Nonlocal Drift Turbulence: coupled DriftwaveZonal flow description Sergey Nazarenko, Warwick, UK

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Nonlocal Drift Turbulence coupled
Drift-wave/Zonal flow descriptionSergey
Nazarenko, Warwick, UK

• Published in 9 papers with collaborators over
  1988-1994.
• Approach adopted as a Low-to-High confinement
  transition paradigm (review by Diamond et al
  2005).
• Strong additional supporting evidence in recent
  experiments of Michael Shats group at ANU.

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Relevant publications

• Kolmogorov Weakly Turbulent Spectra of Some Types
  of Drift Waves in Plasma (A.B. Mikhailovskii,
  S.V. Nazarenko, S.V. Novakovskii, A.P. Churikov
  and O.G. Onishenko) Phys.Lett.A 133 (1988)
  407-409.
• Kinetic Mechanisms of Excitation of
  Drift-Ballooning Modes in Tokamaks (A.B.
  Mikhailovskii, S.V. Nazarenko and A.P. Churikov)
  Soviet Journal of Plasma Physics 15 (1989) 33-38.
• Nonlocal Drift Wave Turbulence (A.M.Balk,
  V.E.Zakharov and S.V. Nazarenko) Sov.Phys.-JETP
  71 (1990) 249-260.
• On the Nonlocal Turbulence of Drift Type Waves
  (A.M.Balk, S.V. Nazarenko and V.E.Zakharov)
  Phys.Lett.A 146 (1990) 217-221.
•  On the Physical Realizability of Anisotropic
  Kolmogorov Spectra of Weak Turbulence (A.M.Balk
  and S.V. Nazarenko) Sov.Phys.-JETP 70 (1990)
  1031-1041.
• A New Invariant for Drift Turbulence (A.M.Balk,
  S.V. Nazarenko and V.E. Zakharov) Phys.Lett.A 152
  (1991) 276-280.
• On the Nonlocal Interaction with Zonal Flows in
  Turbulence of Drift and Rossby Waves (S.V.
  Nazarenko) Sov.Phys.-JETP, Letters, June 25,
  1991, p.604-607.
• Wave-Vortex Dynamics in Drift and beta-plane
  Turbulence (A.I. Dyachenko, S.V. Nazarenko and
  V.E. Zakharov) Phys,Lett.A 165 (1992) 330-334.
• Nonlinear interaction of small-scale Rossby waves
  with an intense large-scale zonal flow. (D.Yu.
  Manin and S.V. Nazarenko) Phys. Fluids. A 6
  (1994) 1158-1167.

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Drift waves in fusion devices

• Rossby waves in atmospheres of rotating planets

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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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Ubiquitous features in Drift/Rossby turbulence

• Drift Wave turbulence generates zonal flows
• Zonal flows suppress waves
• Hence transport barriers, Low-to-High confinement
  transition

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Drift wave zonal flow turbulence paradigm

• Drift turbulence is a tragic superhero who
  carries the seed of its own destruction.
• No turbulence ?no transport ?improved confinement.

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Drift wave zonal flow turbulence paradigm

• Local cascade is replaced by nonlocal (direct)
  interaction of the DW instability scales with ZF.

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Zonal flow generation the local turbulence view.

• CHM becomes 2D Euler equation in the limit ß?0,
  k??8. Hence expect similarities to 2D turbulence.
• Inverse energy cascade and direct cascade of
  potential enstrophy.

• Inverse cascade leads to energy condensation at
  large scales.
• These are round(ish) vortices in Euler.
• Why zonal flows in CHM?

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Anisotropic cascades in drift turbulence

• CHM has a third invariant (Balk, Nazarenko,
  1991).
• 3 cascades cannot be isotropic.

• Potential enstrophy Q and the additional
  invariant F force energy E to the ZF scales.
• No dissipation at ZF ? growth of intense ZF ?
  breakdown of local cascades.
• Nonlocal direct interaction of the
  instability-range scales with ZF.

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Cartoon of nonlocal interaction in DW/ZF system
No Galilean invariance gt N ?E/?r

• DW wavenumber k grows via shearing by ZF
• DW action Nk2E is the potential enstrophy
  (Dyachenko, Nazarenko, Zakharov, 1992).
• N is conserved gt energy EN/k2 is decreasing
• Total E is conserved, hence E is transferred from
  DW to ZF

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DW-ZF evolution in the k-space

• Energy of DW is partially transferred to ZF and
  partially dissipated at large ks.
• 2 regimes random walk/diffusion of DW in the
  k-space (Balk, Nazarenko, Zakharov, 1990),
• Coherent DW modulational instability (Manin,
  Nazarenko, 1994, Smolyakov et at, 2000).
• Coherent interaction corresponds to faster LH
  transition.

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Weakly nonlinear drift waves with random phases?
wave kinetic equation (Zakharov, Piterbarg, 1987)
Resonant three-wave intractions
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Breakdown of local cascades

• Kolmogorov cascade spectra (KS) nk kx?x kyvy.
• Exact solutions of WKE if local.
• Locality corresponds to convergence in WKE
  integral.
• For drift turbulence KS obtained by Monin
  Piterbarg 1987.
• All Kolmogorov spectra of drift turbulence are
  proven to be nonlocal (Balk, Nazarenko, 1989).
• Drift turbulence must be nonlocal, - direct
  interaction with ZF scales

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Evolution of nonlocal drift turbulence

• Diffusion along curves
• Ok ?k ßkx conts.
• S ZF intensity

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Drift-Wave instabilities
Different ways to access the stored free
energy Resistive instability, Electron
Temperature Gradient (ETG), Ion Temperature
Gradient (ITG)

• Maximum on the kx-axis at k? 1.
• ?0 line crosses k0 point.

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Initial evolution

• Solve the eigenvalue problem at each curve.
• Max eigenvalue lt0 ? DW on this curve decay.
• Max eigenvalue gt0 ? DW on this curve grow.
• Growing curves pass through the instability
  scales

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ZF growth

• DW pass energy from the growing curves to ZF.
• ZF accelerates DW transfer to the dissipation
  scales via the increased diffusion coefficient.

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ZF growth

• Hence the growing region shrink.

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Steady state

• Saturated ZF.
• Jet spectrum on a k-curve passing through the
  maximum of instability.
• Suppressed intermediate scales (Shats
  experiment).
• Balanced/correlated DW and ZF (Shats experiment).

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Shats experiment

• L-H transition
• ZF generation
• DW suppression

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Shats experiment

• Suppression of inermediate scales by ZF
• Scale separation
• Nonlocal turbulence

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Shats experiment

• Instability scales are strongly correlated with
  ZF scales
• Nonlocal scale interaction

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Fast mode modulational instability of a coherent
drift wave.

• Two component description ? ?L ?S.
• Small-scale DW sheared by large-scale ZF.
• Large-scale ZF pumped by DW via the ponderomotive
  force.

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Coupled Large-scale small-scale motions
(Dyachenko, Nazarenko, Zakharov, 1992)
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Shear flow geometry
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Modulational Instability Manin, Nazarenko,
1994, Smolyakov, Diamond, Shevchenko, 2000.

• QQ0Q, NN0N, ?L?,
• QQ,N,? exp(?t i?y).
• Unstable if 3Q02 lt P02 ?-2.

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Nonlinear development of MIfinite-time
singularity

• Formation of intense narrow Zonal jets
• Internal Transport Barriers?

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Summary

• CHM model contains all basic mechanisms of the
  DW/ZF interactions.
• Drift turbulence creates its own killer ZF.
• Are these processes universal for all fusion
  devices? For the edge as well as the core
  plasmas?
• When should we expect generation of ZF by random
  DWs and when by coherent DWs?