# Presentation: 934707

## Stellarator Theory Teleconference

- in Stellarators

- January 26, 2006

**Don Spong**
**Oak Ridge National Laboratory**

- Acknowledgements: Hideo Sugama, Shin Nishimura, Jeff Harris, Andrew Ware, Steve Hirshman, Wayne Houlberg, Jim Lyon, Lee Berry, Mike Zarnstorff, Dave Mikkelson, Joe Talmadge

## Plasma flow properties provide an additional dimension to the stellarator transport optimization problem

-  Ripple reduction, quasi-symmetry, isodynamicity, omnigeneity
- However, within these devices a variety of parallel momentum transport characteristics are present
-  poloidal/toroidal velocity shearing
-  two-dimensional flow structure within magnetic surfaces

## Plasma flow characteristics vary significantly across stellarator configurations - opens opportunities for a range of physics issues to be explored:

- Relevance to turbulence suppression/enhanced confinement regimes
- Experiments/simulations show importance of E x B shearing
- Reduced neoclassical viscosity lowers zonal flow electric field damping rate
- K. C. Shaing, Phys. Plasmas **12**, (2005) 082508
- Magnetic perturbations shielding - island suppression
- Toroidal flows shield resonant magnetic perturbations at rational surfaces
- A. Reiman, M. Zarnstorff, et al., Nuc. Fusion **45**, (2005) 360
- Possible source of “self-healing” mechanism in tokamaks
- Impurity accumulation/shielding analysis
- S. Nishimura, H. Sugama, Fusion Science and Tech. **46** (2004) 77
- New hidden variable in the international stellarator transport scaling database?
-  Productive area for experiment/theory comparison
-  stellarator analog of NCLASS code
- Improved bootstrap current/ambipolar electric field analysis

## Stellarator flow damping:

-    Reference: K. C. Shaing, J. D. Callen, Phys.Fluids 26, (1983) 3315.

- weights viscous stress-tensor components differently than cross-field transport
- removes linear dependencies (from symmetry-breaking effects)

##  Advances in stellarator optimization have allowed the design of 3D configurations with magnetic structures that approximate: straight helix/tokamak/connected mirrors:

-  **Advances in stellarator optimization have allowed the design of 3D configurations with magnetic structures that approximate: straight helix/tokamak/connected mirrors:**

- U

- Ufinal

- U||

- U

- Ufinal

- U||

- U||

- Ufinal

- U

- HSX
- <a>=0.15 m
- <R> = 1.2 m

- NCSX
- <a> = 0.32 m
- <R> = 1.4 m

- QPS
- <a> = 0.34 m
- <R> = 1 m

- Quasi-helical symmetry
- |B| ~ |B|(m - n)

- Quasi-toroidal symmetry
- |B| ~ |B|()

- Quasi-poloidal symmetry
- |B| ~ |B|()

## In addition, LHD and W7-X achieve closed drift surfaces by inward shifts (LHD) and finite plasma effects (W7-X)

- U

- Ufinal

- U||

- U

- Ufinal

- U||

- LHD
- <a> = 0.6 m
- <R> = 3.6 m

- W7-X
- <a> = 0.55 m
- <R> = 5.5 m

##  Based on anisotropic pressure moment of distribution function rather than momentum or particle moments
Less affected by neglect of field particle collisions on test particles
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 has related viscosities to Drift Kinetic Equation Solver (DKES) transport coefficients
Moments method, viscosities related to D11 D31, D33
       M. Taguchi, Phys. Fluids B4 (1992) 3638
      H. Sugama, S. Nishimura, Phys. of Plasmas 9 (2002) 4637
DKES: D11 (diffusion of n,T), D13 (bootstrap current), D33 (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

- Less affected by neglect of field particle collisions on test particles
- 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 has related viscosities to Drift Kinetic Equation Solver (DKES) transport coefficients
- Moments method, viscosities related to D11 D31, D33
-        M. Taguchi, Phys. Fluids B4 (1992) 3638
-       H. Sugama, S. Nishimura, Phys. of Plasmas **9** (2002) 4637
- DKES: D11 (diffusion of n,T), D13 (bootstrap current), D33 (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

- Development of stellarator moments methods

## 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:

- • 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 relations

**Leads to coupled equations that can be solved for <u****||a****B>, <q****||a****B>, ********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)

- 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        self-consistent
- energy fluxes and bootstrap currents

- Parallel mass and energy flows:

## Parallel Environment for Neoclassical Transport Analysis (PENTA)

- flux
- surface 1

- flux
- surface 2

- flux
- surface 3

- flux
- surface 4

- flux
- surface n

**. . .**

- processor 1

- processor 2

- processor 3

- processor 4

- processor n

- DKES Transport coefficient
- Code: D11, D13, D33
- (functions of v, Er/v)

- results vs. v, Er/v
- concatenated together

- DKES results supplemented
- at low/high collisionalities
- using asymptotic forms

- Energy integrations, parallel force balance
- relations, ambipolarity condition solved, profiles
- obtained for: Er, ie, qi, qe, <ui>, <ui>, JEBS

**Parallel Environment for Neoclassical Transport Analysis (PENTA)**

- delta-f Monte Carlo
- superbanana effects (i.e., limits on 1/ regime)
- better connection formulas
- DKES extensions
- convergence studies
- E effects
- Magnetic islands

- Work in progress:

## The neoclassical theory provides <u||B> and the ambipolar electric field Es (from solving ie). The final term needed is U, the Pfirsch-Schlüter flow.

- <U2> can be obtained by:
- solving this equation directly (with damping to resolve singularities at rational surfaces)
-  matching to high collisionality DKES coefficient: <U2> = 1.5D11v/ (for large )

- Integrating             leads to an equation for U:

## The flow model will be applied to two parameter ranges with radially continuous/stable electric field roots

- ECH regime:
- n(0) = 2.5  1019 m-3, Te(0) = 1.5 keV, Ti(0) = 0.2 keV
- ICH regime
- n(0) = 8  1019 m-3, Te(0) = 0.5 keV, Ti(0) = 0.3 keV
- Roots chosen to give stable restoring force for electric field perturbations through:

- ion > elec

- ion < elec

- ion < elec

- ion > elec

- ECH: electron root

- ICH: ion root

- Er/E0

- Er/E0

- ionelec

- ionelec

## Electric field profiles for ICH (ion root) and ECH (electron root) cases

- ICH
- parameters

- ECH
- parameters

## The neoclassical parallel flow velocity can vary significantly among configurations. The lowest levels are present in W7-X. Higher u|| flows characterize HSX/NCSX

- The neoclassical parallel flow velocity can vary significantly among configurations. The lowest levels are present in W7-X. Higher u|| flows characterize HSX/NCSX

- ICH regime

## 2D flow variation within a flux surface:

- Visualization of flows in real (Cartesian space)
- 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

- diamagnetic
- and E x B

- neoclassical
- parallel flow

- Pfirsch-Schlüter
- - for

-     1/B

- B

- fB

- (variation within a flux surface)

## HSX 2D-flow streamlines

- |B| variation

## LHD 2D-flow streamlines

- |B| variation

## W7-X 2D-flow streamlines

- |B| variation

- W7-X 2D-flow streamlines

## QPS 2D-flow streamlines

- |B| variation

## NCSX 2D-flow streamlines

- |B| variation

- NCSX 2D-flow streamlines

## Plasma flow velocity - averages

- Components taken:
- Reduction to 1D - flux surface average:
- Due to 1/B2 variation of Jacobian, averages will be influenced by whether 1/B (EB, diamagnetic) or B (neo. Parallel, PfirshSchlüter) terms dominate

- diamagnetic
- and E x B

- neoclassical
- parallel flow

- Pfirsch-Schlüter
- - for

## Comparison of flux-averaged toroidal flow components (contra-variant) among devices indicates NCSX and HSX have the largest toroidal flows

- ICH parameters

- ECH parameters

## Comparison of flux-averaged poloidal flow components (contra-variant) among devices indicates QPS has largest poloidal flows

- ECH parameters

## Comparison of shearing rates from ambient flows with ITG growth rates:

- ITG = (CS/ LT)LT/R)
- where 0 <  < 1, CS = sound speed
- from J. W. Connor, et al., Nuclear Fusion **44** (2004) R1

**Recent stellarator DTEM-ITG**
**growth rates**
- from G. Rewoldt, L.-P. Ku, W. M.
- Tang, PPPL-4082, June, 2005

- }

- QPS, NCSX, W7-X, LHD
- 0.2 to 1.6  105 sec-1

- HSX
- 4  105 sec-1

**Transport barrier condition: shearing rate > ********ITG**

- Even without external
- torque, QPS, NCSX,
- LHD, W7-X shearing
- rates can exceed
- DTEM-ITG at edge region

- |d<vEB>/d|, ITG

## Flow variations within flux surfaces can also impact MHD ballooning/interchange thresholds:

- QPS flow shearing along **B**

- Maximum flow shearing rates are ~0.5 of Alfvén time
- Could influence MHD stability thresholds

- inboard

- outboard

- outboard

## Er is modified by parallel momentum source(40 keV H0 beam, F||(i,e) = 0.1, 0.2, 0.5, 0.7 Nt/m3)

- QPS-ICH

- Er is modified by parallel momentum source(40 keV H0 beam, F||(i,e) = 0.1, 0.2, 0.5, 0.7 Nt/m3)

- NCSX-ICH

- Same viscosity coefficients that
- relate gradients in n, T, to u||
- also imply that changes in u||
- modify 

## Conclusions

- A self-consistent ambipolar model has been developed for the calculation of flow profiles in stellarators
- Ambipolar electric field solution with viscous effects
- Predicts profiles of Er, <u||>, <u>, and <u>; 2D flow variations
- Magnetic field structure influences flows in quasi–symmetric stellarators
- QPS: poloidal flows dominate over toroidal flows
- W7-X: poloidal flows dominate, but flows reduced from other systems
- NCSX: toroidal flows dominate except near the edge
- HSX: flows are dominantly helical/toroidal
- LHD: toroidal flows dominate for ICH case, poloidal flows for ECH case

## Conclusions (contd.)

- Ambient flow shearing rates approach levels that could suppress turbulence
- ITG, ballooning
- Further stellarator-specific turbulence work needed, but with 2D neoclassical flow fields
- Collaborations initiated with LHD, HSX to further apply model and look for correlations with confinement data
- 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
- Stellarator optimizations/flexibility studies -> should use E  B shear (turbulence suppression to complement neoclassical transport reduction