Reflectometry Simulations : Choices to do, and ...charles/fusion/exposes/expose... · Heuraux €...

1 $. Heuraux € AE Fusion INRIA Paris

Reflectometry Simulations : Choices to do, and applications to current issues

S. Heuraux$, F. da Silva£, E. Gusakov€, A. Popov€, N. Kosolapova€, K. Syisoeva€, F. Clairet¥, R. Sabot¥

$Institut Jean Lamour-Faculté des Sciences-Nancy-Université UMR 7198 CNRS, BP 70239 F-54506 Vandoeuvre cedex ( [email protected] ) €Ioffé Institute Politekhnicheskaya 26, 194021 St.Petersburg, Russia £Associação EURATOM/IST IPFN, Instituto Superior Técnico, 1046-001 Lisboa Portugal ¥Association Euratom-CEA_Cadarache 13108 St Paul-lez-Durance – France

Schedule: -recall of basic mechanisms involved in turbulence characterisation, -how to choice a relevant set of equations for a given simulation, -current issues: stochastic against Bragg backscattering processes,

beam spreading - impact on the measurements, new concepts of diagnostics, …

conclusions and status of the ERCC 3D-code and Thanks to INRIA for the invitation to this Fusion Summer School


  Principle

source

detector

ν(t)

φ(t)

A(t) [φ(t)] ν(t'), φ(t')

[ν1; ν2] (∂ν /∂t) ν [8-155 GHz] λο [mm-cm]

Heuraux et al., Rev. Sci. I (2003) 74, 1501, L. Vermare et al, Plasma Phys Cont Fusion 47, 1895 (2005) T. Gerbaud et al 77, 10E928 (2006), S. Heuraux et al Aims J. Discrete and Cont. Dyn. Sys. - Series S to be published.

Reflectometry : basic principles (1)

δφ

Experimental data from Tore Supra Reflectomter

!(" ) =!WKB (" )+#!(" ,E2 (" ,#n))

A(! ) = AWKB (! )+"A(! ,E2 (! ,"n))

Unperturbed density profile

δn density fluctuations

or !(" )

or A(! )


Reflectometry : basic principles (2)

!"(!n(# ),E 2 (# ,!n),!V,N(# )) = !"source +

!"N +!"!n +!"E2+!" !V +!"!T +!"!

!B +....

In experiments , what should be taken into account ?

!"source

!"N

!"!n

!"E2

!" !V!"!T

!"!!B

+.... Mode conversion, cross-polarisation, multi-reflections,

Hardware noise, thermal phase shift of hardware, plasma emission, … Macroscopic variations of the background plasma parameters, Tranverse gradients effects, Doppler shift if time dependent, Refraction effects, scattering contribution (for- and back-ward)

Damping of the probing wave , resonances (wave trapping), …

Plasma motion inducing Doppler shift => phase shift by n variations, …

Through relativistic corrections on the e- mass, thermal effects …

Magnetic fluctuations -> scattering, magnetic shear effects, …

In addition Numerical errors if a code is used to compute transfer function !


Bragg BackScattering rule

κf

κi κs

Basic mechanisms involved in turbulence characterisation

!"!t

=!F!t

N(F, s)ds+!sed

sc! F !!t

N(F, s)dssed

sc!"#$%&'

fixed probing frequency

!

"F"t

= 0

fast sweep probing frequency

!

"N"t

= 0

(BBS)

! i = 2k(xr )

Measurements based on phase variations

Fluctuations reflectometry or frequency hopping systems Radial and poloidal correlation reflecto.

Density profile reflectometry

Forward scattering Doppler shift (dn motion)

F. Da Silva et al Nuclear Fusion 46, S816 (2006); F. Da Silva et al & A. Popov et al IRW8 reflectometry meeting 2-4 May 2007

Reflection at cut-off layer

NO,X (xc)= 0

Amplitude modulation

Destructive interference

Doppler shift n(t)


!

< "# 2 > = $ k02 LN 2

% f eff

< "n2 >nc

2

Basic mechanisms involved in turbulence characterisation

More efficient

Closer to the cut-off

Gusakov et al. PPCF 44 (2002) 2327 Schubert et al IRW9 Lisbon (2009)

!

< "# 2 > = $ k02 dx G(x)

x edge

x C (k o )

% < "n2(x) >nc

2

Inhomogeneous turbulence case These methods assume E2(r) known

=> New method for the density fluctuation profile reconstruction

Fanack et al PPCF 38 (1996) 1915


•  Assumptions versus set of equations describing wave propagation

A. Casiti et al PRL 2009


Reflectometry simulations : How to do it ? (1)

First question : how to choose the relevant equation for the simulation ? need to know exactly the physical system to simulate that is to say what are the relevant assumptions for this system?

A proposal to define it corresponds to a set of equations:

time dependent or not ?

refraction effects only ?

doppler effects there or not?

single polarisation description or not?

Ray-Beam tracing, Helmholtz Wave Eq, Maxwell' Eqs + plasma dynamics

Ray-Beam tracing

Ray-Beam tracing, Helmholtz Wave Eq, Maxwell' Eqs + plas. dyn.

Single description

Coupled equations Coupling to model


From ray tracing to wave equation (1) Ray tracing

Single mode description D(w,k,r,t)=0

Set of coupled Odes to solve

!

"! r "#

= $"D(%,

! k ,! r ,t)

"! k

"! k "#

= "D(%,! k ,! r ,t)

"! r

&

' ( (

) ( (

!

"t"#

="D($,

! k ,! r ,t)

"$"$"#

= %"D($,

! k ,! r ,t)"t

&

' ( (

) ( (

Can be extended to Gaussian beam propagation by one ODE associated to amplitude or using stationnary phase method

Numerical Tools needed for ITER plasma position studies

Quasi-optic description without scattering

RungeKutta 4th order

Hyp. WKB : !!!

!!!dk

dx « k2, !!!!

!!!!d2k

dx2 « !!!

!!!dk

dx k

D. G. Swanson "Plasma Waves", 2nd Ed IoP 2003,ch6.5, ISBN 0 7503 0927 X

G. V. Pereverzev Phys. Plasmas 5, 3529 (1998) R.A. Cairns, V. Fuchs Nucl. Fusion 50 (2010) 095001

A. Richardson, P. Bonoli, J. Wright, PoP 17 (2010) 052107 Or eikonal method with wavepacket for the amplitude description (as quantum phy.)


From ray tracing to wave equation (2)

Helmholtz's equation (full-wave)

Hyp: monochromatic wave, steady state plasma (∆t or lcorr >> 4rc/c)

Single mode description: Computation of the index N(r)

!

"! E + N 2(! r )

! E = 0

Be careful in multi dimensional case, possible cross derivatives more complicated to solve especially for X-mode No Doppler O-mode other method see L-M. Imbert this conférence

Monochromatic and single polarisation probing system

Finite Difference 4th order (Numerov)

C. Fanack, PhD Thesis or et al PPCF 38, 1915 (1996) S. Heuraux, F. da Silva DCDS_S 5, 307 (2012)


Finite Element Method

Actually only few developments on FEM with dispersive media:

In plasma only using equivalent dielectric (Ph Lamalle for ICRH or F. Braun & L. Colas) for ICRH (HFSS or COMSOL multi-Physics) including boundary sheath conditions L. Colas, D. Milanesio, E. Faudot#, M. Goniche, A. Loarte J. N. Mat 390-391 (2009) 959-962, O. Meneghini, S. Shiraiwa, R. Parker PoP, 16 (2009) 090701 Accurate method in vacuum and in complex geometry (commercial software) ALCYON was ICRH code based on functionals, if will be replaced by EVE code developed by R. Dumont (CEA_cadarache) and needs a lot of memory (~10-20 Gbytes) In the case of reflectometry possible ? Yes M. Irzak, et al Nuc Fus 35 1341 (1995)

Monochromatic multi-polarisation probing system

EVE R. Dumont Nuc Fus 49 075033 (2009)

seeTalk R. Dumont friday

Resonances generated by Bragg resonant perturbation with ≠ sources



Shrödinger like's equation (full-wave)

Hyp: quasi-monochromatic wave quasi steady state plasma (∆t or lcorr >> 4rc/c)

Single mode description: Computation of the index N(r)

!

i"t

! E + #

! E + N 2(! r )

! E = 0

Lin et al, Plasma Phys. Cont. Fusion 40 L1 (2001)

!

" >> #t

Restriction on dispersion effects, Quasi-paraxial approximation

Quasi steady state plasma

Parabolic



wave equation (quasi-steady state medium) Hyp: (tf, ∆t or lcorr >> 4rc/c)

!

"t2! E # c 2$

! E +% pe

2 (! r )! E = 0

O-mode or isotrpic plasma

Set of coupled partial differential equations associated to X-mode

Time dependent physical processes or probing system

Hacquin et al, J. of Computational Physics 174, 1 (2001),

Cohen et al, Plas. Phys. Cont Fusion 40, 75 (1998),

V = V/VD where VD=Eo/Bo

and E= E/Eo

Finite Difference + wp

2E rewritting

[!t2 " c2 (!x

2 +!y2 )+! pe

2 ]Ex + c2!x#.!

!E = "! pe

2 vyBo / c

[!t2 " c2 (!x

2 +!y2 )+! pe

2 ]Ey + c2!y#.!

!E =! pe

2 vxBo / c

me!tvx = "e(Ex + vyBo )

me!tvy = "e(Ey " vxBo )

$

%

&&&

'

&&&



wave equation (time-depend medium)

Hyp: single mode polarisation

!t2!E ! c2"

!E +" pe

2 (!r, t)!E = e

#o

!v!tn

!t!v = ! e

me

!E

#

$

%%

&

%%

Just to add ∂tn in the set of coupled partial differential equations associated to X-mode

O-mode or isotrpic plasma

Fast gradient motion, up or down frequency shift amplitude variation

Frequency upshift with ∂tn

Turbulence dynamics, fast events

Finite Difference + wp

2E rewritting + RK45

J. Mendoça, New Journal of Phys.,11(2009) 013029


Cross polarisation simulations

!t2Ez ! c

2!x2Ez +" pe

2 (x, t)Ez =COX Ex,Ey( )!t2Ex +" pe

2 (x, t)Ex = !" pe2 (x, t)vy +CXOx Ez( )

!t2Ey ! c

2!x2Ey +" pe

2 (x, t)Ey =" pe2 (x, t)vx +CXOy Ez( )

!t!v = ! e

me

!E ! e

me

!v "!B

#

$

%%%

&

%%%

1D Case: O-mode and X-mode

O-mode X-mode

dB measurements

Finite difference

N. Katsuragawa, H. Hojo, A. Mase J. Phys. Soc. Jpn. 67 (1998) 2574-2577

©€


Full description: Maxwell's equations

Hyp: linear response of the plasma

!

".! B = 0

".! E = #

$o

" %! E = &'

! B 't

" %! B = µo

! j + 1

c 2'! E 't

(

)

* * *

+

* * *

r total density of charges j current density Associated model fluid or kinetic

Radial direction

60 GHz

Polo

idal

dir

ecti

on

F. da Silva et al , J Plasma Phys. 72 1205 (2006), and Rev. Sci Instr. 79, 10F104 (2008) C. Lechte, IEEE TPS 37 (2009) 1099.

TE and TM modes are usually treated separately

Velocity field mapping, Shear layer detection

Yee's algorithm + J solver

x/lo

50 cm

Eqs Maxwell Eqs + Vlasov or PIC, Required too much computation time for reflectometry simulations


On the role of E2 spatial distribution


The amplification factor of the scattered wave (blue dots or black stars), the squared amplification factor of the probing wave in the cavity (red dots) and the analytical eq. versus the phase F ofδnq

Heuraux IEEE trans Plasmas Sci. 38 (2010) 2150 Gusakov et al. Phys. Cont. Fusion 51 (2009) 065018

Enhanced scattering induced by a resonant cavity

δnf/nc= 3.3·10-5 δnf/nc= 1.6·10-3

δnf

δnq

enhanced scattering factor Ees/Es

amplification factor of the probing wave

Eres/Enp

18 $. Heuraux € AE Fusion INRIA Paris E. Gusakov, S. Heuraux et al Physica Scripta 84 (2011) 04504.

X

Z

Gaussian probing beam

Radial position of the Bragg resonant dn

1

-ao ao 0

Enhanced scattering induced by a resonant cavity(2) In the previous case, evolution of the

scattered field versus dn/nc with and without resonance

Modification of the probing field structure

Role of E2 distribution


Competition between resonant mechanism and Bragg back scattering in turbulent plasma

Density gradient length L= 160 lo, dntur/nc= 5 %

E/Eo

x(pts)

E2 ≠ Cte


Density gradient length L= 160 lo, homogneneous turbulence dntur with cut-off

Competition between resonant mechanism and Bragg back scattering (1) Reflectometry case

dntur/nc= 5 % dntur/nc= 10 %

Average value (over 104 samples) of electromagnetic flux of probing wave

Resonances win and permit to the probing wave to reach the cut-off

Bragg BackScattering wins and reduce the probing depth

Gusakov et al. plasma phys EPS conf dublin 2010 Heuraux et al Cont Plasma Phys in press

The theory is based on a photon diffusion equation only valid at moderate fluctuation level

<E2(r)> vg ~ Cte

<E2(r)> vg ≠ Cte


2000 4000 6000 8000 10000

-4

-2

0

2

4

61

2000 4000 6000 8000 10000

0.2

0.4

0.6

0.8

1

2000 4000 6000 8000 10000

0.2

0.4

0.6

0.8

1

Removing the resonances Taking into the resonances

Simulations with experimental density fluctuation profile

Good news while the path in high turbulence level stays small


Spectrum effects on the probing field structure

0 2000 4000 6000 8000 100000.0

0.5

1.0

1.5

2.0

Ln= 1000 lo for dntur/nc=1% with nmax = 0.8 nc and the same width for the density fluctuations (~75lo) Wavenumbers are resonant from the middle to the top of the density profile

<E2 RM

S/E2

io>

r/lo

0 2000 4000 6000 8000 10000�3

�2

�1

0

1

2

32BBS zone

<E2(r)> vg

T and R are a function of S(kf) and can change at dntur/nc= Cte


Studies on forward scattering


Simulation of Forward scattering

Probing frequency 40 GHz Spectrum without Bragg resonant wavenumber

Spectrum of the backscattered signal frequency shift

backscattered signal simulated


!

"# = 2kpolVpol$ (%Q)2

2 +$ (%Q)2$ &'( & ( ( "# ) kpolVpol

Theory and simulation in agreement more or less for ∆w

2cyl Q= r radius of the beam

2 2

2 28ln 1

2

0.577

cc cx

cxc

n xx l

lc n! " #

$ %%

#

& '( )= * ++ ,- .

/ 01 23

But a factor of the order of two exist for the scattered power and at high level of density fluctuations

Why ?


Reflectometry Computation Requirements

To describe the forward scattering effects (long wavelength contribution)

To recover the theoretical results of the forward scattered power much larger mesh size is required

Usefulness of the testing of the code by using analytical results Be careful on the choice of the turbulence generator: modes summation, burst superposition, …. or coming from turbulence code BUT has intrinsic limitations (few grid points)

2π

0.02

Gives more than a factor of 2 to be verified in 2D but the needs for computations are very high (underway)

Gaussian spectrum

Ln=0.58 m n =47 GHz lc = 7.5 mm

dn/nc=0.001


Direct evaluation of the beam width using averaging in time and samples(8) Cross-Correlation length of E(y)E(y')

a=20°

a=20°

a=20°

Other parameters to characterize forward scattering effects


Power spectrum using sliding FFT Max of the peaks of Sliding FFT

Scattered signal behaviour for pure forward scattering (Gaussian S(k))

Simulation help to know more


Frequency power spectrum using sliding FFT Max of the peaks of Sliding FFT

Scattered signal behaviour with Bragg backscattering Kolmogorov spectrum

With Bragg backscattering


Electric field structure at a given time in purely forward scattering and Kolmogorov cases for Doppler reflectometry on O-mode

In the previous conditions with dn/nc= 10% (RMS)

To see the Doppler shift a long time series is needed ~ 8 105 time steps Left correct simulation but right some scattering contribution are cut Bigger mesh is required for the right field structure (size should be 3000x5000 pts)


2D wave propagation in moving turbulent plasma

wave trapping

local resonance

multi-scattering

beam spreading

How to obtain local resonances ?

turbulence motion inducing Doppler shift

access to Vtur velocity


Conclusions Depending on the Physics you wont to describe:

choose the adapted set of equations For an exact interpretation of the measurements, the E2 should be known especially if resonances exist or make an average to recover <E2>, which looks like a unperturbed case ( photon diffusion equation), valid up to moderate dn/nc Reflectometry Simulation permits to identify the basic mechanisms involved in a given configuration of the reflectometer. Simulation is an essential tool to improve the interpretation of the reflectometry data but also the theoretical studies (role of resonances,…) Theory-simulations are in agreement for each single mechanism (resonance, multi-scattering) However these agreements require to well define the numerical conditions to obtain it, which is not trivial or reachable especially for 2D and 3D cases. So we need help to improve numerical methods to solve PDEs associated to wave propagation


ERCC activities on 3D-Code in EFDA ITM framework

G.D.Conway (IPP), E.Blanco (CIEMAT), S.Hacquin (CEA), S.Heuraux(IJL),

C.Lechte (IPF), A.Sirinelli (CCFE), F.da Silva (IST), S.Soldatov(FZJ)

ERCC European reflectometry code consortium ITM integrated tokamak modeling

EFDA European fusion development agreement


ERCC: 3D full-wave code status!

Source"

Antenna"

Vacuum" Plasma"pol"

rad"tor"

20 - 50cm" 10-15cm"

20-30cm"

Emitting"plane"

3D cartesian grid"

FDTD comp. domain: l/20 grid"

Large physical domain forces use of mixed scheme = source + vac + FDTD "ne(r)‏"

20cm"

Cutoff"+ turb."

fo = 10 – 180 GHz ® lo ~ 3.0 - 0.17cm"


ERCC: erc3d code module status!

Kernel"

Source"

Profiles"

Turbulence"

Initialize stage!

Antenna CPO"

Edge ne CPO"

Receiver/ IQ"

Equi CPO"

Turb. CPO"

Loop stage! Output stage!

Source step"

Data CPO"

HDF5 files"

Vacuum step"

Turb. step"

Kernel step"

Vac. back-step"

Dectection "& IQ step"

HDF5 files"

Interpolation of profiles &"turb onto siml 3D grid"


ERCC: erc3d code – example results!

Input density profile from “coreprof” CPO"

Cutoff"

Ey field "– xy plane"

Ey field "– xz plane"

x (radial) in grid points"

 Launch fo = 70.245 GHz " ∆x = l/20 = 0.2134 mm" Gaussian beam: wo ~ 4 mm" Grid size: 240 x 240 x 240 points"


Critical issues!  Large domain: e.g. 32GB RAM ® 6 x108 grid points ® 13 field components

SP= 17cm cube grid! Need lots of memory!

  Time: 1 CPU = 6000 hrs for 2048 snap shots (extrapolated from 2D code) ® Need lots of CPUs!

  Parallelization: “snap-shot” (easy) but... “domain” (hard but necessary) Need expertise/manpower

Next step

  Validation and verification against both 2D code & experimental data

  Synthetic turbulence – soon.

  Numerical turbulence coming from turbulence code need effort for data exchange

  Any help are welcomed, thank you.

ERCC: erc3d code – numerical requirements and near future!

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