Role of Upper Hybrid Resonance and Diffraction Effects at Electron Cyclotron Heating . In Tokamaks

V.Vdovin RRC Kurchatov Institute, Nuclear Fusion Institute Abstract. We present modelling results of basic Electron Cyclotron Heating scenarios in tokamaks performed with newly developed 3D full wave STELEC (stellarator_ECH, tokamaks included as particular case) code. Code includes all basic wave physics as interference, diffraction, wave tunnelling, mode conversion at Upper Hybrid resonance to electron Bernstein waves and appropriate boundary conditions. Code operates in real 3D magnetic geometry and uses massive parallel terabyte computers and firstly permitted solution of above problem. Modelled are fundamental and second harmonics O-mode and X-mode scenarios in T-10, FTU, DIII-D, JT-60U, RTP tokamaks and in ITER at fundamental harmonic. The Upper Hybrid resonance plays important role, leading to strong broadening of power deposition profiles at fundamental at O-mode RF power launch. This is partly supported by experiments on DIII-D, RTP, T-10 and JT-60U. Diffraction effects are investigated at second harmonic and these are shown to be important even at moderate plasma densities. Code discovered important new effect of remarkable RF power reflection on second harmonic resonance layer in poloidally oblique EC power launch scenarios, intended to be used for NTM suppression. The O-X-B scheme for over dense plasma is also explored.

Email of V.Vdovin: vdov@pike.pike.ru

Introduction

Electron Cyclotron Resonance plasma heating (ECH) and current drive (CD) inFusion plasma research in tokamaks and stellarators plays a key role in investigation of basic wave - plasma interaction physics like electrons heating, local transport coefficients behaviour, CD and current profile tailoring, Internal Transport Barrier (ITB) creation, Neoclassical Tearing Modes (NTM) stabilization, etc. The method, initially supposed to be very local in space, is an essential and very expensive tool in present fusion machines and ITER Contrary, theoretical and modelling support during last 30 years was performed by so called very simple geometric optic approach, named as “ray tracing”, dropping important physics like reflection, interference, diffraction, wave tunnelling, mode conversion to electron Bernstein waves and appropriate boundary conditions. Reason for that simple ray tracing approach over all world use was due to non sufficient power of world computers in the past for ECH method modelling to solve exact problem in 3D plasma with small scale waves. Respectively planning, interpretation and prediction of ECH/CD were unsatisfactory ones as shows below full wave 3D code modelling.

1. STELEC Code

Code [1] solves 3D full wave equation

rotrot

E

4ik0

c(j p

j ext ) k0

2E (1.1)

in flux coordinates with appropriate boundary conditions, in 3D stellarator geometry, on real VMEC code equilibrium, using poloidal and toroidal harmonic expansions over respective angles and center difference scheme over radial (psi) coordinate ( k0 = /c, c - speed of light). In eq.(1.1) jext is an imposed antenna current density and jp is the RF plasma current being evaluated using the small Larmor radius approximation by procedure starting from the results of [2], in which the Vlasov equation is solved in plane-stratified geometry, and assuming that the vector form of jp in this limit holds also in 3D toroidal geometry.Zero plasma response is given by dielectric components :

(1.2)

,

and finite electron Larmor radius wave induced current is given as

(1.3)

( , in 3D magnetic geometry, with metric tensor, both given by VMEC) and expressed through Shkarovsky functions Fq (n) [3]:

(1.4)

, where parallel wave number is expressed through contravariant

components of equilibrium 3D magnetic fields and toroidal and poloidal modes numbers n, m [4]:

Huge matrix resulting after discretization is solved by “Progonka” Algorithm (Gaussian inversion specialized for three diagonal block system) for parallel processors computer option, distributing matrices across processors, and uses the ScaLapack library and message passing interface (MPI) for parallel matrix operations. The individual blocks are distributed across all processors. The compact storage methods with out of core solver can also be used for serial code option. Antenna is located in vacuum layer between plasma and conducting wall and modeled by current straps. Currents’ generally elliptical polarization is imposed according to ECH O-mode or X-mode polarization at plasma separatrix, calculated through the local dispersion relation. Antennae currents generally are corrected by additional multipliers, requested by Maxwell equations solutions in vacuum layer surrounding plasma. Faraday toroidal and poloidal screens options are also provided, when needed, for linear polarization of an antenna.

2. Similarity Laws

Analysis of above equations shows that only combinations of parameters are used( , - plasma and electron cyclotron frequencies, - electron temperature)

, , , (magnetic equilibrium)

Thus keeping these fixed (similar) ones and providing multi mode regime for excited

ECH waves ( , - vacuum wave length, a – plasma minor radius)

,

it is possible to model ECH scenarios in large scale fusion machines at reduced frequencies thus diminishing requirements on computer memory capabilities (important for ITER like machines).

3. FTU and DIII-D Tokamaks Fundamental Harmonic Modeling

FTU tokamak with major radius R=93 cm uses quasi perpendicular O-mode outside launch at 140 GHz. Contour plots of total wave electrical field are displayed in Fig.1a. The axis magnetic field Bo=4.667 T , plasma density 4.5x1019 m-3 (parabolic density

profile, αn=1), Te(0) =9.2 kV (αT=2), Ip=0.72 MA, N//(0)=0.027. Upper Hybrid resonance layer manifests itself, contrary to ray tracing, by bright “mirror” broadly

FIGURE 1a |E|total in FTU at Ne=4.5x1019 m-3 FIGURE 1b Power dep. Pe FIGURE 1c 2D Pe

radiating Electron Bernstein Waves (EBW) and slow X-waves to cold ECR side. Averaged over flux surface power deposition, in Fig.1b, shows peaked power deposition mainly responding to large electrical fields near UHR layer (X=3 cm). This is confirmed by 2D power deposition in Fig.1c. Thus RF power is absorbed far away from cold EC resonance (X=-6cm). This ECH picture summarizes early record ECH Te results from middle and small tokamaks TM-4, T-10, TFR, RTP etc. obtained at low plasma densities (~1019 m-3) when distance between ECR and UHR resonances ΔX~R n(0) is small one. Location of EC resonance at magnetic axis at Bo=5.0 T moves UHR outside (|real(E_psi)| contours in Fig.2a) with respective off axis power deposition: Fig.2b. Above “Arbalet-like” wave pictures show also EC X-wave cut off and remarkable

FIGURE 2a |real(E_psi)| in FTU at Bo=5.0 T FIGURE 2b Pe in FTU at Bo=5.0 T

parallel fields amplitudes (Fig.1a) generated by used O-mode antenna. Doubling plasma density (9x1019 m-3) leads in FTU to more serious broad power deposition as is shown by Fig.2c. That effect is much more pronounced at oblique power launch for CD goals due to larger classical Doppler broadening effect influence. Contour plots of |E_minus| wave electrical field in DIII-D plasma are displayed in Fig.3a at fundamental harmonic O-

mode 44.6 GHz launch: N//(0) = 0.227, Ne(0) = 0.48´1019 m-3, Nes=0.12´1019 m-3, Te(0) =2.7 kV (αn=0.2, αT=2), Bo=1.47 T, Ip=0.657 MA. Plasma elongation is κ=1.65,

FIGURE 2c Pe in FTU at Ne(0) = 9x1019 m-3

FIGURE 3a |E_minus| in DIII-D FIGURE 3b Pe in DIII-D at N//(0) = 0.227

Again, power deposition is dominated by UH resonance position due to large amplitudesof converted waves. One pays attention that RF power is absorbed by fast electrons.

4. Second Harmonic ECH Scenarios in T-10

Majority of present tokamaks and stellarators operate at second harmonic and at for not so large plasma densities UHR is absent one. Second harmonic X-mode launch in T-10: |real(E_psi)|, Fig.4a at N=90 (N//(0)=0.02), F=140 GHz, Ne(0)=4.5´1019 m-3, Te(0)=8.7 kV, (αn=1, αT=2), Bo=2.5 T, Ip=300 kA (relativistic effects are included) and power deposition in Fig.4b. Diffraction effects are more severe at density Ne(0)=8´1019

m-3 and are given in Fig.5a for |real(E_psi)| and and in Fig.5b for 1D and 2D power depositions.

triangularity δ= 0.5. The EC resonance is far inside displaced one. Power deposition peaks in Fig.3b may be responsible for temperature filamentation. The benefits off centre inside EC resonance scenario are supported by T-10 experience (K.Razumova, private communication).

FIGURE 4a |real(E_psi)|,in T-10 at Ne(0)=4.5´1019 m-3 FIGURE 4b Pe in T-10 at 2ωce F=140 GHz

FIGURE 5a |real(E_psi)|,in T-10 at Ne(0)=8´1019 m-3 FIGURE 5b 1D&2D Pe in T-10 at Ne=8´1019 m-

3

At plasma density Ne(0)=9´1019 m-3 diffraction is more strong one (Figs.6a,b)..

FIGURE 6a |real(E_psi)|,in T-10 at Ne(0)=9´1019 m-3 FIGURE 6b 1D&2D Pe in T-10 at Ne=9´1019 m-

3

The X-mode cut off near plasma axis appears at density Ne(0)=10´1019 m-3 : Figs.7a,b.

FIGURE 7a |real(E_psi)| in T-10 at Ne(0)=1020 m-3 FIGURE 7b 1D&2D Pe in T-10 at Ne(0)=1020 m-3

5. Second Harmonic ECH Scenarios in DIII-D at Oblique Launch

Second harmonic X-mode oblique 4 MW launch in DIII-D: plasma elongation κ=1.65, triangularity δ= 0.5, N=920 (N//(0)=0.29), F=89.2 GHz, Te0=7.2 kV, Ne(0)=1.2´1019

m-3 , Ip=720 kA, q(0)=0.82, q(a)=3.5 is displayed in Figs.8a,b: by |real(E_minus)| and |real(E_psi|) fields components

FIGURE 8a |real(E_minus) |in |in DIII-D at at 2ωce FIGURE 8b |real(E_psi)| in DIII-D at at 2ωce

Radial power depositions calculated by STELEC and GENRAY-GA (R.Prater [6]) for elliptically polarized X-antennae are given by Figs.9a,b. One can see that STELEC gives up to 3 times broader power deposition than predicts ray tracing code.

FIGURE 9a Pe(ρ) in DIII-D calculated by STELEC FIGURE 9b Pe(ρ) in DIII-D due to GENRAY-GA

6. Fundamental Harmonic Scenario at JT-60U

Modelled was JT-60U regime with fundamental harmonic O-mode equatorial 4 MW launch in plasma with Ro=3.5 m, elongation κ=1.25, triangularity δ= 0.34, N=360 (N//(0)=0.11), F=44.6 GHz, Te0=10.8 kV, Ne(0)=4.8´1018 m-3 , (αn=0.2, αT=2), Bo=1.41 T, Ip=634 kA, q(0)=1.41, q(a)=4.4 is displayed in Figs.10a,b: by |real(E_minus)| One can see High Field Side EC resonance is at X=-40cm and UH resonances at

X=-5cm. Power deposition is in Fig.10b again indicates crucial role of UH resonance in

FIGURE 10a |real(E_minus)| in JT-60 FIGURE 10.b Pe(ρ) in JT-60U calculated by STELEC

broadening of EC power localization.

6. Fundamental Harmonic Scenario in ITER-like Machines Modelled was ITER regime for NTM stabilization, relying on above similarity laws, with fundamental harmonic O-mode equatorial 20 MW launch into plasma with Ro=6.2 m, elongation κ=1.85, triangularity δ=0.45, N=380 (N//(0)=0.13), F=22.6 GHz, Te0=25.8 kV, Ne(0)=1.2´1018 m-3 , (αn=0.2, αT=2), Bo=0.646 T, Ip=1.27 MA, q(0)=0.88, q(a)=4.2. Classical Doppler broadening was assumed. Results are displayed in Figs.11a,b: by |real(E_minus)| and Pe(r). One can see HFS EC resonance at X=-117cm and UH resonances at X=-62cm. Power deposition is in Fig.11b again indicates crucial role of UH resonance in broadening of EC power localization, leading to large deposited power outside of supposed NTM localization.

FIGURE 11a FIGURE 11a |real(E_minus)| in ITER FIGURE 11b FIGURE 11b Power deposition Power deposition Pe(ρ) in ITER

7. OXB Scenarios For Over Dense Plasma

These schemes were theoretically proposed in frame of 1D geometry to exploit O-mode conversion to X-mode with subsequent its conversion to EB waves (O-X-B scheme)

which have no density cut off and can propagate at plasma densities larger than cut off densities for e.m. EC waves. We modeled this OXB scheme at fundamental harmonic in

FIGURE 12a |real(E_psi)| in Tore Supra FIGURE 12b Pe(ρ) in TS at Ne(0) = 1.7 Ncrit

Tore Supra tokamak with Ro=2.38 m, a=0.61 m, N=80 (N //(0)=0.125), F=12.85 GHz, Te0=9.5 kV, Ne(0)=6´1017 m-3 , thus Ne(0) = 1.7 Ncrit , (αn=2.5, αT=0.1), Bo=0.37 T, Ip=103 kA, q(0)=1.35, q(a)=3.65. STELEC code found that really O-X-B 2D scheme operates in regime with far inside EC resonance location as demonstrated by Fig.12a for |real(E_psi)| contour plots and by Fig.12b for radial power deposition Pe(r). However we did not succeed at second harmonic O-mode launch in DIII-D and Tore Supra. More work is needed.

8. Fundamental harmonic ECH power deposition in RTP

The RTP tokamak (major radius R=72 cm) [8] used oblique O-mode fundamental harmonic outside launch at 60 GHz in Co-Counter ECCD scenarios. Contour plots of |real(Eψ )| wave electrical field, 2D and 1D power depositions are displayed in Fig.13. The axis magnetic field is Bo=1.98 T , central plasma density 1.65x1019 m-3 (parabolic density profile), Te(0) =2.9 kV (αT=2), Ip=40 kA, N//(0)=0.35. One sees that EB waves play crucial role in creation of EC power deposition profile. Classical Doppler effects

FIGURE 13 |real(Eψ )| and Power depositions: 1D Pe (ρ) and 2D Pe (ρ,θ) in RTP

dominate in this RTP scenario. Counter CD scenario with N//(0)= -0.35 reveals significant amount of RF power being deposited broadly near half of plasma minor radius, consistent with reduced counter CD efficiencies observed in RTP experiments [8].

9. Discussion

Thus full wave code discovered crucially new physics result at fundamental EC harmonic: Upper Hybrid Resonance appearance at scenarios with O-mode radiating antenna, contrary to ray tracing (based on local dispersion relation) philosophy. This UHR layer fires practically as a hole, occupying remarkable part of plasma and radiateslarge amplitude Electron Bernstein Waves and Slow X-waves. These strong waves deposit dominant part of RF input power at LFS to energetic part of electron distribution with account to relativistic and classic Doppler effects. At low plasma densities in smalland middle size fusion machines (with small major radius R) a space separation between cold EC resonance and UH resonance, being proportional to Ne•R, is small one and RF power, being absorbed near UHR position in area of its intersection with equatorial plane,is absorbed sufficiently locally thus recovering majority of best electron heating world results for near on axis ECR used scenarios in tokamaks and stellarators. However, in large machines like JT-60U, TS, ITER at increased densities, required by fusion gain, that resonances pair is well separated and STELEC modeling manifests loss of EC power localization at fundamental harmonic. Fortunately, at second harmonic, typically, UHR is absent in interesting for fusion scenarios and localization still works. There are interesting experimental confirmations of UHR appearance at fundamental O-mode RF power launch: 1) Similar EC heating efficiencies at fundamental harmonic in DIII-D for O-mode and X-mode launches were reported [6] and recently confirmed again [7]. Nice explanation argued to X-mode reflection from X’s cut off and conversion to O-mode during reflection from the walls. Full wave code treats this “conversion” effect automatically and knows something about O or X modes only through antenna polarization. 2) Experimental evidence of UHR existence in RTP tokamak: at outside 60 GHz O-mode launch they observed two orders rise of X-mode signal in central receiving inside chamber horn vs position of ECR layer and central electron density, when UHR according to simple formula, appears at plasma core [8]. They stressed sawtooth modulation of X-signal, thus arguing that signal was coming from plasma core.3) The measured by quasi perpendicular oriented antenna radiation temperatures in RFP MST plasma with Ip=250 kA, <Ne>=4 1018 m-3 for X and O polarizations are comparable ones and attribute to conversion efficiencies from EBW to O-mode and X-mode [7]. They argue that reciprocal principle is valid and both types of plasma heating may be applied at fundamental harmonic.4) Transformation of wave polarization due to 2D plasma non uniformity [9]:- first cylindrical model of theoretical and experimental insight to the problem of wave polarization conversion due to axial plasma density gradient or to wave incident power axial localization. These [9] results are directly relevant to ECH in stellarators and mirrors. So natural is our similar idea of transformation wave polarization due to POLOIDAL non uniformity of toroidal plasmas in tokamaks to understand code’s news.5) The WEGA stellarator electron heating with O- mode fundamental resonance out of plasma [5] is qualitatively confirmed by STELEC modeling.

Conclusions In toroidal bounded plasmas O-mode and X-mode are coupled ones through space inhomogenity and boundary conditions. This modes coupling effect in toroidal plasmas is especially strong one at fundamental harmonic scenarios due to singular layer createdby UHR resonance. This effect leads to strong broadening of EC power deposition at fundamental and must be accounted through full wave codes. In second harmonic scenarios the modes coupling is more weak one. Ray tracing/bi-tracing technique may be still used, neglecting by X and O-modes coupling, for power deposition evaluations, with corrections from full wave codes at high densities for self consistent account of modes coupling and diffraction effects. Full wave codes are powerful tools and must be used, when available, for fundamental harmonic scenarios in small, large and ITER machines REFERENCIES1. V.Vdovin Electron Cyclotron Heating modelling in tokamaks with full wave 3D code, XXXIII International Conf. on Plasma Phys. and Contr. Fusion, 13 – 17 February 2006, Zvenigorod, Moscow, Russia2. M. Brambilla Plasma Physics Controlled Fusion 31 (1989), 7233. I. Shkarovsky, J. Plasma Phys. 35, part2, 319-331 (1986)4. V.Vdovin, T.Watari, A.Fukuyama, Proc.ICPP 96 (Nagoya 1996),1070; NIFS- 4695. Y.Y. Podoba “Radio frequency heating on the WEGA stellarator”, Ph. D. These Ernst-Moritz-Arndt-University Greifswald, 20066. R. Prater et al, Proc. of the EPS25 (Praha) Vol. 22C (European Physical Society, 1998) p. 14107. R. Prater (private communication).8. A.Donne, A.Oomens, EC-9, Proc. of 9th Joint Workshop on ECE and ECH, Borrego Springs, California, 23-26 January 1995, 645-650, F.Schuller et al,. Proc. of 4th Int. Workshop Strong Microwaves in Plasma, NN Russia, 19999. D.Falconer, A.Messian, P.Vandenplas Physics of Fluids, 17 (1974), 1566-1573