EXTENSIONS OF NEOCLASSICAL ROTATION THEORY & COMPARISON WITH EXPERIMENT W.M. Stacey 1 & C. Bae, Georgia Tech Wayne Solomon, Princeton TTF2013, Santa Rosa, CA (2013)

EXTENDED NEOCLASSICAL ROTATION THEORY (Nuclear Fusion 53, ,2013) Solve the FSA toroidal and poloidal momentum balances for the toroidal and poloidal velocities. Solve the sinθ and cosθ weighted FSA poloidal momentum balances for the poloidal density asymmetries. Relate the ion density asymmetries to the poloidal velocity asymmetries using the ion continuity equations, to the toroidal velocity asymmetries using the ion radial momentum balances, and to the electrostatic potential asymmetries using the electron momentum balance. Use the Braginskii decomposition of the viscosity tensor extended to curvilinear geometry and evaluated on the Miller model flux surface geometry. Use the Braginskii gyroviscous coefficient but Shaing’s (NF25, 463, 1985) parallel viscosity coefficient extended to account for trapped particle effects. For an ion-impurity model, 8 coupled non-linear equations.

3 Equilibrium Models  Earlier work (e.g. PoP13,062508,2006 ) based on Circular Model  But, equilibrium flux surfaces are D-shaped  Present work based on Miller Equilibrium Model  Circular model is a special case of Miller geometry Miller Model Circular Model Shot

 Two ELMing H-mode shots  Both are strong rotation shots Comparison w/DIII-D Experiments  Counter-Injection(CTR) Shot  Co-injection(CO) Shot Divertor X-point

Prediction vs. Experiment  CTR USN Shot : Toroidal Velocities (CCW positive)  CTR USN Shot : Poloidal Velocities (positive upward at outer midplane) 5

 CO LSN Shot : Toroidal Velocities (CCW positive)  CO LSN Shot : Poloidal Velocities (positive downward at outer midplane) 6

7  DENSITY ASYMMETRIES  CTR USN Shot  CO LSN Shot

 CIRCULAR VS. MILLER MODEL RESULTS  Higher accuracy achieved with the Miller model geometry  Due to accuracy improvement in the poloidal asymmetry calculation  Shot (Ctr Upper SN) 8

CONCLUSIONS Extended neoclassical theory accounts quite well for the poloidal and toroidal rotation measurements, except in the edge,, in 2 strongly rotating DIII-D shots. Calculating the poloidal asymmetries in densities and velocities to evaluate gyro and parallel viscosity leads to neoclassical predictions of toroidal and poloidal, respectively, rotation that are within about 10% of experiment (except in the edge where further model improvements—recycling neutrals, anti- symmetric B viscosity, small rotation ordering--are necessary). An accurate representation of the poloidal asymmetry in the flux surface geometry (e.g. the Miller model) is important for an accurate calculation of the rotation velocities. Further improvements (X-point effect representation, up-down flux surface asymmetries, etc.) are possible.

BACKGROUND ON NEOCLASSICAL ROTATION THEORY Kaufman (PF3,610,1960), Braginskii (Rev. Plasma Phys. 1, 205,1965) and Mikhailovski & Tsypin (Sov. J. Phys. 10, 51, 1984) worked out a theory for viscosity (parallel, perpendicular and gyro) in collisional plasmas. Rosenbluth, et al. (IAEA, 1, 495,1971), Tsang & Frieman (PF19, 747, 1976) extended parallel and perpendicular viscosity to include trapped particle effects, which were significant for parallel viscosity but not for perpendicular viscosity (which underpredicted measured toroidal momentum damping rates by two orders of magnitude). Hazeltine (PF17, 961, 1974) presented a theory for poloidal rotation driven by ion temperature gradient, including trapped particle effects. Hirshman & Sigmar (Nucl. Fus. 21, 1079, 1981) presented a fluid moments formulation of viscosity, with coefficients derived from kinetic theory, and of rotation theory.

BACKGROUND--continued Stacey & Sigmar extended the Braginskii viscosity to curvilinear flux surface geometry (PF28, 2800, 1985) and demonstrated that gyroviscosity depended on up-down asymmetries in flows and densities, which they estimated from experiment to be the right magnitude to allow gyroviscosity to account for observed momentum damping. Hinton & Wong (PF28, 3082, 1985) and Connor, et al. (PPCF29, 919, 1987) derived rotation theories using a formal gyro-radius ordering, finding that gyroviscosity entered at second (not leading) order in gyro-radius. Both confirmed that gyroviscosity depended on up-down asymmetries in flows and densities, which they assumed to be negligible, because they entered the development at higher order in gyroradius, despite the large multiplicative gyroviscosity coefficient and the contrary experimental evidence. Hsu & Sigmar (PPCF32, 499, 1990) and Stacey, et al. (NF25, 463, 1985, PFB4, 3302, 1992; PFB5, 1828, 1993; PoP8, 158, 2001; PoP9, 1622, 2001; PoP8, 4450, 2001; PoP13, , 2006) subsequently calculated up-down poloidal asymmetries in flows and densities and found them, while small, to be almost large enough to enable gyroviscosity to account for observed momentum damping rates in several tokamaks when multiplied by the large gyroviscosity coefficient. This finding motivated the present work, in which the poloidal asymmetries are calculated using the Miller equilibrium model to represent the poloidal dependence of the flux surface geometry (instead of the 1+ep*cosθ used in previous calculations).

BACKGROUND--continued The Hirshman-Sigmar theory for poloidal rotation was further elaborated by Kim et al. (PFB3, 2050, 1991) and implemented in the NCLASS code (POP4, 3230, 1997). Stacey (PoP15, , 2008) recently compared this theory with the similar Stacey-Sigmar theory which includes also the calculation of poloidal asymmetries and that is used in the present work. Wong and Chan (PoP11, 3432, 2004) extended the Hinton-Wong rotation theory to a strong rotation ordering, but do not mention gyroviscosity. The Mikhailovski rotation theory in the weak rotation ordering has been elaborated by Catto and Simakov (PoP11, 90, 2004; 12, , 2005) and by Ramos (PoP12, , 2005) to include an additional term in the gyroviscosity tensor proportional to the heat flux. This term would contribute additive corrections to the present work that might account for the disagreement of the present theory with experiment in the edge plasma. Callen, Cole and Hegna (NF49, , 2009) introduced a toroidal viscosity due to toroidally anti-symmetric magnetic fields that might account for the disagreement of the present theory with experiment in the edge plasma.