Field Line Equations Research Articles

Hamiltonian formulation of magnetic field line equations

A Hamiltonian formulation of magnetic field line equations is derived for arbitrary magnetic field configurations in orthogonal curvilinear coordinate systems. The canonical equivalence of the different descriptions thus obtained are explicitly demonstrated and action-angle forms are given in cases where there is a geometrical symmetry. Examples of Hamiltonians are discussed for usual plasma machines like the tokamak and the levitron. Finally, generalisation to non-orthogonal coordinate systems is worked out.

Equation for the field lines of an axisymmetric magnetic multipole

Summary. An exact equation is derived for the magnetic field lines of the general axisymmetric magnetic multipole of arbitrary degree (n). This new result has important applications in studies of the possible nature of solarterrestrial physics during geomagnetic polarity reversals. In the limiting case of a magnetic dipole (n = l), the equation for the magnetic field lines of the general axisymmetric magnetic multipole simplifies correctly to the wellknown dipolar form, which is used extensively in geomagnetism, magnetospheric physics and cosmic-ray physics as a first-order approximation to the actual configuration of the geomagnetic field. It is also shown theoretically that suites of similar magnetic field lines of the general axisymmetric multipole attain their maximum radial distances from the origin on a set of circular conical surfaces, with coincident vertices at the centre of the Earth; this set includes the equatorial plane if the degree (n) of the multipole is odd. The magnetic field is horizontal everywhere on all these surfaces. Palaeomagnetic studies have suggested that during some polarity reversals the magnetic field in the inner magnetosphere can be represented approximately either by a single, non-dipolar, low-degree (2 < n < 4), axisymmetric magnetic multipole or by a linear combination of such multipoles. In this situation, the equation for the field lines of an axisymmetric magnetic multipole of low degree (2 < n < 4) would be as fundamental to a proper understanding of magnetospheric, ionospheric and cosmic-ray physics during polarity reversals as is the equation for dipolar field lines in the case of the contemporary geomagnetic field.

Stochastic stability of a plasma torus

The hamiltonian equations for magnetic field lines in a toroidal plasma are derived from a variational principle; we find an equation for critical fluctuations ▪ n by assuming a marginal overlap of adjacent resonances throughout the small cross section for the torus. Comparison with the flux surfaces of JET as obtained from a static equilibrium code leads to critical fluctuations of the order of δB crit ≲ B p x10 −4 for n ≳ 10, where B p is the poloidal equilibrium field. Finally, we solve the equations for magnetic field lines in the case of a set of nested flux surfaces perturbed by critical fluctuations; the picture of a completely destroyed torus has been verified with a stochasticity parameter √3, but no stable integration scheme for a stochastic magnetic line has been found.

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.

Hamiltonian Approach to Magnetic Fields with Toroidal Surfaces

The equivalence of magnetic field line equations to a one-dimensional time-dependent Hamiltonian system is used to construct magnetic fields with arbitrary toroidal magnetic surfaces I = const. For this purpose Hamiltonians H which together with their invariants satisfy periodicity constraints have to be known. The choice of H fixes the rotational transform η(I). Arbitrary axisymmetric fields, and nonaxisymmetric fields with constant η(I) are considered in detail. Configurations with coinciding magnetic and current density surfaces are obtained. The approach used is not well suited, however, to satisfying the additional MHD equilibrium condition of constant pressure on magnetic surfaces.

An example for solar flares caused by magnetic field non-equilibrium

Using a photospheric magnetogram of the solar activity complex HR 16862, 16863, 16864 at the end of May 1980 as boundary data a sequence of three-dimensional force-free magnetic fields with spatially constant α (α defined by the equation ▽ × B = αB) is calculated, including numerical field-line tracing. The field is assumed not to be frozen-in and ¦α¦, which is a measure of the magnetic free energy, is assumed to increase with time due to some dynamo mechanism. Variation of α, starting from α = 0, produces a catastrophe-like change of the topology of a field-line system corresponding to an arch filament system connecting the leading spot of HR 16863 with the trailing spot of HR 16862. This topological change is interpreted as causing a series of large homologous flares with synchronous flaring in the two spots. The catastrophe-like behaviour of the field topology is attributed to the nonlinear field-line equations.

Diffusion coefficient and Kolmogorov entropy of magnetic field lines

A diffusion equation for magnetic field lines of force in a turbulent magnetic field, which describes both the random walk of a single field line and how two nearby lines separate from each other, has been obtained using standard statistical techniques. Starting from such an equation, a closed set of equations for the moments may be obtained, in general, with suitable assumptions. From such a set of equations the Kolmogorov entropy may be explicitly calculated. The results have been applied to the most interesting examples of magnetic field geometries.

Toroidal Resonators and Waveguides of Arbitrary Cross Section

After introducing a new method to solve Maxwell's equations using a complex electromagnetic field vector F, a rotational coordinate system xi, Theta, psi is introduced. In this coordinate system, the field vector components F/sub xi/, F/sub Theta/ may be expressed by F/sub psi/. This component can be obtained from a two-dimensional Hehlmholtz equation. Specifying xi, Theta by cylindrical coordinates r, z the complex Maxwell equation curl F= gamma F is solved for the axisymmetric case (/spl part///spl part/psi = 0) and for the nonsymmetric case. The differential equations for magnetic field lines are solved and surfaces on which the normal component of B and the tangential components of E vanish are recognized as metallic walls of toroidal resonators of various arbitrary cross sections. In the Appendix the results of the new method are compared with well known results for circular cylindrical waveguides.

Non-force-free solar magnetic fields in magnetohydrostatic equilibrium

The objective of the paper is to examine a class of non-force-free fields analytically. Specifically, magnetic fields in magnetohydrostatic equilibrium with a plasma in a gravitational field are treated in an approximation of two independent variables but three vector components. Spherical coordinates are emphasized, although the formal results for cylindrical and Cartesian coordinates are presented in the appendices. Formal solutions for the magnetic-field components are obtained in terms of the plasma variables, and field line equations are derived. A final equation governing the plasma variables is then obtained. Procedures are developed for analyzing this equation and obtaining sets of self-consistent particular solutions to the governing equations; a number of such sets of solutions are presented. As an example, one solution set is examined, illustrating the application of the results to the analysis of solar observational data.

A Dynamic Equation for Stochastic Magnetic Field Lines in the Galaxy

view Abstract Citations (16) References (11) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS A Dynamic Equation for Stochastic Magnetic Field Lines in the Galaxy Jones, Frank C. Abstract It has been pointed out by previous authors that the turbulent motion of the ionized interstellar gas will cause a given line of force of the galactic magnetic field to wander in a random manner even though the average field may be quite regular. We derive here an equation that describes the development of the probability density of a given field line as a function of distance along the average field. We explain why a Fokker-Planck type of equation is quite unsuitable in this case and show that the answer must be sought in the theory of stationary Gaussian processes. We also derive the distribution function for the "endpoints" of a field line that starts on the galactic central plane. Endpoint in this context means the point where the field line first wanders so near the surface of the galactic disk that it can no longer confine cosmic rays. Publication: The Astrophysical Journal Pub Date: November 1971 DOI: 10.1086/151163 Bibcode: 1971ApJ...169..477J full text sources ADS | Related Materials (1) Erratum: 1972ApJ...172..525J

Destruction of magnetic surfaces by magnetic field irregularities

A perturbation theory analogous to that of quasi-linear theory is developed to treat the behavior of magnetic surfaces, in particular to estimate their stability against field irregularities. The method exploits the similarity between the Vlasov equation and the Liouville equation for field lines. We find that a very important role is played by field resonances, that isolated resonances have a limited effect, extended over a finite width which we estimate, but as soon as resonances overlap, a very rapid destruction of flux surfaces may be expected.