# Publications¶

In the following list, a conference proceedings paper is included only if the paper was peer-reviewed.

17. A. H. Hakim. G.W Hammett and E. L. Shi, “On discontinuous Galerkin
discretizations of second-order derivatives”, *Journal of
Computational Physics*, Submitted (2014)

**Abstract** Some properties of a Local discontinuous Galerkin (LDG)
algorithm are demonstrated for the problem of evaluting a second
derivative \(g = f_{xx}\) for a given \(f\). (This is a
somewhat unusual problem, but it is useful for understanding the
initial transient response of an algorithm for diffusion equations.)
LDG uses an auxiliary variable to break this up into two first order
equations and then applies techniques by analogy to DG algorithms for
advection algorithms. This introduces an asymmetry into the solution
that depends on the choice of upwind directions for these two first
order equations. When using piecewise linear basis functions, this
LDG solution \(g_h\) is shown not to converge in an \(L_2\)
norm because the slopes in each cell diverge. However, when LDG is
used in a time-dependent diffusion problem, this error in the second
derivative term is transient and rapidly decays away, so that the
overall error is bounded. I.e., the LDG approximation
\(f_h(x,t)\) for a diffusion equation \(\partial f / \partial
t = f_{xx}\) converges to the proper solution (as has been shown
before), even though the initial rate of change \(\partial f_h /
\partial t\) does not converge. We also show results from the Recovery
discontinuous Galerkin (RDG) approach, which gives symmetric solutions
that can have higher rates of convergence for a stencil that couples
the same number of cells.

16. Thomas G Jenkins, Travis M Austin, David N Smithe, John Loverich,
and Ammar H Hakim, “Time-domain simulation of nonlinear radiofrequency
phenomena”, *Physics of Plasmas*, **20**, 012116 (2013)

**Abstract** Nonlinear effects associated with the physics of
radiofrequency wave propagation through a plasma are investigated
numerically in the time domain, using both fluid and particle-in-cell
(PIC) methods. We find favorable comparisons between parametric decay
instability scenarios observed on the Alcator C-MOD experiment
[J. C. Rost, M. Porkolab, and R. L. Boivin, Phys. Plasmas 9, 1262
(2002)] and PIC models. The capability of fluid models to capture
important nonlinear effects characteristic of wave-plasma interaction
(frequency doubling, cyclotron resonant absorption) is also
demonstrated.

15. A. H. Hakim, T.D. Rognlien, R.J. Groebner, J. Carlsson, J.R. Cary,
S.E. Kruger, M. Miah, A. Pankin, A. Pletzer, S. Shasharina,
S. Vadlamani, R. Cohen and T. Epperly. “Coupled Core-Edge Simulations
of H-Mode Buildup Using the Fuson Application for Core-Edge Transport
Simulations (FACETS) Code”, *Physics of Plasmas*, **19**,
032505 (2012)

**Abstract** Coupled simulations of core and edge transport in the
DIII-D shot number 118897, after the L-H transition but before the
first edge localized mode (ELM), are presented. For the plasma core
transport, a set of one dimensional transport equations are solved
using the FACETS:Core solver. The fluxes in this region are calculated
using the GLF23 anomalous transport model and Chang- Hinton
neoclassical model. For the plasma edge transport, two-dimensional
transport equations are solved using the UEDGE code. Fluxes in the
edge region use static diffusivity profiles based on an interpretive
analysis of the experimental profiles. Simulations are used to study
the range of validity of the selected models and sensitivity to
neutral fueling. It has been demonstrated that the increase of neutral
influx to the level that exceeds the level of neutral influx obtained
from analysis simulations with the UEDGE code by a factor of two
results in increased plasma density pedestal heights and plasma
density levels in the scrape-off-layer region. However, the additional
neutral influx has relatively weak effect on the pedestal width and
plasma density profiles in the plasma core for the DIII-D discharge
studied in this research.

14. B. Srinivasan, A. Hakim and U. Shumlak. “Numerical Methods for
Two-Fluid Dispersive Fast MHD Phenomena”, *Communications in
Computational Physics*, **10**, Pg. 183-215, 2011.

**Abstract** The high-resolution wave propagation method and the
Runge-Kutta discontinuous Galerkin (RKDG) method are studied for
applications to balance laws describing plasma fluids. The plasma
fluid equations explored are dispersive and not dissipative. The
physical dispersion introduced through the source terms leads to the
wide variety of plasma waves. The dispersive nature of the plasma
fluid equations explored separates the work in this paper from
previous publications. The linearized Euler equations with dispersive
source terms are used as a model equation system to compare the wave
propagation and RKDG methods. The numerical methods are then studied
for applications of the full two-fluid plasma equations. The two-fluid
equations describe the self-consistent evolution of electron and ion
fluids in the presence of electromagnetic fields. It is found that the
wave propagation method, when run at a CFL number of 1, is more
accurate for equation systems that do not have disparate
characteristic speeds. However, if the oscillation frequency is large
compared to the frequency of information propagation, source splitting
in the wave propagation method may cause phase errors. The Runge-Kutta
discontinuous Galerkin method provides more accurate results for
problems near steady-state as well as problems with disparate
characteristic speeds when using higher spatial orders.

13. John Loverich, Ammar Hakim, Sudhakar Mahalingam, Peter Stoltz, Sean C.D. Zhou, Michael Keidar, M. Jandrapu, TaiSen Zhuang, Jason Cassibry and Richard Hatcher. “Simulation of labratory accretion disk and weakly ionized hypersonic flows using Nautilus”, American Institute of Aeronautics and Astronautics. Paper number AIAA 2011-4012, 2011.

**Abstract** A proposed experiment at HyperV technologies in
partnership with Los Alamos National Lab will explore the use of
merging, high velcocity (50km/s) plasma jets as a laboratory model for
astrophysical accretion disks. Similarly, researchers at George
Washington Uni- versity (GWU) are currently developing an experiment
for investigating weakly ionized hypersonic flow in magnetic
fields. The two experiments cover a broad range of hypersonic
conducting fluid flows which can be explored using the shock capturing
plasma fluid code, Nautilus. In this paper, progress on modeling both
experiments is presented along with associated physics and numerical
techniques.

12. Ammar H. Hakim and John Loverich. “Robust, positivity and divergence preserving code for multi-fluid, multi-species electromagnetics and plasma physics applications”, American Institute of Aeronautics and Astronautics. Paper number AIAA 2011-4011, 2011.

**Abstract** In this paper we will present algorithms implemented in
Nautilus, a new code for the solution of a wide variety of plasma and
neutral fluid equations. Nautilus can simulate fully and weakly
ionized flows and has been applied to fast magnetohydrodynamics (MHD)
devices like Z-pinches and dense-plasma focuses, chemically reacting
hypersonic flows and plasma jet propagation and merging for the
creation of high energy-density states in the laboratory. We present
details of new positivity preserving fluid algorithms as well as
collocated Maxwell equation solvers that preserve divergence and work
on general geometries. Current state of Nautilus is described and
future directions outlined.

11. John Loverich, Ammar Hakim and Uri Shumlak. “A Discontinuous
Galerkin Method for Ideal Two-Fluid Plasma Equations”, *Communications
in Computational Physics*, **9** (2), Pg. 240-268, 2011.

**Abstract** A discontinuous Galerkin method for the ideal 5 moment
two-fluid plasma system is presented. The method uses a second or
third order discontinuous Galerkin spatial discretization and a third
order TVD Runge-Kutta time stepping scheme. The method is benchmarked
against an analytic solution of a dispersive electron acoustic square
pulse as well as the two-fluid electromagnetic shock and existing
numerical solutions to the GEM challenge magnetic reconnection
problem. The algorithm can be generalized to arbitrary geometries and
three dimensions. An approach to maintaining small gauge errors based
on error propagation is suggested.

10. A H Hakim, J R Cary, J Candy, J Cobb, R H Cohen, T Epperly, D J
Estep, S Krasheninnikov, A D Malony, D C McCune, L McInnes, A Pankin,
S Balay, J A Carlsson, M R Fahey, R J Groebner, S E Kruger, M Miah, A
Pletzer, S Shasharina, S Vadlamani, D Wade-Stein, T D Rognlien, A
Morris, S Shende, G W Hammett, K Indireshkumar, A Yu Pigarov, H
Zhang. “Coupled whole device simulations of plasma transport in
tokamaks with the FACETS code”, *SciDAC 2010: J. Physics:
Conf. Series*, 2010.

**Abstract** The FACETS project aims to provide computational tools
for whole device simulation of tokamak transport for use in fusion
applications. The framework provides flexibility by allowing users to
choose the best model for a given physics target. Our goals are to
develop accurate transport solvers using neoclassical and turbulent
fluxes with varying degree of fidelity and computational complexity,
including embedded gyrokinetic models. Accurate sources using both
ICRH wave absorption and neutral beam injection, using parallel source
components, are included. Modeling of the plasma edge using a fluid
based component, UEDGE, is performed and coupled to the core
solver. The core region is simulated using a newly developed parallel,
nested iteration based nonlinear solver while the UEDGE uses nonlinear
solves from the PETSc/SNES solver package. As a first application we
present coupled core-edge simulations of pedestal buildup in the DIIID
tokamak.

9. John Loverich, Ammar Hakim. “Two-Dimensional Modeling of Ideal
Merging Plasma Jets”, *Journal of Fusion Energy*, **29** (6),
Pg. 532-539, 2010. Journal link

**Abstract** Idealized merging argon plasma jets are simulated in 2D
using both gas dynamic and MHD models. Results indicate that peak
pressures of several hundred kilobar can be achieved for high Mach
number jets. Including a simple optically thin Brehmstrahlung
radiation model and plasma targets shows that extremely high densities
and magnetic fields can be achieved during jet merging on the order of
~1000 times the initial density/field. Further investigations should
include detailed ionization processes and more accurate radiation
modeling to properly capture the radiation transport and subsequent
target compression.

8. John R. Cary, Ammar Hakim, Mahmood Miah, Scott Kruger, Alexander Pletzer, Svetlana Shasharina, Srinath Vadlamani, Alexei Pankin, Ronald Cohen, Tom Epperly, Tom Rognlien, Richard Groebner, Satish Balay, Lois McInnes, Hong Zhang, “FACETS - a Framework for Parallel Coupling of Fusion Components”, The 18th Euromicro International Conference on Parallel, Distributed and Network-Based Computing. Pisa, Italy. 2010.

**Abstract** Coupling separately developed codes offers an attractive
method for increasing the accuracy and fidelity of the computational
models. Examples include the earth sciences and fusion integrated
modeling. This paper describes the Framework Application for Core-Edge
Transport Simulations (FACETS).

Note

This is one of the three papers that describe the proto-Fusion Simulation Programs (proto-FSP). The Facets project aims to tightly couple (with implicit methods) the core-edge and wall regions of the tokamak to perform integrated simulations. The other two proto-FSPs CSWIM and CPES also study coupled systems but they use file/in-memory loose coupling.

7. B. Srinivasan, U. Shumlak and A. Hakim, “Comparisons and Applications of Two-Fluid Plasma Algorithms”, American Institute of Aeronautics and Astronautics. Paper number AIAA 2008-3787, 2008.

This paper describes a study of the five-moment two-fluid plasma model and the asymp- totically approximated fluid model that is derived from it. The two models are compared for applications of an electromagnetic plasma shock, magnetic reconnection and an axisym- metric Z-pinch. The physics captured is compared between these fluid models to determine the regime of applicability. These models are explored for their ability to capture small- scale physics with two-fluid effects that are not sufficiently captured by ideal-MHD.

6. Peter H. Stoltz, Brian Granger, Ammar Hakim, Scott W. Sides and
Seth A. Veitzer, “Effects of Sputtering of and Radiation by Aluminum
on Magnetized Target Fusion Plasmas”, *Journal of Fusion Energy*,
**27** (1-2), 2008. Pages 119-122.

We estimate numerically the rate of radiation by aluminum impurities for parameters relevant to magnetized target fusion (MTF) plasmas. We demonstrate that the coronal equilibrium is appropriate for expected MTF plasma parameters. Using the coronal equilibrium, we estimate the power radiated per impurity ion is 0.25-0.5:math:10^{16} MW for temperatures and densities relevant to present plasma parameters taken from the FRX-L experiment at Los Alamos National Laboratory and is approximately 75.0:math:10^{16} MW for temperatures and densities relevant to anticipated MTF plasmas. We calculate the sputtering rate of aluminum by thermal deuterium and tritium plasma ions is a few percent assuming an impact angle of 45 degrees. Finally, we estimate that with aluminum impurity levels of a few percent, the impurity radiation power density would be approximately 25 kW/cm3 for FRX-L conditions and 2.5 GW/cm3 for anticipated conditions in a MTF plasma. While we have assumed a sputtering model of impurity generation, the results for the power density apply for impurity levels of a few percent, regardless of the generation mechanism.

5. Ammar H. Hakim, “Extended MHD Modelling with the Ten-Moment
Equations”, *Journal of Fusion Energy*, **27** (1-2), 2008. Pages
36-43.

**Abstract** High-order moment fluid equations for simulation of
plasmas are presented. The ten-moment equations are a two-fluid model
in which time dependent equations are used to advance the pressure
tensor. With the inclusion of the full pressure tensor Finite Larmor
Radius (FLR) effects are captured. Further, Hall-effects are captured
correctly by including the full electron momentum equation. Hall and
FLR effects are important to understand stability of compact toroids
like Field Reversed Configurations (FRCs) and also to detailed
understanding of small scale instabilities in current carrying
plasmas. The effects of collisions are discussed. Solutions to a
Riemann problem for the ten-moment equations are presented. The
ten-moment equations show complex dispersive solutions which come
about from the source terms. The model is validated with the GEM fast
magnetic reconnection challenge problem.

4. A. Hakim, U. Shumlak, “Two-fluid physics and field-reversed
configurations”, *Physics of Plasmas*, **14**, 055911, 2007.

**Abstract** In this paper, algorithms for the solution of two-fluid
plasma equations are presented and applied to the study of
field-reversed configurations (FRCs). The two-fluid model is more
general than the often used magnetohydrodynamic (MHD) model. The model
takes into account electron inertia, charge separation, and the full
electromagnetic field equations, and it allows for separate electron
and ion motion. The algorithm presented is the high-resolution wave
propagation scheme. The wave propagation method is based on solutions
to the Riemann problem at cell interfaces. Operator splitting is used
to incorporate the Lorentz and electromagnetic source terms. The
algorithms are benchmarked against the Geospace Environmental Modeling
Reconnection Challenge problem. Equilibrium of FRC is studied. It is
shown that starting from a MHD equilibrium produces a relaxed
two-fluid equilibrium with strong flows at the FRC edges due to
diamagnetic drift. The azimuthal electron flow causes lower-hybrid
drift instabilities (LHDI), which can be captured if the ion
gyroradius is well resolved. The LHDI is known to be a possible source
of anomalous resistivity in many plasma configurations. LHDI
simulations are performed in slab geometries and are compared to
recent experimental results.

3. A. Hakim, J. Loverich, U. Shumlak, “A high resolution wave
propagation scheme for ideal Two-Fluid plasma equations”, *Journal of
Computational Physics*, **219**, 2006.

**Abstract** Algorithms for the solution of the five-moment ideal
Two-Fluid equations are presented. The ideal Two-Fluid model is more
general than the often used magnetohydrodynamic (MHD) model. The model
takes into account electron inertia effects, charge separation and the
full electromagnetic field equations and allows for separate electron
and ion motion. The algorithm presented is the high resolution wave
propagation method. The wave propagation method is based on solutions
to the Riemann problem at cell interfaces. Operator splitting is used
to incorporate the Lorentz and electromagnetic source terms. To
preserve the divergence constraints on the electric and magnetic
fields two different approaches are used. In the first approach
Maxwell equations are rewritten in their mixed-potential form. In the
second approach the so-called perfectly hyperbolic form of Maxwell
equations are used which explicitly incorporate the divergence
equations into the time stepping scheme. The algorithm is applied to a
one-dimensional Riemann problem, ion-acoustic soliton propagation and
magnetic reconnection. In each case Two-Fluid physics described by the
ideal Two-Fluid model is highlighted.

2. Ammar H. Hakim, Brian D. Piening, and Norman J. McCormick,
“Near-asymptotic angle dependence of ocean optical radiance”, *Applied
Optics*, **43** (31), 2004.

**Abstract** The approach of ocean optical radiance to an approximate
asymptotic dependence with increasing depth in spatially uniform
waters is numerically examined for a variety of sea surface
illumination conditions.

1. Ammar H. Hakim and Norman J. McCormick, “Ocean optics estimation
for absorption, backscattering, and phase function parameters”,
*Applied Optics*, **42** (6), 2003.

**Abstract** We propose and test an inverse ocean optics procedure
with numerically simulated data for the determination of inherent
optical properties using in-water radiance measurements. If data are
available at only one depth within a deep homogeneous water layer,
then the single-scattering albedo and the single parameter that
characterizes the Henyey-Greenstein phase function can be
estimated. If data are available at two depths, then these two
parameters can be determined along with the optical thickness so that
the absorption and scattering coefficients, and also the
backscattering coefficient, can be estimated. With a knowledge of
these parameters, the albedo and Lambertian fraction of reflected
radiance of the bottom can be determined if measurements are made
close to the bottom. A simplified method for determining the optical
properties of the water also is developed for only three
irradiance-type measurements if the radiance is approximately in the
asymptotic regime.