Gyrokinetic simulations including the centrifugal force in a strongly

Gyrokinetic simulations including the
centrifugal force in a strongly rotating
tokamak plasma
F.J. Casson, A.G. Peeters, Y. Camenen,
W.A. Hornsby, A.P. Snodin, D. Strintzi, G.Szepesi
CCFE Turbsim, July 2010
Outline
• GKW: The numerical tool
• When is strong rotation important?
• Gyro-kinetics in the rotating frame
– Coriolis force (previous work)
– Centrifugal force (this talk)
• New drift, enhanced trapping, modified
equilibrium
• Consequences for drift waves
[This talk is mostly
contained in a paper
submitted to PoP (2010)]
• Consequences for heat and particle transport
• Momentum transport (earlier work, if time)
GKW: The numerical tool
• Gyro-kinetic flux tube code
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Spectral (perpendicular), local δf
Non-linear, 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 non-linear gyro-kinetic 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:
Gyro-kinetics in a rotating frame
• Elegant formulation using the co-moving
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)
Comments
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 co-moving 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
Consequences of strong rotation (1)
• Centrifugal drift

Peeters PRL 2007
• Coriolis drift
Coriolis drift influences momentum transport (Peeters PRL 2007)
Not discussed here as 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)
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 electro-static potential which is a function of the
poloidal angle in order to satisfy quasi-neutrality
•
Assume Maxwellian in velocity
A.G. Peeters et al. Physics of Plasmas 16, 042310, (2009)
Consequences of strong rotation (3)
• Solve the equilibrium equation
•
Find density is not a flux surface quantity
Centrifugal energy
(species dependant)
“Centrifugal 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
GA-STD 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
(2 species plasma)
“Centrifugal 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
GA-STD case, D plasma:
ITG
Dispersion relation shows
TEM dominates at larger
scales with rotation
TEM
Nonlinear heat fluxes
Zonal flow response
(GAM) Rosenbluth-Hinton
GA-STD 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”
•
Threshold of dominant
linear mode does not
translate exactly to
nonlinear case
•
ITG mode seems more
resilient in nonlinear phase
– Zonal flow reduction
benefits ITG more than
TEM?
•
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
Thermodiffusion
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
coefficent
Locate the null flux
(Back to the GA-STD 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
Strong Rotation: Future work
• Quantify predictions of impurity transport
with rotation and compare to experiment
– Well resolved non-linear 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
– ETG - MAST relevance
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
• These effects could be observed on MAST ?
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
(Dominguez 93,
Waltz PoP 07,
Casson PoP 09)
up-down asym. of
the MHD equil.
(Camenen PRL 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
higher for TEM
magnitude
lower with ExB
shearing due to
toroidal rotation
ε, β, ΤΕΜ/ΙΤΓ, θ, R/Ln, mag. shear, s , s , mag. shear,
main dep. / tested µαγ. σηεαρ, Ρ/Λν, e, q, Te/Ti, R/LT , B j
gE, u'
Ρ/ΛΤι, νεφφ
n ef f , TEM/ITG
q, mag. shear,
R/Ln, TEM/ITG,
sB, sJ
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)
End
Reformulation of Brizard’s equations yields

Finite ρ∗ expansion
Parallel velocity
ExB drift
Coriolis
Force
Centrifugal
force
Curvature drift
Coriolis drift
Grad-B drift
Centrifugal drift