Magnetic Field Line Flow Research Articles

Distinguishing between regular and chaotic orbits of flows by the weighted Birkhoff average

This paper investigates the utility of the weighted Birkhoff average (WBA) for distinguishing between regular and chaotic orbits of flows, extending previous results that applied the WBA to maps. It is shown that the WBA can be super-convergent for flows when the dynamics and phase space function are smooth, and the dynamics is conjugate to a rigid rotation with Diophantine rotation vector. The dependence of the accuracy of the average on orbit length and width of the weight function width are investigated. In practice, the average achieves machine precision of the rotation frequency of quasiperiodic orbits for an integration time of O(103) periods. The contrasting, relatively slow convergence for chaotic trajectories allows an efficient discrimination criterion. Three example systems are studied: a two-wave Hamiltonian system, a quasiperiodically forced, dissipative system that has a strange attractor with no positive Lyapunov exponents, and a model for magnetic field line flow.

Improved accuracy in degenerate variational integrators for guiding centre and magnetic field line flow

First-order-accurate degenerate variational integration (DVI) was introduced in Ellison et al. (Phys. Plasmas, vol. 25, 2018, 052502) for systems with a degenerate Lagrangian, i.e. one in which the velocity-space Hessian is singular. In this paper we introduce second-order-accurate DVI schemes, both with and without non-uniform time stepping. We show that it is not in general possible to construct a second-order scheme with a preserved two-form by composing a first-order scheme with its adjoint, and discuss the conditions under which such a composition is possible. We build two classes of second-order-accurate DVI schemes. We test these second-order schemes numerically on two systems having non-canonical variables, namely the magnetic field line and guiding centre systems. Variational integration for Hamiltonian systems with non-uniform time steps, in terms of an extended phase space Hamiltonian, is generalized to non-canonical variables. It is shown that preservation of proper degeneracy leads to single-step (one-step) methods without parasitic modes, i.e. to non-uniform time step DVIs. This extension applies to second-order-accurate as well as first-order schemes, and can be applied to adapt the time stepping to an error estimate.

On heat conduction in an irregular magnetic field. Part 1

Anisotropic heat conduction in a plasma embedded in a magnetic field with irregular, possibly chaotic, field lines is discussed. If the collisional mean free path exceeds the electron gyroradius, the heat conductivity is much larger along the field lines than across them, and this enhances the transport across a domain where good flux surfaces do not exist. Recognising that anisotropic heat conduction may be cast in a variational form, and by constructing increasingly sophisticated trial functions that are based on invariant and almost-invariant structures under the magnetic field-line flow, bounds are derived on this enhancement and on the temperature variation along the magnetic field. In this way, remarkably accurate approximations for the temperature can be rapidly constructed without solving the diffusion equation, even in the small perpendicular-diffusion limit when the solution for the temperature is dominated by the fractal structure the magnetic field lines.

Degenerate variational integrators for magnetic field line flow and guiding center trajectories

Symplectic integrators offer many benefits for numerically approximating solutions to Hamiltonian differential equations, including bounded energy error and the preservation of invariant sets. Two important Hamiltonian systems encountered in plasma physics—the flow of magnetic field lines and the guiding center motion of magnetized charged particles—resist symplectic integration by conventional means because the dynamics are most naturally formulated in non-canonical coordinates. New algorithms were recently developed using the variational integration formalism; however, those integrators were found to admit parasitic mode instabilities due to their multistep character. This work eliminates the multistep character, and therefore the parasitic mode instabilities via an adaptation of the variational integration formalism that we deem “degenerate variational integration.” Both the magnetic field line and guiding center Lagrangians are degenerate in the sense that the resultant Euler-Lagrange equations are systems of first-order ordinary differential equations. We show that retaining the same degree of degeneracy when constructing discrete Lagrangians yields one-step variational integrators preserving a non-canonical symplectic structure. Numerical examples demonstrate the benefits of the new algorithms, including superior stability relative to the existing variational integrators for these systems and superior qualitative behavior relative to non-conservative algorithms.

Addendum: Debye potentials and Hamiltonians for magnetic field line flow [J. Math. Phys. 36, 3413–3418 (1995)

Addendum: Debye potentials and Hamiltonians for magnetic field line flow [J. Math. Phys. <b>36</b>, 3413–3418 (1995)

Debye potentials and Hamiltonians for magnetic field line flow

It is shown that any divergence‐free vector field invariant under a group of volume‐preserving transformations can be expressed locally in terms of two scalar potentials which depend on two variables only. It is also shown that the corresponding field line equations can be written in Hamiltonian form with one of these potentials as the Hamiltonian.

Magnetic Field Line Diffusion at the Onset of Stochasticity

The Hamiltonian equations of a particle in a random set of waves just above the stochasticity threshold are considered both theoretically and numerically. First we derive the diffusion coefficient and the autocorrelation time perturbatively without using the thermodynamic limit, and we discuss the relevance of the H am iltonian problem for particle acceleration and magnetic field line flow. Then we integrate the equations for an ensemble of magnetic field lines numerically for a model problem [15] and show the time evolution of moments and correlations. Twice above the threshold we observe diffusive behaviour from the beginning, but the diffusion coefficient includes also the non-resonant modes. Just at threshold we find first a short phase o f free acceleration, later a diffusion which is slower than predicted by the theoretical formula. The best way to analyze the problem is in terms of cumulants, but a reliable com parison with any theory requires also a time integration of the corresponding kinetic equations.

Magnetic field line flow as a Hamiltonian problem

The stochastic stability of a plasma torus with nested magnetic surfaces of arbitrary cross-section is investigated. The equations for field lines are put into Hamiltonian form, exhibiting both gauge and canonical symmetry. A careful application of Chirikov's (1979) overlap criterion leads to a formula for critical resonant fluctuations which should be valid for the whole cross-section and for arbitrary mode numbers m, n.

Topological stability of finite-length magnetic flux tubes

It has been suggested that the activity of cosmical magnetic fields is a consequence of a general topological nonequilibrium in the neighbourhood of magnetostatic equilibria. Evidence for this suggestion can be obtained from the Kolmogorov-Arnold-Moser theorem of classical mechanics, applied to the magnetic field line flow as a Hamiltonian system. A finite-length magnetic flux tube, however, always possesses two independent sets of flux surfaces - or, equivalently, the corresponding Hamiltonian system two independent first integrals - and is topologically stable if in the volume occupied by the tube there are no singular (null) points of the magnetic field and the normal field component does not change its sign on the end faces of the tube. Therefore, the concept of nonequilibrium due to flux surface destruction is not applicable to solar atmospheric loops with each end situated in the interior of one polarity of the photospheric normal field component. Further, it seems unlikely that the tearing-mode mechanism can play a role in such loops.

Noncanonical Hamiltonian mechanics and its application to magnetic field line flow

A noncanonical Hamiltonian theory of dynamical systems is presented and applied to magnetic field line flow. The theory allows all of the theorems of Hamiltonian mechanics (most importantly, Noether's theorem, relating symmetries and invariants) to be applied to the magnetic field line system. The theory is not restricted to any particular geometry. An elementary derivation of noncanonical Hamiltonian perturbation theory, based on Lie transforms, is also presented. As an example, the perturbation theory is applied to magnetic field line flow in nearly azimuthally symmetric geometry. Other applications are to the adiabatic motion of charged particles.

Noncanonical Hamiltonian mechanics and its application to magnetic field line flow: John R. Cary. Institute for Fusion Studies, University of Texas, Austin, Texas 78712; and Robert G. Littlejohn. Department of Physics, University of California, Los Angeles, California 90024

Noncanonical Hamiltonian mechanics and its application to magnetic field line flow: John R. Cary. Institute for Fusion Studies, University of Texas, Austin, Texas 78712; and Robert G. Littlejohn. Department of Physics, University of California, Los Angeles, California 90024