Reconciliation of almost-invariant tori in chaotic systems
Creators: name: Gibson, Ashley Martin
Publisher: name: The Australian National University
Year: 2012
Subjects: subject: Chaotic behavior in systems
subject: Torus (Geometry)

description: Magnetic field lines within toroidal magnetic confinement systems can be described as orbits of one-and-a-half-degree-of-freedom Lagrangian and Hamiltonian systems. In axisymmetric devices such as ideal tokamaks in equilibrium, all field lines lie within a smoothly nested set of invariant tori (magnetic surfaces) foliating the plasma volume, but this integrability is lost within non-axisymmetric devices such as stellarators. That is, not all field lines lie within magnetic surfaces and thus they cannot be described by conventional action-angle coordinates. However, according to the Kolmogorov-Arnold-Moser (KAM) theorem, some invariant tori, covered ergodically by quasiperiodic orbits, survive perturbation away from integrability. Furthermore, by the Poincare-Birkhoff theorem, two kinds of periodic orbits (closed field lines), called the action-minimax and action-minimising orbits, survive perturbation, and they can be incorporated into families of pseudo-orbits that provide best approximations to invariant tori within nonintegrable systems. The three candidates for such almost-invariant tori discussed in this thesis are quadratic-flux-minimising (QFMin) tori, which minimise the integral of the square of the action gradient over the poloidal and toroidal angles, action-gradient-minimising (AGMin) tori, which minimise the square of the action gradient over each periodic pseudo-orbit, and ghost tori, which join the action-minimax and action-minimising orbits via an action-gradient flow. Although none of these almost-invariant tori has any direct physical interpretation, it has been shown by Hudson and Breslau (Phys. Rev. Lett., 100, 095001 (2008)) that ghost tori are in very close correspondence with temperature isotherms. There is also a very close relationship between QFMin, AGMin and ghost tori, which suggests that they could be made equivalent to each other or ""reconciled"" by making an appropriate coordinate transformation. By using the standard map as a discrete-time model for magnetic field lines via the kicked rotor, it is demonstrated using Mathematica that QFMin and ghost tori can be reconciled with each other up to at least k = 1.0 for all rational rotational transforms with denominator less than or equal to 13, where k denotes the nonlinearity parameter of the standard map. This is accomplished by expanding the coordinate transformation as a Fourier series and formulating a variational principle, which is used to numerically construct a set of almost-invariant curves that have rotational transforms equal to continued-fraction convergents of two minus the golden mean (i.e., 0.381966...). By calculating the flux between the curves, it is shown via a preliminary investigation that the reconciled QFMin-ghost tori are consistent with Greene's residue criterion and the existence of KAM tori within the standard map for k &lt; 0.971635..., which is the value at which the last KAM curves are destroyed. It is also shown via the construction of a rigidity principle that the reconciled QFMin-ghost tori have also been reconciled with AGMin tori, by appealing to the fact that discrete-time dynamical systems are merely just Poincare sections of continuous-time dynamical systems, for which the rigidity principle was formulated. This provides an important first step towards constructing an almost-straight-field-line coordinate system for magnetic islands.