Discrete effects in Wave Turbulence

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Discrete effects in Wave Turbulence

• Sergey Nazarenko (Warwick)
• Collaborators Yeontaek Choi, Colm Connaughton,
  Petr Denissenko, Uriel Frisch, Sergei Lukaschuk,
  Yuri and Victor Lvov, Elena Kartashova, Dhuba
  Mitra, Boris Pokorni, Andrei Pushkarev, Vladimir
  Zakharov.

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What is Wave Turbulence?

• WT describes a stochastic field of weakly
  interacting dispersive waves.

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Other Examples of Wave Turbulence

• Sound waves,
• Plasma waves,
• Waves in Bose-Einstein condensates,
• Kelvin waves on quantised vortex filaments,
• Interstellar turbulence solar wind,
• Waves in Semi-conductor Lasers,
• Spin waves.

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How can we describe WT?

• Deterministic equations for the wave field.
• Weak nonlinearity expansion.
• Statistical averaging.
• Large-box limit.
• Long-time limit.

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The order of the limits is essential

• Nonlinear resonance broadening must be much wider
  than the spacing of the discrete (because of the
  finite box) Fourier modes.

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Drift waves in plasma and Rossby waves in GFD
Charney-Hasegawa-Mima equation

• ? -- electrostatic potential
  (stream-function)
• ? -- ion Larmor radius (by Te) (Rossby radius)
• ß -- drift velocity (Rossby
  velocity)
• x -- poloidal arc-length (east-west)
• y -- radial length
  (south-north)

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Weakly nonlinear drift waves with random phases?
wave kinetic equation (Longuet-Higgens Gill,
1967)
Resonant three-wave interactions
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Characteristic evolution times

• Deterministic T1/?,
• Stochastic T1/?2 due to cancellations of
  contributions of random-phased waves.
• What happens if the box is finite and the
  resonance broadening is of the order or smaller
  than the spacing of discrete k-modes?

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3 possibilities

• Exact frequency resonances are absent in for
  discrete k on the grid e.g. capillary waves
  (Kartashova 91).
• Some (usually small) number of resonances
  survives e.g. deep water surface waves
  (Kartashova 94,07, Lvov et al 05).
• All resonances survive e.g. Alfven waves
  (Nazarenko 07)

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Capillary waves (Connaughton et al 2001)

• Quasi-resonant interactions.
• d depends on the wave intensities.
• Exists dcrit for solutions to appear.

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Turbulent cascades

• Start with a set of modes at small ks.
• Find quasi-resonances and add the new modes to
  the original set.
• Continue like this to build further cascade
  steps.

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Cascade stages
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Critical intensity for initiating the cascade to
infinite ks.

• Cascade dies out in a finite number of steps if
  dlt 2nd crit value, and it continues to infinite k
  otherwise.

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Deep water gravity waves (Choi et al 2004,
Korotkevich et al 2005)

• Surviving resonances cause k-space intemittency
• Low k modes are more intermittent discrete
  effects.
• Theory in Choi et al predicts power-law PDF
  tails.

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Only one critical d once started the cascade
proceeds to infinity
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Evolving turbulence VERY weak field (Rossby-
Kartashova et al 89, water - Craig 90s, Alfven
Nazarenko06)

• Only modes which are in exact resonance are
  active
• Sometimes sets of resonant modes are finite no
  cascade to high k, deterministic recursive
  (periodic or chaotic) dynamics e.g. Rossby
  (Kartashova Lvov 07).
• Sometimes resonant triads (or quartets) form
  chains leading to infinite ks e.g. Alfven or
  NLS cascades possible poorly studied.
• But
• Most interesting question is how
  discrete/deterministic evolution at low k gets
  transformed into continuous/random process at
  high k. Mesoscopic turbulence. Some ideas
  suggested but a lot of work remains to be done.

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Mesoscopic turbulence sandpile model (Nazarenko
05)

• Put weak forcing at low ks.
• Originally, the wave intensity is weak
• -gt no quasi-resonances
• -gt accumulation of wave energy at low k until d
  reaches the critical value
• -gt initiation of cascade as an avalanche spill
  toward larger k.
• -gt value of d drops to its critical value
• -gt repeat the process

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Weak forcing -gt critical spectrum

• d dcrit -gtsandpile - E(?) ?-6.
• For reference
• Phillips spectrum E(?) ?-5
• Zakharov-Filonenko - E(?) ?-4.
• Kuznetsov (modified Phillips) - E(?) ?-4.

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Wave-tank experiments(Denissenko et al 06,
Falcon et al 06).

• Paris setup -gt 4cm to 1cm gravity range
  capillary range
• Hull setup -gt 1m to 1cm of gravity-wave range.

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Spectra

• Non-universality Steeper spectra for low
  intensities

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Exponents

• Agreement with the critical spectrum at low
  intensity.
• Phillips and Kuznetsov spectra at higher
  intensities (but not ZF)
• Forcing-independent intensity avalanches?

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Cascade sandpiles in numerics (Choi et al 05)

• Flux(t) at two different ks.

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Alfven wave turbulence

• Weak weak dynamics of only modes who are in
  exact resonances -gt enslaving to the 2D component
  (Nazarenko 06).
• Strong weak classical WT (Galtier et al 2000).
• Intermediate weak two component system?
  Avalanches?

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Summary

• Discreteness causes selective dynamics of k-modes
• -gt intermittency
• -gt anisotropic avalanches
• -gt need better theory, numerics and experiment
  for mesoscopic wave systems