**NERSC Greenbook Technical Description**

**Stellarator Design and Physics**

**D. A. Spong (spongda@ornl.gov) - ORNL**

**Overview**

Stellarators are three-dimensional toroidal plasma confinement devices
that rely on a numerically determined plasma surface shape in order to
achieve optimized plasma confinement, stability and steady state
operation. Stellarator physics and simulation is of increasing interest
in the U.S. Fusion program both due to the possibility that new compact
\[see Figs. 1(a) and (b)\] experimental facilities will be sited at
several U.S. research centers. Interest is also driven by the existence
of a large (\$1 billion class) experimental stellarator facility in
operation now in Japan \[LHD - Fig. 1(c)\] and a second one under
construction in Europe \[W7-X - Fig. 1(d)\]. These devices have the
potential of supplanting the tokamak as a path to the development of
fusion power. Due to their inherently three-dimensional nature,
stellarators rely heavily on numerical simulation and computation. The
optimized design and physics analysis of these devices would not be
possible without high performance computing.

\(a\) (b)

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  ![](media/image1.jpeg){width="1.9305555555555556in"   ![](media/image2.jpeg){width="2.201388888888889in"
  height="1.9305555555555556in"}                        height="1.8125in"}

  ![](media/image3.jpeg){width="2.122916666666667in"    ![](media/image4.jpeg){width="2.0395833333333333in"
  height="2.185416666666667in"}                         height="2.0993055555555555in"}
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(c) \(d\)

Figure 1 - (a), (b) Top and side views of QOS compact stellarator
magnetic flux surfaces and filamentary magnet coils (color contours show
magnetic fiield strength (in Tesla), top views of (c) LHD and (d) W7--X
magnetic flux surfaces.

**Plasma Equilibrium, Reconstruction, and Optimization**

Calculation of the plasma equilibrium is the fundamental starting point
for all other physics and optimization studies of the stellarator. In
general, mathematically smoothly varying equilibria do not exist for
three-dimensional configurations and some degree of magnetic island
formation will always be present. In some cases this can be neglected,
as is done in the VMEC code, but in general it must be taken into
account (e.g. as in the PIES, HINT, and NSTAB codes), especially for
understanding plasma edge transport and nonlinear MHD instability
phenomena. This characteristic of stellarator equilibria is expected to
demand high numerical resolution and needs for large amounts of memory.

Two of the more numerically intensive uses for stellarator equilibrium
codes are in the reconstruction of experimental equilibrium
characteristics from magnetic measurements external to the plasma and in
the optimized design of new stellarator experimental devices. Both of
these calculations involve a nonlinear root finding/optimization loop
external to the equilibrium code. This loop typically varies the plasma
shape and profiles of some of the internal plasma physics parameters in
order to minimize a χ^2^ function (either the physics properties of an
optimized design or the goodness of fit to experimental data). A
Levenberg-Marquardt algorithm has been used for this minimization
process; this has recently been adapted to parallel architectures by
using a bank--queueing model with MPI language calls for interprocessor
communication. These calculations involve moderate levels of
communication between processors and would benefit from a tightly
coupled processor architecture.

**Plasma Confinement**

Plasma confinement in stellarators includes transport both of the
thermal plasma component as well as various energetic tail components
which are utilized for plasma heating. Although transport of the thermal
component may in many cases be dominated by anomalous losses from
microturbulence, it is desirable to accurately evaluate the neoclassical
component in order to understand enhanced confinement regimes (where the
microturbulence is suppressed) and transport properties parallel to the
magnetic field lines (which are not as influenced by microturbulence as
is the perpendicular transport). Due to the three-dimensional nature of
stellarators, the calculation of classical transport rates is a much
more non-trivial problem that it has been in the case of axisymmetric
tokamaks. To address this need, we have developed the drift-kinetic
solver (DKES) code that currently solves for the plasma distribution
function in two spatial dimensions and one velocity dimension. However,
to better address the physics of particle drifts at low collisionality,
this code will be extended first to 4 dimensions (three spatial and one
velocity) and eventually 5 dimensions (three spatial and two velocity).
The current code utilizes OpenMP parallelization on a single SMP node.
We envision the 4D and 5D models as mappings of the current 3D model
across multiple SMP nodes; i.e., each node will contain a diagonal block
matrix that looks like a separate 3D model. This calculation will
benefit from very dense nodes of tightly coupled processors.

The alternative approach to these direct solutions of the drift kinetic
equation is in the Monte Carlo or particle simulation approach. This
method is expected to remain of interest particularly for relatively
collisionless energetic tail populations used in heating, such as
neutral beams, RF produced tails and alpha particles from the DT fusion
reaction. Most Monte Carlo calculations to date have been nearly
trivially parallel; both parallelization over particle groups (DELTA5D
code) and over domains (GTC code) have been used based on MPI calls.
However, as higher degrees of self-consistency are incorporated into
these models, greater degrees of interprocessor communication will be
necessary throughout the calculation. Therefore, it expected these
models also should work best on very dense nodes of tightly coupled
processors.

**Plasma Heating**

The primary methods of plasma heating for stellarators consist of RF
(radio-frequency), neutral beams, and, in reactors, alpha populations.
The latter two methods are probably best addressed with Monte Carlo
techniques or direct solution of the plasma kinetic equation, as
described above. RF heating requires the solution of the electromagnetic
wave propagation equations, self-consistently coupled with the plasma
kinetic response to these waves. RF theory and simulation have been
developed over a number of years for tokamaks and have yielded many
important insights not only for efficient methods of heating, but also
for current drive (needed for steady state operation) and flow drive
(needed for turbulence suppression) techniques. Initial experimental
results and modeling of ion cyclotron RF heating on the LHD stellarator
in Japan have indicated a number of anomalies (e.g., centrally peaked
electron absorption) that cannot be explained by the 2D RF absorption
models which have been applied. It is clear that developing an adequate
understanding of the science of RF heating in stellarator devices will
require building new 3D models. Adapting RF theory to stellarators will
introduce major new computational challenges. These calculations
generally lead to large dense linear systems and will require large
amounts of memory and dense tightly coupled processors.

**Plasma Stability and Turbulence**

This topic includes both large scale resistive and kinetically driven
MHD instabilities as well as the smaller scale microturbulence that
drives anomalous transport in magnetically confined plasmas. With the
anticipated advent of new stellarator experiments in this country, it is
expected that activity in this area will increase; however, current work
in this area is still at an embryonic stage. Nonlinear MHD analysis will
become of more importance after stellarator experiments have been in
operation for several years and begin to push at the limits of plasma
pressure and current density. Kinetic MHD is of importance in high
temperature regimes of operation and in extrapolation to fusion reactor
configurations. Microturbulence modeling will be of importance in
understanding plasma transport in all regimes of operation. The impact
of understanding stellarator microturbulence is that it can lead to
turbulence suppression techniques; this can ultimately lead to smaller,
more economical fusion reactors. These topics have been developed to a
high degree of sophistication for axisymmetric tokamaks and it would be
expected that much of this work could be adapted to three-dimensional
stellarator configurations, albeit with much greater computational
performance challenges. The MHD calculations typically involve the
inversion and time-stepping of large sparse linear systems. These codes
generally have difficulty maintaining linear speed improvements when
parallelized over increasing numbers of highly distributed processors;
however, they should benefit from tightly coupled dense nodes of
processors. They also tend to have large memory needs per processor when
run at their highest resolutions.

The types of computational models used to study microturbulence
generally use either particle simulation techniques (δf gyrokinetic
models) or more fluidlike techniques (gyrofluid models); the former
codes may be expected to share similar computing needs with the Monte
Carlo calculations described above. The latter codes will be more
similar in their needs to the nonlinear MHD models as described in the
above paragraph.