saturation rules for etg transport in quasilinear...

DIFFER is part of andDIFFER huisstijl presentatie 30 september 2016

Saturation rules for ETG transport in quasilineartransport models

1 FOM Institute DIFFER, PO Box 6336, 5600 HH, Eindhoven, The Netherlands2CEA, 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

J. Citrin1, C. Bourdelle2, N. Bonanomi3, T. Goerler4, P. Mantica3

Jonathan Citrin Madrid Gyrokinetic Theory Workshop, 2016 2

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?


Outline

• Phenomology of single scale ETG simulations

• Phenomology of multi-scale simulations

• ETG saturation rules in quasilinear models


Case study of agreement of single-scaleETG simulation with experimental fluxes

• 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

Electron heat flux: nonlinear GENE single scale ETG

Single scale (adiabatic ions) ETG nonlinear simulation of JET 78834, with strong electron heatingLx /nkx /min 𝑘𝑘𝑦𝑦𝜌𝜌𝑒𝑒/nky /nz /nw / nv = 200/256/0.05/24/48/48/12

Mantica, Bonanomi, Citrin, Goerler et al, IAEA 2016


• 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/𝐿𝐿𝑇𝑇𝑇𝑇studied in electron scale simulations, ion heat flux always agrees

Ion scale simulations: Miller, electromagnetic, collisions, kinetic electrons, carbon imp, fast ions, Zeff~1.9. Lx /nkx /ky 𝜌𝜌𝑖𝑖min/nky /nz /nw /nv = 100/128/0.05/24/32/48/8

Case study of agreement of single-scaleETG simulation with experimental fluxes

Ion heat flux (same discharge)


Ingredients needed for reasonable flux in single scale electron scale simulations

• Avoid electron scale zonal flows

• Avoid crazy radial streamers


Radial streamers can be controlled byrotation 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: Not saturated unrealistic level

γExB=0.007 𝑐𝑐𝑒𝑒/𝑅𝑅(∼ × 4 exp level)

Saturated comparable to expSaturation mechanism? Drift-wave,

drift-wave coupling?

Φ(x,y)Φ(x,y)

Φ(x,y)

~ x50 exp qe γExB=0.001 𝑐𝑐𝑇𝑇/𝑅𝑅


No convergence with box size: electronscale zonal flows enters the picture

𝛾𝛾𝐸𝐸 = 0, 𝜏𝜏 ≡ 𝑍𝑍𝑒𝑒𝑒𝑒𝑒𝑒𝑇𝑇𝑒𝑒𝑇𝑇𝑖𝑖

= 2.2, 𝑅𝑅𝐿𝐿𝑇𝑇𝑒𝑒

= 6.5 (just above linear threshold). Lx ~ 366, Ly ~ 84

Φ contour plotStrong ZF

• Linearly unstable, but then saturates strongly due to zonal flows.

• ETG “Dimits shift” regime

Increasing the radial box size destabilizes electron scale zonal flows


• 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

𝛾𝛾𝐸𝐸 = 0, 𝜏𝜏 = 2.2, 𝑅𝑅𝐿𝐿𝑇𝑇𝑇𝑇

= 7.0 . Lx ~ 366, Ly ~ 84 Test sensitivity of ZF saturation to 𝑅𝑅/𝐿𝐿𝑇𝑇𝑇𝑇

From lots of dedicated tests, hard to find convergence of Dimits shift thresholdStrong regime sensitivity between i) ZF dominance, ii) streamer dominance, iii) “reasonable” finite flux on: 𝐿𝐿𝑥𝑥/𝐿𝐿𝑦𝑦, nx, kinetic or adiabatic ions (low 𝑘𝑘𝑥𝑥𝜌𝜌𝑇𝑇 is on ion scales), 𝛽𝛽, 𝛾𝛾𝐸𝐸, collisionality. See also Colyer et al., 2016

No convergence with box size: electronscale zonal flows enters the picture


Outline





Multi-scale simulations show multiple regimes of multi-scale interaction

N Howard NF 2016

• High 𝑎𝑎/𝐿𝐿𝑇𝑇𝑇𝑇 , interaction with ion-scale ZF leads to weak ETG• Low 𝑎𝑎/𝐿𝐿𝑇𝑇𝑇𝑇 , weak ion-scale ZF, significant strengthening of ETG

• Indications that at lower (stable) 𝑎𝑎/𝐿𝐿𝑇𝑇𝑇𝑇, ETG significantly reduces again dueto emergence of electron scale ZF (i.e. back to regime as in Colyer et al)


Multi-scale simulation of our JET electronheated discharge

Mantica, Bonanomi,Citrin, Goerleret al, IAEA 2016

(PhD thesis, Nicola Bonanomi)

Multiscale GENE, local, 𝛿𝛿𝛿𝛿

• ETG linearly unstable• At nominal parameters, ion-

scale eddies kill ETG• Stretching 𝑅𝑅/𝐿𝐿𝑇𝑇𝑇𝑇 and 𝑅𝑅/𝐿𝐿𝑇𝑇𝑒𝑒

to edges of error bars seemto recover ETG heat flux as in single-scale case


Tentative conclusions from nonlinearsimulation 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-scalesimulations where electron scale zonal flows are not included?

• What then matters is whether ion-scale eddies saturate the electronscale, or are weak enough to allow electron scale to sature by DW-DW saturation.

• If we know when this occurs, can we then use single-scalesimulations to tune and validate quasilinear ETG saturation rules?

• Of course, this is all with electrostatic simulations. Furthercomplications observed to occur in EM (Maeyama PRL 2015)


Outline





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)

‘Zonal flow mixing’ term.Can couple high 𝑘𝑘𝑦𝑦 ETG withion scale ZF 𝑘𝑘𝑥𝑥

2 terms in quadratic nonlinearity that claim to lead to saturation

𝛾𝛾𝐷𝐷𝐷𝐷𝐷𝐷: ‘Drift wave mixing’ term


𝜙𝜙(𝑘𝑘𝑥𝑥,𝑘𝑘𝑦𝑦) ∼𝛾𝛾𝑚𝑚𝑚𝑚𝑚𝑚𝑒𝑒𝑚𝑚

𝑘𝑘𝑦𝑦2

Main idea of the model: the 𝛾𝛾 usedin the ETG mixing length rule shouldbe small when ZF mixing dominates, and equal to 𝛾𝛾𝐷𝐷𝐷𝐷𝐷𝐷 (saturationmechanism) when ZF is weak

Model for 𝑉𝑉𝐷𝐷𝑍𝑍 based on ion scale linearspectrum, and tuned to GYRO simulations

Same parameters as GYRO multi-scale case: high 𝑎𝑎/𝐿𝐿𝑇𝑇𝑇𝑇

Same parameters as GYRO multi-scale case: low 𝑎𝑎/𝐿𝐿𝑇𝑇𝑇𝑇

Can recover GYRO multi-scale electron heat fluxes (see Staebler PoP 2016)

Carry out Lorentzian broadening for final 𝛾𝛾𝑚𝑚𝑚𝑚𝑚𝑚𝑒𝑒𝑚𝑚

(Staebler PoP 2016)

First multi-scale ETG quasilinear transport model saturation rule now in TGLF-SAT1


ETG saturation rule in QuaLiKiz gyrokineticquasilinear tranport model

𝛿𝛿𝛿𝛿𝑠𝑠 𝜔𝜔,𝑘𝑘 =𝐹𝐹𝐷𝐷𝑇𝑇𝑠𝑠

1 −𝜔𝜔𝑘𝑘 − 𝑛𝑛𝜔𝜔𝑠𝑠∗

𝜔𝜔𝑘𝑘 − 𝑘𝑘∥𝑣𝑣∥ − 𝑛𝑛𝜔𝜔𝑠𝑠𝐷𝐷𝑇𝑇𝑠𝑠𝜙𝜙𝑘𝑘

∑𝑠𝑠 ∫ 𝑑𝑑3𝑣𝑣𝑑𝑑3𝑥𝑥 𝛿𝛿𝛿𝛿𝑠𝑠𝑇𝑇𝑠𝑠𝜙𝜙𝑘𝑘∗ = 0

Linearized Vlasovwith harmonicperturbations

Weak form forquasineutrality to close dispersion relation

𝜕𝜕𝛿𝛿𝑠𝑠𝜕𝜕𝜕𝜕

+ 𝒗𝒗 ⋅ 𝛻𝛻rfs + es𝐄𝐄 ⋅ 𝛻𝛻𝑣𝑣𝛿𝛿𝑠𝑠 = 0Electrostatic Vlasov(collisionless here for simplicity)

QuaLiKiz: a gyrokinetic quasilinear turbulent transport model (Bourdelle PoP 2007, PPCF 2016, Citrin PoP 2012)


𝐷𝐷 𝜔𝜔 = �𝑠𝑠

�𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑛𝑛𝑠𝑠𝑇𝑇𝑠𝑠2

𝑇𝑇𝑠𝑠1 −

𝜔𝜔𝑘𝑘 − 𝑛𝑛𝜔𝜔𝑠𝑠∗

𝜔𝜔𝑘𝑘 − 𝑘𝑘∥𝑣𝑣∥, 0 + 𝑖𝑖𝑖𝑖 − 𝑛𝑛𝜔𝜔𝑠𝑠𝐷𝐷𝐽𝐽02 k⊥ 𝜌𝜌𝑠𝑠,𝛿𝛿𝑠𝑠 𝛿𝛿𝜙𝜙(𝑑𝑑,𝑑𝑑) 2 = 0

Dispersion relation: 𝑘𝑘∥𝑣𝑣∥ for passing ions and electronsbounce average for trapped ions and electrons (𝑘𝑘∥𝑣𝑣∥ = 0)collisions only for trapped electrons

𝑘𝑘∥ = 𝑘𝑘𝜃𝜃𝑠𝑠𝑞𝑞𝑅𝑅 𝑥𝑥

From eikonal: 𝛿𝛿𝛿𝛿, 𝛿𝛿𝜙𝜙 ∝ e−in(φ−q r 𝜃𝜃)

x≡distance from q surface

𝜙𝜙 eigenfunction solved from high 𝜔𝜔 expansion of D(𝜔𝜔) and Gaussian ansatz

𝜔𝜔 ≡ 𝜔𝜔𝑟𝑟 + 𝑖𝑖𝛾𝛾 is the only unknown in the above equation. Root finding in upper complex plane (instabilities only)

Sketch of QuaLiKiz model construction


Transport fluxes for species j: carried by ExB radial drifts

Γ𝑗𝑗 ,𝑄𝑄𝑗𝑗 ,Π𝑗𝑗 ∝�𝑘𝑘

𝛿𝛿𝑛𝑛𝑗𝑗 ,𝛿𝛿𝑇𝑇𝑗𝑗 ,𝛿𝛿𝑣𝑣∥ × 𝑆𝑆𝑘𝑘𝛿𝛿𝜙𝜙𝑘𝑘

Use moments of linearized 𝛿𝛿𝛿𝛿𝑠𝑠 evaluated at theinstabilities, i.e. from solutions of 𝐷𝐷 𝜔𝜔𝑘𝑘

Spectral form factor 𝑆𝑆𝑘𝑘 and saturated amplitude of 𝛿𝛿𝜙𝜙 2 are unknowns. Their model, validated by nonlinear simulations, is the “saturation rule”

+ finite 𝑘𝑘𝑥𝑥 corrections at low-s from nonlinearphysics (JC, PoP 2012)

𝑆𝑆𝑘𝑘 ∝ �𝑘𝑘−3 𝛿𝛿𝑓𝑓𝑑𝑑 𝑘𝑘 > 𝑘𝑘𝑚𝑚𝑚𝑚𝑥𝑥𝑘𝑘 𝛿𝛿𝑓𝑓𝑑𝑑 𝑘𝑘 < 𝑘𝑘𝑚𝑚𝑚𝑚𝑥𝑥

𝑘𝑘𝑚𝑚𝑚𝑚𝑥𝑥 𝑖𝑖𝑠𝑠 𝑘𝑘 𝑎𝑎𝜕𝜕max𝛾𝛾𝑘𝑘𝑘𝑘⊥2

𝛿𝛿𝜙𝜙𝑘𝑘 2 = 𝐶𝐶𝑆𝑆𝑘𝑘 max𝛾𝛾𝑘𝑘𝑘𝑘⊥2

C is scalar factor set by matching heat fluxes in single NL simulation (for ion and electron scales separately)

Setting quasilinear fluxes with a nonlinearsaturation rule

Casati NF 09, PRL 09

Jonathan Citrin Madrid Gyrokinetic Theory Workshop, 2016

QuaLiKiz reproduces nonlinear fluxes

GA-standard s-scanG

B-flu

x

4 6 8 10 12 14-10

-5

0

5

10

15

20

25

R/LT

χ eff /

χ GB

χiχeDp

GA-standardR/LTi scan vs GYRO

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

Scans for “GA-standard case” parameters (numerous other scans and comparisons have also been successfully carried out)


From Bonanomi et al. EPS 2015 ICRH heated JET discharge 78834

• 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

ETG contribution in QuaLiKiz fluxes based on recent work on JET

GENE simulations

QuaLiKiz GA-STD s-scan with new ETG contribution


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 75225and 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

23

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!

Agreement excellent in all channels for 𝜌𝜌 > 0.5

For 𝜌𝜌 < 0.5, Ti underpredictiondue to lack of EM effects in QLK

Main result: JETTO-SANCO integrated modellingAgreement excellent in ALL channels for ρ>0.5

First ever 4-channel flux driven QuaLiKiz simulation. ~100 CPUh

Pr ~0.5


Sensitivity to ETG model in JET hybridscenario integrated modelling

Comparison with and without ETG model

ETG scales can be important for agreement, but sensitive to e.g. boundary conditions

Original fit andboundary conditions

Fit with reduced 𝑇𝑇𝑒𝑒, 𝑇𝑇𝑇𝑇boundary conditions at 𝜌𝜌 = 0.8 by ~20%


JETTO-QLK also validated by comparisonto a JET-ILW baseline scenario

ILW baseline scenarioJET 87412 (3.5MA/3.35T)Comparison with and without ETG-scales

Time window averaged between 10-10.5s

• Boundary condition at 𝜌𝜌 = 0.85

• Stable for 𝜌𝜌 < 0.2. No sawtooth model

• Assuming core measurements Ti=Tedue to poor core CX

• NTV torque due to NTMs flatten profile? Quality of core CX for 𝑉𝑉𝑡𝑡𝑚𝑚𝑟𝑟?

• Interesting interplay between momentum transport and profiles obtained without ETG. Under investigation

Good agreement inAll channels apart from Vtor


Summary

• Electrostatic multi-scale nonlinear simulations seem to show 3 regimes of ETG transport:i) Ion-scales stable extensive ETG Dimits shift regimeii) 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?

saturation rules for etg transport in quasilinear...

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