**NIMROD Team
Meeting Minutes**

**May 1-2, 2009**

**Boulder,
Colorado**

__Performance Development and
Scaling__

Carl Sovinec described the implementation of Newton-Krylov iteration in NIMRODÕs two-fluid implicit leapfrog advance. This iteration centers the algebraically nonlinear momentum-density advection term in the velocity advance and the nonlinear Hall term in the magnetic field advance. ÔAlgebraically nonlinearÕ here means nonlinear with respect to the change over each time-step; most of the physically nonlinear contributions are explicit. The temporal staggering of the implicit leapfrog and the bilinear character of the two terms facilitate computation of the Jacobian for each Newton step, and the implementation is a modification of previously existing code. The nonlinear residual also has a simple form with the bilinear nature of these terms, and iteration tolerance is checked with the norm routine from NIMRODÕs 3D linear solver. Initial tests of the nonlinear iteration do not show significant changes, but the algorithm may improve nonlinear stability.

Jeong-Young Ji
gave a presentation on closure theory and emphasized that use of a transport
equation to close a moment equation is not consistent. In the general moment approach, all
equations have the same general form.
Terms associated with derivatives of the background Maxwellian
distribution lead to drives in a matrix equation for the moments of the kinetic
distortion. He has considered
Braginskii (small gyro-orbit), neoclassical (high to low collisionality with
flux-surface averages over assumed nested surfaces), and general closure
without flux-surface averaging and with the full linearized Coulomb collision
operator. Recent work includes a
simple fit for the kernel function that agrees with Chang and Callen in the
collisionless limit. The simple
fit appears to be very accurate and should help make nonlocal closure
computations more practical.

Travis Austin
described algebraic multigrid (AMG) and its possible application to
NIMROD. The motivation is scalability
as MG has been the only matrix algorithm to show *O*(N^{1}) performance for Poisson problems on
the largest parallel machines available today. The geometric version determines restriction and
prolongation operations geometrically, but this has problems with stretched
meshes. Algebraic versions
determine strongly connected points in the matrix graph and use the strong
connections to define coarse approximations. This is challenging for high-order finite elements, because
the matrices are not M-matrices. His approach is to use linear elements to precondition, based
on ideas from Paul Fischer.
Important algebraic modes within each high-order element are determined
by separate minimization problems.
This form of problem-reduction could also be used with direct solvers
when they are applied to preconditioning.

Jianhua Cheng
presented a hybrid algorithm that uses delta-f on the full kinetic equation for
ions. It adds the delta-B-parallel
response and is expected to be useful for edge physics and for the MHD
ordering. The electrons presently
use fluid modeling, assuming an isothermal response for now, and displacement
current is neglected. The
algorithm is implicit for compressional-wave physics at high beta, and Cheng
has made the advance second-order accurate using iteration for the
particle/electric field coupling.
Test problems considered so far include compressional and shear Alfven
waves, the ion acoustic wave, and the Whistler wave. Work on Harris-sheet reconnection is starting. Modifications for drift and gyrokinetic
electrons are also being considered.

__Project Improvement__

Scott Kruger
gave a presentation and a demonstration of regression testing, which uses the
txtest program written by David Wade-Stein. Its main purpose is to ensure that development in one area
does not affect other parts of the code.
It is used for both automated testing, which is run nightly, and
manually by developers as they are working. Test cases should be short and coarsely resolved, as tests
that take a long time discourage use.
However, this means that we may be looking for changes at high precision
(to a controllable tolerance via h5diff), so deciding when a discrepancy is
acceptable or not can be tricky, as in the ORNL tearing-mode case. More tests are needed, including cases
from SchnackÕs V&V list. The
test cases are placed in separate SVN repositories that are organized by
machine and require 1) a shell script to run the case, 2) input files, and 3)
an edit to makefile.am. There is a
nimrod-regress mailing list for those who would like to see all output.

__Applications__

Charlson Kim
described simulation and visualization work for ICCs at the PSI-Center. He has been simulating the Caltech
spheromak/plasma jet experiment.
The simulated injector specifies toroidal field as a boundary condition
and has an insulating layer above it.
Visualization uses the Visit package from LLNL. It is used to show the formation of
closed-flux regions during decay.
It has also been used to show 3D visualization capability.

Ryoji Takahashi
gave a presentation on kinetic effects of fast particles with resistive MHD
instability. Results from JET
suggest that large hot-particle beta breaks the empirical rho-star scaling of
the 2/1 mode from smaller machines.
Linear computations show that the stabilizing effect of hot particles is
strongest in the resistive regime (with respect to the ratio of beta-normal and
internal inductance) of this mode, so itÕs not an ideal resonance effect. The study will first try to explain the
linear effects then move to nonlinear.
However, the lack of nlayer parallelization with particles is a
hindrance. An **action item** is for Takahashi and Kim to first review the
implementation to verify that characteristics and weight-evolution equations
are correct for nonlinear conditions.

Dalton Schnack
discussed efforts to model giant sawteeth in DIII-D. The work involves collaboration with Alan Turnbull at GA and
seeks to model the effects reported in the PoP paper by Choi. Fast particles stabilize the internal
kink, but slow changes in the magnetic shear lead to less stabilization over
the cycle. Thus, the crash results
from a changing kinetic effect and not from changes to MHD stability; though,
the hot-particle beta also changes.
The high-energy tail is actually from RF kicking beam-driven particles
to 100s of keV, and these particles are the important ones according to
Turnbull. The NIMROD hot-particle
equilibrium presently does not model this part of the distribution function,
and extremely energetic particles will lead to very small timestep. [Brute forcing with better
parallelization may be possible, at least linearly.] Schnack noted that the analytical relation for kink
stabilization/fishbone destabilization seems to require two-fluid effects. Also, NIMRODÕs two-fluid advance
appears to be bypassed completely when particles are used. A list of **action items** for this study is:

1. Confirm
that we are doing an appropriate and tractable case. (Schnack)

2. Correct
particle acceleration and field advance for running particle with 2-fluid.
(Brennan, Takahashi, and Kim)

3. Modify
the distribution function for an RF tail. (Kim)

4. Improve
particle parallelization, possibly through Eric HeldÕs code for closures. (Kim,
Kruger, and Held)

Tom Jenkins
described his progress on 2/1 island saturation subject to a localized
(RF-like) source term. If it is
placed in the right location in the poloidal plane, the localized source can
completely stabilize the island.
However, rational surfaces move in response to the source. This is an important effect that
warrants a more thorough investigation than it was given in the Pletzer-Perkins
paper. Another consideration is
the prompt response of the profile (after several Alfven times) vs. the long-time
response after resistive diffusion times when the source drives net
current. The former is likely more
important for active steering systems that are being used in experiment. Jenkins also discussed progress in
coupling with the GENRAY ray-tracing code through use of the Ôplasma stateÕ
that has been defined as part of the SWIM project.

Valerie Izzo
described her progress in simulating runaway electrons during tokamak
disruption. In her modeling, the
seed electrons are nonrelativistic, and disruption is induced by a modeled gas
jet. The electron trajectories are
now computed at runtime through the use of code developed by Eric Held for
nonlocal closure, instead of separately as postprocessing. The computations compare profiles with
low and high elongation; the high elongation cases have more nonlinear
coupling. Next steps will include
perpendicular drifting, so that orbits need not be perfectly tied to
field-lines, and synchrotron radiation effects.

Ping Zhu gave
an update on his two-fluid ballooning computations. With some unexplained fix of memory use on Franklin, he is
again able to run linear computations with high resolution. From equilibria generated from ESC,
NIMROD produces ideal ballooning modes with little dissipation across most of
the spectrum. However, Zhu finds
that particle hyperdiffusivity alone does not provide enough dissipation to
avoid numerical instability for all n; some thermal conduction is also
important. At very low n, the
computations can still produce slowly growing noise in n in the transition
region between small and large elements.
With two-fluid effects in OhmÕs law and gyroviscosity, the modes show
up-down asymmetry. In these cases,
Zhu is not expecting FLR stabilization, just a reduction in growth rate. Here, the very low n behavior seems to
be okay.

Bonita Burke
(formerly Squires) reviewed verification work for ideal ballooning behavior
through a benchmark with GATO and ELITE.
Her recent sets of computations have no resistivity gradient in the
pressure pedestal, and mass density has a hyperbolic-tangent profile. Burke emphasized the need to scan
S-values in the core and halo regions separately for each case. The existence of a halo is most
important for n-values of around 5 or 6.
For the equilibria labeled dens8 (the more unstable case), NIMROD shows
good quantitative agreement with GATO for the n-values where it has been
applied (n>11). ELITEÕs
predictions are close and depend on whether parallel compression energy is
used. Burke reported that NIMROD
also finds the correct point of marginality between n=8 and n=9 for the less
unstable dens6 case (GA did not provide their spectra); though, ÔfalseÕ modes
with small non-convergent growth-rate can appear in computations below
marginality. A new equilibrium is
being used for peeling/ballooning tests, and Burke has computed an ideal
spectrum. However, with a
realistic Spitzer resistivity profile, a peeling peak is not evident, possibly
due to faster resistive ballooning at high n.

Mark Schlutt
gave an update on his application of NIMROD to straight stellarators. The objective of the study is to
determine what limits beta, how it occurs, and what influences gradual magnetic
topology change. An important
aspect is the pressure-induced Pfirsch-Schluter current. The NIMROD results will be compared
with analytical work for 3D islands with finite parallel thermal
conductivity. Schlutt is
considering two different problems, one that applies nonuniform heating to a 2D
magnetic configuration and the other that adds 3D magnetic perturbations. The computations start with nested flux
surfaces, and induced resonant harmonics change the topology. Schlutt compared low and high heating
power, and high heating produces poloidally elongated configurations that can
split into m=2, n=2 islands.

Sovinec gave
an overview of helicity-injection related computations by students Tom Bird,
John OÕBryan, and Eric Howell.
Bird had presented his work at the 2009 U-WI Undergrad Symposium. His simulations of current injection
and relaxation in the Pegasus ST use the toroidally localized version of the ad
hoc source used in JenkinsÕ study.
The latest shows helical current paths distorting from self-induced
field overcoming vacuum field, but full merging and current multiplication have
yet to occur with a basic resistive-MHD model. New computations are using temperature-dependent
coefficients. OÕBryan is presently
performing 2D modeling of flux compression from an assumed, relaxed state. The computations use T-dependent
coefficients with magnetization and Ohmic heating. Computations with realistic parameters are showing more
heating than current drive. Howell
is studying the linear properties of decaying spheromak profiles, which can now
be generated systematically with NIMEQ.
He is first checking the S-scaling of resonant modes with resistive MHD
and will then consider the influence of two-fluid effects.

Jacob King
reported on progress with two-fluid tearing and dynamo. The present set of computations provide
scans over normalized sound gyroradius with uniform pressure to avoid equilibrium
diamagnetic effects for now.
Single-helicity cases started from paramagnetic pinch profiles produce a
saturated island width that is approximately independent of sound gyroradius. The out-of-phase components of the
magnetic perturbations do change, however, and this is important for the Hall
dynamo contribution. The dynamo
spatial structure is more complicated in the two-fluid cases and shows
significant cancellation of the Hall and MHD dynamo contributions away from the
rational surface. The two-fluid
cases also show slow rotation after saturation. King also described some preliminary results for
single-helicity axis states produced in RFX. The modeling will investigate the importance of
temperature-dependent transport effects.