Abstract
In the study of a magnetic confinement fusion device such as a tokamak, physicists need to understand the topology of the flux (or magnetic) surfaces that form within the magnetic field. Among the two distinct topological structures, we are particularly interested in the magnetic island chains which correspond to the break up of the ideal rational surfaces. Different from our previous method [13], in this work we resort to the periodicity analysis of two distinct functions to identify and characterize flux surfaces and island chains. These two functions are derived from the computation of the fieldlines and puncture points on a Poincaré section, respectively. They are the distance measure plot and the ridgeline plot. We show that the periods of these two functions are directly related to the topology of the surface via a resonance detection (i.e., period estimation and the common denominators computation). In addition, we show that for an island chain the two functions possess resonance components which do not occur for a flux surface. Furthermore, by combining the periodicity analysis of these two functions, we are able to devise a heuristic yet robust and reliable approach for classifying and characterizing different magnetic surfaces in the toroidal magnetic fields.
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References
Bagherjeiran, A., Kamath, C.: Graph-based methods for orbit classification. In: SIAM International Conference on Data Mining. SIAM, PA (2005)
Chen, G., Mischaikow, K., Laramee, R.S., Pilarczyk, P., Zhang, E.: Vector field editing and periodic orbit extraction using morse decomposition. IEEE Trans. Visual. Comput. Graph. 13(4), 769–785 (2007)
D’haeseleer, W.D., Hitchon, W.G., Callen, J.D., Shohet, J.L.: Flux Coordinates and Magnetic Field Structure, A Guide to a Fundamental Tool of Plasma Theory. Springer, New York (1991)
Gerhard, D.: Pitch extraction and fundamental frequency: History and current techniques. Technical report, 2003–06
Green, J.M.: Locating three-dimensional roots by a bisection method. J. Comput. Phys. 98, 194–198 (1992)
J. M. Greene.: Vortex nulls and magnetic nulls. In A. T. H.K. Moffatt (ed.) Topological fluid mechanics: proceedings of the IUTAM Symposium, pp. 478-484. Cambridge Univ. Press (1990)
Hale, J., Kocak, H.: Dynamics and Bifurcations. Springer, New York (1991)
Laramee, R., Hauser, H., Zhao, L., Post, F.H.: Topology based flow visualization: The state of the art. In: Topology-Based Methods in Visualization (Proceedings of Topo-in-Vis 2005), Mathematics and Visualization, pp. 1–19. Springer, Berlin (2007)
Löffelmann, H., Kucera, T., Gröller, M.E.: Visualizing poincare maps together with the underlying flow. In: International Workshop on Visualization and Mathematics ’97, pp. 315–328. Springer, Berlin (1997)
Peikert, R., Sadlo, F.: Visualization methods for vortex rings and vortex breakdown bubbles. In: Museth, A.Y.K., Möller, T. (eds.) Proceedings of the 9th Eurographics/IEEE VGTC Symposium on Visualization (EuroVis’07), pp. 211–218, May 2007
Peikert, R., Sadlo, F.: Flow topology beyond skeletons: Visualization of features in recirculating flow. In: Hege, H.-C., Polthier, K., Scheuermann, G. (eds.) Topology-Based Methods in Visualization II, pp. 145–160. Springer, Berlin (2008)
Post, F.H., Vrolijk, B., Hauser, H., Laramee, R.S., Doleisch, H.: The state of the art in flow visualization: Feature extraction and tracking. Comput. Graph. Forum 22(4), 775–792 (2003)
Sanderson, A., Chen, G., Tricoche, X., Pugmire, D., S. Kruger, S., Breslau, J.: Analysis of recurrent patterns in toroidal magnetic fields. IEEE Trans. Visual. Comput. Graph. 16, 1431–1440 (2010)
Theisel, H., Weinkauf, T., Seidel, H.-P., Seidel, H.: Grid-independent detection of closed stream lines in 2D vector fields. In: Proceedings of the Conference on Vision, Modeling and Visualization 2004 (VMV 04), pp. 421–428, Nov 2004
Wischgoll, T., Scheuermann, G.: Detection and visualization of closed streamlines in planar fields. IEEE Trans. Visual. Comput. Graph. 7(2), 165–172 (2001)
Wischgoll, T., Scheuermann, G.: Locating closed streamlines in 3D vector fields. In: Proceedings of the Joint Eurographics – IEEE TCVG Symposium on Visualization (VisSym 02), pp. 227–280, May 2002
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This work was supported in part by the DOE SciDAC Visualization and Analytics Center for Emerging Technology and the DOE SciDAC Fusion Scientific Application Partnership.
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Sanderson, A., Chen, G., Tricoche, X., Cohen, E. (2012). Understanding Quasi-Periodic Fieldlines and Their Topology in Toroidal Magnetic Fields. In: Peikert, R., Hauser, H., Carr, H., Fuchs, R. (eds) Topological Methods in Data Analysis and Visualization II. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23175-9_9
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DOI: https://doi.org/10.1007/978-3-642-23175-9_9
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