Vitalii D. Shafranov

Reviews of plasma physics

Mechanisms of transverse conductivity and generation of self-consistent electric fields in strongly ionized magnetized plasma V. Rozhansky. 1. Introduction.- 2. Conductivity tensor in partially ionized plasma.- 3. Main mechanisms of perpendicular conductivity in fully ionized plasma.- 4. Acceleration of plasma clouds in an inhomogeneous magnetic field.- 5. Alfven conductivity.- 6. Perpendicular viscosity, radial current, and radial electric field in an infinite cylinder.- 7. Current systems in front of a biased electrode (flush-mounted probe) and spot of emission.- 8. Currents in the vicinity of a biased electrode that is smaller than the ion gyroradius.- 9. Neoclassical perpendicular conductivity in a tokamak.- 10. Transverse conductivity in a reversed field pinch.- 11. Modeling of electric field and currents in the tokamak edge plasma.- 12. Mechanisms of anomalous perpendicular viscosity and viscosity-driven currents.- 13. Transverse conductivity in a stochastic magnetic field.- 14. Electric fields generated in the shielding layer between hot plasma and a solid state.-- Correlations and anomalous transport models O.G. Bakunin. 1. Introduction.- 2. Turbulent diffusion and transport.- 3. Non-local effects and diffusion equations.- 4. The Corrsin conjecture.- 5. Effects of seed diffusivity.- 6. The diffusive tracer equation and averaging.- 7. The quasi-linear approximation.- 8. The diffusive renormalization.- 9. Anomalous transport and convective cells.- 10. Stochastic instability and transport.- 11. Fractal conceptions and turbulence.- 12. Percolation and scalings.- 13. Percolation and turbulent transport scalings.- 14. The temporal hierarchy of scales and correlations.- 15. The stochastic magnetic field and percolation transport.- 16. Percolation in drift flows.- 17. Multiscale flows.- 18. Subdiffusion and traps.- 19. Continuous time random walks.- 20. Fractional differential equations and scalings.- 21. Correlation and phase-space.- 22. Conclusion.

Plasma equilibrium in a Tokamak

The paper summarizes the basic information on the equilibrium of a toroidal plasma column in systems of the Tokamak type. It considers the methods of maintaining a plasma in equilibrium with the help of a conducting casing, an external maintaining field and the iron core of a transformer. Attention is paid to the role of the inhomogeneity of the maintaining field. It is shown in particular how the shape of the column cross-section depends on the curvature of the lines of force of the maintaining field. For the case (which has practical importance) weak asymmetry of the field distribution in the transverse cross-section, this paper describes a uniform method of consideration, which takes into account the influence of different factors on the equilibrium position of the column. This method is used for calculating plasma equilibrium in a Tokamak model with a conducting casing. Account is here taken of the effect of gaps in the casing and of finite electrical conductivity. Some cases of plasma equilibrium which are outside the standard Tokamak scheme are also considered, such as equilibrium in a conducting shell having the shape of a racetrack, equilibrium where the whole current is transferred by relativistic runaway electrons and equilibrium at high plasma pressure βI ~R/a.

Improved alpha-particle confinement in stellarators with poloidally closed contours of the magnetic field strength

The possibility of fulfilling the condition of poloidal closure of the contours of the second adiabatic invariant for all reflected particles is studied for systems with poloidally closed contours of the magnetic field B on the magnetic surfaces through computational stellarator optimization. It is shown that by adjusting the geometry this is possible in a major fraction of the plasma volume. The most salient characteristic (as compared to previous quasi-isodynamic configurations) is a magnetic axis whose curvature vanishes in all cross sections with an extremum of B on the magnetic axis and renders possible a three-dimensional structure of B with unprecedentedly high collisionless α-particle confinement.

Stable current profile in a stellarator with shear

The authors have studied the stability of helical modes in a current carrying stellarator, ignoring the curvature of the axis and plasma pressure. They show that it is possible to fully stabilize a current column with a given arbitrary current value against both ideal and tearing modes by superimposing an external stellarator rotational transform with a sufficiently large value of its difference along the radius, |Δth| = |th(a) − th(0)|, in configurations of two types, namely with the total rotational transform increasing or decreasing with the radius. The value of the difference |Δth| needed for stabilization increases with the value of the total current.

EQUILIBRIUM OF A PLASMA COLUMN THAT DOES NOT POSSESS AXIAL SYMMETRY

We calculate the equilibrium position of a plasma column of round cross section inside an ideally conducting housing for the case when the axis of the plasma column (or the housing) represents a space curve. For calculation we use the model of a plasma column with a sharp boundary and a surface current. Solution is presented in the form of a Fourier-series expansion over the are length of the column axis. For the case when the curvature is a smooth function of the are length, simple formulas have been obtained for displacement of the column axis with respect to the axis of the housing, the validity of which, for the case when the housing has the shape of a racetrack, is confirmed by numerical calculation of the displacement by general formulas. Use of the same method of Fourier-series expansion enables one to solve even the problem of the distortion of the plasma column caused by transverse slits in the housing that are necessary for the introduction of the electric field into the discharge volume where plasma is created. Calculation results are given of distortions by slits in a toroidal housing.

Hydromagnetic stability of a plasma in stellarators

It is shown that, with the help of the sufficient criterion for hydromagnetic stability of a plasma proposed by Solovev, It is possible to calculate for stellarators a non-trivial plasma pressure limit at which there is hydromagnetic plasma stability with respect to arbitrary perturbations. In the simplest case of a two-turn stellarator with a circular magnetic axis this limit has the form β ≤ (δU0/U0)(ι/2π)2.

Mixing of plasma columns in a tokamak

The possibility of a new electrotechnical method of plasma heating in tokamak-like toroidal systems is discussed. The method is based on the use of the external poloidal magnetic field energy which was spent on the creation of equilibrium systems of a few separate toroidal columns with parallel currents. The energy of interaction of the separate currents transforms into plasma particle energy during the process of mixing of the plasma columns to form one column. Dissipation takes place because of the generation of a current layer, the reconnection of magnetic field lines and the change of the topological connection of the magnetic surfaces. Qualitative estimates are given of the power of heating and of final-state parameters.

Pressure insensitivity of a high-β quasi-isodynamic stellarator

The sensitivity of a high-β quasi-isodynamic stellarator equilibrium with respect to changes of the plasma pressure is investigated. It is shown that a plasma boundary near to the high-β boundary exists with the property that the magnetohydrodynamic and drift-kinetic properties of the associated low-β plasma are similar.

Plasma stability in Heliac-like quasi-helically symmetric stellarators

An attempt is made to find an optimal quasi-helically symmetric (QHS) four period magnetic configuration with a Heliac-like structure of the magnetic surface cross-sections. This means that the cross-sections rotate in phase with the magnetic axis principal normal in contrast to the Helias-like systems where the magnetic surface cross-sections rotate half as fast as the principal normal. The equilibrium and local stability are studied with the VMEC and TERPSICHORE codes. It is shown that, in the configuration found, the equilibrium beta limit is about 9%, the Mercier beta limit is about 4% and the ballooning modes restrict the beta value to 1%. The accuracy of the fulfilment of the QHS condition is given by X~3 (i.e. the largest amplitude of the magnetic field strength B harmonic that violates the quasi-symmetry is a third that of the main helical harmonic of B in Boozer co-ordinates)

Quasi-helical symmetry at finite aspect ratio

Computational stellarator optimization is used to create a configuration which is quasi-helically symmetric at finite aspect ratio. For the aspect ratio per period chosen (∼2), this procedure results in benign properties throughout the plasma volume.