L. Kos

Extension of the Bissell–Johnson plasma-sheath model for application to fusion-relevant and general plasmas

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&J) [Phys. Fluids 30, 779 (1987)] 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&L) [Phys. Rev. 34, 876 (1929)] 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 repre...

Cutting optimization with variable-sized stock and inventory status data

Many production environments require economical cutting of one-dimensional items according to bills of materials from objects of several standard lengths. However, even with optimized cutting substantial trim loss may occur. This trim loss should not be regarded as waste. It is returned to store and can be reused in future optimizations. Optimization of packing linear items into standard lengths is presented for items that cannot be packed into available lengths from inventory status data. The core of the proposed optimization tackles the variablesized bin packing problem (VBPP). The article presents a hybrid genetic algorithm that packs items into both available objects from the inventory and variablesized objects from the stock. The algorithm tries to minimize waste. Large trimloss items are returned as remnants to the inventory for subsequent optimizations.

Comprehensive kinetic analysis of the plasma-wall transition layer in a strongly tilted magnetic field

The magnetized plasma-wall transition (MPWT) layer at the presence of the obliquity of the magnetic field to the wall consists of three sub-layers: the Debye sheath (DS), the magnetic pre-sheath (MPS), and the collisional pre-sheath (CPS) with characteristic lengths λD (electron Debye length), ρi (ion gyro-radius), and l (the smallest relevant collision length), respectively. Tokamak plasmas are usually assumed to have the ordering λD≪ρi≪l, when the above-mentioned sub-layers can be distinctly distinguished. In the limits of eDm(λD/ρi)→0 and emc(ρi/l)→0 (“asymptotic three-scale (A3S) limits”), these sub-layers are precisely defined. Using the smallness of the tilting angle of the magnetic field to the wall, the ion distribution functions are found for three sub-regions in the analytic form. The equations and characteristic length-scales governing the transition (intermediate) regions between the neighboring sub-layers (CPS – MPS and MPS – DS) are derived, allowing to avoid the singularities arising from t...

The ionization length in plasmas with finite temperature ion sources

The ionization length is an important quantity which up to now has been precisely determined only in plasmas which assume that the ions are born at rest, i.e., in discharges known as “cold ion-source” plasmas. Presented here are the results of our calculations of the ionization lengths in plasmas with an arbitrary ion source temperature. Harrison and Thompson (H&T) [Proc. Phys. Soc. 74, 145 (1959)] found the values of this quantity for the cases of several ion strength potential profiles in the well-known Tonks–Langmuir [Phys. Rev. 34, 876 (1929)] discharge, which is characterized by “cold” ion temperature. This scenario is also known as the “singular” ion-source discharge. The H&T analytic result covers cases of ion sources proportional to exp(βΦ) with Φ the normalized plasma potential and β=0,1,2 values, which correspond to particular physical scenarios. Many years following H&T’s work, Bissell and Johnson (B&J) [Phys. Fluids 30, 779 (1987)] developed a model with the so-called “warm” ion-source tempera...

Potential profile near singularity point in kinetic Tonks-Langmuir discharges as a function of the ion sources temperature

A plasma–sheath transition analysis requires a reliable mathematical expression for the plasma potential profile Φ(x) near the sheath edge xs in the limit ɛ≡λD/l=0 (where λD is the Debye length and l is a proper characteristic length of the discharge). Such expressions have been explicitly calculated for the fluid model and the singular (cold ion source) kinetic model, where exact analytic solutions for plasma equation (ɛ=0) are known, but not for the regular (warm ion source) kinetic model, where no analytic solution of the plasma equation has ever been obtained. For the latter case, Riemann [J. Phys. D: Appl. Phys. 24, 493 (1991)] only predicted a general formula assuming relatively high ion-source temperatures, i.e., much higher than the plasma-sheath potential drop. Riemann’s formula, however, according to him, never was confirmed in explicit solutions of particular models (e.g., that of Bissell and Johnson [Phys. Fluids 30, 779 (1987)] and Scheuer and Emmert [Phys. Fluids 31, 3645 (1988)]) since “the...

A unified analysis of plasma-sheath transition in the Tonks-Langmuir model with warm ion source

The paper presents a comprehensive kinetic theory of the famous Tonks–Langmuir model of a plane symmetric discharge, taking into account the thermal motion of ion source particles. The ion kinetics is governed by the ionization of neutrals at electron impacts. The plasma consisting of Boltzmann distributed electrons and singly charged ions is in contact with the absorbing negative wall. The derivations are performed in the frame of the “asymptotic two-scale” approximation, when the ionization mean-free path Li is much larger than the electron Debye length λD. In the limit (λD/Li)→0, the plasma-wall transition (PWT) layer can be split into two sublayers: a quasineutral presheath (PS) (with the scale-length Li) and the Debye sheath (DS) (with the scale λD). Such a subdivision of the PWT layer allows to investigate these sublayers separately and simplify the analysis of the influence of the ion source thermal motion (this has been neglected in the major part of publications up to now). The uniform descriptio...

Unified Bohm criterion

Recent decades have seen research into the conditions necessary for the formation of the monotonic potential shape in the sheath, appearing at the plasma boundaries like walls, in fluid, and kinetic approximations separately. Although either of these approaches yields a formulation commonly known as the much-acclaimed Bohm criterion (BC), the respective results involve essentially different physical quantities that describe the ion gas behavior. In the fluid approach, such a quantity is clearly identified as the ion directional velocity. In the kinetic approach, the ion behavior is formulated via a quantity (the squared inverse velocity averaged by the ion distribution function) without any clear physical significance, which is, moreover, impractical. In the present paper, we try to explain this difference by deriving a condition called here the Unified Bohm Criterion, which combines an advanced fluid model with an upgraded explicit kinetic formula in a new form of the BC. By introducing a generalized polytropic coefficient function, the unified BC can be interpreted in a form that holds, irrespective of whether the ions are described kinetically or in the fluid approximation.

Closure of the hierarchy of fluid equations by means of the polytropic‐coefficient function (PCF)

The continuity and momentum equations of a fluid plasma component may be viewed as four scalar evolution equations for the four scalar fluid variables n(x,t) (density) and u(x,t) (fluid velocity), which are zeroth‐ and first order velocity moments of the velocity distribution function (VDF). However, the momentum equation in addition contains the gradient of the pressure p(x,t), which is a second‐order velocity moment for which another equation, the “closure equation”, is needed. In the present work, closure by means of the polytropic‐coefficient function (PCF) is discussed which, by analogy with the well‐known polytropic coefficient (also called the “polytropic index” or “polytropic exponent”) in macroscopic thermodynamic systems, is formally defined by γ(x,t) = (nDp/Dt)(pDn/Dt) = (n/p)(Dp/Dn), with D/Dt = ∂/∂t+u⋅∂/∂x, which amounts to the closure equation if γ(x,t) is known. In fluid problems, however, the PCF is usually unknown and hence must be assumed or guessed, but in kinetic problems it can...

Euforia Integrated Visualization

ITER is the next generation of fusion devices and is intended to demonstrate the scientific and technical feasibility of fusion as a sustainable energy source for the future. To exploit the full potential of the device and to guarantee optimal operation for the device, a high degree of physics modelling and simulation is needed. The European project Euforia contributes to the development of a platform which enables the coupling of different physics codes. The aim of this platform is to simulate the different coupled phenomena which take place in a tokamak at different space and time scales. In particular, the visualization is an issue when the data are generated by codes developed at different places by different people. This paper presents a set of unified and open source visualization tools that have been integrated into this framework. Some of these tools are used for the post-processing purpose, others are directly integrated into the workflow to allow visualization and monitoring of the results during its execution.

Visualization support for code development in EUROfusion integrated modelling

The need for diverse visualization tools within integrated modelling is based on the requirement of physics codes to be coupled under the tailored database model from which these visualization tools read data. We discuss these visualization requirements for fusion integrated modeling, and the existing visualization tools developed within the scope of the European Integrated Modelling (EU IM) framework. The datastructure model provides fundamental description for data exchange between codes that are developed in a variety of programming languages. From this description in XML schema definition one can generate required language interfaces and database access layer routines. The persistent storage database has to efficiently support codes running on HPC with memory caching mechanisms. Visualization tools can be treated as one of the codes running in-situ or as post-process. The complexity of the fusion datastructure and several visualization tools requires that scientists describe standard visualizations in XML schema directly in order to instruct visualization tools what are meaningful visualizations available in the very database. Besides standard visualizations there are specific visualizations that cannot be easily described (as mapping or axes linking) without algorithm. For the custom visualizations we provide Python snippets collected in user-shared library. Interfaces to scientific workflow engine Kepler for visualization with VisIt visualization tool and Matplotlib are presented.