__Particle Accelerators__.
Particle accelerators are ubiquitous experimental tools in the physical
sciences, with applications in a variety of science programs at DOE.
The goal is to generate and control high-intensity charged particle beams
that are nearly collisionless, but for which self-field effects are
substantial. To simulate such problems, a key component is the
solution of the initial value problem for
for Vlasov's equation, a partial differential
equation for the phase space density distribution function
of the particles, combined with various field
equations for the electromagnetic forces (including self-forces) on the
beam. For RF-based accelerators, one transforms the problem to a
coordinate system moving with the beam, and the self-fields can be
represented using Poisson's equation. For plasma-wakefield accelerators,
both Vlasov-Poisson and Vlasov-Maxwell are used. For such acceleartors, it
is also necessary to simulate a variety of fluid-dynamical problems,
such as transient flows in gas jet injectors, the expansion
of a gas jet into near-vacuum conditions, and the interaction of a
gas jet with the laser, including effects such as ionization and shock
formation.

__Magnetic Fusion__.
One of the central design issues for the next generation of magnetic fusion
devices is the macroscopic stability of burning plasmas. In order to
understand this problem, it will be necessary to develop a collection
of tools for fully nonlinear, time-dependent simulations. The dynamics of the
the internal kink mode (or m=1 mode) is a key example.
A hot plasma conducts electricity better than a
cold plasma. During normal tokamak operation, a uniform toroidal electric
field is applied to the torus through magnetic induction. The current
tends to peak in the center of the cross section where the temperature
is highest, making it even hotter, causing the current to peak more.
When the current density becomes sufficiently peaked, it becomes
unstable
to an instability where the center of the plasma kinks up into a helical
distortion.
In the linear (small amplitude) phase, this instability is confined into
the center of the cross-section, inside the q=1 surface. Large current
layers form at this surface due to the helical distortion.
In the nonlinear phase, plasma resistivity causes magnetic reconnection
to occur at these surfaces where the current spikes are. This
reconnection causes "magnetic islands" to form.
The islands can grow to be so large that they eventually occupy
the entire plasma area that used to be inside the q=1 surface. The
plasma can then resymmetrize, and the current will again start to
peak, and this process may repeat. Alternatively, the nonlinear phase
may disrupt the plasma containment.
This whole cycle is affected by plasma pressure, as well as the size
of the particle orbits, and is different if there is a energetic
background species present, as would be the case in a fusing plasma.

In order to design such reactors, a hierarchy of plasma physics models are being developed [27]. The simplest such models are time-dependent nonlinear MHD models, in which the electron and all of the ion species are treated as fluids. In order to model the intense small-scale currents that arise, it is necessary to include non-ideal resistive effects, including nonlinear effects such as Hall resistivity. As plasma burning becomes more important, it is necessary to model energetic alpha particles kinetically, since they are only weakly collision coupled with the rest of the plasma, but contribute substantially to the energetics of the problem.

__Combustion__.
Turbulent combustion is a critical process for both energy
and transportation.
The detailed properties of turbulent chemistry interaction play a
key role in both efficiency of the combustion process and emissions.
In spite of its importance,
high-fidelity modeling of turbulent combustion remains an elusive target.
Detailed simulation of turbulent combustion processes involves
the modeling of three-dimensional turbulent flow as well as the
complexities associated with the
thermo-chemical behavior of the system, such as reaction mechanisms,
thermodynamic properties and transport properties.
These additional complexities not only make combustion simulation
a challenging computational task, they also introduce a dependence on
the existing experimentally determined characterization of
the underlying physical processes.
Our target in combustion to develop new predictive
tools for modeling turbulent combustion processes.
Specifically, our goal is to model fluid-chemistry
interactions with sufficient fidelity to predict not only
the basic energetics but also more detailed
aspects of the behavior such as the formation of pollutants.