**NIMROD Team Meeting Minutes**

**August 13-15, 2013**

**Boulder, Colorado**

__Computational
topics:__

Carl Sovinec presented analysis and results on projecting parallel vorticity and flow-divergence errors in finite-element computations for first-order in time systems. The method extends the application of projecting from another formulation that has a special expansion for vector and scalar fields that responds to divergence at all scales. Projecting divergence onto a limited set of incomplete polynomials while retaining standard continuous expansions for physical fields is more practical and is nearly as effective in stabilizing ideal interchange in physically stable conditions. Linear tests with the NIMROD implementation confirm results from analysis, and a nonlinear interchange turbulence problem run with diffusive projection avoids noise that accumulates without projection.

Ben Sturdevant is investigating sub-cycling and orbit averaging in the CU plasma simulation code with implicit fluid electrons and kinetic ions. Sub-cycling moves particles multiple times for each electromagnetic field advance, which is suitable for low-frequency dynamics. The fields are interpolated in time (and space) as the particles are moved, and particle/field prediction-correction is retained. Tests confirm orbit evolution accuracy and Landau damping. Orbit-averaging filters high-frequency noise by depositing particle density and current contributions for field-solves over multiple particle steps.

Scott Kruger described Tech-X efforts to receive a significant allocation of leadership-class computing time through the INCITE process. The proposal addresses computation for modeling edge harmonic oscillations (EHO) and RF tearing-mode stabilization in tokamaks. Last year's proposal received favorable reviews with respect to computation, but was rated less well on physics, possibly due to reviewers being unfamiliar with recent MHD simulation work. This year's proposal includes some changes to achieve a higher rating.

__Physical
model development and analysis:__

Eric Held reviewed his continuum-based kinetics modeling for macroscopic dynamics. The formulation is similar to Hazeltine's but allows different approximations to incorporate the Ramos model and others. One special case is the NEO approximation, which assumes that dynamics occur at v_drift<<v_parallel and E<<E_Dreicer. The bootstrap benchmark with NEO for a large R/a cylindrical cross-section is excellent, but a more efficient computation of collisions is needed for shaped cross-sections. The improvements have been implemented, and the collision operator is now computed directly in the nodal GLL expansion for pitch-angle. Held also presented his hot-particle modeling and benchmark computations for the internal-kink/fishbone, noting the issue of kinetic modifications to the equilibrium. This modeling has been applied to linear computations of giant sawtooth modes, and turnaround times for these challenging parallel linear computations are just one hour. Held also discussed improvements for parallel computation with the continuum kinetic modeling.

Jeong-Young Ji reviewed closure theory
and described research to develop a new toroidal closure theory that is based
on his general moment method for plasma kinetics. He distinguished
Braginskii and neoclassical closure theory. He has recently published a
collisional theory (PoP **20**, 2013) that improves upon Braginskii to
consider all magnetization values (*x* = W_{gyro}*t*_{coll}) and ion charge states. His new
toroidal theory can be viewed as a combination of his collisional theory and
neoclassical. The small parameter is the ratio of gyroradius and gradient
length-scale. He has also completed an integral closure theory that
improves on Chang&Callen by addressing the full collision operator.
Regarding numerics, Ji has implemented and tested parts of his 21-moment system
and has applied it to the kinetic ion-acoustic wave.

Peter Norgaard described his efforts to implement the Shumlak dynamic neutrals model in NIMROD. He first described the system, noting where neutral-plasma interactions provide sources and collisional effects. His plan includes stages of implementation where different effects can be verified. The verification tests will consider neutral dynamics isolated from any plasma response and then the coupling effects. The model considers rates to be functions of temperature only. Norgaard's development has plasma and neutral density advanced simultaneously for the general case where coupling between the two is strong. He also described possible variants of the implementation, such as advancing a total center-of-mass velocity instead of the plasma center-of-mass velocity.

Andrea Montgomery summarized motivations and the status of resistive-wall modeling using the Green's function approach for the external vacuum region. Her implementation now uses general coupling coefficients that are suitable for arbitrary vacuum-region geometry computation from Alex Pletzer's GRIN code. She has tested the general-geometry implementation on a cylindrical external kink and has found that she can converge to within 2% of analytical results while varying the plasma-vacuum gap distance and the wall resistivity. Future tests will include nonlinear applications with interacting modes of different helicity and computations of driven tearing and flow evolution in NSTX and DIII-D geometry. Jacob King is continuing the development of the GRIN code for an ITER contract, and contributions to the NIMROD coupling are anticipated.

__Applications:__

Ping Zhu presented his study of plasma
response to resonant magnetic perturbation (RMP) in DIII-D shots 142603 and
126006, which are subjects of a community benchmarking exercise. He uses
2D equilibrium profiles and applies the RMP perturbations through asymmetric
boundary conditions on **B**. The distribution and harmonic content of
the perturbations represent DIII-D I-coils and have been provided by Valerie
Izzo; she used them in her 2008 study of RMP. Zhu's recent study includes
2-fluid Ohm's law effects and gyroviscous stress, and rotation will be studied
in the future. Zhu shows that large-n modes are unstable in both
profiles, and while two-fluid effects reduce growth rates relative to MHD,
complete stabilization does not occur. Linear evolution with the RMP
perturbations shows saturation for n=3. Whether profile evolution in full
nonlinear evolution stabilizes the large-n modes that grow linearly awaits
further investigation.

Jacob King summarized his study of
drift-tearing behavior in slab geometry, including analytics and NIMROD
computation. He distinguished two-fluid reconnection as conditions in the
plasma-beta/ion skin-depth (*d _{i}*) parameter space where tearing
without drift is accelerated by ion-electron decoupling. He noted that Coppi's
drift-tearing and his own previous work are in low beta regimes where it is
difficult to achieve sufficiently large drift to affect tearing growth
rates. More realistic conditions are at larger beta and

Joshua Sauppe is investigating drift tearing in an equilibrium with trigonometric-function profiles. He is collaborating with Vladimir Mirnov in an effort to reproduce the tearing stabilization effect published by the Maryland group. The equilibrium pressure gradient is supported by variation in the guide field, and magnetic-shear and density scales can be varied independently. As the shear, hence stability are varied, NIMROD computations show that growth rates peak away from the ideal-stability threshold due to non-constant perturbed-flux effects across the resonance. Sauppe compares Lundquist-number scalings for different values of the density length-scale. The S-scalings change only weakly, but quantitative differences occur. As the density gradient is increased, the tearing eigenmodes show a boomerang structure, implying strong phase variation over the inhomogenous direction. In several computations, a second mode that does not show resonant behavior also appears. Mirnov has used a WKB approach to show that resistive drift is unstable in the conditions of the simulation, and Sauppe is trying to verify whether this second mode is a resistive drift-wave instability.

Jugal Chowdury presented his efforts to simulate global tearing modes with the GEM code, which solves a system that models electrons with drift kinetics and models ions with gyrokinetics. The objective of GK global tearing represents a new and numerically very challenging multi-scale problem for fusion. The geometry of the computations is toroidal with a circular/annular cross-section. He has considered three tearing-mode test problems. The first case is from the Furth-Rutherford-Selberg paper and has no pressure gradient. Here, there is an unstable mode, but the growth-rate becomes independent of S at large S-values, and the eigenmode does not show fine-scale structure near the resonance. The second case is from the Holmes paper, and clear tearing behavior is not apparent. The third case has a prescribed profile to make the parallel-current gradient large at the mode resonance. Here, the mode shows fine-scale structure near the resonance, but the rate of growth suggests ideal instability.

Val Izzo presented a validation effort, where experiments on DIII-D were motivated by her NIMROD simulation results on localized massive gas injection (MGI) for disruption mitigation. She noted that symmetric injection is commonly considered to be ideal, but simulations show that radiation is not symmetric, anyway, due to 1/1 dynamics pushing just a limited region of the hot core into injected impurities. Her results also show that externally imposed magnetic perturbations can influence the 1/1 evolution, including total radiation, and that the Ne impurity distribution is very important. However, the location of gas injection ultimately controls the phase of the mode. The experimental series includes 17 shots with varied perturbation phase relative to the MGI valve and differing coil activation times prior to gas injection. Changing the I-coil timing has a greater effect on measurements far from the gas jet than on those closer, indicating a possible difference in Ne spreading due to rotation changes. The I-coil phases affect the time at which the TQ MHD event occurs, and the total energy radiated at a particular toroidal location varies by a factor of 2 as a function of phase. Thus, there is some evidence of the effects predicted by simulation.

Simon Woodruff, James Stuber, and Sam Schetterer presented objectives and development efforts for studying tokamak disruption and disruption mitigation with support from an ORNL ITER contract. The numerical tools at hand include Corsica, DCON, ray tracing, and others, in addition to NIMROD. Initial simulation work aims to reproduce Kruger's published study of DIII-D shot 87009. The obtained evolution is similar to the published result, and the group is investigating the remaining discrepancies. The second benchmark is to reproduce Strauss's results on VDE in ITER geometry and equilibria. WSI has design information for the first wall and equilibria and has modified Fluxgrid to be able to mesh these structures (approximately) to the vacuum vessel. NIMROD has been modified to advance only magnetic field beyond the first wall, with velocity boundary conditions applied there. WSI is also studying mesh optimization strategies and considering whether mesh adaptivity can be incorporated dynamically. The group is defining a validation plan and aims to study the shattered pellet approach to disruption mitigation.

Dylan Brennan led a discussion of disruption modeling based on presentations and discussion at the Theory and Simulation of Disruptions workshop held at PPPL in July. He noted that a highlights document has been posted on the USBPO website. An overall theme is the growing appreciation for transport time-scale nonlinear evolution that occurs with disruptions in modern tokamaks. Four topical areas, wall forces and halo/hiro currents, the physics of the halo current-layer width, interaction of islands and rotation, and runaway electron generation and evolution were considered. NIMROD modeling can contribute to each of these areas. The team discussed different ideas on boundary conditions for flow, the stability effects of halo currents, and modeling runaway electron generation. An action item for purposes of coordination and planning is to hold a conference call among NIMROD team members who are interested in disruption modeling.

Jonathan Hebert gave an update of his CTH (compact toroidal hybrid) modeling with temperature evolution, temperature-dependent resistivity, Ohmic heating, and anisotropic thermal conduction. Large viscosity in his earlier results prevented MHD activity. As he lowers viscosity to achieve a magnetic Prandtl number of 1000, number density becomes noisy. This has been addressed with hyper-diffusivity in the continuity equation. New computations with lower viscosity encounter a non-resonant (7,5) distortion. Discussion included sensitivity with respect to transport coefficients.

John O'Bryan gave an update on his modeling of current injection for non-inductive startup in the Pegasus spherical tokamak. He noted that the filamentary structure lasts well into the field-reversal stage, and reconnection in this phase releases current rings into the flux-amplified region, further reinforcing the building tokamak fields. He is investigating two-fluid effects, motivated by the fact that the current channel is small with respect to the ion skin depth. Qualitatively, fluctuations and current buildup with two-fluid modeling are similar to MHD results. There are quantitative differences, and O'Bryan finds that the Hall dynamo contributes to the mean electric field during relaxation events. When injection is removed, large-n perturbations decay faster in the two-fluid simulation, leading to a larger volume of closed magnetic flux before resistive decay. New work is examining flux compression, which is typically used together with current injection in Pegasus.

Bick Hooper summarized modeling of transient co-axial helicity injection in NSTX, work that has recently been submitted for publication in PoP. He listed a number of caveats on the modeling, such as not including absorber coils and either simplified or no radiation modeling. His simulations show closure of the magnetic flux shortly after the injection voltage is reduced by his circuit model. Analysis of magnetic pressures identifies the mechanism for reconnection onset as a drop in toroidal field within a narrow poloidal-field null. The resulting decrease in magnetic pressure leads to force densities that drive convergent poloidal flows into the null.