Full Waves, Faster Runs for AORSA
DOE’s magnetic fusion energy research program aims to produce energy by the same process that takes place in the sun. However, on Earth a fusion reaction requires that the fuel be heated to hundreds of millions of degrees, even hotter than in the sun. At these astronomical temperatures, the fuel atoms are torn apart into their constituent electrons and nuclei, forming a state of matter called plasma.
One of the ways to get the fuel this hot is to use intense electromagnetic waves, much as a microwave oven is used to heat food. Waves are also used for other important purposes such as to drive electric currents affecting the plasma magnetic configuration, to force mass flow of the plasma, affecting its stability, and for other plasma control tasks. Therefore, it is important to have a good theoretical understanding of wave behavior and to be able to calculate it accurately. The goal of the SciDAC project “Terascale Computation of Wave-Plasma Interactions in Multidimensional Fusion Plasmas” is to develop the computational capability to understand the physical behavior of waves in fusion plasmas and to model these wave phenomena accurately enough that their effect on other plasma processes, such as stability and transport of particles and energy, can be understood.
According to project leader Don Batchelor of Oak Ridge National Laboratory, “Because the particles are so hot, they move at speeds almost the speed of light and can travel a distance comparable to a wavelength in the time of a few oscillations of the wave. This motion makes it difficult to calculate how the plasma particles will respond to the waves and how much heating or electric current they will produce.”
“Another challenge,” Batchelor adds, “is that at a given frequency, several kinds of plasma waves can exist with very different wavelengths and polarizations. A wave launched into the plasma can in a short distance completely change its character to another type of wave, a process called mode conversion. To study these effects, the computer model must have very high resolution to see the small-scale structures that develop, which means that very large computers are needed to solve for the very large number of unknowns in the equations. Also, the computers must be extremely fast in order to obtain the solutions in a reasonable time.”
Until now researchers wishing to calculate the effects of waves in plasma have been forced by computational feasibility to make a difficult choice between restricting consideration to a single dimension or simplifying the physics model. The first choice, treating the plasma geometry as one-dimensional, is akin to tunnel vision. The wave is computed along a single line through the plasma, but one does not get a picture of what is occurring in the whole plasma cross-section. The second choice eliminates from consideration many of the most important wave processes for today’s fusion experiments, which require high frequencies and can have very short wavelengths in some regions of the device.
In a partnership between plasma physicists and computer scientists, this project has increased the resolution and speed of plasma wave solvers to the point that it is possible for the first time to study the mode conversion process in the complicated geometry and the full scale of real fusion devices. Collaborators have developed new wave solvers in 2D and 3D called All-Orders Spectral Algorithm (AORSA) solvers based on a more general formulation of the physics called an integral equation. It is now possible to compute plasma waves across an entire plasma cross-section with no restriction on wavelength or frequency (Figure 1). In this approach the limit on attainable resolution comes only from the size and speed of the available computer.
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| Figure 1. Mode conversion in a plasma cross section computed with AORSA. From E. F. Jaeger et al., “Sheared poloidal flow driven by mode conversion in tokamak plasmas,” Phys. Rev. Lett. 90, 195001 (2003). |
AORSA uses a fully spectral representation of the wave field in a Cartesian coordinate system. All modes in the spectral representation are coupled, and so computing the solution requires calculating and inverting a very large dense matrix. For example, with 272 × 272 Fourier modes in two dimensions or 34 × 34 × 64 modes in three dimensions, the system processes about 220,000 coupled complex equations that require about 788 GB of memory. AORSA-3D took about 358 minutes to run on 1,936 processors of NERSC’s Seaborg computer. The efficient computation achieved over 60 percent of available peak performance (over 1.9 teraflop/s or 1,900 billion operations per second).
A new formulation of AORSA-3D transforms the linear system from a Fourier basis to a representation in the real configuration space. This alternative representation presents new opportunities to reduce the overall number of equations used. In one example of 34 × 34 × 64 modes, the number of equations was reduced by more than a factor of 5—from 220,000 to about 40,000. The new reduced system required only 26 GB of memory and was completed in about 13 minutes, with a 100-fold speedup in the linear solver and a 27-fold speedup in overall runtime. The efficiency gained from the real space formulation of AORSA-3D allows higher resolution and more design analysis for future experiments such as the Quasi-Poloidal Stellarator.
“Funding from the SciDAC project has enabled us to assemble a team from multiple institutions to work on a unified set of goals in a way that was not possible before SciDAC,” Batchelor says. “Working with computer scientists funded to work directly with our project has made it feasible to study new regimes of wave physics which previously could only be treated in one dimension or with approximate techniques, and to carry out multiple code runs at high resolution for scientific case studies where before it was only possible to perform a single run at minimal resolution.”
“Working with applied mathematicians,” he added, “we are developing new mathematical representations for the wave fields that are more data efficient than the traditional spectral techniques presently employed and offer promise to substantially reduce the size of the computational load to solve plasma wave problems. And we are making scientific progress on problems of importance to the fusion program by applying the codes we have developed and improved.”
Research funding: FES, SciDAC, ORNL