Toroidal Magnetic Surfaces Research Articles

On the existence of toroidal magnetohydrodynamic equilibria with poloidally closed magnetic field lines

The present work explores the existence of nonaxisymmetric toroidal magnetohydrodynamic equilibria when all lines of force of the magnetic field close after a single revolution about a given magnetic axis. It is assumed that the magnetic axis is a closed curve with arbitrary curvature and zero torsion, implying that it is constrained to lie in a plane surface. In addition, it is assumed that the closed magnetic field lines lie in planes that are orthogonal to the magnetic axis. Subject to these conditions, the existence of toroidal magnetic surfaces, F(r)=const, is explored. The governing equilibrium equations of magnetohydrodynamics place a constraint on the function F(r) in the form of two nonlinear partial differential equations that must be simultaneously satisfied. It is demonstrated that this is not always possible without axisymmetry. Two specific cases where toroidal magnetic surfaces do not exist are identified: (I) isodynamic configurations, which implies that the magnetic field strength is constant on each magnetic surface, and (II) configurations with ‘‘bumpy’’ surfaces, which implies that the size but not the geometrical shape of the poloidal section containing the closed field lines depends on s.

Construction of Monte Carlo operators in collisional transport theory

A Monte Carlo approach for investigating the dynamics of quiescent collisional magnetoplasmas is presented, based on the discretization of the gyrokinetic equation. The theory applies to a strongly rotating multispecies plasma, in a toroidally axisymmetric configuration. Expressions of the Monte Carlo collision operators are obtained for general v-space nonorthogonal coordinates systems, in terms of approximate solutions of the discretized gyrokinetic equation. Basic features of the Monte Carlo operators are that they fullfill all the required conservation laws, i.e., linear momentum and kinetic energy conservation, and in addition that they take into account correctly also off-diagonal diffusion coefficients. The present operators are thus potentially useful for describing the dynamics of a multispecies toroidal magnetoplasma. In particular, strict ambipolarity of particle fluxes is ensured automatically in the limit of small departures of the unperturbed particle trajectories from some initial axisymmetric toroidal magnetic surfaces.

Stochastic broadening of the separatrix of a tokamak divertor.

The plasma in a modern tokamak is bounded by a separatrix between magnetic field lines that form toroidal magnetic surfaces, on which the plasma is confined, and open field lines that divert the plasma exhaust to so-called divertor plates. This separatrix is sharp in an ideal tokamak, but we show that magnetic perturbations create a stochastic region of open field lines inside the ideal separatrix which contains approximately 6 times the magnetic flux as does the strike points of these lines on the divertor plates. Since magnetic field lines are a 11/2 degree of freedom Hamiltonian system, the behavior of the field lines in a tokamak divertor is archetypal for the similar Hamiltonian systems.

Comparison between experiment and theory

In the idealized theory of toroidal magnetic confinement, the plasma is supposed to be embedded in a nested set of topologically toroidal magnetic surfaces. The plasma is assumed to be stable so that all transport processes, such as thermal conduction, are determined only by Coulomb collisions between the particles. In this paper, we first compare the experimental results with the predictions of this theory. The actual energy transport rate is found to be much higher than that predicted. This is believed to be due to instability processes in the plasma although there remains the possibility that the idealized or classical theory is not complete. The role of electrostatic and of magnetic instabilities is discussed. Finally, the consequences of a simple model, in which the form of the temperature profile is fixed, are discussed.

Degenerate toroidal magnetohydrodynamic equilibria and minimum B

It is shown that there is a unique configuration of toroidal magnetic surfaces. This configuration corresponds to more than one plasma equilibrium. It is the configuration of isodynamic equilibria. These equilibria include toroidal minimum-B equilibria and the distinction between these unstable systems and the stable minimum-B mirror systems is discussed.

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.

On the congelation of toroidal magnetic surfaces in a moving plasma

The magnetic field freezing in a moving plasma may manifest itself by the existence of the magnetic surfaces. A corollary arising from a known topological theorem on the toroidal surfaces was established and has shown that the magnetic surfaces must be toroidal and nested. The phenomenon of magnetic surfaces freezing give rise to a normal motion defined as a induced drift. Both of these properties require a finite plasma conductibility.

Magnetohydrodynamic theory of plasma equilibrium and stability in stellarators: Survey of results

The main advantage of a stellarator is its capability of steady-state operation. It can be exploited as a reactor if stable plasma confinement can be achieved with β∼10%. Therefore, this limiting pressure value is a key factor in stellarator development. This paper contains a survey of current ideas on the magnetohydrodynamic equilibrium and stability properties of stellarators with sufficiently high pressure. Here, any system of nested toroidal magnetic surfaces generated by external currents is considered a stellarator. Systems produced by helical or equivalent windings, including torsatrons and heliotrons, will be called ordinary stellarators, in contrast to those with spatial axes. It is shown that adequate confinement can be achieved.

Symmetry-breaking toroidal perturbations of a stellarator magnetic field

Analytic expressions for the magnetic field of a toroidal stellarator produced by a set of finite-size helical conductors wound on a torus have been obtained by a direct application of the Biot-Savart law. The effects of toroidal curvature have been included as toroidal perturbations which break the helical symmetry of a corresponding straight system. On expressing the problem of field line behaviour as the problem of a conservative Hamiltonian dynamical system under small non-stationary perturbations, approximate toroidal magnetic surfaces have been obtained by an asymptotic method. Numerical examples for simple cases are given to illustrate the effects of toroidal curvature.

Present and Future Toroidal Confinement Experiments at Princeton Plasma Physics Laboratory

A broad program of research on the confinement of plasma in nested toroidal magnetic surfaces is being carried out at the Princeton Plasma Physics Laboratory. Tokamak research at PPPL is progressing into the phase of technology development, in preparation for the support of a large-scale reactor-oriented effort. World-wide experimental research has brought the stellarator concept to the point of plasma-physical demonstration; at PPPL, the theoretical study of reactor-compatible stellarator design is yielding promising new results. Initial spheromak formation experiments have been successful. The confinement properties of the spheromak will begin to be studied experimentally during the next year.

A fast method to calculate flux surfaces, applied to study helical island structures

Recently two technically simple methods to control plasma-wall interaction in tokamaks have been suggested; one being the helical divertor, which relies on the production of well-defined and well-positioned helical islands on a resonant q- surface , the other being the field line ergodization in the plasma boundary region only. For both methods resonant helical magnetic fields from external helical coils with currents typically 0.01 of the plasma current have to be applied. A fast numerical method is discribed which allows one to calculate toroidal magnetic flux surfaces more efficiently. This method is applied to map the above mentioned magnetic islands and ergodic regions.

Torsatron without toroidal field coils II. configuration ℓ = 2

As a continuation of a previous study, the magnetic configuration of the torsatron with ℓ = 2 helical windings and no toroidal field coils is investigated. A useful volume with closed toroidal magnetic surfaces exists only for sufficiently large pitch angles of the helical windings (> 45°). With increasing pitch angle the rotational transform on the magnetic axis decreases, and high shear is obtained (Θ ∼ 1). In this case, the mirror ratio along the field lines is relatively high (Bmax/Bmin > 1.5). If we require low mirror ratio (Bmax/Bmin < 1.5) the ℓ = 2 configuration is limited to low shear, in contrast to the previously studied case ℓ = 3 which has considerable shear even when the mirror ratio is low.

Negative V″ System with Deep Magnetic Well

Closed nested toroidal magnetic surfaces with very strong negative V″ property are obtained for the combinations of simple toroidal field, bumpy poloidal field, and conventional l = 1 helicalone. Well depth is estimated to be more than 80%.

Field line trajectories in a Skornyakov trap

Skornyakov has proposed a magnetic trap in which a spiral conductor connects diametrally opposite points on the inside surface of a superconducting sphere. He has argued that such a system will have nested toroidal magnetic surfaces and that these surfaces will have favourable stability properties both in terms of ‘average minimum B’ and of ‘shear’. Further he has maintained that the field distribution in such a trap can be closely reproduced by substituting an outer return spiral for the superconducting sphere and traps of this technically simpler type have been constructed. In this note we discuss the results of numerical calculations of the trajectories of magnetic field lines in such a two-spiral trap. We find that there are always some field lines originating near the centre of the system which leave it after one or two transits across the machine. Other lines are contained but trace out surfaces which are not topologically toroidal. Thus traps of the two-spiral type will not contain plasma effectively.

Motion of Toroidal Magnetic Surfaces and Diffusion of a Plasma Ring

The time derivative of the magnetic field can always be represented by a velocity field in a simply connected domain. However, in a toroidal domain the condition for such a representation with a single valued velocity is very stringent. The case considered here is that of initially nested magnetic surfaces in which a multivalued velocity is allowed provided that certain conditions are satisfied. These conditions imply that the component of the velocity normal to the magnetic surfaces is single valued. In this situation there is a convenient representation of the time derivative of the magnetic field. This representation can be applied to the description of resistive diffusion.

Possible Stationary Motion of a Toroidal Plasma

For a given hydromagnetic equilibrium with nested toroidal magnetic surfaces there is a family of divergenceless vectors V such that ▿×(V×B) = 0. This velocity can be interpreted as a mass motion which slightly distorts the equilibrium.

Equilibrium of a three-dimensional plasma filament in a longitudinal magnetic field under steady conditions

Equilibrium conditions are determined for plasmas in toroidal vessels of the figure-of-eight type when the plasma containment time exceeds the time for penetration of the magnetic field through the solenoid windings. In this case some of the magnetic lines of force pass through the solenoid windings and return outside it. The cross-sectional radius of the extreme toroidal magnetic surface lying entirely within the vessel is therefore always less than the cross-sectional radius of the solenoid. When the plasma pressure exceeds some critical value there are no toroidal magnetic surfaces and containment of the plasma is in principle impossible.

Equilibrium of a spatial plasma pinch in a longitudinal magnetic field under steady-state conditions

The equilibrium conditions of plasma in toroidal chambers of the figure-of-eight type are determined for the case in which the plasma is preserved for lpnger than the magnetic field takes to pass through the windings of the solenoid. In this case, some of the magnetic lines of force pass through the solenoid windings, closing outside it. Hence, the radius of the cross section of the extreme toroidal magnetic surface lying as a whole inside the chamber is always smaller than the radius of cross section of the solenoid. For a plasma pressure exceeding a certain critical value, toroidal magnetic surfaces are absent, and containment of the plasma is in principle impossible.

Some questions on the topology of the magnetic field

General requirements are considered for the topology of the field of a magnetic trap which is intended to realize a lengthy thermal insulation of plasma. The author studies the topologic structure of magnetic fields whose lines of force do not leave the boundaries of a finite volume. The necessary condition for the existence of such a field is the presence, among the integral surfaces of the field, of a smooth surfaoe homeomorphous to a torus, where the magnetic field strength does not vanish in any single point. Such fields can, in principle, be realized by means of a superconductor which bounds a toroidal cavity.The stability of the magnetic-field topology has been investigated with respect to small perturbations. The necessary and sufficient condition for the stability of the topology is that the rotation number change in the layer of toroidal magnetic surfaces. A method is proposed for establishing a trap with a magnetic-field strength increasing toward the periphery.Details of a mathematical character have been omitted since they are contained in the author's published papers of which the present paper is fundamentally a short summary.

Magnetic Differential Equations

A necessary and sufficient condition is derived for a magnetic differential equation B·▿r = 0 to have a single-valued solution r, where B is the field of a magnetohydrostatic equilibrium state or, more generally, and field with a system of toroidal magnetic surfaces.