In this paper the general problem of linking fluid and kinetic plasma parameters, with special attention devoted to the plasma boundaries where, due to strong deviations from thermodynamic equilibrium, there are intrinsic difficulties regarding the closure of the hydrodynamic equations, is considered. This problem is demonstrated by means of two examples for which the solutions of the kinetic equations are known. These examples are the collision-free Tonks-Langmuir model [Phys. Rev. 34, 876 (1929)] and Riemann’s presheath model [Phys. Fluids 24, 2163 (1981)] dominated by charge-exchange collisions. It is found that in the vicinity of the sheath edge the “polytropic” coefficient γ(x) shows an unexpected behavior that contradicts the commonly used hydrodynamic approaches assuming γ=const. In spite of all differences, the two models investigated exhibit quite similar behavior of the hydrodynamic quantities and of the polytropic coefficient in the presheath and sheath regions. This rises to hopes that the results presented in this paper can be generalized to models characterizing other physical scenarios of plasma production and confinement. In particular, the basic findings presented here will, in suitably adopted form, be of importance, e.g., in properly formulating boundary conditions for fluid codes simulating bounded plasmas.

In a recent paper by Kuhn et al. [Phys. Plasmas 13, 013503 (2006)], it has been demonstrated that in a plasma, the polytropic coefficient γ is a spatially varying quantity rather than a global constant as usually assumed in fluid theory. Assuming cold ion sources and using the asymptotic two-scale approximation (in which the ratio of the Debye length over the characteristic presheath length, ε, is set to zero), it was found that the γ profile exhibits a sharp peak (with values roughly between 6 and 8) at the plasma-sheath boundary. In the present paper, it is shown that in a finite ε approach, this sharp peak is smoothed to a regular maximum, which for increasing ε (e.g., decreasing plasma density) decreases and finally disappears. In any case, assumptions like γ=1 and/or γ=3, which are customarily encountered in the context of fluid approaches, are disproved. Although the present results were obtained for collisionless plasmas, it is reasonable to assume that the behavior uncovered holds qualitatively for any plasma with cold ion sources. In addition to calculating time-independent theoretical solutions by means of an analytic-numerical approach, we primarily employ particle-in-cell (PIC) computer simulations, which intrinsically represent a time-dependent approach. It is confirmed that, although extreme care is required to separate physical from numerical effects, the PIC simulation method is a highly suitable tool also for future investigations of more demanding physical scenarios assuming, e.g., warm ion sources and other more complex aspects that cannot be treated realistically by analytic-numerical means alone. In addition, the extension from time-independent analytic-numerical calculation to time-dependent simulation permits us also to investigate the effect of collective plasma oscillations on the ion velocity distribution function (VDF). Although the ion VDFs obtained in our PIC simulations visibly differ in some details from the time-independent theoretical ones, the related ion-temperature and γ profiles turn out to fit their theoretical counterparts very well over significant parameter ranges. Hence, the two methods may to some extent be applied as alternative ones in future investigations on the plasma-sheath transition.

The positive ion temperature effect in magnetized electronegative plasma sheath with two species of positive ions Phys.This article presents an approach to solving a special Fredholm-type integral equation of the first kind with a particular kernel containing a modified Bessel function for applications in plasma physics. From the physical point of view, the problem was defined by Bissell and Johnson ͑B&J͒ ͓Phys. Fluids 30, 779 ͑1987͔͒ as a task to find the potential profile and the ion velocity distribution function in a plane-parallel discharge with a Maxwellian ion source. The B&J model is a generalization of the well-known Tonks-Langmuir ͑T&L͒ ͓Phys. Rev. 34, 876 ͑1929͔͒ discharge model characterized by a "cold" ion source. Unlike the T&L model, which can be readily solved analytically, attempts to solve the B&J model with a "warm" ion source have been done only numerically. However, the validity of numerical solutions up to date remains constrained to a rather limited range of a crucial independent parameter of the B&J integral equation, which mathematically is the width of a Gaussian distribution and physically represents the ion temperature. It was solved only for moderately warm ion sources. This paper presents the exact numerical solution of the B&J model, which is valid without any restriction regarding the above-mentioned parameter. It is shown that the ion temperature is very different from the temperature of the ion source. The new results with high-temperature ion sources are not only of particular importance for understanding and describing the plasma-sheath boundary in fusion plasmas, but are of considerable interest for discharge problems in general. The eigenvalue of the problem, found analytically by Harrison and Thompson ͓Proc. Phys. Soc. 74, 145 ͑1959͔͒ for the particular case of a cold ion source, is here extended to arbitrary ion-source temperatures.