Unified Theory of Ghost and Quadratic-Flux-Minimizing Surfaces Robert L. DEWAR, Stuart R. HUDSON 1) and Ashley M. GIBSON Plasma Research Laboratories, Research School of Physics & Engineering, The Australian National University, Canberra ACT 0200, Australia $^{1)}$Princeton Plasma Physics Laboratory, PO Box 451, Princeton NJ 08543, USA November 18, 2018 A generalized Hamiltonian definition of ghost surfaces (surfaces defined by an action-gradient flow) is given and specialized to the usual Lagrangian definition. Numerical calculations show uncorrected quadratic-flux- minimizing (QFMin) and Lagrangian ghost surfaces give very similar results for a chaotic magnetic field weakly perturbed from an integrable case in action-angle coordinates, described by L = L$_{0}$ + GLYPH<15> L$_{1}$ , where L$_{0}$ ( ˙ GLYPH<18> ) (with ˙ GLYPH<18> denoting d GLYPH<18>= d GLYPH<16> ) is an integrable field-line Lagrangian and GLYPH<15> is a perturbation parameter. This is explained using a perturbative construction of the auxiliary poloidal angle GLYPH<2> that corrects QFMin surfaces so they are also ghost surfaces. The di GLYPH<11> erence between the corrected and uncorrected surfaces is O ( GLYPH<15> $^{2}$), explaining the observed smallness of this di GLYPH<11> erence. An alternative definition of ghost surfaces is also introduced, based on an action- gradient flow in GLYPH<2> , which appears to have superior properties when unified with QFMin surfaces. Keywords: Toroidal magnetic fields, Hamiltonian dynamics, Lagrangian dynamics, almost-invariant tori 1. Introduction Recent calculations [1] of heat di GLYPH<11> usion along chaotic field lines show that the isotherms correspond very closely with the “approximate” magnetic surfaces, associated with magnetic island chains, known as ghost surfaces [2]. These surfaces include the “X-point” and “O-point” closed field lines of their associated islands. (By “O-point” field line we mean either the elliptically stable field line at the center of an island or its hyperbolically unstable continuation if it has undergone a period-doubling bifurcation.) Closed field lines extremize the magnetic action, H A GLYPH<1> dl , the hyperbolic X-point field lines in the chaotic separatrices being minima and the O-point field lines being minimax or saddle points of the action. Ghost surfaces are constructed by interpolat- ing smoothly between these two closed-field-line classes by evolving the O-point field lines into the X-point field lines along paths of steepest descent of action, thus gen- erating a family of “pseudo-orbits,” i.e. paths that come close to extremizing action. An alternative approach to defining approximate mag- netic surfaces passing through magnetic islands, is to use the quadratic-flux-minimizing (QFMin) surfaces intro- duced by Dewar, Hudson and Price [2, 3]. Ghost surfaces have nice mathematical properties but are di GLYPH<14> cult to con- struct and have no obvious physical interpretation. QFMin surfaces, on the other hand, have the computational attrac- tion of being easy to construct, and the physical attraction of being defined in terms of a measure of the magnetic flux transport through the surface, but have been found to ex- hibit undesirable distortions in some circumstances. However the definition of QFMin surfaces is not unique, as it depends on the choice of an auxiliary poloidal angle GLYPH<2> ; which raises the question: Can GLYPH<2> be chosen so that QFMin surfaces coincide with ghost surfaces? This question has recently [4] been answered in the a GLYPH<14> rmative, provided we modify the definition of ghost surface slightly, there being some freedom also in the definition of action- gradient flow. This raises the further question, accepting that ghost surfaces need to be redefined to achieve unifica- tion with QFMin surfaces, is the definition used in Ref. 4 optimal, or is there a still better class of unified ghost and QFMin surfaces? In Secs. 2–4 we review the QFMin and ghost surface concepts, and in Sec. 5 summarize the unification proposed in Ref. 4. In Sec. 6 we summarize its perturbative imple- mentation in a test problem. In Sec. 7 we propose a new definition of ghost surfaces based on an action-gradient flow in the auxiliary angle GLYPH<2> , rather than the flow in the unperturbed canonical coordi- nate GLYPH<18> used in Ref. 4. The new definition is not only sim- pler than that in Ref. 4, in that it does not require a variable rate of flow along pseudo field lines, it also leads naturally to straight pseudo-field-line orbits in the GLYPH<2> - GLYPH<16> plane. This is illustrated perturbatively for the same case used in Sec. 6. Fig. 1 A sketch of the general curvilinear toroidal coordinate system described in the text. author’s e-mail: robert.dewar@anu.edu.au arXiv:1001.0483v3 [physics.plasm-ph] 2 Aug 2011