POWER DENSITY ABSORPTION PROFILE IN TOKAMAK PLASMA WITH ICRH
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1 Dedicated to Professor Oliviu Gherman s 80 th Anniversary POWER DENSITY ABSORPTION PROFILE IN TOKAMAK PLASMA WITH ICRH N. POMETESCU Association EURATOM-MECTS Romania, University of Craiova, Faculty of Physics, Str. A. I. Cuza, No.13, Craiova, Romania npomet@yahoo.com Received July 30, 2010 The problem of radial profile of power density absorption in multispecies plasma (D-T plasma) by Ion Cyclotron Resonance Heating (ICRH) in tokamak fusion device is analyzed. The profile of power absorption is of great importance both for the heating efficiency and the transport. This profile is actually not a simple Gaussian, as is sometimes approximated, but a more complex function. Analytic expressions are proposed here for Tritium and Deuterium for a well description of the power deposition profile. 1. INTRODUCTION The use of radio-frequency (rf) power in magnetic fusion devices has many important goals: heating plasma, current drive and profile control. Using rf power in the ion cyclotron range of frequencies (ICRF) to drive poloidal flow with radial velocity shear can stabilize microturbulence and improve plasma confinement, see for example 1. Resonant interaction between ions and waves in the ion cyclotron range of frequencies leads both to heating the plasma and spatial transport of the resonating ions. The current and torque provided by ICRH depend on the concentrations of heated species and also on the positions of the resonance layers. Recently works analyses the resonant ion cyclotron interactions in tokamaks and influence of the radio frequency ponderomotive force on anomalous impurity transport in tokamaks, see for example 2, and 3. The resonance layer position and the profile of power absorption plays an important role for stability and confinement properties of the plasma. In the following we analyze the profile of power absorption in a tokamak fusion device by Tritium and Deuterium in a deuterium-tritium plasma heated by ICRF when the frequency of the launched rf wave is close to the second harmonic resonance frequency for Tritium. In many works this profile is considered for simplicity as a Gaussian, but experiment measurements and numerical simulation reveal a more complex profile. In the section 2 we describe the geometry of the ICRH in a torus with circular cross-section of equilibrium magnetic surface. A realistic power depo- Rom. Journ. Phys., Vol. 55, Nos. 9 10, P , Bucharest, 2010
2 2 Power density absorption profile in tokamak plasma with ICRH 1043 sition profile is described in section 3 and an analytic expression is proposed here for a well reproduction of the power deposition profile. 2. ICRH OF A TOROIDAL PLASMA We assume here, for simplicity, an axisymmetric model of the equilibrium magnetic field for a tokamak with circular concentric magnetic surfaces ( R 0 B = B 0 r ) e θ + e ζ (1) R q s R 0 where R 0 is the major radius, r the toroidal radial coordinate, R = R 0 + r cosθ the cylindrical radial coordinate and q s (r) = rb ζ /R 0 B θ the safety factor (see for example 4). The magnitude of the equilibrium magnetic field is then, see also 5: B (r) = B 0 (r) 1 + η cosθ = B η 1 + η cosθ 2 q 2 (2) with ɛ η ɛ, η = r/r 0, ɛ = a/r 0 the inverse aspect ratio and B 0 = B (r = 0). In the numerical application we use a = 200 cm, R 0 = 620 cm, B 0 = 5.3 T, that correspond to the ITER project design. The cyclotron resonance condition is described by the δ-function δ ( ω nω ci k v ) where ω is the frequency of the externally launched radio-frequency wave. For a given k v the cyclotron resonance is localized on some surface B (R) = constant, which in the present model of magnetic field is a vertical cylindrical surface - see figure 1. In the case k v = 0 the distance r 0 from the toroidal magnetic axis to the resonant surface corresponds to ω = nω ci,0 where Ω ci,0 = e ib m i In the direction with θ = 0 we obtain, using (2), 1 + r0 2 ω = nω R 2 0 q 2 0,i 1 + r 0 R0 1 (3)
3 1044 N. Pometescu 3 where Fig. 1 Geometry for ICRH in a torus with circular crossection. Ω 0,i = e ib 0 m i For a given ω, the radial coordinate r 0i,n of the resonant surface is obtained as solution of (3). For large aspect ratio tokamak we can disregard the dependence on the safety factor q and (3) is replaced by with solution ω nω 0,i 1 + r 0i,n R 1 0 ( ) nω0,i r 0i,n = R 0 ω 1 For more explicit we analyze the case of radio-frequency heating waves with ω = 106 π MHz which correspond to ion cyclotron resonance heating (ICRH) for the second harmonics (n = 2) of Tritium in a plasma with D (46%) and T (46%) - see 7. For Tritium the solution obtained with n = 2 from (3) is r 0T = cm and from (5) is r 0T ap = cm. For Deuterium we obtain r 0D = cm and r 0D ap = cm. From this results we conclude that the solution (5) give a good approximation of the position of the resonant surface. As we can see the resonant surface for Tritium is near the centre and for Deuterium is outside of the plasma. The resonant layer width is determined by the Doppler shift of thermal particles. The resonance condition is then given as (4) (5) ω = nω ci + k v,i (6)
4 4 Power density absorption profile in tokamak plasma with ICRH 1045 with Ω ci = e ib (r i ) m i where r i is the radial position of the resonant surface for a given k v,i moved by X i from r 0,i : r i = r 0,i + X i In numerical application we consider k = 27/R with the major radius measured in meter. Since resonant wave-ion particle interactions at the cyclotron frequency increase the average perpendicular energy of the resonant ions without appreciably affecting their parallel energy we can approximate v v th,i = T i /m i, where v thi is the ion thermal velocity. In the following we name X th,i, as usual, resonant layer width. For solving (6) we need to know the temperature profile of the ion species. In numerical application we will consider ( ) 2 T i = T i,0 1 r2 a 2 (7) with T i,0 = 10 kev for both Tritium and Deuterium. Solving (6) for Tritium we obtain r T = cm and for Deuterium r D = cm. So, we have X th,t = cm and X th,d = cm. With assumption rr0 1 1 we can represent the width X in first order approximation as (see 6) X 0,i = R k Ti,0 (8) nω 0,i m i But using (8) we obtain X 0,T = 4.5 and X 0,D = 3 cm. So, in the case of T the approximation is good, but in the case of D is not. This is because not D is primarily heated by ICRH, the resonant layer being outside the plasma. 3. MODEL FOR POWER DENSITY ABSORPTION PROFILE Analyzing the experimental results we observe that the rf heating power is distributed between plasma species and has specific localization. We analyze here the case of multispecies plasma with the following composition: electrons (100%), T (46%), D (46%), alpha particles (4%) and other impurities (4%) - see 7. Using numerical data we plot the power density absorbed by Tritium (see figure 2) and Deuterium (see figure 3). We give the plots in separate figure because of the magnitude discrepancy. To obtain the real magnitudes of the power density the relative units must be multiplied by factor W/m 3 per MW.
5 1046 N. Pometescu 5 In the following we aim to give analytical expressions which can approximate the data obtained in numerical simulations. In the case of Maxwellian ion velocity distribution, the intensity of absorption within the layer has a Gaussian profile and the flux surface averaged absorbed rf power density, P i, read as P i = P 0,i exp (r r 0,i) 2 2( X 0,i ) 2 where θ res is the poloidal angle corresponding to the central vertical axes of the resonance layer and a the minor radius. (9) P Tritium r cm Fig. 2 Power density absorption for Tritium. The relative units of P-Tritium must be multiplied by W/m 3 per MW. P Deuterium r cm Fig. 3 Power density absorption for Deuterium. The relative units must be multiplied by W/m 3 per MW. But, from the experiments evidences the ion velocity distribution is not Max-
6 6 Power density absorption profile in tokamak plasma with ICRH 1047 wellian and the intensity of absorption has not a Gaussian profile. In the case of Tritium the profile for P can be well described through a superposition of Gaussian functions centered in the range of layer resonance, see 8, but in the case of Deuterium we have seen that the layer resonance is outside of plasma. Now, instead of the Gaussian function from (9) we will use (r r 0,i ) 2 exp 2( X 0,i f 1 f 2,i ) 2 f 3,i with f 1 = B 0 B T i T i,o and f 2,i, f 3,i different for Tritium (main heated species) and Deuterium. Also we take into account the Shafranov shift d S. In numerical evaluation we use d S = 20 cm. Pm Tritium r cm Fig. 4 Power density absorption for Tritium approximated by analytical expression. The power density absorption for Tritium given as analytically modelled expression is P m,t = 0.06exp (r r 0T ) 2 ( 2(2f 1 X 0,T ) sin 2 2π (10r/a) 2) (10) exp (r r 0T d S /2) 2 2(2f 1 X 0,T ) exp 3r/asin 2 2π (3r/a) 2 and is plotted in figure 4. For Deuterium the power density is approximated as
7 1048 N. Pometescu 7 Pm Deuterium r cm Fig. 5 Power density absorption for Deuterium given by analytic approximation. (r r 0,D ) 2 ( P m,d = 5exp 2(f 1 X 0,D ) 2 (10 r/a) sin 2 2π (3r/a) 2) (11) (r r 0,D d S /2) 2 ( +exp 2(f 1 X 0,D ) 2 (1.88 r/a) sin 2 2π (5r/a) 2) and is represented in figure CONCLUSIONS The problem of radial profile of power density absorption in multispecies plasma by ICRH is analyzed. The fusion device is a tokamak with circular concentric magnetic surfaces and the ion species of the plasma are T (46%), D (46%), alpha particles (4%) and other impurities (4%). The frequency of the heating waves is ω = 106 π MHz and corresponds to the second harmonics of Tritium cyclotron resonance. The data from numerical simulation are used to construct analytical expression for the power density absorption (surface averaged) P for the main two ion species of the plasma: Tritium and Deuterium. We show that the simple Gaussian profile, used in some works as a zero order approximation, is not suitable to describe the profile of P and a more complex analytical expressions, (10) and (11) are proposed in order to describe the radial dependence of the power density absorption. Despite the fact that the main resonant layer absorption is situated out of the plasma, there is an absorption of heating power in a secondary resonant layer situated into the plasma and the power absorbed by D is about 9 times less than that absorbed by T. The approximation given here can be used in analytic analyses, both in kinetic and fluid transport theories. This model will be improved in next future by analyz-
8 8 Power density absorption profile in tokamak plasma with ICRH 1049 ing other profiles obtained by numerical methods or experimental measurements for other tokamak parameters. Acknowledgements. Professor Oliviu Gherman is acknowledged for the support given in my all activity. Dr. Ernesto Lerche from ERM/KMS is acknowledged for the data of numerical simulations of power deposition profile. This work was supported by EURATOM-MECTS Romania. REFERENCES 1. B. P. LeBlanc et al, Phys. Rev. Lett. 82, 331 (1999). 2. T. Johnson, T. Hellsten and L.-G. Eriksson, Nucl. Fusion 46, S433 (2006). 3. H. Nordman, R. Singh, T. Fülop, L.-G. Eriksson, R. Dumont, J. Anderson, P. Kaw, P. Strand, M. Tokar and J. Weiland, Phys. Plasmas 15, (2008). 4. G. L. Falchetto, J. Vaclavik and L. Villard, Phys. Plasmas 10, 1424 (2003). 5. R. Balescu, Transport Processes in Plasmas, vols 1 and 2, North-Holand, Amsterdam, M. Brambilla, Kinetic Theory of Plasma Waves, Clarendon Press Oxford, R. Koch, Phys. Letters A, 157 (6,7), 399 (1991) 8. N. Pometescu, Physics AUC 19, (2009)
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