Theories of the electric field effect on toroidal plasma confinement are reviewed with the emphasis placed on recent progress in the areas of anomalous transport, structural formation and bifurcation, research which has been motivated by the discovery of improved confinement. Topics include single-particle physics, such as particle orbits or collisional transport, turbulent transport, transport matrix, structural formation and dynamics, bifurcation, and improved confinement.

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The role of the electric field in confinement

Wendelstein 7-AS was the first modular stellarator device to test some basic elements of stellarator optimization: a reduced Shafranov shift and improved stability properties resulted in β-values up to 3.4% (at 0.9 T). This operational limit was determined by power balance and impurity radiation without noticeable degradation of stability or a violent collapse. The partial reduction of neoclassical transport could be verified in agreement with calculations indicating the feasibility of the concept of drift optimization. A full neoclassical optimization, in particular a minimization of the bootstrap current was beyond the scope of this project. A variety of non-ohmic heating and current drive scenarios by ICRH, NBI and in particular, ECRH were tested and compared successfully with their theoretical predictions. Besides, new heating schemes of overdense plasmas were developed such as RF mode conversion heating—Ordinary mode, Extraordinary mode, Bernstein-wave (OXB) heating—or 2nd harmonic O-mode (O2) heating. The energy confinement was about a factor of 2 above ISS95 without degradation near operational boundaries. A number of improved confinement regimes such as core electron-root confinement with central Te ≤ 7 keV and regimes with strongly sheared radial electric field at the plasma edge resulting in Ti ≤ 1.7 keV were obtained. As the first non-tokamak device, W7-AS achieved the H-mode and moreover developed a high density H-mode regime (HDH) with strongly reduced impurity confinement that allowed quasi-steady-state operation (τ ≈ 65 · τE) at densities (at 2.5 T). The first island divertor was tested successfully and operated with stable partial detachment in agreement with numerical simulations. With these results W7-AS laid the physics background for operation of an optimized low-shear steady-state stellarator.

Major results from the stellarator Wendelstein 7-AS (Review Article)

The three-dimensional (3D) magnetohydrodynamic (MHD) equilibrium solver HINT is improved as a new HINT code, 'HINT2' having various useful features by introducing modern computing techniques. By using the algorithm inherent to HINT2, the treatment of the peripheral plasma corresponding to the experimental results is addressed in the Large Helical Device (LHD).

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Development and application of HINT2 to helical system plasmas

An overview is given of physics differences between stellarators and tokamaks, including magnetohydrodynamic equilibrium, stability, fast-ion physics, plasma rotation, neoclassical and turbulent transport and edge physics. Regarding microinstabilities, it is shown that the ordinary, collisionless trapped-electron mode is stable in large parts of parameter space in stellarators that have been designed so that the parallel adiabatic invariant decreases with radius. Also, the first global, electromagnetic, gyrokinetic stability calculations performed for Wendelstein 7-X suggest that kinetic ballooning modes are more stable than in a typical tokamak.

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Stellarator and tokamak plasmas: a comparison

In the Large Helical Device (LHD), the highest operational averaged beta value has been expanded from 3.2% to 4% in the last 2 years by increasing the heating capability and exploring a new magnetic configuration with a high aspect ratio. Although the magneto-hydrodynamic (MHD) stability properties are considered to be unfavourable in the new high aspect configuration, the heating efficiency due to neutral beams and the transport properties are expected to be favourable in a high-beta range. In order to clarify the effect of the global ideal MHD unstable mode on the operational regimes in helical systems, especially the beta gradients in the peripheral region and the beta value, the MHD analysis and the transport analysis are performed in a high-beta range of up to 4% in LHD. In a high-beta range of more than 3%, the maxima of the observed thermal pressure gradients at a low order rational magnetic surface in the peripheral region are marginally unstable to the low-mode-number ideal MHD instability. Though a gradual degradation of the local transport in the region has been observed as beta increases, a disruptive degradation of the local transport does not appear in the beta range up to 4%.

https://iopscience.iop.org/article/10.1088/0029-5515/45/11/004/pdf

Effects of global MHD instability on operational high beta-regime in LHD

It is shown that solitary and shock structures can appear in tokamaks, when a poloidal flow velocity becomes close to e c s , where c s is a sound velocity and e is a small inverse aspect ratio. To describe asymptotic behavior of the flow velocity v p , a KdV (Korteweg de Vries)-Burgers equation with a sinusoidal force term is derived from the two-fluid MHD equations with small resistivity and parallel viscosity. A solitary flow velocity structure appears inside of the torus (at the poloidal angle θ=π) in the non-dissipative, dispersive case. The supersonic flow ( v p > e c s ) parallel to the poloidal magnetic field and the subsonic anti-parallel flow ( v p < e c s ) are stable. In both cases the solitary density perturbation is negative. For the non-dispersive, dissipative case or the one-fluid MHD limit the forced Burgers equation becomes appropriate to describe the flow dynamics. This case corresponds to an appearance of shock structure. By solving an initial value problem it is shown that the shock p...

Solitary and Shock Structures Induced by Poloidal Flow in Tokamaks

Recently, dramatic progress has been achieved in the study of helical systems with high-beta experiments. Discharges with more than 3% beta plasmas have been achieved in Large Helical Device (LHD) and Wendelstein 7-AS (W7-AS). Although magnetohydrodynamic (MHD) instabilities affect local pressure gradients, the global transport property does not seem to limit the achieved beta value in either device. We summarize the LHD high-beta properties in MHD stability, equilibrium, and transport, and we show the relationship between the experimentally achieved parameters and theoretical predictions. We contrast the LHD results with the W7-AS high-beta properties. In both devices, stationary discharges in the definitely MHD unstable region have not been observed. We mention the key issue for achievement of the beta values >5%.

Progress of High-Beta Experiments in Stellarator/Heliotron

Results of numerical analyses of edge and divertor magnetic field structure are presented for the Large Helical Device (LHD). For the analyses, a field line tracing code has been developed to include the finite beta effect and cross-field particle diffusion, simulating the anomalous transport. Precise analyses of the helical divertor under finite beta conditions were performed for the first time. It is found that, in the magnetic axis shift operation from 3.6 m to 3.9 m, striking points of divertor legs shift more than 250 mm on target plates. Meanwhile very small shift less than 15 mm can be seen with increased beta up to ∼2%, in spite of the large magnetic axis shift more than 300 mm. Although the shift of striking point is small under finite beta conditions, distortion in the fine structure of a divertor leg is taken place, which may affect the edge particle transport during high beta discharges.

Characteristics of Edge Magnetic Field Structure in LHD Heliotron

A core density collapse (CDC) phenomenon is a rapid collapse events observed in super dense core (SDC) plasma with internal diffusion barrier (IDB) in the Large Helical Device (LHD). By CDC, the central beta is decreased by up to 50%. The collapse starts from the edge region of the plasma. CDCs appear with plasma parameters where the high–n ballooning modes are unstable at ϱ ∼ 0.8. With less collisional conditions, m = 1 type oscillations are observed with similar beta profile. The origin of the m = 1 oscillations is not clarified (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Density Collapse Events Observed in the Large Helical Device

Radial electric field, E r , is obtained by imposing the ambipolar condition or solving the electric field diffusion equation in the transport code for Heliotron E. The latter approach usually gives a little smaller | E r | than the former one. In the context of neoclassical transport theory toroidal and poloidal ion flow are also calculated and compared consistently with measured ion flow velocities at r / a ≃0.7∼0.8 in Heliotron E ECRH experiments. Positive electric field is realized in the low density regime \(\bar{n}\lesssim1\times10^{19}\) m -3 with a high heating power. E r changes from positive to negative according to the increase of the density. It is pointed out that the edge anomalous transport is essential to obtain the positive E r in the stationary state, which is different from Kovrizhnykh's conjecture that E r is negative or small positive within the neoclassical diffusion theory.

Kiyomasa Watanabe

Papers

Solitary and Shock Structures Induced by Poloidal Flow in Tokamaks

Progress of High-Beta Experiments in Stellarator/Heliotron

Characteristics of Edge Magnetic Field Structure in LHD Heliotron

Density Collapse Events Observed in the Large Helical Device

Transport Study for Radial Electric Field and Poloidal Ion Flow in Heliotron E