Chain Of Magnetic Islands Research Articles

An Analytic Approach to Developing Transport Threshold Models of Neoclassical Tearing Modes in Tokamaks

Transport threshold models of neoclassical tearing modes in tokamaks are investigated analytically. An analysis is made of the competition between strong transverse heat transport, on the one hand, and longitudinal heat transport, longitudinal heat convection, longitudinal inertial transport, and rotational transport, on the other hand, which leads to the establishment of the perturbed temperature profile in magnetic islands. It is shown that, in all these cases, the temperature profile can be found analytically by using rigorous solutions to the heat conduction equation in the near and far regions of a chain of magnetic islands and then by matching these solutions. Analytic expressions for the temperature profile are used to calculate the contribution of the bootstrap current to the generalized Rutherford equation for the island width evolution with the aim of constructing particular transport threshold models of neoclassical tearing modes. Four transport threshold models, differing in the underlying competing mechanisms, are analyzed: collisional, convective, inertial, and rotational models. The collisional model constructed analytically is shown to coincide exactly with that calculated numerically; the reason is that the analytical temperature profile turns out to be the same as the numerical profile. The results obtained can be useful in developing the next generation of general threshold models. The first steps toward such models have already been made.

Fast recovery of vacuum magnetic configuration of the W7-X stellarator using function parametrization and artificial neural networks

W7-X, a five-period, fully optimized stellarator, currently under construction at IPP-Greifswald, Germany, is built with superconducting coils to show the steady state capability of stellarators. However, the steady state needs continuous equilibrium information for monitoring and controlling the discharge. Although the timescales are long compared with tokamaks, the computational effort for calculating three-dimensional magnetohydrodynamic equilibria is also orders of magnitude higher. This has led us to start the development of a fast equilibrium recovery for W7-X. As a starting point and also for investigating the richness of magnetic configurations, of which only nine physically interesting examples have been examined till now, a fast recovery of vacuum magnetic configurations, described by the flux surface geometry and profile parameters, is carried out using the function parametrization (FP) and artificial neural networks (ANN) methods. Additionally, we parametrize the detectable major magnetic island chains (5/6, 5/5, 5/4) in the form of their locations, , and their width, w(is). The quality of FP recovery is compared with ANN, where the vacuum parameters are non-linearly regressed in terms of linear combinations of the coil currents. The results show that a quadratic FP model is generally sufficient for a good recovery of the parameters which are related to the magnetic axis. However, a cubic model is necessary for modelling accurately the magnetic island-related parameters. ANN models offer no improvement over the cubic FP model.

Formation of nonlinear magnetic islands via trapped electrons in the lower-hybrid range

Coherent nonlinear stationary state is studied in the frequency and spatial domain of electromagnetic lower-hybrid waves in a beam-plasma configuration that is subjected to the oblique linear Buneman instability. In the presence of magnetic shear, a solution is found in the form of a nonlinear chain of magnetic islands. Such equilibrium may represent the saturated state of fast collisionless magnetic reconnection that is reached after the reconnection has ended. It is demonstrated that the topological constraints that hamper the reconnection of magnetic field lines in the fluid representation are efficiently removed by the nonlinear currents and space charges of trapped electrons. Two distinct trapping mechanisms are distinguished. Besides the electrostatic trapping of particles in the self-consistent minima of potential energy, a different type of magnetic trapping is identified that is responsible for the creation of thin current sheets at the separatrices between open and closed magnetic field lines. Combined effects of the resulting two types of nonlinear phase-space vortices provide for a topological transformation of a sheared magnetic field into a single chain of magnetic islands coupled with a double chain of hydrodynamic vortices.

Magnetic Reconnection Mediated by Lower-Hybrid Phase-Space Vortices

Steady state collisionless magnetic reconnection is studied in the frequency and spatial range of lower-hybrid waves that are linearly unstable due to the Buneman instability. We demonstrate that nonlinear currents and charges associated with electron phase-space vortices can efficiently mediate a topological transformation of a sheared magnetic field, yielding a single chain of magnetic islands coupled with a double chain of hydrodynamic vortices.

Improved evolution equations for magnetic island chains in toroidal pinch plasmas subject to externally applied resonant magnetic perturbations

An improved set of island evolution equations is derived that incorporates the latest advances in MHD (magnetohydrodynamical) theory. These equations describe the resistive/viscous-MHD dynamics of a nonlinear magnetic island chain, embedded in a toroidal pinch plasma, in the presence of a programmable, externally applied, resonant magnetic perturbation. A number of interesting example calculations are performed using the new equations. In particular, an investigation is made of a recently discovered class of multiharmonic resonant magnetic perturbations that have the novel property that they can lock resonant island chains in a stabilizing phase.

Control of tearing modes in toroidal fusion experiments using “designer” error fields

It is demonstrated that a magnetic island chain formed by a saturated tearing instability in a toroidal magnetic fusion device can lock to a special class of externally generated magnetic perturbation in a stabilizing phase. The theoretical apparatus needed to design such perturbations is outlined. These special perturbations—which are termed “designer” error fields—could be used to control the amplitudes of tearing modes in toroidal magnetic fusion experiments without the requirement of fast phase modulation. This type of control would be far more feasible in a reactor environment than conventional active feedback control via external magnetic perturbations.

Nonlinear dynamics of feedback modulated magnetic islands in toroidal plasmas

An analysis is presented of the dynamics of a helical magnetic island chain embedded in a toroidal plasma, in the presence of an externally imposed, rotating, magnetic perturbation of the same helicity. Calculations are carried out in the large aspect-ratio, zero-β, resistive magnetohydrodynamical limit, and incorporate a realistic treatment of plasma viscosity. There are three regimes of operation, depending on the modulation frequency (i.e., the difference in rotation frequency between the island chain and the external perturbation). For slowly modulated islands, the perturbed velocity profile extends across the whole plasma. For strongly modulated islands, the perturbed velocity profile is localized around the island chain, but remains much wider than the chain. Finally, for very strongly modulated islands, the perturbed velocity profile collapses to a boundary layer on the island separatrix, plus a residual profile which extends a few island widths beyond the separatrix. Analytic expressions are obtained for the perturbed velocity profile, the island equation of motion, and the island width evolution equation in each of these three regimes. The ion polarization correction to the island width evolution equation, which has previously been reported to be stabilizing, is found to be destabilizing in all three regimes.

Magnetic islands in a magnetized plasma with electron flow

A system of two coupled nonlinear equations describing magnetic electron modes in a magnetized inhomogeneous plasma with a spatially dependent electron flow is derived. For a homogeneous basic state electron concentration, the two equations can be decoupled, and a nonlinear solution for the magnetic field in the form of a traveling stationary vortex chain of magnetic islands is found.

Plasma transport near the separatrix of a magnetic island

The simplest nontrivial model of transport across a magnetic island chain in the presence of collisionless streaming along the magnetic field is solved by a Wiener–Hopf procedure. The solution found is valid provided the boundary layer about the island separatrix is narrow compared to the island width. The result demonstrates that when this assumption is satisfied the flattened profile region is reduced by the boundary layer width. The calculation is similar to the recent work by Fitzpatrick [Phys. Plasmas 2, 825 (1995)] but is carried out in the collisionless, rather than the collisional, limit of parallel transport, and determines the plasma parameters on the separatrix self-consistently.

Nonlinear magnetic chains associated with plasma flow

Nonlinear equations describing magnetic electron mode in a nonuniform, unmagnetized plasma, with a space-dependent flow are derived. A set of two coupled equations, for the self-generated magnetic field and electron temperature perturbation is obtained, and two kinds of solutions, in the form of a moving single and double chain of magnetic islands are found.

Nonlinear tearing mode and vortex chains

We study the nonlinear stage of a tearing mode, whose island width exceeds the tearing layer thickness, and the wavelength is of the order of collisionless skin depth. A coherent solution is found in the form of a moving vortex chain. It is the result of a self-organization process, which adjusts the profile of the sheared poloidal magnetic field and excites a localized perpendicular sheared plasma flow, consisting of three counterstreaming jets. A numerical solution shows a twin chain of plasma vortices, coupled with a single chain of magnetic islands, whose width is of the order of collisionless skin depth. Adiabatic evolution of the vortex chain in the presence of small viscosity reveals its finite lifetime. The chain destruction may occur either directly, or through a sequence of bifurcations (corresponding to abrupt changes of the vortex chain parameters) to magnetic field stochastization within a layer of the collisionless skin depth scale, which occurs before the magnetic island overlapping takes place. This provides a new mechanism for the anomalous transport.

Hamiltonian Maps for Heliac Magnetic Islands

Magnetic islands in toroidal heliac stellarator vacuum fields are explored with Hamiltonian chaos theory and the associated area-preserving maps. Magnetic field line island chains are examined first analytically, with perturbation theory, and then numerically to produce Poincaré sections, which are compared with H−1 Heliac stellarator puncture plot diagrams. Rotational transform profiles are chosen to permit the comparison of twist map and nontwist map predictions with field line behaviour computed by a field line tracing computer program and observed experimentally.

Nonlinear vortex chain associated with tearing mode

A nonlinear stage in the development of the tearing mode instability is studied analytically using a two-fluid plasma description. A coherent nonlinear solution is constructed in the form of a moving single chain of magnetic islands, coupled with a double chain of hydrodynamic vortices. The adiabatic theory of the chain evolution in the presence of finite electron viscosity indicates its finite lifetime and the possibility of intermittency, which provides a new mechanism for the magnetic field stochastization within a layer of a collisionless skin depth scale. This process occurs before the magnetic island overlapping and the global stochastization takes place.

Stability of coupled tearing and twisting modes in tokamaks

A dispersion relation is derived for resistive modes of arbitrary parity in a tokamak plasma. At low mode amplitude, tearing and twisting modes which have nonideal magnetohydrodynamical (MHD) behavior at only one rational surface at a time in the plasma are decoupled via sheared rotation and diamagnetic flows. At higher amplitude, more unstable ‘‘compound’’ modes develop which have nonideal behavior simultaneously at many surfaces. Such modes possess tearing parity layers at some of the nonideal surfaces, and twisting parity layers at others, but mixed parity layers are generally disallowed. At low mode number, ‘‘compound’’ modes are likely to have tearing parity layers at all of the nonideal surfaces in a very low-β plasma, but twisting parity layers become more probable as the plasma β is increased. At high mode number, unstable twisting modes which exceed a critical amplitude drive conventional magnetic island chains on alternate rational surfaces, to form an interlocking structure in which the O points and X points of neighboring chains line up.

Field line stochasticity during the m=1 reconnection

A full three-dimensional nonlinear magnetohydrodynamic (MHD) simulation using a high number of modes supports a model of the m=1 asymmetric resistive reconnection in tokamaks. The current sheet associated with the internal kink mode exhibits a toroidal modulation mainly due to a finite toroidal pressure effect that leads to an asymmetric m=1 magnetic island. The main effect of the modulation is to drive the growth of (m±1,m) magnetic islands in the plasma on a time scale imposed by the m=1 reconnection process. As the reconnection proceeds, different magnetic island chains with m of order 10 overlap, leading to an annular stochastic region around the m=1 separatrix. The diffusion in this stochastic layer depends exponentially on the current state of the reconnection, and can reach a value high enough to expel the electronic temperature in less than about 100 μsec if the modulation is greater than about 1.0, that is, for low-shear and/or high-pressure gradient equilibrium values on the q=1 surface.

Chaotic diffusion across a magnetic island due to a single low-frequency electrostatic wave

It is shown that the guiding center motion around a single chain of magnetic islands in a sheared slab can become chaotic in the presence of a single prescribed low-frequency electrostatic wave. This process leads to radial diffusion across the islands without magnetic braiding. The chaotic diffusion appears to be selective in velocity space. Using realistic values of the physical parameters for a fusion plasma, it is concluded that this process can be effective in usual conditions: with the observed islands, and electronic field values corresponding to measured density fluctuations in the TFR Tokamak, this diffusion concerns ions with velocities higher than thermal, and almost all of the electron population. The consequences for radial diffusion are discussed.

Nonlinear evolution of magnetic island in a sheared magnetic field with uniform plasma background

The paper is devoted to the nonlinear evolution of the single tearing type perturbation in a plasma configuration with magnetic shear. The primary effect in the study of the nonlinear dynamics of the magnetic perturbation is the modification of the orbits of resonant particles caused by the chain of magnetic islands, moving along the magnetic surface due to drift effects. The structure of a parallel electric field and its influence on ion dynamics, including the so-called 'integral' effects dealing with the finiteness of the Larmor radius, are incorporated in the nonlinear analysis. Nonlinear growth rates and saturation widths of the magnetic islands are evaluated. The drift tearing mode is found to have metastable properties: the perturbations of finite amplitude which are damped in the linear approximation appear to be nonlinearly unstable. The authors discuss the energetics of such a hard regime of instability onset and the nonlinear stabilization mechanisms. The influence of a uniform stationary plasma background (BG) is considered. The presence of the cold dense BG slows down the growth of aperiodical instability and decreases the threshold of its nonlinear saturation. The field-aligned oscillation of BG electrons has a strong stabilizing effect on the drift tearing mode, even if their density is relatively small. When the density of the BG plasma exceeds the density of the current-carrying plasma, drift tearing modes cannot exist. The presence of the BG plasma strengthens the metastable properties of configurations with magnetic shear. The nonlinear growth rates and thresholds of linear and nonlinear stabilization are evaluated for a wide range of parameters (densities, temperatures) of current-carrying and BG plasmas.

Relaxation of magnetotail plasmas with field‐aligned currents

Recently, equilibria of low free energy have been constructed for magnetotail plasmas (Bhattacharjee, 1987). This model is extended to obtain axisymmetric states which have the structure of flux ropes containing field‐aligned currents. An interesting feature of these equilibria is a chain of magnetic islands along the tail axis. Theoretical results are compared with observations from ISEE 1 and ISEE 2 on flux ropes (Elphic et al., 1986). A sufficient condition for stability is formulated.

Relaxation of magnetotail plasmas

A quasi‐thermodynamic model is presented for the relaxation of magnetotail plasmas during substorms, followed by quiet times. It is proposed that the plasma relaxes to a state of low‐potential energy subject to a small number of global constraints. The constraints are exactly preserved by all ideal motions and, approximately, by a wide class of nonideal motions of the plasma undergoing magnetic reconnection. A variational principle which minimizes the free energy predicts the relaxed state. Exact, two‐dimensional solutions of the relaxed state are obtained. A universal feature of the exact solutions is a chain of magnetic islands along the tail axis. Sufficient conditions for the stability of relaxed states are obtained from the second variation of the free‐energy functional.

Reconnection studies in field-reversed configurations

Internal magnetic field measurements are used to study the reconnection process near the ends of a low-compression field-reversed theta pinch. The measurements indicate that the addition of independently driven magnetic mirror coils at the ends are effective in inducing rapid reconnection and in minimizing plasma losses out the ends. Fast reconnection leading to the formation of a chain of magnetic islands along the length of the coil is observed on a time scale of a 100 nsec. These islands coalesce on a time scale of 1–2 μsec. Measured values of the resistivity are two orders of magnitude greater than classical during the implosion phase. The resistivity during the equilibrium phase is four times the classical value. Axial contraction and the n=2 rotational instability are not observed to occur in these experiments. Sudden termination of the configurations is thought to be caused by rapid loss of the internal field. This flux dissipation may be a consequence of electron cooling by the crossing of a radiation barrier. An oxygen impurity concentration of 2% is enough to account for the power dissipated during the collapse.