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EFTC-11 : Aix-en-Provence NEOCLASSICAL TOROIDAL ANGULAR MOMENTUM TRANSPORT IN A ROTATING IMPURE PLASMA S. Newton P. Helander This work was funded jointly by the UK Engineering and Physical Sciences Research Council and by the European Communities under the contract of Association between EURATOM and UKAEA OVERVIEW • Motivation • Current observations • Neoclassical transport • Current predictions • Impure, rotating plasma • Calculating the transport • Results • Summary MOTIVATION • Internal Transport Barriers (ITBs) observed in tokamaks - steep pressure & temperature gradients - low radial transport - believed to be caused by sheared electric field • Toroidal velocity related to radial electric field dVs ps' Er B ms ns V s es n...s E Vs B R s ps Bp ns es Bp dt Er (r) determined by angular momentum transport CURRENT OBSERVATIONS If turbulence is suppressed in an ITB neoclassical angular momentum transport should play key role in ITB formation / sustainment Bulk ion thermal diffusivity - observed at neoclassical level in ITB core plasma - prediction assumes bulk ions in low collisionality, ‘banana’ regime Bulk ion viscosity determines angular momentum confinement - measured toroidal viscosity order of magnitude higher than neoclassical prediction angular momentum transport anomalous NEOCLASSICAL TRANSPORT • Gyroradius r << perpendicular length scale Lr - follow the motion of the guiding centre - determined by magnetic field structure and Coulomb collisions • Parallel friction neoclassical cross-field diffusion - classical transport from perpendicular friction • Transit frequency wt = vT / qR • Two extreme collisionality regimes - n >> wt : Pfirsch-Schlüter regime - n << wt : banana regime NEOCLASSICAL TRANSPORT Pfirsch-Schlüter Banana - parallel motion is diffusive - trapped particle orbits form - radial drift step width - step width ~ 1/2 r p D PS n q r 2 - trapped particle effects dominate 2 DB n D 1 ~ 3 / 2 q2r 2 3/ 2 wt /n Parallel friction neoclassical cross-field diffusion CURRENT PREDICTIONS 1971, Rosenbluth et al calculate viscosity in pure plasma: - bulk ions in banana regime, slow rotation - scales as nii q 2r 2 - expected for Pfirsch-Schlüter regime 1985, Hinton and Wong, 1987, Catto et al: - extended to sonic plasma rotation: - still no enhancement characteristic of banana regime - transport is diffusive, driven by gradient of toroidal velocity - plasma on a flux surface rotates as a rigid body - angular velocity determined by local radial electric field INTERPRETATION Er V Bp ExB • Trapped particles collide: - change position, toroidal velocity determined by local field - no net transfer of angular momentum • Angular momentum transported by passing particles: - same toroidal velocity as trapped particles due to friction - typical excursion from flux surface ~ q r momentum diffusivity ~ nii q 2r 2 IMPURE ROTATING PLASMA Typically Zeff > 1 - plasma contains heavy, highly charged impurity species - mixed collisionality plasma 1976, Hirshman: particle flux ~ Pfirsch-Schlüter regime heat flux ~ banana regime Rotating Plasma - centrifugal force pushes particles to outboard side of flux surface - impurity ions undergo significant poloidal redistribution - variation in collision frequency around flux surface 1999, Fülöp & Helander: particle flux typical of banana regime IMPURE ROTATING PLASMA Zeff MAST discharge #8321 V ~ 300 km s-1 R[m] Rotating Plasma - centrifugal force pushes particles to outboard side of flux surface - impurity ions undergo significant poloidal redistribution - variation in collision frequency around flux surface 1999, Fülöp & Helander: particle flux typical of banana regime IMPURE ROTATING PLASMA Typically Zeff > 1 - plasma contains heavy, highly charged impurity species - mixed collisionality plasma 1976, Hirshman: particle flux ~ Pfirsch-Schlüter regime heat flux ~ banana regime Rotating Plasma - centrifugal force pushes particles to outboard side of flux surface - impurity ions undergo significant poloidal redistribution - variation in collision frequency around flux surface 1999, Fülöp & Helander: particle flux typical of banana regime CALCULATING THE TRANSPORT Hinton & Wong: - transform kinetic equation to rotating frame - consider effects occurring on different timescales - expansion in d ri / Lr f f 0 f1 f 2 ... f1 ~ d f 0 • Cross-field transport second order in d - evaluate using flux-friction relations - relate flux to collisional moment of f1: Angular momentum flux: mi d 3 v mi R 2 v2 CiL f1 2e • Linearised collision operator ~ • Separate classical and neoclassical contributions: f1 f1 f1 CALCULATING THE TRANSPORT ~ • f1 determined by Hinton & Wong: - valid for any species, independent of form of Ci • f1 obtained from the drift kinetic equation - this can be cast as a variational problem - requires an assumed trial function • Alternative method of solution - subsidary expansion of drift kinetic equation in small ratio of ion collision to bounce frequency - adopt a model for the collision operator - the drift kinetic equation may be solved analytically COLLISION OPERATOR • Neglect ion-electron collisions • Impurity concentration typically nii ~ niz • Explicit form of collision operator: Ci = Cii + Ciz Cii : Kovrizhnykh model operator for self collisions Ciz : disparate masses analogous to electron - ion collisions mi Ciz n iz , L f1 v Vz , f 0 Ti • Parallel impurity momentum equation used to determine Vz mz nz Vz Vz nz ze p R nz zeVz B TRANSPORT MATRIX Represent the fluxes in matrix form L11 q L21 L 31 L12 L22 L32 L13 d ln N iTi dr L23 d (ln Ti ) dr L33 d ln w dr e ~ miw 2 R 2 • ni N i exp 2Ti Ti • Homogeneous non-rotating plasma: - spin-up as a rigid body dni dTi dw 0 dr dr dr - centrifugal potential between flux surfaces drives transport • Slow rotation: Ni ni and Ni Ti pi TRANSPORT MATRIX Represent the fluxes in matrix form L11 q L21 L 31 L12 L22 L32 L13 d ln N iTi dr L23 d (ln Ti ) dr L33 d ln w dr • Off-diagonal terms are non-zero due to the presence of impurities ~ • Each Lij is sum of classical, Lij , and neoclassical, Lij , contribution • Restricted to subsonic rotation to calculate neoclassical terms • Angular momentum transport coefficients L31, L32 and L33 • L33 usual measure of toroidal viscosity RESULTS r ( ) • Not restricted to circular flux surfaces - minor radius r ( ) and inverse aspect ratio ( ) functions of poloidal angle • Zeff = 1 recover: Braginskii - classical contributions to fluxes Hinton & Wong - neoclassical part of L33 • Classical angular momentum flux ~ Enhanced transport - larger outboard step size ri ~ 1/B - larger angular momentum miw R2 No dominant driving force R2 R2 ni 2 nz 2 B B NEOCLASSICAL COEFFICIENTS i , n L31 ... i pi' Ti ' w' L31 L32 L33 2 pi Ti w miw R pi I 2 iz1 mi i2 2 b 2 1 f nr c 2 n nr f nr 2 f c t b2 b2 1 f c n - flux surface average nz n nz ft - ‘fraction of trapped particles’ Zeff Zeff R2 r 2 R 2 • Off-diagonal only iz appears • n within average effect of redistribution B2 b 2 B 2 NEOCLASSICAL COEFFICIENTS Most experimentally relevant limit: - conventional aspect ratio, << 1 - strong impurity redistribution, n cos ~ 1 , n nz nz L31 ~ L32 ~ n iz q r 2 2 3/ 2 2 M z2 q 2 r 2 n iz 3 / 2 L33 ~ M i z • Enhancement of -3/2 over previous predictions - effectiveness of rotation shear as a drive increased by small factor - radial pressure and temperature gradients dominate strong density and temperature gradients sustain strong Er shear NEOCLASSICAL COEFFICIENTS Numerical evaluation using magnetic surfaces of MAST - = 0.14 Zeff 2 L31 0.015 Zeff 0.010 0.005 previous level Zeff 0.00024 0.1 0.2 0.3 0.4 ion Mach number 0.5 • increase with impurity content • increase with Mach number • Transport 1.5 as~impurity 10 times redistribution previous increases predictions - weak effect for small : 1 n cos ~ 0.04 NEOCLASSICAL COEFFICIENTS I mi 3 2 2 L miw R 2 Rzi|| 2 d v mi R v Ci f1 eB 2e • Large aspect ratio and strong impurity redistribution ( 2) 2 - convective angular momentum transport dominates • Angular momentum transport enhanced by enhanced particle flux I Rzi|| eB • Small w - impurities rotate poloidally to minimise friction: Rzi|| 0 • Large w - poloidal impurity rotation reduced due to centrifugal localisation enhanced friction NEOCLASSICAL COEFFICIENTS • Large L31 , L32 spontaneous toroidal rotation may arise: w' 1 p' T' L31 L32 w L33 p T • Rotation direction depends on edge boundary condition • L23 relates heat flux to toroidal rotation shear: L11 q L21 L 31 L12 L22 L32 L13 d ln pi dr L23 d (ln Ti ) dr L33 d ln w dr Co-NBI shear heat pinch Sub-neoclassical heat transport… SUMMARY • Experimentally, angular momentum transport in regions of neoclassical ion thermal transport has remained anomalous • In a rotating plasma impurities will undergo poloidal redistribution • Including this effect, general expressions for particle, heat and angular momentum fluxes derived for mixed collisionality plasma • At conventional aspect ratio, with impurities pushed towards outboard side, angular momentum flux seen to increase by a factor of -3/2 now typical of banana regime • Radial bulk ion pressure and temperature gradients are the primary driving forces, not rotation shear strong density and temperature gradients sustain strongly sheared Er • Spontaneous toroidal rotation may arise in plasmas with no external angular momentum source REFERENCES [1] S. I. Braginskii, JETP (U.S.S.R) 6, 358 (1958) [2] P. J. Catto et al, Phys. Fluids 28, 2784 (1987) [3] T. Fülöp & P. Helander, Phys. Plasmas 6, 3066 (1999) [4] C. M. Greenfield et al, Nucl. Fusion 39, 1723 (1999) [5] P. Helander & D. J. Sigmar, Collisional Transport in Magnetized Plasmas (Cambridge U. P., Cambridge, 2002) [6] F. L. Hinton & S. K. Wong, Phys. Fluids 28, 3082 (1985) [7] S. P. Hirshman, Phys. Fluids 19, 155 (1976) [8] W. D. Lee et al, Phys. Rev. Lett. 91, 205003 (2003) [9] M. N. Rosenbluth et al, Plasma Physics & Controlled Nuclear Fusion Research, 1970, Vol. 1 (IAEA, Vienna, 1971) [10] J. Wesson, Nucl. Fusion 37, 577 (1997), P. Helander, Phys. Plasmas 5, 1209 (1998)