From Laminar Flow to Wave Turbulence in Holographic Superfluid - - PowerPoint PPT Presentation

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From Laminar Flow to Wave Turbulence in Holographic Superfluid - - PowerPoint PPT Presentation

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Motivation and introduction Laminar-turbulent transition in holographic superfluids Wave turbulence in holographic superfluids Conclusion and discussion From Laminar Flow to Wave Turbulence in Holographic Superfluid University of Chinese

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Motivation and introduction Laminar-turbulent transition in holographic superfluids Wave turbulence in holographic superfluids Conclusion and discussion

From Laminar Flow to Wave Turbulence in Holographic Superfluid

Yu Tian (田雨) University of Chinese Academy of Sciences 中国科学院大学

(mainly based on the work in finalizing with Shan-Quan Lan, Wen-Biao Liu, Hong Liu and Hongbao Zhang)

March 2, 2018 Gravity and Cosmology 2018

Yukawa Institute for Theoretical Physics, Kyoto University

Yu Tian (田雨) From Laminar Flow to Wave Turbulence in Holographic Superfluid

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Motivation and introduction Laminar-turbulent transition in holographic superfluids Wave turbulence in holographic superfluids Conclusion and discussion

1 Motivation and introduction 2 Laminar-turbulent transition in holographic superfluids 3 Wave turbulence in holographic superfluids 4 Conclusion and discussion

Yu Tian (田雨) From Laminar Flow to Wave Turbulence in Holographic Superfluid

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Motivation and introduction Laminar-turbulent transition in holographic superfluids Wave turbulence in holographic superfluids Conclusion and discussion

1 Motivation and introduction 2 Laminar-turbulent transition in holographic superfluids 3 Wave turbulence in holographic superfluids 4 Conclusion and discussion

Yu Tian (田雨) From Laminar Flow to Wave Turbulence in Holographic Superfluid

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Motivation and introduction Laminar-turbulent transition in holographic superfluids Wave turbulence in holographic superfluids Conclusion and discussion

Turbulence: one of the most important scientific problems

Figure: Turbulence in classical fluids

Yu Tian (田雨) From Laminar Flow to Wave Turbulence in Holographic Superfluid

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Motivation and introduction Laminar-turbulent transition in holographic superfluids Wave turbulence in holographic superfluids Conclusion and discussion

Turbulence: characteristics

  • no consensus definition of turbulence
  • spatially complex
  • aperiodic in time
  • spanning several orders of magnitude in spatial extent and

temporal frequency

  • chaotic

sensitive to initial conditions (unpredictable)

Yu Tian (田雨) From Laminar Flow to Wave Turbulence in Holographic Superfluid

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Motivation and introduction Laminar-turbulent transition in holographic superfluids Wave turbulence in holographic superfluids Conclusion and discussion

Quantum turbulence

  • turbulence in quantum fluids (superfluid Heliums,

Bose-Einstein condensates, superconductors, etc)

  • expected to be similar to classical turbulence at large scales
  • expected to be distinct from classical turbulence at small

scales set by the healing length, which is the characteristic size of local defects (basically vortices) of the order parameter

Yu Tian (田雨) From Laminar Flow to Wave Turbulence in Holographic Superfluid

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Motivation and introduction Laminar-turbulent transition in holographic superfluids Wave turbulence in holographic superfluids Conclusion and discussion

Vortex turbulence vs wave turbulence

  • eddies in classical turbulence and vortices in quantum

turbulence eddy (or vortex) as the fundamental characteristic of the traditional turbulence (vortex turbulence)

  • wave turbulence: a type of turbulence different from the

traditional one, where eddies (or vortices) exist but do not dominate the physics

  • decomposition: vortex (incompressible) and wave

(compressible) v = vi + vc ∇ · vi = 0, ∇ × vc = 0

  • vortex dominant vs wave dominant

Yu Tian (田雨) From Laminar Flow to Wave Turbulence in Holographic Superfluid

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Motivation and introduction Laminar-turbulent transition in holographic superfluids Wave turbulence in holographic superfluids Conclusion and discussion

Quantum wave turbulence: experiments and numerics

[Navon et al, Nature 539, 72 (2016)]

  • onset of 3D turbulence in BEC by shaking
  • numerical modeling using Gross-Pitaevskii equation

i∂tϕ =

  • − ∇2

2m + V(t, x) + g|ϕ|2 − µ

  • ϕ

with excellent agreement with experimental measurements

  • wave dominant (from numerics) isotropic steady turbulent

state

Yu Tian (田雨) From Laminar Flow to Wave Turbulence in Holographic Superfluid

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Motivation and introduction Laminar-turbulent transition in holographic superfluids Wave turbulence in holographic superfluids Conclusion and discussion

Turbulence: open problems

  • onset mechanism of (classical and quantum) turbulence
  • control parameter of quantum turbulence (like the Reynolds

number in classical turbulence)

  • (non-)universal properties (Kolmogorov scaling law,

Kolmogorov-Zakharov scaling law, energy cascade, etc)

Yu Tian (田雨) From Laminar Flow to Wave Turbulence in Holographic Superfluid

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Motivation and introduction Laminar-turbulent transition in holographic superfluids Wave turbulence in holographic superfluids Conclusion and discussion

Why applied holography (AdS/CFT)?

  • Quantum systems can be dually described by classical

gravitational theories.

  • Far-from-equilibrium dynamics as well as near-equilibrium

transport processes can be easily realized.

  • Dissipation at finite temperature is naturally included by

putting a black hole in the bulk.

Yu Tian (田雨) From Laminar Flow to Wave Turbulence in Holographic Superfluid

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Motivation and introduction Laminar-turbulent transition in holographic superfluids Wave turbulence in holographic superfluids Conclusion and discussion

Facts about (applied) AdS/CFT

  • Finite temperature field theory with finite chemical potential

is dual to a charged black hole in the bulk AdS: Temperature ← → Hawking temperature Conserved charge ← → Charge Chemical potential ← → Electric potential Energy dissipation ← → Energy accretion

[YT, X.-N, Wu and H. Zhang, arXiv:1407.8273]

Yu Tian (田雨) From Laminar Flow to Wave Turbulence in Holographic Superfluid

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Motivation and introduction Laminar-turbulent transition in holographic superfluids Wave turbulence in holographic superfluids Conclusion and discussion

1 Motivation and introduction 2 Laminar-turbulent transition in holographic superfluids 3 Wave turbulence in holographic superfluids 4 Conclusion and discussion

Yu Tian (田雨) From Laminar Flow to Wave Turbulence in Holographic Superfluid

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Motivation and introduction Laminar-turbulent transition in holographic superfluids Wave turbulence in holographic superfluids Conclusion and discussion

  • Action of the simplest holographic superfluid model

[Hartnoll, Herzog and Horowitz, arXiv:0803.3295]

I =

  • M

d4x√−g(−1 4FABFAB − |DΨ|2 − m2|Ψ|2).

  • Background metric

ds2 = L2 z2 [−f (z)dt2 − 2dtdz + dx2 + dy2], f (z) = 1 − z3 z3

h

.

  • Heat bath temperature

T = 3 4πzh .

Yu Tian (田雨) From Laminar Flow to Wave Turbulence in Holographic Superfluid

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Motivation and introduction Laminar-turbulent transition in holographic superfluids Wave turbulence in holographic superfluids Conclusion and discussion

  • The hairless-hairy phase transition of the black hole occurs at

the critical electric potential (chemical potential) µc = 4.06.

  • The above transition is interpreted as the normal-superfluid

phase transition of the boundary system.

Yu Tian (田雨) From Laminar Flow to Wave Turbulence in Holographic Superfluid

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Motivation and introduction Laminar-turbulent transition in holographic superfluids Wave turbulence in holographic superfluids Conclusion and discussion

The laminar-turbulent transition by shaking (periodic driving)

  • Shaking a holographic superfluid in a periodic box of length L

with an appropriate frequency ω ux = A sin ωt

  • Random initial perturbations
  • The laminar-turbulent transition observed at the shaking

amplitude A = Ac

Yu Tian (田雨) From Laminar Flow to Wave Turbulence in Holographic Superfluid

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Motivation and introduction Laminar-turbulent transition in holographic superfluids Wave turbulence in holographic superfluids Conclusion and discussion

  • The case of laminar flow

Figure: Superfluid velocity fields for the shaking amplitude A < Ac at around the twentieth shaking cycles which changes direction and at the twentieth shaking cycles it is zero.

Yu Tian (田雨) From Laminar Flow to Wave Turbulence in Holographic Superfluid

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Motivation and introduction Laminar-turbulent transition in holographic superfluids Wave turbulence in holographic superfluids Conclusion and discussion

  • The case of turbulent flow

Figure: Superfluid velocity fields for the shaking amplitude A > Ac before, at and after the twentieth shaking cycles where the total net velocity changes its direction. For the middle panel, the total net velocity is approximately zero.

Yu Tian (田雨) From Laminar Flow to Wave Turbulence in Holographic Superfluid

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Motivation and introduction Laminar-turbulent transition in holographic superfluids Wave turbulence in holographic superfluids Conclusion and discussion

Figure: The configurations of |ψ| after 1, 2, 3, 4, 5 and 25 shaking cycles for A > Ac.

Yu Tian (田雨) From Laminar Flow to Wave Turbulence in Holographic Superfluid

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Motivation and introduction Laminar-turbulent transition in holographic superfluids Wave turbulence in holographic superfluids Conclusion and discussion

  • Characterization of the laminar-turbulent transition by the

total kinetic energy Ekin(t) = 1 2u2|ψ|2d2x at integral shaking cycles

  • Characterization of the laminar-turbulent transition by vortex

formation

Yu Tian (田雨) From Laminar Flow to Wave Turbulence in Holographic Superfluid

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Motivation and introduction Laminar-turbulent transition in holographic superfluids Wave turbulence in holographic superfluids Conclusion and discussion

1 Motivation and introduction 2 Laminar-turbulent transition in holographic superfluids 3 Wave turbulence in holographic superfluids 4 Conclusion and discussion

Yu Tian (田雨) From Laminar Flow to Wave Turbulence in Holographic Superfluid

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Motivation and introduction Laminar-turbulent transition in holographic superfluids Wave turbulence in holographic superfluids Conclusion and discussion

Kinetic energy spectra and direct energy cascade

  • Kinetic energy spectra:

Ekin(t) = ∞ ǫkin(t, k)dk

  • Direct energy cascade in holographic superfluids:

log 10 k log 10 ϵkin(k)

  • 1.0
  • 0.5

0.5 1.0

  • 1

1 2 3 4 log 10 k log 10 ϵkin(k)

  • 1.0
  • 0.5

0.5 1.0

  • 1

1 2 3 4 log 10 ϵkin(k) log 10 k

  • 1.0
  • 0.5

0.5 1.0

  • 1

1 2 3 4 log 10 k log 10 ϵkin(k)

  • 1.0
  • 0.5

0.5 1.0

  • 1

1 2 3 4

Figure: ǫkin(k) after 2, 3, 4 and 6 shaking cycles, respectively.

Yu Tian (田雨) From Laminar Flow to Wave Turbulence in Holographic Superfluid

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Motivation and introduction Laminar-turbulent transition in holographic superfluids Wave turbulence in holographic superfluids Conclusion and discussion

Kolmogorov −5/3 scaling law in vortex turbulence

  • The turbulent dynamics is assumed to be characterized by the

energy dissipation rate per unit mass ε at one end of an inertial range k− < k < k+.

  • Dimensional analysis simply gives

ǫkin(k) = Cε2/3k−5/3 (ǫkin : [L3T −2], ε : [L2T −3], k : [L−1]) in the inertial range, where C is a dimensionless constant.

  • Kolmogorov −5/3 scaling law and direct energy cascade in

vortex turbulence (relaxation) of holographic superfluids

[Chesler, Liu and Adams, 2012; Lan, YT and Zhang, 2016]

Yu Tian (田雨) From Laminar Flow to Wave Turbulence in Holographic Superfluid

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Motivation and introduction Laminar-turbulent transition in holographic superfluids Wave turbulence in holographic superfluids Conclusion and discussion

The scaling law in shaken holographic superfluids

log10 ϵkin(k ) log10 k

  • 2.94 x + 4.38
  • 1.64 x + 3.69
  • 1.5
  • 1.0
  • 0.5

0.5 1.0 1 2 3 4 log 10 ϵkin(k)

  • 2.49 x + 4.78

log 10 k

  • 1.0
  • 0.5

0.0 0.5 1.0 1 2 3 4

Figure: The kinetic energy spectra ǫkin(k) for vortex relaxation (left) and shaking (right) in holographic superfluids. The left panel shows two scaling law, one the Kolmogorov −5/3 scaling law in the inertial range and the other the −3 scaling law characterizing free vortices. The right panel shows only one scaling law, which is different from −5/3 or −3. Wave turbulence?

Yu Tian (田雨) From Laminar Flow to Wave Turbulence in Holographic Superfluid

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Motivation and introduction Laminar-turbulent transition in holographic superfluids Wave turbulence in holographic superfluids Conclusion and discussion

Wave turbulence in holographic superfluids

  • decomposition: vortex (incompressible) and wave

(compressible) u = ui + uc ∇ · ui = 0, ∇ × uc = 0 Ei,c(t) = 1 2u2

i,c|ψ|2d2x =

∞ ǫi,c(t, k)dk ǫkin(t, k) = ǫi(t, k) + ǫc(t, k)

Yu Tian (田雨) From Laminar Flow to Wave Turbulence in Holographic Superfluid

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Motivation and introduction Laminar-turbulent transition in holographic superfluids Wave turbulence in holographic superfluids Conclusion and discussion

  • compressible to incompressible ratios:

ϵc (k )  ϵi (k ) log10 k

  • 1.5
  • 1.0
  • 0.5

0.5 1.0 5 10 15 20 25 30 log10 k ϵc (k )  ϵi (k )

  • 1.0
  • 0.5

0.5 1.0 5 10 15 20 25

Figure: The typical ratios of compressible to incompressible energy spectra ǫc(k)/ǫi(k) for vortex relaxation (left) and shaking (right). The left panel shows that waves mainly live at small k and vortices dominate the large k regime. The right panel is a telltale signature

  • f wave turbulence.

Yu Tian (田雨) From Laminar Flow to Wave Turbulence in Holographic Superfluid

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Motivation and introduction Laminar-turbulent transition in holographic superfluids Wave turbulence in holographic superfluids Conclusion and discussion

1 Motivation and introduction 2 Laminar-turbulent transition in holographic superfluids 3 Wave turbulence in holographic superfluids 4 Conclusion and discussion

Yu Tian (田雨) From Laminar Flow to Wave Turbulence in Holographic Superfluid

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Motivation and introduction Laminar-turbulent transition in holographic superfluids Wave turbulence in holographic superfluids Conclusion and discussion

Conclusion

  • The lamina-turbulent transition is observed by shaking 2D

holographic superfluids.

  • The turbulent state of the shaken holographic superfluids has

a ∼ −2.5 scaling law.

  • The turbulent state of the shaken holographic superfluids is

identified as a wave turbulence.

Yu Tian (田雨) From Laminar Flow to Wave Turbulence in Holographic Superfluid

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Motivation and introduction Laminar-turbulent transition in holographic superfluids Wave turbulence in holographic superfluids Conclusion and discussion

Discussion

  • Onset mechanism of 2D quantum turbulence (in holographic

superfluids)?

  • Chaotic behavior from linear analysis (Lyapunov exponents)?
  • More physical insights from the holographic point of view?

Yu Tian (田雨) From Laminar Flow to Wave Turbulence in Holographic Superfluid

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Motivation and introduction Laminar-turbulent transition in holographic superfluids Wave turbulence in holographic superfluids Conclusion and discussion

元宵节快乐！

Happy Lantern Festival!

Yu Tian (田雨) From Laminar Flow to Wave Turbulence in Holographic Superfluid

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Motivation and introduction Laminar-turbulent transition in holographic superfluids Wave turbulence in holographic superfluids Conclusion and discussion

Thanks for your attention!

Yu Tian (田雨) From Laminar Flow to Wave Turbulence in Holographic Superfluid