S.C. Jardin

Dynamic modeling of transport and positional control of tokamaks

We describe a numerical model of a free boundary axisymmetric tokamak plasma and its associated control systems. The plasma is modeled with a hybrid method using two-dimensional velocity and flux functions with surface-averaged MHD equations describing the evolution of the adiabatic invariants. Equations are solved for the external circuits and for the effects of eddy currents in nearby conductors. The method is verified by application to several test problems and used to simulate the formation of a bean-shaped plasma in the PBX experiment.

An iterative metric method for solving the inverse tokamak equilibrium problem

A method is presented for solving the toroidal magnetohydrodynamic equilibrium equation in a coordinate system based on the magnetic field lines. Both fixed boundary (conducting shell) and free boundary (external coil) boundary conditions are considered. A comparison with a special analytic solution is made. The method is useful for obtaining equilibria to use in tokamak stability and transport calculations.

Numerical determination of axisymmetric toroidal magnetohydrodynamic equilibria

Numerical schemes for the determination of stationary axisymmetric toroidal equilibria appropriate for modeling real experimental devices are given. Iterative schemes are used to solve the elliptic nonlinear partial differential equation for the poloidal flux function psi. The principal emphasis is on solving the free boundary (plasma-vacuum interface) equilibrium problem where external current-carrying toroidal coils support the plasma column, but fixed boundary (e.g., conducting shell) cases are also included. The toroidal current distribution is given by specifying the pressure and either the poloidal current or the safety factor profiles as functions of psi. Examples of the application of the codes to tokamak design at PPPL are given.

IDEAL MHD STABILITY LIMITS OF LOW ASPECT RATIO TOKAMAK PLASMAS

The ideal magnetohydrodynamic (MHD) stability limits of low aspect ratio tokamak plasmas are computed numerically for plasmas with a range of cylindrical safety factors q*, normalized plasma pressures beta , elongations kappa and central safety factors q(0). Four distinct regimes are optimized, namely: (a) low-q* plasmas with q(0)=1.1 with and without a stabilizing wall, (b) low-q* plasmas with no wall and 1.1<q(0)<2, (c) high- beta , high bootstrap fraction plasmas at moderate kappa requiring a wall and edge current drive and (d) high- beta , very high bootstrap fraction plasmas with moderate to high kappa requiring a stabilizing wall but little external current drive. A stable equilibrium is found at an aspect ratio of A=1.4 and an elongation of kappa =3.0, with 99.3% of the current provided by the plasma pressure and beta =45%. Special attention is paid to the issues of numerical convergence and the proper definition of bootstrap current fraction

Dynamic modelling of lower hybrid current drive

A dynamic computational model of lower hybrid current drive in the presence of an electric field is described and some results are given. Details of geometry, plasma profiles and circuit equations are treated carefully. Two dimensional velocity space effects are approximated in a one dimensional Fokker-Planck treatment. The model is unable to approximate experimental results in some cases characterized by low density, low current, high aspect ratio and a launched spectrum at high phase velocity relative to the thermal velocity. In other cases, qualitative agreement with measurements is found. A simple formula already in the literature appears to determine whether agreement of model and experiment will be good or poor. Application to an experimental discharge in which q(0) is raised above unity shows an appropriate time-scale. Computation of a planned effort for high beta suggests potential success

THE PROSPECTS FOR MAGNETOHYDRODYNAMIC STABILITY IN ADVANCED TOKAMAK REGIMES

Stability analysis of advanced regime tokamaks is presented. Here advanced regimes are defined to include configurations where the ratio of the bootstrap current, IBS, to the total plasma current, Ip, approaches unity, and the normalized stored energy, βN* = 80π〈p2〉1/2a/IpB0, has a value greater than 4.5. Here, p is the plasma pressure, a the minor radius in meters, Ip is in mega‐amps, B0 is the magnetic field in Tesla, and 〈⋅〉 represents a volume average. Specific scenarios are discussed in the context of Toroidal Physics Experiment (TPX) [Proceedings of the 20th European Physical Society Conference on Controlled Fusion and Plasma Physics, Lisbon, 1993, edited by J. A. Costa Cabral, M. E. Manso, F. M. Serra, and F. C. Schuller (European Physical Society, Petit‐Lancy, 1993), p. I‐80]. The best scenario is one with reversed shear, in the q profile, in the central region of the tokamak. The bootstrap current obtained from the plasma profiles provides 90% of the required current, and is well aligned with the...

TSC simulation of ohmic discharges in TFTR

The Tokamak Simulation Code (TSC) has been used to model the time dependence of several Ohmic discharges in the TFTR experiment. The semi-empirical thermal conductivity model and the sawtooth model in TSC have been refined so that good agreement between the simulation and the experiment is obtained in the electron and ion temperature profiles and in the current profiles for the entire duration of the discharges. Neoclassical resistivity gives good agreement with the measured surface voltage and the rate of poloidal flux consumption

Stability of high-beta tokamaks to ballooning modes

Fixed-boundary ballooning modes are found to possess a second globally stable regime for high-beta flux-conserving equilibria. This confirms a conjecture of several authors based on local analysis of the instability in the vicinity of the magnetic axis. The range of unstable beta values depends on the details of the equilibrium and, in particular, on shear. Very high shear can decrease the width of the unstable region.

A high-order implicit finite element method for integrating the two-fluid magnetohydrodynamic equations in two dimensions

We describe a new method for solving the time-dependent two-fluid magnetohydrodynamic (2F-MHD) equations in two dimensions that has significant advantages over other methods. The stream-function/potential representation of the velocity and magnetic field vectors, while fully general, allows accurate description of nearly incompressible fluid motions and manifestly satisfies the divergence condition on the magnetic field. Through analytic manipulation, the split semi-implicit method breaks the full matrix time advance into four sequential time advances, each involving smaller matrices. The use of a high-order triangular element with continuous first derivatives (C1 continuity) allows the Galerkin method to be applied without introduction of new auxiliary variables (such as the vorticity or the current density). These features, along with the manifestly compact nature of the fully node-based C1 finite elements, lead to minimum size matrices for an unconditionally stable method with order of accuracy h4. The resulting matrices are compatible with direct factorization using SuperLU_dist. We demonstrate the accuracy of the method by presenting examples of two-fluid linear wave propagation, two-fluid linear eigenmodes of a tilting cylinder, and of a challenging nonlinear problem in two-fluid magnetic reconnection.

Non-linear saturation of the internal kink mode

A numerical study shows that in a cylindrical tokamak the internal kink mode (m = 1) develops non-linearly into a helical equilibrium state that possesses a singular current sheet. In the large-aspect-ratio limit, the neighbouring equilibria obtained agree well with the asymptotic analytic theory of Rosenbluth et al.