Physics of the Power Threshold Minimum for L-H Transition M.A. Malkov University of California, San Diego 1 Collaborators: P.H. Diamond , K. Miki 2, J.E. Rice 3 and G.R. Tynan 1 Supported by US DoE 1 / 18 Outline 1 1 Motivation P - the LH transition Power Threshold -Micro - Macro connection → How does physics set Pthr ? thr 2 3 4 5 Key Questions Reduced Model Basic Structure Te and Ti equations Anomalous Coupling Model Studies Recovering the Minimum Towards the Anomalous Regime (Preliminary, if time allows) Conclusions 1 N.B. No discussion of hysteresis, back transition due to time limitation 2 / 18 Motivation and brief history of LH studies L→H transition is a 33 (!) year-old story (Wagner, revolutionized connement physics central to ITER ignition Underlying ideas dimensional analysis (e.g. scalings in general Pthr ∝ nBS et al 1982) Connor and Taylor, 1977) and simple early phenomenology (t) Pthr ∝ n0.7 - inconsistent with the minimum in Pthr (n) connection of the power threshold to the edge parameters (Fukuda et al 1988): evolving story Mechanism: shear suppression paradigm (Biglari, Diamond and Terry, 1990 ++) 3 / 18 Emerging Scenario LH-triggering sequence of events Q ↑ =⇒ etc. ñ, ṽ ↑=⇒ < ṽr ṽϑ >; < ṽr ṽϑ > d < v >/dr ↑ =⇒ =⇒ ∇Pi | ↑ =⇒ lock in transition (Tynan |ñ|2 ↓, 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 → ∇ hPi i builds up at the edge, accompanied by electric eld shear ∇ hPi i → hVE i0 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) 4 / 18 Some Questions: How does the scenario relate to the Power Threshold? Is Pthr (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 Pthr (n)? Energy coupling? Will Pmin persist in collisionless, electron-heated regimes (ITER)? 5 / 18 Further Questions and important Clue: J. Hughes, Y. Ma, J. Rice, 2011,12 density at LOC/SOC transition (1020/m3) 1.4 H-mode threshold density minimum 1.2 Is P set only by local properties at the edge? (Common wisdom) Is P minimum related to collisional energy transfer? i.e. ν n (Te − Ti ). Low n branch couples to ions, enables ∇Pi ? P (n) minimum correlates with n 'LOC-SOC' transition ⇒ i.e. min power related to collisional inter-species transfer Threshold is controlled by global transport processes!? thr FT 1.0 2.8 < q < 3.8 0.8 C-Mod thr TCV 0.6 1/R 0.4 TEXT ASDEX, AUG JFT-2M T-10 0.2 JT-60 Tore Supra DIII-D JET 0.0 0.0 0.5 1.0 1.5 2.0 R (m) Rice et al., 2009 2.5 3.0 3.5 thr 6 / 18 Scenario (inspired partly by F. Ryter, 2013-14) ∇Pi |edge essential to 'lock in' transition to form ∇Pi at low n, etc. need (collisional) energy transfer from electrons to ions ∂ Te 1 ∂ 2m + rΓ = − (T − Ti ) + Qe ∂t r ∂r e Mτ e 1 ∂ 2m ∂ Ti + rΓ = + (T − Ti ) + Qi ∂t r ∂r i Mτ e suggests that the minimum is due to: Pthr decreases due to increasing heat transfer from electrons to ions Pthr increases (stronger edge∇Pi 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 Tr = τEe /τei - most important in pure e-heating regimes Tr 1 somewhat equivalent to direct ion heating Tr 1 ions remain cold → no LH transition (or else, it's anomalous!) 7 / 18 Predator-Prey Model Equations Based on 1-D numerical 5-eld model (Miki & Diamond++ 2012,13+) Currently operates on 6 elds (+Pe ) 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): ∂ Pe 1 ∂ 2m + r Γ = − (Pe − Pi ) + Qe − γCTEM · I ∂t r ∂r e Mτ 1 ∂ 2m ∂ Pi + r Γi = (P − Pi ) + Qi + γCTEM · I + γZFdiss · I ∂t r ∂r Mτ e √ 2 2 √ 2 ∂P ∂ E0 ∂ E0 Γ = − (χneo + χt ) , γZFdiss = γvisc + γHvisc ∂r ∂r ∂r 2 I and E0 - DW and ZF energy (next VG), plasma density and the mean ow, as before 8 / 18 Equations cont'd; Anomalous Heat Exchange in high Te 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 ∂I = γ − ∆ω I − α0 E0 − αV hVE i02 I + χN I , χN ∼ ω∗ Cs2 ∂t ∂r ∂r Driver : γ = γITG +γCTEM +NL ZF Dissip less Pi Heat (currently balanced ) ZF energy: ∂ E0 = ∂t α0 I − γdamp E0 , γdamp = γcol + γZFdiss · I /E0 1 + ζ0 hVE i02 √ 2 2 √ 2 γZFdiss = γvisc ∂ ∂ rE0 + γHvisc ∂ ∂ r 2E0 - toy model form (work in progress) 9 / 18 Model studies: Transition (Collisional Coupling) ion heat dominated transition Hi /(i +e ) = 0.7 0.005 strong pre-transition uctuations of all quantities well organized post-transition ow strong Pe edge barrier 0.004 0.003 0.002 0.001 000 10 10 000 5 000 0 0 0.2 0.4 0.6 0.8 0.0025 6 0.002 0.0015 4 0.001 0.0005 0 1 5 000 10 000 0.5 5 000 2 00 15 000 0.2 10 000 0.4 0.6 0.8 0 0 5 000 1 0 10 / 18 Model Studies: Control Parameters Heating mix Hi /(i +e ) ≡ Qi Qi + Qe (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 Hi /(i +e ) 11 / 18 Pth n, Hi /(i +e ) scans: Recovering the Minimum Relate Hi /(i +e ) and n by a monotonic Hi /(i +e ) (n) 0.008 0.006 0.004 0.000 2.5 0.0 0.2 2.0 0.4 Hmix 1.5 0.6 1.0 0.8 0.5 1.0 P thr n 0.010 1.0 0.008 0.8 Hi/(i+e) 0.006 0.6 0.0 Hi /(i +e ) , n - Pthr(n) min 0.004 Hmix 0.002 Pthr Pthr 0.4 electron heating at lower densities ion heating at higher densities 0.002 0.000 0.2 0.5 P thr 1.0 20 -3 1.5 2.0 0.0 n/10 m (n) min recovered! 12 / 18 Summary of collisional coupling results P (n) grows monotonically in both pure ion Hi /(i +e ) = 1 and pure electron Hi /(i +e ) = 0 heating regimes with collisional coupling thr The descending (low-density) branch, followed by a distinct minimum, results from a combination of: 1 2 increase in electron-to-ion collisional heat transfer and growing fraction of heat Hi /(i +e ) ↑ deposited to ions (relative to total heat) The later upturn of P (n) is due to increase of the shear ow damping The heating mix ratio Hi /(i +e ) 6= 0 is essential for the heat transport from the core to build up the ion pressure gradient at the edge, ∇Pi , which is the primary driver of the LH transition There are many possibilities to render Hi /(i +e ) 6= 0 thr 13 / 18 Anomalous Regime (Preliminary) Anomalous Regime: νei n (Te − Ti ) < γ '78; Zhao, PD, 2012; Garbet, 2013) −eicoupl anom ·I (Manheimer, Anomalous regime, strong electron heating (ITER) n scaling coupling =⇒ Anomalous coupling ∂ Te 1 ∂ 2m + r Γ = − (Te − Ti ) + Qe ∂t r ∂r e Mτ ∂ Ti 1 ∂ 2m + r Γ = + (Te − Ti ) + Qi ∂t r ∂r i Mτ Anomalous coupling dominates scaling + intensity dependence =⇒coupling ∂ Te 1 ∂ + r Γ = Qe + hE · Je i → (< 0) ∂t r ∂r e ∂ Ti 1 ∂ + r Γ = Qe + hE·Ji i → (> 0) ∂t r ∂r i 14 / 18 LH transition: Anomalous Transfer Dominates Extreme limit to illustrate temperature relaxation: Pure electron heating, νei → 0 % turbulence & ions Is Pthr set only by local properties at the edge? CTEM →Heat Exch 0.8 0.6 2 1.5 0.4 1 0.5 e − i -temperature equilibration front Pi ↑ globally→strong ∇Pi at the edge 0.2 0 0 0.2 0 0.4 0.6 0.8 1 → LH transition 0.25 0.1 0.2 2 0.05 1.5 1 0.5 0 0 0.2 0.4 0.6 0.8 1 0 0.15 2 0.1 1 0.05 0 0 0.2 0.4 0.6 0.8 0 1 15 / 18 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 → Pi Results so far transition with anomalous heat exchange happens! requirements for LH transition in high Te 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 2014. Subcritical ∇Te ↑ states at ultra-low density 16 / 18 Conclusions 1 2 3 4 5 density minimum in Pthr (n) is recovered in the extended model Pthr decrease: due to e → i heat transfer and ion heating increase Pthr increase: due to increase in ow damping ion heat channel (direct or indirect ⇔ through electrons) is ultimately responsible for LH transitions The role of Tr = τEe /τequil (global quantity!) in LH is crucial: a2 /DGB τequil 1 - no electron-heated LH transition a2 /DGB τequil 1 - LH trans. originated by electron heat. is possible Threshold physics requires, but is not limited, to edge physics anomalous heat exchange important in low collisionality, anomalous coupling regimes (collisional e − i heat coupling negligible) Anomalous exchange ⇔ Fluctuation intensity dependent CTEM driven turbulence dissipation → ion heating ITG driven turbulence dissipation → ion heating ZF dissipation → ion heating 6 Density minimum is TBD 17 / 18 Future (ongoing) work continue exploration of anomalous regime explore eects of ZF anomalous spreading back transitions, hysteresis fate of minimum in anomalous regime what are relevant global parameters? toroidal rotation geometry/conguration (builds on Fedorczak, PD, et al. 2012) → Collisionless saturation/damping of CTEM-driven ZF is fundamental issue 18 / 18