On the Collisionless Damping and Saturation of Zonal Flows

Transcription

1 On the Collisionless Damping and Saturation of Zonal Flows P.H. Diamond CMTFO, CASS and Dept. of Physics, UCSD, USA Collaborators: M. Malkov 1, P.-C. Hsu 1, S.M. Tobias 2,A. Ashourvan 1, G. Tynan, 1 O.D. Curcan 3 1 UCSD, 2 Leeds, 3 Ecole Polytechnique 1 / 31

2 Caveats Time limit reduction in scope relative to abstract Works in Progress 2 / 31

3 Outline Zonal Flows: Some Things We Know Major Unresolved Issue: Damping at Low Collisionality Something General: Non-Perturbative Approaches to the Structure of the Reynolds Stress. Something Specic: A Second Look at Reynolds Work (R T ) and What it (really) Means. Something Relevant: L H Threshold of Low Collisionality? Discussion 3 / 31

4 Advertisement Zonal Flows and Pattern Formation O. D. Gurcan and P.D. J. Phys. A, in press. emphasized real space approach, in contrast to PPCF. enlarged discussion of non-mfe connections. 4 / 31

5 I.) Zonal Flows: Some Things We know sheared n = m = 0 E B ows minimal inertia (Hasegawa) minimal damping (NMR) no radial transport k: nonlinear coupling drive: modulation, parametrics, etc. (see D I ) Better: Space (GD, 2015) inhomogeneous PV mixing drives PV: q = 2 ψ + βy (QG) ṽ r ñ = ṽ r 2 φ q = n 2 φ (HW) = r ṽ r ṽ ϑ = Flow 5 / 31

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7 Predator-Prey Paradigm Drift Waves Prey (P.D., et. al 1994) Zonal Flow Predator N/ t = γn αv 2 N ωn 2 +? V 2 / t = αnv 2 γ d V 2 [ γ NL ( N, V 2 ) V 2] drag dam- NL damping ν i,i ping (?!) drag regulates system! sets uctuation levels, etc. what of ν i,i 0, γ d 0 limit? need confront nonlinear damping and feedback! 7 / 31

8 II.) Major Unresolved Issues: Collisionless Damping Usual rejoinder to ν i,i 0, R/L T E/L Te? Kelvin Helmholtz Tertiary (Rogers) } linear instability of strongly sheared ZF. - n, T i, V driven (beyond classic Rayleigh) Complications: magnetic shear imposes signicant constraint - linear instability ν e?? 8 / 31

9 Issues cont'd... numerical results controversial, inconclusive KH must be accompanied by feed back to uctuation intensity (not addressed). no work on CTEM-driven ZFs N.B. : at -q, weak-shear de-stiened mode (ala' JET) is relevant improved core connement regime q 0 removes ŝ constraint ZF stability? (c.f. Z. Lu, et. al. ; TTF 2015) Question: How weak must ŝ be? 9 / 31

10 Issues cont'd... More interestingly: linear stability can't be only feedback mechanism in nonlinearly coupled system ambient uctuations scatter u eective viscosity? Related: models perturbative - often capture only lowest order eddy tilting eects non-perturbative approaches? general models of vorticity ux? impact of feedback on evolution? i.e. see what happens / 31

11 Something general: Non-Perturbative Approaches to the Reynolds Stress P.-C. Hsu, P.D., S.M. Tobias, Phys. Rev. E, 2015 P.-C. Hsu, P.D., PoP / 31

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19 Something Relevant: L H Threshold at Low Collisionality M. Malkov, P.D., et. al.; PoP 2015 P.-C. Hsu, P.D., S.M. Tobias, Phys. Rev. E, 2015 M.M., P.D., et. al. TTF 2015 (Invited) 19 / 31

20 Emerging Scenario for L H LH-triggering sequence of events Q = ñ, ṽ = < ṽ r ṽ ϑ >; < ṽ r ṽ ϑ > d < v >/dr = ñ 2, etc. = P i = lock in transition (Tynan et al. 2013) T etc. drives turbulence that generates low frequency shear ow via Reynolds stress Reynolds work coupling collapses the turbulence thus reducing particle and heat transport Transport weakens P i builds up at the edge, accompanied by electric eld shear P i V E locks in L H transition: (see Hinton,Staebler 1991, 93) Complex sequence of Transition Evolution and Alternative End States (I-mode) possible (D. Whyte et al. 2011) 20 / 31

21 Some Questions: How does the scenario relate to the Power Threshold? Is P thr (n) minimum recoverable? Ryter et al 2013 Micro-Macro connection in threshold, if any? How does micro-physics determine threshold scalings? What is the physics/origin of P thr (n)? Energy coupling? Will P min persist in collisionless, electron-heated regimes (ITER)? 21 / 31

22 Scenario (inspired partly by F. Ryter, ) P i edge essential to 'lock in' transition to form P i at low n, etc. need (collisional) energy transfer from electrons to ions T e + 1 t r r rγ e T i t + 1 r r rγ i suggests that the minimum is due to: = 2m Mτ (T e T i ) + Q e = + 2m Mτ (T e T i ) + Q i P thr decreases due to increasing heat transfer from electrons to ions P thr increases (stronger edge P i driver needed) due to increase in shear ow damping Power and edge heat ux are not the only crit. variables: also need the ratio of electron energy conf. time to exceed that of e i temp. equilibration T r = τ Ee /τ ei - most important in pure e-heating regimes T r 1 somewhat equivalent to direct ion heating T r 1 ions remain cold no LH transition (or else, it's anomalous!) 22 / 31

23 Predator-Prey Model Equations Based on 1-D numerical 5-eld model (Miki & Diamond ,13+) Currently operates on 6 elds (+P e ) with self-consistenly evolved transport coecients, anomalous heat exchange and NL ow dissipation (MM, PD, K. Miki, J. Rice and G. Tynan, PoP 2015) Heat transport, + Two species, with coupling, i,e (anomalous heat exchange in color): P e + 1 t r r rγ e = 2m Mτ (P e P i ) + Q e γ CTEM I P i t + 1 r r rγ i = 2m Mτ (P e P i ) + Q i + γ CTEM I + γ ZFdiss I ( ) Γ = (χ neo + χ t ) P 2 ( r, γ E0 2 E 0 ZFdiss = γ visc + γ Hvisc r r 2 I and E 0 - DW and ZF energy (next VG), plasma density and the mean ow, as before ) 2 23 / 31

24 Equations cont'd; Anomalous Heat Exchange in high T e low n regimes (pure e-heating) the thermal coupling is anomalous (through turbulence) ZF dissip. (KH?) supplies energy to ions, and returns energy to turbulence DW turbulence: ( ) I t = γ ωi α 0 E 0 α V V E 2 I + χ N r I I r, χ N ω Cs 2 Driver : γ = γ ITG +γ CTEM +NL ZF Dissip less P i Heat (currently balanced) ZF energy: ( ) E 0 α 0 I = t 1 + ζ 0 V E γ 2 damp E 0, γ damp = γ col + γ ZFdiss I /E 0 ( ) 2 ( γ ZFdiss = γ E0 visc r + 2 ) E 2- γhvisc 0 r toy model form (work in 2 progress) 24 / 31

25 Model studies: Transition (Collisional Coupling). ion heat dominated transition. strong pre-transition uctuations of all Hi /(i +e ) = 0.7 quantities well organized post-transition ow. strong Pe edge barrier / 31

26 Model Studies: Control Parameters Heating mix H i/(i+e) Q i Q i + Q e (aka H mix ) Density (center-line averaged) is NOT a control parameter. It is measured at each transition point Related control parameter is the reference density given through BC and fueling rate There is a complicated relation between density and ref. density Other control parameters: fueling depth heat deposition depth and width, etc. they appear less critical than H i/(i+e) 26 / 31

27 P th (n, H i/(i+e) ) scans: Recovering the Minimum P thr H mix P thr ( Hi/(i+e), n ) n 1.5 electron heating at lower densities ion heating at higher densities Pthr Relate H i/(i+e) and n by a monotonic H i/(i+e) (n) H i/(i+e) min n/10 20 m -3 P thr (n) P thr (n) min recovered! Hmix 27 / 31

28 Anomalous Regime (Preliminary) Anomalous Regime: ν ei n (T e T i ) < γ anom eicoupl I (Manheimer, '78; Zhao, PD, 2012; Garbet, 2013) Anomalous regime, strong electron heating (ITER) n scaling coupling = Anomalous coupling T e + 1 t r r rγ e T i t + 1 r r rγ i Anomalous coupling dominates = 2m Mτ (T e T i ) + Q e = + 2m Mτ (T e T i ) + Q i scaling + intensity dependence = coupling T e + 1 t r r rγ e = Q e + E J e (< 0) T i t + 1 r r rγ i = Q e + E J i (> 0) 28 / 31

29 LH transition: Anomalous Transfer Dominates Extreme limit to illustrate temperature relaxation: Pure electron heating, νei 0. CTEM 0.8 Is Pthr Exch % & turbulence ions set only by local properties at the edge? Heat. 1 e i -temperature equilibration front Pi globally strong Pi at the edge LH transition / 31

30 Anomalous Regime: Issues An Issue: Predator-Prey Shear Flow Damping Anomalous regime: collisional drag problematic Low collisionality what controls heat exchange? NL damping mediated by ZF instability (i.e. KH, tertiary; Rogers et al 2000; Kim, PD, 2003) hyperviscosity, intensity dependent Returns ZF energy to turbulence P i Results so far transition with anomalous heat exchange happens! requirements for LH transition in high T e regimes when the collisional heat exchange is weak: ecient ion heating by CTEM turbulence energy return to turbulence by ZF damping (caused by KH instability?!) may be related to Ryter Subcritical T e states at ultra-low density 30 / 31

31 Fundamental Problems Identied: ZF stability and saturation in CTEM regimes General theory of Reynolds work. 31 / 31