Feedback Loop in 2D: Formation of Zonal Jets and Suppression of Turbulence

Sergey Nazarenko, Warwick, UK

• Balk, Nazarenko & Zakharov, 1990

•LH transitions in plasmas.•Barotropic governor.

Drift waves in fusion devices

Rossby waves in atmospheres of rotating planets

Charney-Hasegawa-Mima equation

• Ψ – streamfunction (electrostatic potential).• ρ – Deformation radius (ion Larmor radius).• β – PV gradient (diamagnetic drift). • x – east-west (poloidal arc-length)• y – south-north (radial length).

Baroclinic instability and ITG

• Close analogy between the baroclinic instability and the ion-temperature gradient instability in tokamak.

• GFD: two-layer model.

• Plasma: Hasegawa-Wakatani model for plasma potential and density.

Low-to-High confinement transitions in fusion plasmas

• LH transition discovered in the ASDEX tokamak (Wagner,1982) and now routinely observed in most tokamaks and stellarators.

• Left: Heliac data (Shats et al, 2004)

• ZF generation• DW suppression

LH transition paradigm

• Small-scale turbulence causes anomalous transport, hence L-mode.• Negative feedback loop.• Suppressed turbulence →no transport →improved confinement & H-mode.

Balk, SN and Zakharov 1990

Barotropic governor in GFG• James and Gray’ 1986

Zonal flow generation: the local turbulence view.

• CHM becomes 2D Euler equation in the limit β→0, kρ→∞. Hence expect similarities to 2D turbulence.

• Inverse energy cascade and direct cascade of potential enstrophy (FjØrtoft’53 argument).

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

Ubiquitous features in Drift/Rossby turbulence

• Condensation into zonal jets in presence of β.

Rhines scale crossover

• Nonlinear=linear → Rhines scale.

• “Lazy 8” separates vortex-dominated and wave-dominated scales (Rhines’75, Valis & Maltrud’93, Holloway’84)

• Outside of lazy-8: Kraichnan’s isotropic inverse cascade.

• Inside lazy-8 the cascade is anisotropic and dominated by triad wave resonances.

Weakly nonlinear drift waves with random phases→ wave kinetic equation (Longuet-Higgens &Gill, 1967)

Resonant three-wave interactions.

Anisotropic cascades in drift turbulence

• CHM has a third invariant (Balk, SN, Zakharov, 1990).

• 3 cascades cannot be isotropic.

•Potential enstrophy Q and the additional invariant Φ 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.

Cartoon of nonlocal interaction

• Eddy scale L decreases via shearing by ZF• Potential enstrophy Z is conserved.• => Eddy energy E =Z L2 is decreasing• Total E is conserved, => E is transferred from the eddy to ZF• Wrong! Both smaller and larger L’s are produced. The energy of the

eddy is unchanged. (Kraichnan 1976).

Victor P. Starr,Physics of Negative Viscosity Phenomena (McGraw Hill Book Co., New York 1968).

Small-scale energy conservation

• Energy in SS eddies is conserved if they are initially isotropic (Kraichnan 1976)

• 1. Dissipation: ellipse cannot get too thin.• 2. Nonisotropic eddies: Modulational

Instability (Lorenz’72, Gill’74, Manin, Nazarenko, 1994; Manfroi, Young, 1999; Smolyakov et al, 2000)

• 3. Breaking of the scale separation due to inverse cascade

Nonlocal 2D turbulence

• Condensate forms – interaction of scales becomes nonlocal (Smith & Yakhot’93, Maltrud &Valis’93, Borue’94, Laval, SN & Dubrulle’99).

• Small-scale spectrum changes to E~s-1ε k-1. (Kraichnan 1974, SN & Laval 2000; Connaugton et al 2007).

Connaugton et al 2007Nazarenko & Laval 2000

Condensate coupled with turbulence1) Instability forcing:

ε(t) ~ γ(kf) E(kf) kf 2) Spectrum of small-scale

turbulence: E(kf) ~ s-1ε kf

-1

3) Condensate energy: Ec ~ Vc

2/2 ~ s2L2 ~ ∫ε(t) dt,4) 1,2 =>

(i) E(kf) =0, - suppression of turbulence by jets;

(ii) s ~ γ(kf) – saturation of the jets.

2D turbulence governor

Feedback loop in 2D turbulence

• Instability generates small-scale turbulence.• Inverse cascade leads to energy condensation

(into jets in presence of beta).• Jets kill small-scale turbulence and saturate.• LH transition: this is why ITER must work.• Barotropic governor and other GFD mechanisms.

Modulational Instability Manin, Nazarenko, 1994; Manfroi, Young, 1999;

Smolyakov et al, 2000;Ongoing numerics: Connaughton, Nadiga, SN, Quinn.

• Unstable if

3ky2 < kx

2 +ρ-2

Nonlinear development of MI:narrow zonal jets

• Formation of intense narrow Zonal jets.• Transport/mixing Barriers. Analog of LH transition in fusion plasmas.• Secondary instability preferentially breaks westward jets (consistent

with linear condition β-uyy <0 ?).• Irregular multiple jets with westward preference • Rhines spectrum: E ~ β2 k-5. Chekhlov et al’95.

Evolution in the k-space

• Energy of WP is partially transferred to ZF and partially dissipated at large k’s.

• 2 regimes: random walk/diffusion of WP in the k-space (Balk, Nazarenko, Zakharov, 1990),

• Coherent wave – modulational instability (Manin, Nazarenko, 1994, Smolyakov et at, 2000).

Fast mode: modulational instability of a coherent drift wave.

• Two component description Ψ = ΨL +ΨS.

• Small-scale Rossby wave sheared by large-scale ZF.

• Large-scale ZF pumped by RW via the ponderomotive force.

Evolution of nonlocal drift turbulence:retain only interaction with small k’s and Taylor-expand the

integrand of the wave-collision integral; integrate.

• Diffusion along curves

Ωk = ωk –βkx =conts.

• S ~ZF intensity

Drift-Wave instabilities

• Maximum on the kx-axis at kρ ~ 1.

• γ=0 line crosses k=0 point.

Different ways to access the stored free energy:Resistive instability, ITG, ETG.

Initial evolution

• Solve the eigenvalue problem at each curve.

• Max eigenvalue <0 → DW on this curve decay.

• Max eigenvalue >0 → DW on this curve grow.

• Growing curves pass through the instability scales

ZF growth

• DW pass energy from the growing curves to ZF.

• ZF accelerates DW transfer to the dissipation scales via the increased diffusion coefficient.

ZF growth

• Hence the growing region shrink.

• DW-ZF loop closed!

Steady state

• Saturated ZF.• Jet spectrum on a k-

curve passing through the maximum of instability.

• Suppressed intermediate scales (Dimits shift).

• Balanced/correlated DW and ZF

• (Shats experiment).

Shats experiment

• Suppression of inermediate scales by ZF

• Scale separation

• Nonlocal turbulence

Shats experiment

• Instability scales are strongly correlated with ZF scales

• Nonlocal scale interaction

Saturation of zonal flow

• Different expressions for random 3-wave (low γ) and coherent (high γ) regimes

• Intermediate range with Uzf ~ V*.• Only weak ZF damping dependence (important γ is at ρk~1).• No oscillatory behaviour. ZF cannot fall below the crit value because

it’d be immediately pumped due to renewed instability.

Summary

• Self-regulating DW-ZF system.• Drift turbulence creates ZF.• ZF kills drift turbulence and switches the

forcing off (cf Dimits shift). • For large grad T small scales reappear because ZF gets KH unstable.

• Predictions for the saturated ZF, scale separation, jet-like spectrum of drift turbulence.

• Experimental evidence in Heliac. Tokamaks?

Breakdown of local cascades

• Kolmogorov cascade spectra (KS) nk ~kxνx ky

vy. • 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

Coupled large-scale & small-scale motions (Dyachenko, Nazarenko, Zakharov, 1992)

Shear flow geometry

• North-Pacific zonal jets at 1000 m depth as seen in 58-year simulation with ECMWF climotological forcing (Nakano and Hasumi, 2005)

Co-authors and 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.

Ubiquitous features in Drift/Rossby turbulence

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

Drift wave – zonal flow turbulence paradigm

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

Zonostrophy invariant

• Extra quadratic invariant of CHM (Balk, Nazarenko, Zakharov, 1991).

• Conserved by triad interactions.

• At high k’s it is essentially Kraichnan’s isotropic inverse cascade.

• Anisotropy only occurs when Rhines scale is reached.

• Weak/wave turbulence theory inside lazy-8: Triad wave resonances.

Typical numerical result (from Naulin 2002)

Zonal flow formation

Plan

•Dual-cascade behavior.•Condensation into zonal jets:

• Zonostrophy invariant, •modulational instability,

•Rhines spectrum. •Transition to nonlocal interaction:

•Suppression of turbulence by zonal jets,•Reduced turbulence transport,

•LH transition.