Full-wave test of analytical theory for fixed frequency fluctuation reflectometry

• Motivation• Full-wave codes• Analytical theory (1D) compared• Fluctuation profile reconstruction• Conclusion

M.Schubert

CEA, IRFM, F-13108 Saint-Paul-lez-Durance, France

05/05/2009

TORE SUPRA

2

Formula (1) valid, if

• dnrms/nc is small

• dnrms is homogeneous

Problems :

• keff = ?

• the case keff 0

Local fluctuationmeasurement ?

c

rms

eff0rms

nn

k

Lk N d

d2

BA A ≠ B

dnonlocal

(Eq.1)

3

Example: fast-hopping reflectometer on Tore-Supra (simulation)

f1

f2f2

t

f1

tii tff 2),( )(rms dd

Profile shape not well recovered if using (1) fordrms dnrms

Using < ne > to obtain the mapping k0( fi ) x .

(input)

Simulation result :

4

Arbitrary profiles of the fluctuation level

)(c

edge c

)( )( )(0

2

2200

2kx

x n

xnxGxkk ddd

2ceff

n

nk

Lk N

220

2 2 dd

xc cut-off of vacuum wavenumber k0(f) in the average N2 profile

G(x) weighting function: turbulence properties and N2(x) profile

Gusakov et al. PPCF44 (2002) 2327

(Eq.2)

5

Numerical Tools(developped at LPMIA and IST)

• 1D, time-independent (“Helmholtz”)– Very fast, good to do statistics d k dnk

– Stationary state: fixed frequency, snapshot of turbulence– used on TS (DREFLUC), e.g. theses L.Vermare, T.Gerbaud

• Spatio-temporal– Wave and turbulent dynamics– Dedicated to Ultra-fast sweep (projet ANR)– Simulation of dynamic phase measure-

ment in presence of secondary cut-off

• 2D (space) + time– Beam diffraction, Doppler reflectometry– O-mode OK , X-mode to come soon …

plasma density

random superposition

Heuraux et al. RSI 74 (2003) 1501.

Modelling Turbulence

wavenumber resolution

2/k longest wavelength of turbulence mode

i snaphots of turbulence (~ t)irandom phases [0 .. 2]

, i = 1…M

7

use analytic expression, given lC is known

Restrictions :

• linear gradient of ne (O-Mode) with characteristic length L = LN2

• homogeneous turbulence, small correlation length lC << L

• small fluctuation level, no secondary cut-offs

• Bragg backscattering negligible (k02 / L)1/3 >> (lC )-1

Application of 1D code “Helmholtz”

Gaussian spectrum

Gusakov et al. PPCF44 (2002) 2327

710

8 2.

ln

c

c1-eff l

L

πl

k N

8

Effect of long wavelength fluctuations

scan of k

47 GHzL = 58 cm

lC = 0.75 cm

dn/n = 0.001

( Gaussian spectrum , <dn2> = const. )

Convergence !

9

Full-scale comparison : theory – code result

Cross hair

f = 47 GHz, L = 58 cm,

lC = 3 cm, dn/n = 0.001

Scan single parameters,

keeping the others const.

x

< d>Gusakov

< d

>nu

mer

ic

+ lC = 1, 4.5 cm

dn/n = 0.0005 ... 0.1

f = 24, 58 GHz

L = 32, 100 cm

10

Inhomogeneous fluctuation profile

For certain spectra G can be derived accurately : e.g. Gaussian

wavenumber spectrum, O-Mode. If N2 changes linearly near the

cut-off, Gnear is exact.

2

2

02

2 222 2

c

c

c

cnear

)(

)(exp

l

xxI

l

xxLG

N

modified Bessel function ( )

|x-xc| < lc :

|x-xc| > lc :

(Eq.2)

11

1. Input profilesfluctuation level

2. Select set of fi

evaluate cut-offpositions x(fi)

3. Full wave code :2(fi) o(~ “measurement“)

4. Make a guess for fluctuation level profile dn(xi)

5. Use Eq.2, evaluate <d2>i +

6. Minimize least squares, dn(xi) are free parameters

12

Result of Least-squares minimization

- High fluctuation level (non-linear scattering)

- N2 does not change linearly near cut-off

- Grid of dn(xi) is not fine enough

Remaining deviation :

13

Numerical study of error propagation using Eq.2 :

Large error bar on <d2> “measurement“ at the plasma edge (30% rel.err)

Effect on the reconstructed fluctuation profile: error propagates.

Measurement in the plasma center is robust.

Fluctuation measurement in the plasma center of Tore-Supra :

2D Simulation

• Single path transmission

• Profile of fluctuation amplitude

• Single frozen pattern of the turbulence

• Can the 1D (“Helmholtz”) result be applied ?• Large edge fluctuation level: What happens to Gaussian beam (2D slab) probing the center ?

Launch 40 GHz,

FWHM ~ 7cm

Fluctuation level

at plasma edge

(x = 0) ~10%

Contour plot of

electric field

Phase coherence is intact

Proc. 3rd France-Russia Seminar, Metz (2007)

16

Summary

Successful validation of analytical formula for the absolute density fluctuation, using full-wave code.

Sucessfully tested an inversion algorithm for the fluctuation level profile, based on integral expression for the phase fluctuations.

2D simulations in Tore Supra geometry : Beam shape and phase fronts remain intact when crossing the strongly fluctuation plasma edge.

Validation is being extended to X-mode.

17

Contributors :

• F.Clairet, R.Sabot (CEA Cadarache)• E.Gusakov, A.Popov (Ioffe Institute St.Petersburg)• F.Da Silva (IST Lisbon)• T.Gerbaud (JET)• S.Heuraux (LPMIA / IJL Nancy)

Support:

RFBR grants 06-02-17212, 07-02-92162-CNRS,

ANR grants FI-071215-01-01, 06-blan0084

18

Motivation / possible applications

• Understanding of plasma turbulence, particle and energy transport fluctuation measurement

• Systematic investigation of the density fluctuation level as function of external parameters(grad T, grad n, IP, * etc)

• Measurement of the propagation velocity of a turbulent wave front generated by heat pulse

• Comparison : Sk Skr

19

Motivation - I

M.Schubert, LPMIA (2007), code by F.daSilva, IST

source

lens

plasma

Strong fluctuations (plasma density)

propagationin vacuum

Validate the reflectometry measurement in case of :

• strong fluctuations of ne on

the beam path non-linear scattering

20

Motivation - II

x

ne

L

t + t

t k0

Drawing: C.Fanack, UHP Nancy (1997)The Theory of Plasma Waves, T. Stix (1962)

L k0

Exact solution

usual approximation

large fluctuation(H-mode/ELM, Blob) L k0 0

• non-linear phase shift from cut-off reflection

21

If there are :

• strong fluctuations of ne on the beam path

(non-linear scattering)

• strong fluctuations at the cut-off(non-linear phase shift of reflected wave)

Only full-wave codes can validate the reflectometry measurement.

22

Far from the cut-off G can be approximated as

wherecan take into account an arbitrary density profile.

Fig.2: Weighting factor G,linear density gradient L= 50 cm, turbulence with Gaussian spectrum, correlation length lc=1cm. x-axis: distance to the cut-off, normalised to lc

Black: exact solution Gnear

Red: approximation Gfar.

modified Bessel function ( )