Criteria useful for the investigation of the stability of a system of charged particles are derived from the Boltzmann equation in the small m/e limit. These criteria are obtained from the examination of the variation of the energy due to a perturbation, when subject to the general constraint that all regular, time‐independent constants of the motion (including the energy) have their equilibrium values.The first‐order variation of the energy vanishes trivially, and the second‐order variation yields a quadratic form in the displacement variable ξ (which may be introduced because of the well‐known properties of this limit). The positive‐definiteness of this form is a sufficient condition for stability.Several theorems are stated comparing stability under the present theory with that under conventional hydromagnetic fluid theories where heat flow along magnetic lines of force is neglected. Generalizations can be made to systems where the effect of collisions is included.

On the Stability of Plasma in Static Equilibrium

The area of energetic particle (EP) physics in fusion research has been actively and extensively researched in recent decades. The progress achieved in advancing and understanding EP physics has been substantial since the last comprehensive review on this topic by Heidbrink and Sadler (1994 Nucl. Fusion 34 535). That review coincided with the start of deuterium?tritium (DT) experiments on the Tokamak Fusion Test Reactor (TFTR) and full scale fusion alphas physics studies.Fusion research in recent years has been influenced by EP physics in many ways including the limitations imposed by the ?sea? of Alfv?n eigenmodes (AEs), in particular by the toroidicity-induced AE (TAE) modes and reversed shear AEs (RSAEs). In the present paper we attempt a broad review of the progress that has been made in EP physics in tokamaks and spherical tori since the first DT experiments on TFTR and JET (Joint European Torus), including stellarator/helical devices. Introductory discussions on the basic ingredients of EP physics, i.e., particle orbits in STs, fundamental diagnostic techniques of EPs and instabilities, wave particle resonances and others, are given to help understanding of the advanced topics of EP physics. At the end we cover important and interesting physics issues related to the burning plasma experiments such as ITER (International Thermonuclear Experimental Reactor).

Energetic particle physics in fusion research in preparation for burning plasma experiments

In the first four years of the LHD experiment, several encouraging results have emerged, the most significant of which is that MHD stability and good transport are compatible in the inward shifted axis configuration. The observed energy confinement at this optimal configuration is consistent with ISS95 scaling with an enhancement factor of 1.5. The confinement enhancement over the smaller heliotron devices is attributed to the high edge temperature. We find that the plasma with an average beta of 3% is stable in this configuration, even though the theoretical stability conditions of Mercier modes and pressure driven low-n modes are violated. In the low density discharges heated by NBI and ECR, internal transport barrier (ITB) and an associated high central temperature (> 10 keV) are seen. The radial electric field measured in these discharges is positive (electron root) and expected to play a key role in the formation of the ITB. The positive electric field is also found to suppress the ion thermal diffusivity as predicted by neoclassical transport theory. The width of the externally imposed island (n/m = 1/1) is found to decrease when the plasma is collisionless with finite beta and increase when the plasma is collisional. The ICRF heating in LHD is successful and a high energy tail (up to 500 keV) has been detected for minority ion heating, demonstrating good confinement of the high energy particles. The magnetic field line structure unique to the heliotron edge configuration is confirmed by measuring the plasma density and temperature profiles on the divertor plate. A long pulse (2 min) discharge with an ICRF power of 0.4 MW has been demonstrated and the energy confinement characteristics are almost the same as those in short pulse discharges.

https://iopscience.iop.org/article/10.1088/0029-5515/43/12/013/pdf

Recent advances in the LHD experiment

Recent experimental results in the Large Helical Device have indicated that a large pressure gradient can be formed beyond the stability criterion for the Mercier (high-n) mode. While the stability against an interchange mode is violated in the inward-shifted configuration due to an enhancement of the magnetic hill, the neoclassical transport and confinement of high-energy particle are, in contrast, improved by this inward shift. Mitigation of the unfavourable effects of MHD instability has led to a significant extension of the operational regime. Achievements of the stored energy of I MJ and the volume-averaged beta of 3% are representative results from this finding. A confinement enhancement factor above the international stellarator scaling ISS95 is also maintained around 1.5 towards a volume-averaged beta, (beta), of 3%. Configuration studies on confinement and MHD characteristics emphasize the superiority of the inward-shifted geometry to other geometries. The emergence of coherent modes appears to be consistent with the linear ideal MHD theory; however, the inward-shifted configuration has reduced heat transport in spite of a larger amplitude of magnetic fluctuation than the outward-shifted configuration. While neoclassical helical ripple transport becomes visible for the outward-shifted configuration in the collisionless regime, the inward-shifted configuration does not show any degradation of confinement deep in the collisionless regime (nu* < 0.1). The distinguished characteristics observed in the inward-shifted configuration help in creating a new perspective of MHD stability and related transport in net current-free plasmas. The first result of the pellet launching at different locations is also reported.

Configuration flexibility and extended regimes in Large Helical Device

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

The linear MHD stability problem for 3D plasmas with noninteracting hot electron layers and nested magnetic flux surfaces in the absence of dissipation mechanisms is formulated in variational form. Boozer magnetic coordinates are extended to anisotropic pressure plasmas with nested magnetic flux surfaces for this purpose. A modified energy principle in which the hot electron species current is imperturbable is considered. Plasma incompressibility is imposed and thus a simplified kinetic energy is employed to determine the conditions of marginal stability. The vacuum region surrounding the plasma is treated as a shearless, pressureless and massless pseudoplasma. Fourier decomposition of the perturbations in the periodic angular variables of the magnetic coordinates is applied and a finite element discretization scheme reduces the stability problem to a special block pentadiagonal matrix eigenvalue equation which is amenable to solution with an inverse vector iteration technique. The formulation described is useful to evaluate the linear MHD stability properties of 3D plasma configurations with rigid hot electrons to global internal and external modes. The conditions for linear MHD stability to local modes are determined with the application of the ballooning mode representation to the internal plasma potential energy to derive the corresponding ballooning mode equation. The asymptotic analysis of this equation yields the Mercier criterion.

http://iopscience.iop.org/article/10.1088/0741-3335/34/6/009/pdf

A Cooper

Papers

Variational Formulation of the Linear Mhd Stability of 3d-Plasmas with Noninteracting Hot-Electrons