Unified Theory of Ghost and Quadratic-Flux-Minimizing Surfaces

Abstract

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 $L=L_0+\epsilon L_1$, where $L_0(\dot{\theta })$ (with $\dot{\theta }$ denoting $d\theta /d\zeta$) is an integrable field-line Lagrangian and $\epsilon$ is a perturbation parameter. This is explained using a perturbative construction of the auxiliary poloidal angle $\Theta$ that corrects QFMin surfaces so they are also ghost surfaces. The difference between the corrected and uncorrected surfaces is $O({\epsilon }^2)$, explaining the observed smallness of this difference. An alternative definition of ghost surfaces is also introduced, based on an action-gradient flow in $\Theta$, which appears to have superior properties when unified with QFMin surfaces.

4 pp, 3 figs. In Proc. of 7th General Scientific Assembly of Asia Plasma and Fusion Association (APFA2009) and Asia-Pacific Plasma Theory Conference (APPTC2009), Aomori, Japan, October 27-30 2009 http://www.jspf.or.jp/JPFRS/index_vol9-4.html . v2: corr. ref. to eq (9) on p 4 to eq (8); in l. below eq (19) changed /δθ to /δΘ. v3: 3rd l. above eq (7), now η-> 0 not \infty