Allen Sanderson, Guoning Chen, Xavier Tricoche, and Elaine Cohen
1 Introduction
In magnetic confinement fusion devices such as a tokamak, magnetic fields are used to confine a burning plasma (Fig.1a). In the study of such devices, physicists need to understand the topology of the flux surfaces that form within the magnetic fields.
Flux surfaces come in a rational and irrational form and are defined by periodic and quasi-periodic fieldlines, respectively. Our focus is the break up of the rational surfaces into the irrational ones, specifically those that form magnetic island chains because they are where the plasma escapes and will damage the wall of the fusion reactor.
To distinguish the magnetic island chains from other flux surfaces, we study their behavior in a Poincar´e map computed by intersecting fieldlines with a plane perpendicular to the axis of the torus. A naive geometric test was proposed to identify different magnetic surfaces from the Poincar´e section in our previous work [13]. Unfortunately, it does not make use of the periodicity properties of the flux surfaces as we will show later and hence it is computationally expensive and not reliable. In this work we describe a more robust approach that analyzes the distinct periodic behaviors of flux surfaces and island chains, which helps us achieve more efficient characterization of these two structures. More specifically, we estimate the fundamental periods of two functions stemming from the fieldline tracing and the
A. Sanderson ()G. Chen
SCI Institute, University of Utah, Salt Lake City, UT 84112, USA e-mail:allen@sci.utah.edu;chengu@sci.utah.edu
X. Tricoche
Purdue University, West Lafayette, IN 47907, USA e-mail:xmt@purdue.edu
E. Cohen
University of Utah, Salt Lake City, UT 84112, USA e-mail:cohen@cs.utah.edu
R. Peikert et al. (eds.), Topological Methods in Data Analysis and Visualization II, Mathematics and Visualization, DOI 10.1007/978-3-642-23175-9 9,
© Springer-Verlag Berlin Heidelberg 2012
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Fig. 1 (a) Profile of the DIII-D Tokamak and a single quasi-periodic magnetic fieldline (the red curves). (b) A magnetic fieldline (blue) that intersects the poloidal plane (gray) and the toroidal plane (gold)
puncture point computation, respectively. These two functions are formed via the distance measure plot and the ridgeline plot. We show that the periods of these two functions, when coupled are directly related to the topology of a surface. We should note that in our previous work we have described the ridgeline plot but only used it to complement the geometric test. With the addition of the distance measure plot we are now able to couple them together to form a magnetic surface characterization framework which has a more rigorous foundation.
One of the key components in the classification of rational and irrational surfaces, is the safety factor. The safety factor is the limit of the ratio of the winding pair, i.e., the number of times a fieldline traverses around the major axis of the torus (toroidal winding) for each traversal around the minor axis of the torus (poloidal winding) (Fig.1b). Although based on the above description the poloidal winding may not be integer (Sect.4), in the later discussion we consider only integer pairs for the winding pairs. Choosing a good winding pair is of paramount importance because it will determine how we connect the discrete puncture points to get the contiguous representation of the surfaces [13]. In what follows, we will describe in detail how we combine the results of the period estimation of the two functions and other metrics to form a heuristic framework and obtain the desired winding pairs for both geometry construction and topological characterization of a fieldline. We will also explain how the detection of resonance components in the period analysis of an island chain can help us identify this structure in an early stage.
2 Related Work
Magnetic fields are described in terms of vector fields. While a rich body of visualization research has focused on the extraction of features of interest in vector fields [8,12] starting from [7], the identification of the invariant structures such as periodic orbits from the flow is most relevant to our work.
Within the fusion community researchers have located periodic fieldlines using numerical approaches such as those by [5]. Wischgoll and Scheuermann were the
first in the visualization community to present an algorithm for detecting periodic orbits in planar flows [15]. Their method examines how a fieldline re-enters a cell and re-connects. They have also extended their technique to 3D vector fields [16].
In the meantime, Theisel et al. [14] presented a mesh independent approach to compute periodic orbits. Recently, Chen et al. [2] proposed an efficient algorithm to extract periodic orbits from surface flows using Morse decompositions.
We also note the work of L¨offelmann et al. [9] who integrated 2D Poincar´e plots with the original 3D flow for visualization purposes. Recently, an analysis technique for divergence free flow fields has also been proposed by Peikert and Sadlo [10,11] who introduced a divergence cleaning scheme to study vortex breakdown flow patterns through their long-term Poincar´e plot. Others have used a graph-based approach combined with the machine learning technique to classify the fieldlines [1].
3 Background
In this section, we briefly review some important concepts of Poincar´e map and toroidal magnetic fields. More details can be found in [13].
3.1 Poincar´e Map
Consider a vector fieldV on a manifoldM (e.g., a triangulation) with dimension n(nD3in this work), which can be expressed as an ordinary differential equation
dx
dt D V .x/. The set of solutions to it gives rise to a flow' onM. Let be a trajectory (integral curve) of a vector fieldV. LetS be a cross section of dimension n1 (e.g., a plane perpendicular to the major axis of the torus) such that' is everywhere transverse toS.S is referred to as a Poincar´e section. An intersection of withS is called a puncture point, denoted byli 2S \ (i 2Nshows the intersection order). The Poincar´e map is defined as a mapping inS P WRS ! S that leads a puncture pointlito the next positionliC12S \ following.
3.2 Toroidal Magnetic Fields
A toroidal magnetic field is a 3D vector field where the magnetic fieldlines exhibit helical behavior and wind around the major (toroidal) circle and minor (poloidal) circle of the torus (Fig.1b).
The safety factor of a fieldline,q, is defined as the number of times a fieldline goes around the toroidal circle for each rotation around the poloidal circle, and is computed as:
qD lim
#T!1
#T
#P (1)
Fig. 2 The evolution of a magnetic surface at different times of the fusion simulation: (a) an irrational flux surface at time step 22; (b) a 5,2 island chain at time 23; (c) the growing 5,2 island chain at time 24
where #T is the toroidal winding count (rotations about the toroidal circle) and #P is the poloidal winding count (rotations about the poloidal circle). We define the two winding counts when expressed as rational numbers as a winding pair. Because we cannot integrate to infinity, in practice we estimate the safety factor of a surface based on a finite number of integrations (Sect.4.1).
By definition, the safety factorqcan be either rational or irrational. A rational qimplies that the fieldline is periodic (or closed in finite distance). Such a fieldline lies on a rational surface. Such surfaces are found in a fusion device. However, they are unstable and sensitive to the magnetic perturbation. Among them, the ones with lower-orderqare the first to break down into island chains [3]. Figure2provides an example of such a topology change of a magnetic surface due to the magnetic perturbations. An irrationalq implies that the fieldline is quasi-periodic. Such a fieldline lies on an irrational surface and spreads out over it. This type of flux surfaces is our focus in this work. They have two distinct topological structures shown in a Poincar´e section, a single closed curve or multiple closed curves.
A single closed curve typically represents a magnetic flux surface when it encloses the center of the magnetic field. Multiple closed curves represent a magnetic island chain which is usually associated with a reduction in magnetic confinement [13].
There are two types of critical points for a magnetic island chain, commonly referred to as X (unstable or saddle) and O (stable or center) points. They correspond to the locations where there is no poloidal magnetic flux [6]. In the case of anOpoint, it is located at the magnetic center of an island. While for an Xpoint, it is located where two flux surfaces appear to cross and form a separatrix around the magnetic islands. In this work, we focus on only the characterization of fieldlines. The detection of critical points can be found in [13].
According to [13], an irrational surface consists of #T winding groups in the Poincar´e section. The geometrically neighboring groups need not be neighbors in the puncture point ordering. In order to construct a valid geometry representation
without self-intersections for the surface, a proper winding pair is greatly desired.
In the following, we describe how we achieve so.