Simulations on RF antenna-plasma coupling: RF sheath ... · PDF fileSimulations on RF antenna-plasma coupling: RF sheath rectification process taken into transverse currents in

1NMCF april 2009 S. Heuraux-E. Faudot

Simulations on RF antenna-plasma coupling:RF sheath rectification process taken into

transverse currents in Ωci range

Eric FAUDOT, Stéphane HEURAUX, Alain NgadjeuLPMIA Nancy

Laurent COLASIRFM CEA Cadarache

-Schedule:- Assumptions associated to the modelling, theory, SEM code- Applications to ITER- Reassessment of L//

NMCF april 2009 S. Heuraux-E. Faudot

ionic current

electronic current

Δj/jisat

ϕ Vfl

2004

B0

Flux tube

RF sheaths

IR pictures of Tore Supra ICRH antenna

!="linefield

RFdlEV//RF

ΦRF= VRF/Te

ICRH Antenna

Magnetic field line

Sheath I(Φ)characteristic

Context


!

jperppol

=Lpara"ci

2#ci

$

$t%&

j para j paraB0

!

z toroidal( )

!

x radial( )!

y poloidal( )

The rectified potential (x,y,t) is computed self-consistently with currentsaccording to this set of equations

RF/2 RF/2

Flux tube sketch

ICRF Antenna

Lpara


!

"Iperp =1# exp $0 #$( )

!

1

"ci

2

#2

#t 2$Iperp +$Iperp =

Lpara%ci2"ci

& i +#

#t

'

( )

*

+ , $-

Current conservation

Momentum equationapplied to transversecurrent

!

"0=" fl +ln cosh

"RF

2

#

$ %

&

' (

#

$ %

&

' ( Rectified potential without

transverse currents

Assumptions

Φ(x,y,t) is computed in a 2D map perpendicular to B0 and is normalized to Te Φ is constant along the flux tube Density n is constant all over the map Collisional, parallel and transverse RF displacement currents are integrated in the model Computation have been completed only with RF transverse currents (other components negligible) ‏ Electrostatic model

!

' " #t , " " #

t

2


Linear modelling in space (for SEM validation)

!

" x( ) # L2$" x( )="0x( )

Assumptions :

!

"Iperp <1#L2($)%RF

x0

2<1

!

L2 "( )=

Lpara

eff #ci

2

i" /$ci

1%" 2/$ci

2

!

"Iperp Transverse current normalized to 2 jisat

Characteristic length for RF currents and xo characteristic length for the potential structure

!

L "( )

!

"RF=" 1+

x0

2

L2

#

$ %

&

' (

linear/non linear limit :

L2/x02


spatial resolution of linear modelling

!

" x( )= "0# x'( )G x $ x'( )dx'

L is a complex number

if ||L||>x0 then the structure of

potential is broadened

if ||L||<x0 the structure remains

unchanged

if Im(L)>Re(L) the potential

structure oscillates along x

Green's function

Solution :

!

" x( )="0

x0

x0

2 # L2x0e

# x| |x0 # Le

# x| |L

$

%

& & & &

'

(

) ) ) )


SEM (Sheath Effect Modelling) code

Parameters

RF Potential evanescent

Simulation in a poloïdal-radial plane

Map width 10 cm (100 cellules radiales, Δx=1 mm)

time step ∆t=10-12 s

Simulation time : 3 to 4h.

RF potential profileperpendicular to B0

Code SEM

RF rectified potential (x,t) ‏Time average

DC rectified potential profile (x) ‏

Conditions de bord d'un Tokamak :

Te = Ti = 20 eV

N0=1018 m-3

Ωci = 32 MHz (Deuterium plasma)

F= 53 MHz

L = [ 1 - 30 ] m

ΦRF= [ 10 - 150 ]

Simulation steps

!

para

eff


2D fluid code: SEM

potential = Cste along magetic field line

Algorithm : finite différences, implicit in space andexplicit in time ΔI

x

yφ(x0,y0,t)‏

!

A=L//"ci

2#ci

!

1" exp # fl +ln cosh#RF (xi , yi , t)

2

$

% &

'

( )

$

% &

'

( ) "#i, j, t

*

+ ,

-

. / +

A

0t

#i+1, j, t " 2#i, j, t +#i"1, j, t

0x!+#i, j+1,t " 2#i, j, t +#i, j"1,t

0y!

*

+ ,

-

. /

( ) ( )[ ] ( ) ( )

( ) ( ) ( )[ ] ( ) ( )

( ) ( ) ( )[ ] ( ) ( )

( ) ( ) ( ) ( )[ ] ( ) ( )

( ) ( ) ( ) ( )[ ] ( ) ( )

( ) ( ) ( ) ( )[ ] ( ) ( )

( ) ( ) ( ) ( )[ ] ( )

( ) ( ) ( ) ( )[ ] ( )

( ) ( ) ( ) ( )[ ]

!!!!!!!!!!!!!!!!

"

#

$$$$$$$$$$$$$$$$

%

&

!!!!!!!!!!!!!!!!!!!

"

#

$$$$$$$$$$$$$$$$$$$

%

&

'

'

'

'

'

'

'

'

'

'

'

''''

'''''

'''''

''''''

''''''

''''''

'''''

'''''

''''

...

...

...

...

2............

2.........

...2......

...2...

......2......

.........2

............2

...............2

..................2

11,

11,

1

11,

11,

2222

22222

22222

222222

222222

222222

22222

22222

2222

+t+ji,

+tj,+i

+tj,i,

+tj,i

+tji,

ö

ö

ö

ö

ö

Äy+ÄxÄxÄy

ÄxÄy+ÄxÄxÄy

ÄxÄy+ÄxÄxÄy

ÄyÄxÄy+ÄxÄxÄy

ÄyÄxÄy+ÄxÄxÄy

ÄyÄxÄy+ÄxÄxÄy

ÄyÄxÄy+ÄxÄx

ÄyÄxÄy+ÄxÄx

ÄyÄxÄy+ÄxÄt

A

( )tj,i,RF öt),j,(i,öf=r

Matrix penta-diagonal solve at each time step

φ(x0,y0+1,t)‏

z

!

A

"t

#i+1, j, t+1 $ 2#i, j, t+1+#i$1, j, t+1

"x!+#i, j+1,t+1 $ 2#i, j, t+1+#i, j$1,t+1

"y!

%

& '

(

) * =


Numerical scheme for explicit time computation

Then Φt+1 can be deduced from

Stable numerical scheme + results validated by 2D Particle in cell code


!

"0x,t( )="0

t( )e#x

x0

RF potential profilesperpendicular to B0

x0= Skin depth --> 5 mm

B0

X (radiale) ‏

RF potential induced by the slow wave


Time average

DC rectified potential Φ(x)‏

DC potential profile

!

L / x0= 2

!

L / x0= 0.6

!

L / x0= 6

The non linear broadening is smaller than the one expected by linear modelling


The combination of 80 simulations (8 values for φRF and 10 for L) gives a surface for Lnl as afunction of linear L and φRF.

Lnl increases quasi-linearly with Lexcept for high values of φRF

Lnl exponentially decreases with φRF!

Lnl=

x0

ln " 0( )( ) # ln " x0( )( )

logarithmic slope

Non linear L as a function of linear L and the RF potential

The increase of φRF tend to makeLnl decrease down to a limit valueof 2 cm for these parameters.

--> How justify this value ?

--> it means : high RF potentials = low broadening


Projection on the (Lnl,L) plane

!

L0= x

0+

Lpara

eff "ci

2

Each iso-curve represents a value of φRF

!

Lnl~ x

0+L

!

Lnl < L0 for "RF < 50

Lnl evaluation

L0 is a good approximation for a upper value of the structure broadening.


Application to ITER


TOPICA Torino

SEM code

Nancy

2D map E//RF

@ antenna

2D map DC

potential

2D map

density

Cells code

Cadarache

2D map Q//

Integration //,

Slow Wave

evanescence

RF antenna

3 phasings

Field line

Pitch angle

T profiles

4 scenarios

2D map RF

potentialvx

T profiles, L// profiles, B0

ne

profiles

4 scenarios

TOPICA Torino

SEM code

Nancy

2D map E//RF

@ antenna

2D map DC

potential

2D map

density

Cells code

Cadarache

2D map Q//

Integration //,

Slow Wave

evanescence

RF antenna

3 phasings

Field line

Pitch angle

T profiles

4 scenarios

2D map RF

potentialvx

T profiles, L// profiles, B0

ne

profiles

4 scenarios

Modelling Flowchart


Parameters

4 scenarios have been simulated with 3 phasing for the 4 straps of the ITER antenna: 00ππ, 0π0π, 0ππ0

The tabular below summerizes simulation parameters : density, temperature, magnetic connexionlength in the SOL for each scenario made by Alberto Loarte (Turin).

RF field maps have been computed with TOPICA (D Milanesio) coupled with FELICE for the absorption ofthe slow wave. Electric field are calculated for a 1V feeder and are normalized to 20 MW of coupled power.

Sc2 short Sc2 long Sc4 short Sc4 long

B0 (T) 4 4 4 4

tilt angle (°) 15 15 9 9

f (MHz) 53 53 53 53

lx (m) 0,1 0,1 0,1 0,1

ly (m) 3,5 3,5 3,5 3,5

n0 (m-3)

Lpara (m) 2,8 & 20 2,8 & 20 2,8 & 35 2,8 & 35

Te (eV) 10 10 20 20

Ti (eV) 20 20 60 60

dt (s)

1018 1018 1018 1018

10-12 10-12 10-12 10-12


Density profiles for each scenario


RF potential map in the radial-poloidal plane

Phasing

Scenarios

!

VRF = EparadlLsol"


DC rectified potential map after SEM code run (Lpara=2.8m) ‏

Phasing

Scenarios


DC map Potential profiles : average over y

Profil RF

Profil DC Lpara= 2.8m Profil RF Lpara=20 & 35 m

!

"max

<"

RF

2

!

"max

<"

RF

2


Normalized potential profiles

Profil DCLpara=2.8 m

Profil RF

Sc long

Sc short

For long SOL scenarios the skin depth is shorter than L and then the profile is broadened bytransverses RF currents. On the contrary for short SOL the effect of transverses currents is notvisible (x0=L).


vE×B

( ) ( )

2

0

0

0

//0/

B

Vv

xnnDn

DC

x

Bv

v

!"+=

=+"#"

$$

$$$$ %

Sc. 4 long[0pp0]

w/o SEMwith SEM

• VDC amplitude magnitude of vE×B

• VDC decay length cells extention• => Local density @ mouth

Utility to take rectified potential in front of the antenna

M. Bécoulet et al, Plasma Physics 9, 2619 (2002)

Then energy flux deposition can be computed => Q= e n(VDC) Cs VDC

CELLS Code(CEA)

Hot Spots

CELLS SEM


Double probe model applied to a flux tube exchanging RF transverse currents

--> linear modelling for solving DC rectified potential structures

--> potential amplitude condition [Faudot2006] :

--> broadening condition ( Green function ) : - si ||L|| < x0 --> x0

- if ||L|| > x0 --> ||L||with

--> non linear modelling--> SEM code

--> Validation of the amplitude for DC potentials--> Rectification criterion : linear/non linear--> Validation of the broadening with the parameter L, L0 is a upper value for the non

linear broadening :

--> broadening criterion: linear/non linear--> non linear effect make increase the DC amplitude and decrease the broadening

--> Evaluation of an effective connexion length for transverse RF currents with experimental probe measurements :

Conclusion

!

" x( ) <"

RFx( )

2

!

L "( )=Lpara

eff #ci

2

i" /$ci

1%" 2/$ci

2

!

L0= x

0+

Lpara

eff "ci

2

!

Lpara

eff< 2m Needs 3D



2-D map of Vfloat versus (δr,ZQ5) connected side for PQ5 = 1.5 MW.Dashed lines: side limiter radial position, and antenna box verticalextension. Arrows: sketch of vExB induced by Vfloat

L. Colas et al. / Journal of Nuclear Materials 363–365 (2007) 555–559

2-D around Tore Supra ICRF antennas using reciprocating Langmuir probes

reciprocating Langmuir probes very useful tool


!

Lperp = x0+

Lpara

up "ci

2

!

Lperp = x0+

Lpara

mean"ci

2

i# /$ci

1%# 2/$ci

2Upper value for Lpara

1 cm 13 cm 46 cm

1.5 cm 55 cm

Lperp

Lpara

mean Lpara

up

1.85 m

Mean value for Lpara

Shot 41869

Evaluation of the effective Lpara from probe measurements

With B0=3 Tx0=5 mmTe=Ti=20 eVf=53 MHz

connection length = 7 m

limiter

Probe connected toantenna (J. Gunn) ‏

Magnetic line ICRFantenna

J. P. Gunn et al 22nd IAEA(2008).


Sample of a 2D DC rectified potential map in front of an ICRF antenna

x (m) ‏

y (m

‏(

60

50

40

30

20

10

<φ>t

!

VRF = EparadlLsol"

Computation of the RF potential map

B0


Code SEM

RF rectified potential profile Φ(x,t) ‏

1D RF rectified potential

!

L2 "( )= 5.10#4m2

!

L2 "( )= 5.10#3m2

Transient time during which the AC amplitude decreases

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Simulations on RF antenna-plasma coupling: RF sheath ... · PDF fileSimulations on RF antenna-plasma coupling: RF sheath rectification process taken into transverse currents in