Three-dimensional Toroidal Geometry Research Articles

Influence of toroidal rotation on nonlinear evolution of tearing mode in tokamak plasmas

The nonlinear evolution of the tearing mode instability with toroidal rotation is investigated using a three-dimensional toroidal geometry magnetohydrodynamic code (M3D). It is found that toroidal rotation plays a significantly stable role in reducing the width of magnetic island, which is independent of the direction of toroidal rotation. Compared with the stabilizing effect in the linear phase, the stabilizing effect of toroidal rotation is much stronger in the nonlinear phase. The magnetic island is almost suppressed by large enough rotation, even though the linear growth rate is weakly reduced in the linear phase. It is observed that the flow shear has a stabilizing effect, which is significant when the rotation frequency is large. However, it is shown that the magnitude of rotation plays a dominant role in stabilizing the magnetic island when the rotation frequency is small. The Coriolis force, rather than the centrifugal force, is found to play a dominant role in stabilizing the tearing mode in the nonlinear phase. When and tearing modes exist simultaneously, the sheared toroidal rotation is shown to effectively suppress the overlap of magnetic islands and field stochasticity caused by the coupling of neighboring islands.

The linear tearing instability in three dimensional, toroidal gyro-kinetic simulations

Linear gyro-kinetic simulations of the classical tearing mode in three-dimensional toroidal geometry were performed using the global gyro-kinetic turbulence code, GKW. The results were benchmarked against a cylindrical ideal MHD and analytical theory calculations. The stability, growth rate, and frequency of the mode were investigated by varying the current profile, collisionality, and the pressure gradients. Both collisionless and semi-collisional tearing modes were found with a smooth transition between the two. A residual, finite, rotation frequency of the mode even in the absence of a pressure gradient is observed, which is attributed to toroidal finite Larmor-radius effects. When a pressure gradient is present at low collisionality, the mode rotates at the expected electron diamagnetic frequency. However, the island rotation reverses direction at high collisionality. The growth rate is found to follow a η1∕7 scaling with collisional resistivity in the semi-collisional regime, closely following the semi-collisional scaling found by Fitzpatrick. The stability of the mode closely follows the stability analysis as performed by Hastie et al. using the same current and safety factor profiles but for cylindrical geometry, however, here a modification due to toroidal coupling and pressure effects is seen.

A δf particle method for gyrokinetic simulations with kinetic electrons and electromagnetic perturbations

A δf particle simulation method is developed for solving the gyrokinetic-Maxwell system of equations that describes turbulence and anomalous transport in toroidally confined plasmas. A generalized split-weight scheme is used to overcome the constraint on the time step due to fast parallel motion of the electrons. The inaccuracy problem at high plasma β is solved by using the same marker particle distribution as is used for δf to evaluate the βm i/ m e A ∥ term in Ampere’s equation, which is solved iteratively. The algorithm is implemented in three-dimensional toroidal geometry using field-line-following coordinates. Also discussed is the implementation of electron–ion collisional effects which are important when kinetic electron physics is included. Linear benchmarks in toroidal geometry are presented for moderate β, that is, β≪1, but βm i/ m e≫1. Nonlinear simulation results with moderate β are also presented.

The split-weight particle simulation scheme for plasmas

An efficient numerical method for treating electrons in magnetized plasmas has been developed. The scheme, which is based on the perturbative (δf) gyrokinetic particle simulation, splits the particle electron responses into adiabatic and nonadiabatic parts. The former is incorporated into the gyrokinetic Poisson’s equation, while the latter is calculated dynamically with the aid of the charge conservation equation. The new scheme affords us the possibility of suppressing unwanted high-frequency oscillations and, in the meantime, relaxing the Courant condition for the thermal particles moving in the parallel direction. It is most useful for studying low-frequency phenomena in plasmas. As an example, one-dimensional drift wave simulation has been carried out using the scheme and the results are presented in this paper. This methodology can easily be generalized to problems in three-dimensional toroidal geometry, as well as those in unmagnetized plasmas.

Gyro-Bohm Scaling of Ion Thermal Transport from Global Numerical Simulations of Ion-Temperature-Gradient-Driven Turbulence

The ion gyroradius scaling of ion thermal transport caused by ion-temperature-gradient-driven turbulence is studied with a global fluid simulation code in three-dimensional toroidal geometry. It is found that the effective conductivity scales like the ion gyroradius (gyro-Bohm scaling).

Existence and calculation of sharp boundary magnetohydrodynamic equilibrium in three-dimensional toroidal geometry

The problem of sharp boundary, ideal magnetohydrodynamic equilibria in three-dimensional toroidal geometry is addressed. The sharp boundary, which separates a uniform pressure, current-free plasma from a vacuum, is determined by a magnetic surface of a given vacuum magnetic field. The pressure balance equation has the form of a Hamilton–Jacobi equation with a Hamiltonian that is quadratic in the momentum variables, which are the two covariant components of the magnetic field on the outer surface of the plasma. The condition of finding a unique solution on the outer surface is identical with finding phase-space tori in nonlinear dynamics problems, and the Kolmogorov–Arnold–Moser (KAM) theorem guarantees that such solutions exist for a wide band of parameters. Perturbation theory is used to calculate the properties of the magnetic field just outside the plasma. Special perturbation theory is needed to treat resonances and it is explicitly shown that there are bands of pressure where there are no solutions.

Variational moment method for computing magnetohydrodynamic equilibria

A fast yet accurate method to compute magnetohydrodynamic equilibria is provided by the variational moment method, which is similar to the classical Rayleigh-Ritz-Galerkin approximation. The equilibrium solution sought is decomposed into a spectral representation. The partial differential equations describing the equilibrium are then recast into their equivalent variational form and systematically reduced to an optimum finite set of coupled ordinary differential equations. An appropriate spectral decomposition can make the series representing the solution converge rapidly and hence substantially reduces the amount of computational time involved. The moment method was developed first to compute fixed boundary inverse equilibria in axisymmetric toroidal geometry, and was demonstrated to be both efficient and accurate. The method since has been generalized to calculate free boundary axisymmetric equilibria, to include toroidal plasma rotation and pressure anisotropy, and to treat three-dimensional toroidal geometry. In all these formulations, the flux surfaces are assumed to be smooth and nested so that the solutions can be decomposed in Fourier series in inverse coordinates. These recent developments and the advantages and limitations of the moment method are reviewed. The use of alternate coordinates for decomposition is discussed.