Nonlocal Drift Turbulence: coupled Drift-wave/Zonal flow description Sergey Nazarenko, Warwick, UK

1. 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.

2. 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.A133 (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.A146 (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.A165 (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. A6 (1994) 1158-1167.

3. Drift waves in fusion devices Rossby waves in atmospheres of rotating planets

4. Charney-Hasegawa-Mima equation • Ψ -- electrostatic potential (stream-function) • ρ -- ion Larmor radius (by Te) (Rossby radius) • β -- drift velocity (Rossby velocity) • x -- poloidal arc-length (east-west) • y -- radial length (south-north)

5. Ubiquitous features in Drift/Rossby turbulence • Drift Wave turbulence generates zonal flows • Zonal flows suppress waves • Hence transport barriers, Low-to-High confinement transition

6. 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.

7. Drift wave – zonal flow turbulence paradigm • Local cascade is replaced by nonlocal (direct) interaction of the DW instability scales with ZF.

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

9. 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 Φforce 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.

10. Cartoon of nonlocal interaction in DW/ZF system • DW wavenumber k grows via shearing by ZF • DW action N=k2E is the potential enstrophy (Dyachenko, Nazarenko, Zakharov, 1992). • N is conserved => energy E=N/k2 is decreasing • Total E is conserved, hence E is transferred from DW to ZF No Galilean invariance => N ≠E/ωr

11. DW-ZF evolution in the k-space • Energy of DW is partially transferred to ZF and partially dissipated at large k’s. • 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.

12. Weakly nonlinear drift waves with random phases→ wave kinetic equation (Zakharov, Piterbarg, 1987) Resonant three-wave intractions

13. Breakdown of local cascades • Kolmogorov cascade spectra (KS) nk ~kxνx 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

14. Evolution of nonlocal drift turbulence • Diffusion along curves Ωk = ωk –βkx =conts. • S ~ZF intensity

15. 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, Electron Temperature Gradient (ETG), Ion Temperature Gradient (ITG) …

16. 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

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

18. ZF growth • Hence the growing region shrink.

19. 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).

20. Shats experiment • L-H transition • ZF generation • DW suppression

21. Shats experiment • Suppression of inermediate scales by ZF • Scale separation • Nonlocal turbulence

22. Shats experiment • Instability scales are strongly correlated with ZF scales • Nonlocal scale interaction

23. Fast mode: modulational instability of a coherent drift wave. • Two component description Ψ = ΨL +ΨS. • Small-scale DW sheared by large-scale ZF. • Large-scale ZF pumped by DW via the ponderomotive force.

24. Coupled Large-scale & small-scale motions (Dyachenko, Nazarenko, Zakharov, 1992)

25. Shear flow geometry

26. Modulational Instability Manin, Nazarenko, 1994, Smolyakov, Diamond, Shevchenko, 2000. • Q=Q0+Q̃,N=N0+Ñ,ΨL=Ψ̃, • Q=Q̃,Ñ,Ψ̃ ~ exp(λt +iκy). • Unstable if 3Q02 < P02 +ρ-2.

27. Nonlinear development of MI:finite-time singularity • Formation of intense narrow Zonal jets • Internal Transport Barriers?

28. 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 DW’s and when by coherent DW’s?