Saturation rules for ETG transport in quasilinear
transport models
J. Citrin1, C. Bourdelle2, N. Bonanomi3, T. Goerler4, P. Mantica3
1 FOM
Institute DIFFER, PO Box 6336, 5600 HH, Eindhoven, The Netherlands
2CEA, IRFM, F-13108 Saint Paul Lez Durance, France
3Istituto di Fisica del Plasma CNR, 20125 Milano, Italy
4Max Planck Institute for Plasma Physics, Boltzmannstr. 2, 85748 Garching, Germany
DIFFER huisstijl presentatie
30 september 2016
DIFFER is part of
and
Motivation
•
Significant ETG fluxes reported at experimental conditions in recent multi-scale
simulations (T Görler et al, PRL 2008, N. Howard et al, PoP 2014, NF 2016)
•
Complex multi-scale physics, with strong dependence on ion turbulence level
and zonal flows, and electromagnetic effects (Maeyema PRL 2015)
•
We desire accurate ETG saturation levels in quasilinear transport models, for
full profile and discharge evolution prediction
•
Tuning a quasilinear ETG saturation rule from
multi-scale nonlinear simulations is not everyone’s
cup of tea
•
What can single-scale simulations teach
us nonetheless? Can we find a cheaper
way to tune the quasilinear models?
Madrid Gyrokinetic Theory Workshop, 2016
Jonathan Citrin
2
Outline
•
Phenomology of single scale ETG simulations
•
Phenomology of multi-scale simulations
•
ETG saturation rules in quasilinear models
Madrid Gyrokinetic Theory Workshop, 2016
Jonathan Citrin
3
Case study of agreement of single-scale
ETG simulation with experimental fluxes
Single scale (adiabatic ions) ETG nonlinear simulation of JET 78834, with strong electron heating
Lx /nkx /min 𝑘𝑘𝑦𝑦 𝜌𝜌𝑒𝑒 /nky /nz /nw / nv = 200/256/0.05/24/48/48/12
Electron heat flux: nonlinear GENE single scale ETG
•
ETG saturation reached using 𝛾𝛾𝐸𝐸 to
break up radial streamers
•
Ion scales (TEM/ITG) alone cannot
explain the exp. qe flux and the exp.
qe stiffness
•
~50% of the electron flux from
electron scale
Mantica, Bonanomi, Citrin,
Goerler et al, IAEA 2016
Madrid Gyrokinetic Theory Workshop, 2016
Jonathan Citrin
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Case study of agreement of single-scale
ETG simulation with experimental fluxes
Ion scale simulations: Miller, electromagnetic, collisions, kinetic electrons, carbon imp,
fast ions, Zeff~1.9. Lx /nkx /ky 𝜌𝜌𝑖𝑖 min/nky /nz /nw /nv = 100/128/0.05/24/32/48/8
Ion heat flux (same discharge)
• qi,gB in good agreement with
experimental power balance
• Slight sensitivity of ion heat flux on
𝑅𝑅/𝐿𝐿𝑇𝑇𝑇𝑇 (trapped electron drive)
• However, within range of 𝑅𝑅/𝐿𝐿𝑇𝑇𝑇𝑇
studied in electron scale simulations,
ion heat flux always agrees
Madrid Gyrokinetic Theory Workshop, 2016
Jonathan Citrin
5
Ingredients needed for reasonable flux in
single scale electron scale simulations
•
Avoid electron scale zonal flows
•
Avoid crazy radial streamers
Madrid Gyrokinetic Theory Workshop, 2016
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Radial streamers can be controlled by
rotation shear. Proxy for ion scale eddies?
𝐿𝐿𝑥𝑥 ∼ 𝐿𝐿𝑦𝑦 (box sizes) , no destabilization of electron scale ZF
From 𝛾𝛾𝐸𝐸 2 times exp value, stabilizes ETG to experimentally relevant levels.
ETG flux level remains roughly constant for higher 𝛾𝛾𝐸𝐸 (encouraging)
• What does this mean? A proxy for ion scale eddies?
γExB=0.007 𝑐𝑐𝑒𝑒 /𝑅𝑅
(∼ × 4 exp level)
γExB=0: Not saturated  unrealistic level
Saturated  comparable to exp
Saturation mechanism? Drift-wave,
drift-wave coupling?
~ x50
exp qe
Φ(x,y)
γExB=0.001 𝑐𝑐𝑒𝑒 /𝑅𝑅
•
•
Φ(x,y)
Φ(x,y)
Madrid Gyrokinetic Theory Workshop, 2016
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No convergence with box size: electron
scale zonal flows enters the picture
Increasing the radial box size destabilizes electron scale zonal flows
𝑇𝑇
𝛾𝛾𝐸𝐸 = 0, 𝜏𝜏 ≡ 𝑍𝑍𝑒𝑒𝑒𝑒𝑒𝑒 𝑇𝑇𝑒𝑒 = 2.2,
Lx ~ 366, Ly ~ 84
𝑖𝑖
𝑅𝑅
𝐿𝐿𝑇𝑇𝑇𝑇
= 6.5 (just above linear threshold).
• Linearly unstable, but then
saturates strongly due to
zonal flows.
• ETG “Dimits shift” regime
Madrid Gyrokinetic Theory Workshop, 2016
Φ contour plot
Strong ZF
Jonathan Citrin
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No convergence with box size: electron
scale zonal flows enters the picture
𝛾𝛾𝐸𝐸 = 0, 𝜏𝜏 = 2.2,
𝑅𝑅
𝐿𝐿𝑇𝑇𝑇𝑇
Test sensitivity of ZF saturation to 𝑅𝑅/𝐿𝐿𝑇𝑇𝑇𝑇
= 7.0 . Lx ~ 366, Ly ~ 84
•
Increasing 𝑅𝑅/𝐿𝐿𝑇𝑇𝑇𝑇 by just 0.5 leads to
huge streamers that aren’t stabilized
by ZF.
•
“Out of Dimits shift zone”. This
effectively means very high stiffness
•
Open question: are these electron
scale zonal flows ever experimentally
relevant? Do ion scale eddies short
them out? Perhaps relevant only
when ion scales fully suppressed
From lots of dedicated tests, hard to find convergence of Dimits shift threshold
Strong regime sensitivity between i) ZF dominance, ii) streamer dominance, iii)
“reasonable” finite flux on: 𝐿𝐿𝑥𝑥 /𝐿𝐿𝑦𝑦 , nx, kinetic or adiabatic ions (low 𝑘𝑘𝑥𝑥 𝜌𝜌𝑒𝑒 is on ion
scales), 𝛽𝛽, 𝛾𝛾𝐸𝐸 , collisionality. See also Colyer et al., 2016
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Outline
•
Phenomology of single scale ETG simulations
•
Phenomology of multi-scale simulations
•
ETG saturation rules in quasilinear models
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Multi-scale simulations show multiple
regimes of multi-scale interaction
N Howard NF 2016
•
•
•
High 𝑎𝑎/𝐿𝐿 𝑇𝑇𝑇𝑇 , interaction with ion-scale ZF leads to weak ETG
Low 𝑎𝑎/𝐿𝐿 𝑇𝑇𝑇𝑇 , weak ion-scale ZF, significant strengthening of ETG
Indications that at lower (stable) 𝑎𝑎/𝐿𝐿 𝑇𝑇𝑇𝑇 , ETG significantly reduces again due
to emergence of electron scale ZF (i.e. back to regime as in Colyer et al)
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Multi-scale simulation of our JET electron
heated discharge
Mantica,
Bonanomi,
Citrin, Goerler
et al, IAEA
2016
(PhD thesis,
Nicola
Bonanomi)
Multiscale
GENE, local, 𝛿𝛿𝛿𝛿
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Madrid Gyrokinetic Theory Workshop, 2016
ETG linearly unstable
At nominal parameters, ionscale eddies kill ETG
Stretching 𝑅𝑅/𝐿𝐿 𝑇𝑇𝑇𝑇 and 𝑅𝑅/𝐿𝐿 𝑇𝑇𝑇𝑇
to edges of error bars seem
to recover ETG heat flux as
in single-scale case
Jonathan Citrin 12
Tentative conclusions from nonlinear
simulation phenomology
•
Electron scale zonal flows only matter when ion-scale turbulence is
completely stabilized (e.g. spherical tokamak with high rotation)
•
Therefore, if in regime where ion-scale unstable, OK for single-scale
simulations where electron scale zonal flows are not included?
•
What then matters is whether ion-scale eddies saturate the electron
scale, or are weak enough to allow electron scale to sature by DWDW saturation.
•
If we know when this occurs, can we then use single-scale
simulations to tune and validate quasilinear ETG saturation rules?
•
Of course, this is all with electrostatic simulations. Further
complications observed to occur in EM (Maeyama PRL 2015)
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Outline
•
Phenomology of single scale ETG simulations
•
Phenomology of multi-scale simulations
•
ETG saturation rules in quasilinear models
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First multi-scale ETG quasilinear transport
model saturation rule now in TGLF-SAT1
TGLF: a gyrofluid quasilinear turbulence transport model (Staebler
PoP 2007). Now with multi-scale saturation rule (Staebler PoP 2016)
2 terms in quadratic nonlinearity that claim
to lead to saturation
‘Zonal flow mixing’ term.
Can couple high 𝑘𝑘𝑦𝑦 ETG with
ion scale ZF 𝑘𝑘𝑥𝑥
Madrid Gyrokinetic Theory Workshop, 2016
𝛾𝛾𝐷𝐷𝐷𝐷𝐷𝐷 : ‘Drift wave
mixing’ term
Jonathan Citrin 15
First multi-scale ETG quasilinear transport
model saturation rule now in TGLF-SAT1
Main idea of the model: the 𝛾𝛾 used
in the ETG mixing length rule should
be small when ZF mixing dominates,
and equal to 𝛾𝛾𝐷𝐷𝐷𝐷𝐷𝐷 (saturation
mechanism) when ZF is weak
𝛾𝛾𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚
𝜙𝜙(𝑘𝑘𝑥𝑥 , 𝑘𝑘𝑦𝑦 ) ∼
𝑘𝑘𝑦𝑦2
Carry out Lorentzian broadening for final 𝛾𝛾𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚
Model for 𝑉𝑉𝑍𝑍𝑍𝑍 based on ion scale linear
spectrum, and tuned to GYRO simulations
Same parameters as GYRO
multi-scale case: high 𝑎𝑎/𝐿𝐿 𝑇𝑇𝑇𝑇
(Staebler PoP 2016)
Same parameters as GYRO
multi-scale case: low 𝑎𝑎/𝐿𝐿 𝑇𝑇𝑇𝑇
Can recover GYRO multi-scale electron heat
fluxes (see Staebler PoP 2016)
Madrid Gyrokinetic Theory Workshop, 2016
Jonathan Citrin 16
ETG saturation rule in QuaLiKiz gyrokinetic
quasilinear tranport model
QuaLiKiz: a gyrokinetic quasilinear turbulent transport
model (Bourdelle PoP 2007, PPCF 2016, Citrin PoP 2012)
𝜕𝜕𝑓𝑓𝑠𝑠
+ 𝒗𝒗 ⋅ 𝛻𝛻r fs + es 𝐄𝐄 ⋅ 𝛻𝛻𝑣𝑣 𝑓𝑓𝑠𝑠 = 0
𝜕𝜕𝜕𝜕
Electrostatic Vlasov
(collisionless here
for simplicity)
Linearized Vlasov
𝐹𝐹𝑀𝑀
𝜔𝜔𝑘𝑘 − 𝑛𝑛𝑛𝑛𝑠𝑠∗
𝛿𝛿𝑓𝑓𝑠𝑠 𝜔𝜔, 𝑘𝑘 =
1−
𝑒𝑒𝑠𝑠 𝜙𝜙𝑘𝑘 with harmonic
𝑇𝑇𝑠𝑠
𝜔𝜔𝑘𝑘 − 𝑘𝑘∥ 𝑣𝑣∥ − 𝑛𝑛𝜔𝜔𝑠𝑠𝑠𝑠
perturbations
∑𝑠𝑠 ∫
𝑑𝑑3 𝑣𝑣𝑑𝑑3 𝑥𝑥
𝛿𝛿𝑓𝑓𝑠𝑠 𝑒𝑒𝑠𝑠 𝜙𝜙𝑘𝑘∗
Madrid Gyrokinetic Theory Workshop, 2016
=0
Weak form for
quasineutrality to close
dispersion relation
Jonathan Citrin 17
Sketch of QuaLiKiz model construction
Dispersion relation:
𝑘𝑘∥ 𝑣𝑣∥ for passing ions and electrons
bounce average for trapped ions and electrons (𝑘𝑘∥ 𝑣𝑣∥ = 0)
collisions only for trapped electrons
𝑛𝑛𝑠𝑠 𝑒𝑒𝑠𝑠2
𝜔𝜔𝑘𝑘 − 𝑛𝑛𝑛𝑛𝑠𝑠∗
𝐷𝐷 𝜔𝜔 = � � 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
1−
𝐽𝐽02 k ⊥ 𝜌𝜌𝑠𝑠 , 𝛿𝛿𝑠𝑠
𝑇𝑇𝑠𝑠
𝜔𝜔𝑘𝑘 − 𝑘𝑘∥ 𝑣𝑣∥ , 0 + 𝑖𝑖𝑖𝑖 − 𝑛𝑛𝜔𝜔𝑠𝑠𝐷𝐷
𝑠𝑠
𝑠𝑠
From eikonal: 𝛿𝛿𝛿𝛿, 𝛿𝛿𝛿𝛿 ∝ e−in(φ−q r 𝜃𝜃)
𝑥𝑥
𝑘𝑘∥ = 𝑘𝑘𝜃𝜃
x≡distance from q surface
𝑞𝑞𝑞𝑞
𝛿𝛿𝛿𝛿(𝑟𝑟, 𝜃𝜃)
2
=0
𝜙𝜙 eigenfunction solved from high 𝜔𝜔 expansion of D(𝜔𝜔) and Gaussian ansatz
𝜔𝜔 ≡ 𝜔𝜔𝑟𝑟 + 𝑖𝑖𝑖𝑖 is the only unknown in the above equation.
Root finding in upper complex plane (instabilities only)
Madrid Gyrokinetic Theory Workshop, 2016
Jonathan Citrin 18
Setting quasilinear fluxes with a nonlinear
saturation rule
Transport fluxes for species j: carried by ExB radial drifts
Γ𝑗𝑗 , 𝑄𝑄𝑗𝑗 , Π𝑗𝑗 ∝ � 𝛿𝛿𝑛𝑛𝑗𝑗 , 𝛿𝛿𝑇𝑇𝑗𝑗 , 𝛿𝛿𝑣𝑣∥ × 𝑆𝑆𝑘𝑘 𝛿𝛿𝜙𝜙𝑘𝑘
𝑘𝑘
Use moments of linearized 𝛿𝛿𝑓𝑓𝑠𝑠 evaluated at the
instabilities, i.e. from solutions of 𝐷𝐷 𝜔𝜔𝑘𝑘
Spectral form factor 𝑆𝑆𝑘𝑘 and saturated amplitude of 𝛿𝛿𝛿𝛿 2 are unknowns.
Their model, validated by nonlinear simulations, is the “saturation rule”
𝑘𝑘 −3
𝑆𝑆𝑘𝑘 ∝ �
𝑓𝑓𝑓𝑓𝑓𝑓 𝑘𝑘 > 𝑘𝑘𝑚𝑚𝑚𝑚𝑚𝑚
𝑘𝑘 𝑓𝑓𝑓𝑓𝑓𝑓 𝑘𝑘 < 𝑘𝑘𝑚𝑚𝑚𝑚𝑚𝑚
Casati NF 09,
PRL 09
Madrid Gyrokinetic Theory Workshop, 2016
𝛾𝛾𝑘𝑘
𝑘𝑘𝑚𝑚𝑚𝑚𝑚𝑚 𝑖𝑖𝑖𝑖 𝑘𝑘 𝑎𝑎𝑎𝑎 max 2
𝑘𝑘⊥
𝛿𝛿𝜙𝜙𝑘𝑘
2
= 𝐶𝐶𝐶𝐶𝑘𝑘 max
𝛾𝛾𝑘𝑘
𝑘𝑘⊥2
C is scalar factor set by matching heat fluxes in single
NL simulation (for ion and electron scales separately)
+ finite 𝑘𝑘𝑥𝑥 corrections at low-s from nonlinear
physics (JC, PoP 2012)
Jonathan Citrin 19
QuaLiKiz reproduces nonlinear fluxes
Scans for “GA-standard case” parameters
(numerous other scans and comparisons have also been successfully carried out)
25
GA-standard s-scan
χi
20
χe
χeff / χGB
GB-flux
15
GA-standard
R/LTi scan vs GYRO
Dp
10
5
0
-5
-10
4
6
8
R/LT
10
12
14
Validation against experimental fluxes:
e.g. Tore Supra (Casati PhD 2009, Villegas PRL 2010),
JET (Baiocchi NF 2015, J. Citrin Varenna 2016, S.Breton, C. Bourdelle)
Continuous comparison of QLK to both nonlinear and experiment “part of our culture”
For transport studies, trivial parallelization of code over wavenumbers and radii
Madrid Gyrokinetic Theory Workshop, 2016
Jonathan Citrin
ETG contribution in QuaLiKiz fluxes
based on recent work on JET
From Bonanomi et al. EPS 2015
ICRH heated JET discharge 78834
QuaLiKiz GA-STD s-scan
with new ETG contribution
GENE
simulations
•
GENE single-scale NL simulation with 𝛾𝛾𝐸𝐸 to break apart streamers and avoid box
effects. ~50% of electron power balance in agreement with observation. Used to
tune scalar prefactor in QuaLiKiz ETG nonlinear saturation rule. Corresponds to
regime with drift-wave drift-wave coupling saturation mechanism?
•
Impact shown on GASTD case magnetic shear scan. Up to 50% of 𝑞𝑞𝑒𝑒 in some cases
Madrid Gyrokinetic Theory Workshop, 2016
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Extensive coupling work to JETTO-SANCO
JETTO – flux driven transport solver with sources and equilibrium [1,2]
SANCO – impurity density and charge state evolution, radiation
• Includes Pereverzev and G. Corrigan numerical treatment for stiff transport
• Neoclassical transport from NCLASS or NEO
1s of JET plasma takes ~20h walltime with QuaLiKiz on 16 CPUs (2.33GHz)
(Note: this is with rotation. Without rotation, around × 4 quicker due to symmetry
in 2D integration)
Extensive testing done on well diagnosed and studied hybrid scenario 75225
and baseline scenario 87412
First QuaLiKiz integrated modelling simulations with impact of rotation on
turbulence, multiple ions, and momentum transport
[1] G. Cenacchi, A. Taroni, JETTO: A free-boundary plasma transport code, JET-IR (1988)
[2] M. Romanelli et al., 2003, 23rd International Toki Conference
Madrid Gyrokinetic Theory Workshop, 2016
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Main result: JETTO-SANCO integrated modelling
Agreement excellent in ALL channels for ρ>0.5
First ever 4-channel flux driven QuaLiKiz
simulation. ~100 CPUh
JET 75225 (C-wall hybrid scenario)
Time window from 6-7s
C impurity in SANCO  D and C
modelled separately
Boundary condition at 𝜌𝜌 = 0.8
Includes rotation (𝜌𝜌 > 0.5) and
momentum transport!
Pr ~0.5
Agreement excellent in all
channels for 𝜌𝜌 > 0.5
For 𝜌𝜌 < 0.5, Ti underprediction
due to lack of EM effects in QLK
23
Sensitivity to ETG model in JET hybrid
scenario integrated modelling
Comparison with and without ETG model
Original fit and
boundary conditions
Fit with reduced 𝑇𝑇𝑒𝑒 , 𝑇𝑇𝑖𝑖
boundary conditions at
𝜌𝜌 = 0.8 by ~20%
ETG scales can be important for agreement, but sensitive to e.g. boundary conditions
Madrid Gyrokinetic Theory Workshop, 2016
Jonathan Citrin 24
JETTO-QLK also validated by comparison
to a JET-ILW baseline scenario
ILW baseline scenario
JET 87412 (3.5MA/3.35T)
Comparison with and without ETG-scales
Time window averaged between 10-10.5s
Good agreement in
All channels apart from Vtor
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Madrid Gyrokinetic Theory Workshop, 2016
Boundary condition at 𝜌𝜌 = 0.85
Stable for 𝜌𝜌 < 0.2. No sawtooth model
Assuming core measurements Ti=Te
due to poor core CX
NTV torque due to NTMs flatten
profile? Quality of core CX for 𝑉𝑉𝑡𝑡𝑡𝑡𝑡𝑡 ?
Interesting interplay between
momentum transport and profiles
obtained without ETG.
Under investigation
Jonathan Citrin 25
Summary
• Electrostatic multi-scale nonlinear simulations seem to show 3 regimes of
ETG transport:
i) Ion-scales stable  extensive ETG Dimits shift regime
ii) Ion-scales weak, significant ETG flux (saturated by DW-DW coupling?)
iii) Ion-scales strong, ETG scales are suppressed
• Can thus avoid electron-scale ZF in single-scale ETG when ion-scale is active?
• TGLF saturation rule has model to transition between regimes (ii) and (iii),
based on model of ion-scale ZF advection of ETG modes. Recovers GYRO
multi-scale runs
• QuaLiKiz saturation rule seems only to recover regime (ii), no multi-scale
apart from implicit ignoring of ETG zonal flow. Nevertheless, seems to
improve agreement in limited validation set.
Future work: simple model for determining regime change between (ii) and
(iii). Then can validate saturate rule for (ii) cases on single-scale ETG?
Madrid Gyrokinetic Theory Workshop, 2016
Jonathan Citrin 26