ETFP Krakow, 11.9.2006

Edge plasma turbulence theory:the role of magnetic topologyAlexander Kendl Bruce D. Scott

Institute for Theoretical Physics Max-Planck-Institut für Plasmaphysik,University of Innsbruck, Austria Garching, Germany

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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

„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)

Electromagnetic gyrofluid model: two-moment GEM3 equations (B. Scott)

Electrons

Ions

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Poisson + Ampere equation:

Geometric factors:

i ~ 1/B

|| = bz ∂zB || B

-1= ∂z bz

Metric representation in a field-aligned system

Differentiation operators: expression in general curvilinear coordinates

- Preferentially: field-aligned (flux-tube) coordinates (u1,u2,u3) = () = () = (x,y,z)

- General definitions:

- Laplacian:

- Perp. Components:

- Parallel grad components:

- Parallel divergences:

Metric representation: magnetic field strength |B|

|B| acts mainly as scaling factor for some terms:

- B(z): in vE ~ 1/B , and i / FLR-effects contained in gyrofluid polarisation equation

- bz = 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)

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

Metric representation: normal and geodesic curvature

Definition: magnetic curvature:

- low beta:

- normal curvature: with

- geodesic curvature: with

- flux-tube:

Metric representation: dependence on flux-surface shaping

- Metric quantities gxx(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)

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. 106 grid point, 105 time steps(run into saturated, equilibrated state)

grid resolution: x,y: ~ mm (drift scale), z ~ m t ~ 0.05 Ln/cs < µs

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

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 kx smaller)

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

Computational results: zonal flows, Reynolds stress and geodesic transfer

eddies

vy(x)vy(x)V0

V0: mean flowvxvy: Reynolds stressBxBy: Maxwell stressn sin z: geodesic transfer

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

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

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