[en] Global stability of the field-reversed configuration (FRC) has been investigated numerically using both three-dimensional magnetohydrodynamic and hybrid (fluid electron and δf particle ion) simulations. The stabilizing effects of velocity shear and finite ion Larmor radius (FLR) on the n=1 internal tilt mode in the prolate FRCs have been studied. Sheared rotation is found to reduce the growth rate, however a large rotation rate with Mach number of M∼>1 is required in order for significant reduction in the instability growth rate to occur. Kinetic effects associated with large thermal ion orbits have been studied for different kinetic equilibria. The simulations show that there is a reduction in the tilt mode growth rate due to FLR effects, but complete linear stability has not been found, even when the thermal ion gyroradius is comparable to the distance between the field null and the separatrix. The instability existing beyond the FLR theory threshold could be due to the resonant interaction of the wave with ions whose Doppler shifted frequency matches the betatron frequency

[en] Global stability of the oblate (small elongation, E<1) Field-Reversed Configuration (FRC) has been investigated numerically using both three-dimensional magnetohydrodynamic (MHD) and hybrid (fluid electrons and kinetic ions) simulations. For every nonzero value of the toroidal mode number n, there are three MHD modes that must be stabilized. For n=1, these are the interchange, the tilt and the radial shift; while for n>1 these are the interchange and two co-interchange modes with different polarization. It is shown that the n=1 tilt mode becomes an external mode when E<1, and it can be effectively stabilized by close-fitting conducting shells, even in the small Larmor radii (MHD) regime. The tilt mode stability improves with increasing oblateness, however at sufficiently small elongations the radial shift mode becomes more unstable than the tilt mode. The interchange mode stability is strongly profile dependent, and all n≥1 interchange modes can be stabilized for a class of pressure profile with separatrix beta larger than 0.035. Our results show that all three n=1 modes can be stabilized in the MHD regime, but the stabilization of the n>1 co-interchange modes still remains an open question

[en] Global stability of the Field-Reversed Configuration (FRC) has been investigated numerically using both 3D MHD and hybrid (fluid electron and delta f particle ion) simulations. The stabilizing effects of velocity shear and large ion orbits on the n = 1 internal tilt mode in the prolate FRCs have been studied. Sheared rotation is found to reduce the growth rate, however a large rotation rate with Mach number of M greater than or approximately equal to 1 is required in order for significant reduction in the instability growth rate to occur. Kinetic effects associated with large thermal ion orbits have been studied for different kinetic equilibria. These simulations show that there is a reduction in the tilt mode growth rate due to finite ion Larmor radius (FLR) effects, but complete linear stability has not been found, even when the thermal ion gyroradius is comparable to the distance between the field null and the separatrix. The instability existing beyond the FLR theory threshold could be due to the resonant interaction of the wave with ions whose Doppler shifted frequency matches the betatron frequency

[en] A new approach for solving the 3D MHD equations in a strongly magnetized toroidal plasma is presented which uses high-order 2D finite elements with C1 continuity. The vector fields use a physics-based decomposition. An efficient implicit time advance separates the velocity and field advance. ITAPS (SCOREC) adaptivity software and TOPS solvers are used

[en] A general coordinate-independent expression for Braginskii's form of the ion gyroviscosity in the two-dimensional potential field representation is presented, and is implemented in a full two-dimensional, two-fluid extended magnetohydrodynamic (MHD) numerical model. The expression for the gyroviscous force requires no field to be differentiated more than twice, and thus is appropriate for finite elements with first derivatives continuous across element boundaries (C¹ finite elements). From the extended MHD model, which includes the full gyroviscous stress, are derived linear dispersion relations of a homogeneous equilibrium and of an inverted-density profile in the presence of gravity. The treatment of the gravitational instability presented here extends previous work on the subject [M. N. Rosenbluth, N. A. Krall, and N. Rostoker, Nucl. Fusion Suppl. 1, 143 (1962); K. V. Roberts and J. B. Taylor, Phys. Rev. Lett. 8, 197 (1962)]. Linear and nonlinear simulations of the gravitational instability are presented. Simulations are shown to agree closely with the derived dispersion relations in the linear regime. The 'gyroviscous cancellation' effect is demonstrated, and some limitations of the v-vector sign_* approximation are discussed

[en] The eigenvalue equations describing a cylindrical ideal magnetohydrodynamics plasma interacting with a thin resistive wall are presented in the standard mathematical form, A·x=λB·x, without discretizing the vacuum regions surrounding the plasma. This is accomplished by using a finite-element basis for the plasma perturbations, and by coupling the plasma surface perturbations to the perturbed electrical current in the wall using a Green's-function approach. The perturbed wall current introduces a single additional degree of freedom into the system, which, together with an auxiliary variable, u=ωξ, allows the system to take the standard linear form. The standard form allows the use of linear eigenvalue solvers, without additional iterations, to compute the complete spectrum of plasma modes in the presence of a surrounding resistive wall at arbitrary separation. Standard results are recovered in the limits of (i) an infinitely resistive wall (no wall), and (ii) a zero resistance wall (ideal wall)

[en] Here we describe a technique for solving the four-field extended-magnetohydrodynamic (MHD) equations in two dimensions. The introduction of triangular high-order finite elements with continuous first derivatives (C¹ continuity) leads to a compact representation compatible with direct inversion of the associated sparse matrices. The split semi-implicit method is introduced and used to integrate the equations in time, yielding unconditional stability for arbitrary time step. The method is applied to the cylindrical tilt mode problem with the result that a nonzero value of the collisionless ion skin depth will increase the growth rate of that mode. The effect of this parameter on the reconnection rate and geometry of a Harris equilibrium and on the Taylor reconnection problem is also demonstrated. This method forms the basis for a generalization to a full extended-MHD description of the plasma with six, eight, or more scalar fields

[en] The sawtooth instability is one of the most fundamental dynamics of an inductive tokamak discharge such as will occur in ITER [R. Aymar et al., Plasma Phys. Controlled Fusion 44, 519 (2002)]. Sawtooth behavior is complex and remains incompletely explained. The Center for Extended MHD Modeling (CEMM) SciDAC project has undertaken an ambitious campaign to model this periodic motion in a small tokamak as accurately as possible using the extended MHD model. Both M3D [W. Park et al., Phys. Plasmas 6, 1796 (1999)] and NIMROD [C. R. Sovinec et al., Phys. Plasmas 10, 1727 (2003)] have been applied to this problem. Preliminary nonlinear MHD results show pronounced stochasticity in the magnetic field following the sawtooth crash but are not yet fully converged. Compared to the MHD model, extended MHD predicts plasma rotation, faster reconnection, and reduced field line stochasticity in the crash aftermath. The multiple time and space scales associated with the reconnection layer and growth time make this an extremely challenging computational problem. However, these calculations are providing useful guidelines to the numerical and physical requirements for more rigorous future studies

[en] In the present paper, the numerical calculation of transonic equilibria, first introduced with the FLOW code in Guazzotto et al.[Phys. Plasmas 11, 604 (2004)], is critically reviewed. In particular, the necessity and effect of imposing explicit jump conditions at the transonic discontinuity are investigated. It is found that “standard” (low-β, large aspect ratio) transonic equilibria satisfy the correct jump condition with very good approximation even if the jump condition is not explicitly imposed. On the other hand, it is also found that high-β, low aspect ratio equilibria require the correct jump condition to be explicitly imposed. Various numerical approaches are described to modify FLOW to include the jump condition. It is proved that the new methods converge to the correct solution even in extreme cases of very large β, while they agree with the results obtained with the old implementation of FLOW in lower-β equilibria.

[en] The sawtooth instability is one of the most fundamental dynamics of an inductive tokamak discharge such as will occur in ITER. The sawtooth occurs when the current peaks in a tokamak, creating a region in the core where the safety factor is less than unity, q<1. While this instability is confined to the center of the plasma in low-pressure, low-current, large-aspect-ratio discharges, under certain conditions it can create magnetic islands at the outer resonant surfaces or set off a sequence of events that leads to a major disruption. Sawtooth behavior is complex and remains incompletely explained. The SciDAC Center for Extended MHD Modeling (CEMM) has undertaken an ambitious campaign to model this periodic motion as accurately as possible using the most complete fluid-like description of the plasma - the Extended MHD model. The multiple time and space scales associated with the reconnection layer and growth time make this an extremely challenging computational problem. The most recent simulation by the M3D code used over 500,000 elements for 400,000 partially implicit time steps for a total of 2x10¹¹ space-time points, and there still remain some resolution issues. However, these calculations are providing insight into the nonlinear mechanisms of surface breakup and healing. We have been able to match many features of a small tokamak and can now project to the computational requirements for simulations of larger, hotter devices such as ITER