William D’haeseleer

Fuzzy-Set Decision Support for a Belgian Long-Term Sustainable Energy Strategy

1 Expertise Group ‘Society and Policy Support’ (SPS), Belgian Nuclear Research Centre (SCK•CEN), Boeretang, Belgium elaes@sckcen.be, gmeskens@sckcen.be, druan@sckcen.be 2 Faculty of Information Technology, University of Technology, Sydney (UTS), Broadway, NSW, Australia jielu@it.ets.edu.au, zhangg@it.ets.edu.au, fengjiew@it.ets.edu.au 3 Energy Institute, Faculty of Applied Mechanics and Energy Conversion, University of Leuven (KULeuven), Heverlee, Belgium william.dhaeseleer@mech.kuleuven.be 4 Brussels EU Chapter, Club of Rome (CoR-EU) raoul.weiler@skynet.be

The Clebsch-Type Coordinate Systems

The idea of a stream function (or flow function) is borrowed from hydrodynamic theory (see, e.g., Milne-Thomson (1968)). A stream function is any function that is constant along a stream line. In plasma physics, we are interested in stream functions for the magnetic field. Hence, in our context, a stream function is any function that is constant along afield line. In three-dimensional space, the equation Ф(u1, u2, u3) ≡ Ф(R) = constant represents a surface. A magnetic-field line can thus be identified with a curve lying at the intersection of two surfaces belonging to two different stream functions.

Vector Algebra and Analysis in Curvilinear Coordinates

In this chapter we derive and compile some fundamental relations concerning the 3-D vector algebra and calculus in curvilinear coordinates. More detailed treatments (containing additional material which is less important for our needs) can be found in standard works on vectors and tensors such as Borisenko and Tarapov (1968), Wrede (1963), and Spiegel (1959). Classic textbooks on mathematical physics such as Morse and Feshbach (1953), Margenau and Murphy (1976), Mathews and Walker (1970) and Butkov (1968), contain some instructive chapters on the subject matter. Also worth mentioning are the treatments in well-known physics and engineering books such as Symon (1971, Mechanics), Stratton (1941, Electromagnetic Theory), Milne-Thomson (1968, Hydrodynamics) and Bird, Stewart and Lightfoot (1960, Transport Phenomena).

The shifted reflective boundary for the study of two-phase systems with molecular dynamics simulations

When studying fluids with molecular dynamics simulations, periodic boundaries are usually used to model the infinite bulk fluid surrounding the primary cell. For homogeneous systems this is, as a rule, the most appropriate way. For inhomogeneous systems, e.g. systems with a fluid–vapour interface, periodic boundaries have some disadvantages. Therefore, an alternative for periodic boundaries, called the shifted reflective boundary, is proposed for modelling such systems. From a computational point of view, this type of boundary is no more difficult to implement than periodic boundaries. It is shown that the shifted reflective boundary results in a stable spatial fluid–vapour configuration with one fluid–vapour interface, while retrieving the same numerical results for the thermodynamic properties, e.g. the surface tension, as molecular dynamics simulations with periodic boundaries. Molecular dynamics simulations with shifted reflective boundaries also need fewer particles than corresponding simulations with periodic boundaries.

Toroidal Flux Coordinates

Toroidal flux coordinates are a set of poloidal and toroidal “angles” θf and ζf chosen such that the equation of a field line is that of a straight line in those coordinates. It is common to say that the magnetic-field lines appear as straight lines, or that the magnetic field is straight in (θf , ζf ).

The Dynamic Equilibrium of an Ideal Tokamak Plasma

To acquire a good understanding of the relationship between the different flux measures in a tokamak (Ψtor, Ψ pol d , Ψ pol r ), a careful investigation of the time evolution of the flux surfaces in an ideal plasma (with σ → ∞) is called for. Plasma and flux surfaces will be shown to move in response to a time-varying transformer flux.

Establishment of the Flux-Coordinate Transformation; A Summary

The material in this section on finding the transformation from a particular coordinate system to flux coordinates has to a large extent already been discussed above, although the details were scattered throughout several sections. Here we summarize briefly the methods used and provide some practical references.

Alternative Derivations of the Divergence Formula

In this chapter, we discuss two alternative derivations for the divergence formula of (2.6.39). The one found most frequently in the applied mathematics literature involves Christoffel symbols. This derivation is rather remote from the usual plasma physics calculations, however. Nevertheless, for completeness and cohesiveness of our treatment, we include it as a last case. The method given first is the most natural.

Conversion from Clebsch Coordinates to Toroidal Flux Coordinates

In this Chapter, we discuss the conversion formulae from toroidal flux coordinates to Clebsch functions. First, we deal with the (ϱ, v,l) system, and after that, we discuss the Boozer-Grad (ϱ, v,x) system.

The Relationship Between ∫dl/B and dV/dΨtor

In many elementary discussions it is “proved” in an intuitive manner that the specific volume \( \dot V(\Psi _{tor} ) \equiv dV/d\Psi _{tor} \) equals the closed line integral ∮dl/B. That this is true only in an approximate sense, and that a distinction is to be made between rational and irrational surfaces is discussed in the authoritative review paper by Solov’ev and Shafranov (1970). For completeness, we reproduce their arguments in this Chapter albeit converted to our notation.