Simulation of Microturbulence in Magnetic Fusion Experiments

One of the fundamental grand challenge problems in magnetic fusion
energy research is the understanding and control of turbulent transport
of energy observed in the core of many fusion experiments. Drift-wave
turbulence has been identified experimentally as a primary mechanism in
degrading energy confinement in tokamak core plasmas. For some years
there has been a large effort in the fusion community to simulate
drift-wave.^1^ This simulation activity has lead to a suite of
three-dimensional, toroidal simulation codes that have been used by a
national collaboration (first under the auspices of the Numerical
Tokamak Turbulence Project and subsequently as the Plasma
Microturbulence Project). These models have been extensively benchmarked
against independent linear calculations of the basic underlying
microinstabilities and nonlinearly against one another to obtain results
for the nonlinear saturation of drift-type instabilities in current
experiments, e.g., Princeton's TFTR and General Atomics DIII-D. The
simulation results have been used to calibrate reduced models of the
turbulent transport and to derive scaling relations for use in comparing
and predicting experimental results with increasing success. Specific
features associated with the moderation of the turbulent transport by
means of externally imposed and self-generated velocity shear have been
illustrated and demonstrated with the simulations. The elucidation of
the physics of shear-flow inhibition by simulation and theory, and its
confirmation in experiments exhibiting internal transport barriers have
led to major advances in producing tokamak plasmas with improved energy
confinement in a more predictable and repeatable fashion.

Three related models have been used, all of which solve for the
self-consistent electric or electromagnetic fields and the associated
nonlinear plasma response. The three models solve the coupled Maxwell
and Vlasov equations for plasmas supporting drift-type
microinstabilities in three spatial dimensions and two velocity-space
variables (the third velocity-space variable, the gyrophase angle in the
applied magnetic field, has been analytically removed by gyro-averaging
the equations). The three models are gyrokinetic particle codes
(Lagrangian description), gyrofluid codes with Landau closure, and
gyrokinetic continuum codes (Eulerian description). All three models
have been parallelized and run on the NERSC T3E and other massively
parallel platforms. By developing and applying three different
approaches to microturbulence simulation, the magnetic fusion community
has been able to carefully explore the oomparative computational
efficiencies of the three approaches and perform important code
cross-checks on the nonlinear simulations (which has been essential for
debugging the codes and determining the reliability of the physics
results^2^). Because the three approaches differ significantly in their
algorithms, their diversity also has been useful for understanding how
to optimize code efficiency for the specific architecture of the host
supercomputer with benefits ensuing to other scientific disciplines
faced with similar computational challenges.

We have been very successful in developing codes for modeling plasma
turbulence for which computer run-time and problem size scale well with
the number of processors on massively parallel machines (Fig. 1).^3^ We
have experience in being able to make successful use of new platforms at
NERSC and to fully utilize NERSC resources. To date our simulations have
been limited by computing resources that constrain us to simulate
experiments with either smaller plasmas or plasmas with less than the
optimal spatial resolution, or to undertake fewer simulations and limit
parameter studies, or to target an annular region of a tokamak
experiment albeit with realistic parameters (Fig. 2), whose
computational requirements are generally less stringent than those for a
full global simulation (although we also routinely undertake global
simulations, Fig. 3). Another important limitation on research progress
due to limited computer resources has been the turnaround time for a
researcher to be able to undertake a series of simulations addressing a
parameter scan, which profoundly impacts the pace of physics progress.
Thus, upgrades at NERSC addressing both capability and capacity
simultaneously are vital to taking the next steps in increasingly
realistic physical simulations in magnetic fusion research on
microturbulence and in all disciplines of programmatic interest to DOE.
To illustrate this, we consider a full-device particle simulation of a
next-step tokamak capable of achieving ignition or high fusion gain.
Such a device would have a minor radius of the order of 103 times ρi
(the ion Larmor radius). Using a magnetic field-aligned co-ordinate
system, a full-device simulation of ITG turbulence, including
wavelengths down to the ρi scale, would require a grid with NRadial≈
1000, NPoloidal ≈ 3000, and N\|\| ≈ 64 for a total of about 2×108 grid
points. Our experience indicates that 16 particles/grid cell (for a
total of 3.2×109 particles) is adequate to provide a long interval of
fully developed turbulence without excessive discrete particle noise.
About 50,000 time steps will be required to simulate 100 turbulent
decorrelation times. Scaling from our experience on the T3E (a 0.5 TFlop
machine) we find that a 10 TFlop computer can be expected to achieve a
speed of 2×10-9 sec/particle/time-step. Hence, a full-device simulation
of ITG turbulence in an ignition experiment will take about 90 hours on
a 10 TFlop computer. Presently, our codes require about 0.3kbytes of
memory/particle. However, the larger grid required for these simulations
will may require domain-decomposition in 2 or 3 dimensions (presently we
only decompose in 1-D), leading to somewhat larger memory
requirements/particle. Taking a conservative estimate of
1 kbyte/particle, we conclude that we this simulation would require
about 3 Tbytes of RAM distributed among the processors. The disk storage
requirements, which are dominated by restart dumps, would be about 100
times the memory size, or 300 Tbytes.

We conclude that full-device simulations of ITG turbulence in an
ignition-scale magnetic fusion device is an ambitious, but achievable on
a 10 TFlop computer. Such simulations would allow detailed investigation
of scientific questions regarding the role of meso scales in ITG
turbulence, the dynamics of spectral transport, and the formation and
evolution of transport barriers. The 10 Tflop computer will also make
easier the inclusion of the more complete and better physics models that
we have developed that include, for example, kinetic electron effects
and electromagnetic coupling of the drift waves to kinetic shear-Alfven
waves that modify the microinstabilities in the plasma at finite plasma
pressures.

1\. B.I. Cohen, D. C. Barnes, J. M. Dawson, *et al* Comp. Phys. Commun.
**87**, 1 (1995).

2\. A.M. Dimits, G. Bateman, M.A. Beer, B.I. Cohen, *et al.*, Phys.
Plasmas **7**, 969 (1999).

3\. Z. Lin, T.S. Hahm, W.W. Lee, W.M. Tang, and R.B. White, *Science*
**281**, 1835 (Sept. 1998); A.M. Dimits, T.J. Williams, J.A. Byers, and
B.I. Cohen, *Phys. Rev. Lett.* **77**, 71, (1996); J. Kepner, S. Parker
and V. Decyk, *SIAM News* **30**, 1 (1997); R. E. Waltz, G.M. Staebler,
W. Dorland, G.W. Hammett, M. Kotschenreuther, and J.A. Konings, *Phys.
Plasmas* **4**, 2482 (1997).

Figure 1. Scaling of massively parallel code performance of the GTC
global gyrokinetic particle code. (Z. Lin, Princeton U.)

![](media/image1.pct){width="5.7in" height="5.810416666666667in"}

Figure 2. Flux-tube gyrokinetic simulation of ion-temperature-gradient
drift-wave microturbulence. Contours of density fluctuations. (Dimits,
Williams, Shumaker, Cohen, and Nevins, LLNL)

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height="5.315972222222222in"}

Figure 3. Global simulation of ion-temperature-gradient turbulence in a
tokamak showing the influence of sheared flow on moderating the
turbulence. Contours of density fluctuations .(Z. Lin, T.S. Hahm, W.W.
Lee, W.M. Tang, and R.B. White. Princeton U.)

![](media/image3.pct){width="6.04375in" height="5.022222222222222in"}

Figure 4. Global simulation of ion-temperature-gradient turbulence
modeling the DIII-D tokamak. Contours of density fluctuations (J.-N.
Leboeuf, UCLA; R. Sydora, U. Alberta)

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height="2.8291338582677166in"}![](media/image5.png){width="2.8291338582677166in"
height="2.8291338582677166in"}