**NIMROD Team
Meeting Minutes**

**August 27-28, 2008**

**San
Diego, California**

__Model Development__

Carl Sovinec started this session reporting on two development projects. The first adds a numerically implicit hyper-diffusive operator to the continuity equation. The associated particle flux is proportional to third derivatives of number density, and like hyper-viscosity in fluid turbulence computations, it allows strong smoothing near the node spacing with little effect on scales of interest. This has been helpful for 3D two-fluid simulations. The second project is the development of a Grad-Shafranov solver using the NIMROD computational framework—work being done by U-WI grad student Eric Howell. The solver is working and getting benchmarked on simple equilibria. Further tests are needed for cases with pressure profiles and a separatrix. Development work for writing dump files based on the GS solution is underway.

Eric Held gave
an overview of Jeong-Young JiÕs work on his generalized velocity-moment method
as a kinetic description of plasmas.
It is based on complete orthogonal polynomials and the system of
equations is derived with the exact linear Coulomb collision operator. Ji is publishing separate manuscripts
on derivations and results for heat flux and stress. One significant physical effect from the results is how the
ion heat flux depends on collisions with electrons when *T _{e}*<

Held also
presented recent work and new ideas related to drift kinetic equation (DKE)
calculations. He and his group are
trying to make better use of data from integral computations (to deposit
accumulated information at more locations than just the launch point for a
characteristic), but new work for solution of the DKE in differential form
shows promise. The differential
DKE computations are motivated by success with the mixed finite element method
(MFEM), where the parallel component of the diffusive heat flux density is
solved simultaneously with temperature.
This results in impressive efficiency improvements over standard
high-order FE with large conductivity ratios. The differential DKE computations solve for coefficients of
a polynomial expansion of the kinetic distortion simultaneously with *T*, with some simplifications wrt to collisions and
speed dependence for feasibility testing.
Held showed the results of heat flux over an island in the collisional
limit, and they already show reasonable agreement with the fluid closure. He will be considering temporal
centering and how to incorporate a finite differencing grid or FE approximation
for the speed independent variable.

Charlson Kim
reported on recent progress and plans for the energetic particle model. The full-kinetic model is an option,
along with drift kinetics, and can use orbit averaging. More work is going into the use of
high-order fields for the particle push and using high-order shape functions
for the particle deposition. The
particle shapes are defined with respect to physical position and deposited
onto all nearby quadrature points where the shape is nonzero. This will affect domain decomposition
for parallel computation—more needs to be done, and the decomposition for
particles will likely need to be separate from that of the fluid
algorithm. Kim has also modified
the lagr_1D routine to use a table for the coefficients, which improves
performance. Schnack and Kruger
expressed interested in the updates for application to the giant sawtooth
problem.

Scott Kruger
reported on code coupling for the slow-MHD part of the SWIM project and on
diagnostics created by Dylan Brennan for tearing-mode analysis. Kruger reviewed the status of the
coupling project at the time of their review earlier this summer. [Phase 0: symm. ad hoc
drive—done, Phase 1: asymm. ad hoc drive—in progress, Phase 2: pass
NIMROD fields to GENRAY—in progress, Phase 3: use quasi-linear
diffusivity in F_{RF} computation—formulated.] The coupling is going through EQDSK
files, and Kruger has modified NIMROD to write EFIT-format files (called ÔfakeÕ
because they may not represent an accurate GS solution). Here, the challenging part is going
through different spatial meshes.
This part is working. Tom
Jenkins is working to understand GENRAY so that the quasi-linear computation
can be returned to NIMROD. This is
also nontrivial, because the data is generated along rays, which deposit
unevenly across the NIMROD mesh, an issue similar to making better use of the
integral closure information, which is mention above. Through numerical trials, it has been found that the
standard QSHEP2 routine is inadequate for this application. The tearing diagnostic determines the
poloidal Fourier components of NIMROD data using a straight field-line angle
and flux-coordinate components. It
is now working reliably and is showing phase differences between different
rational surfaces, in addition to the amount of reconnected field. Kruger also showed equilibria where
packing nodes during initialization helps field-line quality.

Val Izzo
presented development work that uses NIMFL to assess the acceleration of
superthermal electrons into runaways that could be damaging to a device with
the stored magnetic energy of ITER.
She uses existing NIMROD solutions to determine **E** and **B**
for each of a set of particles.
The present computations use drift kinetic orbits and a drag term, but
they can be generalized to include FLR effects. The modifications are mostly in int_segment; though, it was
suggested that the integration can be done by modifying the derivative-function
routine that is called by LSODE.
Other planned development includes drag from synchrotron radiation and
to automate the computations over multiple dump files.

__Performance Development and
Scaling__

Ping Zhu
presented information on porting NIMROD to the Cell Broadband Engine processors
that are used in IBM blade systems, such as Roadrunner, and in PS3
computers. The environment is
inhomogeneous with the power processor (PPE) being similar to other IBM power
chips but the SPEs being accelerators that have no memory. To take advantage of the SPEs, one has
to call libraries, such as a special CBE BLAS library, that use them. In an effort to test NIMROD, Zhu has
focused on the matrix-vector multiplication routine, which is used intensively
if SuperLU is not called for preconditioning. So far, there is no systematic improvement in speed, and
this may be due to the fact that the SPEs have been developed for
single-precision computation, while the NIMROD computations use
double-precision.

Sovinec
presented recent work on improving preconditioning for the two-fluid magnetic
field advance. He reviewed the
fact that the Fourier representation for the toroidal coordinate provides a
natural physics basis for preconditioning magnetic-confinement computations;
perturbations are small, so the diagonal-in-*n* block is naturally dominant.
This has been the basis for preconditioning all of NIMRODÕs 3D
solves. The use of a
preconditioner based on planes at different toroidal angles was expected to be
complementary, but it suffered from lack of diagonal dominance and was not
effective even in combination with what is normally done. Using selected off-diagonal couplings
has proven to be more effective; tests show that the iteration count does not
grow with increasing toroidal resolution.
The importance of using some electron inertia was also discussed. Finally, a new weak scaling study on
Franklin shows that the new approach works well with SuperLU_DIST for the diagonal-in-*n* inversions up to more than 1000 processors. Some data and looping rearrangement is
expected to further improve the Fourier scaling part.

Held reviewed
processor scaling for the integral closures computations. He and Kruger had developed code to
separate closure-calculation processors from fluid-calculation processors, and
this approach scales reasonably well to a few thousand processors. Synchronization between the separate
groups is the primary issue for further scaling. Running an entire parameter scan simultaneously is another
approach to using large numbers of processors, and this was used for heat flux
computations that were presented at the RMP workshop.

Kim discussed
scaling issues for the energetic particle model. The present bottleneck is load balancing with NIMRODÕs
standard domain decomposition being based on grid blocks and Fourier
layers. The particles can be
decomposed separately but may need duplication of the field data that is used
for the particle push. An __action
item__ from this discussion is to do a literature search on different domain
decomposition strategies that are used in PIC computations to bring in as many
ideas as possible before further development.

__Applications__

Bonita Squires
presented computations of peeling/ballooning based on the toroidal circular
cross section TOQ equilibrium developed by Scott Kruger and Phil Snyder for
benchmarking. For this case, ideal
plasma behavior is reproduced for S³5«10^{7}. Results also asymptote when S for the
edge region is below unity. [There
was discussion concerning length scales for the edge.] To see a growth-rate maxima at low *n* (associated with peeling), the resistivity gradient
needs to be very narrow and fit between the pressure pedestal and the rational
surface of the externally resonant mode.
Squires also described cases that show unusual linear numerical
behavior. For example, when the
resistivity gradient is broad and has some extent across the pressure gradient,
growth rates computed from magnetic and kinetic energies do not remain at the
same value. A nonlinear
demonstration computation that uses a fixed resistivity profile was also
presented. An __action item__
related to this work is to send the TOQ equilibrium and specification to Steve
Jardin, who is doing similar testing with M3D-C1.

Zhu presented
ballooning computations that go into the late nonlinear stage (defined by
perturbation amplitude scaling with respect to the *k*-parallel/*k*-perp ratio). They are
helped by the new particle hyper-diffusivity and are being used in comparison
with recent analytical work. The
equilibrium has circular cross section and is generated by the ESC code. There is no dissipation except for the
particle hyper-diffusivity, and the nonlinear computations go a factor of 10
further in amplitude than they did with the FickÕs law particle flux. Development to track displacement is
underway.

Chris Hegna
presented work from graduate student Mark Schlutt that applies NIMROD to 3D
cylindrical stellarator equilibria.
The motivation is the existence of experimental results beyond the
conditions where interchange-type and low-*n*
instability is expected. The
computations start with helically asymmetric magnetic perturbations that are
frozen in the wall. Computational
results show that when perturbed with shearing motions, the 3D equilibrium
remains intact. A source of heat
is then added with anisotropic thermal conduction to develop pressure
gradients. A Chapman-Enskog-like
closure will eventually be added to the study.

Hegna also
described a study that graduate student Andrea Montgomery is starting. The general topic is the resistive wall
mode (RWM) and field-error penetration in a cylinder. It is motivated by a Õ95 publication from J. Finn that
considered how resonant modes can provide the damping that is needed for
rotational stabilization.

Kruger
presented tearing-mode applications by Brennan (with some help from
Kruger). He reviewed the different
categories of neoclassical tearing modes (NTMs) outlined in a paper by Brennan. To assess different simulations, the D¢-matrix
theory is being applied. The
importance of rotation profiles is also being considered, and the new
diagnostics described earlier are being used to examine relative phases of
perturbations at different resonant surfaces. In the case of an *n*=2
mode, it appears as a secondary instability (being made linearly unstable
during the course of a nonlinear computation) but is not driven directly by the
primary mode. There is no evidence
of secondary islands at this point, but the response when viscosity is changed
is not understood.

Sovinec
presented two-fluid reconnection and tearing work by graduate student Jacob
King and himself. He reviewed some
history of two-fluid 1/1 internal kink computations. Recent NIMROD simulation results use the new preconditioning
approach and have been spatially converged with S=10^{6}, Pm=0.1, *d _{i}*=0.11, and

Izzo presented
her recent study of runaway electrons in DIII-D and comparing with conditions
for ITER. The runaway electrons
form when electric field exceeds the Rosenbluth *E _{crit}*, and relative to this criterion, ITER will have
safer conditions than DIII-D.
Massive gas injection is too slow to penetrate, so the new technique is
dilution cooling with D

Dalton Schnack
presented Tom JenkinsÕ recent work for RF stabilization of slow-MHD; Jenkins
participated remotely. Schnack
reviewed the cartoon model for asymmetric ad hoc current drive, and Jenkins has
worked out the analytics for the dissipation of shear Alfven waves excited by a
local source. The symmetric *n*=0 mode is the only one that is not damped, which
seems reasonable, but there was debate on the spatial scale that would appear
in the decay rates. When using the
ad hoc drive in toroidal tearing computations, cases with varying integrated
drive suggest that a small spot size (less than island width) for the drive is
not very effective. However in a
series with fixed ad hoc-induced current, the smaller spot size is actually
more effective in reducing gamma (measured by resetting and running the linear
phase of the mode with the ad hoc current-modified profile). Heating associated with ECRH can also
be modeled.

Charlson Kim
reviewed computations being performed at PSI-Center. The center is returning to the nimdevel version of the
code. Their NIMROD applications
include FLR in RFP tearing, HIT-SI, LDX, Swarthmore SSX modifications, FRC work
for thrusters, and ZAP. The LDX
computations are using a heat source to generate equilibria from the vacuum
fields. They have progressed to
getting a global interchange linearly unstable.

Sovinec
presented graduate student Chris CareyÕs MHD study related to astrophysical
jets. The 3D simulations retain
symmetry in regions where the current profile is very collimated, and linear
initial-value computations indicate that viscosity and axial flow are not the
stabilizing effect. The azimuthal
rotation is key. Carey has done
eigenvalue analysis in both Lagrangian and Eulerian references frames and found
that rigid rotation stabilizes pinches where either scalar pressure or magnetic
pressure provide force-balance with centrifugal acceleration. With the perturbed mass term added to
NIMRODÕs linear advective force, the initial-value computations agree
quatitatively. [This change makes
the linear behavior consistent with ÔfullÕ continuity in nonlinear computations
instead of Ôfix profileÕ continuity.]
A paper describing this linear behavior is in preparation.

Sovinec
provided an overview on helicity injection computations at Wisconsin. He reviewed results from Adam Bayliss
on simulation of HIT-II in weakly relaxing conditions and direct comparison
with experimental results on plasma current and injector current as a function
of toroidal field strength. There
is good quantitative agreement, but experiment and computation deviate somewhat
and in different directions from a simple equilibrium estimate. For flux compression on Pegasus,
graduate student John OÕBryan has developed the thermal conduction model to
include demagnetization in BraginskiiÕs fluid model. In this application, the low-temperature plasma outside the
current channel is very collisional, so the modeling reverts to what is usually
parallel conduction, which is smaller than perpendicular in these conditions. Undergraduate student Tom Bird is using
a toroidally localized ad hoc current drive to induce a helix of current, as in
Pegasus miniature gun injection.
He will be investigating relaxation as more current is induced; however,
greater resolution is needed than in the current helix result that was shown.

Schnack
presented BaylissÕ recent work to simulate Cary ForestÕs new plasma dynamo
experiment, which has rings of permanent magnets and electrodes that spin
plasma from the surface. A
small-scale experiment uses cylindrical geometry, and this series of
computations model that experiment.
The effect of the electrodes is modeled with a boundary condition on
tangential electric field. While
the dynamics near the surface are complicated, the flow transitions to rigid
rotor at the interior of the midplane.
The flows excite an *n*=1
perturbation in 3D computations.

__Project Improvement__

Related to the
website, the following __action items__ were discussed:

á
Provide updates to the publication list to either
Sovinec or Kruger.

á
Modify the opening page so that the links go to text on
the same page.

á
The team member list needs updating.

á
Send pdfs of this meetingÕs presentations to Sovinec.

Related to discussions with Steve Jardin:

á
Send him the peeling/ballooning benchmark equilibrium.

á
ITER wants computational information on structural
forces during disruption.

á
We should consider modeling the Zakharov wall mode.

Rostom Dagazian
told us that OFES is looking for a new computational milestone for FY
2010. We discussed possibilities
related to sawteeth and RMP.

The final
segment included a discussion of using workflow software to automate jobs and
manage data. There are different
opinions on how useful this would be for NIMROD, but it may provide some
political benefits. The Doxygen
software can be useful for documentation.
It automatically extracts comments from code and puts them in linked
html pages. A minor modification
in the writing of nimrod.out will allow it to be used for generating a
nimrod.in file. Finally, there was
a request to establish a new release version of nimdevel.