Purely helical absolute equilibria and chirality of (magneto)fluid turbulence
aa r X i v : . [ n li n . C D ] J u l Under consideration for publication in J. Fluid Mech. Purely helical absolute equilibria and chirality of(magneto)fluid turbulence
Jian-Zhou Zhu, Weihong Yang and Guang-Yu Zhu , WCI Center for Fusion Theory, National Fusion Research Institute, 169-148 Gwahak-ro, Daejeon, Korea Department of Modern Physics, University of Science and Technology of China, 230026 Hefei, China Life and Chinese Medicine Study Center, 47 Bayi Cun, 366025 Yong’an, Fujian, China(Received ?; revised ?; accepted ?. - To be entered by editorial office)
Purely helical absolute equilibria of incompressible neutral fluids and plasmas (electron, single-fluid and two-fluid magnetohydrodyanmics) are systematically studied with the help of helical(wave) representation and truncation, for genericities and specificities about helicity. A uniquechirality selection and amplification mechanism and relevant insights, such as the one-chiral-sector-dominated states, among others, about (magneto)fluid turbulence follow.
Key words:
1. Background, technique, and basic ideas
Helical modes are basic in electromagneto- and hydrodynamics (see, e.g. , Moses 1971). Theyare left- or right-handed, signaturing chirality † which may be quantified by helicity and its rele-vant derivatives, such as the relative helicity, important for the statistical dynamics.1.1. Helical turbulence and absolute equilibrium
Recognizing the importance of helicity in hydrodynamic turbulence is relatively new, thoughHelmholtz-Kelvin theorem is old (Moffatt 2008). Indeed, in a communication with C.-C. Lin in1945, L. Onsager noticed that the coefficients of the Fourier modes of hydrodynamic velocityfield are “‘momentoids’ in the sense of Boltzmann, and the theorem of equipartition would applyif their number were finite. Since this is not the case, we get a ‘violet catastrophe’ instead.” ‡ Statistical absolute equilibrium (AE) energy equipartition among each Fourier modes was laterexplicitly formulated by T.-D. Lee (1952) for both pure hydrodynamics (HD) and magneto-hydrodynamics (MHD). Neither Onsager nor Lee (who, interestingly, as well-known, howeversoon suggested with Yang in 1956 the chiral “world” — parity is violated in the weak interac-tions!) considered the invariance of helicity ¶ which makes the flow field lose mirror symmetryand which can also be involved in the generalized equipartitions (Kraichnan 1973; Frisch et al.1975). Now, tremendous progresses with helical representation/decomposition have been made(see, e.g., Yang, Su & Wu 2010, and references therein): Nature of the triadic interactions canbe exposed more clearly (Waleffe 1992) and be exploited to understand better the fluctuations,for instance, of electron MHD (EMHD) and Hall MHD (Galtier & Bhattacharjee 2003; Galtier † This notion is widely used in chemistry, physics and (origin of) life sciences and was called dis-symmetry , which is still occasionally used, before Kelvin (1904) and various attempts have been made tomathematically quantify it (see, e.g. , the review by Petitjean 2003). ‡ This remarkable comment adds more to Onsager’s ignored legacy on hydrodynamics than that exposedby Eyink & Sreenivasan (2006, private communication in 2008) who reproduced the letter. ¶ Later, Betchov (1961) first tried to explore invariant helicity’s role in turbulence, contrasting a box ofnails to screws with reflexion asymmetry.
J.-Z. Zhu & W. Yang and G.-Y. Zhu i.e. , one-chiral-sector-dominated states (OCSDSs) with severe chiral symmetry breaking, i.e. , imbalance of positive and negative helicity, along scales have been explicitly demonstrated( e.g. , Meneguzzi, Frisch & Pouquet 1981; Brandenburg, Dobler & Subramanian 2002, see moredetailed discussions in §2.3.1) but want a corresponding theoretical understanding, as we will of-fer. Recently, Meyrand & Galtier (2012) studied Hall MHD new chirality symmetry breaking inthe sense of domination by whistler or ion-cyclotron waves defined by the linear wave dispersionrelation, which is different to the chirality signature coming from the helical representation usedin this paper; see §2.3.2 for more remarks.The equilibrium-statistical-mechanics approach to investigate turbulence had been somewhatesoteric, but Kraichnan (1967, 1973, hereafter K67 and K73) established in a more explicitand complete way the AE for both 2D and 3D incompressible HD. Fourier modes beyond [ k min , k max ] being discarded (Galerkin truncation), certain rugged quadratic invariants — forsolutions regular enough to bear no dissipative anomaly (see, e.g. , Eyink 2008) — such as thekinetic energy ( E K ) and enstrophy (for 2D) or helicity ( H K for 3D), are still conserved. With theconstraints of these rugged invariants, Kraichnan obtained the respective energy spectral densi-ties for 2D and 3D: U K ( k ) = 1 / ( α + βk ) and U K ( k ) = 2 α/ ( α − β k ) respectively. For symbolic convenience, from now on the vector argument k will be replacedwith its module k by isotropicity consideration. U K ( k ) for 3D, for instance, can be derivedimmediately from the Gibbs distribution ∼ exp {− ( α E K + β H K ) } , where α is the temperature parameter associated with energy and β with helicity (enstrophyin 2D). † K67 showed that low enstrophy state in 2D corresponds to a negative α , indicatingcondensation of energy at smallest k with a roughly (smoothed) x shape spectral density: c.f. ,similar MHD figures in Frisch et al. (1975). In 3D, there is no such negative temperature state( α > due to the realizability condition from the positive definiteness of the quadratic form α E K + β H K ) but only y shape spectral density, and low helicity state corresponds to vanishing β and equipartition of energy. By statistical consideration of the tendency of the interactingmodes to relax towards the equilibrium state, inverse energy cascade was then argued for 2Dbut disputed for 3D. Note also that, as argued by L’vov et al. (2002) for 2D turbulence, in some † We adopt K73 notations and definitions: E K = ˜ P k U K ( k ) → ˜ R dk πE K ( k ) and H K = ˜ P k Q K ( k ) → ˜ R dk πH K ( k ) in the continuous- k limit, where ˜ • implies restricting to the sub-set of surviving modes and E ( k ) = k U K ( k ) and H ( k ) = k Q K ( k ) are the 1D spectra. We will alwaysuse α for energy related temperature parameter and β for helicity. Self-evident indexes, such as M for“magnetic”, when necessary for discrimination, will be added to β , U , Q , E and H etc. And, for simplicitywe will always use Gibbs ensemble calculation and will not repeatedly formulate and explain it. The generalresults of this paper are not affected by the differences between an infinite domain and a finite cyclic box, sowe may switch between these two descriptions, depending on which one is more convenient. Difference ofa factor of may arise, depending on how one treats the realizability condition (see below) and the invari-ant(s) (summation over the whole or half of the wavenumber domain etc.), and yet another freedom aboutthe sign of helicity is of one’s free choice. Also, spectra in this paper may be obtained in other approaches,such as finding the stationary solution of the master equation with the properties of vertices relevant to theconservation laws, which may avoid explicitly resorting to the Gibbs distribution (Private communicationwith E. A. Kuznetsov). hiroid absolute equilibria and turbulence entropic force towards the maximum entropy state.1.2. Helical (wave/mode) representation
For a 3D transverse vector field, such as the velocity u , vorticity ω = ∇ × u and the transversecomponent of vector potential A of magnetic field B † etc., in a cyclic box of volume, say, V = (2 π ) , the helical mode/wave representation in Fourier space reads (Moses 1971) v = X k ˆ v ( k ) e ˆ i k · r = X c v c = X k ,c ˆ v c ( k ) e ˆ i k · r = X k ,c ˆ v c ( k )ˆ h c ( k ) e ˆ i k · r . (1.1)Here ˆ i = − and c = 1 for the chirality indexes c = “+” or “-”. The helical mode bases(complex eigenvectors of the curl operator) have the following properties ˆ i k × ˆ h c ( k ) = ck ˆ h c ( k ) , ˆ h c ( − k ) = ˆ h ∗ c ( k ) = ˆ h − c ( k ) and ˆ h c ( k ) · ˆ h ∗ c ( k ) = δ c ,c , the Euclidean norm. A relation usedfor numerical computation, such as the numerical experiments with the (pseudo-)spectral methodusing the various truncation schemes to be discussed in §2.2, is ˆ v c ( k ) = ˆ v ( k ) + c ˆ i k × ˆ v ( k ) /k (see, e.g. , Lesieur 1990; Melander & Hussain 1993).Here the new element in the theoretical formulation of the absolute equilibrium problem liesin viewing the system a gas of pure helical modes ˆ v c ( k ) , representing ˆ v c ( k )ˆ h c ( k ) e ˆ i k · r + c.c. for simplicity, i.e. , the chiroids a la Kelvin (1904), as the working ‘momentoids’. Correspondingdensities can be defined accordingly: For instance, mean magnetic energy E M = ˜ P c, k U cM ( k ) and helicity H M = ˜ P k ,c Q cM ( k ) with U cM ( k ) = ckQ cM ( k ) = h| ˆ B c ( k ) | i / , (1.2)with a reversed factor of k for the kinetic case due to the difference between magnetic and kinetichelicities by definition, where h•i denotes the mean, per unit volume or in the statistical sense.Inserting Eq. (1.1) back into the rugged invariants, E and H , constraining the statistical ensemble,the Gibbsian one used for our calculations, we can obtain the chirally split densities U c ( k ) and Q c ( k ) which present finer physical structures than the mixed ones U ( k ) = U + ( k ) + U − ( k ) and Q ( k ) = Q + ( k ) + Q − ( k ) : (1.3)We remark that this result should be perceived in two perspectives. One is that the AE spec-tra of pure helical modes of each chiral sector present separately, independent of the existenceof the other one, i.e. , whether the other sector is truncated or not for some k s, since the trun-cation can be performed arbitrarily on the chiroids, except that Hermitian symmetry should bekept like the classical Fourier truncation; the other perspective is that the spectra are chirallydecomposed into two sectors, if both exist, i.e. , the truncations for both sectors are symmet-ric. Note that | Q M ( k ) | U M ( k ) /k , so the purely helical mode is called maximally helical (see, e.g. , Kraichnan 1973). Other derivatives such as the relative helicity | kQ M ( k ) | / [ U M ( k )] = k P c Q cM ( k ) / P c U cM ( k ) (Kraichnan 1973, with a reversed factor of k as in Eq. 1.2) can beused for quantifying the degree of chirality. Only when the modes have the same wavelength and † Note that we have used Coulomb gauge ˆ A · k = 0 , so ˆ i k × [ˆ i k × ˆ A ( k )] = k ˆ A ( k ) = ˆ i k × ˆ B ( k ) : Thelongitudinal component of A with whatever gauge is not involved in the relevant calculations, so Coulombgauge is not really necessary but brings symbolic convenience. Note also that helical wave , with the timeargument included, is truly chiral in the sense of Barron (see, e.g. , Cintas & Viedma 2012). J.-Z. Zhu & W. Yang and G.-Y. Zhu are homochiral , i.e., all with same handedness, can we see the physical-space field resulting fromtheir superposition is Beltrami, i.e. , ∇ × v = γ v with constant γ , force free for magnetic field,in which case, nonlinearity is typically depleted completely.1.3. Statistical ensembles of truncated chiroids
When the system is reduced to the dynamics of the helical modes, one can then consider var-ious truncations directly on such chiroids (Waleffe 1992; Biferale, Musacchio & Toschi 2012).Detailed triadic interactions of the helical Fourier modes in various hydrodynamic-type mod-els, such as HD, MHD, EMHD and Hall MHD etc., have been closely looked into by differentauthors ( e.g. , Waleffe 1992; Lessinnes, Plunian & Carati 2009; Galtier & Bhattacharjee 2003;Galtier 2006). And, relevant details of the AE calculation have been well described in the lit-erature and do not require any further elaboration here. For instance, it is routinary to checkLiouville theorem and rugged conservation properties after Galerkin truncation of the helicalmodes, which is true for all the models studied here (for HD, c.f. , equations 7 and/or 9 of Waleffe1992). To be a bit more definite but without loss of generality, using indexed y s for the vari-ables related to the real and imaginary parts of the active chiroids ˆ v c ( k ) , we can write down thedynamical equations as ∂ t y n = X l,m Y lmn y l y m , (1.4)with Y lmn satisfying some specific symmetries to assure the detailed conservation of energy andrelevant helicity(ies) and Liouville theorem. Note that for cases with linear terms of the originalvariables on the right-hand side, such as the 3D gyrokinetics (Zhu & Hammett 2010), a simplelinear transformation of variables reduces them to this form which is formally the same as thewell-known classical Fourier Galerkin truncated Euler case. Readers can go to the Appendixesfor more discussions on the detailed conservation laws, dynamical and topological aspects, and,tacit assumptions about ergodicity or mixing etc. in the statistical considerations .1.4. Plan
We progressively perform a minimal but systematic investigation, with different emphases, ofEMHD with formally pure magnetic field dynamics in §2.1, HD in §2.2, and, single-fluid (§2.3.1)and two-fluid (§2.3.2) MHDs. §2.1 discusses mainly the natural chiral selection for inverse mag-netic helicity transfer; §2.2 concentrates on the chirally asymmetric truncation effects, §2.3.1 onnew insights to the classical dynamo issue, §2.3.2 on the two-fluid effects. Although many of thediscussions in the (sub)subsections, such as the asymmetric truncations of the two chiral sectorsfor some k s in §2.2, can be carried over to other (sub)subsections, mutatis mutandis , to get somerelevant new insights, we won’t detail such obvious points. The general purpose is to lay out thebasic AE as a first step to explore some fundamentals of turbulent transfers, especially for a com-prehensive basic understanding of the relevant helicity effects. It should be pointed out that therehave been many other interesting AE-relevant investigations for different dynamical models andon various specific physical issues, the important one of which is relevant to a space uniformmagnetic field and anisotropic fluctuations and has been continuously attacked in the last severaldecades. Extra particular studies should be done, though k = 0 mode can formally be includedin the calculations and brief relevant remarks will be offered at the appropriate circumstances.The focus is the most basic new insights attached to the chirally decomposed AE, with which wewill revisit the most relevant studies, rather than any other specific turbulence (closure) theories,such as the wave turbulence theories studied by Galtier and collaborators, though our results maybe used as benchmarks of relevant analytical or numerical treatments.In summary, an incompressible hydrodynamic-type system can be reduced to the dynamicsof pure helical modes, the “chiroids” a la Kelvin, with helical wave/mode representation. Left- hiroid absolute equilibria and turbulence i.e. , unichirally with asymmet-ric Galerkin truncation. One sector can dominate around its positive pole(s) with correspondingnet helicity, providing a unique chirality selection and amplification mechanism. Chirally trun-cated systems preserve Moffatt’s topological interpretation of helicity due to the detailed incom-pressibility for each chiroid. We obtain new insights about chirality selection and amplification,and, spectral transfers of turbulence of various neutral-fluid and plasma systems. For instance,one-chiral-sector-dominated states are naturally supported by magnetohydrodynamics (MHD)absolute equilibria with magnetic helicity, and homochiral Euler system allows negative temper-ature states which were excluded by Kraichnan for the non-homochiral case. A major purposeis to make a systematic comparison of the effects of various helicities, for finding genericitiesand specificities, and we clarify the special role of magnetic helicity for turbulent inverse mag-netic helicity transfer/cascade by analyzing the electron MHD, with only magnetic field, and thetwo-fluid MHD, with the combination of various helicities in a symmetric way.As said, it is not necessary to elaborate the calculations repeatedly, thus the following presenta-tion will mainly consist in a set of brief backgrounds and theoretical frameworks, discussions ofour results with careful comments on relevant studies and new explanations of documented data.Readers are suggested to go directly to the (sub)subsection(s) for the interested model(s)/topicsfirst and then, before trying to read the analyses of others, to §3 for further discussions, wherenot only the major results are summarized, but also the genericities and differences are extractedby comparisons across different models.
2. One-chiral-sector-dominated absolute equilibrium and turbulence states
Basic (rules of) notations, definitions and calculation techniques follow §1. Readers are as-sumed to be familiar with the relevant ideas and techniques of K67 and K73 summarized there.We do not repeat the further detailed simple calculations of the spectra following K73 andFrisch et al. (1975) which will actually be found to be greatly simplified, since much, for single-and two-fluid MHDs, or all, for EMHD and HD, of the diagonalization work and the solenoidalconstraints, have already been performed from the beginning with the helical representationwhile constructing our ensemble. 2.1.
Pure magnetic fluid
EMHD equation ∂ t B + ∇ × (cid:2) ( ∇ × B ) × B (cid:3) = 0 formally involves only the magnetic field and may be called pure magneto-dynamics. This fluidmodel corresponds to the small electron skin depth d e ≪ limit of the more general case, whichwe will discuss later, and is relevant to helicons or whistler waves in solid conductor, includingneutron star’s solid crust, atmosphere etc (see, e.g., Biskamp et al. 1999; Galtier & Bhattacharjee2003, and references therein). Note that B is “frozen in”, by definition, to the electron fluidvelocity u e = −∇ × B . Rugged invariants are magnetic energy and helicity: E M = 12 V Z B d r = 12 X k ,c | ˆ B c ( k ) | and H M = 12 V Z A · B d r = 12 X k ,c c | ˆ B c ( k ) | /k. The two chiral sectors of the AE spectral densities ( c.f. , §1) of energy and helicity are then U cM ( k ) = k/ ( cβ + αk ) and Q cM ( k ) = 1 / ( β + cαk ) = cU cM ( k ) /k. (2.1)From the above spectral relations, just as K67, but with energy playing the role of enstrophythere, a low energy state corresponds to condensation of Q at smallest wavenumbers, close to the J.-Z. Zhu & W. Yang and G.-Y. Zhu positive pole k p = − cβ/α of one of the chiral sectors, say, c = + , with β < < α . The implica-tion for turbulence is inverse helicity and forward energy transfers. In principle, as long as thereis net helicity with β = 0 , one chiral sector can dominate at large scales, i.e. , OCSDS around thepositive pole, though commonly done in experiment to provoke EMHD turbulence is to imposea background magnetic field (see, e.g., Stenzel 1999) which breaks the skin effect and guides thewaves. If helicity is injected at some intermediate k , we should see dominant inverse helicity andforward energy transfers. If the transfers in these two regimes are approximable by self-similarlocal cascades and that suitable for simple dimensional analysis, energy spectra follow k − / and k − / scalings: Biskamp et al. (1999) first proposed and presented in slightly different situationssuch scalings, and their Fig. 8b with electron skin depth d e ≪ does correspond to the forwardenergy cascade of our case. The scale separation between the dominant dynamics of the twocascade quantities of EMHD is weaker than 2D fluid turbulence, in the sense that the spectralratio of each chiral sector is at most k , instead of k of the latter. So, at finite Reynolds numbers,cascade of either definitely is accompanied by stronger (than 2D turbulence) leaking of the other.The subdominant energy transfer, accompanying the inverse helicity transfer and vanishing athigh Reynolds number limit, should not be recognized to be the genuine inertial cascade. Oneshould be particularly careful for the decaying case which is exactly the nice simulation by Cho(2011), who, by “inverse energy cascade”, meant merely the backward shift of the peak of his en-ergy spectrum, which is not genuine in connect to the conventional notion of inertial cascade andwhich is not in conflict with our statement of forward energy cascade (Private communication).Our result indicates a largest-scale nearly force-free magnetic fields. For the discrete- k case,the smallest- k modes contain most of the energy, so the whole global structure may appear tobe roughly Beltrami, with smaller-scale “turbulent” fluctuations. Note that completely force-freefields, instead of ours with the scale-dependent degree of chirality measurable by the relativehelicity (Kraichnan 1973), were obtained with several variational formulations in 1950s: We willcome back to this in §2.3.1 for single-fluid MHD.The basic feature, concerning the issue of magnetic helicity inverse transfers/cascades, of theEMHD results in the above is also central to other MHD models with magnetic field. Some briefremarks for the finite- d e (which is used for scale normalization here) general EMHD model, ∂ t ( ∇ B − B ) + ∇ × (cid:2) ( ∇ × B ) × ( ∇ B − B ) (cid:3) = 0 , are in order. The “frozen-in” generalized vorticity is ∇ × P e = ∇ × u e − B with P e = u e − A . The rugged invariants are now total energy and generalized helicity(Biskamp et al. 1999) E = 12 V Z ( B + u e ) d r , H G = 12 V Z ∇ × P e · P e d r , resulting in U cM ( k ) = k/ { ( k + 1)[ cβ ( k + 1) + αk ] } , which complicates the quantitative transitional spectral behaviors ( c.f. , §2.3.2 for more generaldiscussions for similar situations in two-fluid MHD). In the other limit regime of scales muchsmaller than d e , or in another word, when k ≫ and that k + 1 can be replaced by k , k U cM ( k ) → / ( cβk + α ) , the same as the pure HD case, supporting both energy and helic-ity cascading forwardly ( c.f. , Fig. 8a of Biskamp et al. 1999); † Magnetic helicity concentrating † Note that, for convenience, in this regime one may want to study magnetic enstrophy W definedthrough W ( k ) = k U cM ( k ) and the other quantity S , which one might want to call magnetic helistrophy ,defined through S ( k ) = k Q cM ( k ) . Such an attempt however is conceptually not very appropriate, sinceneither of them are conserved quantities. hiroid absolute equilibria and turbulence c = + sector (Waleffe 1992, and see §2.2 for the discussions). This should not be surprising,since electron kinetic fluid flow dominates in this limit.2.2. Pure neutral fluids
For the classical incompressible HD, i.e. , ∂ t u + u · ∇ u = −∇ p, where the pressure p can be eliminated by ∇ · u = 0 , the rugged invariants are kinetic energyand helicity E K = 12 V Z u d r , H K = 12 V Z ∇ × u · u d r , which lead to the densities of separate chiral sectors: U cK ( k ) = 1 / ( α + cβk ) , Q cK ( k ) = ckU cK ( k ) . (2.2)Note that the above spectra can not be considered to be simply the decomposition (into twochiral sectors) of K73 densities ( c.f. , §1) which are not valid when there is asymmetric truncationbetween the two chiral sectors of some k , that is, when only one of the chiral sectors of some k is truncated to be unichiral. For example, if there is no cancelation at some k , one can not derivefrom Eq. (2.2) α = [ U − K ( k ) + U + K ( k )] / [2 U + K ( k ) U − K ( k )] , β = [ U − K ( k ) − U + K ( k )] / [2 kU + K ( k ) U − K ( k )] (2.3)which are in particular not true for any k in the homochiral system with only one, say, the pos-itive chiral sector. In such homochiral case with c = + , α > is not required by realizabilitycondition (§2.2) and Eq. (2.2) shows that the low helicity state corresponds to a negative α witha sharp peak at the lowest modes k min > k p = | α/β | close to the positive simple pole k p .Such x -shape energy spectral density, just as K67, indicates inverse energy and forward helicitydual transfers (Waleffe 1992, who also aruged for this with his “instability assumption”) withKraichnan’s argument of the tendency of relaxation towards equilibrium, as numerically real-ized by Biferale, Musacchio & Toschi (2012) with remarkable quality. Note however that suchHD scenario does not work in vanishing- d e EMHD in §2.1, neither for other more complicatedMHD models as will be studied in the next section, which, for instance, when truncated to be ho-mochiral, presents no drastic change of transfer/cascade direction; this is because the large-scalemagnetic helicity concentration of OCSDS with symmetric truncation and that of the homochiralstate coincide.2.2.1.
OCSCSs in the sense of fluxes: “Second order” OCSDSs
If both sectors present for every k , only y shape spectral density dominated by one of thesector around the peak is allowed by the realizability conditions α > and k max < α/ | β | from the positive definiteness of the quadratic form: c.f. , §1. Large-scale condensation mecha-nism is absent, thus inverse cascade in HD generally needs other special treatments as reviewedby Yang, Su & Wu (2010) and Biferale, Musacchio & Toschi (2012). Although, unlike at largescales, normal dissipation would devastatingly ruin such small-scale explicit OCSDS AE struc-ture, some residuals of such intrinsic nonlinearity effects may persist. Indeed, the fluxes of thetwo chiral sectors reported by Chen, Chen & Eyink (2003) do show systematic differences: Ac-cording to the working conditions, their Figs. 2–5 correspond to the case with positive helicity,that is, the positive pole belongs to the positive sector with negative β . Dominance of the positivesector of AE spectrum indicates that nonlinearity should support the transfer of this sector to J.-Z. Zhu & W. Yang and G.-Y. Zhu be more persistent, consistent with the results and analyses of Chen, Chen & Eyink (2003). Wemay also call them “implicit” OCSDSs, or “second order” OCSDSs, in the sense of dominanceof energy and helicity fluxes of one sector, as signatured by the lower panels of their Figs. 4and 5. Such second-order OCSDSs may be viewed as yet another special evidence, besides theisotropization (Lee 1952) and bottleneck (Frisch et al. 2008), of the persistence of thermaliza-tion around the end of inertial range. The large- k viscous effect efficiently restores the reflexionsymmetry, and the degree of local-in- k chirality measured by relative helicity vanishes as k − throughout the inertial range with accurate k − / scaling exponents for both energy and helicity(Kraichnan 1973). The large- k pole effect of one chiral sector nevertheless provides a prototypefor other similar possible physics ( c.f. , §2.3.2) in more complex situations, furthermore it mightbe possible to find its stronger activity in a non-Newtonian fluid such as the (dilute) solution ofchiral polymers, say, deoxyribonucleic acid (DNA), in a normal fluid where some kind of res-onance between one chiral sector of the fluid motion and the polymer’s chiral structure/activitycould happen; see more relevant discussions in §3.1.2.2.2. OCSDSs with special truncation schemes and “smooth” transitions of energy transferdirections
Negative temperature state emerges with dramatic physical indications for the homochiralcase. Now, if we add just one pair of conjugate modes, with opposite helicity, negative tem-perature is then excluded, by the nonnegativity of U cK . Naturally, a kind of “phase transition” ishappening in the temperature parameters, since the negative temperature must jump to some pos-itive value. Then with superficial impression from the above analysis and from K73 one wouldtend to expect similarly a sharp transition of inverse energy cascade to forward energy cascade.But, such a superpowerful potential of a single alien chiroid would be shocking. Thus we need tolook into the corresponding absolute equilibrium states by starting with a clean pool of homochi-ral c = + modes and put aliens into it. It turns out that there should be no phase-transition-like behavior concerning cascades and energy can “smoothly” switch from completely-inverseto partly-inverse-and-partly-forward and to completely-forward cascades, depending on how(many) aliens are put into the pool. For the general truncations of chiroids with asymmetry butnot homochirally, K73’s argument for excluding inverse energy cascade still does not simplywork, and large-scale concentration of energy, indicating inverse energy transfer in turbulence,can exist without a negative temperature state.
To see this, suppose we have only one alien chi-roid, i.e. , one negative-helicity mode, at the objective condensation wavenumber , k c = k min much smaller than the injection wavenumber k in , and that negative temperature state is excludedfor implying a conventional inverse energy transfer/cascade argument a la K67. Would energyabruptly turn to cascade forwardly, or there should be a transitional behavior? To get illumi-nation, one may apply to the arguments and thought experiment presented at the end of §1.1.Consider Eqs. (2.2) and (2.3) with α > and β > , and suppose energy is injected at someintermediate k in . This alien mode may help transfer extra positive helicity nonlocally to smallscales, by which, though, its own amplitudes of (negative) helicity and energy would have toincrease; this is because for the excitation of any mode in k > k in the injected helicity at k in isnot enough to support it, just by the relation Q K ( k ) = kU K ( k ) for pure helical mode, thus someextra positive helicity should be provided from k < k in , which is facilitated by the excitation ofnegative helicity in the alien mode. The growth of this alien mode is allowed, even in the senseof complete AE, with α/β approaching k + c , i.e. , from above: α/β → k + c and that U − K ( k c ) → ∞ , turning k c into the pole k p . (2.4)Note that k can be larger than α/β even though U − K ( k ) = 1 / ( α − βk ) , since the alien mode in thissector is restricted below k c , actually, as said, the c = − sector of Eq. (2.2) being only for k c = k min now, unlike the traditional symmetrically truncated system with α/β > k max as discussed hiroid absolute equilibria and turbulence α/β approach k c fromabove, because he had other larger- k alien modes which otherwise would have negative energyfor k > α/β . Thus, adding such a single alien excludes the negative-temperature state, butthe single alien can carry the energy condensating there as a particular form of OCSDS.
The k -distribution of energy of such a state does not differ from that of the homochiral negativetemperature state too much. And, one probably can carefully design a simulation by adding analien to the smallest k of Biferale, Musacchio & Toschi (2012)’s simulation and still get inversetransfer of energy. When we gradually increase the number of aliens starting from k min to thecondensation wavenumber k c > k min , there may or may not be a nonzero forward nonlocaltransfer of energy to small scales in the infinite Reynolds number limit; or, in another word,depending on the strength of the nonlocal kicking of the alien modes, the energy accompanyingthe forward helicity transfer may or may not vanish as the wavenumber goes to infinity: It may notbe impossible that, on average, a single alien could remotely “kick” the small scales by “stirring”up vortex stretching over the field, making the solution rougher and that causing some finiteenergy dissipation. With more aliens, forward helicity transfer will be more. The injected energymust gradually be partitioned to be transferred to two opposite directions simultaneously in theinfinite Reynolds number limit. And, to transit to a state with all energy completely cascadingforwardly, sufficient aliens must be added to the regime of k larger than k in , since if all aliens areadded only to the regime below k in , the nonlocal interaction across k in for transferring helicityto larger k must be accompanied with some transfer of energy to k < k in , as is also indicated byanalyzing the corresponding absolute equilibria [ c.f. , statement (2.4) for k c < k in ]. Regarding thecascades and/or nonlocal transfers in the infinite Reynolds number limit, the approach we havetaken is tuning the degree of the regularity of the solutions to be appropriately “dissipative” bymanipulating the population of the helical modes . To our best knowledge, for the full 3D Navier-Stokes, actually even for 2D, there is not yet satisfying mathematical theory for the cascadestatements in the infinite Reynolds number limit. But, it appears that we may use the absoluteequilibrium states to obtain at least some fine and clear physical-picture intuitions for such anapproach.2.2.3. HD summary
We have come from magneto-dynamics in §2.1 to this hydro-dynamics which, with the greatsubstantiation for transition to magneto-hydro-dynamics , needs a summary:( a ) Kraichnan (1973)’s argument for forward energy and helicity cascades can be refinedto find the relevance of his absolute equilibrium with the energy/helicity fluxes reported byChen, Chen & Eyink (2003), as a kind of “second order” OCSDS;( b ) as shown by our chirally split spectra, his argument however is not for the homochiraltruncation where the negative temperature state, indicating inverse energy transfer/cascade, isallowed;( c ) such a “sharp” change of cascade scenario is not shared by the pure magneto-dynamics ;( d ) The cascade transition is not really “sharp” but has a “smooth” dependence on how alienmodes are added to the homochiral pool.2.3. Magnetised fluids
In principle, there can be many fluid models for describing different subsets of the kinetic phasespace of plasma dynamics (see, e.g. , a very limited list in a review by Schekochihin et al. 2009).Here we study the classical single-fluid and the most general two-fluid MHDs. From the plasmaphysics point of view, two-fluid MHD is for a more complete description of the kinetic effects forthe dynamics/scales between those of EMHD and single-fluid MHD. Two-fluid MHD presentsvarious helicities in a unified way.0
J.-Z. Zhu & W. Yang and G.-Y. Zhu
Single-fluid MHD
As introduced in §1, Meneguzzi, Frisch & Pouquet (1981) found with direct numerical sim-ulations that “the large-scale B is mostly force free and produces only very little large-scalemotion,” with the relative magnetic helicity density | kQ M ( k ) /U M ( k ) | being close to , nearlymaximally helical; and, recently, Brandenburg, Dobler & Subramanian (2002) explicitly pointedout, by their Fig. 21 with postprocessing using helical decomposition, that the simulation withsimilar setup also present such OCSDSs. We now turn to explain such findings with the classicalsingle fluid MHD equations ∂ t u = − ( u · ∇ ) u + ( B · ∇ ) B − ∇ ( p + B / ,∂ t B = − ( u · ∇ ) B + ( B · ∇ ) u , where ∇ · u = 0 and ∇ · B = 0 . Rugged invariants are three (see, e.g. , Woltjer 1959; Frisch et al. 1975), the energy, magnetichelicity and cross helicity n E = 12 V Z ( u + B ) d r , H M = 12 V Z A · B d r and H C = 12 V Z u · B d r o , which, together with Eq. (1.1), leads to ( c.f. , §1, also for notations) U cK ( k ) = 4( α k + cβ M )(4 α − β C ) k + c α β M , U cM ( k ) = 4( α k )(4 α − β C ) k + c α β M , (2.5) Q cM ( k ) = ck U cM ( k ) , Q cK ( k ) = ckU cK ( k ) , Q cC ( k ) = − β C k (4 α − β C ) k + c α β M . (2.6)Similar to the statement in §2.2 in comparing our spectra to those of Kraichnan (1973), ourspectra can not be considered to be simply the decomposition of those of Frisch et al. (1975)which are not valid when there is asymmetric truncation of the two chiral sectors of some k .When there is no asymmetric truncation, our spectra chirally split those of Frisch et al. (1975)following whom we start with the case of null cross helicity with β C = 0 for discussions: Q cM ( k ) with sgn ( cβ M ) = − is responsible for the condensation of Q M at small k , around the positivesimple pole k p = − cβ M /α . When the dynamics is dominated by the c = + ( c = − ) sector, it issimply to say that large positive (negative) magnetic helicity state corresponds to a negative (pos-itive) β M with a x shape spectral density is favored: Frisch et al. (1975) plotted such spectra,the spectral densities multiplied by k , in their Figs. 1 and 2 for illustration, by choosing severaltypical temperature parameters. The other sector’s pole has the opposite sign and is not reach-able, thus, without such a mechanism of large-scale attraction, the energy would be transferredto small scales or simply less excited. When β C (or H C ) is nonzero, the prefactor before k in thedenominators quantitatively changes, but the qualitative picture is not altered. As pointed out in§2.1, the large-scale nearly maximally helical state predicted by AE is close (see also next para-graph) but different to the purely force-free one intensively studied in 1950s (see references inWoltjer 1959). The common feature with that of Woltjer (1959) is that the invariant cross helicitydoes not essentially change the large-scale nearly maximally helical physics. Recently, some au-thors argue that cross-helicity, signature of the imbalance along and opposite to the backgroundmagnetic field, may be important to determine the (reduced) MHD forward cascade inertial rangescaling exponent (see, e.g. , Perez et al. 2012, and references therein), which is beyond the scopeof this study, although a spacially uniform B can be formally included in our calculation andanalysis.Thus, the OCSDSs in Meneguzzi, Frisch & Pouquet (1981) and Brandenburg, Dobler & Subramanian(2002) are related to our AE spectra. Actually, Pouquet, Frisch & Léorat (1976) had carried outsystematic study of eddy-damped quasi-normal Markovian (EDQNM) MHD turbulence and non-linear dynamo. The data of their Figs. 4 and 5 around k = 0 . , actually starting from the be- hiroid absolute equilibria and turbulence k = 0 . , alreadyclearly presented OCSDSs as can be seen from the values of the relative helicities computed fromtheir figures. Obviously, as EDQNM shares the conservation properties of the original system, itsatisfies the AE spectra and that the OCSDS arguments also work. We won’t go too far into muchmore details, but just remark that the pertinent discussion of Pouquet, Frisch & Léorat (1976, p.345, second paragraph) can also be elaborated: For instance, the large-scale ensemble can beunderstood by AE with positive magnetic helicity, with finer chiral-sector dynamics for the dif-ferent chiralities in separate scale regimes. We only want to remark that relaxation of magnetichelicity to the largest scales does not necessarily indicate local cascade, nonlocal transfer also ispossible (see, e.g. Brandenburg & Subramanian 2005); and, there is nothing in conflict with themore mechanical reasoning, such as the alpha effect.Conceptually, a reader may quickly question whether we really learn anything more fromchiral decomposition than that if helicity is large, one chirality must dominate. Isn’t that ob-vious? The simple pole mentioned under Eqs. (2.5) and (2.6) is already present in the spectraof Frisch et al. (1975) and indeed accounts for the accumulation of magnetic helicity at largescales. Chiral decomposition is not needed to reach this conclusion? The answers are “no”s. Tounderstand these problems, one must first understand that large net helicity does not necessarilymean OCSDS or big relative helicity. Now, suppose the spectra were not chirally decomposedand that the small- k pole might contribute to both left- and righ-handed helicities. In such anambiguous situation, one would not be able to conclude OCSDS as the scale approaching thepole, which was exactly what happened in the past studies mentioned in the above, sounding likejust a hair’s breadth though. Interestingly, even in the extremely strong sense of “non-helical”state with Q M ( k ) = Q K ( k ) = 0 , i.e. , the two chiral sectors of both magnetic and velocityfields balancing at each wavenumber, AE seems to still support the so-called non-helical tur-bulent dynamo. The reason is that, in this situation, while energy, either kinetic or magnetic, isequipartitioned into each helical mode, magnetic helicities of both sectors with opposite signsare “attracted” by the same pole k p = 0 . Note that unlike EMHD in §2.1, magnetic energy itselfhere is not conserved and kinetic energy can be transformed to it to ease the inverse magnetic he-licity transfers for the two chiral sectors simultaneously. Note also that Pouquet, Frisch & Léorat(1976) and Meneguzzi, Frisch & Pouquet (1981), and other isotropic MHD simulations with unitPrandtl number, found a slight excess of magnetic energy at small scales, which may be due tosuch “attraction” from large scales. Without decomposition, as net helicity at any k is seen tobe zero, researchers traditionally have not seriously thought about the simultaneous backwardtransfer of both sectors, to our best knowledge.2.3.2. Two-fluid MHD
Two-fluid effects, the decoupling between electrons and ions, are important in many laboratoryand astrophysical situations (see, e.g. , Yamada et al. 2002; Brandenburg & Subramanian 2005).The ideal incompressible two-fluid MHD states that the generalized vorticities ∇ × P s , withcanonical momenta P s = m s u s + q s A for each species s , are “frozen in” ( c.f. §2.1) to therespective flows (see, e.g. , Ruban 1999, and references therein) m s n s d u s dt = q s n s ( E + u s × B ) − ∇ p s , where E is the electric field vector and q s and m s are charge and mass. Since this model hasvery rich physics, to remind ourselves the relevant context and the weights of physical quantitiesin quantifying chirality, instead of being purely geometrical as discussed in Petitjean (2003),we now use the normalization which keeps some physical parameters explicitly, unlike those inEMHD and single-fluid MHD. The dynamics is constrained by three rugged invariants, i.e. , the2 J.-Z. Zhu & W. Yang and G.-Y. Zhu total energy and self-helicities: E = 12 V Z (cid:2) E + B + X ˜ s m ˜ s n ˜ s u s (cid:3) d r and H s = 12 V Z ∇ × P s · P s d r . Here, two-fluid effects are in the extra terms in the invariants compared to single-fluid MHD.With Eq. (1.1), we are led to the following AE spectra densities ( c.f. , §1, also for notations) Q cM ( k ) = ck U cM ( k ) , U cM ( k ) = kT ( k ) D c ( k ) , U csK ( k ) = N cs ( k ) D c ( k ) , Q csC ( k ) = q s β s L ¯ s ( k ) − D c ( k ) /k ,Q csK ( k ) = ckU csK ( k ) and Q cs ( k ) = m s Q csK ( k ) + 2 m s q s Q csC ( k ) + q s Q cM ( k ) (2.7)with D c ( k ) = αk [ T ( k )+( P ˜ s q s n ˜ s m ¯˜ s ) Q ˜ s β ˜ s ]+ cαO Q ˜ s n ˜ s , L s ( k ) = α n s + cβ s m s k , m s N cs ( k ) = k [ αL ¯ s ( k )+ q s m ¯ s Q ˜ s β ˜ s ]+ cn ¯ s O , O = α P ˜ s q s β ˜ s and T ( k ) = Q ˜ s L ˜ s ( k ) where ¯ s means the otherspecies than s and where the index C is for the “cross” helicity as defined in §2.3.1. We summa-rize the following points: † First . The poles, i.e. , roots of the third order polynomials D c ( k ) , of the two sectors are ofopposite signs, as c appears in the second and zeroth order terms. Second . The relevant spectrum may be of x shape, with a positive pole on the left. This issimilar to the single-fluid MHD (§2.3.1) case with OCSDS of inverse magnetic helicity transfer.Like the discussion in the end of §2.1 for the regime of scales much smaller than d e , y shapespectral density as in HD (§2.2) may be relevant to the scenario of both energy and helicitiescascading forwardly. Now there can be other larger positive poles, from the same chiral sector ornot, which may also be physically relevant to the possible persistence or emergence of chiralitiesdue to the dominance of different physical processes at different scale regimes. Note that likeHall MHD in Meyrand & Galtier (2012), ion MHD (IMHD) or EMHD can be identified in ourtwo-fluid model by putting the fluid velocity of one species to zero: Asymmetries between therespective dynamics, with the opposite chirality in their sense of whistler or cyclotron waves,may present at different scale regimes, due to, say, the different m s of the two species (unlikeelectron-positron plasma). ‡ Third . A F shape spectral density [ Q M ( k ) , say] may be confined inbetween two distinct pos-itive poles, belonging to the same chiral sector or not, which presents cross-scale, e.g. , goingacross the ion or electron skin depthes etc., behavior: The general EMHD in §2.1 with finiteelectron inertial can already have such a feature as is clearly seen from U cM ( k ) given there.The general effects can be understood with the combination of the two cases in the above sec-ond item. And, the large- k peak may also be relevant to small-scale field generation (see, e.g. Brandenburg & Subramanian 2005, in the context of battery mechanisms) or indicating the “sec-ond order” or “implicit” OCSDS as discussed in §2.2 .
Last . If O = 0 in the zeroth order terms underlined below Eq. (2.7), magnetic helicity does † The result unavoidably appearing a bit complicated, it may be helpful for readers to focus on U cM ( k ) and Q cM ( k ) first. Further simplification of these formulae can be made for some situations. For instance,electron-positron plasma with mass equivalence and charge conjugation enables us to take all masses andcharges be normalized unity. But, for our purposes here, taking D c as polynomials of k with the funda-mental theorem of algebra and Vieta’s formulas in mind suffices. Note that we have general formula for theroots whose nature is determined by the discriminant. In the cases discussed below, the parameter regimes(constrained by the realizability condition) can be obtained with such basic knowledge by some simple buttedious manipulations and are omitted here. Electric energy distribution is omitted for two reasons: One isthat it can be neglected in usual cases; and, the second is that it is decoupled from the others. ‡ The chirality of Meyrand & Galtier (2012) designated by their magnetic polarization does not dependon our sense of chiral sectors in the IMHD or EMHD linear dispersion relations exploited by them, but onlyon the signs of the charges. In our work, chirality refers to the definite right- or left-handed sector in thehelical representation, which is more basic and works for any models, and which may be used to similarlyinterpret strong turbulence (not limited to the linear wave dispersion relation) as well. hiroid absolute equilibria and turbulence P ˜ s β ˜ s H ˜ s ( c.f. , §1). Note that k = 0 does notbecome a pole for this situation, as every spectrum in Eq. (2.7) has a factor of k to cancel it. Sincenow Q ˜ s β ˜ s < , we can see that this case is similar to the last situation considered by Kraichnan(1967, p. 1423), with our [ T ( k )] − > from the realizability conditions being eligible to actthe role of k there; see also ¯ U K ( k ) in §1. This corresponds to a y shape spectral density. Thepole for large-scale concentration is gone now, since only Q cM , by definition, has k = 0 as theasymptotic pole as in single-fluid MHD. We thus can infer from this point that magnetic helicityconstraint under two-fluid framework is still crucial for large-scale concentration of magneticfields, as in single-fluid MHD where it is conserved.
Careful analyses with appropriate choices of the physical parameters can be made for detailedillustration and more subtle implications. Similarly is for other intermediate models such as HallMHD and general EMHD with finite electron inertial in §2.1 whose result is a bit simpler, withthe denominators being polynomials of second order. From the plasma physics point of view, itis very interesting to spell out all detailed effects of each physical element (skin depths, massratio effects etc.), which however is not the focus of this paper. We have to refrain from treatingthese cases in too great a detail.
3. Further discussions
General remarks
Turbulence statistics can be sharpened with helical representation, which has been well discernedsince Moffatt (1970) and K73 (Kraichnan 1973) and has become fully workable since Moses(1971) and Waleffe (1992). We have merely focused on the most basic AE aspect concerningthe direction of spectral transfer as well as the selection and amplification of chirality. The keypoint is that although the dynamics of the two chiral sectors are in general coupled, the absoluteequilibrium spectra are cleanly split, with poles of opposite signs. Not only that the finer phys-ical structures offer new insights about the “(near-)racemic mixture” ( c.f. , the last paragraph of§2.3.1), but also that one should keep away from the misconception that the chirally decomposedquantities derived in this paper never appear in the AE ensemble by themselves, but always incombinations giving inviscid invariants. Actually, OCSDSs may emerge in natural systems dueto mechanisms relevant to what we have discussed or one can work with samples of “enantiopurecompounds” in §2.2. Concerning partial fraction decomposition, our results physically assuresthe decomposability of the spectra of the traditional Fourier modes from the hydrodynamic-typemodels studied here and practically solves the mathematical problem, giving also the nice “con-jugate” mathematical structures in the spectra of the opposite chiral sectors, at least to the degreeof two parts with poles of exactly opposite signs, which is not trivial for some models such asthe two-fluid MHD in §2.3.2. Naive attempts to perform the “post” decomposition of the tradi-tional spectrum could be formidable and confusing, for lack of physical motivation: For instance,one could think of trying further to decompose the already chirally decomposed two-fluid MHDspectra.For the absolute equilibria themselves, both K73 and Frisch et al. (1975)’s insights were al-most here as we are now. Especially K73 explicitly discussed the interactions of pure helicalmodes. Curiously, K73 however did not † study the chiroids absolute equilibria to which his † Kraichnan (1973) might not have been motivated to systematically examine the dynamics in helical–mode representation, especially the Liouville theorem and detailed conservation laws, the latter of whichonly appeared two decades later (Waleffe 1992). There is a conceptual issue here as we will elaborate alittle bit. He wrote in the second page of that paper: “The two helical waves provide an alternative to theusual Fourier decomposition into plane-wave components.” In the usual Fourier representation, with con-sideration of isotropy (but lack of reflexion symmetry) as he was considering, there is no reason and noway to distinguish special components of the 3D spectra or spectra of special components of the Fourier J.-Z. Zhu & W. Yang and G.-Y. Zhu traditional mixed ones can not be reduced by taking any limits of any of the temperature param-eters, in which sense we mean, in §2.2, his results are not valid for asymmetrically truncatedsystems. Such a piece of thin “window paper” was not pierced probably due to the fact that con-ventionally the relevance of the traditional chirally symmetric Galerkin truncation were madeto the classical chirally symmetric viscosity or resistivity following Lee (1952, footnote 2) andKraichnan (1973, footnote 8). For such physical considerations, see also some recent works(Frisch et al. 2008; Zhu & Taylor 2010) where other dissipation models lead to convergence tothe classical Galerkin truncations in some sense. However, with our HD in §2.2 results, we haverefined Kraichnan’s argument to reveal an interesting feature of helical turbulence beyond thenon-existence of negative temperature states that Kraichnan emphasized. Just as Kraichnan usedthe AE to suggest directions of energy transfer, so that a pile-up of energy at large scales inthe 2D case can suggest the flux of energy to large scales far from equilibrium, we suggest thatthe preferential transfer into one chiral sector observed by Chen, Chen & Eyink (2003) might berelated to our observation about his helical AE. Furthermore, if we can somehow introduce chi-rally asymmetric dissipation and/or resistance in (magneto)fluids ‡ , then the small-scale dampingas discussed in §2.2 would not be simply only for chiral symmetry restoration and that explicitsmall-scale chirality selection and amplification similar to those at large scales could also present.Actually, in general plasma dynamics, such as cyclotron damping, essentially a 3D analog of theclassical 1D Landau damping, and plasma heating (see, e.g. , Chaps. 10, 11 and 17 of Stix 1992),our result may be of stronger connection, since the ion and electron cyclotron resonances are ofopposite chiralities and at different scales, addressable by two-fluid MHD model in §2.3.2.3.2. Comparisons: for genericities, specificities and beyond
A major purpose of this work is to find the genericities and differences of various helicity effectsby comparisons of the different hydrodynamic-type models. The subject in the center of thecomparison is that of magnetic versus kinetic helicities. The pure magnetodynamic result forOCSDS of magnetic helicity (transfer) at large scales, as represented in the vanishing- d e EMHD,generically lies in the core also of other MHD models. Two-fluid MHD has the most general andcomplete elements of helicities and show convincingly the crucial role of magnetic helicity forlarge scales. It appears to be nothing deep but simply due to the mathematical relations Q cK ( k ) = kU cK ( k ) and Q cM ( k ) = U cM ( k ) /k by definitions, by which, one can practically assume equipartition between kinetic and magneticenergy of same chiral sector at some intermediate scale and find that magnetic helicity of thatsector belongs more to larger scales. One “artificial” way to look at it is the following: Thegyrofrequency of a charged particle’s helical motion around B is Ω = qB/m , which means coefficients, since they are all statistically identical. And, now the helical representation, as he noted, is onlyan alternative to the usual Fourier representation, concerning the degrees of freedom, thus he might omitthe important distinguishability of the spectra between the opposite sectors. ‡ It might be possible to work with some chiral (conducting) polymers; or, for classical magnetofluids,some special electromagnetic techniques would be wanted. Note that conventional study of elastic polymereffects (such as Procaccia, L’vov & Benzi 2008; Steinberg 2009) have not paid attention to the chirality, thatis, the possibility of a third chiral time scale τ cθ , over which the (chiral) torque is to be balanced, besidesthe transverse and longitudinal ones ( τ ⊥ and τ k in Hatfield & Quake 1999) of the extended coil/helix (suchas DNA), and that in a simple dilute polymer solutions dynamical model (Fouxon & Lebedev 2003, whosenonlinear dynamics is exactly the same as the classical 3D single fluid MHD studied in §2.3.1!) only asingle relaxation time τ is used for all modes of B , the so-called “tau approximation”. It is possible thatthe 3D chiral property of the polymers have non-neglectable rheological effects, especially in the turbulentstates where small-scale helical modes are excited. hiroid absolute equilibria and turbulence B -line as “kinetic vorticity” Ω -line, macroscopically; magneticfield is indeed a pseudovector , like fluid vorticity, allowing the well-known dynamical analogybetween them as initiated by Batchelor (see, e.g. , Moffatt 2008). This then gives various helicitiesa kind of unified description. Magnetic helicity is thus related to the more “intrinsic” plasmaparticle motion. There are also other supports of the robustness of magnetic helicity (see, e.g.,Brandenburg & Subramanian 2005), such as analysis with more general context (Berger & Field1984) and measurements (Ji 1999).The HD and the large- d e EMHD situations are different to the others, in the sense that therealizable AE spectra can only have positive pole at large k s which regime however is subjectto dissipation in real physical systems. That is, reflexion symmetry breaking and restorationmechanisms meet at the same battlefield and they reach another kind of equilibrium balancewhich is far from our statistical absolute equilibrium, whose implications and residuals con-cerning chirality in conventional fluids can still be identified with careful analyses as shown in§2.2 by refining Kraichnan’s argument. Restricting to the homochiral situations (Waleffe 1992;Biferale, Musacchio & Toschi 2012), HD kinetic energy or EMHD magnetic helicity accumulat-ing at or transferring to large scales becomes possible as indicated by the negative- α state withsmall- k pole, and we further find that, as long as the asymmetry is strong enough, adding aliensto exclude the negative temperature state does not necessarily drastically change the transfer pic-ture. It is possible to control the smooth transition from completely inverse to partly inverse andpartly forward and to completely forward transfers.Concerning turbulence cascade in the infinite Reynolds number limit or some kind of ther-modynamic limit in the sense of k max → ∞ in the conventional Galerkin truncation, full 3DNavier-Stokes’ energy and helicity both cascading to small scales indicates that the solution issingular. Note that such an indication however has not yet found rigorous mathematical supportand that there is still space for opposite conjectures, such as a solution as some kind of “di-rectional limit” without such dissipative anomalies (Zhu & Taylor 2010). Careful examination ofchiroids absolute equilibria as partly illustrated in §2.2 turns out to be able to give fine and clearintuitive pictures about the roughness of the solutions.
Now, for homochiral 3D Navier-Stokeswith, say, c = + , it is expected that only helicity, but not energy, is transferred to small scales,which indicates that the solution is slightly less singular, with the H ¨ o lder exponent h in δu ( ℓ ) ∼ ℓ h be some value inbetween / and / (Eyink 2008), of course in some statistical sense as themultifractal spectrum of h spans over a wide range in realizations (Frisch 1995). The self-similarpure kinetic helicity cascade spectrum would go as (Brissaud et al. 1973; Waleffe 1992) H + K ( k ) ∼ k − / which, unlike Kraichnan’s 2D enstrophy spectrum ∼ k − , is convergent when integrated over k . Note that now h = 2 / . This convergence so far does not bring any troubles: Unlike 2DEuler, where finite enstrophy ensures the smoothness of the solution and that in principle ensuresan equilibrium statistical mechanics without truncation (Miller 1990; Robert 2003), there is nomathematical theorem to assure conservation of helicity with its finiteness (see, e.g. , Eyink 2008).Of course, such cascade still may have spacial intermittency, in the sense of Onsager (Eyink2008) that the helicity dissipation appears “spotty”, in which case an anomalous part of thedissipation may be considered to be curdling with infinite density on some fractal sets of zerovolumes (Mandelbrot 1974), described by some Dirac delta function supported by the fractal,as an extremal limit. Coming back to full Navier-Stokes and supposing both energy and helicitycascading forwardly as k − / , we immediately see from Eqs. (1.2) and (1.3) that H cK ( k ) = cC k − / + C k − / J.-Z. Zhu & W. Yang and G.-Y. Zhu with C and C being constants (Ditlevsen & Giuliani 2001). That is, the two sectors of helicityboth present ultraviolet divergences, indicating more singular solutions, which is consistent withthe absolute equilibrium spectra showing poles at large k , compared to the small- k poles for thehomochiral case. Thus, as indicated by the smooth transition from backward, to bi-direction andto forward cascades by manipulating the helical modes in §2.2 , how the added “alien” helicalmodes increase the nonlinearity to roughen the solutions is intriguing and may be relevant to theintermittency property in the sense of Onsager and Mandelbrot as mentioned above.3.3. Conclusion and prospects
Since an incompressible hydrodynamic-type system can be reduced to the dynamics of chiroids a la
Kelvin, it is natural that one reduces the statistical dynamics to what is based on them and“hopes that one can get some insight into the nature of more general viscous flows and even, per-haps, a deeper understanding of turbulence.” (Moses 1971) We have studied the chirality issue,starting from and essentially based on the chiroids absolute equilibria. The equilibrium spectracan also be used to guide and benchmark numerical experiments with truncation schemes suchas those discussed in §2.2 with asymmetric truncation between the two chiral sectors. Hints forfurther theoretical considerations may also be inferred. For instance, the clear OCSDS for large-scale magnetic helicity could imply some clues to dynamical dynamo model. Looking furtherinto anisotropic fluctuations with a background magnetic field (for such discussions under theframework of 3D gyrokinetics, see, e.g. , Zhu & Hammett 2010) and to more realistic laboratorysituations is also a reasonable step towards a more comprehensive theory for multi-scale plasmadynamics. And, due to the cross-disciplinary popularity of the notion of chirality as the legacyof Pasteur and Lord Kelvin (see, e.g. , Barron 1997), one may not be able to resist a thought ex-cursion into other fields, such as biochirality (see, e.g. , Blackmond 2010) among others, which isthe reason why we choose the terminology “chirality” instead of “parity”, which may be thoughtto be associated with the symmetry of fundamental physical laws, or pure geometrical symmetry, i.e. , mirror symmetry, of objects, or “polarization” which is used more for (linear) waves.In conclusion, Appendix A is written thanks to a referee who raised the questions, believedin their popularity among readers and asked for explicit answers in the paper, and who is alsoacknowledged for the decompression of the early version of the manuscript. This work waspartially supported by the Fundamental Research Funds for the Central Universities of China andby the WCI Program of the NRF of Korea [WCI 2009-001]. We thank for the discussions with U.Frisch and S. Kurien on helical hydrodynamic closures and with M. Taylor on direct numericalsimulations of helical absolute equilibria back to half a solar cycle ago, with Z.-B. Guo, Z.-W. Xia and D.-D. Zou on plasma waves, with X.-P. Hu and Z. Lin on plasma heating, and thecorrespondences with L. Biferale, C.-K. Chan, P. Diamond, D. Escande, M. Faganello, S. Galtier,J. Miller, M. Petitjean and V. P. Ruban, A. Schwartz during the course of this work. is gratefulfor the hospitality of the International Institute for Fusion Science, Université de Provence, andfor the workshop “The Solar Course, the Chemic Force, and the Speeding Change of Water” atNORDITA (2011).
Appendix A. Very basic aspects of the truncated system
A.1.
On the detailed conservation laws for the pure helical modes in each interacting triad
Let us outline here, for the HD case, a direct verification ( e.g. , Waleffe 1992) and a proof ( e.g. ,Kraichnan 1973) of the detailed conservation laws for energy and helicity of the pure helicalmodes among each interacting triad. Both of them are simply carried over from those for thetraditional Fourier modes. The direct verification starts from the dynamical equation of the purehelical modes, with the interactions restricted among only one triad as given by Eq. (9) of Waleffe(1992). As he shows, simple algebras by the definitions of energy and helicity using this equation hiroid absolute equilibria and turbulence proof also needs only to changethe objects of the classical Fourier modes, in the third paragraph in p. 748 of Kraichnan (1973),to pure helical modes. The idea is simply that the overall energy and helicity are formally con-served by the original dynamics without explicit truncation and the truncated modes’ energy andhelicity are constantly zero, due to the facts that their amplitudes are set to be nulls by defini-tion of truncation and that the ‘energy and helicity expressions are quadratic and diagonal in thewave-vector amplitudes.’ Note that the expression being diagonal in the wave-vector amplitudesis also important: Suppose it is not diagonal and that the convolution involves the multiplicationof modes in the truncated and un-truncated domains, then the change rate of it is not assuredto be constantly zero, since the change rate constitutes a component from the multiplication ofthe time derivative of a mode in the truncated domain with another mode in the un-truncated do-main, both of which can be non-zero; an example is the quadratic invariants of 2D gyrokinetics inthe Fourier-Hankel/Bessel representation and truncation, where the phase-space “wave-vector”is extended from the conventional wave-vector to include a component from the spectral repre-sentation of the velocity variable and where those quadratic expressions not being diagonal in theextended wave-vector are not ruggedly conserved, i.e. , not invariant after Fourier-Hankel/Besseltruncation (see pp. 3–4 of Zhu 2011).As some readers may feel easier to start with a degenerate trivial case to get somewhat moreconcrete grasp, let’s suppose first that we retain only contributions from the region of wave-number space | k | < K (Galerkin truncation), and that we start with a single triad of pure helicalmodes (chiroids), using indexed c to denote the chirality of the leg, { [ ± k , c k ]; [ ± p , c p ]; [ ± q , c q ] } with k + p + q = 0 and K/ < | k | , | p | , | q | < K , so that harmonics generated by the Eulerequations are eliminated under this truncation which we can think of as providing some kindof ‘artificial dynamics’ and which is denoted by the index “ g ”. Let u g ( x , t ) and ω g ( x , t ) = ∇ × u g ( x , t ) be the velocity and vorticity fields evolving under this artificial dynamics. Thenthe claim is that the mean kinetic energy E g = < u g / > and mean helicity H g = < u g · ω g > are invariant in time. Yet another degenerate case is the Arnold-Beltrami-Childress (ABC)flow, composed of three conjugate pairs of pure helical modes with the same wavelength, whichhas been used by many authors to study kinematic dynamo and which can be generalized tocontain more conjugate pairs of pure helical modes of same wavelengths and that presumably tobecome more chaotic, in the Lagrangian/streamline sense (see, e.g. , Arnold & Khesin 1998, andreferences therein). A.2. Dynamical and topological aspects
By definition, the Galerkin-truncation dynamics of vorticity is ∂ t ω g = [ ∇ × ( u g × ω g )] g . (A 1)We are not sure whether the fact that H g is constant assures a fictitious meaningful ˜ v to solve ∂ t ω g = ∇ × (˜ v × ω g ) , i.e. , making an analogue of the Kelvin-Helmholtz or “frozen-in” theorem(which is sufficient but not necessary for the conservation law.) Definitely, ˜ v = u g is not thesolution, otherwise the last index “ g ” on the right-hand side of Eq. (A 1) would have no effect.Note that the topological interpretation of the helicity as the degree of (average) knottednessand/or linkage (Moffatt 1969) of (closed) field line(s) formally carries over to the Galerkin trun-cated case. Actually, the interpretation itself has not much to do with the dynamics but simplyworks for fields satisfying some basic properties by the definition of Gauss linking number,which has a lot of subtleties and complications when generalized to continuous fields and gen-eral boundary conditions (see, e.g. , Moffatt 1969; Berger & Field 1984; Arnold & Khesin 1998,among many other references cited therein and appearing later). The Galerkin-truncated dynam-ics is formally changed much in physical space, as partly shown in the last paragraph, while8 J.-Z. Zhu & W. Yang and G.-Y. Zhu formally unchanged (except for truncation) in Fourier space concerning triadic interactions. Oneformally unchanged thing, besides those such as the quadratic invariants, that clearly bridgesthe physical- and Fourier-space representation, is the preservation of the incompressibility ofthe fields, which is also due to the fact that the orthogonality between their chiroids and thewavevectors holds in detail, i.e. , for each chiroid k · ˆ h c ( k ) = 0 , and which justifies the definitionof flux tube(s) as the key to the topological interpretation (Moffatt 1969). Topological, especiallyknot-theory, approaches to the statistical dynamics of course deserves further pursue, which ishowever beyond the scope of this note.A.3. On the tacit assumptions in the statistical consideration
It is difficult and quite open to establish mathematically rigorous conditions for justifying the ap-plication of statistical mechanics. For example, according to literatures (see, e.g. , Eyink & Sreenivasan2006, and references therein), ergodicity, being sufficient, may not be trivially satisfied but mayneither be necessary; and, the mixing time scale could be hard to estimate for evaluating thecloseness of physical relevance of the equilibrium ensemble. However there is a trivial bottomline that is assumed to be met, that is, all modes should be directly or indirectly connected byforming the interacting triads to define a system. For example, now suppose instead that we startwith two triads { [ ± k j , c k j ]; [ ± p j , c p j ]; [ ± q j , c q j ] } with j = 1 , and all these wave-vectorsin the spherical annulus ( K/ , K ) , as in Appendix A.1, and suppose that these triads are non-interacting. Let the energies in the triads be E g and E g respectively, and the helicities H g and H g . Then it is not eligible to use the ensemble defined by the total energy E g = E g + E g andthe total helicity H g = H g + H g for the union of these two isolated systems, not to mentionthat sufficient number of modes are necessary for a statistical consideration, and in particular theapplication of Gibbs ensemble. Actually, in practice, when the number modes is large it is hardlypossible for any triad to be isolated from others. In performing the calculations as in the maintext, such tacit assumptions are made to exclude cases not describable by the canonical ensem-ble. In the 1970s, people already performed many numerical simulations, mostly for 2D cases(Orszag 1977; Kraichnan & Montgomery 1981), to study the ergodicity and mixing properties,to measure the difference between microcanonical and canonical ensembles, to find how manymodes would be needed to reach the Gibbs state and to finally verify the corresponding energyspectrum. Matthaeus and collaborators, among others, have many followup studies, especiallyfor 2D and 3D MHD absolute equilibrium ensembles. We cannot exhaust the list of referenceshere, but would like to remark that there can be new findings by revisiting numerical check of thetacit assumptions relevant to the application of Gibbs ensembles with the new special truncationson the new freedoms of chiroids. R E F E R E N C E SA
RNOLD , V. I. & K
HESIN , B. A. 1998 Topological Methods in Hydrodynamics. Springer.B
ARRON , L. D. 1997 From cosmic chirality to protein structure and function: Lord Kelvin’s legacy. Q. J.Med. ERGER , M. A. & F
IELD , G. B. 1984 The topological properties of magnetic helicity. J. Fluid Mech. ,133–148.B
ETCHOV , R. 1961 Semi-isotropic turbulence and helicoidal flows. Phys. Fluids , 925–926.B IFERALE , L., M
USACCHIO , S. & T
OSCHI , F. 2012 Inverse energy cascade in three-dimensional isotropicturbulence. Phys. Rev. Lett. , 104501–104504.B
ISKAMP , D., S
CHWARZ , E., Z
EILER , A. C
ELANI & D
RAKE
J. F. 1999 Electron magnetohydrodynamicturbulence. Phys. Plasmas , 751–758.B LACKMOND , D. G. 2010 The Origin of Bilogical Homochirality Cold Spring Harb Perspect Biol ,a002147.B RANDENBURG , A., D
OBLER
W. & S
UBRAMANIAN , K. 2002 Magnetic helicity in stellar dynamos: newnumerical experiments. Astron. Nachr. , 99–123. hiroid absolute equilibria and turbulence B RANDENBURG , A. & S
UBRAMANIAN , K. 2005 Astrophysical magnetic fields and nonlinear dynamotheory. Physics Reports , 1–209.B
RISSAUD , A., F
RISCH , U., L
EORAT , J., L
ESIEUR , M. & M
AZURE , M. 1973 Helicity cascades inisotropic turbulence. Phys. Fluids , 1366–1367.C HEN , Q. N., C
HEN , S. Y. & E
YINK , G. L. 2003 The joint cascade of energy and helicity in three-dimensional turbulence. Phys. Fluids (2), 361–374.C HEN , Q. N., C
HEN , S. Y., E
YINK , G. L. & H
OLM , D. 2005 Resonant interactions in rotating homoge-neous three-dimensional turbulence. J. Fluid Mech. (2), 139–163.C HO , J. 2011 Magnetic Helicity Conservation and Inverse Energy Cascade in Electron Magnetohydrody-namic Wave Packets. Phys. Rev. Lett. , 191104–191107.C INTAS , P. & V
IEDMA , C. 2012 On the Physical Basis of Asymmetry and Homochirality. Chirality ,894–908.D ITLEVSEN , P. D. & G
IULIANI , P. 2001 Dissipation in helical turbulence. Phys. Fluids , 3508–3509.E YINK , G. L. 2008 Dissipative anomalies in singular Euler flows. Physica D , 1956–1968.E
YINK , G. L. & S
REENIVASAN , K. R. 2006 Onsager and the theory of turbulence. Review of ModernPhysics , 87–135.F OUXON , A. & L
EBEDEV , V. 2003 Spectra of turbulence in dilute polymer solutions. Phys. Fluids ,2060–2072.F RISCH , U., P
OUQUET , A., L
EORAT , J. & M
AZURE , A. 1975 Possibility of an inverse magnetic helicitycascade in magnetohydrodynamic turbulence. J. Fluid Mech. , 769–778.F RISCH , U., K
URIEN , S, P
ANDIT , R., P
AULS , W., R AY , S., W IRTH , A. & Z HU , J.-Z. 2008 Hyperviscos-ity, Galerkin Turncation and Bottleneck of Turbulence. Phys. Rev. Lett. , 114501–114504.F RISCH , U. 1995 Turbulence: The Legacy of Kolmogorov. Cambridge University Press.G
ALTIER , S. 2006 Wave turbulence in incompressible Hall magnetohydrodynamics. J. Plasma Physics ,721–769.G ALTIER , S. & B
HATTACHARJEE , A. 2003 Anisotropic weak whistler wave turbulence in electron mag-netohydrodynamics. Phys. Plasmas , 3065–3076.H ATFIELD , J. W. & Q
UAKE , S. R. 1999 Dynamic Properties of an Extended Polymer in Solution. Phys.Rev. Lett. , 3548–3551.J I , H. 1999 Turbulent dynamos and magnetic helicity. Phys. Rev. Lett. , 3198–3191.K ELVIN , W. T
HOMSON
RAICHNAN , R. H. 1967 Inertial ranges in two-dimensional turbulence. Phys. Fluids , 1417–1423.K
RAICHNAN , R. H. 1973 Helical turbulence and absolute equilibrium. J. Fluid Mech. , 745–752.K RAICHNAN , R. H. 1958 Irreversible Statistical Mechanics of Incompressible Hydromagnetic Turbulence.Phys. Rev. , 1407-1422.K
RAICHNAN , R. H. & M
ONTGOMERY , D. 1980 Two-dimensional turbulence. Reports Progr. Phys. ,547-619.L EE , T.-D. 1952 On some statistical properties of hydrodynamic and hydromagnetic fields. Q. Appl. Math. , 69–74).L ESSINNES , T., P
LUNIAN
F. & C
ARATI , D. 2009 Helical shell models for MHD. Theor. Comput. FluidDyn. , 439–450.L ESIEUR , M. 1990 Turbulence in Fluids. 2nd ed. Kluwer Academic, Dordrecht, The Netherlands.L’
VOV , V. S., P
OMYALOV , A. &
AND P ROCACCIA , I. 2002 Quasi-Gaussian Statisitcs of HydrodynamicTurbulence in / ǫ Dimensions. Phys. Rev. Lett , 064501–064504.M ANDELBROT , B. B. 1974 Intermittent turbulence in self-similar cascades: divergence of high momentsand dimension of the carrier. Journal of Fluid Mechanics , 331–358.M ELANDER , V., H
USSAIN , F. 1993 Polarized vorticity dynamics in a vortex column. Phys. Fluids A ,1992–2003.M ENEGUZZI , M, F
RISCH , U
AND P OUQUET , A 1981 Helical and nonhelical turbulent dynamos. Phys.Rev. Lett. , 1060–1063.M EYRAND , M. & G
ALTIER , S. 2012 Spontaneous Chiral Symmetry Breaking of Hall Magnetohydrodyn-mic Turbulence. Phys.Rev. Lett. , 194501-1–5.M
ILLER , J. 1990 Statistical mechanics of Euler equations in two dimensions. P hys. Rev. Lett. , 2137-2140.M OFFATT , H. K. 1969 The degree of knottedness of tangled vortex lines. J. Fluid Mech. , 117–129. J.-Z. Zhu & W. Yang and G.-Y. Zhu M OFFATT , H. K. 1970 Turbulent dynamo action at low magnetic Reynolds number. J. Fluid Mech. ,435–452.M OFFATT , H. K. 2008 Vortex Dynamics: The Legacy of Helmholtz and Kelvin. In Hamiltonian Dynamics,Vortex Structures, Turbulence (ed. A.V. Borisov, V.V. Kozlov, I.S. Mamaev & M.A. Sokolovskiy), pp.113–138. Springer.M
OSES , H. E. 1971 Eigenfunctions of the curl operator, rotationally invariant Helmholtz theorem andapplications to electromagnetic theory and fluid mechanics. SIAM (Soc. Ind. Appl. Math.) J. Appl.Math. , 114–130.P EREZ , J. C., M
ASON , J., B
OLDYREV , S. & C
ATTANEO , F. 2012 On the Energy Spectrum of StrongMagnetohydrodynamic Turbulence. Phys. Rev. X , 041005.P ETITJEAN , M. 2003 Chirality and Symmetry Measures: A Transdisciplinary Review. Entropy , 271-312.P OUQUET , A., F
RISCH , U. & L
ÉORAT , J. 1976 Strong MHD helical turbulence and the nonlinear dynamoeffect. J. Fluid Mech. , 321–354.P ROCACCIA , I., L’
VOV , V. & B
ENZI , R. 2008 Colloquium: Theory of Drag Reduction by Polymers inWall Bounded Turbulence. Rev. Mod. Phys. , 225–247.O RSZAG , S. A. 1977 Statistical Theory of Turbulence, in Fluid Dynamics, Les Houches 1973, 237-374,eds. R. Balian & J.L. Peube. Gordon and Breach, New York.R
OBERT , R. 2003 Statistical Hydrodynamics (Onsager Revisited). in: Handbook of mathematical fluiddynamics, Volume 2, Eds. Susan Friedlander, Denis Serre, pp. 1–54. Gulf Professional Publishing.S
TEINBERG , V. 2009 Elastic stresses in random flow of a dilute polymer solution and the turbulent dragreduction problem. C. R. Physique , 728–739.S TIX , T. H. 1992 Waves in Plasmas. American Institute of Physics.R
UBAN , V. P. 1999 Motion of magnetic flux lines in magnetohydrodynamics. Sov. Phys. JETP , 299–310.S CHEKOCHIHIN , A. A., C
OWLEY , S. C., D
ORLAND , W., H
AMMETT , G. W., H
OWES , G. G., Q
UATAERT ,E. & T
ATSUNO , T. 2009 Astrophysical gyrokinetics: Kinetic and fluid cascades in magnetized weaklycollisional plasmas. The Astrophysical Journal Supplement Series , 310–377.S
TENZEL , R. L. 1999 Whistler waves in space and laboratory plasmas. J. Geophys. Res. , 14379–14395.W ALEFFE , F. 1992 The nature of triad interactions in homogeneous turbulence. Phys. Fluids A , 350–363.W OLTJER , L. 1959 Hydromagnetic equilibrium II. Stability in the variational formulation. Proc. Nat. Acad.Sci. U.S.A. AMADA , H., K
ATANO , T., K
ANAI , K., I
SHIDA , A. & S
TEINHAUER , L. 2002 Equilibrium analysis of aflowing two-fluid plasma. Phys. Plasmas , 4605–4614.Y ANG , Y.-T., S U , W.-D. & W U , J.-Z. 2010 Helical-wave decomposition and applications to channelturbulence with streamwise rotation. J. Fluid Mech. , 91–122.Z HU , J.-Z. 2011 Fourier-Hankel/Bessel space absolute equilibria of 2D gyrokinetics. arXiv:1109.2511v1[nlin.CD]: under revision for later journal publication.Z HU , J.-Z. & H AMMETT , G. W. 2010 Gyrokinetic absolute equilibrium and turbulence. Phys. Plasmas ,122307-1–13.Z HU , J.-Z. & T AYLOR , M. 2010 Intermittency and Thermalization of Turbulence. Chinese Phys. Lett.27