| Feedback Loop in 2D: Formation
of Zonal Jets and Suppression of Turbulence Sergey Nazarenko, Warwick, UK |
| Drift waves in fusion devices |
| Rossby waves in atmospheres of rotating planets |
| Charney-Hasegawa-Mima equation |
| 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. |
| Barotropic governor in GFG |
| James and GrayÕ 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 (Fj¯rtoftÕ53 argument). |
| Ubiquitous features in Drift/Rossby turbulence |
| Condensation into zonal jets in presence of β. |
| Rhines scale crossover |
| Nonlinear=linear → Rhines scale. | |
| ÒLazy 8Ó separates vortex-dominated and wave-dominated scales (RhinesÕ75, Valis & MaltrudÕ93, HollowayÕ84) | |
| Outside of lazy-8: KraichnanÕs 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) |
| Anisotropic cascades in drift turbulence |
| CHM has a third invariant (Balk, SN, Zakharov, 1990). | |
| 3 cascades cannot be isotropic. |
| 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 LÕs are produced. The energy of the eddy is unchanged. (Kraichnan 1976). |
| 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 (LorenzÕ72, GillÕ74, 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 & YakhotÕ93, Maltrud &ValisÕ93, BorueÕ94, Laval, SN & DubrulleÕ99). | |
| Small-scale spectrum changes to E~s-1ε k-1. (Kraichnan 1974, SN & Laval 2000; Connaugton et al 2007). |
| Condensate coupled with turbulence |
| Instability forcing: ε(t) ~ γ(kf) E(kf) kf | |
| Spectrum of small-scale turbulence: E(kf) ~ s-1ε kf-1 | |
| Condensate energy: Ec ~ Vc2/2 ~ s2L2 ~ ºε(t) dt, | |
| 1,2 => | |
| (i) E(kf) =0, - suppression of turbulence by jets; | |
| (ii) s ~ γ(kf) – saturation of the jets. | |
| 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 < kx2 +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 alÕ95. |
| Evolution in the k-space |
| Energy of WP is partially transferred to ZF and partially dissipated at large kÕs. | |
| 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 kÕs 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. |
| 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 itÕd 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 kyvy. | |
| Exact solutions of WKE É 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..." |
| 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 kÕs it is..." |
| At high kÕs it is essentially KraichnanÕs isotropic inverse cascade. | |
| Anisotropy only occurs when Rhines scale is reached. | |
| Weak/wave turbulence theory inside lazy-8: Triad wave resonances. | |
| 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. | |