Gyrokinetic turbulence in the presence of strong rotation
F.J. Casson, A.G. Peeters, C. Angioni, Y. Camenen,
W.A. Hornsby, A.P. Snodin, G.Szepesi
Varenna, September 2010
This talk is contained in a paper just accepted in PoP (2010):
“Gyrokinetic simulations including the centrifugal force in a rotating tokamak plasma”
Outline
• GKW: The numerical tool
• When is strong rotation important?
• Gyro­kinetics in the rotating frame
– Coriolis force (previous work)
– Centrifugal force (this talk)
• New drift, enhanced trapping, modified equilibrium
• Consequences for drift waves
• Consequences for heat and particle transport
“Gyrokinetic simulations including the centrifugal force in a rotating tokamak plasma”
GKW: The numerical tool
• Gyro­kinetic flux tube code
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Spectral (perpendicular), local δf
Non­linear, kinetic electrons
Fully electromagnetic perturbations
General geometry (with CHEASE)
Collisions, multiple species
MPI scales to 8192+ processors
Formulated in rotating frame
http://gkw.googlecode.com/
A.G. Peeters, Y. Camenen, F.J. Casson, W.A. Hornsby, A.P. Snodin, D. Strintzi, G. Szepesi
“The non­linear gyro­kinetic flux tube code GKW” Comp Phys Comm, 180, 2650, (2009)
Strong rotation
For strong rotation, keep the centrifugal force
(Going round a corner throws you sideways)
– Rotation is “strong” when the toroidal velocity of the plasma is of the order of the thermal velocity Mach Number: Caution
:
Gyro­kinetics in a rotating frame
• Elegant formulation using the co­moving system [1]
Guiding centre velocity
Parallel velocity Inertial term • A rigid body toroidal rotation is assumed Toroidal background rotation Angular frequency [1] A.J. Brizard PoP (1995)
Application in local model
Toroidal background rotation Angular frequency 
Choose the rotation of the frame to be the rotation of the plasma on the flux surface we are modelling

In a local model the co­moving system yields compact equation similar in form to the non rotating system. One does not have to deal with a large flow across the grid (the large ExB velocity of strong toroidal rotation is transformed away)
Not suited for a global description since a gradient in the rotation would lead to a time dependent metric
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A.G. Peeters et al. Physics of Plasmas 16, 042310, (2009)
Consequences of strong rotation (1)
• Centrifugal drift • Coriolis drift
Coriolis drift influences momentum transport (Peeters PRL 2007)
Not discussed further here; it appears at “weak rotation” when
• Enhanced trapping
Unlike the Coriolis drift the centrifugal drift has a component along the field and therefore accelerates the particles outward Energy equation:
A.G. Peeters et al. Physics of Plasmas 16, 042310, (2009)
Potential transform and ordering
Consequences of strong rotation (2)
• Modified equilibrium equation
Normal trapping. Any distribution (like the Maxwell) which is isotropic is an equilibrium distribution Mass dependent centrifugal force different for electrons and ions Must retain a background electro­static potential which is a function of the poloidal angle in order to satisfy quasi­neutrality (Hinton + Wong, Phys Fluids, 1985)
Assume Maxwellian in velocity
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Consequences of strong rotation (3)
• Solve the equilibrium equation
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Find density is not a flux surface quantity
Centrifugal energy
(species dependant)
Background potential found by solving for quasineutrality
(Numerically for a plasma with more than 2 species)
•Density gradient also varies
Choice for density definition location
(Here we use LFS)
Choose where to define the density
• GKW gives two choices for R0:
– Low field side (used for all results later)
– Magnetic axis
Physics is independent of this choice, but it is only possible to hold R / Ln constant at one location when scanning over rotation
Consequences of strong rotation (4)
• Density and rotation not independent parameters
– Some care required in interpreting results GA­STD case
Inertial Terms (full)
(2 species plasma)
Source term ­ comments
• In order that the density gradient at R0 has the intuitive meaning of being in the radial direction, the radial derivative of R0 (at constant theta) must be kept in the source.
• The other parts of the derivative of the density exponential cancel with parts of the term Strong rotation: Enhanced trapping
Background potential traps electrons, but detraps ions. For both species:
Velocity space plot for a trapped electron mode (TEM):
Trapped region enlarges with rotation,
so mode is enhanced
Modified threshold for TEM
Growth rate spectrum
ITG
TEM
GA­STD case, D plasma:
ITG
Dispersion relation shows TEM dominates at larger scales with rotation
TEM
Nonlinear heat fluxes
Zonal flow response (GAM) Rosenbluth­Hinton
GA­STD case
• For ITG turbulence dominated case, heat transport increases with rotation
• Not included here: Stabilising effect of background ExB shear from sheared toroidal rotation
Behaviour near TEM “threshold”
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Threshold of dominant linear mode does not translate exactly to nonlinear case
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ITG mode seems more resilient in nonlinear phase
– Zonal flow reduction benefits ITG more than TEM?
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Coexistence and interaction of ITG / TEM modes
Particle and Impurity transport
• The Mach number for impurities is high even at low Mach number for the bulk ions:
Diffusion
Thermo­
diffusion
Convection
(inward)
C.Angioni, A.G.Peeters, PRL 96, 095003 (2006)
– Impurities feel the centrifugal force more strongly
Linear analysis of a single mode may be misleading:
• No interplay of scales
• No interplay of TEM / ITG modes
• All coexist in the nonlinear state
• Balance is important for particle flux
[ Angioni PoP 2009 ]
– Choice of R/Ln needed to define diffusion coefficient
Locate the null flux
(Back to the GA­STD case)
• Null flux state independent of choice of R0
• Null flux state is a balance of scales, at each scale:
– Inward contribution from slow trapped electrons
– Outward contribution from fast trapped electrons
• Effect expected to be stronger for heavy impurities
[ Angioni PoP 2009 ]
– More important to locate the null flux state due to strong density redistribution
Summary and Conclusions
• The flux tube model in the rotating frame naturally allows the inclusion of the centrifugal force
• Density gradient and rotation become coupled
• The centrifugal force leads to an enhanced trapping which promotes trapped electron modes
• In strong TEM regimes, increased electron heat transport is expected with strong rotation
• Particle pinch for ITG dominated cases examined
– Increased fraction of slow trapped electrons
– Stronger effect expected for impurity transport
– More nonlinear simulations needed
End
Strong Rotation: Future work
• Quantify predictions of impurity transport with rotation and compare to experiment
– Well resolved non­linear simulations, collisions
– Ideal experiment: Rotation controlled independently from heating with dual NBI.
• Include E x B shear with strong rotation
– Regimes with less ITG dominance
Reformulation of Brizard’s equations yields 
Finite ρ ∗ expansion Parallel velocity Curvature drift Grad­B drift ExB drift Coriolis drift Centrifugal drift Coriolis Force
Centrifugal force Momentum transport (local)
symmetry breaking mechanism
type / direction
toroidal rotation gradient (Peeters PoP 05)
toroidal rotation (Peeters PRL 07, Peeters PoP 09)
ExB advection in Coriolis drift the background (+ kin. electrons)
ExB sheared flow up­down asym. of (Dominguez 93, the MHD equil. Waltz PoP 07, (Camenen PRL 09, Casson PoP 09)
Camenen PoP 09)
ExB shearing
asymmetry of perp. drifts and k┴
diagonal part
pinch
residual stress
residual stress
outward
generally inward
inward/outward
inward/outward
­1 to ­4
0.4 to 0.8
0 to 1
[small param. range explored]
[linear sim. only]
Pr = 0.8 to 1.2
magnitude
higher for TEM
lower with ExB shearing due to toroidal rotation
ε, β, ΤΕΜ/ΙΤΓ, θ, R/Ln, mag. shear, s , s , mag. shear, q, mag. shear, R/Ln, TEM/ITG, main dep. / tested µαγ. σηεαρ, Ρ/Λν, e, q, Te/Ti, R/LT , B j
gE, u'
sB, sJ
Ρ/ΛΤι, νεφφ
n ef f , TEM/ITG
Symmetry breaking by E x B shear
Toroidal rotation Modification to diffusivity
Reduction in effective diffusivity (opposite for negative magnetic shear)
F.J.Casson, A.G.Peeters et Al., Phys. Plasmas, 16, 092303 (2009)