Improving Predictive Transport Model

C. Bourdelle 1), A. Casati 1), X. Garbet 1), F. Imbeaux 1), J. Candy 2), F. Clairet 1), G. Dif-Pradalier 1), G. Falchetto 1),

T. Gerbaud 1), V. Grandgirard 1), P. Hennequin 3), R. Sabot 1), Y. Sarazin 1), L. Vermare 3), R. Waltz 2)

1) CEA, IRFM, F-13108 Saint-Paul-lez-Durance, France 2) General Atomics, P.O. Box 85608, San Diego, California 92186-5608, USA

3) Laboratoire de Physique et Technologie des Plasmas, CNRS-Ecole Polytechnique, 91128 Palaiseau Cedex, France

Guideline

• Goal: To improve predictions on turbulent fluxes need physics based transport models

• Context: – Nonlinear gyrokinetic electromagnetic simulations still too costly

in terms of computing tim– Interestingly, quasi-linear approximation seems to retain the

relevant physics • Work on quasi-linear fluxes in two parts:

– quasi-linear weight : phase and amplitude, follows well non-linear predictions

– electrostatic potential: based on both non-linear simulations and turbulence measurements

• Integrated in QuaLiKiz where flux agrees with non-linear one when ranging from Ion Temperature Gradient (ITG) to Trapped Electron Modes (TEM)

general approach for quasi-linear model, QuaLiKiz [Bourdelle PoP07]

• fluctuating distribution function linearly responds to the fluctuating electrostatic potential through Vlasov equation computed by eigenvalue code Kinezero [Bourdelle NF02]

• Example for particle flux:

• No information on the saturation of the fluctuating electrostatic potential in terms of its amplitude or on its spectral shape versus the wave number and the frequency

2

,,

22

2~

0,

1Im

2

3

1

~~

k

k

nsDss

s

s

s

n

s

ss

innT

TR

n

nRen

d

Br

q

R

n

B

ikn

Accounting for the « non-resonant terms »

• Resonance Broadening Theory: non negligible finite +i0+=+i linked to irreversibility through mixing of the particles orbits in the phase space. Moreover in the limit →0 the particle fluxes are not ambipolar

• intrinsic frequency spectral shape of the fluctuating potential

• In QuaLiKiz, =0+ and :

equivalent to RBT where =k and• Nevertheless shape and width choices arbitrary. ongoing

measurements vs nonlinear simulations

2*

,

22*

,

2 ImIm ns

s

nn

s

s

ns ni

nS

dn

ni

nn

dk

k

k

k

22

kk

kk

S

kkS

Frequency spectrum: non-linear simulations vs measurements

k = k + Cst*kwith

reproduce widths of the frequency spectra observed from

GYRO simulations and measurements. Ongoing…

102

10310

-2

10-1

100

101

k [m-1

]

k

= 2.1809

n/n

0

= 2.2834

|n()|2

|nn/n0()|2

=2.3=2.2

Antar PPCF 1999

GYRO Backscattering on Tore Supra

Saturation rule: mixing length

• In QuaLiKiz, flux = sum over all unstable modes each weighted by corresponding k

•

as [Jenko, Dannert, Angioni 2005] adding

maxmaxmax

2

2max

k

k

k

ns

ss

sks

seff

kT

en

B

k

n

R

n

RD

k

2222 1 skk

kr spectrum: non-linear simulation vs measurements

• nonlinear GYRO compared with fast-sweeping reflectometer [Casati TTF08]

10-1

10010

-3

10-2

10-1

100

101

102

kr

s

1/

*2 |

n /

n|2

exp

= -2.8115

sim

= -2.9734

Fast-sweepingreflectometryGYRO

#39596, r/a=0.7

r,exp = -2.8

r,sim = -3.0

max,k

0

S

2

,kkn

ndkk rr

k spectrum: non-linear simulations vs measurements

-1

1-

cm 1

cm 0

S2

,

kkn

ndkk rr

•nonlinear GYRO compared with Doppler reflectometer [Casati TTF08]

10-1

10010

-4

10-2

100

102

k

s

1/

*2 |

n /

n|2

, [

a.u

.]

= -3.9946

DopplerreflectometryGYRO

#39596, r/a=0.7

k spectrum isotropy

• Isotropy found in some GYRO simulationsOngoing…

• Apparent (k,kr) anisotropy due to Doppler instrumentalintegration domain

• Hence, actual choice: from 0 to kmax:

and from kmax to infinity:

32~sn k

32~ sn k

quasi-linear weights

• in the case of an eigenvalue approach, the fluxes can not be unequivocally divided by

Therefore, discussion limited to most unstable mode• no simple tool allowing testing the validity of the quasi-

linear approach for subdominant modes yet developed

tr

k

rkk

k

tr

trQtrw

,

2

,

,,~

,,,,,

2~kn

Amplitude of the weight: QL/NL~1.5

• local and global simulations : systematic over-prediction QL vs NL around 1.5

• QL/NL ratio stays reasonably constant when changing plasma parameters, especially at low k scales

• Reason of this over

prediction to be assessed

0 0.5 1 1.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

ks

wQ

L /

wN

L

GYROGYSELA

adiabatic electrons, r/a=0.4, R/LTi=8.28, *=1/256

Phase of the weight: OK for ITG, fails for ITG-TEM

• Test introduced for TEM by [Jenko 2005-2008] extended to ITG and ITG-TEM cases

• Good QL/NL phase matching for ITG cases: particle and energy

• But close to ITG/TEM transition QL phase from most unstable mode fails for particle whereas energy OK

Cross-phase density vs potential

angle [rad]

k s

-3 -2 -1 0 1 2 3

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

x 10-3Cross-phase density vs potential

angle [rad]

k s

-3 -2 -1 0 1 2 3

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.005

0.01

0.015

0.02

0.025

R/LTi=9R/LTe=9R/Ln=3

R/LTi=6 R/LTe=9R/Ln=3

quasilinear fluxes vs nonlinear predictions

• test quasi-linear fluxes computed by actual version of QuaLiKiz versus nonlinear GYRO ion and electron energy fluxes and particle fluxes for various parameter scans ranging from ITG to TEM dominated cases

• only one renormalisation factor, C0, has been used in order to get the best fit to the nonlinear fluxes

R/LT scan

4 6 8 10 12 14-10

-5

0

5

10

15

20

25

QuaLiKiz (all unstable modes) versus GYRO

R / LT

eff /

G

B

Ion energyelectron energy particle effective diffusivities

GYRO (diamonds) QuaLiKiz (lines)

for R/LTi=R/LTe scan with R/Ln=3

* scan

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-3

-2

-1

0

1

2

3

4

5

6

7

8

QuaLiKiz versus GYRO: * scan on Tore Supra

ei [cs/a]

eff /

G

B

Based on Tore Supra * experimentIn agreement with experimental obs.GYRO (diamonds) QuaLiKiz (lines)Ion energyelectron energy particle r/a=0.5R/LTi=8R/LTe=6.5 R/Ln=2.5

Ti/Te scan

DIII-D Ti/Te scanPRL Petty 99Qualitative agreementwith experimentGYRO (diamonds) QuaLiKiz (lines)Ion energyelectron energy particle r/a=0.3R/LTi=6.5R/LTe=4.6R/Ln=1.4

0.5 1 1.5 2-2

-1

0

1

2

3

4

5

Ti / T

e

eff /

gB

eff,i

eff,e

Deff

Summary

• Assuming a linear response of the transported quantities to the fluctuating potential works rather well: phase OK if one unstable mode, amplitude over-estimated

• Moreover, when coupling the choices for electrostatic potential with the quasi-linear response, find quasi-linear fluxes agreeing well to nonlinear predictions for energy and particle fluxes over a wide range of parameters, from ITG to TEM dominated cases

Discussion

• A number of challenging issues remain to be tackled:

– quasi-linear approach known to fail : far from the threshold, onset of zonal flows, etc. Hence, domain in which it can be applied should be better understood

– choices for the electrostatic potential deserve more comparisons with nonlinear simulations and experimental measurements. In Tore Supra, presently comparing density fluctuations k and frequency spectra from Doppler and fast-sweeping measurements versus GYRO and GYSELA

– Finally, only integration of QuaLiKiz in a transport code such as CRONOS will allow testing in situ the predictive capabilities

Cross-phase ion energy vs potential

angle [rad]

k s

-3 -2 -1 0 1 2 3

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1

2

3

4

5

6

7

8

9

10x 10

-3 Cross-phase ion energy vs potential

angle [rad]

k s

-3 -2 -1 0 1 2 3

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1

2

3

4

5

6

7

8

9

10

x 10-3

Cross-phase electron energy vs potential

angle [rad]

k s

-3 -2 -1 0 1 2 3

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1

2

3

4

5

6

7

8x 10

-3 Cross-phase electron energy vs potential

angle [rad]

k s

-3 -2 -1 0 1 2 3

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1

2

3

4

5

6

x 10-3

R/LTi=9R/LTe=9R/Ln=3

R/LTi=6 R/LTe=9R/Ln=3