Magnetic Field Line Equations Research Articles

A General Curvilinear Magnetic Field‐Line‐Following Coordinate System for Ionosphere‐Plasmasphere Modeling

AbstractWe propose a simple way to define a field‐line‐following, general curvilinear coordinate system for a general magnetic field. This definition of field‐line‐following coordinate system reduces to a usual definition of dipole coordinate system when the magnetic field is approximated by an axisymmetric dipole. In this way, it can facilitate the numerical implementation by enabling validation of various metric terms computed numerically against those defined analytically in the case of the dipole field. Steps involved in grid generation are also sketched. Highly accurate results are obtained using the high‐order ordinary differential equation (ODE) solver to solve the general magnetic field line equations. The accuracy and consistency of the numerical implementation are validated against analytical results in the case of a dipole field. Numerical results show that this field‐line‐following coordinate system for the general magnetic field, like the coordinates for the dipole field, is also an Euler potential or Clebsch type coordinate system.

Structure of magnetic field lines

In this paper the Hamiltonian structure of magnetic lines is studied in many ways. First it is used vector analysis for defining the Poisson bracket and Casimir variable for this system. Second it is derived Pfaffian equations for magnetic field lines. Third, Lie derivative and derivative of Poisson bracket is used to show structure of this system. Finally, it is shown Nambu structure of the magnetic field lines.

Calculation of the magnetic surface function gradient in stellarators with broken stellarator symmetry

The computation of the gradient of the magnetic surface function, ∇ψ, plays an essential role in plasma physics, e.g., for investigations of plasma equilibrium currents or transport fluxes in stellarators. The evaluation of ∇ψ becomes more complicated if the magnetic field B does not exhibit stellarator symmetry. Here, a scheme for computation of ∇ψ for magnetic configurations which do not show stellarator symmetry is presented. The proposed method is based on computations of gradients of integrals of magnetic field line equations. This new technique for ∇ψ calculations is applied to Uragan-2M [O. S. Pavlichenko for the U-2M group, Plasma Phys. Controlled Fusion 35, B223 (1993)]. Taking into account the influence of current feeds and detachable joints of the helical winding the magnetic configuration does not exhibit stellarator symmetry. Computations of ∇ψ, the effective ripple ϵeff, and the geometrical factor λb for the bootstrap current in the 1/ν transport regime are performed.

Integrable maps with non-trivial topology: application to divertor configurations

We explore a method for constructing two-dimensional area-preserving, integrable maps associated with Hamiltonian systems, with a given set of fixed points and given invariant curves. The method is used to find an integrable Poincaré map for the field lines in a large aspect ratio tokamak with a poloidal single-null divertor. The divertor field is a superposition of a magnetohydrodynamic equilibrium with an arbitrarily chosen safety factor profile, with a wire carrying an electric current to create an X-point. This integrable map is perturbed by an impulsive perturbation that describes non-axisymmetric magnetic resonances at the plasma edge. The non-integrable perturbed map is applied to study the structure of the open field lines in the scrape-off layer, reproducing the main transport features obtained by integrating numerically the magnetic field line equations, such as the connection lengths and magnetic footprints on the divertor plate.

Tokamak magnetic field lines described by simple maps

The magnetic field line structure in a tokamak can be obtained by direct numerical integration of the field line equations. However, this is a lengthy procedure and the analysis of the solution may be very time-consuming. Otherwise we can use simple two-dimensional, area-preserving maps, obtained either by approximations of the magnetic field line equations, or from dynamical considerations. These maps can be quickly iterated, furnishing solutions that mirror the ones obtained from direct numerical integration, and which are useful when long-term studies of field line behavior are necessary (e.g. in diffusion calculations). In this work we focus on a set of simple tokamak maps for which these advantages are specially pronounced.

Topology of toroidal helical fields in non-circular cross-sectional tokamaks

The ordinary differential magnetic field line equations are solved numerically; the tokamak magnetic structure is studied on Hefei Tokamak-7 Upgrade (HT-7U) when the equilibrium field with a monotonic q-profile is perturbed by a helical magnetic field. We find that a single mode (m, n) helical perturbation can cause the formation of islands on rational surfaces with q = m/n and q = (m +/- 1, +/- 2, +/- 3,...)/n due to the toroidicity and plasma shape (i.e. elongation and triangularity), while there are many undestroyed magnetic surfaces called Kolmogorov-Arnold-Moser (KAM) barriers on irrational surfaces. The islands on the same rational surface do not have the same size. When the ratio between the perturbing magnetic field B-r(r) and the toroidal magnetic field amplitude B(phi)0 is large enough, the magnetic island chains on different rational surfaces will overlap and chaotic orbits appear in the overlapping area, and the magnetic field becomes stochastic. It is remarkable that the stochastic layer appears first in the plasma edge region.

The structure of chaotic magnetic field lines in a tokamak with external nonsymmetric magnetic perturbations

We consider the effects of external nonsymmetric magnetostatic perturbations caused by resonant helical windings and a chaotic magnetic limiter on the plasma confined in a tokamak. The main purpose of both types of perturbation is to create a region in which field lines are chaotic in the Lagrangian sense: two initially nearby field lines diverge exponentially through many turns around the tokamak. The equilibrium field is obtained from the equations of magneto-hydrodynamic equilibrium written down in a polar toroidal coordinate system. The magnetic fields generated by the resonant helical windings and the chaotic magnetic limiter are obtained through an analytical solution of Laplace equation. The magnetic field line equations are integrated to give a Hamiltonian mapping of field lines that we use to characterize the structure of chaotic field lines. In the case of resonant windings, we obtained the map by both numerical integration and a Hamiltonian formulation. For a chaotic limiter, we analytically derived a symplectic map by using a Hamiltonian formulation.

Chaotic magnetic field lines in a Tokamak with resonant helical windings

We use a Hamiltonian description for magnetic field lines in a large aspect ratio Tokamak for describing the effect of resonant helical windings in a perturbative way, taking into account the toroidal correction. We consider the formation of chains of primary (main and satellites) and secondary islands, analysing their interactions to describe the generation of chaotic magnetic field lines within a portion of the plasma column. This chaotic structure is responsible for disruptive instabilities that may spoil plasma confinement in the Tokamak. Results are compared with numerical integration of magnetic field line equations.

Application of the separatrix map to study perturbed magnetic field lines near the separatrix

In this paper a generalized separatrix map to study perturbed magnetic field lines in a confined magnetic system with the separatrix is discussed. The separatrix map is an iteration transform for the magnetic flux and a toroidal angle at a certain cross section after one period of the unperturbed trajectory. The map can be derived from magnetic field line equations and can be shifted along the toroidal angle, depending on the point of interest. Different asymptotic forms of the shifted separatrix map, with respect to the distance between the cross section of the toroidal angle and the X point, are considered, and their invariant properties are discussed in regard to the renormalization transformation of the perturbation amplitude. A method to find the map’s parameters for any specific magnetic system is proposed. It is shown that the shifted separatrix map can be directly applied to obtain magnetic footprints at the divertor plates, and to study chaotic scattering of field lines.

Comments on the magnetic field generated by an infinite current grid

The magnetic field generated by an infinite current grid is a useful result in Tokamak physics, but also an interesting electromagnetism problem at the undergraduate level. The authors outline the solution of the boundary value problem in rectangular coordinates and comment about the Hamiltonian nature of the magnetic field line equations.

On Hamiltonian formulations of magnetic field line equations

The authors show the equivalence between the two canonical formulations in any coordinate system of magnetic field line equations of Janaki and Ghosh (1987), and of Cary and Littlejohn (1983), through the expression of the magnetic field in terms of the vector potential.

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.

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.

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.

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.