)



E(k) ∼ ε2/3k−5/3





vortices (incompressible approximation)I ≈ scale invariant in inertial range

I d = 3 spatial dimensions: (approximate)flux of energy (ε) from IR to UVdissipative scale: direct cascade


E(k) ∼ ε2/3k−5/3





vortices (incompressible approximation)I ≈ scale invariant in inertial rangeI d = 3 spatial dimensions: (approximate)



E(k) ∼ ε2/3k−5/3





vortices (incompressible approximation)I ≈ scale invariant in inertial rangeI d = 3 spatial dimensions: (approximate)



E(k) ∼ ε2/3k−5/3



d = 2: Inverse Cascade

I d = 2 normal turbulence insteadhas inverse cascade:energy convected UV→IR

“Andrew”

I enstrophy conservation: 0 =d

dt

∫d2x ω2. ω = εij∂ivj .

[Kraichnan, Physics of Fluids 10 1417 (1967)]


d = 2: Inverse Cascade

I d = 2 normal turbulence insteadhas inverse cascade:energy convected UV→IR

“Andrew”

I enstrophy conservation: 0 =d

dt

∫d2x ω2. ω = εij∂ivj .

[Kraichnan, Physics of Fluids 10 1417 (1967)]


A (Holographic) Inverse Cascade

[Adams, Chesler, Liu, Physical Review Letters 112 151602 (2014)]

Superfluid Turbulence 7

Principles

I vortex annihilation =⇒enstrophy no longer conserved

I what causes annihilation?

I is a direct cascade now possible?

I is SF turbulence with annihilationcharacterized by Kolmogorov scaling?


Principles






Principles






Principles






Holography

AdS/CFT calculation: energy → UV; driven by vortexannihilation

[Chesler, Liu, Adams, Science 341 368 (2013)]


Holography

AdS/CFT calculation: energy → UV; driven by vortexannihilation

[Chesler, Liu, Adams, Science 341 368 (2013)]


Gross-Pitaevskii: Inverse Cascade

dynamics of T = 0 superfluid using GPE

[Simula, Davis, Helmerson, arXiv:1405.3399]


Experiments: Cold Atomic Gases

I vortex annihilation in cold atomic turbulent BEC of 23Naatoms: [Kwon, Moon, Choi, Seo, Shin, arXiv:1403.4658]

I experiment can’t distinguish between ±1 vortices.gyroscope proposal to do this [Powis, Sammut, Simula,

arXiv:1405.4352]


Experiments: Cold Atomic Gases

I vortex annihilation in cold atomic turbulent BEC of 23Naatoms: [Kwon, Moon, Choi, Seo, Shin, arXiv:1403.4658]

I experiment can’t distinguish between ±1 vortices.gyroscope proposal to do this [Powis, Sammut, Simula,

arXiv:1405.4352]

Effective Theory 12

Our Approach

could an effective theory of vortex dynamicscapture this phenomenology?

I dilute limit: (intervortex spacing) r � ξ (vortex core size)

I leading order dynamics of Gross-Pitaevskii equation (e.g.):point-vortex dynamics:

Xni = V n

i = −∑

m6=n

~κmm

εij(Xnj −Xm

j )

|Xn −Xm|2

I corrections from sound?

I finite temperature?

I long, controversial history: HVI equations [e.g. Sonin, Physical

Review B55 485 (1997); Thompson, Stamp, Physical Review Letters 108

184501 (2011)]

Effective Theory 12

Our Approach




Xni = V n

i = −∑

m6=n

~κmm

εij(Xnj −Xm

j )

|Xn −Xm|2





184501 (2011)]

Effective Theory 12

Our Approach




Xni = V n

i = −∑

m6=n

~κmm

εij(Xnj −Xm

j )

|Xn −Xm|2





184501 (2011)]

Effective Theory 12

Our Approach




Xni = V n

i = −∑

m6=n

~κmm

εij(Xnj −Xm

j )

|Xn −Xm|2





184501 (2011)]

Effective Theory 12

Our Approach




Xni = V n

i = −∑

m6=n

~κmm

εij(Xnj −Xm

j )

|Xn −Xm|2





184501 (2011)]

Effective Theory 12

Our Approach




Xni = V n

i = −∑

m6=n

~κmm

εij(Xnj −Xm

j )

|Xn −Xm|2





184501 (2011)]

Effective Theory 13

T = 0: Effective Action Techniques

I leading order sol’n to GPE: (κn = ±1: winding number)

ψPV = eχ+iθ, χ =∑

n

χn(x−Xn)

︸︷︷︸near-core density correction

, θ =∑

n

κnθn(x−Xn)

︸︷︷︸long-range phase fluctuations

I effective action:

eiSeff [Xn] =

∫Dδψ eiS[ψPV(Xn)+δψ]

integrate out smooth (no-vortex) fluctuations δψ

I Gaussian path integral: effective action valid to O(ξ2/r2).


Effective Theory 13




n

χn(x−Xn)


, θ =∑

n

κnθn(x−Xn)


I effective action:

eiSeff [Xn] =





Effective Theory 13




n

χn(x−Xn)


, θ =∑

n

κnθn(x−Xn)


I effective action:

eiSeff [Xn] =





Effective Theory 14

T = 0: The Effective Action for Vortices

I typical HVI answer:

δL =∑

logL

ξX2n?

I we find a very complicated answer: mixing kinetic terms(Xn · Xm), 2,3,4-body terms...

I log divergences cancel on shell:

δL = logL

rtyp

(∑κnXn

)2+ log

rtyp

ξ

∑(Xn −Vn

)2+ · · ·

I no radiation! vortices moving through pure SF

Effective Theory 14



δL =∑

logL

ξX2n?



δL = logL

rtyp

(∑κnXn

)2+ log

rtyp

ξ

∑(Xn −Vn

)2+ · · ·


Effective Theory 14



δL =∑

logL

ξX2n?



δL = logL

rtyp

(∑κnXn

)2+ log

rtyp

ξ

∑(Xn −Vn

)2+ · · ·


Effective Theory 14



δL =∑

logL

ξX2n?



δL = logL

rtyp

(∑κnXn

)2+ log

rtyp

ξ

∑(Xn −Vn

)2+ · · ·


Effective Theory 15

T = 0: Vortex-Antivortex Pair

I inverse cascade? check whether pair annihilates!

I exact effective action! conserved energy

E

πρ0=

1

mlog|∆X|ξ− ξ2

m|∆X|2+

∆X2

4µ+µ

4

(εij

P j

πρ0+ ∆Xi

)2

.

I expand around zeroth order (const. velocity 1/mr0):fluctuations ≈ described by harmonic oscillation!

r0

I fluctuation velocities ∼ ξ2/mr30 – do not leave regime of

validity

I no instability to vortex annihilation =⇒ inverse cascade

Effective Theory 15




E

πρ0=

1

mlog|∆X|ξ− ξ2

m|∆X|2+

∆X2

4µ+µ

4

(εij

P j

πρ0+ ∆Xi

)2

.


r0


validity


Effective Theory 15




E

πρ0=

1

mlog|∆X|ξ− ξ2

m|∆X|2+

∆X2

4µ+µ

4

(εij

P j

πρ0+ ∆Xi

)2

.


r0


validity


Effective Theory 15




E

πρ0=

1

mlog|∆X|ξ− ξ2

m|∆X|2+

∆X2

4µ+µ

4

(εij

P j

πρ0+ ∆Xi

)2

.


r0


validity


Effective Theory 15




E

πρ0=

1

mlog|∆X|ξ− ξ2

m|∆X|2+

∆X2

4µ+µ

4

(εij

P j

πρ0+ ∆Xi

)2

.


r0


validity


Effective Theory 16

T > 0: Normal Fluid

I exchange of energy/momentum only at vortex core andsymmetry =⇒ first order HVI equation [Ambegaokar,

Halperin, Nelson, Siggia, Physical Review Letters 40 783 (1978)]:

~ρs

mκnεij

(Xnj − V n

j

)

︸︷︷︸Magnus force

= −η(Xni − Uni )− η′κnεij(Xn

j − Unj )︸︷︷︸

vortex drag force

I η, η′ microscopic coefficients beyond EFT; vanish as T → 0

I (normal) sound radiated from momentum exchangesuppressed faster than r−1

I (normal) sound from vortex annihilation suppressed fasterthan r−4; consistently set Un = 0.

I no single fluid continuum description: η is not “viscosity” –it is κ-dependent friction

Effective Theory 16

T > 0: Normal Fluid



~ρs

mκnεij

(Xnj − V n

j

)



j − Unj )︸︷︷︸

vortex drag force





Effective Theory 16

T > 0: Normal Fluid



~ρs

mκnεij

(Xnj − V n

j

)



j − Unj )︸︷︷︸

vortex drag force





Effective Theory 16

T > 0: Normal Fluid



~ρs

mκnεij

(Xnj − V n

j

)



j − Unj )︸︷︷︸

vortex drag force





Effective Theory 16

T > 0: Normal Fluid



~ρs

mκnεij

(Xnj − V n

j

)



j − Unj )︸︷︷︸

vortex drag force





Effective Theory 17

The Magnus Force

consider vortex at rest in background flow V:

R

V

Fi =

∮dxjΠij =

∮dxj (ρsvivj + µδij)

v = V +κ~m

φ

r

Integrate over circle of large radius R:

Fi = −εijκ~ρs

mVj .

Galilean invariance: boost to frame withvortex moving

[Sonin, Physical Review B55 485 (1997)]

Effective Theory 17

The Magnus Force


R

V

Fi =

∮dxjΠij =


v = V +κ~m

φ

r


Fi = −εijκ~ρs

mVj .



Effective Theory 17

The Magnus Force


R

V

Fi =

∮dxjΠij =


v = V +κ~m

φ

r


Fi = −εijκ~ρs

mVj .



Effective Theory 18

The Equations of Motion

I if l = characteristic length, rescale

Xn = lXn, t =ml2

~t.

EOM l-independent: exact scale invariance

I after rescaling t to remove η′:

κnεij

(Xnj − V n

j

)= −ηeffX

ni .

a single dimensionless parameter ηeff ∼ η.

I ηeff > 0. system dissipates energy into normal fluid.vortex/anti-vortex pairs annihilate in finite time.

I length scales: “core size” ξ (numerical deletion of vortices);torus size L; initial conditions

Effective Theory 18



Xn = lXn, t =ml2

~t.



κnεij

(Xnj − V n

j

)= −ηeffX

ni .




Effective Theory 18



Xn = lXn, t =ml2

~t.



κnεij

(Xnj − V n

j

)= −ηeffX

ni .




Effective Theory 18



Xn = lXn, t =ml2

~t.



κnεij

(Xnj − V n

j

)= −ηeffX

ni .




Inverse Cascade 19

ηeff = 0.0025: Inverse Cascade

Inverse Cascade 20

Kolmogorov Scaling: Theory

I how to properly define energy at length scale 1/k?

I energy at wavelength k when kζT & 1: “project wavefunction onto k modes, measure H: 〈ψ|PkHPk|ψ〉:”

Equ(k) =

∫dkθ

1

2|kψ|2 ≈

∫dkθ

1

2

∣∣∣(veiθ

)(k)∣∣∣2.

I T > 0 SF loses phase coherence when kζT . 1. here useclassical definition:

Ecl(k) =

∫dkθ

1

2|v(k)|2.

I Ecl most common in literature.ignoring quantum phenomena!

Inverse Cascade 20




Equ(k) =

∫dkθ

1

2|kψ|2 ≈

∫dkθ

1

2

∣∣∣(veiθ

)(k)∣∣∣2.


Ecl(k) =

∫dkθ

1

2|v(k)|2.


Inverse Cascade 20




Equ(k) =

∫dkθ

1

2|kψ|2 ≈

∫dkθ

1

2

∣∣∣(veiθ

)(k)∣∣∣2.


Ecl(k) =

∫dkθ

1

2|v(k)|2.


Inverse Cascade 20




Equ(k) =

∫dkθ

1

2|kψ|2 ≈

∫dkθ

1

2

∣∣∣(veiθ

)(k)∣∣∣2.


Ecl(k) =

∫dkθ

1

2|v(k)|2.


Inverse Cascade 21

Classical Kolmogorov Scaling: Inverse Cascade

N(t) = 3590. N(0) = 4000. ηeff = 0.0025.

10−3 10−2 10−1 100102

103

104

105

106

k�5/3

k�1

k

k

E

1

Counting Vortices 22

Experimental Results

I experiment easily counts vortices at time t, N(t)

dN

dt= −Γ1N − Γ2N

2.

vortices leave imaging region

I they measured Γ2 ∼ T 2/µL2.[Kwon, Moon, Choi, Seo, Shin, arXiv:1403.4658]


Experimental Results

I experiment easily counts vortices at time t, N(t)

dN

dt= −Γ1N − Γ2N

2.

vortices leave imaging region

I they measured Γ2 ∼ T 2/µL2.[Kwon, Moon, Choi, Seo, Shin, arXiv:1403.4658]


Smoke Ring Dynamics

I our equations can be exactly solved for± pair of vortices

I given initial separation r0, annihilationin finite time

tann =r2

0m

4~1 + η2

eff

ηeff

I uniform “gas”: r0 ∼ LN−1/2 =⇒global vortex count N(t) obeys

dN

dt≈ −Γ2N

2, Γ2 ≡8~ηeff

mL2

N(t) =N(0)

1 +N(0)Γ2t.


Smoke Ring Dynamics



tann =r2

0m

4~1 + η2

eff

ηeff


dN

dt≈ −Γ2N

2, Γ2 ≡8~ηeff

mL2

N(t) =N(0)

1 +N(0)Γ2t.


Smoke Ring Dynamics



tann =r2

0m

4~1 + η2

eff

ηeff


dN

dt≈ −Γ2N

2, Γ2 ≡8~ηeff

mL2

N(t) =N(0)

1 +N(0)Γ2t.


Simulations: Random Mixture

ηeff = 0.1

102 10310−2

10−1

100

N 2

two-body fit

N

��dN

dt

��


The Inverse Cascade

large same-sign clusters form. annihilation only at edges ofclusters (a curve of fractal dimension df = 4/3 [Bernard, Boffetta,

Celani, Falkovich, Nature Physics 2 124 (2006)])

dN

dt≈ −

∫d2x

~ηeff

mn+n− ∼ −Γ2

N2

L2× fractal area

finite size: fractal on length scalesr < x < L.

fractal area ∼ r2 ×(L

r

)df

r ∼ LN−1/2:

dN

dt∼ −N1+df/2 ∼ −N5/3


The Inverse Cascade



dN

dt≈ −

∫d2x

~ηeff

mn+n− ∼ −Γ2

N2

L2× fractal area



r

)df

r ∼ LN−1/2:

dN

dt∼ −N1+df/2 ∼ −N5/3


The Inverse Cascade



dN

dt≈ −

∫d2x

~ηeff

mn+n− ∼ −Γ2

N2

L2× fractal area



r

)df

r ∼ LN−1/2:

dN

dt∼ −N1+df/2 ∼ −N5/3


The Inverse Cascade



dN

dt≈ −

∫d2x

~ηeff

mn+n− ∼ −Γ2

N2

L2× fractal area



r

)df

r ∼ LN−1/2:

dN

dt∼ −N1+df/2 ∼ −N5/3


Simulations: Inverse Cascade

ηeff = 0.02 (fairly large...).

300 600 1000 300010−2

10−1

100

N 5/3

N

��dN

dt

��

Conclusions 27

Our Model

I T = 0: point vortex dynamics acceptable – not destroyedby sound!

I first-order HVI equations are the effective theory of T > 02d SF turbulence:

κnεij

(Xjn − V j

n

)= −ηeffX

ni

I classical turbulence in the small ηeff limitI understand ηeff large? beyond inverse cascadeI thermodynamic limit?

I predictions for experiment:

dN

dt

∣∣∣∣rand

≈ −8~ηeff

mL2N2.

dN

dt

∣∣∣∣inv

∼ −N5/3.

I 23Na experiment: ηeff ∼ 0.01− 0.05

Conclusions 27

Our Model



κnεij

(Xjn − V j

n

)= −ηeffX

ni



dN

dt

∣∣∣∣rand

≈ −8~ηeff

mL2N2.

dN

dt

∣∣∣∣inv

∼ −N5/3.


Conclusions 27

Our Model



κnεij

(Xjn − V j

n

)= −ηeffX

ni



dN

dt

∣∣∣∣rand

≈ −8~ηeff

mL2N2.

dN

dt

∣∣∣∣inv

∼ −N5/3.


Conclusions 27

Our Model



κnεij

(Xjn − V j

n

)= −ηeffX

ni



dN

dt

∣∣∣∣rand

≈ −8~ηeff

mL2N2.

dN

dt

∣∣∣∣inv

∼ −N5/3.


Conclusions 27

Our Model



κnεij

(Xjn − V j

n

)= −ηeffX

ni



dN

dt

∣∣∣∣rand

≈ −8~ηeff

mL2N2.

dN

dt

∣∣∣∣inv

∼ −N5/3.


Looking Forward 28

ηeff = 0.1: Direct Cascade???

Looking Forward 29

Quantum Turbulence???

a novel regime of “quantum turbulence”?

I dominated by vortex annihilation physics

I breakdown of inverse cascade, classical 5/3 scaling

I novel collective scaling in quantum energy spectrum???

I connects with high T regime found in holography

more to come!

Looking Forward 29







more to come!

Looking Forward 29







more to come!

Looking Forward 29







more to come!

Looking Forward 29







more to come!

Looking Forward 29







more to come!

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