Physics of Non-Diffusive Turbulent Transport of Momentum and the Origins of Spontaneous Rotation in Tokamaks

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1 1 Physics of Non-Diffusie Turbulent Transport of Momentum and the Origins of Spontaneous Rotation in Toamas P.H. Diamond 1), C. McDeitt 1), O.D. Gurcan 1), K. Mii 1), T.S. Hahm ), W. Wang ), G. Rewoldt ), I. Holod 3), Z. Lin 3), V. Naulin 4), R. Singh 5) 1) Uniersity of California at San Diego, La Jolla, CA USA ) Princeton Plasma Physics Laboratory, Princeton, NJ USA 3) Uniersity of California at Irine, Irine, CA USA 4) Association EURATOM-Risø DTU, Rosilde, DK-4000 Denmar 5) Institute for Plasma Research, Bhat, Gandhinagar, 3848 India contact of main author: phd@mamacass.ucsd.edu Abstract. The theory of turbulent transport of toroidal momentum is discussed in the context of the phenomenon of spontaneous/intrinsic rotation. We reiew the basic phenomenology and surey the fundamental theoretical concepts. We then proceed to an in-depth discussion of the radial flux of toroidal momentum, with special emphasis on the off-diagonal elements, namely the residual stress (the portion independent of V) and the pinch. A simple model is deeloped which unifies these effects in a single framewor and which recoers many of the features of the Rice scaling trends for intrinsic rotation. We also discuss extensions to finite beta and the effect of SOL boundary conditions. Seeral issues for future consideration are identified. 1. Summary of Phenomenology a) Intrinsic Rotation Basics Intrinsic (spontaneous) toroidal rotation obsered in nearly all toamas H-mode phenomenology demonstrates clear empirical trends, L-mode phenomenology remains mury and complex In H-mode: i. rotation typically co-current ii. " W / I p, M A N [1] iii. no apparent scalings with, i. offset in torque scan matches intrinsic rotation[] Obserations appear consistent with edge phenomena originating with transition i. Obsered co-current elocity builds inward from periphery [3] ii. rotation direction inerts at L-H mode transition b) Indications of Off-Diagonal Momentum Flux Historically, " i [4], yet many deiations from Pr 1 obsered Pi drien momentum pinch suggested by inductie analysis[5] Perturbation Experiments From JT-60U [6] i. ripple loss + pulsed beams > pulsed torque ii. inward V clearly indicated V V V obsered in β-scan on JT-60U [7] residual measured perturbation Vresidual coincident with region of steep Pi c) Boundary Condition Effects Strong SOL flows obsered with i. strong ballooning particle flux outboard mid-plane source ii. SOL symmetry breaing (LSN, USN) SOL flow correlated with " increment in L-mode i.e. C-Mod [8]

2 LSN USN V toward X-point " co B V away from X-point " counter But: in H-mode, " is always co. Addressing the Phenomenology B i. Focus: Off-Diagonal Momentum Flux in Electrostatic Drift Wae Turbulence INWARD COMPONENT? ii. Beyond Diffusion and Conection d n Particle number consered " n D + V n dr pinch is only off-diagonal for particles but: wae-particle momentum exchange possible # " n n (1) r, r + r resid r $ %" + V + # r,, () $ r residual stress/flux possible and distinct from pinch residual stress acts with boundary condition to generate intrinsic rotation iii. Key Theoretical Issues Flux of wae momentum? Origins of symmetry breaing? Boundary conditions? a) Wae Momentum [9] Momentum Budget: Resonant+Non-Resonant Particles + Fields Non-Resonant Waes Wae momentum flux crucial for fluid-lie DWT wae b) Calculating r, Necessary to compute radial flux of parallel mom. # W " grx N In simplest scenario, finite momentum flux requires: Radial wae flux 0 grx symmetry breaing 0 Wae Momentum Flux Proceed ia Chapmen-Ensog expansion (radiation hydrodynamics in large optical depth limit) in Wae Kinetics in short mean free path limit, expansion parameter gien by: # " c, ( gr / LI )," c, E. (3) Lowest order: ( ) 0 > saturated spectrum due to wae interactions C N

3 3 (a) (b) Next order, yields: FIG.1 (a) Time eolution of " and " / i, and (b) that of radial profile of u normalized by th N " & N #% c, gr + % c, $ E (4) r r 1 st term c, / ln N, nd term " c, E Wae momentum flux: - & ' N ( ' N # * r, + d % 0r N ). c, gr +. c, gr, E " (5) $ ' r ' r Second term radiatie diffusion of quanta requires gradient in turbulence intensity profile (uniersally increasing) related to momentum flux from edge? Third term refraction induced wae population imbalance crucial for regimes of strong shear flow most actie near edge, or ITB sensitie to L-H mode transition, local steepening in P * mode dependence ia Mechanisms of symmetry breaing: Moments of W.K.E. 1. Influx: radial inflow of wae momentum potentially critical in edge region captures possible influx of momentum from SOL. Wind-up: mode sheared by poloidal elocity ala spiral arm requires magnetic shear, i.e. critical in barrier regions, either pedestal or ITB, but not limited to these 3. Growth asymmetry enters due to parallel elocity shear - unliely 4. Refraction due to GAMs refractie force largely unexplored liely to be most important near edge N

4 4 c) Boundary Condition Effects Communication between SOL and Core across LCFS poorly understood In some present day discharges, neutral drag strong at edge (DIII-D, TCV) no slip boundary condition reasonable in H-mode Important: WON T be true for ITER, and other plasmas with high neutral opacity Area for ongoing research 3. Physics of Off-Diagonal Momentum Flux resid r $ %" + V + # r, $ r " turbulent iscosity (largely understood) V (inward) conectie elocity pinch resid r, " residual stress (nonlinear wae particle momentum deposition) a) Is there an Inward Flux? Gyroinetic turbulence dries off-diagonal and diffusie momentum transport from GTS simulations (FIG. 1) A robust, large inward " is found in post-saturation phase of ITG turbulence Core plasma spins up with u few % of th (no momentum source at edge) Smaller " in long-time steady state is liely diffusie with effectie " / i on order of unity, consistent with experiments and early ITG theory GTC simulations of Toroidal Momentum Transport Constant angular elocity (rigid rotation case) (FIG. (a)) : Inward momentum flux (pinch); Redistribution of momentum (spinning up towards the center) Sheared rotation case (FIG. (b)) Flux separation: subtracting pinch contribution from the total flux gies diffusie flux # $ diff % i Pr / $ " Quasilinear estimates: D( ) ' ' QL + QL i 1 # x - 1 n ( R ), c d( J0 ( ) ( y i B V %$ )+ &* ),, ( $ dr D( ) f ( ) " () 1 m d D( ) f ( ) n T $ Due to smaller elocity weight " / < 1 if ratio of particle s resonant energy to thermal energy is larger than unity QL QL QL Based on measured fluctuation spectra: Pr # $ / # 0. 7 [10] i " i b) Physics of Residual Stress " Key Point: E 0 conerts poloidal shear into toroidal shear ia: asymmetry in wae particle momentum deposition Residual Stress Finite +generic acoustic coupling E

5 5 shifted spectral enelope (FIG. 3) special case of wind-up asymmetry 0 [11] imbalance in acoustic populations momentum deposition by ion Landau damping Underlying physics for ITG drien off-diagonal momentum transport is related to zonal flow shear Self-generated zonal flow is quasi stationary in global ITG showing existence of toroidal zonal flow Mechanism: generation of residual stress due to symmetry breaing induced by quasi-stationary ZF shear [extending the mechanism due to mean ExB shear[1]] 1 ( nq # m) $ " mn ( r) qr $ " 0 Residual stress causes rotation buildup in pedestal (FIG.4) Sharp gradients cause a torque density p asymmetry residual stress E mn Leading to a net rotation due to the off-diagonal term effectie local source " V$ + r, $ ( a # ) 0 " t H-mode spin-up c) Physics of Momentum Pinch Key point: # " E$B 0 in torus TEP-lie pinch N.B.: Symmetry Breaing Ballooning Structure Turbulent Equipartition of Magnetically Weighted Quantities Turbulence Equipartition Pinch (TEP) of density has been demonstrated ia simple model with nonuniform B [13] & n # # tn + " ( ne ) 0, # " E 0, () t + E ( ') $ 0 % B " Extended to trapped electrons in toamas[14] Turbulence Mixing Relaxation towards canonical profiles [15] Inward Pinch in the obsered field n as a consequence of a tendency towards homogenization of the locally consered field n/b For angular momentum density[16] ' nu R $ # t ( nu R) + " ( nu RE ) 0, # " E 0, (* t + E )() % 0 " & B # Inward Pinch in obsered quantity nu R is a consequence of a tendency towards Homogenization of the locally consered quantity nu R/B Curature drien toroidal momentum pinch Curature drien toroidal momentum pinch[16] hae two parts: 1) TEP pinch: mode-independent, inward ) Thermo-electric pinch: mode-dependent For the model we consider TEP part only. TEP pinch of Angular Momentum:

6 6 diffusion of the magnetically weighted field leading to an effectie conectie flow in the obsered field is part of the general deriation from conseratie gyroinetic equations [16]. can be shown to correspond to local conseration of magnetically weighted angular momentum[16,17] L / B Pinch of momentum from diffusion of magnetically weighted momentum Homogenization (mixing) of the locally consered quantity nu R/B occurs ia diffusion of the magnetically weighted angular momentum. d & nu R # ( MWA * r * ( nu R / B )...quasilinear calc. ') MWA $ dr % B " Separating the d / dr(1/ B ) drie from d / dr ( nu R ) the drie, we get Ang d / dr(1/ B ) + Vpinch ( nu R)] / [" B with V / # B d / dr(1/ B ) " / R pinch Ang Inward Pinch in obsered quantity nu R is a consequence of tendency towards a canonical profile with "( nu R / B ) 0 Values from theories are in the range of experimental releance for NSTX [18] The two candidates: [16,19] Perturbatie Momentum Studies using Magnetic Braing Pinch at arious radii Can L n dependence be discriminated? 4. Toward a Simple, Tractable Model R Key Points: profile 1) " due r, rotation at L-H transition "( P) W p ) V TEP due # " E 0 peaing on axis couple to simple L-H transition model (a la Hinton) R N.B. " decays with slower than ", i, D r, fix on boundary Simple Self Consistent Model Conseration Laws: $ n 1 $ + ( r% n) Sn $ t r $ r $ P 1 $ + ( rq) H, where $ t r $ r $ L 1 $ + ( r# ) " $ t r $ r Algebraic Relations: E E n ( n % * n " D0 " D1+ & + Vrn# r ' r $ L0 ( L0 ) 0 "/ 0 "/ 1+ Vr L r & + ' r ( - P % E S " +. ( r) y & 1" P r # ' 0 $ r P P Q ", 0 ", 1+ r r 0 % # + S $

7 7 + E y 1 µ. + n + P... r / n0p, r + r # 0 # [1 + " ( + / + r) ] E y * L$ ( r, t) ' $ ( r, t) ( % ( r) ) n( r, t) & TEP (wea ballooning case): " 1 V r # 1 (# 1 ) " ( E # 1 ) 0 " r B.C. s: " ( r) % (1 $ r 0 / R 0 ( r) # (1 + r / R ) 0 L a) ( a) P( a) 0 ( n( a) n a ) Implications of the Model and Scaling Trends Dimensional analysis estimate for pedestal flow elocity suggests a width scaling: / ( / L )( w / a) " th # s s ped This is width scaling for pressure profile, but the simple model lins width to height wp. " s Ti nped C s C L n R L s s T [ n TV ] e ped i p T [ n V / R] Where n pedtv i p Wp : Stored Energy i te I p scaling not obious other pedestal physics? Model is not quantitatiely accurate. but yields self-consistent H-mode. predicts a scaling of the pedestal toroidal elocity with the pedestal width. Comments re: Model eolution not addressed. Eidence for anomaly and for significant role in exists (JET, C-Mod...) results insensitie to (a) b.c., but core-sol coupling must be addressed pedestal physics controls spontaneous rotation 5. Forefront Topics wped, w ped Pped easy to recoer Wp scaling I p scaling elusie a) Electromagnetics and Saturation Resonant component of turbulent momentum flux is proportional to E inclusion of inductie component allows for reduction/enhancement of E For large aspect ratio, a quasilinear calculation yields resonant component [0]: ES tot " " + AA ( 1 Re # ) p

8 8 AA Re " ( qr / Ln ) either high β or steep density gradients lead to significant EM impact For drift waes: Re AA > 0 ES noel means of quenching for high β or steep density grad. For ITG: Re AA < 0 ES slight enhancement of aboe leel predicted by ES prediction Non-resonant component qualitatiely similar, with important exception that only offdiagonal terms are modified to lowest order Alfen waes proide alternate channel for momentum transport aside from well studied limit of ES microturbulence B.P. releant Off-diagonal component of momentum flux requires finite E KSAWs proide natural candidate for transport of parallel momentum dispersie corrections introduce a radial group elocity/finite E mode conersion of TAEs at resonant surfaces proide robust generation mechanism Residual stress for each branch computed ia a quasilinear calculation imbalance in Elsasser populations required for finite leels of off-diagonal transport symmetry breaing liely induced by asymmetry in energetic particle drie b) Boundary Effects L-mode edge + asymmetry > SOL circulation > > 0, or < 0 impact on L-H power threshold Key Question: How can SOL flow influence core plasma? Tail wags the dog? Key Quantity: S (r) flow speed profile ds ( r) > 0, away from r sep dr suggested by particle balance confirmed by C-Mod, DIII-D Possible Mechanisms: inward turbulent diffusie momentum flux (any SOL mechanism) parallel shear flow instability: s. n is ey competition 6. Looing Ahead Open issues a) MFE Experiment dual perturbation ( ", P) experiments R disentangle V, " r, explore synergy of intrinsic rotation with transition, pedestal physics. Slow transitions helpful here? " (ped) analogue of Rice plot " s. P ped? I p scaling? neutral opacity scans "

9 9 intrinsic rotation on electron dominated regimes (ITER releant) CTEM as transport agent explore mode dependency (IOC?) intrinsic rotation synergy with ITB (c.f. JT-60U) b) MFE Theory Numerous technical details: Blob ejection [1] recoil? H-mode? Major Unnown: Poloidal Momentum many cases of deiation from standard neoclassical need: improed neoclassical turbulent flux > V, Π E structure across separatrix r, strongly coupled AE critical in Burning Plasmas role of field momentum? role of cross-scale coupling? c) Thoughts on the Big Picture General Circulation in the Toama as a fascinating and critical problem? Ocean/Atmosphere Rotation, continents, solar heating, eddys, jets, Hadley cells, annular modes, and western boundary layer Toama B-geometry, boundary, heat flux, drift-itg, zonal flows, poloidal flows, KH of zonal flows, and pedestal Hierarchical structure of global flow pattern? Reference: [1] Rice, J.E., et al., Phys. Plasmas 7 (1999) 185. [] Solomon [3] Ince-Cushman, Rice et al., [4] Scott, S.D., et al., Phys. Re. Lett. 64 (000) 531, Mattor, N., and Diamond, P., Phys. Fluids 31 (1988) [5] Ida, K., et. al., Phys. Re. Lett. 86 (001) [6] Yoshida, M., Plasma Phys. Control. Fusion 48 (006) [7] Yoshida, M., (008) [8] LaBombard, B., Nucl. Fusion 44 (004) [9] Diamond, P.H., et al., Phys. Plasmas 15 (008) [10] Holod, I., and Lin, Z., Phys. Plasmas, (008) (in press). [11] Itoh, S.-I., Phys. Fluids B 4, (199) 796, Dominguez et al., Phys. Fluids B 5 (1993) 3876, Diamond, P.H., (1994) [1] Gurcan et al., Phys. Plasmas 14 (007) [13] Yano (1994), Naulin (1998).

10 10 [14] Isicheno et al., (1997), Baer-MNR (1998). [15] Garbet (005). [16] Hahm, T.S., Phys. Plasmas (007). [17] Gürcan et al., Phys. Re. Lett. (008) [18] Solomon et al., Phys. Re. Lett. 101 (008) [19] Peeters et al., (007). [0] McDeitt, C., Diamond, P.H., submitted to Phys. Plasmas (008). [1] Myra, J.R., et al., Phys. Plasmas 13 (006)