An Explanation for the High-Beta Runaway: the Non-Zonal Transition - PowerPoint PPT Presentation

An Explanation for the High-Beta Runaway: the Non-Zonal Transition M.J. Pueschel many thanks to : P .W. Terry, F. Jenko, D.R. Hatch, W.M. Nevins, T. G¨ orler, D. Told, A.E. White 55th Annual Meeting of the APS Division of Plasma Physics Denver, November 12, 2013

What is the runaway? Zonal Flow Dynamics Field Line Decorrelation 1 What is the runaway? 2 Zonal Flow Dynamics 3 Field Line Decorrelation M.J. Pueschel The Non-Zonal Transition

What is the runaway? Zonal Flow Dynamics Field Line Decorrelation Gyrokinetics and G ENE G ENE : gene.rzg.mpg.de Gyrokinetics : eliminate nonlinear gyrokinetic the fast gyrophase from equations the equations of motion radially local and ⇒ significant speed-up nonlocal modes δ f approach general geometry equilibria linear eigenvalue solver ⇒ gyrokinetic Vlasov , collisions, field equations electromagnetic ( A � , B � ) M.J. Pueschel The Non-Zonal Transition

What is the runaway? Zonal Flow Dynamics Field Line Decorrelation Electromagnetic Cyclone Standard parameter set: Cyclone Base Case (Dimits 2000) ⇒ three linear regimes: ITG, TEM, KBM Also : no sudden changes among subdominant/stable modes Nonlinearly: saturation often only temporary at high β ⇒ turbulent amplitudes tend to grow to extremely large values M.J. Pueschel The Non-Zonal Transition

What is the runaway? Zonal Flow Dynamics Field Line Decorrelation Nomenclature the high- β runaway is a consequence of the non-zonal transition (NZT) M.J. Pueschel The Non-Zonal Transition

What is the runaway? Zonal Flow Dynamics Field Line Decorrelation The High- β Runaway Cyclone Base Case: above β NZT = 0 . 8 % , no simple saturation crit See : Nevins APS 2009, Waltz 2010, Pueschel 2013 PRL/PoP simulations with β > β NZT experience (possibly long) crit transient saturation , then continue to grow with γ ITG high- Q regime only relevant for qualitative arguments, no physical meaning (fluxes too high � δ f approximation) high- Q : nonlinear frequency matches linear, ω NL ≈ ω ITG lin M.J. Pueschel The Non-Zonal Transition

What is the runaway? Zonal Flow Dynamics Field Line Decorrelation KBM behavior Kinetic Ballooning Mode: highly detrimental to confinement Does γ KBM linger subdominantly near zero at lower β ? ⇒ No KBM influence expected below the linear threshold nonlinear β KBM same as linear (Pueschel 2008, 2010) crit Waltz 2010 : tertiary subcritical KBM excitation possible Pueschel 2013 : subcritical KBM not the reason for runaway M.J. Pueschel The Non-Zonal Transition

What is the runaway? Zonal Flow Dynamics Field Line Decorrelation NZT Occurrences Robust phenomenon , re- Also occurs for other parame- ter regimes: pure-ITG case produced by multiple codes: (Pueschel 2010), GA-std (Waltz 2010): shown: G ENE , G YRO , GKW Also : extensive numerical tests underscore no runaway observed physical nature for TEM turbulence M.J. Pueschel The Non-Zonal Transition

What is the runaway? Zonal Flow Dynamics Field Line Decorrelation 1 What is the runaway? 2 Zonal Flow Dynamics 3 Field Line Decorrelation M.J. Pueschel The Non-Zonal Transition

What is the runaway? Zonal Flow Dynamics Field Line Decorrelation Zonal Flow Activity β < β NZT β > β NZT crit : strong ZFs crit : ZFs break up Zonal flows can no longer saturate ITG ⇒ non-zonal transition (NZT) M.J. Pueschel The Non-Zonal Transition

What is the runaway? Zonal Flow Dynamics Field Line Decorrelation Secondary Instability Secondary instability analysis: get zonal flow growth rate Usual procedure : three modes (ITG, zonal flow, sideband) Here : need extended mode structure to resolve linear physics half of all k x connected to linear mode, other half used as sideband freeze linear mode at constant amplitude let nonlinearity channel energy to zonal flow ⇒ no discontinuity near β NZT crit , no weakened ˆ γ ZF , thus zonal flow growth cannot be the cause of the NZT M.J. Pueschel The Non-Zonal Transition

What is the runaway? Zonal Flow Dynamics Field Line Decorrelation Impact of Magnetic Perturbations Radial fluctuations break flux surfaces, short-circuit ZFs Rosenbluth-Hinton residual residual with resonant B x ↔ ZF impact (non-resonant: no impact) magnetic fluctuations erode residual quadratic erosion/“decay”, with t Φ= 0 ∝ B − 1 x M.J. Pueschel The Non-Zonal Transition

What is the runaway? Zonal Flow Dynamics Field Line Decorrelation Analytical Theory Rosenbluth-Hinton with resonant B x (Terry 2013) ⇒ look at electrons peeling of flux surface due to B x ∂ f e k y A � A � k y eF 0 k x Φ = ˆ S Φ ∂ t + v � k x f e − v � e δ ( t ) B 0 T e B 0 ( electron source replaces nonlinearity: energy pulse at t = 0 ) After a lengthy calculation, arrive at Φ( t ) n e ( t = 0 ) / Φ( t = 0 ) � α 2 t 2 = 1 − � Φ R 0 /ǫ 1 / 2 k 2 x ρ 2 1 + 1 . 6 q 2 s t With α = A � k x k y v th , e / B 0 ≪ t − 1 , get (in normalized units) m i T e B 2 x t 2 Φ R − Φ( t ) ∝ Φ R m e T i M.J. Pueschel The Non-Zonal Transition

What is the runaway? Zonal Flow Dynamics Field Line Decorrelation NZT Energetics Look at energetics of saturation below, above β NZT crit : � ∗ � � d z d v � d µ T j 0 F j 0 � E k = g j k + q j χ j k g j k F j 0 T j 0 j Nonlinear transfer from 0.4 β =0 . 7% ITG, moderated by ZF : β =0 . 9% y ( NZ ) | 0.2 β < β NZT crit : zonal x ,k ′ 0.0 y ( Z ) / | <N > k ′ flows facilitate −0.2 saturation −0.4 x ,k ′ β > β NZT <N > k ′ crit : little net −0.6 impact due to zonal −0.8 flows −1.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 k y ρ s M.J. Pueschel The Non-Zonal Transition

What is the runaway? Zonal Flow Dynamics Field Line Decorrelation 1 What is the runaway? 2 Zonal Flow Dynamics 3 Field Line Decorrelation M.J. Pueschel The Non-Zonal Transition

What is the runaway? Zonal Flow Dynamics Field Line Decorrelation Field Line Decorrelation B x decomposes into (flux-surface-breaking) resonant and into non-resonant part What if field line decorrelates along its poloidal trajectory? ⇒ if half-turn displacement ∆ r 1 / 2 exceeds correlation length λ Bxx , non-resonant part contributes to stochasticity ! M.J. Pueschel The Non-Zonal Transition

What is the runaway? Zonal Flow Dynamics Field Line Decorrelation Other Examples Other physical parameters confirm decorrelation theory : GA standard case: TEM case (Pueschel 2010): ( direct confirmation ) ( indirect confirmation ) Note : density-gradient-driven TEM relies on zonal flows for transport regulation ⇒ expect no runaway , but could have NZT (diminished zonal flows → higher flux) M.J. Pueschel The Non-Zonal Transition

What is the runaway? Zonal Flow Dynamics Field Line Decorrelation Gradient Dependencies NZT very sensitive to background gradients ω T , n = R 0 / L T , n ⇒ transition quickly disappears at values below CBC, is only observed for large gradients (limit cycle?) M.J. Pueschel The Non-Zonal Transition

What is the runaway? Zonal Flow Dynamics Field Line Decorrelation Summary new critical β : non-zonal transition of ITG turbulence at β > β NZT crit , ITG mode no longer saturated by zonal flows not observed for TEM turbulence, although some quantitative impact possible due to reduced ZFs zonal flows critically weakened by resonant B x (captured by residual flow simulations, analytical theory) field lines decorrelate from field at half turn, non-resonant B x becomes flux-surface-breaking β NZT strongly dependent on background gradients crit NZT may be related to linear → saturated Ohmic confinement transition (as one among multiple causes) ≈ 8 . 6 / ( q 0 χ 1 / 2 preliminary : simple estimate β NZT crit /β KBM eff ) crit preliminary : transport time in system with NZT ∝ ( τ ITG lin ) 1 / 2 see : M.J. Pueschel et al. PRL & 2 × PoP 2013, P .W. Terry et al. 2013 M.J. Pueschel The Non-Zonal Transition