Good news: The amount of spurious heating caused by N-point gyroaveraging in our guiding center simulations is significantly smaller than what was shown in Figs. 14(i)–14(x) of the original published paper.1
We have corrected the error and the new results are shown in Figs. 1(i)–1(x) of this Erratum. One can see from the good conservation of the rotating frame energy that spurious heating caused by gyroaveraging is now ignorable in our working example on the timescale of a few milliseconds.
(Replaces Fig. 14 of the original published paper.1) Comparison of results from simulations using the full orbit model (a)–(d), the GC model (e)–(h), and the GC model with N-point gyroaveraging (i)–(x). We followed co-passing deuterons, and the nonnormal mode of case (c) was used as a perturbation. Results are presented as Poincaré plots (columns 2 and 4) and time traces of the rotating frame energy (columns 3 and 5) for scenarios (i) and (ii) with electrostatic and electromagnetic perturbation, respectively. The first column shows schematically the differences between the three methods for representing gyration in the (R, z) plane. In the full orbit case, particles travel along a helix, whose poloidal projection is drawn here as a red circle of radius around the guiding center (black dot). In the GC model, fields are evaluated at the GC only. Gyroaveraging is done by placing satellite particles (small red circles) on the gyrocircle and taking the average of the fluctuating fields E and at those locations to push the GC. Here we compare results obtained with and different satellite positions. As a visual aid, the Poincaré plots are overlaid with a pair of vertical dashed lines, which indicate the limits of the resonance's separatrix in the full orbit simulations.
(Replaces Fig. 14 of the original published paper.1) Comparison of results from simulations using the full orbit model (a)–(d), the GC model (e)–(h), and the GC model with N-point gyroaveraging (i)–(x). We followed co-passing deuterons, and the nonnormal mode of case (c) was used as a perturbation. Results are presented as Poincaré plots (columns 2 and 4) and time traces of the rotating frame energy (columns 3 and 5) for scenarios (i) and (ii) with electrostatic and electromagnetic perturbation, respectively. The first column shows schematically the differences between the three methods for representing gyration in the (R, z) plane. In the full orbit case, particles travel along a helix, whose poloidal projection is drawn here as a red circle of radius around the guiding center (black dot). In the GC model, fields are evaluated at the GC only. Gyroaveraging is done by placing satellite particles (small red circles) on the gyrocircle and taking the average of the fluctuating fields E and at those locations to push the GC. Here we compare results obtained with and different satellite positions. As a visual aid, the Poincaré plots are overlaid with a pair of vertical dashed lines, which indicate the limits of the resonance's separatrix in the full orbit simulations.
Counterexample showing that some forms of gyroaveraging can cause an accumulation of spurious heating. Here, we used a constant gyroradius instead of given by Eq. (2). The left column shows Poincaré plots as functions of normalized canonical toroidal angular momentum , poloidal angle ϑ, and normalized rotating frame energy (see the original paper1 for details and definitions). The right column shows time traces of . The results in panels (a) and (b) were obtained with the same spatial resolution as in the published paper: for constructing the mode, and in the guiding center simulation. The results in panels (c) and (d) were obtained with twice as many grid points: , and . Along the toroidal direction, we used the particle-in-Fourier method. The number of satellites was as in Figs. 1(w) and 1(x). The results seem to be converged for . A comparison between panels (b) and (d) shows that higher resolution reduces the spurious heating, as expected from Eq. (4). The only exception seems to be the fifth sample from the top, which has been colored black for emphasis. This may have various reasons, one of which is that the form and location of the resonant island differs somewhat between panels (a) and (c), so that we are effectively sampling different portions of the resonance. This is most obvious if one looks at the fourth sample from the top (colored magenta), which lies outside the separatrix in (a) and inside the separatrix in (c).
Counterexample showing that some forms of gyroaveraging can cause an accumulation of spurious heating. Here, we used a constant gyroradius instead of given by Eq. (2). The left column shows Poincaré plots as functions of normalized canonical toroidal angular momentum , poloidal angle ϑ, and normalized rotating frame energy (see the original paper1 for details and definitions). The right column shows time traces of . The results in panels (a) and (b) were obtained with the same spatial resolution as in the published paper: for constructing the mode, and in the guiding center simulation. The results in panels (c) and (d) were obtained with twice as many grid points: , and . Along the toroidal direction, we used the particle-in-Fourier method. The number of satellites was as in Figs. 1(w) and 1(x). The results seem to be converged for . A comparison between panels (b) and (d) shows that higher resolution reduces the spurious heating, as expected from Eq. (4). The only exception seems to be the fifth sample from the top, which has been colored black for emphasis. This may have various reasons, one of which is that the form and location of the resonant island differs somewhat between panels (a) and (c), so that we are effectively sampling different portions of the resonance. This is most obvious if one looks at the fourth sample from the top (colored magenta), which lies outside the separatrix in (a) and inside the separatrix in (c).
Additional corrections. The minus signs of and were missing in Fig. 16 of the published paper.1 Note that .
ACKNOWLEDGMENTS
A.B. thanks Chang Liu (PPPL) for stimulating discussions that led to the discovery of the above-mentioned coding error.