Post on 03-Oct-2018
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
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
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
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
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)]
Classical Turbulence 4
d = 3: Direct Cascade
I what is turbulence?I strongly nonlinear, chaotic motion of
vortices (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)]
Classical Turbulence 4
d = 3: Direct Cascade
I what is turbulence?I strongly nonlinear, chaotic motion of
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
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)]
Classical Turbulence 4
d = 3: Direct Cascade
I what is turbulence?I strongly nonlinear, chaotic motion of
vortices (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)]
Classical Turbulence 4
d = 3: Direct Cascade
I what is turbulence?I strongly nonlinear, chaotic motion of
vortices (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)]
Classical Turbulence 5
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)]
Classical Turbulence 5
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)]
Classical Turbulence 6
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?
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?
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?
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?
Superfluid Turbulence 8
Holography
AdS/CFT calculation: energy → UV; driven by vortexannihilation
[Chesler, Liu, Adams, Science 341 368 (2013)]
Superfluid Turbulence 9
Holography
AdS/CFT calculation: energy → UV; driven by vortexannihilation
[Chesler, Liu, Adams, Science 341 368 (2013)]
Superfluid Turbulence 10
Gross-Pitaevskii: Inverse Cascade
dynamics of T = 0 superfluid using GPE
[Simula, Davis, Helmerson, arXiv:1405.3399]
Superfluid Turbulence 11
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]
Superfluid Turbulence 11
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
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
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
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
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
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 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).
[Lucas, Surowka, arXiv:1408.5913]
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).
[Lucas, Surowka, arXiv:1408.5913]
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).
[Lucas, Surowka, arXiv:1408.5913]
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
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
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
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 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
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
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
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
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 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
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
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
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
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 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
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
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 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
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
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
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
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
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
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
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 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]
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]
Counting Vortices 23
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.
Counting Vortices 23
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.
Counting Vortices 23
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.
Counting Vortices 24
Simulations: Random Mixture
ηeff = 0.1
102 10310−2
10−1
100
N 2
two-body fit
N
����dN
dt
����
Counting Vortices 25
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
Counting Vortices 25
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
Counting Vortices 25
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
Counting Vortices 25
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
Counting Vortices 26
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
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
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
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
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
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
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
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
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
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!