Scaling of the Density Peak with Pellet Injection in ITER

Plasma Science and Technology, Vol.14, No.12, Dec. 2012 Scaling of the Density Peak with Pellet Injection in ITER P. KLAYWITTAPHAT and T. ONJUN School of Manufacturing Systems and Mechanical Engineering, Sirindhorn International Institute of Technology, Thammasat University, Pathum Thani 12121, Thailand Abstract Scalings of the density peak and pellet penetration length in ITER are developed based on simulations using 1.5D BALDUR integrated predictive modeling code. In these simulations, the pellet ablation is described by the Neutral Gas Shielding (NGS) model with grad-b drift effect taken into account. The NGS pellet model is coupled with a plasma core transport model, which is a combination of an MMM95 anomalous transport model and an NCLASS neoclassical transport model. The BALDUR code with a combination of MMM95 and NCLASS models, together with the NGS model, is used to simulate the time evolution of plasma current, ion and electron temperatures, and density profiles for ITER standard type I ELMy H-mode discharges during the pellet injection. As a result, the scaling of the density peak and pellet penetration length at peak density can be established using this set of predictive simulations that covers a wide range of ITER plasma conditions and pellet parameters. The multiple regression technique is utilized in the development of the scalings. It is found that the scaling for density at center is sensitive to both the plasma and pellet parameters; whereas the scalings for density and location of the additional peak are sensitive to the pellet parameters only. Keywords: plasma, tokamak, fusion, pellet injection, ITER PACS: 52.55.Fa, 52.65.-y DOI: 10.1088/1009-0630/14/12/01 1 Introduction Several simulations of the International Thermonuclear Experimental Reactor (ITER) demonstrate an increase of plasma density at the center once pellets are injected into plasma. In addition, another density peak can form in the outer region of the plasma, which is a result of the density of the ablated plasma [1,2]. The central density peak can contribute to plasma performance enhancement [3 20]. The new density peak can influence plasma behaviors, such as sawtooth oscillation [21 24], and stabilization of high n ballooning modes, and low n external kink modes [25]. Due to the large impact of pellets on plasmas [26], it is interesting to develop scalings that adequately describe the plasma density and pellet peaking location, which will be useful for fusion study, especially for ITER. In this work, scalings for density at the plasma center and at the additional peak, and the peak location of the additional peak for ITER are developed based on simulations using the 1.5D BALDUR integrated predictive modeling code. In these simulations, the pellet ablation is described using the neutral gas shielding (NGS) model with grad-b drift effect taken into account. The NGS pellet model is coupled with a plasma core transport model, which is a combination of an MMM95 anomalous transport model and an NCLASS neoclassical transport model. The scalings of density peaks and pellet penetration can be established using this set of simulations that covers a wide range of ITER plasma conditions, such as plasma current [9 15 MA], electron density [0.8 10 20 1.2 10 20 m 3 ] and pellet parameters, such as the pellet s radius [1 5 mm] and velocity [100 500 m/s]. The multiple regression technique is utilized in the development of these scalings. This paper is organized as follows: a brief description of relevant components of the BALDUR code, including the MMM95 anomalous transport model and the pellet ablation model, is presented in section 2; predictions of ITER for standard type I ELMy H-mode with pellet injection and the scaling for density at the peak, the peak location and the scaling for density at the center are described in section 3; and the conclusion is given in section 4. 2 Description of codes The BALDUR integrated predictive modeling code [27] is self-consistent for sources (such as heating from neutral beam injection and fusion reactions), sinks (such as impurity radiation), transport (neoclassical and anomalous), equilibrium shapes of the magnetic surfaces, the effects of large-scale instabilities (such as sawtooth oscillations or tearing modes), and boundary conditions (such as gas puffing and recycling). It uses nuclear reaction rates and a Fokker Planck pack- supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission, and the Low Carbon Scholarship for SIIT Graduate Students

Plasma Science and Technology, Vol.14, No.12, Dec. 2012 age to compute the slowing down spectrum of fast alpha particles on each flux surface in the plasma. The fusion heating component of the BALDUR code also computes the rate of production of thermal helium ions and the rate of depletion of deuterium and tritium ions within the plasma core. BALDUR simulations have been intensively compared against various plasma experiments, which yield an overall agreement of 10% RMS deviation [28 38]. In this work, the boundary conditions are described using a predictive pedestal module, one of the best pedestal models in Ref. [39] is chosen. This pedestal model was developed by combining the theoretical-based pedestal width model based on magnetic and flow shear stabilization ( ρ i s 2 ) [39] with the pressure gradient limits imposed by ballooning mode instability [40]. This pedestal temperature model yields agreement with Root Mean Square Error (RMSE) in the range of 30% for predicting pedestal temperature when its predictions were compared against type I ELMy H-mode discharges from various present-day tokamaks [41]. This pedestal model is taken from the National Transport Code Collaboration library [42 49]. 2.1 Pellet injection module The pellet injection can be described as two simultaneous processes, which are pellet ablation and mass relocation (plasmoid drift). We simulate the pellet injection with the pellet penetration model combined with the ablation model and the mass relocation model. There are few pellet ablation models capable of satisfactorily describing the pellet ablation and relevant experimental light emissions [50], among which the most popular one, the neutral gas shielding (NGS) [51], is selected for this work. In these models the usual hydrodynamic conservation equations for mass, momentum and energy are supplemented by sets of expressions that define the heat deposition in the cloud due to collisions of the energy carriers with the outstreaming neutral ablatant particles. These equations describe a spherically symmetric expansion. According to this model, immediately after the pellet is subjected to the direct impact of energetic plasma electrons, a protective cloud of neutral particles forms around the pellet. Through slowing down the inelastic and elastic collision process and the scattering of the incoming plasma electrons by the neutral particles presented in the cloud, the pellet lifetime is significantly prolonged. In other words, the ablated cloud serves as a stopping medium for the incoming plasma electrons. However, the cloud differs from the usual stopping medium of a solid target because the density of the stopping medium is governed by the hydrodynamics of the expanding cloud. The expansion of the cloud depends, in turn, on the way the energy of the incoming electrons is deposited at the pellet surface. For this reason, one might expect that the precise vaporization process occurring at the pellet surface would affect the expansion process of the ablatant, which might then influence the energy absorption process of the cloud. The ablation rate of this model can be expressed in terms of power functions as follows: dn dt = 5.2 1016 n 0.333 e T 1.64 e r 1.333 p M 0.333 p, (1) where N, n e, T e, r p, and M p are the number of particles in a pellet, the electron density, the electron temperature, the pellet size, and the atomic mass number of the pellet, respectively. The number of atoms in the pellet, N, is expressed as: ) N = 2n s ( 4πr 3 p 3, (2) where n s is the molecular density of the solid hydrogen (n s = 3.12 10 28 m 3 ). A pellet injected into the tokamak is rapidly ablated due to the high temperature of the hot plasma. The ablated particles are immediately ionized and then drift in a high field side (HFS) direction, caused by the variation of the magnetic field in tokamaks [52,53]. To include this drift effect, a scaling model of pellet drift displacement, based on the grad-b induced pellet drift [54], has been taken into account for the pellet injection. 2.2 Multi-mode core transport model The multi-mode model [44,45] is an extension of the Weiland model [46,47]. It uses the Weiland model, but adds the effects of drift-resistive [48,49] and kinetic ballooning modes [16]. The expressions of transport coefficients in MMM95 are: χ i = 0.8χ iitg&tem + 1.0χ irb + 1.0χ ikb, (3) χ e = 0.8χ eitg&tem + 1.0χ erb + 1.0χ ekb, (4) D H = 0.8D HITG&TEM + 1.0D HRB + 1.0D HKB, (5) D Z = 0.8D ZITG&TEM + 1.0D ZRB + 1.0D ZKB, (6) where χ i is the ion diffusivity, χ e is the electron diffusivity, D H is the particle diffusivity, D Z is the impurity diffusivity, χ ITG&TEM is the thermal diffusivity of the ion temperature gradient and trapped electron mode, χ RB is the resistive ballooning thermal diffusivity, and χ KB is the kinetic ballooning thermal diffusivity. 3 Simulation results and discussion A series of self-consistent ITER simulations are carried out for different plasma conditions and pellet parameters using a 1.5D BALDUR integrated predictive modeling code. Pellets are applied at 600 s after the plasma reaches its quasi-steady state for a long enough time. It is assumed in this work that there are four plasma species: two working gas species (deuterium and tritium) and two impurity species (helium and beryllium). It is worth noting that the effect of ELMs is not considered in this work. 1036

P. KLAYWITTAPHAT et al.: Scaling of the Density Peak with Pellet Injection in ITER In Fig. 1, the electron temperature and deuterium density profiles obtained from the ITER simulation for a full-current standard type I ELMy, H-mode conditions (R = 6.2 m, a = 2.0 m, Ip = 15 MA, BT = 5.3 T, κ95 = 1.7, 95 δ95 = 0.33, Zeff,edge = 1.4, and nl = 1.0 1020 m 3 ) during pellet injection are shown as a function of normalized minor radius. Note that in this simulation, a deuterium pellet size of 3 mm with a velocity of 300 m/s is used. It can be seen in the simulations that after pellet injection, the shape of the electron temperature profile in the outer region of the plasma drops slightly, due to the energy exchange between the hot plasma and cold particles from the pellet. On the other hand, the temperature in the plasma center increases slightly. This is due to the reduction of total ion thermal diffusivity in the region closed to the plasma center, as can be seen in Fig. 2. The deuterium densities in the outer region of the plasma increase significantly and form another density peak located at the normalized radius r/a 0.63. Fig.2 Total ion thermal diffusion coefficient, shown as a function of minor radius at the time before and during the pellet injection (color online) Fig.3 The evolution of line averaged electron density (top), central electron density (bottom) Fig.1 Plasma electron temperature profile (top) and deuterium density profiles (bottom), shown as functions of minor radius at the time before and during pellet injection. In this simulation, a deuterium pellet with a radius of 3 mm, and a velocity of 300 m/s is used (color online) In Fig. 3 (top) the line-average density is increased from 1.04 1020 m 3 to 1.1 1020 m 3 after pellet injection. Then, it decays to a quasi-steady value of 1.04 1020 m 3 after 150 ms. The central density also increases after the pellet injection. However, it reaches the peak value slower than the line average density. This is due to the fact that the pellet ablation occurs mostly in the outer region of the plasma. It takes some time for the density from the pellet to diffuse to the center. It is known that the increased bootstrap current reduces the need of current drive. As a result, the formation of a bootstrap current is crucial for the steady state nuclear fusion operation. In this work, the prediction of bootstrap current profiles is shown in Fig. 4, It can be seen that the bootstrap current increases after pellet injection, and the increase is due to the strong density gradient. In the simulation, it is found that the bootstrap current fraction can increase up to 4% due to the pellet. Note that a series of pellets can result in more density peaks, which can further contribute to the bootstrap current formation. Scalings of density at the additional peak (ne,peak ) and at the center (ne0,peak ) are developed based on the BALDUR simulation results. In this work, the follow1037

Plasma Science and Technology, Vol.14, No.12, Dec. 2012 the large standard deviation, we do not have confidence in the prediction of this scaling. Further study should be carried out to enhance the confidence in predictive capability of this scaling. Table 1. Ranges of simulation parameters used for developing the scale of density peak and pellet penetration length Fig.4 Bootstrap current density, shown as a function of minor radius at the time before and during the pellet injection (color online) Parameters Range of value Pellet radius (rp ) (mm) Pellet velocity (vp ) (km) Plasma current (Ip ) (MA) Pellet species (Mp ) (AMU) Density (nl ) ( 1020 m 3 ) 1, 2, 3, 4, 5 0.1, 0.2, 0.3, 0.4, 0.5 11, 13, 15, 17, 19 2, 3 0.8, 0.9, 1.0, 1.1, 1.2 ing form of density peaking due to pellet injection is assumed based on both plasma and pellet parameters: ne,peak = Crpa vpb Ipc Mpd, nl (7) when ne,peak is the electron density at the peak due to pellet injection; nl is the electron line average density, Ip is the plasma current, rp is the effective spherical pellet radius, vp is the pellet velocity, and Mp is the atomic mass number of pellet. The multiple regression analysis technique is utilized to find the constant a, b, c, and d. In this work, the following scaling density peak scaling is found: ne,peak = 0.544rp0.306 vp0.215 Ip 0.013 Mp0.024. nl (8) Note that the ranges of simulation parameters used in this work are shown in Table 1. It can be seen that the scaling of density at the additional peak is sensitive to the pellet parameters, but not the plasma parameters. The comparison between the density peak from scaling using Eq. (8) and the density peak from the pellet simulations is shown in Fig. 5. It is found that the standard deviation and correlation between the prediction from the scaling and simulation are 25.4% and 0.64, respectively. The same method is used to find the plasma peaking density scaling at the plasma core. We found that the peaking density scaling at plasma core can be written in the following from: ne0,peak = 0.316rp1.92 νp0.1140 Ip 0.4090 Mp0.7533. nl (9) It can be seen that the scaling of density at the center is sensitive to both pellet and plasma parameters. The comparison between the density peak at the center from the scaling shown in Eq. (9) and the density peak at the center in simulations is shown in Fig. 6, from which a large scatter of results can be seen. The standard deviation and correlation between the prediction from the scaling and simulation are 48.2% and 0.69, respectively, in which the standard deviation is significantly worse than that for the density at the additional peak Eq. (8), but the correlation is about the same. Due to 1038 Fig.5 The comparison between the values of density peak from scaling developed in Eq. (8) and from simulations Fig.6 The comparison between the values of density peak at center from scaling developed in Eq. (9) and from simulations For the scaling of the pellet peak location, the following form is assumed: λp = Crpa vpb Ipc Mpd nel, (10) a when λp is the penetration length from the edge to the peak position and a is the minor radius. The multiple regression analysis technique is utilized to find constants a, b, c, d, and e. In this work, the following scaling for pellet penetration length at the density peak is found to be: λp = 0.146rp0.862 vp0.558 Ip 0.022 Mp0.009 n0.009. (11) l a

P. KLAYWITTAPHAT et al.: Scaling of the Density Peak with Pellet Injection in ITER It is found that the standard deviation and correlation for this scaling at the density peak and simulation are 12.2 % and 0.87, respectively. It can be seen that this scaling is sensitive to the pellet parameters including the pellet radius and pellet velocity but not to the plasma parameters, as in the case of the scaling of the density peak at the additional peak. The comparison between the scaling for the pellet penetration length at the density peak using Eq. (11) and the pellet penetration length at the density peak from simulations is shown in Fig. 7. Note that in this work pellet penetration is calculated using the NGS model. The relationship in the NGS model between the pellet surface erosion rate ṙ, the effective spherical pellet radius r p, the molecular density of solid hydrogen n m, the atomic mass number of the ablatant M p, the background plasma electron density n e and temperature T e, is given approximately by: dr dt n1/3, (12) Mp 1/3 n m rp 2/3 e Te 5/3 where n m = 2.12 10 28 + 6.30 10 27 M p 8.66 10 26 Mp 2 m 3 [55], which can approximate the penetration length between pellet penetration (λ) and pellet velocity (v p ), with integration of Eq. (12) by assuming that the plasma electron density and temperature profiles can respectively be expressed by n e (x) = n e0 (1 x) αn, and T e (x) = T e0 (1 x) α T where n e0 and T e0 designate values at the minor axis of the plasma (magnetic axis), x = r/a 0 is a dimensionless measure of the plasma minor radius and a 0 is the distance from the magnetic axis to the outer plasma edge in the midplane. Time can be removed by relating it to the location of the pellet in the plasma, assuming constant pellet velocity: dt = a 0 dx/v p, where v p is the pellet velocity (injection normal to the plasma from the outside midplane, typical of most experiments). The penetration depth is derived through conversion from dt to dx. Integration over the life time of the pellet (dr from an initial radius of r p0 to 0, and dx from pellet entry at x = 1 to 1 λ/a 0, where λ is the final penetration depth) gives: λ ( v pn m M 1/3 a 0 a 0 n 1/3 p r 5/3 p0 e0 T 5/3 e0 ) βngs = (C p r p M p v p ) βngs, (13) where β ngs 3/(3 + α n + 5α T ) for linear profiles (α n + α T = 1)β ngs = 1/3 as noted by PARKS and TURNBULL [51] which shows that λ is directly proportional to vp 0.333, rp 0.333 and Mp 0.111. Then, it is not surprising that the scaling for the pellet penetration length at density peak λ p is proportional to vp 0.558, rp 0.862 and Mp 0.009. 4 Conclusions Scalings of density peak and pellet penetration length in ITER are developed based on the simulations carried out using a 1.5D BALDUR predictive modelling Fig.7 The comparison between the values of pellet penetration length from scaling developed in Eq. (11) and from simulations code integrated with a combination of MMM95 and NCLASS models, together with NGS pellet ablation models. Using a multiple regression technique on a dataset that covers a wide range of plasma and pellet parameters, the scale can be found based on engineering controlled parameters. It is found that the scaling for density at the center is sensitive to both plasma and pellet parameters, whereas the scaling of the density and location of the additional peak arising from pellet injection is sensitive to pellet parameters only. References 1 Wisitsorasak A, and Onjun T. 2011, Plasma Phys. Rep., 37: 1 2 Leekhaphan P, and Onjun T. 2011, Plasma Phys Reps., 37: 321 3 Kesner J, Conn R W. 1976, Nucl. Fusion, 16: 397 4 Perkins L J, Spears W R, Galambos J D, et al. 1991, ITER Parametric Analysis and Operational Performance, ITER Documentation Series No. 22, IAEA, Vienna 5 Mikkelsen D R, Budny R V, Bell M G, et al. 1993, in Plasma Physics and Controlled Nuclear Fusion Research, 1992 (Proc. 14th Int. Conf. Wiirzburg, 1992), Vol.3, IAEA, Vienna. p.463 6 Pomphrey N. 1992, Bootstrap Dependence on Plasma Profile Parameters. Tech. Rep. PPPL- 2854, Princeton Plasma Phys. Lab., NJ 7 Meade D M, Arunasalam V, Barnes C W, et al. 1991, in Plasma Physics and Controlled Nuclear Fusion Research, 1990 (Proc. 13th Int. Conf. Washington, DC, 1990), Vol.1, IAEA, Vienna. p.9 8 Ishida S, Kikuchi M, Hirayama T. in Plasma Physics and Controlled Nuclear Fusion Research, 1990, Washington (International Atomic Energy Agency, Vienna, 1991), Vol.I, p.195 9 Navratil G A, Gross R A, Mauel M A, et al. in Plasma Physics and Controlled Fusion Research, 1990, (IAEA, Vienna, Austria 1991), Vol.1, p.209 10 Sabbagh S A, Gross R A, Mauel M A, et al. 1991, Phys. Fluids B, 3: 2277 11 Politzer P A. 1994, Phys. Plasmas, 1: 1545 1039

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