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Gyrokinetic statistical absolute equilibrium and turbulence
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37.We acknowledge Dr. T.-S. Hahm and Dr. W. W. Lee for the interactions which helped write down this terse formula.
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45.Conceptually, as Eq. (6) is actually the Hankel transform, working in Hankel space and doing Hankel/Bessel–Galerkin truncation, seems to be also attractive. However, due to the extra complication of the (modified) Bessel functions (series) and the interplay of and spaces, it is not yet clear how to proceed in that direction. Hankel–Galerkin truncated absolute equilibrium may bring velocity-space insights; but our analysis in Fourier space, with the velocity variable being integrated out, does not depend on the details of the treatment of velocity.
46.The formulation and calculation for the arbitrary (but integrable) velocity field or the continuous limits of the present calculations will involve some subtleties which are not essential to our main results here for a discretized velocity dimension, which is in any case needed for practical comparisons with numerical codes.
47.More general integration algorithms can be represented in this form, as the weights and grid points can be chosen to correspond to high-order Gaussian quadrature (as is done in present continuum codes) (Ref. 72) equivalent to a weighted orthogonal polynomial basis of degree (these weighted basis functions can cover the infinite velocity domain), providing superexponential spectral accuracy with an error that asymptotically scales as for smooth solutions.
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53.Hasegawa and Mima did not include the factor in the expression and thus neglected the enhancement of zonal flows by the lack of electron response to modes. There is ambiguity in how to treat modes in the quasi-2D limit where one does not directly keep track of the spectrum and does not know what fraction of the fluctuation energy is in modes that satisfy so that an adiabatic electron response can be used.
58.If a standard second order finite differencing is used in calculating the parallel electric field, then the finite differencing factor in Ref. 34 is given by , where is the parallel grid spacing, so for well resolved wave numbers. If a pseudospectral method is used to calculate in -space, then .
65.J.Z.Z. is not responsible for all the computations and discussions in this section. Especially, J.Z.Z. believes there are subtleties in taking the continuous limit which can be illustrated with the direct formulation and calculation for the problem with continuous velocity, which is consistent with the present calculation with discretization of (or quantized) velocity, as shown in Ref. 50.
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, B. S. Garbow
, and K. E. Hillstrom
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67.The isotropic energy spectra in these plots is defined as , where is angle-averaged by averaging over all such that . This avoids small fluctuations in the spectrum that could occur because the number of discrete Fourier modes that lie in the band fluctuates around .
68.Note that the -axis in these plots is , in order that the eye can more easily gauge regions of equal contribution to the total energy with a log -axis, since . For example, this extra factor of accounts for the fact that there are ten times as many Fourier modes between and than between and .
69.The gyrokinetic case in Fig. 2 has , very close to the realizability limit . The values of were larger than the initial estimate of . For example, is times this estimate at the corresponding to .
70.For reference, the case in Fig. 4 with the increased by a factor of 400 still has a negative value of , with . The were all within 20% of the initial estimate of .
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