Toroidal Magnetic Surfaces Research Articles

The rapid destruction of toroidal magnetic surfaces

The operation of ITER will require reliable simulations in order to avoid major damage to the device from disruptions. Disruptions are the sudden breakup of magnetic surfaces across the plasma volume—a fast magnetic reconnection. This reconnection can be caused by the growth of perturbations outside of the plasma core causing an ideal perturbation to the core. This causes an increasing ratio of the maximum to the minimum separation, Δmax/Δmin, between neighboring magnetic surfaces. Magnetic reconnection becomes a dominant process when magnetic field lines can quickly interchange connections over a spatial scale ar. This occurs when Δmax/Δmin≳ar/Δd, where Δd is the scale over which non-ideal effects make magnetic field lines indistinguishable. Traditional reconnection theory is fundamentally different. It is a study of the steady-state cancelation of oppositely directed magnetic field components across a thin layer. During more than sixty years, mathematical implications of Faraday's law have been derived that clarify and constrain the physics of fast magnetic reconnection. These are reviewed because they are not commonly known but are needed to understand and to place in context how an ideal magnetic evolution can cause reconnection to quickly become a dominant process no matter how small Δd/ar may be.

Analytical studies of PROTO-SPHERA equilibria

Analytical solutions of the Grad–Shafranov equilibrium equation in simply connected plasma configurations, comprised of toroidal magnetic surfaces and open surfaces connected to electrodes, are reviewed and generalised. The Grad–Shafranov equation is linearised introducing assumptions on plasma current and pressure, which preserve regularity of solutions on the symmetry axis, as required for a simply connected geometry. Particular solutions are found by separation of variables both in cylindrical coordinates and in spherical ones. Equilibria that model local or global features of PROTO-SPHERA plasmas are constructed by combining a few particular solutions.

Symplectic Maps for Diverted Plasmas

Nowadays, divertors are used in the main tokamaks to control the magnetic field and to improve the plasma confinement. In this article, we present analytical symplectic maps describing Poincar\'e maps of the magnetic field lines in confined plasmas with a single null poloidal divertor. Initially, we present a divertor map and the tokamap for a diverted configuration. We also introduce the Ullmann map for a diverted plasma, whose control parameters are determined from tokamak experiments. Finally, an explicit, area-preserving and integrable magnetic field line map for a single-null divertor tokamak is obtained using a trajectory integration method to represent toroidal equilibrium magnetic surfaces. In this method, we also give examples of onset of chaotic field lines at the plasma edge due to resonant perturbations.

Local synthesis of 3D magnetic surfaces

Local synthesis of nested 3D toroidal magnetic surfaces is carried out on the basis of the general theory of surfaces by using magnetic coordinates (generally unknown a priori). An equilibrium magnetic surface is calculated by specifying two functions on the surface (the absolute value of the magnetic field and the distance to the nearest magnetic surface) and three parameters (the rotational transform, pressure gradient, and poloidal current). The choice of the parameters is restricted by the requirement that the surface should be closed toroidally. A method of synthesis of a closed magnetic surface is proposed when two functions—the absolute value of the magnetic field and the major radius—are specified. A set of harmonics of a new type of poloidally preudosymmetric configuration (a toroidal mirror with a large mirror ratio and small rotational transform) is obtained.

Implementation of Hamada principle in calculations of nested 3-D equilibria

Plasma confinement is based on the use of nested toroidal magnetic surfaces. In axisymmetric configurations the nestedness is provided by the existence of a flux function describing the magnetic surfaces explicitly. In the case of a three-dimensional magnetic field, the nested surfaces represent an exception. More typically, magnetic islands are formed on the resonant surfaces. The islands could degrade the plasma performance. The rigorous condition for the existence of nested surfaces without islands was formulated by Hamada (Nucl. Fusion, vol. 2, 1962, pp. 23–37) but was not implemented directly into numerical codes used, for example, for designing the stellarator configurations. This paper introduces a method of implementation of the Hamada principle in numerical algorithms. The proposed approach allows for simple linearized equilibrium equations (LEE) and potentially very efficient three-dimensional calculations of nested equilibria.

Experimental Measurement of Asymmetric Fluctuations of Poloidal Magnetic Field in Damavand Tokomak at Different Plasma Currents

Toroidal and Poloidal magnetic fields have an important effect on the tokomak topology. Damavand Tokomak is a small size tokomak characterized with k = 1.2, B t = 1T, R 0 = 36 cm, maximum plasma current is about 35 KA with a discharge time of 21 ms. In this experimental work, the variation of poloidal magnetic field on the torodial cross section is measured and analyzed. In order to measure the polodial magnetic field, 18 probes were installed on the edge of tokomak plasma with ∆θ = 18°, while a limiter was installed inside the torus. Plasma current, I p, induces a polodial magnetic field, B p, smaller than the torodial magnetic field B t. Magnetic lines B produced as a combination of B t and B p, are localized on the nested toroidal magnetic surfaces. The presence of polodial magnetic field is necessary for particles confinement. Mirnov oscillations are the fluctuations of polodial magnetic field, detected by magnetic probes. Disrupted instability in Tokomak typically starts with mirnov oscillations which appear as fluctuations of polodial magnetic field and is detected by magnetic probes. Minor disruptions inside the plasma can contain principal magnetic islands and their satellites can cause the annihilation of plasma confinement. Production of thin layer of turbulent magnetic field lines cause minor disruption. Magnetic limiter may cause the deformation of symmetric equilibrium configuration and chaotic magnetic islands reveal in plasma occurring in thin region of chaotic field lines close to their separatrix. The width of this chaotic layer in the right side of poloidal profile of Damavand Tokomak is smaller than the width in the left side profile because of Shafranov displacement. Ergodic region in the left side of profile develops a perturbation on the magnetic polodial field lines, B p, that are greater in magnitude than that in the right side, although the values of B p on the left side are smaller than that on the right side of the profile. The Left side of profile is close to the principal magnetic axis and the right side is away from Z axis of Tokamak.

Analytical solutions for Tokamak equilibria with reversed toroidal current

In tokamaks, an advanced plasma confinement regime has been investigated with a central hollow electric current with negative density which gives rise to non-nested magnetic surfaces. We present analytical solutions for the magnetohydrodynamic equilibria of this regime in terms of non-orthogonal toroidal polar coordinates. These solutions are obtained for large aspect ratio tokamaks and they are valid for any kind of reversed hollow current density profiles. The zero order solution of the poloidal magnetic flux function describes nested toroidal magnetic surfaces with a magnetic axis displaced due to the toroidal geometry. The first order correction introduces a poloidal field asymmetry and, consequently, magnetic islands arise around the zero order surface with null poloidal magnetic flux gradient. An analytic expression for the magnetic island width is deduced in terms of the equilibrium parameters. We give examples of the equilibrium plasma profiles and islands obtained for a class of current density profile.

Kinetic axisymmetric gravitational equilibria in collisionless accretion disk plasmas

A theoretical treatment is presented of kinetic equilibria in accretion disks (AD) around compact objects, for cases where the plasma can be considered as collisionless. The plasma is assumed to be axisymmetric and to be acted on by gravitational and electromagnetic fields; in this paper, the particular case is considered where the magnetic field admits a family of toroidal magnetic surfaces, which are locally mutually nested and closed. It is pointed out that there exist asymptotic kinetic equilibria represented by generalized bi-Maxwellian distribution functions and characterized by primarily toroidal differential rotation and temperature anisotropy. It is conjectured that kinetic equilibria of this type can exist which are able to sustain both toroidal and poloidal electric current densities, the latter being produced via finite Larmor-radius effects associated with the temperature anisotropy. This leads to the possibility of existence of a new kinetic effect—referred to here as a “kinetic dynamo effect”—resulting in the self-generation of toroidal magnetic field even by a stationary plasma, without any net radial accretion flow being required. The conditions for these equilibria to occur, their basic theoretical features, and their physical properties are all discussed in detail.

Plasma equilibrium in 3D magnetic confinement systems and soliton theory

Single-valued conformal flux (magnetic) coordinates can always be introduced on arbitrary toroidal magnetic surfaces. It is shown how such coordinates can be obtained by transforming Boozer magnetic coordinates on the surfaces. The metrics is substantially simplified and the coordinate grid is orthogonalized at the expense of a more complicated representation of the magnetic field in conformal flux coordinates. This in turn makes it possible to introduce complex angular flux coordinates on any toroidal magnetic surface and to develop efficient methods for a complex analysis of the geometry of equilibrium magnetic surfaces. The complex analysis reveals how the plasma equilibrium problem is related to soliton theory. Magnetic surfaces of constant mean curvature are considered to exemplify this relationship.

Geometric properties of magnetic field lines on toroidal magnetic surfaces in the context of plasma equilibrium

An analysis of plasma equilibrium in a magnetic confinement system includes studies of how the shape of the magnetic surfaces is distorted with varying magnitude and profile of the plasma pressure. Such studies allow one, in particular, to determine the maximum β value consistent with equilibrium, βeq, i.e., the maximum plasma pressure above which the equilibrium in a confinement system under analysis is impossible. Since the magnetic field lines form magnetic surfaces, their global relationship with equilibrium is obvious. Here, special attention is paid to a local relationship between equilibrium and geometric properties of the magnetic field lines.

Eigenvalue problems for Beltrami fields arising in a three-dimensional toroidal magnetohydrodynamic equilibrium problem

A generalized energy principle for finite-pressure, toroidal magnetohydrodynamic (MHD) equilibria in general three-dimensional configurations is proposed. The full set of ideal-MHD constraints is applied only on a discrete set of toroidal magnetic surfaces (invariant tori), which act as barriers against leakage of magnetic flux, helicity, and pressure through chaotic field-line transport. It is argued that a necessary condition for such invariant tori to exist is that they have fixed, irrational rotational transforms. In the toroidal domains bounded by these surfaces, full Taylor relaxation is assumed, thus leading to Beltrami fields ∇×B=λB, where λ is constant within each domain. Two distinct eigenvalue problems for λ arise in this formulation, depending on whether fluxes and helicity are fixed, or boundary rotational transforms. These are studied in cylindrical geometry and in a three-dimensional toroidal region of annular cross section. In the latter case, an application of a residue criterion is used to determine the threshold for connected chaos.

Possible Global Magneto-Fluid Structure of the Stellar Convection Zone

Structures of the magnetic field and velocity in stars are discussed based on the mean field MHD equations. A special case is presented, where the solution is constructed by the Beltrami solution in the stellar convection zone with the symmetry in the azimuthal direction. Magnetic field lines form concentric toroidal magnetic surfaces. The cross-helicity dynamo mechanism induces a mean flow of plasmas. The structure of this driven flow is also shown to constitute toroidal surfaces. Considering the symmetry and the relation of this toroidal magnetic structure with the polarity, it is shown that the latitudinal component of this flow is pole-ward in the northern as well as southern hemispheres. This gives an insight into the role of the magnetic field for the meridional flow.

Density limit for electron plasmas confined by magnetic surfaces

The density limit for electron plasmas confined by toroidal magnetic surfaces is investigated. In a cylinder, the well-known limit is the Brillouin density, nB≡ϵ0B2∕2me. In an axisymmetric torus, the confining region shifts outward in major radius, and this shift is shown to equal half the plasma radius when n∕nB≈ι2a∕R0, where ι=1∕q is the rotational transform of the magnetic field and a∕R0 is the inverse aspect ratio of the torus. In a nonaxisymmetric torus, electron confinement is found to be lost due to stochasticity effects when n∕nB≈(ι2∕8M2)(a∕R0)2∕δMN. The asymmetry amplitudes δMN are the fractional variations in n∕B2 on a magnetic surface in the poloidal mode number M and the toroidal mode number N≈ιM.

Plasma effects on the location of the outermost magnetic surface

In both tokamaks and stellarators, the plasma boundary is often defined by the separatrix between magnetic field lines that form toroidal surfaces and field lines that strike solid objects such as the chamber walls or divertor plates. Extremely small magnetic perturbations can move this separatrix radially inward. However, in the presence of a plasma, this radial movement of the separatrix causes a complicated spatial distribution of plasma pressure, which gives rise to currents that can greatly modify or even shield out the magnetic perturbations. An estimate is given of the minimum magnetic perturbation that is required to break up toroidal magnetic surfaces while avoiding plasma shielding.

Thermal energy confinement of high-triangularity ELMy H-mode plasmas in JT-60U

High-triangularity ELMy H-mode experiments (δx~0.45) have been carried out in JT-60U. In the low-density regime (n̄e/nGW<0.5), the discharges are characterized by type-I ELMs and high-triangularity plasmas produced higher H-mode pedestal pressure than low-triangularity plasmas (δx~0.20). In the high-density regime (n̄e/nGW>0.5), at fixed neutral beam heating power type-III ELMs are generated and the pedestal pressure reduces gradually with density whereas the ELM frequency at low triangularity is much lower at a given density. Compared to low-triangularity plasmas at low densities, higher pedestal temperature was obtained at high triangularity and the core temperature was also improved by almost the same factor. The profiles of temperature are stiff in the sense that there seems to exist a minimum scale length of temperature gradient. Since plasma shaping affects strongly the peripheral region of toroidal magnetic flux surfaces, the improvement of the edge stability by triangularity leads to higher pedestal temperature, which in turn raises the core temperature.

Some theoretical problems of the toroidal plasma equilibrium

Two problems of toroidal magnetic plasma confinement systems are considered. The first one is connected with advanced helical systems. The peculiarity of geometry of these stellarator systems with the improved confinement is discussed. Some additional ways for further optimization are marked too. The second problem is concerned with the island structure in toroidal systems. From the very beginning, the theoretical research on plasma equilibrium in stellarators was based mainly on the assumption that the toroidal magnetic surfaces are nested. It means that there is a unique magnetic axis which serves as the edge of the azimuthal coordinate surface θ = const. To avoid the difficulties with such a coordinate in the case of many magnetic axes that result from islands, the tokamak-like representation of the magnetic field B which does not use the coordinate θ is proposed.

Analytical models of axisymmetric toroidal magnetic fields withnon-circular flux surfaces

Axisymmetric toroidal magnetic configurations relevant to current andfuture tokamak experiments are analytically modelled taking into account anarbitrary aspect ratio, noncircularity and up-down asymmetry of fluxsurface shapes. It is shown that equilibrium tokamak magnetic fields can besatisfactorily described using only four geometric parameters, whichdetermine the shape and position of the flux surfaces. These are theShafranov shift, the elongation, and the parameters of triangularity and ofup-down asymmetry. To cover the variety of observed flux surface shapes,three different analytical expansions are provided. Comparison of themodelled with the equilibrium flux surfaces in NSTX (Princeton), JET(Abingdon) and TCV (Lausanne) demonstrates good coincidence.

Collisional Neoclassical Transport for Elliptic Magnetic Surfaces with Triangularity and Grad–Shafranov Shift

An analysis of neoclassical transport for tokamaks in the collisional regime has been carried out for toroidal elliptic magnetic surfaces with triangularity and Grad–Shafranov shift. Several formulas for this general case have been obtained. Several numerical results are presented here with different ellipticity, triangularity and Grad–Shafranov shift. The elliptic elongation becomes an important parameter determining the diffusion of charged particles.

Pfirsch–Schlüter transport for toroidal elliptic magnetic surfaces in tokamak

Transport equations in the collisional regime have been found for very general geometries in tokamaks, using resistive magnetohydrodynamic equations, neglecting the toroidal field Eφ and assuming toroidal axisymmetry. Detailed analytic results have been obtained for families of elliptic magnetic surfaces. In the limit case of equal semiaxis and large aspect ratio, the well known Pfirsch–Schlüter result is obtained.

Hamiltonian approach to the magnetostatic equilibrium problem

The purpose of this paper is to investigate the classical scalar-pressure magnetostatic equilibrium problem for nonsymmetric configurations in the framework of a Hamiltonian approach. Requiring that the equilibrium admits locally a family of nested toroidal magnetic surfaces, the Hamiltonian equations describing the magnetic flux lines in such a subdomain are obtained for a general canonical curvilinear coordinate system. The properties of such a coordinate system are investigated and a representation of the magnetic field is obtained. Its basic feature is that the magnetic field must fulfill suitable periodicity constraints to be imposed on arbitrary rational magnetic surfaces for general nonsymmetric toroidal equilibria, i.e., it is quasisymmetric. Implications for the existence of magnetostatic equilibria are pointed out. In particular, it is proven that a generalized equilibrium equation exists for such quasisymmetric equilibria, which extends the Grad–Shafranov equation to fully three-dimensional configurations. As an application, a representation is obtained for generalized helically symmetric equilibrium, extending the definition given by Nührenberg and Zille [Phys. Lett. A 129, 113 (1988)]. Since the new representation overcomes the inconsistency exhibited by the previous representation near the magnetic axis, pointed out by Garren and Boozer [Phys. Fluids B 3, 2805, 2822 (1991)], it appears potentially useful to interpret the numerical findings of quasihelical equilibria obtained so far.