Single Helicity Research Articles

Nonlinear tearing mode: Rutherford regime and global characteristics.

Experiments on the TEXT tokamak show that the temporal evolution of a single helicity tearing mode is dominated by the Rutherford regime. Further analysis of the Mirnov signal and sawtooth activity combined with numerical simulation emphasizes the global characteristics of the tearing mode. 4 figs., 6 refs.

Nonlinear Coupling of the Resistive Tearing Modes under the Unperturbed Shear Flow

The influence of the unperturbed shear flow on the nonlinear evolution of the tearing mode is studied. In the case of single helicity, the shear flow activates the unstable mode which finally saturates to a rigid rotor state. In the case of multiple helicity, a variety of flow patterns is created depending on parameters, and always forms the current bubble soon after the collapse of the 3/2 magnetic island.

Nonlinear behavior of magnetohydrodynamic modes near marginally stable states. I. General formulation and application to the nonresonant kink modes in a reversed field pinch and to the quasi-interchange modes in a tokamak

Two types of nonlinear equations describing the time development of modes near marginally stable states in an inhomogeneous medium are obtained through a general formulation that employs a perturbation expansion around the marginally stable state under the assumption of a single helicity. One type of nonlinear equation has a Hamiltonian form that may be interpreted as the equation of motion for a particle in the potential field of a central force; the other type leads to the Landau equation, which is well known in fluid dynamics. The former equation is obtained when the linear operator is degenerate at the marginally stable state, which corresponds to the case when the linear dispersion relation has a double root for the frequency at the marginally stable state, whereas the latter is obtained when the linear operator is nondegenerate, i.e., the linear dispersion relation has a single root. In the framework of magnetohydrodynamics, the former corresponds to the nonresonant ideal modes, and the latter to the resistive modes. The nonlinear behavior of the nonresonant kink modes in a reversed field pinch and of the quasi-interchange modes in a tokamak are examined with the application of the general formulation. It is shown that new stable helical equilibria bifurcate near the initial axisymmetric equilibrium, so that the plasma nonlinearly oscillates around the new bifurcated equilibrium, which leads to nonlinear saturation of the nonresonant kink modes in a reversed field pinch and of the quasi-interchange mode in a tokamak. The nonlinear stabilizing effects causing bifurcation of the equilibrium are interpreted as quasilinear effects. Compressibility reduces the nonlinear stabilizing effects through changing the quasilinear components and is important even when the modes are near marginally stable states.

Suppression of the tearing mode by energetic ions.

The effect of an energetic ion population on the nonlinear stability of a tearing mode of single helicity is determined using kinetic theory. A dynamical equation is derived for the magnetic-island width. It is shown that the island growth can be suppressed in a tokamak when the energetic ion-density profile peaks just outside the rational surface. Effects due to resistive interchanges and bootstrap currents are included in the model.

An approximate analysis of the interaction between two tearing modes of the same helicity

The interaction between tearing modes with the mode numbersm=1,n=1 andm=2,n=2 is investigated for different initial amplitudes of the modes, using a single helicity approximation, a step-current profile and a time-independent resistivity. Also included are the results on the temporal behaviour of the amplitude of the uncoupled modem=2,n=2 in the situation with identical equilibrium parameters as ours.

Magnetic topology of helical Ohmic steady states

Necessary analytic constraints for a force-free, single harmonic, single helicity, magnetic-field configuration to be an Ohmic equilibrium are derived. These conditions imply that the typical configuration possesses an m=1 helical structure and has only open magnetic flux surfaces. The conclusion is drawn that Taylor’s helical state is not a steady state in the presence of resistive dissipation, no matter how small. Nevertheless, the field configurations that commonly have been associated with Taylor’s minimum energy states retain significance because they may represent either the mean configuration of a temporally fluctuating state or the zero-resistivity limit of a (hypothetical) three-dimensional, Ohmic equilibrium. Examples of single harmonic, single helicity, reversed-field, Ohmic equilibria are presented.

Simulation study of the self-reversal process in the reversed-field pinch based on a non-linearly driven reconnection model

The self-reversal process in the reversed-field pinch is studied in detail by means of a resistive magnetohydrodynamic simulation. It is confirmed that self-reversal can be caused by non-linearly driven reconnection resulting from the m = 1 global kink instability, as was previously proposed by the authors. The dependence of the degree of field reversal on the pinch parameter θ and on the instability mode (resonant and non-resonant) is examined. The results are consistent with theoretical predictions. Taylor's conjecture that the total helicity is a better conserved quantity than the total magnetic energy during the relaxation process is numerically confirmed. It is found that this conjecture can be consistently explained by the non-linearly driven reconnection model. It is also found that the single helicity relaxation process has a definite energy offset from Taylor's minimum energy state. Hence, a totally relaxed state cannot be achieved through the single helicity relaxation process. Finally, the dependence of the reversal process on the resistivity is examined

Analytic theory of the nonlinear m=1 tearing mode

Numerical studies show that the m=1 tearing mode continues to grow exponentially well into the nonlinear regime, in contrast with the slow, ‘‘Rutherford,’’ growth of m>1 modes. A single helicity calculation is presented which generalizes that of Rutherford [Phys. Fluids 16, 1903 (1973)] to the case when the constant-ψ approximation is invalid. As in that theory, the parallel current becomes an approximate flux function when the island size W exceeds the linear tearing layer width. However, for the m=1 mode, W becomes proportional to δB, rather than (δB)1/2 above this critical amplitude. This implies that the convective nonlinearity in Ohm’s law, which couples the m=0 component to the m=1 component, dominates the resistive diffusion term. The balance between the inductive electric field and this convective nonlinearity results in exponential growth. Assuming the form of the perturbed fields to be like that of the linear mode, we find that growth occurs at 71% of the linear rate.

Alfvén continuum with toroidicity

The symmetry property of the magnetohydrodynamic (MHD) wave propagation operator is utilized to express the toroidal eigenmodes as a superposition of the mutually orthogonal cylindrical modes. Because of the degeneracy among cylindrical modes with the same frequency but resonant surfaces of different helicity, the toroidal perturbation produces a zeroth-order mixing of the above modes. The toroidal eigenmodes of frequency ω20 have multiple resonant surfaces, with each surface shifted relative to its cylindrical position and carrying a multispectral content. Thus a single helicity toroidal antenna of frequency ω0 couples strongly to all different helicity resonant surfaces with matching local Alfvén frequency. Zeroth-order coupling between modes in the continuum and global Alfvén modes also results from toroidicity and degeneracy. The perturbation technique used in this study is the MHD counterpart of the quantum-mechanical methods and is applicable through the entire range of the MHD spectrum.

Observation of helical sawtooth oscillations in a tokamak

In the Tokapole II tokamak [Nucl. Fusion 19, 1509 (1979)] the three-dimensional structure of the rapid sawtooth crash that accompanies internal disruptions has been observed. The sawtooth crash is helical, i.e., the decrease in soft x-ray intensity occurs along a q=1 helical contour, in accord with the standard reconnection model. The poloidally asymmetric radial expansion of the crash inward from the q=1 surface provides a measure of the reconnection time consistent with single helicity reconnection from a growing, nonrotating magnetic island.

Numerical Study of Reversed Field Pinch Equilibria

Numerical results on the reversed-field-pinch (RFP) equilibria are presented with use of a partially relaxed state model derived from an energy principle. The relaxation process is assumed to be dominated by the helical tearing mode with a single helicity. The profiles of current densities concentrate more to the magnetic axis in the relaxed state which is dominated by the mode with the lower n toroidal mode number. The maximum available β value with respect to the Suydam criterion is higher in the relaxed state dominated by the higher n mode. Numerical results suggest that experimental RFP plasma has a tendency to relax to the partially relaxed state by the higher n dominant mode.

Crystal Chirality and Helicity of the Helical Spin Density Wave in MnSi. II. Polarized Neutron Diffraction

Recent theoretical and experimental studies have shown the single handed helicity of the helical spin density wave in MnSi. In order to determine unambiguously the sense of the helicity we have carried out detailed experiments using pulsed polarized neutrons. Combining the result with that of the experiment on the crystal chirality, we have determined the left-handed or anticlockwise helicity for the left-handed chirality of the crystal symmetry. This leads to the positive sign of the antisymmetric exchange interaction in the left-handed chirality.

Three-dimensional magnetohydrodynamic studies of the reversed-field pinch

Three-dimensional computational studies of nonlinear resistive megnetohydrodynamic (MHD) instabilities in the reversed-field pinch are presented. It is found that multihelicity mode coupling effects alter the evolution of m=1 modes from that obtained in single helicity. Strong mode coupling to m=0 and m=2 is observed. Magnetic island overlap results in stochastic field lines. At low values of the pinch parameter this is confined to the central region of the pinch; for larger values the stochasticity may extend to the wall. Healing of outer flux surfaces is observed as modes peak and relax. Decay of positive toroidal flux is retarded. Impact on confinement is discussed, and connection is made with experimental observations.

Nonlinear interaction of tearing modes: A comparison between the tokamak and the reversed field pinch configurations

The multiple helicity nonlinear interaction of resistive tearing modes is compared for the tokamak and reversed field pinch configurations using the magnetohydrodynamic equations. Unlike the case of the tokamak disruption, for which this interaction is destabilizing when islands overlap, the nonlinear coupling of the dominant helicities is shown to be a stabilizing influence in the reversed field pinch. The behavior of the coupled instabilities in the two configurations can be understood as a consequence of the stability properties of the nonlinearly driven modes. In the case of the tokamak disruption, quasilinear effects linearly destabilize the dominant driven mode, which then feeds energy to the driving mode. For the reversed field pinch the driven modes remain stable, acting as a brake on the growth of the dominant instabilities. Furthermore, for the reversed field pinch configuration, numerical results indicate that nonlinear coupling of different helicities results in noticeably more rapid saturation of the dominant instabilities than was observed in single helicity studies.

Nonlinear kinetic theory of a single helicity tearing instability

The evolution of a single helicity tearing mode is analyzed by kinetic theory, including modification of the electron orbits by the islands themselves. Using flux coordinates appropriate to island geometry, analytic solutions for the time-dependent island width are obtained. These describe evolution from linear growth to nonlinear saturation in the collisionless limit, and transition from such saturation to slow Rutherford-type growth for increasing finite collisionality. Previous work is thus unified and generalized.

Nonlinear evolution of the resistive interchange mode in the cylindrical spheromak

Results are presented of a study of various aspects of the single helicity nonlinear development of the resistive interchange mode in the cylindrical spheromak. A formulation of the helically symmetric resistive magnetohydrodynamic (MHD) equations that partially separates the ideal MHD characteristics is developed. Mode saturation can occur because of the quasilinear flattening of the pressure profile in the vicinity of the mode rational surface. However, this saturation process is defeated when the plasma overheats and in regions of the plasma where the shear is low. Finite fluid compression has significant, and optimistic, consequences on the long-term nonlinear behavior of this mode. For a tearing mode stable cylindrical spheromak equilibrium configuration with an axial beta value of 6%, complete overlap of the m=1 islands occurs in about 3% of the resistive skin time for a magnetic Reynolds number of S=105. For typical parameters of the S-1 device at Princeton, this time corresponds to nearly one millisecond.

Energy principle with global invariants: Applications

Minimum-energy equilibrium states of a pressureless cylindrical plasma are constructed subject to a recently proposed set of constraints appropriate to turbulent relaxation dominated by a tearing mode of single helicity. Stability to ideal and resistive modes is examined. It is found that the relaxed states of minimum energy are always stable to the dominant mode, as well as a class of other low m and n modes, but that there are typically high n residual instabilities in the case of the pitch branch, and a residual m=3, n=2 resistive instability is found for an otherwise stable narrow window in the tokamak branch. It is argued that the residual instabilities may be compatible with experimental observations.

Ideal magnetohydrodynamic stability limits of a high-β tokamak with superimposed helical fields

The stability of a high-β tokamak with a superimposed single helicity stellarator field is investigated in the context of the sharp boundary surface current model. The analysis treats external ideal magnetohydrodynamic modes including both pressure-driven ballooning effects and current-driven kink instabilities. The results show that the addition of an l = 2 field is unfavorable, lowering the allowable Ohmic heating current and decreasing the maximum stable β. The addition of an l = 3 field, however, is favorable. Both the allowable Ohmic heating current and maximum stable β are increased as the l = 3 amplitude increases.

Energy principle with global invariants

A variational principle is proposed for constructing equilibria with minimum energy in a toroidal plasma. The total energy is minimized subject to global invariants which act as constraints during relaxation of the plasma. These global integrals of motion are preserved exactly for all ideal motions and approximately for a wide class of resistive motions. We assume, specifically, that relaxation of the plasma is dominated by a tearing mode of single helicity. Equilibria with realistic current density and pressure profiles may be constructed in this theory, which is also used here to study current penetration in tokamaks. The second variation of the free energy functional is computed. It is shown that if the second variation of any equilibrium constructed in this theory is positive, the equilibrium satisfies the necessary and sufficient conditions for ideal stability.

Kinetic description of Alfvén wave heating

Heating of tokamak plasmas by Alfvén waves is studied by means of a linearized kinetic model which takes into account electron inertia and Landau damping, finite ion gyroradius, the equilibrium current, and magnetic shear. In cylindrical geometry, a fourth-order set of differential equations in r for the perturbed fields Er and E⊥ is solved numerically for modes driven by a sheet current of single helicity and frequency ω, located between the plasma edge and a conducting wall. Realistic profiles of density, temperature, and safety factor are employed. The energy deposition and density fluctuations as functions of r and the total impedance to be expected in experiments on the pretext tokamak are computed, and optimum conditions for heating are investigated. Mode conversion to the kinetic Alfvén wave and its damping are observed in the computed solutions. The plasma impedance is sensitive to the profiles and mode numbers chosen, and, with two exceptions, is consistent with previous work based on magnetohydrodynamics. Kinetic effects can produce ’’high Q’’ resonant absorption, both for frequencies below the Alfvén continuum (corresponding to discrete stable kink modes) and for frequencies such that the Alfvén resonance approaches the plasma edge (corresponding to normal modes of the kinetic shear wave in cold plasma).