Abstract
Recent gyrokinetic simulations of electron temperature gradient (ETG) turbulence with the global particleincell(PIC) code GTC [Z. Lin et al., Proceedings of the 20th Fusion Energy Conference, Vilamoura, Portugal, 2004 (IAEA, Vienna, 2005)] yielded different results from earlier fluxtube continuum code simulations [F. Jenko and W. Dorland, Phys. Rev. Lett.89, 225001 (2002)] despite similar plasma parameters. Differences between the simulation results were attributed to insufficient phasespace resolution and novel physics associated with global simulation models. The results of the global PIC code are reproduced here using the fluxtube PIC code PG3EQ [A. M. Dimits et al., Phys. Rev. Lett.77, 71 (1996)], thereby eliminating global effects as the cause of the discrepancy. The latetime decay of the ETG turbulence and the steadystate heat transport observed in these PIC simulations are shown to result from discrete particle noise. Discrete particle noise is a numerical artifact, so both these PG3EQ simulations and, by inference, the GTC simulations that they reproduced have little to say about steadystate ETG turbulence and the associated anomalous heat transport. In the course of this work several diagnostics are developed to retrospectively test whether a particular PIC simulation is dominated by discrete particle noise.
We gratefully acknowledge discussions with Frank Jenko regarding his continuum ETG turbulence simulations, discussions with Mike Kotschenreuther regarding the sensitivity of ETG turbulence to noise, John Krommes for useful suggestions regarding improved forms of renormalized dielectric shielding of noise, Bruce Cohen for his interest, advice, and careful editing, and W.W. Lee and Z. Lin for their contributions to increasing interest in the subject matter addressed here.
This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. W7405ENG48, by Princeton Plasma Physics Laboratory under Contract No. DEAC0276CH03073, and by the Center for Multiscale Plasma Dynamics at the University of Maryland under Contract No. DEFC0204ER54784. The simulations described here made use of resources at the National Energy Research Supercomputer Center under Department of Energy Contract No. DEAC0376SF00098.
I. INTRODUCTION
II. SIMULATIONS OF CYCLONEBASECASELIKE ETG TURBULENCE
A. Cyclonebasecaselike ETG turbulence without magnetic trapping
B. Cyclonebasecaselike ETG turbulence with magnetic trapping
C. Latetime ETG potential fluctuations
III. DISCRETE PARTICLE NOISE
IV. CAN DISCRETE PARTICLE NOISE SUPPRESS ETG TURBULENCE?
A. The noise test
V. HOW DISCRETE PARTICLE NOISE SUPPRESSES ETG TURBULENCE
A. Linear stabilization of ETG modes by diffusion
B. Computing and
C. Estimates of
VI. SUMMARY AND CONCLUSIONS
Key Topics
 Turbulent flows
 101.0
 Turbulence generated noise
 48.0
 Particleincell method
 45.0
 Heat transport
 31.0
 Plasma turbulence
 21.0
Figures
(Color online). The linear growth rate in units of for Cyclonebasecaselike ETG modes is plotted vs the wave number in the binormal direction, both with (red curve) and without (green curve) magnetically trapped electrons. For comparison, we also plot an estimate of the damping rate, , that would be associated with noiseinduced diffusion for (blue curve) or (purple curve).
(Color online). The linear growth rate in units of for Cyclonebasecaselike ETG modes is plotted vs the wave number in the binormal direction, both with (red curve) and without (green curve) magnetically trapped electrons. For comparison, we also plot an estimate of the damping rate, , that would be associated with noiseinduced diffusion for (blue curve) or (purple curve).
(Color online). The coefficient of electron thermal transport from a particlenumber and boxsize convergence study of Cyclonebasecaselike ETG turbulence without magnetic trapping including runs in a fluxtube cross section of and two particles∕grid cell (blue curve), four particles∕grid cell (green curve), and eight particles∕grid cell (black curve) and in a fluxtube cross section of with 16 particles∕grid cell (red curve).
(Color online). The coefficient of electron thermal transport from a particlenumber and boxsize convergence study of Cyclonebasecaselike ETG turbulence without magnetic trapping including runs in a fluxtube cross section of and two particles∕grid cell (blue curve), four particles∕grid cell (green curve), and eight particles∕grid cell (black curve) and in a fluxtube cross section of with 16 particles∕grid cell (red curve).
(Color online). The ETG fluctuation spectra in the linear phase (, red curve) and at after saturation from the PG3EQ simulation with a cross section of and eight particles∕grid cell (, black curve) corresponding to the black curve in Fig. 2.
(Color online). The ETG fluctuation spectra in the linear phase (, red curve) and at after saturation from the PG3EQ simulation with a cross section of and eight particles∕grid cell (, black curve) corresponding to the black curve in Fig. 2.
(Color online). The coefficient of electron thermal transport from a particlenumber and fluxtube crosssection convergence study of Cyclonebasecaselike ETG turbulence with magnetic trapping , including runs in a flux tube cross section of with 2 particles∕grid cell (green curve), 4 particles∕grid cell (blue curve), and 16 particles∕grid cell (red curve); and in a fluxtube cross section of with 16 particles∕grid cell (black curve).
(Color online). The coefficient of electron thermal transport from a particlenumber and fluxtube crosssection convergence study of Cyclonebasecaselike ETG turbulence with magnetic trapping , including runs in a flux tube cross section of with 2 particles∕grid cell (green curve), 4 particles∕grid cell (blue curve), and 16 particles∕grid cell (red curve); and in a fluxtube cross section of with 16 particles∕grid cell (black curve).
(Color online). The ETG fluctuation spectra in the linear phase (, red curve) and 30 linear growth times later at (, black curve). Data from the PG3EQ simulation with a cross section of and 16 particles∕grid cell (corresponding to the black curve in Fig. 4).
(Color online). The ETG fluctuation spectra in the linear phase (, red curve) and 30 linear growth times later at (, black curve). Data from the PG3EQ simulation with a cross section of and 16 particles∕grid cell (corresponding to the black curve in Fig. 4).
Greytone rendering of the potential on the outboard midplane at late in the linear phase (top panel) and 30 linear growth times later at (middle panel) show the characteristic ETG streamers. These streamers are absent at very late times, (bottom panel) during the steadystate phase of the simulation. Data from the PG3EQ simulation of the Cyclonebasecaselike ETG turbulence with fluxtube cross section of , magnetic trapping , and 16 particles∕grid cell (black curve of Fig. 4).
Greytone rendering of the potential on the outboard midplane at late in the linear phase (top panel) and 30 linear growth times later at (middle panel) show the characteristic ETG streamers. These streamers are absent at very late times, (bottom panel) during the steadystate phase of the simulation. Data from the PG3EQ simulation of the Cyclonebasecaselike ETG turbulence with fluxtube cross section of , magnetic trapping , and 16 particles∕grid cell (black curve of Fig. 4).
(Color online). (a) The fluctuation spectrum at the outboard midplane averaged over the radial coordinate and the interval is plotted on a semilog scale vs (, black curve), together with the corresponding fully uncorrelated noise estimate (, blue curve) and the selfDebye shield noise estimate (, red curve). The fluctuation spectrum averaged over the radius and the interval (, green curve) and the corresponding selfDebye shielded noise level (, chartreuse curve) are shown for comparison. (b) Fluctuation and noise data for the interval on a linear scale.
(Color online). (a) The fluctuation spectrum at the outboard midplane averaged over the radial coordinate and the interval is plotted on a semilog scale vs (, black curve), together with the corresponding fully uncorrelated noise estimate (, blue curve) and the selfDebye shield noise estimate (, red curve). The fluctuation spectrum averaged over the radius and the interval (, green curve) and the corresponding selfDebye shielded noise level (, chartreuse curve) are shown for comparison. (b) Fluctuation and noise data for the interval on a linear scale.
(Color online). (a) The fluctuation spectrum at the outboard midplane averaged over the binormal coordinate and the interval is plotted on a semilog scale vs (, black curve), together with the corresponding fully uncorrelated noise estimate (, blue curve) and the selfDebye shield noise estimate (, red curve). (b) Same data on a linear scale.
(Color online). (a) The fluctuation spectrum at the outboard midplane averaged over the binormal coordinate and the interval is plotted on a semilog scale vs (, black curve), together with the corresponding fully uncorrelated noise estimate (, blue curve) and the selfDebye shield noise estimate (, red curve). (b) Same data on a linear scale.
(Color online). The fluctuation energy averaged over the outboard midplane (black curve) is compared with the fluctuation intensity from the fully uncorrelated noise spectrum (blue curve) or selfDebye shielded noise spectrum (red curve).
(Color online). The fluctuation energy averaged over the outboard midplane (black curve) is compared with the fluctuation intensity from the fully uncorrelated noise spectrum (blue curve) or selfDebye shielded noise spectrum (red curve).
(Color online). The fluctuation energy is plotted vs time for a PG3EQ simulation of Cyclonebasecase ITG turbulence (black curve). The red curve shows the fluctuation energy expected from discrete particle noise. The corresponding level of thermal transport from this PG3EQ simulation is shown by the blue curve.
(Color online). The fluctuation energy is plotted vs time for a PG3EQ simulation of Cyclonebasecase ITG turbulence (black curve). The red curve shows the fluctuation energy expected from discrete particle noise. The corresponding level of thermal transport from this PG3EQ simulation is shown by the blue curve.
(Color online). Electron heat flux from the “noise test” of Lin and Bolton. The black curve is from the initial simulation. The remaining five curves correspond to simulations initialized with (red curve), (blue curve), (gold curve), (green curve), (chartreuse curve).
(Color online). Electron heat flux from the “noise test” of Lin and Bolton. The black curve is from the initial simulation. The remaining five curves correspond to simulations initialized with (red curve), (blue curve), (gold curve), (green curve), (chartreuse curve).
(Color online). (a) The intensity of the dominant Fourier mode during the linear phase of each run in Fig. 11. The black curve is from the initial simulation (multiplied by ). In the remaining five curves is measured from the time of the restart. For (red curve) and (blue curve) the dominant mode is . For (gold curve), (green curve), and (chartreuse curve) the dominant mode is . (b) The same data replotted to display the weak linear growth of the mode in the restart with (blue curve).
(Color online). (a) The intensity of the dominant Fourier mode during the linear phase of each run in Fig. 11. The black curve is from the initial simulation (multiplied by ). In the remaining five curves is measured from the time of the restart. For (red curve) and (blue curve) the dominant mode is . For (gold curve), (green curve), and (chartreuse curve) the dominant mode is . (b) The same data replotted to display the weak linear growth of the mode in the restart with (blue curve).
(Color online). The real frequency (a) and growth rate (b) for the Cyclonebasecaselike ETG turbulence as a function of the magnitude of the diffusion acting on the nonadiabatic part of the electron distribution function for (black curves), (red curves), (blue curves), (green curves), and (chartreuse curves).
(Color online). The real frequency (a) and growth rate (b) for the Cyclonebasecaselike ETG turbulence as a function of the magnitude of the diffusion acting on the nonadiabatic part of the electron distribution function for (black curves), (red curves), (blue curves), (green curves), and (chartreuse curves).
(Color online). Measured maximum linear growth rate (with error bars and connected by the grey line) after restart from the noise test simulations of Sec. IV is compared with from Eq. (15) with (heavy line). The colors of both data points and model are chosen to indicate the corresponding wave number as (black), (red), and (blue).
(Color online). Measured maximum linear growth rate (with error bars and connected by the grey line) after restart from the noise test simulations of Sec. IV is compared with from Eq. (15) with (heavy line). The colors of both data points and model are chosen to indicate the corresponding wave number as (black), (red), and (blue).
(Color online). The value of from Eq. (16) for Cyclonebasecaselike ETG turbulence with magnetic trapping from a PIC simulation in a fluxtube cross section of with 16 particles∕grid cell is displayed both with (red curve) and without (black curve) the contribution of turbulence, toroidal drifts, and magnetic shear to the decorrelation.
(Color online). The value of from Eq. (16) for Cyclonebasecaselike ETG turbulence with magnetic trapping from a PIC simulation in a fluxtube cross section of with 16 particles∕grid cell is displayed both with (red curve) and without (black curve) the contribution of turbulence, toroidal drifts, and magnetic shear to the decorrelation.
The predicted value of from Eq. (16) is compared with from the noise test described in Sec. IV. The black curves are from the initial simulation (distinguished by large variations) and from Eq. (16) (smooth curve at ). The remaining curves show simulations initialized with (red curves), (blue curves), (gold curves), (green curves), and (chartreuse curves).
The predicted value of from Eq. (16) is compared with from the noise test described in Sec. IV. The black curves are from the initial simulation (distinguished by large variations) and from Eq. (16) (smooth curve at ). The remaining curves show simulations initialized with (red curves), (blue curves), (gold curves), (green curves), and (chartreuse curves).
(Color online). (a) The from the noise test described in Sec. IV (larger variation) and the predictions from Eq. (16) (smaller variation), with (black curves), (blue curves), and (red curves). (b) The predicted value of from Eq. (16) (green squares) and the simulation results (red crosses) from all of the random restart tests in Figs. 16 and 17(a) showing that Eq. (16) predicts the observed scaling well as the average squared weight is varied by a factor of 512.
(Color online). (a) The from the noise test described in Sec. IV (larger variation) and the predictions from Eq. (16) (smaller variation), with (black curves), (blue curves), and (red curves). (b) The predicted value of from Eq. (16) (green squares) and the simulation results (red crosses) from all of the random restart tests in Figs. 16 and 17(a) showing that Eq. (16) predicts the observed scaling well as the average squared weight is varied by a factor of 512.
(Color online). (a) The net linear growth rate vs time for (black curves), (red curves), and (blue curves). The thick (upper) curves use from Eq. (16). The thin (lower) curves show the sensitivity to turbulent decorrelation by setting . (b) The intensity of Fourier modes vs time for (black curve), (red curve), (blue curve), (green curve), and (chartreuse curve).
(Color online). (a) The net linear growth rate vs time for (black curves), (red curves), and (blue curves). The thick (upper) curves use from Eq. (16). The thin (lower) curves show the sensitivity to turbulent decorrelation by setting . (b) The intensity of Fourier modes vs time for (black curve), (red curve), (blue curve), (green curve), and (chartreuse curve).
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