Equations In Toroidal Geometry Research Articles

A divergence-free constrained magnetic field interpolation method for scattered data

An interpolation method to evaluate magnetic fields, given its unstructured and scattered magnetic data, is presented. The method is based on the reconstruction of the global magnetic field using a superposition of orthogonal functions. The coefficients of the expansion are obtained by minimizing a cost function defined as the L2 norm of the difference between the ground truth and the reconstructed magnetic field evaluated on the training data. The divergence-free condition is incorporated as a constraint in the cost function, allowing the method to achieve arbitrarily small errors in the magnetic field divergence. An exponential decay of the approximation error is observed and compared with the less favorable algebraic decay of local splines. Compared to local methods involving computationally expensive search algorithms, the proposed method exhibits a significant reduction of the computational complexity of the field evaluation, while maintaining a small error in the divergence even in the presence of magnetic islands and stochasticity. Applications to the computation of Poincaré sections using data obtained from numerical solutions of the magnetohydrodynamic equations in toroidal geometry are presented and compared with local methods currently in use.

Fast ion driven drift instability in reversed shear plasmas

It is shown that electron drift waves can be destabilized by trapped fast ions produced by a fusion reaction because a significant fraction of those particles reverses their precession direction in reversed shear plasmas toward the electron diamagnetic direction and can resonate with the electron drift wave. We perform a local stability analysis and calculate the consequent quasi-linear transport caused by this new instability using bounce-averaged gyrokinetic equations in toroidal geometry and under fusion reactor conditions. We consider the equilibrium distribution function of fast ions as the slowing down distribution and compare with the equivalent Maxwellian case for illustration. The instability occurs when the temperature profile of fast ions peaks sufficiently compared to the density profile, and the magnetic shear is strongly negative. The resulting particle flux of fast ions is outward, while the particle flux of main hydrogenic ions can be inward.

An efficient interface between the solvers of Maxwell’s equations and of the quasilinear kinetic equation for simulations of ion cyclotron heating in tokamak plasmas

Selfconsistent simulations of ion–cyclotron heating of tokamak plasmas require iterating between a solver of the wave equations in toroidal geometry and a solver of the Fokker–Planck equation describing the evolution of the ion distribution functions. A huge amount of information must be exchanged between the two codes at each iteration. For the package TORIC–SSFPQL, we have developed an interface which substantially reduces the CPU and memory requirements for the storage and transmission of these data. We present this interface here, and we take advantage of its efficiency to compare simulations in which the quasilinear diffusion operator in SSFPQL is evaluated superposing the TORIC results for all relevant toroidal modes excited by a given antenna with simulations using the fields of a single “representative” toroidal mode.

Role of the plasma interaction with magnetic field in the “missing power” problem

It is shown that, at rapid changes of the heating power, the magnetically confined equilibrium plasma almost completely absorbs the injected energy, so that its only small part goes to the magnetic field. The result is obtained within the standard MHD theory with use of exact consequences of the force-balance equations in toroidal geometry. It is assumed that, when heated, the plasma evolves from one equilibrium state to another with the magnetic field frozen-in. Another constraint is the conservation of the toroidal magnetic flux in the plasma-wall vacuum gap. It is shown that the plasma interaction with magnetic field (which is traditionally neglected in the analysis of heat transport in tokamaks and stellarators) is the natural mechanism of fast redistribution of energy in the plasma, observed in some experiments in these devices at switchon of powerful heat sources.

The XTOR code for nonlinear 3D simulations of MHD instabilities in tokamak plasmas

The latest version of the XTOR code which solves a set of the extended magnetohydrodynamic (MHD) equations in toroidal geometry is presented. The numerical method is discussed with particular emphasis on critical issues leading to numerical stability and robustness. This includes the time advance algorithm, the choice of variables and the boundary conditions. The physics in the model includes resistive MHD, anisotropic thermal diffusion and some neoclassical effects. The time advance method used in XTOR is unconditionally stable for linear MHD. First, both the ideal and the resistive MHD parts of the equations are advanced semi-implicitly and then the thermal transport part full-implicitly, using sub-stepping [H. Lütjens, Comp. Phys. Commun. 164 (2004) 301]. The time steps are only weakly limited by the departure of the nonlinear MHD dynamics from the linear one and are automatically defined by a set of nonlinear stability criteria. The robustness of the method is illustrated by some numerically difficult simulations, i.e. sawtooth simulations, the nonlinear destabilization of ballooning instabilities by an internal kink, and the dynamics of a neoclassical tearing mode in International Thermonuclear Experimental Reactor (ITER) [R. Aymar, V.A. Chuyanov, M. Huguet, et al., Nucl. Fusion 41 (2001) 1301] like geometry about its nonlinear stability threshold.

Solving the 3D MHD equilibrium equations in toroidal geometry by Newton’s method

We describe a novel form of Newton’s method for computing 3D MHD equilibria. The method has been implemented as an extension to the hybrid spectral/finite-difference Princeton Iterative Equilibrium Solver (PIES) which normally uses Picard iteration on the full nonlinear MHD equilibrium equations. Computing the Newton functional derivative numerically is not feasible in a code of this type but we are able to do the calculation analytically in magnetic coordinates by considering the response of the plasma’s Pfirsch–Schlüter currents to small changes in the magnetic field. Results demonstrate a significant advantage over Picard iteration in many cases, including simple finite- β stellarator equilibria. The method shows promise in cases that are difficult for Picard iteration, although it is sensitive to resolution and imperfections in the magnetic coordinates, and further work is required to adapt it to the presence of magnetic islands and stochastic regions.

`Quasi-local' wave equations in toroidal geometry with applications to fast wave propagation and absorption at high harmonics of the ion cyclotron frequency

The integral constitutive relation for high frequency waves propagating in toroidal axisymmetric plasmas, obtained by formal integration of the linearized Vlasov equation, is simplified assuming the range of spatial dispersion to be small compared to the linear dimensions of the plasma. We propose to call this the `quasi-local approximation'. A (formally infinite) system of purely differential wave equations is obtained, which should be a good approximation under conditions similar to those which would justify an Eikonal Ansatz for the form of the wave fields. This system is valid to all orders in the Larmor radius, and, in the presence of a poloidal static magnetic field, predicts a different plasma response to each poloidal Fourier component of the h.f. field. Compared to ray tracing based on the Eikonal approximation, these wave equations have the advantage of allowing to take into account periodicity, boundary conditions, and toroidicity-induced coupling between poloidal Fourier modes.As an example, the quasi-local wave equations are used to model propagation and absorption of the compressional wave at frequencies higher than the ion cyclotron frequency in the high-β plasma of the National Spherical Tokamak Experiment in Princeton, USA. Because of the low magnetic field and the tight aspect ratio of this device, large Larmor radius effects and toroidicity play an important role in these experiments. This example, therefore, illustrates well the importance of taking into account these effects, and, in particular, the different response of the plasma to each poloidal Fourier mode.

The Linear Evolution of Tearing Mode in Toroidal Geometry

A set of linearly-reduced MHD equations in toroidal geometry has been solved numerically in flux coordinate with toroidal coupling. In the case of q > 1 on the magnetic axis, where q is the safety factor, the result shows that an unstable 2/1 tearing mode destabilizes both 1/1 and 3/1 modes. The 1/1 and 3/1 modes contribute local perturbations on the resonant surface of q = 2. And the 2/1 mode also contributes a local perturbation on the resonant surface of q = 3.

Reduced Magnetohydrodynamic Equations in Toroidal Geometry

By applying a new assumption of density, i.e. R2ρ = const, thecontinuity equation is satisfied to the order of ε2 withε being the inverse aspect ratio. In the case of largeaspect ratio, a set of reduced magnetohydrodynamic equations in toroidalgeometry are obtained. The new assumption about the density is supportedby experimental observation to some extent.

A note on the toroidal plasma dispersion function

We have found a simple yet accurate approximation for the toroidal plasma dispersion function, which described the effects of parallel space dispersion on the propagation and absorption of waves near ion cyclotron resonances in tokamak plasmas. It can be easily implemented in numerical codes solving the finite Larmor radius wave equations in toroidal geometry, allowing a quantitative discussion of the effects of toroidicity on ion cyclotron damping and on linear mode conversion to ion Bernstein waves.

Tokamak m=1 magnetohydrodynamic calculations in toroidal geometry using a full set of nonlinear resistive magnetohydrodynamic equations

The linear stability and nonlinear evolution of the resistive m=1 mode in tokamaks is studied using a full set of resistive magnetohydrodynamic (MHD) equations in toroidal geometry. The modification of the linear and nonlinear properties of the mode by a combination of strong toroidal effects and low resistivity is the focus of this work. Linearly there is a transition from resistive kink to resistive tearing behavior as the aspect ratio and resistivity are reduced, and there is a corresponding modification of the nonlinear behavior, including a slowing of the island growth and development of a Rutherford regime, as the tearing regime is approached. In order to study the sensitivity of the stability and evolution to assumptions concerning the equation of state, two sets of full nonlinear resistive MHD equations (a pressure convection set and an incompressible set) are used. Both sets give more stable nonlinear behavior as the aspect ratio is reduced. The pressure convection set shows a transition from a Kadomtsev reconnection at large aspect ratio to a saturation at small aspect ratio. The incompressible set yields Kadomtsev reconnection for all aspect ratios, but with a significant lengthening of the reconnection time and development of a Rutherford regime at an aspect ratio approaching the transition from a resistive kink mode to a tearing mode. The pressure convection set gives an incomplete reconnection similar to that sometimes seen experimentally. The pressure convection set is, however, strictly justified only at high beta.

Numerical calculations using the full MHD equations in toroidal geometry

A computer code has been constructed that solves the full magnetohydronamic (MHD) equations in toroidal geometry. The code is applicable to toroidal devices, including tokamaks, stellarators, and reversed field pinches. A fully implicit numerical technique is used that allows linear eigenvalues and eigenfunctions to be found in a very few computational steps. Althought the present work describes the solution of the linearized equations, generalization of the numerical method to the solution of the nonlinear problem is straightforward. Use of the code is illustrated by calculating the n = 1 instability for a tokamak configuration. The results show the structural changes in the eigenfunctions as the plasma pressure is increased.

A pseudospectral algorithm for three-dimensional magnetohydrodynamic simulation

An algorithm for the solution of the three-dimensional resistive magnetohydrodynamic equations in toroidal geometry is presented. The algorithm employs the pseudospectral method for approximation in the two periodic coordinates, and finite differences in the radial direction. Efficient Fast Fourier Transforms are used to communicate between configuration and Fourier space. Leapfrog time advancement is used for advective terms. Diffusion terms are treated implicitly to avoid severe time step restrictions. Sample cases are presented, and a comparison of the method with standard finite difference techniques is presented and discussed.