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3D Equilibrium Reconstruction: There are many basic equilibrium, stability, and confinement issues that are common to 2D tokamak and 3D stellarator configurations, in which a more universal approach, fostering cross-fertilization, is likely to yield unique insights. Stellarators have now achieved quite substantial b values and new hybrid stellarators such as NCSX, QPS, and CTH are designed to have a significant fraction of the rotational transform produced by plasma currents. Operation and interpretation of experimental results from these existing and new stellarators will require accurate and efficient 3D reconstruction of experimental equilibria. Such a reconstruction capability will also be needed for investigation of the important 3D error field effects on experimentally reconstructed equilibrium in tokamaks. In collaboration with ORNL and Auburn University, progress was made toward development of a new 3D equilibrium reconstruction tool, V3FIT. V3FIT is based on the VMEC 3D equilibrium code [Hirshman 1983] and the EFIT response function method [Lao 1985a]. The EFIT response function method was generalized to 3D magnetic geometry and modules developed to provide efficient evaluation of 3D magnetic responses to a diagnostic set [Hirshman 2004]. Both the response function method and the new modules are being applied to support design of magnetic diagnostics for NCSX and CTH and have been incorporated into the 3D stellarator optimization code STELLOPT to provide a prototype 3D reconstruction code to examine various numerical features of the reconstruction process. Various linearization schemes for use with V3FIT to approximate the signal gradients and to accelerate the search for the solution vector based on the efficient 2D EFIT optimization approach have also been formulated and are being tested.
3D Stability and Confinement: The equilibrium and stability of Compact Stellarator reactors is being investigated for the ARIES-CS project using VMEC and TERPSICHORE [Ardelea 1998]. These codes can also be used for investigating non-axisymmetric perturbations arising in tokamaks from instabilities and error fields. A formulation for eliminating islands from a perturbed equilibrium was recently proposed for stellarators by Nuhrenberg and Boozer [Nuhrenberg 2003]. The approach is based on the energy principle for perturbed equilibria. It can be efficiently used either to determine the change in the equilibrium energy from the perturbation or to optimize the equilibrium energy against the external changes. This formulation has significant overlap with the more fundamental work on AIMHD and also fits naturally with the problem of investigating the effect of error fields in tokamaks.
We have also been investigating issues arising from comparisons of tokamak and stellarator experiments. Recent experiments on W7-AS and LHD raise questions as to the applicability of linear ideal MHD stability in stellarators since the predicted stability limits appear to be significantly exceeded; in tokamaks, violation of ideal stability limits is considered to have experimental consequences. Yet tokamaks and stellarators are designed on the basis of the predicted ideal stability limits. Some common assumptions about the experimental discharge equilibria used for the global MHD stability predictions in stellarators may be questionable at finite b. The nonlinear consequences and non-ideal contributions need to be explored. In collaboration with P. Garabedian of NYU, nonlinear stability predictions using the NSTAB [Garabedian 1997] code. Equilibrium limits may set the b limit in stellarators in some cases. This limit is likely to be a soft limit due to breakup into islands, with the system holding pressure much like a sponge holding water; adding more simply increases the leakage.
Also, in collaboration with the H-1 Heliac group at the Australian National University and DIII-D experimentalists, the intriguing role played by rational and nearby irrational surfaces on transport, seen on both DIII-D and in H-1 is under investigation; local transport is closely correlated with the local density of rational surfaces. But the actual mechanism responsible remains elusive. The GYRO code has shown that long time average (quasi-equilibrium) and flux surface average gradients of temperature, density and divergence of poloidal magnetic field (i.e. toroidal current density) as well as divergence of the plasma and energy flows are highly corrugated at low-order rational q surfaces. This may provide a key to resolving this issue.
Compact Toroids and Helicity Injection: Important work was also completed on several compact torus (CT) issues. In collaboration with Swarthmore College, we found a new class of analytical �doublet FRC� equilibria and stability analyses verified that these configurations are more stable than standard FRCs [Parks, 2003]. Using hybrid-particle-in-cell approach, it was also shown that a self-generated bipolar toroidal magnetic field resulting from the Hall effect, coupled with finite Larmor radius effects, nonlinearly stabilizes the FRC tilting instability [Omelchenko 2001]. Spheromak and Spherical Torus equilibria driven by Coaxial Helicity Injection (CHI) and with current on open field lines, were shown to be self-stabilizing [Brennan 2002] in general agreement with experiment. An alternative concept, the Repetitive Merging for Spheromak Sustainment (RMSS) in which the main spheromak plasma is built up by injecting successive small CT rings [Bourke 2002] has also been proposed.
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