1Russian
Research Center
"Kurchatov Institute“___
Nuclear Fusion Institute
2 FOM-Institute
for Plasma Physics
Rijnhuizen,Association EURATOM-FOM,
Trilateral Euregio Cluster, The
Netherlands, www.rijnh.nl
EC Radiation Transport in Fusion ReactorGrade Tokamaks:
Parameterization of Power Loss Density
Profile, Non-Thermal Profile Effects under
ECCD/ECRH conditions
K.V. Cherepanov, A.B. Kukushkin,
L.K. Kuznetsova, E. Westerhof 2
GOAL:
Numerical studies of contribution of electron cyclotron radiation
(ECR) to local energy balance in toroidal magnetically confined
plasmas with
• hot electrons
<Te> > 10 keV
(*)
• strong magnetic field
BT > 5 T
(*)
• highly reflecting walls
Rw > 0.5
(*)
(including the ITER-like conditions).
MOTIVATION:
Spatial profile of net radiated power density, PEC(r), is peaked in
the plasma core and, for D-He3, D-D tokamaks (Te(0) ~ 70 keV), is a
major power loss channel:
Tamor S., Fusion Technol. (1983)
Kukushkin A.B., 14th IAEA (1992)
Albajar F., Bornatici M., Engelmann F., Nucl. Fusion (2002)
PEC(r) becomes dominant electron cooling mechanism in DT
reactor-grade tokamaks:
ASTRA+CYTRAN: ITER steady-state scenarios (Te(0) ~ 30-40
keV) Albajar F., Bornatici M., Cortes G., et al., Nucl. Fusion, 45 (2005) 642
ne=10%; Te(hot) = 2Te (0);
Curves (1)-(3):
code CYNEQ
Curves (4)-(6) :
“localized”
Trubnikov’s
formula.
ITER-FEAT
Te(0)=25 keV
Te(a)= 2 keV
ne(0)=1020 m-3
ne(a)=0.5 ne(0)
[Cherepanov K.V., Kukushkin A.B., EPS-2004]
Comparison of radial profiles of net ECR power loss for
(i) two regimes (*),(**) with bi-maxwellian electrons and (ii)
maxwellian background plasma, which give almost the same value of
total ECR power loss, for wall reflection coefficient RW = 0.6.
ASTRA+CYTRAN
Steady-state 1
Te,0 (keV):
<Te>(keV):
Ti,0 (keV):
<Ti> (keV):
I (MA):
Rwall :
36
17
37
17
9
0.6
43
19
53
21
9
0.6
Steady-state 2
27
8
23
7.5
15
0.6
Inductive
Profile of local energy
balance in ITER for various
scenarios (Albajar, Bornatici
et. al., Nuclear Fusion, 2005)
Numerical code ASTRA is
combined with CYTRAN
(red stripe = PEC(r)).
TASK:
Parameterization of PEC(r) to be used as a simple simulator during
the transport calculations for ITER-like range of parameters
METHOD:
On the basis of calculations of PEC(r) with the code CYNEQ
[Cherepanov K.V., Kukushkin A.B., EPS-2004, IAEA-2004, etc.],
we propose
further simplification of the well-known fast-routine code
CYTRAN [Tamor S., Reps. Science Applications, 1981] with an accent on
satisfactory description of PEC(r) in the core.
Code CYNEQ
(electron CYclotron radiation transport in Non-EQuilibrium hot
tokamak plasmas) is based [Kukushkin A.B., JETP Lett., 1992; 14th
IAEA PPCF conf., 1992] on extending the escape probability methods
developed in the theory of nonlocal transport. The code
• solves semi-analytically the transport problem;
• simplifies semi-analytic approach of the code CYTRAN
[S. Tamor, Reps. Science Applications (1981)]:
hot maxwellian toroidal plasmas of
- non-circular cross-section,
- not too large aspect ratio,
- multiple reflection of waves from the wall,
• retains CYTRAN’s accuracy of approximating the
results of Monte Carlo code SNECTR [S. Tamor, 1978].
CYNEQ was also tested [1992] against well-known benchmark -- numerical
results, and respective analytic fit [Trubnikov B.A., Rev. Plasma Phys., vol. 7,
1979], for total ECR power loss for maxwellian electron plasmas of homogeneous
profiles of temperature and density.
Spatial profile of net radiated power density in CYNEQ
Kukushkin A.B., Proc. 24th EPS Conf. on Contr. Fusion & Plasma Phys.,
Berchtesgaden, 1997, vol. 21A, Part II, p. 849-852.
PEC(r) = Qwave


   dn d (, r )J esc ( )  Q(, r )


r


  
 esc     (, r ) (n,dr )  1
rb

rb – plasma’s boundary
J esc 
k() -
absorption
coefficient
Q() - emission rate
 
 , V  Vesc ( )
n, dS s 1  R, S s    dV  dn , r 
 d  
n

 dV  d n Q, r 

   , n,  

Vesc() is projection of phase space esc (with angleaveraged absorption coefficient ) onto coordinate space.
Simple analytic description of spectral temperature TECR(,K) of
outgoing EC radiation
(1)
K= extraordinary (X) and ordinary (O) waves; Te=Te(); =r/a;
Fitting formulas [Tamor,1981] for normalized absorption coefficients K(,Te) are
slightly modified to avoid the increasing errors at small temperatures.
Characteristic optical thickness  = 6.04103a/B0..
a, one-dimensional minor radius in meters;
B0, central magnetic field in Tesla;
RWK , wall reflection coefficient for K mode;
<Tcut(,K)> = (f Tcut(,K) + (1-f)Te(1)); f=0.6;
Tcut(,K) = Te(cut(,K));
cut(,K) - boundary of optically thick core in the radiation’s reduced phase space
{frequency, radius}
cut /B0 = 2 + DK (1-cut-Kmin),
DK= (Te(0) + Te(1))/2 { ln(ne(0))/CK }2 + AK, AX=0, AO= -1;
Kmin = 0.01; CX = 17.9; CO = 19.7; ne(0) in 1020 m-3.
cut/B0= 2 for  > 1-Kmin,
cut = 0 for /B0 > 2 + DK (1-Kmin).
Analytic description of power loss radial profile
Deviation of spectral temperature of EC radiation from local electron temperature
determines the spectral density of the local ECR power loss in maxwellian
plasmas. The remaining integration over frequency to evaluate PEC(r) has to be
done numerically.
C = 3.9 10-8 MW/m3, B0(Tesla), a(meters)
(2)
The profiles of plasma were taken close to those for one of
ITER regimes predicted by the ASTRA code 1D simulations:
(major/minor radius 6.2/2 m, BT = 5.3 T ,  = r/a )
Polevoi A.R., Medvedev S.Yu., Mukhovatov S.V., et. al., J Plasma Fusion
Res. SERIES, 5 (2002) 82-87.

ne (  )  ne (1)  ne (0)  ne (1)  1   2

0.1
,
ne (0)  10 20 m 3 , ne (1)  0.5  10 20 m 3

Te (  )  Te (1)  Te (0)  Te (1)  1   2

1.5
,
Te(0) = 25 keV, Te(1) = 2 keV, ~ Inductive, ITER scenario 2
Te(0) = 24 keV, Te(1) = 0.3 keV, ~ Steady-state, scenario 4
(analytic, ~ flat PEC(r))
Te(0) = 35 keV, Te(1) = 2 keV, ~ Steady-state (PEC(r) of CYTRAN)
TECR()
~ 2TECR()
__ CYNEQ …. Eq. (1) Te(0) = 25 keV Te(1) = 2 keV RWK = 0.6
X mode, O mode, X+O
TECR()
~ 2TECR()
__ CYNEQ …. Eq. (1) Te(0) = 24 keV Te(1) = 0.3 keV RWK = 0.6
X mode, O mode, X+O
TECR()
~ 2TECR()
__ CYNEQ …. Eq. (1) Te(0) = 35 keV, Te(1) = 2 keV, RWK = 0.6
X mode, O mode, X+O
Te(0) = 25 keV
Te(1) = 2 keV
RWK = 0.6
X mode
O mode
X+O
….. Eq. (2)
___ CYNEQ
_ _ Eq. (2) with CYNEQ’s numerical absorption coefficients 
Te(0) = 24 keV
Te(1) = 0.3 keV
RWK = 0.6
X mode
O mode
X+O
….. Eq. (2)
___ CYNEQ
_ _ Eq. (2) with CYNEQ’s numerical absorption coefficients 
Te(0) = 35 keV
Te(1) = 2 keV
RWK = 0.6
X mode
O mode
X+O
….. Eq. (2)
___ CYNEQ
_ _ Eq. (2) with CYNEQ’s numerical absorption coefficients 
Conclusion. Part 1
Analysis of comparing the CYNEQ and CYTRAN
calculations enabled us to simplify further the fast
routine of CYTRAN and retain reasonable accuracy of
describing the radial profile of EC net radiated power,
PEC(r), in the region of significant contribution of
PEC(r) to the local power balance in fusion reactorgrade tokamaks
(first of all, in the central part of the plasma column).
Acknowledgments. The work is partly supported by the Russian
Federal Agency on Science and Innovations (contract
02.445.11.7505) and the RF grant NSh-9878.2006.2 for Leading
Research Schools.
ECRH 20 MW, O mode, n=1, f=138 GHz,
Perpendicular launch (=0), equatorial plane
ITER-like
Te(0) = 35 keV
T ef ( E )   ln  f ( E ) / E
1
METHOD OF CALCULATION
Beam tracing code TORBEAM [1] calculates a power deposition
profile PTORBEAM(x) and provides w and DN|| on set of flux surfaces.
Fokker-Planck code RELAX [2] takes w and DN|| from TORBEAM
and calculates deposition PRELAX(x) with resonance broadening.
Fokker-Planck code RELAX [2] outputs the distribution function
====================================
[1] E. Poli, et al., Comp. Phys. Commun. 136 (2001) 90.
[2] E. Westerhof, et al., Rijnhuizen Report RR 92-211 (1992).
Formation of Superthermal Electrons Under ECRH/ECCD in ITER-like Tokamak
L.K. Kuznetsova, K.V. Cherepanov, A.B. Kukushkin, E. Westerhof
CONCLUSIONS. Part 2.
1. The EC absorbed power density may attain ~10 MW/m3, for 20
MW total absorbed power and wave beam focusing in the plasma
core.
2. For perpendicular launch (ECRH only), the deviation of the
EDF from the Maxwellian is stronger for the thermal part (Ekin <
Te), with the effective temperature Tef(Ekin) (defined as the
exponential slope of the EDF with respect to energy for a given
electron kinetic energy Ekin) exceeding Te by 10-20% and 20-40%
for, respectively, Te(0)= 25 keV and 35 keV.
3. For oblique launch (ECCD/ECRH), with an injection angle  ~
20, Tef/Te is about twice smaller, but in a substantially broader
energy range, up to Ekin/mec2 ~ 0.5, producing thus a strong enough
fraction of superthermal electrons. Also, formation of a plateau on
the EDF at higher energies is found (Ekin/mec2 ~1, Tef/Te ~2-5) for
both launch geometries.
How the distortions of the electron velocity
distribution (EDF) caused by the absorption of
external intense ECR, injected into the plasma at low
harmonics of the cyclotron frequency (n=1,2) for
ECRH and ECCD, may influence the transport of
ECR, emitted by the plasma itself at all other
harmonics (2<n<15) responsible for formation of the
PEC(r) profile in reactor-grade tokamaks.
Wall reflection
coefficient
RW=0.6
Method of calculation
1. ECRH and ECCD (n=1,2) [1] in ITER-like tokamak
(Scenario 2, Te(0)=25 keV; modified steady state,
Te(0)=35 keV)
code TORBEAM [2] (beam tracing)  [4] 
code RELAX [3] (Fokker-Planck, distribution function
fe(v,r) ) 
2. ECR transport (n>2):
 code CYNEQ [5] (power loss PEC(r))
=======================================
Comparison of radial profiles of net ECR
power loss density:
(1) maxwellian background plasma;
(2) non-maxwellian EDF produced by ECRH;
(3) maxwellian EDF with the same relativistic
mean electron energy.
[1] L. K. Kuznetsova, K. V. Cherepanov, A. B. Kukushkin,
E. Westerhof, Formation of superthermal electrons …
(EC-14 paper 71).
[2] E. Poli, et al., Comp. Phys. Commun. 136 (2001) 90.
[3] E. Westerhof, et al., Rijnhuizen Report RR 92-211
(1992).
[4] L.K. Kuznetsova, Juelich (2002).
[5] Cherepanov K.V., Kukushkin A.B., EPS-2004, IAEA2004.
Influence of ECCD/ECRH-Produced Superthermal Electrons on Transport of Plasma’s Electron
Cyclotron Radiation in Tokamak-Reactor
CONCLUSIONS. Part 3
1. For the same value of total absorbed EC power, the effect of
ECCD/ECRH-produced superthermal electrons on the net ECR
power loss density, PEC(r), is stronger for power absorption at
larger electron velocities. For equatorial plane launch, the effect is
maximal (~20%) for wave beam focusing in the core and toroidal
injection angle ~20. For perpendicular launch, effect is ~few
percents only, similarly to the case when EDF’s distortions are
caused exclusively by the ECR emitted by the plasma (“self
ECR”).
2. In ITER, total power of self ECR inside the chamber (15 MW
for scenario 2, wall reflection coefficient Rw=0.6) will be
comparable to 20 MW ECCD total power. As far as the ECCD is
resulted exclusively from asymmetry of electron velocity
distribution in co- and counter-current directions, the impact of
self ECR (at high harmonics, of total 15 MW) on ECCD may not
be small and has to be evaluated.