Innovative Concepts Workshop Austin, Texas February 13-15, 2006

Don Spong Oak Ridge National Laboratory Acknowledgements: Jeff Harris, Hideo Sugama, Shin Nishimura, Andrew Ware, Steve Hirshman, Wayne Houlberg, Jim Lyon Innovative Concepts Workshop Austin, Texas February 13-15, 2006

3D shaping offers a large design space for stellarator configurations Recent stellarator optimizations have reduced crossfield neoclassical losses << anomalous losses Ripple reduction, quasi-symmetry, isodynamicity, omnigeneity Hybrid with plasma current) devices, aspect ratios down to ~ 2 Within these devices a variety of parallel momentum transport characteristics are present poloidal/toroidal velocity shearing two-dimensional flow structure within magnetic surfaces Parallel transport: a new dimension for stellarator optimizations?

Motivation for parallel transport/flow analysis: Universal nature of E x B shearing for shredding eddies over a wide range of configurations/turbulence mechanisms Tokamak specific drives: grad p Stellarator specific drives: Γ i - Γ e Drives shared in common: momentum, Reynold s stress, orbit loss ITER/DEMO H-mode power thresholds K. Burrell, POP 4 1997) 1499 50 to 200 MW 1999 ITER Basis document) Current design: 40 to 70 MW DEMO: up to 300 MW Will alpha heating power isotropic) count the same way as power anisotropic) on which these scaling estimates are based? Could a stellarator lower these power thresholds by optimizing around E x B shearing rates?

Advances in stellarator optimization have allowed the design of 3D configurations with magnetic structures that approximate: straight helix/tokamak/connected mirrors: HSX <a>=0.15 m <R> = 1.2 m Quasi-helical symmetry B ~ B mθ - nζ) NCSX QPS <a> = 0.32 m <R> = 1.4 m <a> = 0.34 m <R> = 1 m Quasi-toroidal symmetry B ~ B θ) Quasi-poloidal symmetry B ~ B ζ) U U U

Advances in stellarator optimization have allowed the design of 3D configurations with magnetic structures that approximate: straight helix/tokamak/connected mirrors: HSX <a>=0.15 m <R> = 1.2 m Quasi-helical symmetry B ~ B mθ - nζ) U Ufinal U NCSX QPS <a> = 0.32 m <R> = 1.4 m <a> = 0.34 m <R> = 1 m Quasi-toroidal symmetry B ~ B θ) U Ufinal Quasi-poloidal symmetry B ~ B ζ) U U U Ufinal

In addition, LHD and W7-X achieve closed drift surfaces by inward shifts LHD) and finite plasma β effects W7-X) LHD <a> = 0.6 m <R> = 3.6 m W7-X <a> = 0.55 m <R> = 5.5 m U U

In addition, LHD and W7-X achieve closed drift surfaces by inward shifts LHD) and finite plasma β effects W7-X) LHD <a> = 0.6 m <R> = 3.6 m W7-X <a> = 0.55 m <R> = 5.5 m U U U U final U U final

Development of stellarator moments methods Viscosities incorporate all needed kinetic information Momentum balance invoked at macroscopic level rather than kinetic level Multiple species can be more readily decoupled Recent work Moments method, viscosities related to D 11 D 31, D 33 M. Taguchi, Phys. Fluids B4 1992) 3638 H. Sugama, S. Nishimura, Phys. of Plasmas 9 2002) 4637 - has related viscosities to Drift Kinetic Equation Solver transport coefficients DKES: D 11 diffusion of n,t), D 13 bootstrap current), D 33 resistivity enhancement) W. I.Van Rij and S. P. Hirshman, Phys. Fluids B, 1, 563 1989) Implemented into a suite of codes that generate the transport coefficient database, perform velocity convolutions, find ambipolar roots, and calculate flow components D. A. Spong, Phys. Plasmas 12 2005) 056114 D. A. Spong, S. P., Hirshman, et al., Nuclear Fusion 45 2005) 918

Moments Method Closures for Stellarators The parallel viscous stresses, particle and heat flows are treated as fluxes conjugate to the forces of parallel momentum, parallel heat flow, and gradients of density, temperature and potential: & ' r B! r B! r "!# t a r "! $ t a ) ) % a Q a / T a ) * ) $ n a e a BE = BF a1 ) = BF a2 = & M a1 M a2 N a1 N a2 ) M a2 M a3 N a2 N a3 N a1 N a2 L a1 L a2 ' N a2 N a3 L a2 L a3 * & u a B / B 2 ) 2 q a B / B 2 5 p a, 1 - p a n a -s, e -. a -s, -T a ' -s * Analysis of Sugama and Nishimura related monoenergetic forms of the M, N, L viscosity coefficients to DKES transport coefficients Combining the above relation with the parallel momentum balances and friction-flow r r B! "! # t relations " a " BF r r B! "! % t a1 % " ab ab l $ ' = 11 l 12 % Bu b $ ) $ ab ab ' 2 # BF a 2 & b #l 12 l 22 & $ Bq b a # 5p b Leads to coupled equations that can be solved for <u a B>, <q a B>, Γ a, Q a % ' ' &

Using solutions for an electron/ion plasma, the self-consistent electric fields, bootstrap currents and parallel flows can be obtained Appendix C - H. Sugama, S. Nishimura, Phys. Plasmas 9 2002) 4637) Radial particle flows required for ambipolar condition energy fluxes and bootstrap currents self-consistent! e " % $ q e / T ' $ e ' $! i ' = $ ' $ q i / T i ' E # $ J BS &' " $ $ $ $ $ $ # ee L 11 ee L 21 ie L 11 ie L 21 e L E1 ee L 12 ee L 22 ie L 12 ie L 22 e L E 2 ei L 11 ei L 21 ii L 11 ii L 21 i L E1 ei L 12 ei L 22 ii L 12 ii L 22 i L E 2 e L 1E e L 2E i L 1E i L 2E L EE % " ' $ ' $ '! $ ' $ ' $ ' & # $ X e1 X e2 X i1 X i2 X E % ' ' ' ' ' &' where X a1 ) 1 * p a n a *s ) e * a *s, X ) *T a a2 *s, X = BE E / B 2 1/2 Parallel mass and energy flows:! Bu i # # 2 Bq i "# 5 p i $ & & = %&! # "# 1! M i1 M i2 $ B 2 # " M i2 M & i 3 % ' l ii! 11 # ii " 'l 12 ii 'l 12 ii l 22 $ $ && %%& '1 N i1 N i2! $! # " N i2 N & # i 3 % " X i1 X i2 $ & %

The flow model will be applied to two parameter ranges with radially continuous/stable electric field roots ECH regime: n0) = 2.5 10 19 m -3, T e 0) = 1.5 kev, T i 0) = 0.2 kev ICH regime n0) = 8 10 19 m - 3, Te0) = 0.5 kev, T i 0) = 0.3 kev n/n0), T e,i /T e,i 0) 1.2 1 0.8 0.6 0.4 0.2 Roots chosen to give stable restoring force for electric field perturbations through:! " q n/n0) T e,i /T e,i 0) 0 0 0.2 0.4 0.6 0.8 1!/! edge ) 1/2 #E r #t = $ e % $ i E r /E 0 E r /E 0 Γ ion > Γ elec ECH: electron root Γ ion > Γ elec Γ ion < Γ elec Γ ion < Γ elec ICH: ion root Γ ion = Γ elec Γ ion = Γ elec

Electric field profiles for ICH ion root) and ECH electron root) cases 50 ECH parameters 0 ICH parameters HSX 40 W7-X LHD -5 30 E! Volt/cm) 20 QPS E! Volt/cm) -10-15 HSX QPS LHD 10 0 NCSX -20 W7-X NCSX -10 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1!/! edge ) 1/2-25 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1!/! edge ) 1/2

E r is modified by parallel momentum source Same viscosity coefficients that relate gradients in n, T, φ to u also imply that changes in u modify Γ. 40 kev H 0 beam, F i,e) = 0.1, 0.2, 0.5, 0.7 Nt/m 3 ) & ' r B! r B! r "! # t a ) r "! $ t a ) % a Q a / T a ) * = & M a1 M a2 N a1 N a2 ) M a2 M a3 N a2 N a3 N a1 N a2 L a1 L a2 ' N a2 N a3 L a2 L a3 * & u a B / B 2 2 q a B / B 2 5 p a, 1 - p a n a -s, e a ', -T a -s -. -s ) * QPS-ICH NCSX-ICH

The neoclassical parallel flow velocity can vary significantly among configurations. The lowest levels are present in W7-X. Higher u flows characterize HSX/NCSX 5 10 4 ECH Regime ICH regime <U <U 2 > m/sec) B>/<B 2 > 1/2 m/sec) 4 10 4 3 10 4 2 10 4 1 10 4 0-1 10 4 NCSX HSX W7-X LHD QPS <U <U 2 B>/<B 2 > 1/2 m/sec) 0-5000 -1 10 4 HSX QPS LHD W7-X NCSX -2 10 4 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1!/! edge ) 1/2-1.5 10 4 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1!/! edge ) 1/2

2D flow variation within a flux surface - reveals features not present in flux surface averages r V = X 1 eb 2 r!s " r B diamagnetic and E x B u B r X B 1 B 2 e neoclassical parallel flow %U r B where X 1 # $ 1 n Pfirsch-Schlüter - for! r " V r ) = 0 % p %s $ e %& %s 1/B B )B variation within a flux surface) Visualization of flows in real Cartesian space) at 75% flux surface Indicates flow shearing geodesic) within a flux surface over shorter connection lengths than for a tokamak Could impact ballooning, interchange stability, microturbulence Implies need for multipoint experimental measurements and/or theoretical modeling support

LHD 2D-flow variation

HSX 2D-flow variation

W7-X 2D-flow variation

QPS 2D-flow variation

NCSX 2D-flow variation

Plasma flow velocity r V = X 1 eb 2 r!s " r B diamagnetic and E x B u B r X B 1 B 2 e neoclassical parallel flow %U r B where X 1 # $ 1 n Pfirsch-Schlüter - for r! " r V ) = 0 % p %s $ e %& %s Components taken: e! = r "! / r "!, and e # = r "# / r "# Reduction to 1D - flux surface average: K = 1 V! ) $$ d"d# g K

Comparison of flux-averaged poloidal flow components contra-variant) among devices indicates QPS has largest poloidal flows ECH parameters ICH parameters 6 10 4 0 W7-X 5 10 4 QPS -1000 NCSX LHD <u e " > m/sec) 4 10 4 3 10 4 <u e! > m/sec) -2000 HSX 2 10 4-3000 LHD 1 10 4 HSX W7-X NCSX 0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1!/! edge ) 1/2-4000 QPS -5000 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 "/" ) 1/2 edge

Comparison of shearing rates from ambient flows with ITG growth rates:! ITG, d<u e " >/dr sec -1 ) - with diamagnetic terms removed d<v e θ >/dρ, γ ITG γ ITG = C S / L T )L T /R) µ where 0 < µ < 1, C S = sound speed from J. W. Connor, et al., Nuclear Fusion 44 2004) R1 10 6! µ = 0) ITG 10 5 10 4 Transport barrier condition: shearing rate > γ ITG HSX LHD! ITG µ = 1) QPS W7-X NCSX 1000 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 #/# edge ) 1/2 Recent stellarator DTEM-ITG growth rates from G. Rewoldt, L.-P. Ku, W. M. Tang, PPPL-4082, June, 2005 HSX γ = 4 10 5 sec -1 }QPS, NCSX, W7-X, LHD γ = 0.2 to 1.6 10 5 sec -1 Even without external torque, QPS, NCSX, LHD, W7-X shearing rates can exceed γ DTEM-ITG at edge region

DIIID reversed shear w/ NBI momentum): ω ExB ~ γ core confinement transition K. Burrell PoP 1997 QPS: quasi-poloidal symmetry ω ExB ~ γ ITG with ICH no momentum input DIIID ERS mode 9.6 MW NBI DIIID L mode 5.2 MW NBI

Flow variations within flux surfaces can also impact MHD ballooning/interchange thresholds: Maximum parallel flow shearing rates are ~0.5 of Alfvén time Could influence MHD stability thresholds QPS flow shearing along B ) r! A0 b " ) v r where! A0 = R / v A0 outboard 1 Maximum flow shearing 0.1 0.01 0.001 0.0001 ECH ICH inboard outboard 10-5 W7-X HSX NCSX LHD QPS

Conclusions Stellarator ambient no momentum input) shearing rates approach turbulence suppressing levels ITG, ballooning: ω ExB /γ ITG ~ ρ ion /<a>)t ion /T elec ) 1/2 fν *, B) Next step: apply theory to LHD discharges where magnetic structure changes resulted in 2.5 increase in confinement time, also HSX Further stellarator-specific gyro-turbulence work needed, with ω ExB Develop flow-related optimization targets - use E B shear turbulence suppression to complement neoclassical transport reduction The sensitivity of flow properties to magnetic structure is an opportunity for stellarators Experimental tests of flow damping in different directions Possible new hidden variable in confinement scaling data Other stellarator specific tricks Electric field bifurcations via profile control Create steady state island -> localized high viscosity region -> strong flow shear