1. ETFP Krakow, 11.9.2006 Edge plasma turbulence theory: the role of magnetic topology Alexander KendlBruce D. Scott Institute for Theoretical PhysicsMax-Planck-Institut für Plasmaphysik, University of Innsbruck, AustriaGarching, Germany → g  → ?

2. Influence of magnetic topology on plasma edge turbulence Paradigm for plasma edge turbulence: - resistive electromagnetic gyrofluid drift (-Alfven) wave turbulence - driven by pressure gradient: density + temperature gradients (  i ~ 2 in pedestal) - nonlinear drive, saturation and sustainment Toroidal magnetic topology: - axial-symmetric tokamaks - „three-dimensional“ stellarators Flux-surface shaping: elongation, triangularity, Shafranov shift, D-shape, X-point,... → enters into (nonlinear, gyro-fluid) drift wave equations by: Metric description:|B|, mag. shear, curvature, metric tensor,... (preferably in field-aligned coordinates, e.g. flux-tube representation) → computationally determine influence on fully developed turbulence and flows → identify and understand mechanisms → use understanding to optimise tokamaks and stellarators for turbulent transport reduction / transport barrier formation

3. „2D“ and „3D“ toroidal magnetic topology Flux-surface shape: elongation, triangularity, Shafranov shift, D-shape, X-point,... Tokamak: axial symmetry elongation:  = b/a triangularity:  = (c+d)/(2a) Stellarator here Wendelstein 7-X: five-fold periodicity (contours: left: local shear, right: |B| ) Geometric quantities on a flux surface: e.g. here local magnetic shear (left) and |B| (right)

4. Electromagnetic gyrofluid model: two-moment GEM3 equations (B. Scott) Electrons Ions { { Poisson + Ampere equation: Geometric factors: r i ~ 1/B  || = b z ∂ z B  || B -1 = ∂ z b z

5. Metric representation in a field-aligned system Differentiation operators: expression in general curvilinear coordinates - Preferentially: field-aligned (flux-tube) coordinates (u 1,u 2,u 3 ) = (  ) = (  ) = ( x,y,z ) - General definitions: - Laplacian: - Perp. Components: - Parallel grad components: - Parallel divergences:

6. Metric representation: magnetic field strength |B| |B| acts mainly as scaling factor for some terms: - B(z) : in v E ~ 1/B, and  i / FLR-effects contained in gyrofluid polarisation equation - b z = B·  z =  '/(BJ) in parallel derivatives New physics due to B(z): TEM - particle trapping in magnetic field wells (not discussed in this talk) Influence of flux-surface shaping on |B| effects: - toroidicity effects due to scaling by |B| comparable in moderately shaped tokamaks, but may vary significantly for different stellarator configurations - effects of variation of |B| included in curvature terms (see curvature effects)

7. Metric representation: local and global magnetic shear Global magnetic shear: Local magnetic shear: - enters into perp. Laplacian as relation between off-diagonal and radial derivatives: - global and local magnetic shear damping of edge turbulence: Kendl & Scott PRL 03

8. Metric representation: normal and geodesic curvature Definition: magnetic curvature: - low beta: - normal curvature: with - geodesic curvature: with - flux-tube:

9. Metric representation: dependence on flux-surface shaping - Metric quantities g xx (z), |B|(z),... etc for various tokamak plasma shapes: simple circular torus (  = 1,  = 0 ) / shaped torus (  = 2,  = 0.4 ) / AUG (  = 1.7,  = 0.3, LSN Div)

10. Computational set-up: flux-tube approximation Codes: - GEM (B. Scott, IPP Garching): full 6 moments or 2-moment model GEM3 - TYR (V. Naulin, Risoe Denmark): drift-Alfvén Fluxtube approximation of toroidal geometry: - field-aligned coordinate transformation - local approximation of metric, shear-shift transformation (see Scott...) 3D computational grid (flux-tube): radial - perp - parallel x y z 64 256 16-64 ExB convection in (x,y), parallel coupling in z → efficient parallelisation in z (8-128 procs, domain decomposition, MPI) ca. 10 6 grid point, 10 5 time steps (run into saturated, equilibrated state) grid resolution: x,y: ~ mm (drift scale), z ~ m t ~ 0.05 L n /c s < µs

11. Theoretical expectations: Normal curvature: - defines ballooning region:  p·  B > 0 destabilises interchange drive (ITG/ETG) and catalyses resistive drift wave turbulence Geodesic curvature: - determines geodesic transfer: coupling of zonal flows to turbulence (cf. GAM oscillation) (Scott PLA 03; Kendl & Scott PoP 05) - energetics: GT couples energy for edge turbulence out of flows (e.g. Naulin, Kendl et al PoP 05) Local and global magnetic shear (LMS / GMS): - limits ballooning region - twists vortices: nonlinear decorrelation, general damping mechanism for turbulence - enhances zonal flows (Kendl & Scott PRL 03) Elongation:- enhances magnetic shear: LMS stronger at ballooning boundaries, GMS stronger if other parameters fixed - reduces geodesic curvature at upper / lower regions Triangularity: - slight enhancement of LMS at outboard midplane (little influence on ballooning region) Divertor X-point: - stronger LMS, more reduced geod. curvature

12. Computational results: Results from model geometries and realistic tokamak + stellarator MHD equilibria Normal curvature: - catalysing for edge turbulence (phase shift properties); ballooning depends on parameters; linear properties determine only long wavelenghts (Scott PoP 05) - sets with beta the ideal MHD ballooning boundary Geodesic curvature: - geodesic transfer effect (Scott PLA 03) scales with geod. curvature (Kendl & Scott PoP 05) - (strong) elongation and X-point shaping enhances GTE (Kendl & Scott PoP 06) Local and global magnetic shear: - general damping effect (nonlinear decorrelation, smaller vortices, lower transport) - LMS relevant even if GMS=0, e.g. in adv. stellarator (Kendl & Scott PRL 03) - strong shear ( s > 1 ) enhances zonal flows (max ZF k x smaller)

13. General results: flux-surface shaping effects on tokamak edge turbulence - Elongation is always favourable (lower transport, stronger Zfs): simulation transport scaling agrees with empirically found scaling laws (Bateman et al PoP 98)  ~  -4 - Triangularity has only slight effect - X-point shaping similar effects as strong elongation (shear flow enhancement stronger if ITG dynamics is active,  i ~ 2 → role of ITG crit for L-H threshold, if ZF trigger mean flow?) - Stellarator: general statements difficult, specific computations necessary for each configuration (Kendl & Scott PoP 03) - Strong potential for low-transport / strong shear flow optimisation of tokamaks and stellarators! Next steps: - include dynamic equilibrium coupling for realistic shaping - include radial variations of geometry (esp. important near X-point) - annulus simulations of stellarators instead of flux-tube approximation - try transport optimisation of flux surfaces → large number of simulations necessary

15. Computational results: zonal flows, Reynolds stress and geodesic transfer eddies v y (x) V0V0 V 0 :mean flow v x v y :Reynolds stress B x B y :Maxwell stress n sin z:geodesic transfer

16. Computational results: Shear flow generation and energetics (beta dependence) Relative importance of transfer mechanisms: Reynolds stress, Maxwell stress, transfer Flow energy: [ Naulin, Kendl, Garcia, Nielsen, Rasmussen, Phys. Plasmas 12, 052515 (2005) ] 0 5 10 15 20

17. Computational results: Influence of elongation and triangularity Elongation reduces edge turbulence and transport. Major mechanisms: magnetic shear damping and shear flow enhancement - Flux surface shaping effects on tokamak edge turbulence and flows: Kendl, Scott; Phys. Plasmas 13, 012504 (2006) - Plasma turbulence in complex magnetic field structures: Kendl; J. Plasma Phys. 41, (2005); in print

18. Computational results: Influence of X-point shaping on zonal flows X-point shaping enhances zonal flows for ITG turbulence - relevance for L-H transition? (zonal flow triggers mean flow?) - threshold linked to (nonlinear) ITG critical gradient ? - Flux surface shaping effects on tokamak edge turbulence and flows: Kendl, Scott; Phys. Plasmas 13, 012504 (2006) - Plasma turbulence in complex magnetic field structures: Kendl; J. Plasma Phys. 41, (2005); in print