Abstract:
Electrostatic gyrokinetic absolute equilibria with continuum velocity field are obtained through the partition function and through the Green function of the functional integral. The new results justify and explain the prescription for quantization/discretization or taking the continuum limit of velocity. The mistakes in the Appendix D of our earlier work [J.-Z. Zhu and G. W. Hammett, Phys. Plasmas {\bf 17}, 122307 (2010)] are explained and corrected. If the lattice spacing for discretizing velocity is big enough, all the invariants could concentrate at the lowest Fourier modes in a negative-temperature state, which might indicate a possible variation of the dual cascade picture in two-dimension magnetized plasma turbulence.

Abstract:
Two global invariants of two dimensional gyrokinetics are shown to be "rugged" (still conserved by the dynamics) concerning both Fourier and Hankel/Bessel Galerkin truncations. The truncations are made to keep only a finite range of wavenumber $\textbf{k}$ and the Hankel variable $b$ (or $z$ in the Bessel series). The absolute equilibria are used for the discussion of the spectral transfers in the configuration-velocity scale space of kinetic magnetized plasma turbulence. Some interesting aspects of recent numerical results, which were not well understood, are explained with more satisfaction.

Abstract:
The principles of restricted superposition of circularly polarized arbitrary-amplitude waves for several hydrodynamic type models are illustrated systematically with helical representation in a unified sense. It is shown that the only general modes satisfying arbitrary-amplitude superposition to kill the generic nonlinearity are the mono-wavelength homochiral Beltrami mode and the one-dimensional-two-component stratified vorticity mode, which we call the XYz flow/wave; while, there are other special superposition principles for some specific cases. We try to remark on the possible connections with the geo- and/or astro-physical fluid and magnetohydrodynamic turbulence issues, such as the rotating turbulence, dynamo and solar atmosphere turbulence, especially with the introduction of disorder locally frozen in some (randomly distributed) space-time regions. Recent disagreements about exact solutions of Hall and fully two-fluid magnetohydrodynamics are also settled down by such a treatment. This work complements, by studying the modes which completely kill the triadic interactions or the nonlinearities, previous studies on the thermalization purely from the triadic interactions, and in turn offers alternative perspectives of the nonlinearities.

Abstract:
Hydrodynamic helicity signatures the parity symmetry breaking, chirality, of the flow. Statistical hydrodynamics thus respect chirality, as symmetry breaking and restoration are key to their fundamentals, such as the spectral transfer direction and its mechanism. Homochiral sub-system of three-dimensional (3D) Navier-Stokes isotropic turbulence has been numerically realized with helical representation technique to present inverse energy cascade [Biferale et al., Phys. Rev. Lett., {\bf 108}, 164501 (2012)]. The situation is analogous to 2D turbulence where inverse energy cascade, or more generally energy-enstrophy dual cascade scenario, was argued with the help of a negative temperature state of the absolute equilibrium by Kraichnan. Indeed, if the helicity in such a system is taken to be positive without loss of generality, a corresponding negative temperature state can be identified [Zhu et al., J. Fluid Mech., {\bf 739}, 479 (2014)]. Here, for some specific chiral ensembles of turbulence, we show with the corresponding absolute equilibria that even if the helicity distribution over wavenumbers is sign definite, different \textit{ansatzes} of the shape function, defined by the ratio between the specific helicity and energy spectra $s(k)=H(k)/E(k)$, imply distinct transfer directions, and we could have inverse-helicity and forward-energy dual transfers (with, say, $s(k)\propto k^{-2}$ resulting in absolute equilibrium modal spectral density of energy $U(k)=\frac{1}{\alpha +\beta k^{-2}}$, exactly the enstrophy one of two-dimensional Euler by Kraichan), simultaneous forward transfers (with $s(k)=constant$), or even no simply-directed transfer (with, say, non-monotonic $s(k) \propto \sin^2k$), besides the inverse-energy and forward-helicity dual transfers (with, say, $s(k)=k$ as in the homochiral case).

Abstract:
We argue that the constraints on transfers, given in the Letter [G. Plunk and T. Tatsuno, Phys. Rev. Lett. {\bf 106}, 165003 (2011)], but not correctly, do not give the transfer and/or cascade directions which however can be assisted by the absolute equilibria calculated in this Comment, following Kraichnan [R. H. Kraichnan, Phys. Fluids {\bf 102}, 1417 (1967)]. One of the important statements about the transfers with only one or no diagonal component can be shown to be inappropriate according to the fundamental dynamics. Some mathematical mistakes are pointed out.

Abstract:
The cross-correlation $\mathcal{C}_{\theta\zeta}$ between the passive scalar $\theta$ and the two-dimensional (2D) vorticity $\zeta$ of the advecting velocity field is a quadratic invariant. This cross correlation results in a condensation state of passive-scalar energy in the absolute equilibrium spectra, similar to 2D kinetic energy, from which one may infer a genuine inverse transfer or cascade of the variance of $\theta$, at least splitting part of it to large scales, contrary to the conventional knowledge of equilibrium equipartition and turbulent forward cascade. Recent numerical data of intermediate Rossby numbers showing inverse transfers of the vertically-averaged vertical velocity $u_z^2$ may be relevant.

Abstract:
The helical absolute equilibrium of a compressible adiabatic flow presents not only the polarization between the two purely helical modes of opposite chiralities but also that between the vortical and acoustic modes, deviating from the equipartition predicted by {\sc Kraichnan, R. H.} [1955 The Journal of the Acoustical Society of America {\bf 27}, 438--441.]. Due to the existence of the acoustic mode, even if all Fourier modes of one chiral sector in the sharpened Helmholtz decomposition [{\sc Moses, H. E.} 1971 SIAM ~(Soc. Ind. Appl. Math.) J. Appl. Math. {\bf 21}, 114--130] are thoroughly truncated, leaving the system with positive definite helicity and energy, negative temperature and the corresponding large-scale concentration of vortical modes are not allowed, unlike the incompressible case.

Abstract:
Measurement and phenomenological analyses of intermittency growth in an experimental turbulent pipe flow and numerical turbulence are performed, for which working definitions such as degree, increment, and growth rate of intermittency are introduced with the help of quasiscaling theory. The logarithmic-normal inertial scaling model is extended to quasiscaling as the second-order truncation of the Taylor expansion and is used for studying the intermittency growth problem. The extended self-similarity properties are shown to be not consistent with the monotonicity of the third order local quasiscaling exponent and the nonmonotonic behaviour of the intermittency growth rate as a result of bottleneck. Digestions of the results with scale-dependent multifractals are provided.

Abstract:
A dissipation rate, which grows faster than any power of the wave number in Fourier space, may be scaled to lead a hydrodynamic system {\it actually} or {\it potentially} converge to its Galerkin truncation. Actual convergence we name for the asymptotic truncation at a finite wavenumber $k_G$ above which modes have no dynamics; and, we define potential convergence for the truncation at $k_G$ which, however, grows without bound. Both types of convergence can be obtained with the dissipation rate $\mu[cosh(k/k_c)-1]$ who behaves as $k^2$ (newtonian) and $\exp\{k/k_c\}$ for small and large $k/k_c$ respectively. Competition physics of cascade, thermalization and dissipation are discussed with numerical Navier-Stokes turbulence, emphasizing on the intermittency growth.

Abstract:
A paradigm based on the absolute equilibrium of Galerkin-truncated inviscid systems to aid in understanding turbulence [T.-D. Lee, "On some statistical properties of hydrodynamical and magnetohydrodynamical fields," Q. Appl. Math. 10, 69 (1952)] is taken to study gyrokinetic plasma turbulence: A finite set of Fourier modes of the collisionless gyrokinetic equations are kept and the statistical equilibria are calculated; possible implications for plasma turbulence in various situations are discussed. For the case of two spatial and one velocity dimension, in the calculation with discretization also of velocity $v$ with $N$ grid points (where $N+1$ quantities are conserved, corresponding to an energy invariant and $N$ entropy-related invariants), the negative temperature states, corresponding to the condensation of the generalized energy into the lowest modes, are found. This indicates a generic feature of inverse energy cascade. Comparisons are made with some classical results, such as those of Charney-Hasegawa-Mima in the cold-ion limit. There is a universal shape for statistical equilibrium of gyrokinetics in three spatial and two velocity dimensions with just one conserved quantity. Possible physical relevance to turbulence, such as ITG zonal flows, and to a critical balance hypothesis are also discussed.