II. THEORY AND ANALYSIS

Much progress was made in confinement, transport and MHD researches, such as beta dependence of ELMy H-mode confinement, ITB in reversed shear plasma, aspect ratio effect on external MHD modes and magnetic island evolution in rotating plasma. Integrated simulation code for burning plasma analysis is being developed and validated by fundamental researches of JT-60U experiments. Progress has been made in the NEXT project to investigate complex physical processes in MHD and transport phenomena. The behaviors of the collisionless MHD modes in high temperature plasmas, and the effect of MHD modes on current hole formation were shown. The dynamics of the zonal flows and geodesic acoustic modes (GAMs) were understood in reversed shear configuration and a new gyrokinetic Vlasov code was developed. Cross section data for atomic and molecular collisions and spectral data relevant to fusion research have been compiled and produced.

1. Confinement and Transport
1.1 Origin of the Various Beta Dependence of ELMy H-mode Confinement

Dependence of the energy confinement in ELMy H-mode tokamak on the beta has been investigated for a long time, but a common conclusion has not been obtained so far. Recent non-dimensional transport experiments in JT-60U demonstrated clearly the beta degradation. A database for JT-60U ELMy H-mode confinement was assembled. Analysis of this database is carried out, and the strong beta degradation consistent with above experiments is confirmed. Two subsets of ASDEX Upgrade and JET data in the ITPA H-mode confinement database are analyzed to find the origin of the various beta dependences. The shaping of the plasma cross section, as well as the fuelling condition, affects the confinement performance. The beta dependence is not identical for different devices and conditions. The shaping effect, as well as the fuelling effect, is a possible candidate to cause the variation of beta dependence. [1.1-1]

References
1.1-1 Takizuka, T., et al., Plasma Phys. Control. Fusion, 48, 799 (2006).

1.2 Internal Transport Barriers in JT-60U Reversed-Shear Plasmas

Physics of strong internal transport barriers (ITBs) in JT-60U reversed-shear (RS) plasmas has been studied through the modeling on the 1.5 dimensional transport simulation. Key physics to produce two scalings on the basis of the JT-60U box-type ITB database are identified. Figure II.1.2-1 shows the ITB width, (ΔITB, as a function of the ion poloidal gyroradius at the ITB centre, ρpi,ITB. The standard model reproduces the JT-60U scaling (ΔITB ~1.5ρpi,ITB), while other models does not. As a result, as for the scaling for the narrow ITB width proportional to the ion poloidal gyroradius, the following three physics are important: (1) the sharp reduction of the anomalous transport below the neoclassical level in the RS region, (2) the autonomous formation of pressure and current profiles through the neoclassical transport and the bootstrap current, and (3) the large difference between the neoclassical transport and the anomalous transport in the normal-shear region. As for the scaling for the energy confinement inside ITB (εfβp,core ~0.25 where (εf is the inverse aspect ratio at the ITB foot and βp,core is the core poloidal beta value), the value of 0.25 is found to be a saturation value due to the MHD equilibrium. The value of εfβp,core reaches the saturation value, when the box-type ITB is formed in the strong RS plasma with the large asymmetry of the poloidal magnetic field, regardless of details of the transport and the non-inductively driven current [1.2-1].

References
1.2-1 Hayashi, H., et al., Plasma Phys. Control. Fusion, 48, A55 (2006).

2. MHD Stability
2.1 Aspect Ratio Effect on the Stability of the External MHD Mode in Tokamaks

The formulation for solving numerically the two-dimensional Newcomb equation is extended to calculate the vacuum energy integral. This extension realizes the stability analysis of ideal external MHD modes from low to high toroidal mode numbers. According to this extension, an effect of the aspect ratio on the achievable normalized plasma pressure (βN), restricted by ideal external MHD modes whose toroidal mode number is from 1 to 10, is studied. Figure II.2.1-1 shows the decrease of the aspect ratio improves the achievable βN value, and increases the toroidal mode number of the external mode restricting the achievable βN when the conducting wall is placed close to the plasma surface. This aspect ratio effect is confirmed when the safety factor at the plasma surface is between 4 and 5. These represent the importance of the stability of external MHD modes whose toroidal mode number is larger than 3 to determine the achievable βN.

References
2.1-1 Aiba, N., et al., "Analysis of an Aspect Ratio Effect on the Stability of External MHD Modes in Tokamaks with Newcomb Equation," to be published in J.Plasma Phys..

2.2 Role of Anomalous Transport in Neoclassical Tearing Modes

Role of anomalous transport in onset and evolution of neoclassical tearing modes (NTMs) is investigated. A key role in the evolution NTMs belongs to the radial profiles of the perturbed plasma flow, temperature and density which are determined by the conjunction of the longitudinal and cross-field transport. The influence of anomalous perpendicular heat transport and anomalous ion perpendicular viscosity on early stages of NTM evolution are studied.

Several parallel transport mechanisms competitive with anomalous cross-island heat transport in the formation of the perturbed electron and ion temperature profiles within the island are considered. The partial contributions from the plasma electron and ion temperature perturbations in the bootstrap drive of the mode and magnetic curvature effect were taken into account in construction of a generalized transport threshold model of NTMs. This model gives more favourable predictions for NTM stability and qualitatively modifies the scaling law for βonset. The anomalous perpendicular ion viscosity is shown to modify the collisionality dependence of the polarization current effect, reducing it to the low collisionality limit. In its turn a viscous contribution to the bootstrap drive of NTMs is found to be of the same order as a conventional bootstrap drive for the islands of width close to the characteristic one of the transport threshold model. A viscous contribution to the perturbed bootstrap current is destabilizing for the island rotating in the ion diamagnetic drift direction [2.2-1].

References
2.2-1 Konovalov, S.V., et al., Plasma Phys. Control. Fusion, 47, B223 (2005).

2.3 Magnetic Island Evolution in Rotating Plasma

It has been well understood that, in tokamak plasmas, magnetic islands resonant with the low q rational surface deteriorate the plasma confinement. Hence, the suppression and control of the magnetic islands is an urgent subject in a tokamak fusion research. Thus, the time evolution of the magnetic island formed at the tearing stable rational surface by the external magnetic flux perturbation in the plasma with poloidal flow is investigated numerically by using the resistive MHD model. It was found that the magnetic island growth phase is divided into four phases, 1) flow-suppressed phase, 2) rapid growth phase, 3) transient phase, and 4) Rutherford type phase. It was also found that the onset condition of this rapid growth depends on the resistivity, but does not much depend on the viscosity. On the other hand, the time constant of the rapid growth phase is almost independent on both the plasma resistivity and the viscosity. After the rapid growth phase, the island enters a transient phase, which becomes clear in the low resistivity regime. Then, the magnetic island grows slowly. This phase seems to be the Rutherford type phase [2.3-1].

References
2.3-1 Ishii, Y., et al., "Magnetic Island Evolution in Rotating Plasmas," to be published in J. Plasma Phys..

2.4 Mechanism of Rotational Stabilization of High-n Ballooning Modes

It has been clarified that ballooning modes in a shear toroidal rotating tokamak are stabilized by a countably infinite number of crossings among eigenvalues associated with ballooning modes in a static plasma. It was also found that the crossings cause energy transfer from an unstable mode to the infinite number of stable modes; such transfer works as the stabilization mechanism of the ballooning mode [2.4-1]. The method used in this research has been further explored from the view point of regularization of singular eigenfunctions of operators encountered often in plasma physics [2.4-2]. It has been confirmed that the set of regularized eigenfunctions does capture the transient behavior of the original equations of motion with singular operator for a finite time. Thus, this method will resolves the practical dif ficulties in analyzing various MHD phenomena with continuous spectra in tokamak plasmas.

References
2.4-1 Furukawa, M., et al., Phys. Rev. Lett., 94, 175001 (2005).
2.4-2 Furukawa, M., et al., Phys. Plasmas, 12, 072517 (2005).

3. Integrated Simulation
3.1 Integrated Simulation Code for Burning Plasma Analysis

Strategy of integrated modeling for burning plasmas in Japan Atomic Energy Agency is as follows: In order to simulate the burning plasma which has a complex feature with wide time and spatial scales, a simulation code cluster based on the transport code TOPICS is being developed by the integration with heating and current drive, the impurity transport, edge pedestal model, divertor model, MHD and high energy behavior model. Developed integration models are validated by fundamental researches of JT-60U experiments and the simulation based on the first principle in our strategy.

The integration of MHD stability and the transport is progressed for three phenomena with different time scale of NTMs (~τNTM~10-2(~τR), beta limits (~τAlfven) and ELMs (intermittent of (~τE and (~τAlfven). Here, τR, τAlfven and τE are the resistive skin time, the Alfven transit time and the energy confinement time, respectively. Integrated model of the NTM is produced by coupling the modified Rutherford equation with the transport equation. Integrated model of beta limits is developed by the low-n stability analysis of down streaming data from the TOPICS code. Integrated model of ELM is developed by the iterative calculation of the ideal MHD stability code MARG2D and the TOPICS code. These models are being validated by the data of JT-60 experiments and estimate the plasma performance for burning plasmas.

3.2 Transient Behaviour of SOL-Divertor Plasmas after an ELM Crash

An integrated SOL/divertor code is being developed by the JAEA for interpretation and prediction studies of the behavior of plasmas, neutrals, and impurities in the SOL/divertor region [3.2-1]. A code system consists of the 2D fluid code for plasma (SOLDOR), the neutral Monte-Carlo code (NEUT2D), the impurity Monte-Carlo code (IMPMC), and the particle simulation code (PARASOL) as shown in Fig. II.3.2-1. The physical processes of neutrals and impurities are studied using the Monte Carlo (MC) code to accomplish highly accurate simulations. The so-called divertor code, SOLDOR/NEUT2D, has the following features: 1) a high-resolution oscillation-free scheme for solving fluid equations, 2) neutral transport calculation under the condition of fine meshes, 3) successful reduction of MC noise, and 4) optimization of the massive parallel computer. As a result, our code can obtain a steady state solution within 3 ~ 4 hours even in the first run of a series of simulations, allowing the performance of an effective parameter survey. The simulation reproduces the X-point MARFE in the JT-60U. It is found that the chemically sputtered carbon at the dome causes radiation peaking near the X-point. The performance of divertor pumping in the JT-60U is evaluated based on particle balances. In regard to the design of NCT (National Centralized Tokamak, renaming to the JT-60SA Satellite Tokamak at present) divertor [3.2-1, 3.2-2], the simulation indicates that pumping efficiency is determined by the balance between the incident and back-flow fluxes into and from the exhaust chamber, which depends on the divertor geometry and operational conditions.

References
3.2-1 Kawashima, H., et al., Plasma and Fusion Res., 1, 031 (2006).
3.2-2 Kawashima, H., et al., Fusion Eng. Des., 81, 1613 (20 06).

3.3 Transient Behaviour of SOL-Divertor Plasmas after an ELM Crash

Enhanced heat flux to the divertor plates after an ELM crash in H-mode plasmas is a crucial issue for the tokamak reactor. Characteristic time of this heat flux is one of key factors of the influence on the plate. We investigate the transient behavior of SOL-divertor plasmas after an ELM crash with the use of a one-dimensional particle simulation code, PARASOL. Influence of the collisionality and the recycling rate on characteristic times of the fast-time-scale response and of the slow-time-scale response are examined. The fast time scale is further classified into the supra-thermal-electron transit time scale and the thermal-electron-transit time scale. Supra-thermal electrons supplied by an ELM crash induce the large electron heat flux Qe and the high sheath potential φ at the plate soon after the crash, while the time scale of electron temperature Te is governed by the thermal electrons. Extremely large heat transmission factor and higher φ are observed in the low collisionality regime. In the higher collisionality regime, supra-thermal electrons are thermalized and the value of φ becomes proportional to Te as usual. On the other hand, the slow-time-scale characteristic time is governed by the sound speed in the central SOL region, and is insensitive to the collisionality compared with the fast-time-scale one. The slow-time-scale phenomena are affected by the recycling condition in contrast to fast-time-scale behaviors being independent of the recycling. Peaks of particle and heat fluxes, Γ and Q, are delayed by the increase of recycling rate, though the arrival times of Γ and Q are not changed. Large recycling after the arrival of the enhanced _ makes the flow speed small in the central SOL region, and the peaks are forced to be delayed. [3.3-1]

References
3.3-1 Takizuka, T., et al., "Origin of the Various Beta Dependence of ELMy H-mode Confinement Properties," to be published in Contrib. Plasma Phys..

4. Numerical Experiment of Tokamak (NEXT)
4.1 Nonlinear Behaviors of Collisionless Double Tearing and Kink Modes

In high temperature plasmas, the collisionless effects such as the electron inertia and the electron parallel compressibility become important for the magnetic reconnection in MHD modes. Thus, the behaviors of collisionless MHD modes were investigated by gyrokinetic particle simulations. The collisionless double tearing mode (DTM) grows at the Alfven time scale due to the electron inertia, and nonlinearly induces the internal collapse when the helical flux at the magnetic axis is less than that at the outer resonant surface. It was found that, after the internal collapse, the secondary reconnection is induced by the current concentration due to the convective flow, and a new reversed shear configuration with resonant surfaces can be generated [4.1-1]. The collisionless internal kink mode was also studied in the parameter region where the effects of electron inertia and electron parallel compressibility are competitive for magnetic reconnection. Although the linear growth of the mode is dominated by the electron inertia, it was found that the growth rate can be nonlinearly accelerated due to the electron parallel compressibility proportional to the ion sound Larmor radius. The acceleration of growth is also observed in the nonlinear phase of the DTM [4.1-2].

References
4.1-1 Matsumoto, T., et al., Nucl. Fusion, 45, 1264 (2005).
4.1-2 Matsumoto, T., et al., Phys. Plasmas, 12, 092505-1-7 (2005).

4.2 Stability of Double Tearing Mode and its Effects on Current Hole Formation

In tokamak plasmas with negative central current density, so called the current hole formation can be explained by the destabilization of m=1/n=0 resistive kink MHD mode. Here, a strong reversed magnetic shear configuration has two resonant surfaces for low mode numbers, thus DTM could become unstable before the hole formation. However, it was found that the stability of the resistive kink mode, so that the current density gradient to drive the mode, does not change much after a crash by DTM, although the current profile is flatten near the minimum safety factor region [4.2-1].

After a formation of the hole, no MHD activity identified to DTM was observed in experiments, and a resultant profile with two resonant surfaces could have a good stability for DTM. It was also found that the current profile with a strong peak around an inner resonant surface, as shown in Fig.II.4.2-1, is stable for DTM [4.2-2].

References
4.2-1 Tuda, T., et al., "Roles of Double Tearing Mode on the Formation of Current Hole," to be published in J. Plasma Phys..
4.2-2 Tuda, T., et al., "Stability of Double Tearing Mode in Current Hole Configuration," to be published in J. Korean Phys. Soc..

4.3 ZF/GAM Dynamics and Ion Turbulent Transport in Reversed Shear Tokamaks

Zonal flow behavior and its effect on turbulent transport in reversed magnetic shear tokamaks were investigated by global simulations of electrostatic ion temperature gradient driven turbulence. In a high safety factor case (q0=2.2), Fig.II.4.3-1 shows that turbulent heat transport is high in a broad radial region because oscillatory zonal flows or GAMs are dominant. When q0 is reduced from 2.2 to 1.8 with keeping the other parameters unchanged, zonal flows change from the GAMs to stationary flows in the region around t he minimum q surface. As a result of the change in zonal flow behavior, the ion thermal diffusivity is reduced, as shown in Fig.II.4.3-1. This result indicates that the change of zonal flow behavior may trigger the formation of ion transport barriers in the minimum q region.

References
4.3-1 Miyato, N., et al., "Zonal Flow and GAM Dynamics and Associated Transport Characteristics in Reversed Shear Tokamaks," to be published in J. Plasma Phys..

4.4 Development of Gyrokinetic Vlasov CIP Code

A gyrokinetic simulation is an essential tool to study anomalous turbulent transport in tokamak plasmas. Although the δf Particle-In-Cell (PIC) method enabled an accurate gyrokinetic simulation of small amplitude turbulent fluctuations with ~1%, the method has a difficulty in implementing non-conservative effects such as heat and particle sources and collisions are important, which are essential in realistic long time turbulence simulations. To overcome the difficulty, a new gyrokinetic Vlasov code has been developed using the Constrained-Interpolation-Profile (CIP) method. The code is numerically stable and numerical oscillations, which have been a critical issue in the previous Vlasov simulations, are quite small. In the benchmark tests of ion temperature gradient driven (ITG) turbulence simulations, the linear growth rates, the nonlinear saturation levels, the zonal flow structures, and the conservation properties are almost the same between the PIC and CIP codes. In addition, computational costs are almost comparable between two codes. A possibility of a long time turbulence simulation was demonstrated from the viewpoints of numerical properties and a computational cost.

References
4.4-1 Idomura, Y., et al., "Comparisons of Gyrokinetic PIC and CIP Codes," Proc. 32nd EPS Pla sma Physics Conf., p1.044 (2005), to be published in J. Plasma Phys..

5. Atomic and Molecular Data

We have been compiling and producing cross section data for atomic and molecular collisions and spectral data relevant to fusion research [5-1].

Cross sections for 74 processes in collisions of electrons with N2 and N2+ have been compiled [5-2]. In tokamak fusion research, N2 gas has been injected for heat control. The cross sections are plotted as a function of the electron collision energy, and recommended cross sections are expressed by analytic expressions to facilitate practical use of the data. Figure II.5-1 shows an example of the complied cross section data. The data have been included in Japanese Evaluated Atomic and Molecular Data Library (JEAMDL), which is available through the Web at the URL http://www-jt60.naka.jaea.go.jp/english/JEAMDL/index.html. As to the data production, cross sections for various carbon containing molecules, which are produced from carbon-based plasma-facing materials, have been measured [5-3,4]. Charge transfer cross sections of impurity ions produced from the plasma-facing materials: Be, B, C, Cr, Fe and Ni ions, with gaseous atoms and molecules have also been measured [5-5]. Cross sections of state-selective electron capture in collisions of C+4 ions with H* (n=2) atoms have been calculated using a molecular-bases close-coupling method [5-6].

Regarding spectral data, wavelengths, energy levels, oscillator strengths, transition probabilities and ionization energies have been critically compiled for tungsten [5-7, 5-8] and gallium [5-9]. Tungsten is one of the candidate plasma-facing materials for future fusion devises, and gallium is a candidate material for liquid divertors.

References
5-1 Kubo, H., et al., "Atomic and Molecular Data Activities for Fusion Research at JAERI," to be published in J. Plasma Fusion Res..
5-2 Tabata, T., et al., Atomic Data and Nuclear Data Tables, 92, 375 (2006).
5-3 Makochekanwa, C., et al., J. Phys. Chem., 124, 024323 (2006).
5-4 Kusakabe, T., et al., "Cross Sections of Charge Transfer by Slow Doubly-Charged Carbon Ions from Various Carbon Containing Molecules," to be published in J. Plasma Fusion Res..
5-5 Imai, M., et al., "Production and Compilation of Charge Changing Cross Sections of Ion-Atom and Ion-Molecule Collisions," to be published in J. Plasma Fusion Res..
5-6 Shimakura, N., et al., "Electron Capture Processes in Low-Energy Collisions of C4+ Ions with Excited H Atoms Using Molecular-Bases Close-Coupling Method," to be published in J. Plasma Fusion Res..
5-7 Kramida, A.E., et al., "Compilation of Wavelengths, Energy Levels, and Transition Probabilities for W I and W II," to be published in J. Phys. Chem. Ref. Data.
5-8 Kramida, A.E., et al., "Compilation of Wavelengths and Energy Levels of Tungsten, W III through W LXXIV," to be published in J. Plasma Fusion Res..
5-9 Shirai, T., et al., "Spectral Data for Gallium: Ga I through Ga XXXI," to be published in J. Phys. Chem. Ref. Data.