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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
3NMCF april 2009 S. Heuraux-E. Faudot
!
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
4NMCF april 2009 S. Heuraux-E. Faudot
!
"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
NMCF april 2009 S. Heuraux-E. Faudot
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
NMCF april 2009 S. Heuraux-E. Faudot
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
$
%
& & & &
'
(
) ) ) )
NMCF april 2009 S. Heuraux-E. Faudot
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
NMCF april 2009 S. Heuraux-E. Faudot
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!
%
& '
(
) * =
NMCF april 2009 S. Heuraux-E. Faudot
Numerical scheme for explicit time computation
Then Φt+1 can be deduced from
Stable numerical scheme + results validated by 2D Particle in cell code
NMCF april 2009 S. Heuraux-E. Faudot
!
"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
NMCF april 2009 S. Heuraux-E. Faudot
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
NMCF april 2009 S. Heuraux-E. Faudot
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
NMCF april 2009 S. Heuraux-E. Faudot
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.
NMCF april 2009 S. Heuraux-E. Faudot
Application to ITER
NMCF april 2009 S. Heuraux-E. Faudot
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
NMCF april 2009 S. Heuraux-E. Faudot
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
NMCF april 2009 S. Heuraux-E. Faudot
Density profiles for each scenario
NMCF april 2009 S. Heuraux-E. Faudot
RF potential map in the radial-poloidal plane
Phasing
Scenarios
!
VRF = EparadlLsol"
NMCF april 2009 S. Heuraux-E. Faudot
DC rectified potential map after SEM code run (Lpara=2.8m)
Phasing
Scenarios
NMCF april 2009 S. Heuraux-E. Faudot
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
NMCF april 2009 S. Heuraux-E. Faudot
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).
NMCF april 2009 S. Heuraux-E. Faudot
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
NMCF april 2009 S. Heuraux-E. Faudot
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
NMCF april 2009 S. Heuraux-E. Faudot
NMCF april 2009 S. Heuraux-E. Faudot
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
NMCF april 2009 S. Heuraux-E. Faudot
!
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).
NMCF april 2009 S. Heuraux-E. Faudot
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
NMCF april 2009 S. Heuraux-E. Faudot
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