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
→ g → ?
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