W. Park

Plasma simulation studies using multilevel physics models

The question of how to proceed toward ever more realistic plasma simulation studies using ever increasing computing power is addressed. The answer presented here is the M3D (Multilevel 3D) project, which has developed a code package with a hierarchy of physics levels that resolve increasingly complete subsets of phase-spaces and are thus increasingly more realistic. The rationale for the multilevel physics models is given. Each physics level is described and examples of its application are given. The existing physics levels are fluid models (3D configuration space), namely magnetohydrodynamic (MHD) and two-fluids; and hybrid models, namely gyrokinetic-energetic-particle/MHD (5D energetic particle phase-space), gyrokinetic-particle-ion/fluid-electron (5D ion phase-space), and full-kinetic-particle-ion/fluid-electron level (6D ion phase-space). Resolving electron phase-space (5D or 6D) remains a future project. Phase-space-fluid models are not used in favor of δf particle models. A practical and accurate no...

Reconnection rates of magnetic fields including the effects of viscosity

The Sweet–Parker and Petschek scalings of the magnetic reconnection rate are modified to include the effect of the viscosity. The modified scalings show that the viscous effect can be important in high‐β plasmas. The theoretical reconnection scalings are compared with numerical simulation results in a tokamak geometry for three different cases: a forced reconnection driven by external coils, the nonlinear m=1 resistive internal kink, and the nonlinear m=2 tearing mode. In the first two cases, the numerical reconnection rate agrees well with the modified Sweet–Parker scaling when the viscosity is sufficiently large. When the viscosity is negligible, a steady state which was assumed in the derivation of the reconnection scalings is not reached and the current sheet in the reconnection layer either remains stable through sloshing motions of the plasma or breaks up to higher m modes. When the current sheet remains stable, a rough comparison with the Sweet–Parker scaling is obtained. In the nonlinear m=2 teari...

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.

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.

Three-dimensional stellarator equilibrium as an ohmic steady state

A stable three‐dimensional stellarator equilibrium can be obtained numerically by a time‐dependent relaxation method using small values of dissipation. The final state is an Ohmic steady state which approaches an Ohmic equilibrium in the limit of small dissipation coefficients. A method to speed up the relaxation process and a method to implement the B⋅∇p=0 condition are described. These methods are applied to obtain three‐dimensional heliac equilibria using the reduced heliac equations.

Nonlinear simulation studies of tokamaks and STs

The multilevel physics, massively parallel plasma simulation code, M3D has been used to study spherical toris (STs) and tokamaks. The magnitude of outboard shift of density profiles relative to electron temperature profiles seen in NSTX under strong toroidal flow is explained. Internal reconnection events in ST discharges can be classified depending on the crash mechanism, just as in tokamak discharges; a sawtooth crash, disruption due to stochasticity, or high-β disruption. Toroidal shear flow can reduce linear growth of internal kink. It has a strong stabilizing effect nonlinearly and causes mode saturation if its profile is maintained, e.g. through a fast momentum source. Normally, however, the flow profile itself flattens during the reconnection process, allowing a complete reconnection to occur. In some cases, the maximum density and pressure spontaneously occur inside the island and cause mode saturation. Gyrokinetic hot particle/MHD hybrid studies of NSTX show the effects of fluid compression on a fast-ion driven n = 1 mode. MHD studies of recent tokamak experiments with a central current hole indicate that the current clamping is due to sawtooth-like crashes, but with n = 0.

Finite pressure effects on sawtooth oscillations

Nonlinear 3‐D simulation of sawtooth oscillations shows that the helically twisted hot spot has a prominent toroidal bulge at the large major radius side, characteristic of a high‐eβp instability. This bulge drives other m/n islands just outside the mixing radius resulting in a stochastic annular region. A similar effect is also found in shaped plasmas, even at low beta. These effects agree with experimental data, giving detailed support to the Kadomtsev reconnection model. This effect can also be used to explain the experimental sawtooth heat pulse ‘‘anomaly.’’ At high eβp, the stochastic region can essentially fill the whole plasma, and can result in a ‘‘high‐beta disruption.’’

Sawtooth stabilization through island pressure enhancement

Using the compressible resistive magnetohydrodynamic (MHD) equations in a finite aspect ratio cylinder, it is found that the m=1 mode (the sawtooth oscillation) can saturate when the pressure inside the magnetic island is higher than that of the original core plasma. The saturation condition is of the form Δβp≳8e−1q=1(1−q0)2. This saturation effect can be used to actively stabilize sawteeth by heating or raising the density in the island. This mechanism, together with a stabilizing toroidal effect, may also explain recent lower‐hybrid wave‐driven tokamak experiments where the saturation of sawteeth has been observed.

Three-dimensional modeling of the sawtooth instability in a small tokamak

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 extreme...

Simulation of two fluid and energetic particle effects in stellarators

MHD and resistive MHD are inadequate to understand the stability of stellarators properly. Ideal MHD ballooning mode theory predicts β limits substantially below the values that can be expected in experiments. Resistive MHD is even more pessimistic, predicting that many stellarators are completely unstable. Including two fluid effects, ideally and resistively stable stellarator equilibria can be obtained. It may be possible to completely stabilize ballooning modes. The two fluid computations are done with a realistic value of the Hall parameter, the ratio of the ion skin depth to the major radius. Hybrid gyrokinetic simulations with energetic particles indicate that global shear Alfven TAE modes can be more stable in stellarators than in tokamaks. Computations in a two-period compact stellarator obtained a predominantly n = 1 toroidal mode with the expected TAE frequency. The TAE modes are more stable in the two-period compact stellarator than in a tokamak with the same q and pressure profiles. The cause for the stabilization is believed to be the increased damping rate due to 3D geometry. Simulations were performed with the M3D extended MHD code.