Global fullf gyrokinetic simulations of ITG turbulence

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Global full-f gyrokinetic simulationsof ITG
turbulence

• Y. Sarazin , V. Grandgirard, P. Angelino, G.
  Darmet,
• G. Dif-Pradalier, X. Garbet, Ph. Ghendrih
• Association Euratom-CEA, CEA/DSM/DRFC Cadarache,
  France
• G. Latu
• INRIA Scalapplix, LaBRI, Talence, France
• N. Crouseilles, E. Sonnendrücker
• INRIA Calvi, IRMA, Strasbourg, France

Acknowledgements C. Passeron, G. Falchetto
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Physics of global full-f gyrokinetic models

• Global code (? flux-tube)
• Large scale transport events
• Boundary conditions
• Numerical challenge
• Full-f code (? df )
• Equilibrium Fluctuations no scale separation
  assumption
• Profile relaxation ? statistical steady state
  requires
• prescribed driving flux
• Neoclassical theory

• Outline 1. Physics of the gyrokinetic
  equilibrium
• 2. Transport scaling with r with a global
  full-f code
• 3. Towards flux-driven model

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GYSELA-5D a GYrokinetic SEmi-LAgrangian code

• Standard gyrokinetic equation F(r,q,j,v//,m,t)
  f(r,q,j,t)
• Vlasov d/dt F 0
• Quasi-neutrality Se,i ? F d3p 0 (
  adiabatic electrons)

Poster P.Angelino et al., "Effects of plasma
elongation on drift wave-zonal flow turbulence"
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Parallelisation is challenging
Poster G. Latu et al., "Parallelization of a
full-f semi Lagrangian code GYSELA"

• Challenging parallelisation in GYSELA
• Eulerian ? domain decomposition
• semi-Lagrangian ? loss of locality due to
  interpolation
• (domains communicate)
• Much memory per node (10 Go at r2.10-3)
• ? "MPI OpenMP" parallelisation
• Limitating factor communications between procs.
• ? powerful interconnection is essential

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An efficient parallelisation
Poster G. Latu et al., "Parallelization of a
full-f semi Lagrangian code GYSELA"

• Time splitting numerical scheme
• 2 ? 1D advection in v on Dt/2
• 2 ? 1D advection in on j Dt/2
• 1 ? 2D advection in (r,q) on Dt
• Electro-neutrality "Poisson solver"

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Gyrokinetic equilibrium poloidal flows

• Gyrokinetic equilibrium without collisions
• ?F /?t 0 Feq FC(anonical) M(axwellian)
• depends on motion invariants only E, m, Pj

Idomura 03, Angelino 06
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Any local equilibrium maxwellian drives flows

• Starting from Local Maxwellian

Up-down (sin q) electric potential
FL(ocal) M(axwellian) ? FC(anonical)
M(axwellian) d/dt F ? 0 electric potential
f Large scale flow f00 ? r2 t2
Polarization
Well understood physics Analytical results agree
with simulations
Dif-Pradalier 07
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Reduced transport at early time when Finit ? FCM

• IF Finit FLM ? Turbulent transport delayed
• by large scale sheared flows

• Benchmark OK for canonical equilibrium

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Turbulence scaling with r

• Motivations Empirical scaling law wctE ? ??2.8
• ?ITER ? 2.10?3 not accessible to present day
  experiments
• Consistent with
• gyroBohm scaling
• H-mode exp. Ions particles gyroBohm
• McKee 01, Petty02, Hennequin 04

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Present status of gyrokinetic r scaling studies
Lin 02

• Transition Bohm ? gyroBohm
• when r decreases
• Lin 02, Waltz 02

• Open issues Transition threshold in r ?
• Physical mechanism of this transition ?

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Simulating a fraction of torus only

• Lj 2p/p ? modes n0, p, 2p, 3p, , N only
  (periodicity)
• all m
• Ratio (nb resonant modes / total nb modes) Cst

Top view

• Validity scales with r
• n ? kqri / qr ? Nb unstable modes ? (p r)-1

Small fraction of torus (pgtgt1) all the more valid
since r is small
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Case r1/128 ? OK up to Lj ? 2p / 8

• Case r1/128
• Correlation length/time diffusivity start
  deviating for Lj lt 2p / 8

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Transport scaling with r
Snapshots of electric potential cross-section
Lj 2p
Lj 2p / 2
r2.10-2
Lj 2p / 4
r10-2

• Decreasing r

r5.10-3
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lc consistent with Bohm scaling at large r
close to threshold

• c? lc2 / tc ra cBohm

lc (ari)1/2 ? Bohm (?0) lc ri ? gyroBohm
(?1)
tc a/cs and
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GyroBohm-like scaling above the threshold

• lc ri above the threshold
• ? consistent with gyroBohm
• Grandgirard '07

Threshold
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Strategies for driving turbulence
Fixed gradient (scale separation)
Fixed boundaries (thermal bath)
Profile relaxes

• Turbulence / Equilibrium
• decoupled

Most present codes
Equivalent if source term sustains mean profile
at t tE
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Implementing a source term in GYSELA

• Krook operator df /dt -n(f-feq)
• Diffusion in buffer regions df /dt ?r D(r) ?r
  f
• Profile relaxation is reduced

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Towards a statistical steady-state
n D

• Krook / Diffusion both efficient in maintaining
  the profile

• Marginal differences in
• Zonal Flow magnitude

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Similar diffusivity, different dynamics

• D, n ? similar magnitude of diffusivity
• Consistent with CYCLONE benchmark
• Asymmetric PDF of heat flux fluctuations ?Q / ?Q?

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Close to Maxwellian on average

• Issue
• w/o collisions does turbulence enforce
  Maxwellian?
• Given time position ? significant departure
  from Maxwellian
• Temporal or toroidal average ? f
  Maxwellian-like shape

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Flux driven reduced 3D kinetic model a paradigm

• Reduced gyrokinetic model f(E,y,a,t) Depret 00,
  Sarazin 05
• Goal Sustain a statistical steady-state
• Mean Heat source in gyrokinetic equation
• dF /dt S(E,y)
• Bursty ZF-Turbulence predator-prey model

T
Strong ZF ? Weak turbulence ? Core Temperature ?
fZF
df
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A heat source drives several fluid moments

• Decomposition on ortho-normal Laguerre
  polynomials Lk(E)

Lk(E) Fluid description
k0 ? density k1 ? temperature dT k2 ?
... (higher moments)
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Conclusions

• Full-f, global, semi-Lagrangian GYSELA code
• Efficient parallelisation 83 on 4096 proc.
• Canonical equilibrium
• Linear non linear benchmarks (R-H theory,
  CYCLONE, ORB5)
• Large scale sheared flows inherent to FLocal
  Maxwell
• Evolve like (rt)2 saturate at significant
  level
• Delay turbulent transport ? makes results
  difficult to interpret
• Bohm vs. gyroBohm depends on r distance to
  threshold
• Flux-driven gyrokinetics towards statistical
  steady-state

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Various strategies for gyrokinetic codes
Eulerian Particle-in-Cell (PIC) Dissipation Nois
e ? ? high order scheme df optimized loading
Full-f df
Flux-tube Global small scale structures large
scale events
GS2 code
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GYSELA full-f code for toroidal ITG turbulence

• Standard gyrokinetic equation f(r,q,j,v//,m,t)
  f(r,q,j,t)

"Vlasov"
Electro- neutrality

• Linear benchmark with other codes (CYCLONE)

after Dimits 00
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ZF linearly undamped in collisionless regime
E-folding decay GAM oscillations Residual
magnitude Consistent with theory Rosenbluth-Hinto
n 98, Sugama-Watanabe '07
Initial poloidal flow (m,n)(0,0) shielded by
finite orbit width effects
Sideband coupling Undamped GAMs
(m,n)(?1,0) component Landau-damping Zonal Flow
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Transport strongly reduced by poloidal flows

• Prescribing f00(r,t)0 ? Vanishing poloidal
  flows
• zonal (??0) mean (?0) flows
• When poloidal flows are present
• c? reduced by a factor of 3-4
• c? consistent with CYCLONE base case

after Dimits 00
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Poloidal flows significantly reduce lcorr and
tcorr

• Two point correlation length/time of potential
  fluctuations
• lcorr increases by about 30
• tcorr doubles

when prescribing f00(r,t)0
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Any local equilibrium maxwellian drives flows

• Starting from local Maxwellian ? electric
  potential f
• fs10 sinq excited by curvature
• Time evolution recovered analytically
• GAM frequency recovered when accounting for E?B
  drift

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Poor statistics for correlation time tc

• c? lc2 / tc ra cBohm

lc/ri ? r-1/2 ? Bohm (?0) lc/ri
Cst ? gyroBohm (?1)
wctc ? r-1 and
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GyroBohm-like scaling "far" from threshold
Correlation time

• Correlation length

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Flux driven reduced 3D kinetic model a paradigm

• Reduced gyrokinetic model f(E,y,a,t) Depret 00,
  Sarazin 05
• Goal Sustain a statistical steady-state
• Mean Heat source in gyrokinetic equation
• dF /dt S(E,y)
• Bursty ZF-Turbulence predator-prey model

T(?)
S(?)
Darmet 07
?
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GYSELA-3D Flux driven Trapped Ion Modes

• Complex interplay between ZF and Drift Waves
• Non-gaussian statistics of fluctuations

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Parallel trapping