Saturated Tearing Mode Research Articles

Direct prediction of nonlinear tearing mode saturation using a variational principle

It is shown that the variational principle of multi-region relaxed magnetohydrodynamics (MRxMHD) can be used to predict the stability and nonlinear saturation of tearing modes in strong guide field configurations without resolving the dynamics and without explicit dependence on the plasma resistivity. While the magnetic helicity is not a good invariant for tearing modes, we show that the saturated tearing mode can be obtained as an MRxMHD state of a priori unknown helicity by appropriately constraining the current profile. The predicted saturated island width in a tearing-unstable force-free slab equilibrium is shown to reproduce the theoretical scaling at small values of Δ′ and the scaling obtained from resistive magnetohydrodynamics simulations at large Δ′.

Nonlocal transport in toroidal plasma devices

Collisional particle transport is examined in several toroidal plasma devices in the presence of perturbations typical of modes leading up to a disruption, of saturated tearing modes, or of unstable Alfvén modes. The existence of subdiffusive transport for electrons is found to occur in some cases at very low mode amplitudes and to also exist even for ions of high energy. Orbit resonances can produce long time correlations and traps for particle trajectories at perturbation amplitudes much too small for the orbits to be represented as uniformly chaotic. The existence and nature of subdiffusive transport is found to depend on the nature of the mode spectrum and frequency as well as the mode amplitudes.

Nonlocal transport in the reversed field pinch

Several heuristic models for nonlocal transport in plasmas have been developed, but they have had a limited possibility of detailed comparison with experimental data. Nonlocal aspects introduced by the existence of a known spectrum of relatively stable saturated tearing modes in a low current reversed field pinch (RFP) offers a unique possibility for such a study. A numerical modeling of the magnetic structure and associated particle transport is carried out for the RFP experiment at the Consorzio RFX, Padova, Italy. A reproduction of the tearing mode spectrum with a guiding center code (White and Chance 1984 Phys. Fluids 27 2455) reliably reproduces the observed soft x-ray tomography. Following particle trajectories in the stochastic magnetic field shows the transport across the unperturbed flux surfaces to be due to a spectrum of Lévy flights, with the details of the spectrum position dependent. The resulting transport is subdiffusive, and cannot be described by Rechester–Rosenbluth diffusion, which depends on a random phase approximation. If one attempts to fit the local transport phenomenologically, the subdiffusion can be fit with a combination of diffusion and inward pinch (Spizzo et al 2007 Phys. Plasmas 14 102310). It is found that whereas passing particles explore the stochastic field and hence participate in Lévy flights, the trapped particles experience normal neoclassical diffusion.A two fluid nonlocal Montroll equation is used to model this transport, with a Lévy flight defined as the motion of an ion during the period that the pitch has one sign. The necessary input to the Montroll equation consists of a time distribution for the Lévy flights, given by the pitch angle scattering operator, and a distribution of the flight distances, determined numerically using a guiding center code. Results are compared with the experiment. The relation of this formulation to fractional kinetics is also described.

Simulation of saturated tearing modes in tokamaks

A quasi-linear model, which includes the effect of the neoclassical bootstrap current, is developed for saturated tearing modes in order to compute magnetic island widths in axisymmetric toroidal plasmas with arbitrary aspect ratio and cross-sectional shape. The model is tested in a simple stand-alone code and is implemented in the BALDUR [C. E. Singer et al., Comput. Phys. Commun. 49, 275 (1982)] predictive modeling code. It is found that the widths of tearing mode islands increase with decreasing aspect ratio and with increasing elongation. Also, the island widths increase when the gradient of the current density increases at the edge of the islands and when the current density inside the islands is suppressed, such as the suppression caused by the near absence of the bootstrap current within the islands. In simulations of tokamak discharges, it is found that tearing mode island widths oscillate in time in response to periodic sawtooth crashes. The local enhancements in the transport produced by magnetic islands have a noticeable effect on global plasma confinement in simulations of low aspect ratio, high beta tokamaks, where saturated tearing mode islands can occur with widths that are greater than 15% of the plasma minor radius.

Drift stabilization of tearing modes and the L–H transition

The L to H-mode transition is accompanied by a drop in the magnetic signal of a mode with low toroidal mode number, usually n=1. Quasilinear drift stabilization can cause a saturated tearing mode to collapse, under the conditions at the edge of a tokamak plasma. The criterion for stabilization gives a relation of temperature to magnetic field T∝Bα, with 6/7≥α≥2/3. The collapse of the island could trigger the H mode by reducing the stochastic magnetic field layer around the magnetic separatrix. This would improve edge confinement, helping to form a transport barrier. Improvement of electron edge confinement could make the electric field more negative, producing the rotation seen in experiments.

Tokamak error fields and locked modes

The effects of externally applied, resonant magnetic field perturbations on tokamak plasmas are discussed. An analytical model assuming tokamak ordering and narrow islands clarifies the conditions under which error fields are amplified or diminished by self-consistent effects. A strong amplification of error fields near marginal tearing stability is found to broaden the range of parameters over which deleterious tearing effects are seen. Three-dimensional numerical solutions indicate that saturated tearing mode island widths can be sensitive to the presence of small nonaxisymmetric perturbations. It is argued that the theoretical conclusions are consistent with observations in error compensation experiments, and are also consistent with observations of enhanced growth for locked modes.

Saturated tearing modes in toroidal geometry

A quasilinear method is developed for determining saturated tearing mode magnetic island widths in axisymmetric toroidal plasmas with arbitrary cross-sectional shape, aspect ratio, or plasma pressure (beta). The method is applied to compute magnetic island widths in force-free toroidal plasmas with aspect ratio as low as 2.0 and elongation between 1.0 and 2.0. It is found that current suppression within the magnetic island strongly increases the saturated width while current peaking reduces width. The effects of current profile, geometry, and harmonic mixing are also studied.