A Detailed Examination of Anisotropy and Timescales in Three-dimensional Incompressible Magnetohydrodynamic Turbulence
Rohit Chhiber, William H. Matthaeus, Sean Oughton, Tulasi N. Parashar
AAnisotropy and Timescales in 3D MHD Turbulence
A Detailed Examination of Anisotropy and Timescales inThree-dimensional Incompressible Magnetohydrodynamic Turbulence
Rohit Chhiber,
1, 2, a) William H. Matthaeus, Sean Oughton, and Tulasi N. Parashar
1, 4 Department of Physics and Astronomy, Bartol Research Institute, University of Delaware, Newark, DE 19716,USA Heliophysics Science Division, NASA Goddard Space Flight Center, Greenbelt, MD 20771,USA Department of Mathematics and Statistics, University of Waikato, Hamilton 3240, New Zealand School of Chemical and Physical Sciences, Victoria University of Wellington, Kelburn, Wellington 6012,NZ (Dated: 20 May 2020)
When magnetohydrodynamic turbulence evolves in the presence of a large-scale mean magnetic field, an anisotropydevelops relative to that preferred direction. The well-known tendency is to develop stronger gradients perpendicularto the magnetic field, relative to the direction along the field. This anisotropy of the spectrum is deeply connectedwith anisotropy of estimated timescales for dynamical processes, and requires reconsideration of basic issues suchas scale locality and spectral transfer. Here analysis of high-resolution three-dimensional simulations of unforcedmagnetohydrodynamic turbulence permits quantitative assessment of the behavior of theoretically relevant timescales inFourier wavevector space. We discuss the distribution of nonlinear times, Alfvén times, and estimated spectral transferrates. Attention is called to the potential significance of special regions of the spectrum, such as the two-dimensionallimit and the “critical balance” region. A formulation of estimated spectral transfer in terms of a suppression factorsupports a conclusion that the quasi two-dimensional fluctuations (characterized by strong nonlinearities) are not asingular limit, but may be in general expected to make important contributions.
I. INTRODUCTION
In standard Kolmogorov theory, turbulence is assumedto be isotropic so that all relevant correlation functionsand their spectral decompositions are invariant under properrotations.
Magnetohydrodynamics (MHD), and magnetizedplasma turbulence, are different in that their dynamics canbecome anisotropic when a large-scale magnetic field ispresent.
At this point the assignment of characteristictimescales required for turbulence closures (e.g., Refs 7,8),or even the heuristic identification of timescales in turbulencephenomenologies becomes more difficult, and even some-what ambiguous from a theoretical perspective. How aretimescales distributed in wavevector space when the energyis anisotropically distributed? What does scale locality im-ply when turbulence is anisotropic? What are the relevantrelationships among the different theoretically constructedtimescales? Here we will discuss several issues that come intoplay regarding these timescales, favoring physical relevanceover complete mathematical rigor.We begin with the idea of nonlinear timescale τ nl , whichis traditionally defined in terms of the scalar magnitude ofwavevector, in view of isotropy along with Kolmogorov’s as-sumption of scale locality. The structure of the usual esti-mate arguably remains valid, even for anisotropic spectra. Within the inertial range, the time dependence is dominatedtypically by the random sweeping of inertial-range fluctua-tions by large eddies, both in hydrodynamics and inMHD. However, this does not directly influence spectraltransfer or the nonlinear time, since it corresponds, essentially, a) Electronic mail: [email protected] to a local spatial translation.
Another timescale of rele-vance in MHD, and one that has been the subject of intensediscussion, is the Alfvén timescale, associated with small-amplitude propagating waves, or under certain specific con-ditions, large-amplitude fluctuations. More generally thanin the propagating wave scenario, wave-like couplings associ-ated with the Alfvén timescale act to suppress nonlinear be-havior. This occurs due to the potentially rapid changes in thephases of Fourier components that are induced by Alfvéniccouplings.
This effect, which occurs in an anisotropicfashion, reduces the third-order correlations necessary forspectral transfer. The competition that exists between Alfvénand nonlinear effects, as well as the balance of local andnonlocal effects, are issues that have pervaded discussions ofMHD and plasma turbulence for decades.Here we examine these issues directly by employing MHDsimulation. We evaluate the basic timescales throughoutFourier space in a simulation with anisotropy induced by alarge-scale mean magnetic field. For clarity of presentation,just one run, employing the incompressible model, and with afixed uniform externally-supported magnetic field, is the ba-sis for most of what is reported here. However, the results,while not “universal” in any meaningful sense, are expectedto remain relevant for useful and familiar regions of param-eter space, such as low cross helicity, small magnetic helic-ity, near-unit Alfvén ratio, unit magnetic Prandtl number, etc(see Ref. 20 for details). One additional simulation, witha variation in initial data, is discussed in Appendix B. Wealso employ standard methods to estimate quantities relatedto turbulence activity, such as contributions to spectral trans-fer in different regions of wavevector space. This strategywill enable an evaluation not only of the instantaneous kine-matic state of the energy spectrum but also the greater or a r X i v : . [ phy s i c s . p l a s m - ph ] M a y nisotropy and Timescales in 3D MHD Turbulence 2lesser concentrations of dynamic turbulence activity, a subjectless well-described previously. The results obtained with thisstrategy will be employed to inform discussion of several ap-proaches to explaining anisotropic spectral distributions of en-ergy, such as multi-component models, Reduced-MHD(RMHD) models, critical balance, and diffusionmodels, all of which require consideration of elements ofthe classic isotropic model for context. The outline of the paper is as follows. We briefly describethe simulation used in Section II and present basic results onspectra and timescales in Section III. The wavevector distri-bution of energy and the so-called nonlinearity parameter areexamined in Section IV. In Section V we employ these quan-tities to estimate spectral transfer rates for anisotropic MHD,and examine their spectral distribution as well as their distri-bution across the nonlinearity parameter. We conclude withdiscussion in Section VI. Appendix A provides backgroundto motivate our formulation of the anisotropic spectral trans-fer rates while Appendix B briefly describes results from a runinitialized with fully isotropic fluctuations.
II. SIMULATION DETAILS
We employ a pseudospectral three-dimensional (3D) in-compressible MHD code with 1024 resolution in a periodicbox of dimensionless side 2 π for this study, where the unitlength, L , is arbitrary. Our results are primarily obtainedfrom an undriven, freely decaying run whose initial fluctua-tions are toroidally polarized (i.e., in the same sense as lin-earized Alfvén waves), with some, less detailed references toa second run for which the initial fluctuations are isotropi-cally polarized (toroidal and poloidal modes excited). Herewe focus on a single run with a mean magnetic field B = z (in Alfvén units), and initial total energy and Alfvén ratioboth equal to unity. This corresponds to an initial value of δ b / B = δ b is the root mean square magnetic fluctu-ation.Toroidally polarized Fourier modes are excited initially ina band 3 ≤ k ≤ k = k an abbreviated − / k is themagnitude of the wavevector k . Initial cross helicity and mag-netic helicity are both very close to zero. The simulation is runusing a second-order Runge-Kutta scheme for 8000 timestepswith a step size of 2 . × − . Therefore the entire run pro-ceeds for about two large-scale nominal nonlinear times. Thekinematic viscosity ν and resistivity µ are assigned equal val-ues of 7 . × − . These represent reciprocal Reynolds num-ber and magnetic Reynolds number, respectively, at scale L .We emphasize that the results presented here are based ona single snapshot from this simulation. We have repeated theanalysis for a similar run initialized with isotropic fluctuations(toroidal + poloidal polarization) and obtained consistent re-sults, discussed below. Numerous other simulations of a sim-ilar nature were carried out and were found to present simi-lar properties with regard to resolution, energy decay, and in general the dynamical features of decaying, unforced MHD. We therefore believe that the present results are quite typical.However, we caution the reader that true universality is not ex-pected in MHD turbulence due to the multiplicity of control-ling parameters (see Ref. 20), and therefore extrapolation ofthe present observations to widely varying parameters shouldbe undertaken with caution. Further relevant details are pre-sented in the subsequent sections below.
III. SPECTRA AND BASIC TIMESCALES
The context of the following analysis is established byexamining the time history of key global quantities, andwavenumber spectra, which are shown in Figures 1 and 2, re-spectively. The evolution of kinetic and magnetic energies ( E v and E b , respectively), mean-squared electric current density(magnetic enstrophy) (cid:104) j (cid:105) , and the kinetic enstrophy (cid:104) ω (cid:105) areillustrated in Figure 1. The enstrophies both peak near about t = .
5, and accordingly this corresponds to the time of peakdissipation rate. By t = Therefore we carry out theremainder of our analyses at this time,at which the value of δ b / B is about 0.8.Figure 2 shows the omnidirectional wavenumber spec-tra of magnetic energy E b and kinetic energy E v at t = E omni ( k ) ≡ E v ( k ) + E b ( k ) , so that the total energy per unitmass is (cid:82) E omni ( k ) d k . An estimate of the MHD Kolmogorovdissipation wavenumber k η = (cid:104) j + ω (cid:105) / / √ ν is markedon the plot as a vertical line. An inertial range with an approx-imate powerlaw slope of − / k = k =
30, falling short of k η by a factor of2-4 as is typical of fluid models with scalar dissipation coeffi-cients (e.g., Refs. 42–44). The steepening of the inertial-rangespectrum becomes significant at about k =
50, and the spec-trum has dropped substantially, by approximately two ordersof magnitude, prior to reaching the dissipation wavenumbernear k = in contrast to therather sharper termination of the spectrum often seen in the so-lar wind (e.g., Ref. 46). This is because the MHD fluid modelin this paper does not capture the collisionless-plasma effectson nonlinear dynamics that come into play at kinetic scales inthe solar wind (eg., Ref. 47).To proceed we examine the basic timescales. First we con-sider the nonlinear timescale, defined as local in wavenum-ber, following Kolmogorov’s assumption of the dominance oflocal transfer. It is then standard practice to define τ nl ( k ) ≡ [ k δ v ( k )] − , where δ v ( k ) is the characteristic speed at scales (cid:96) ≈ / k . Using the omnidirectional total-energy spectrum,with δ v ( k ) ≈ kE omni ( k ) , yields τ nl ( k ) = / [ k (cid:112) kE omni ( k )] . (1)This definition maintains the plausible approximation thatnisotropy and Timescales in 3D MHD Turbulence 3 FIG. 1. Time evolution of kinetic and magnetic energies, mean cur-rent density, and enstrophy ( E v , E b , (cid:104) j (cid:105) , and (cid:104) ω (cid:105) , respectively).FIG. 2. Omnidirectional spectra of magnetic ( E b ) and kinetic ( E v )energies, and nonlinear timescale τ nl , at t = .
0. Here τ nl ( k ) = / [ k (cid:112) kE omni ( k )] (see text). The dissipation wavenumber k η isshown as a vertical line. Reference lines with k − / and k − / slopesare plotted in dotted green. the nonlinear couplings are mainly local in the magnitude ofwavevector. The dependence on | k | of this standard, scale-local nonlin-ear time is shown in Figure 2 (see also Refs. 53,54). Thereis no angular dependence with this definition, and nonlineartime so-defined is constant on shells with a given modulusof wave vector. At this modest Reynolds number the iner-tial range spans only about one decade, as was seen also inthe energy spectra. Here the expected inertial range behavior τ nl ( k ) ∼ k − / is seen only over a rather narrow approximaterange of k = k =
35. For k >
100 the scale-local nonlineartime sharply increases due to rapidly decreasing spectral en-ergy density. But in any case the approximation of locality is
FIG. 3. Alfvén timescale τ A ( k ) = / | k z B | in the k z – k x plane. For k z = k z = / B . The Higdon curve ( k z = k / x ), the χ = τ A / τ nl = ), and the dissipation wavenumber k η are plottedas dashed cyan, dash-dotted black, and dotted green curves, respec-tively. The latter two contours are constructed from simulation dataat t = questionable as the dissipation scale is approached. Similarly,this definition of local nonlinear timescale is not relevant tothe lowest few wavenumbers at the energy-containing scales,as the granularity of the excited modes comes into play andthe strength of nonlinear couplings depends on the specificenergy distribution in that range.The principal antagonist to nonlinear effects in incompress-ible MHD turbulence with a large-scale magnetic field is theAlfvén propagation effect that interferes – anisotropically –with couplings that drive turbulence. The associated Alfvéntimescale τ A ( k ) = / | k z B | is shown in Figure 3, where thedistribution of τ A ( k ) is illustrated over the k z – k x plane definedby k y =
0. Along the k z = k z = k x axis, bydefinition. The other contours in Figure 3 will be presentlydiscussed below and are included for later referencing. SeeRef. 55 for similar results based on Fourier-space ratios oflinear and nonlinear accelerations, rather than timescales. IV. WAVEVECTOR DISTRIBUTION OF ENERGY ANDNONLINEARITY PARAMETER
A more detailed view of the energy spectrum is affordedby examining the distribution of energy in parallel and per-pendicular components of wavevector. We define the modalenergy spectral density as E mod ( k ) = [ | v ( k ) | + | b ( k ) | ] / ∆ k ,nisotropy and Timescales in 3D MHD Turbulence 4where v ( k ) and b ( k ) are the (periodic domain) Fourier coef-ficients of velocity and magnetic field, respectively, and ∆ k is the fundamental Cartesian volume element for the simula-tion (equal to unity in our case). Figure 4 shows a map of E mod ( k ) in a two-dimensional (2D) cut through the wavevec-tor space, for the simulation at t = .
0. The particular selectedplane, E mod ( k x , k y = , k z ) , spans one perpendicular direction, k x , and the parallel direction, k z . In the Figure, lines of con-stant spectral density are shown, the pattern of distributionfurther emphasized by plotting colors associated with rangesof energy density. The following analyses and figures corre-spond to t = . k x (perpendicular)direction, and spanning about ten orders of magnitude in spec-tral density. The bottom panel shows a close-up of the samedistribution, concentrating on the region closer to the originwith k x ≤
40 and k z ≤
40, covering the inertial range.The distribution is highly anisotropic, extending moredeeply towards the perpendicular direction, as expected giventhe well-known tendency for MHD turbulence to producestronger gradients perpendicular to an externally-supporteduniform magnetic field.
One may note, for example, thatthe contour labeled as 2 . × − extends beyond k x = k z = k x =
0, extends only toabout k z = k z >
25 and indi-cate peaks in the spectral density at k z = k x . Therefore one immediately concludes that the “quasi-2D”modes ( k z (cid:28) k ) are the most energetic, and in some ways maybe dominant. We explore this further below. Note that, ingeneral, for MHD one would define quasi-2D modes as thosewith k z (cid:28) k and τ nl ( k ) (cid:28) τ A ( k ) . In a continuum there maybe quasi-2D modes with nonzero k z (cid:28) k (see Figure 13), butfor our discrete case and our particular simulation parameters,the quasi-2D requirements are satisfied only for the k z = computed nonlinearity parameter, χ ( k ) = τ A ( k ) τ nl ( k ) , (2)takes on the values of unity. This parameter provides a mea-sure of the relative strength of local-in-scale nonlinearity andlocal Alfvén wave propagation (see, e.g., Refs. 10,28,34).This idealized “equal-timescale curve” is computed assum-ing inertial-range Kolmogorov scaling, associated with theHigdon formula k z L ∼ ( k x L ) / , and was later adopted byGoldreich & Sridhar as a central element of the critical bal-ance phenomenology. The contours of k z = α k / x (in dimen-sionless units), associated with τ nl = constant × τ A , are alsoshown for different values of the proportionality constant α (see caption).A significant feature of the spectra in Figure 4 is the lack of any apparent preferred regions in wavevector space. In par-ticular, there is no discernible enhancement of energy density FIG. 4.
Top:
Contours of modal energy spectrum in the k z – k x plane.The Higdon curve, the χ = Bottom:
A blow-up of the region near the ori-gin, covering the inertial range of wavenumbers. The dark-red dot-ted, dashed, and dash-triple-dotted lines show Higdon curves withproportionality constant α equal to 0.5, 1, and 2, respectively. near the Higdon/critical-balance curve, or along any curve ofthat order, such as the k z = ( k x ) / curve that is illustrated(see also discussion of Figures 2 and 3 in Ref. 55). We alsonote that there is no discernible deficit of spectral power alongeither the quasi-2D axis k z = k x = | k | <
7. Therefore, for example, the small-scale quasi-2Dmodes are populated by the cascade on the same timescalesas those over which the rest of the anisotropic spectral distri-bution is formed.nisotropy and Timescales in 3D MHD Turbulence 5
FIG. 5.
Top:
Nonlinearity parameter χ = τ A / τ nl . Bottom:
A blow-up of the region near the origin. In both panels, the dissipationwavenumber and the χ = Figure 5 provides complementary information by illustrat-ing the distribution of the nonlinearity parameter in the same2D cut through k -space. Here the nonlinearity parameter iscomputed from the actual spectral distribution, and not froman assumed Kolmogorov powerlaw. As in the previous figure,this accounts for why the computed χ = − / χ greater than unity near the k z = k -space. Here too the formallyinfinite values of χ on the 2D axis have been replaced withvalues found at the juxtaposed modes with k z =
1. We re-call then that the energy density is substantial in these modes,and actually somewhat larger than modes found at higher k z closer to the χ = χ = χ values. Analysis of that type is provided below. V. NONLINEAR TRANSFER-RATE ESTIMATES
In the previous section we examined the anisotropic distri-bution of energy due to an externally supported mean mag-netic field, and we computed the basic physical timescales atthe instant at which the analysis is performed. To proceed toan understanding of the dynamics, one must go further. Sev-eral options exist.One approach is to compute, based on the known stateof the system, the scale-to-scale transfer rates using a scale-filtering approach. Studies of this type have been done forisotropic MHD; the extension of such studies to theanisotropic case has been less frequently studied. However,for the anisotropic case, there are some models based onphysical assumptions consistent with those adopted here, while examination of transfer across surfaces such as planesand cylinders also led to conclusions broadly consistent withthe ideas adopted here.We note that while computing scale-to-scale transfer di-rectly is extremely valuable for addressing questions suchas the validity of Kolmgorogov locality, it does not imme-diately provide insights into the role of the various physi-cal timescales in producing the computed effects. In par-ticular, if we wish to develop or test phenomenologicaltheories or closures, it will be necessary tounderstand how the fundamental timescales enter the dynam-ics.Similar challenges exist for interpretation of the 3D formof the Kolmogorov–Yaglom–Politano–Pouquet laws. This relationship provides a direct connection between third-order correlations and total energy-transfer across an arbitraryclosed surface in scale space. However third-order laws arenotorious for their slow convergence, and in any case requireadditional analysis to connect their empirically obtained val-ues to physical parameters and timescales. This is particu-larly germane here because the applied magnetic field doesnot change the formal structure of the third-order laws, butrather appears in the hierarchy on the same footing as thefourth-order moments. Consideration of higher-order mo-ments would complicate the present study in which the goal isto quantify the roles of the nonlinear and wave timescales inspectral transfer.In view of these issues, in what follows we pursue a dif-ferent and simpler approach, emphasizing potential insightsinto the role of the fundamental timescales in inducing spec-nisotropy and Timescales in 3D MHD Turbulence 6tral transfer of energy. We employ existing frameworksof turbulence phenomenologies, plausibly generalized to theanisotropic case, to probe relative strength of nonlinear andlinear effects, and to understand the role of the associatedtimescales in controlling spectral transfer.
A. Elementary estimates
The simplest estimate of energy transfer ignores both theinfluence of Alfvénic propagation and the anisotropic distribu-tion of energy in wavevector. In effect one averages the non-linear effects and the energy distribution over angle, leaving adependence on wavenumber alone. Such a direction-averagedestimate of (local) energy spectral-transfer rate may be writtenas δ v ( k ) / τ nl ( k ) = kE omni ( k ) / τ nl ( k ) . In steady state, this rate,in units of energy per unit mass per unit time, will be balancedby the global dissipation rate ε = ν (cid:104) j + ω (cid:105) . This approachis fully equivalent to the classic Kolmogorov theory, since allinformation about anisotropy is lost through averaging. Suchan approach has been successfully applied to solar wind ob-servations to provide useful estimates of rates of cascade anddissipation. It is not necessary to plot this spectral trans-fer rate in regions of wavevector space as has been done abovein Figure 5, since the pattern would consist simply of concen-tric circular regions.A modified estimate, also incomplete, is one that includesanisotropy of the energy distribution, but no explicit accountof the Alfvénic timescale. To take this step, we maintainKolmogorov’s scale-locality in computation of the nonlin-ear timescale and quantify its effect on the anisotropic spec-tral distribution of energy. This estimate of a “local pseudo-cascade rate” may be expressed as δ v ( k ) τ nl ( k ) = ∆ k E mod ( k ) τ nl ( k ) , (3)and is illustrated in Figure 6. This is, in effect, an estimateof spectral transfer acting at a given wavevector if the meanmagnetic field is suddenly extinguished while the spectrum re-mains anisotropic. In the following we denote the turbulencestrength at wavevector k as δ v ( k ) , estimated using Equation(3).Several properties of the quantity plotted in Figure 6 areapparent. The first is that it is anisotropic in the plane, butalso that its anisotropy is identical to that of the modal energyspectrum itself, which was shown in Figure 4. This is due tothe fact that the Kolmogorov nonlinear timescale, as definedin Equation (1), is isotropic. For this reason, the quantity de-picted in Figure 6, even if it has the dimensions of energycascade rate (per unit mass), is not useful as a measure of theanisotropy of the cascade. Such a measure must include influ-ence of the Alfvén propagation effect (see Figure 3), which isknown to suppress spectral transfer in the parallel direction in k -space. FIG. 6.
Top : Estimate of local energy-cascade rate δ v ( k ) / τ nl ( k ) . Bottom : A blow-up of the region near the origin. In both panels, thedissipation wavenumber and the χ = B. Triple Correlations and Introduction of the Alfvén time
An improved estimate of local cascade rates may be con-structed based on timescales. We begin with the formulationof spectral transfer rate in terms of nonlinear time and the life-time of triple correlations as formulated by Kraichnan andsubsequently extended into a “golden rule”. Note thatthe triple correlations are so named since they involve tripleproducts of Elsasser-field components, and are responsible forinducing turbulent energy transfer across the inertial range.The essential statement, formulated for the isotropic case, isnisotropy and Timescales in 3D MHD Turbulence 7that the spectral transfer time may be defined by the relation τ sp ( k ) τ ( k ) = τ ( k ) , which is equivalent to Kraichnan’s ob-servation that in steady state the transfer rate ε ( k ) = τ ( k ) [ δ v ( k )] τ nl ( k ) (4)must be independent of scale. In this expression, the directproportionality on the lifetime of triple correlations τ is aphysical requirement, one that may be amply motivated byexamining the structure of moment hierarchies appearing inclosures such as the Eddy Damped Quasi-Normal MarkovianApproximation. Furthermore, the only other timescaleavailable on the right side of Equation (4) is the nonlinear time τ nl . It follows that the effective spectral transfer rate is τ − = τ / τ . (5)At this point we introduce anisotropic effects. While the non-linear time (assuming scale locality) is a function only of themagnitude | k | , there is a possibility to introduce on physi-cal grounds a directional dependence in τ . This introducesa wavevector directional dependence in the spectral transferrate 1 / τ sp .To proceed, we approximate the lifetime of the triplecorrelations. A standard approach is to assume that the totalrate of decay of the triple correlations is the sum of contri-butions from individual rates.
Here, the available ratesare those derived from nonlinear effects and the Alfvén prop-agation effect. Specifically, allowing for the directionality ofAlfvén propagation, we may write1 τ = τ A + τ nl , (6)which yields τ ( k ) = τ A ( k ) τ nl ( k ) τ A ( k ) + τ nl ( k ) , (7)where as before, τ A ( k ) = ( k · V A ) − = ( k z B ) − for Alfvénspeed V A . This leads to a locally defined anisotropic spectral-transfer rate (see Appendix A):1 τ sp ( k ) = τ ( k ) τ ( k ) = χ ( k ) + χ ( k ) τ nl ( k ) . (8)This way of writing the spectral transfer rate (in terms of thenonlinearity parameter χ ) makes it clear that the combination σ ( k ) = χ ( k ) + χ ( k ) (9)acts as a suppression factor which, when multiplying the non-linear rate 1 / τ nl , reduces the net transfer rate due to the Alfvénpropagation effect. We note that the suppression factor admitsno singularity for any value of applied magnetic field B , butinstead σ → B orfor the quasi-2D regions of the spectrum). The variation ofthe suppression factor in the wavector plane for our examplesimulation is shown in Figure 7. The above approximate formulation of the spectral trans-fer rate 1 / τ sp achieves the desired connection with basictimescales and permits contact with several existing theoret-ical frameworks; it is illustrated in Figure 8, computed fromthe standard simulation described above. It is evident that thevery strong values of the spectral transfer rate are concentratednear the 2D plane defined by k z =
0, consistent with the behav-ior of the suppression factor in that region. In particular, thischaracterization holds for values of perpendicular wavenum-ber that were not excited initially.
FIG. 7.
Top : Suppression factor for anisotropic spectral transfer σ = χ / ( χ + ) . At k z = χ → ∞ , and therefore we set σ equalto unity there. Bottom : A blow-up of the region near the origin. Inboth panels, the dissipation wavenumber and the χ = nisotropy and Timescales in 3D MHD Turbulence 8 FIG. 8.
Top : Anisotropic spectral transfer rate 1 / τ sp = τ ( k ) / τ ( k ) = χχ + τ nl , where τ sp is the spectral transfer time and τ is the triple correlation time (see text). At k z = χ → ∞ , andtherefore we set the factor χχ + equal to one. Bottom : A blow-up ofthe region near the origin. In both panels, the dissipation wavenum-ber and the χ = C. Estimates of local strength of energy transfer
The above anisotropic triple decay time is appropriate forintroduction into estimation of a modal rate of energy transfer,that is semi-local in wavevector space. This will have dimen-sion energy per unit mass per unit time. If constructed in aphysically reasonable way, this would provide a phenomeno-logical (non-rigorous) estimate of the contributions to the totalenergy transfer rate due to the energy residing near a wavevec-tor k . There are several ways to proceed to develop cascade rateestimates that incorporate anisotropic spectral transfer. Themost formal approaches are those based on adaptations ofthird-order laws. An example is Ref. 73, which assumeda two-component slab+2D model of turbulence geometry toestimate such rates. In principle, with sufficient data cover-age, it is also possible to apply the third-order cascade law forarbitrary geometry (see e.g., Refs. 74,75) , provided that thesystem is steady and homogeneous. Such approaches rely onrelatively delicate third-order statistics, and do not necessarilyreveal dependence on underlying physical timescales.In Appendix A, beginning from first principles and adopt-ing a minimal set of approximations, including scale lo-cality, we develop a justification for an estimation of cas-cade strength in an anisotropic inertial range. Two dis-tinct approaches are needed when there is a significant meanmagnetic-field, corresponding first to the case of non-resonanttransfer, and then to the case of resonant transfer.
In mak-ing this distinction it becomes relevant to distinguish two sep-arate roles of modes contributing to any particular triadic in-teraction: participating modes experience an exchange of en-ergy, while a spectator mode acts to drive this exchange whileits energy remains unchanged.
The case of non-resonant transfer, which will also be rele-vant when there is no substantial mean magnetic field, will bebased on the timescales discussed above and a generalizationof the argument leading to Equation (4). The correspondingspectral transfer estimate isˆ ε ( k ) non-res = δ v ( k ) τ sp ( k ) = τ ( k ) τ ( k ) δ v ( k ) = χ ( k ) + χ ( k ) δ v ( k ) τ nl ( k ) . (10)For resonant transfer the argument is modified so that thenonlinear rate, and the triple correlation time of the specta-tor mode is used in the approximation, rather than the corre-sponding timescales associated with the on-shell participatingmode. The spectator modes for resonant transfer are quasi-2D and have approximately zero wave frequency, so the sup-pression factor χ / ( + χ ) → k -shell, but only by the 2D energy near B · k ≈
0. Conse-quently we estimate the resonant energy transfer asˆ ε ( k ) res = δ v ( k ) τ nl2D ( k ) . (11)where τ nl2D ( k ) = [ k δ v ( k )] − is the quasi-2D on-shell non-linear timescale. Here δ v ( k ) is estimated as (cid:112) kE ( k ) where E is the contribution of 2D modes (defined by k z = k -shell (see Figure 13). We call atten-tion to the fact that for resonant transfer, the triple decay timeis equal to the nonlinear time τ ( k ) that drives the process. Inaddition, the resonant transfer is strictly in the perpendiculardirection (see Appendix A).To illustrate these two contributions to energy transfer, weemploy data from the standard simulation used above. First,we plot estimates of the associated nonresonant contributionin Figure 9 in the same planar cut through the wavevectornisotropy and Timescales in 3D MHD Turbulence 9 FIG. 9.
Top : Estimate of nonresonant energy transfer ˆ ε ( k ) non-res = δ v ( k ) / τ ( k ) = χ ( k ) + χ ( k ) δ v ( k ) τ nl ( k ) . At k z = χ → ∞ , and there-fore we set the factor χχ + equal to one. Bottom : A blow-up of theregion near the origin. In both panels, the dissipation wavenumberand the χ = space employed above. In Figure 11 we plot the estimatedresonant energy transfer using the same data and in the samewavevector plane.We observe in Figure 9 that the estimated nonresonantenergy transfer, like other quantities depicted above, isanisotropic with larger values extended in the perpendicu-lar direction. For example, the energy transfer estimate at k x =
100 and k z = k x = k z =
50. Furthermore, the contours of equal estimated energytransfer differ greatly from the χ = τ A < τ nl , and strongerparallel transfer for τ A > τ nl . As a general comment, it is dif-ficult to identify specific features of the estimated nonresonantenergy transfer that correspond to either the χ = FIG. 10. Horizontal slices along (parallel wavenumber) k z of the non-resonant spectral transfer rate of energy ˆ ε ( k ) non-res = χ / ( χ + ) × δ v ( k ) / τ nl , through the k z – k x plane shown in Figure 9, for differentvalues of (perpendicular wavenumber) k x . The locations where theHigdon and χ = ∗ ’and ‘ (cid:5) ’ symbols, respectively. Figure 10 shows profiles of the nonresonant energy-transferrate estimate along lines in the ( k x , k z ) plane. Each curve is fora fixed perpendicular wavenumber k x and is plotted as a func-tion of parallel wavenumber k z . The curves each show max-imum values at k z =
0. It is therefore clear that the strongestestimated spectral transfer is found along the 2D ( k z =
0) planefor all plotted values of perpendicular wavenumber, which arechosen to span the inertial range. In addition, each curve isannotated with a ‘ (cid:5) ’ symbol at the (pair of) values of parallelwavenumber corresponding to the value χ = ∗ ’ symbols are the inter-sections of each line with the Higdon curve (in dimensionlessterms, k z = ± k / x ). It is apparent that the values of estimatednonresonant energy-transfer at these special values of k z aresystematically lower than the estimated energy transfer on the2D axis. In fact, at k x = k x =
15 the energy transferat the 2D plane is approximately five times greater than thatfound at the intersection with χ =
1. At higher perpendic-ular wavenumbers the contrast is less dramatic, but one stillfinds about a factor of two larger estimated energy transfer atzero parallel wavenumber as compared to the position of unitnonlinearity parameter χ = FIG. 11.
Top : Estimate of resonant energy transfer ˆ ε ( k ) res = δ v ( k ) / τ nl2D ( k ) , where τ nl2D ( k ) = / [ k (cid:112) kE ( k )] . Only 2D modescontribute to nonlinear time on the entire k -shell (see text). Bottom :A blow-up of the region near the origin. In both panels, the dissi-pation wavenumber and the χ = To further identify the role of the nonlinearity parameter χ in organizing the turbulence, Figure 12 shows two distribu-tions of turbulence parameters over χ , including modes with0 ≤ k x ≤
200 and − ≤ k y , k z ≤ χ is formally infinite) weretreated as though their χ value is the same as for the nearby k z = χ that have equal width in log-space. The energy in the initially populated energy-containing FIG. 12.
Left:
Barplots show percentage of modal energy (left verti-cal axis) in bins of χ , for two values of high-pass cutoff wavenumber k c ; blue bars for k c > k c >
11. The binsare equally spaced in log-scale. Curves with ‘ ∗ ’ and ‘+’ symbolsshow cumulative percentage of energy (right vertical axis in browncolor) for the two cases. Arrows ( ↑ ) below the lower horizontal axismark the weighted mean of χ (see text). Arrows ( ↓ ) above the up-per horizontal axis mark the weighted median of χ (where the cu-mulative percentage of energy is 50%). Regions shaded pale green,brown, and pink represent regions of weak, moderate, and strongnonlinearity/turbulence, with the boundaries for the three regions setat χ ≤ . , . < χ ≤ , and χ >
2, respectively.
Right:
As above,with the modal energy replaced by the non-resonant spectral energytransfer rate ˆ ε non-res . Green axes at the bottom of each panel showvaues of the suppression factor σ corresponding to selected repre-sentative values of χ . modes is excluded by employing high-pass wavenumber cut-offs, to avoid biasing the distribution with the (isotropic) ini-tial data. Two values of the cutoff, k c > k c >
11, areemployed, and the results for each of these is shown. It is ap-parent that the distribution, while peaked at a value slightlyless than χ =
1, is also asymmetric and skewed towards largervalues. As indicated on the figure, 50% of the energy lies be-low (above) a value of χ that is very close to unity. However,nisotropy and Timescales in 3D MHD Turbulence 11due to the skewness of the distribution, the energy-weightedmean value of χ is approximately 2 and changes very littlefor the two high-pass wavenumber thresholds. This weightedmean is computed as (cid:104) χ (cid:105) = ∑ i χ i w i ∑ i w i , where χ i is the value of χ at the center of the i th bin and w i is the percentage of energyin the bin.A similar picture emerges on inspection of the right panelof Figure 12 that shows the distribution over χ of the localestimates of nonresonant energy transfer ˆ ε non-res , again for thesame two high-pass wavenumber cutoffs. Once more the dis-tribution is peaked near χ = χ in this case is a little greater than 2.This result is qualitatively similar to that obtained by Refs. 32and 34, with some possibly significant differences in the de-tails of the simulations and in the analysis. However, the dis-tributions shown here reveal quantitatively the energy contentand spectral transfer rates prevailing in modes with differentvalues of χ . The present diagnostics suggest that a wide rangeof values of nonlinearity parameter are realized in anisotropicMHD turbulence, with substantial contributions from valuesof χ well above unity. These results are consistent with thosefrom a run with isotropically polarized fluctations (see Ap-pendix B).As a final point we should emphasize that the treatment ofspectral transfer in this section has consisted of estimating ascalar transfer rate: contributions from each point in k -spaceto the total energy transfer due to activity at that point. Wehave not made an attempt to further model this as a directionalvector flux, as defined in Equation (A7) of Appendix A. Thevector flux model would be required to demonstrate steadyflux across scales by integrating the normal component over aclosed surface in wavevector space. See for an anisotropicMHD model of this type based on k -space diffusion. VI. DISCUSSION AND CONCLUSIONS
In this paper we have examined in some detail the relation-ship between spectral anisotropy and physical timescales ina computation of MHD turbulence. This is a topic that hasbeen extensively discussed in other studies (see the Refer-ences) from a variety of perspectives and often with a mo-tivation oriented towards establishing or promoting a partic-ular theoretical framework. Previous works on closely re-lated topics have often adopted approaches that we intention-ally avoid here. First, in the present work we examine thesimulation results in terms of simple estimates of timescalesand transfer rates, reporting the analysis with a minimum ofcritical commentary. This stands in strong contrast to anal-yses that adopt approximations based on assumptions abouttimescales in order to derive particular models of turbulence.Examples of this would include: Montgomery and Turner’sderivation of RMHD, in which dynamical timescales are as-sumed to be slow in comparison with Alfvén timescales; Goldreich and Sridhar’s critical balance theory, which as-serts that turbulence evolves such that Alfvén and nonlineartimescales remain nearly equal; and weak turbulence the- ory, which requires that Alfvén times be shorter than non-linear times. We make none of these approximations here.Second, as noted in earlier sections, we have not adopted themost rigorous available frameworks for quantifying spectraltransfer. For the problem at hand this would correspond to theseveral possible forms of the Kolmogorov–Yaglom–Politano–Pouquet third-order laws that are available for application toMHD simulation.
These are extremely powerful tools butthey do not provide direct information about the availablephysically-relevant timescales. Such information is containedin higher-order correlations. Finally, we remind the reader thatthe present results are based on analysis of a single snapshotfrom a simulation initialized with toroidally polarized fluctu-ations, and we have made no attempt to claim any form ofuniversality concerning the results.To confirm the robustness of our results we repeated ouranalyses for a simulation initialized with isotropic fluctuations(with toriodal + poloidal polarization), and obtained similarresults (see Appendix B). Therefore it is our impression thatthe results presented here are of a fairly typical character forthe class of initial data that we have examined. This corre-sponds, roughly speaking, to low cross-helicity, incompress-ible “Alfvénic” turbulence, and, within limitations of incom-pressible MHD, does not differ greatly from parameters oftenchosen to simulate solar-wind-like conditions at 1 AU (e.g.,Refs. 21,78,79).The main results of the paper are the estimates of theanisotropic distributions of estimated spectral transfer ratesand rates of energy transfer (shown in Figures 8–11), as wellas the distributions of energy and its transfer rate across thenonlinearity parameter χ and the suppression factor σ (shownin Figure 12). As mentioned above, these results pertain onlyto the specific case of homogeneous MHD turbulence with amean magnetic field of moderate strength. The results maybe summarized succinctly with the statement that the distri-butions of estimated transfer rates are relatively featureless.There are no strong enhancements in any part of k -space be-yond a local peak of nonlinear activity near the 2D axis, whichis globally perpendicular to the direction of the externally ap-plied field. This is broadly consistent with expectations basedon derivations of the RMHD model, and, at least super-ficially, appears to stand in contrast to certain interpretationsof critical balance that anticipate a special role for modes hav-ing equal nonlinear and Alfvén timescales. Further exami-nation of the relationship of nonlinear and Alfvén timescaleswas performed by looking at the distributions of energy andspectral energy transfer in bins of nonlinearity parameter χ = τ A / τ nl and the suppression factor σ = χ / ( + χ ) . The dis-tributions are peaked near χ = σ = . χ corresponding tostrong nonlinearity. The broad distributions of nonlinearityparameter admit numerous values and an average (weightedby energy and its spectral transfer rate) that exceeds unity. Itis therefore difficult to argue in favor of theoretical develop-ments, such as certain interpretations of critical balance, thatpostulate that χ = that analysis of anisotropyusing structure functions parallel and perpendicular to locally-computed average magnetic fields gives results with higheranisotropy. It is also established that this effect is sensitiveto phase coherence or intermittency, and disappears when thefields are “gaussianized”. This enhancement is therefore fun-damentally of higher order than classical second-order spec-tra (see Ref. 84). However, our interest in the present paperis in classical spectra (and underlying correlation functions)and these are only well defined in a fixed coordinate system inwhich the preferred reference direction is fixed.The interested reader might find additional useful infor-mation about effects that are studied using the local, ran-dom coordinate system in, e.g., Refs. 30,32,34. The issueof the appropriate mean fields to use when analyzing spec-tral anisotropy has received substantial attention (e.g., Refs.30,31,85–91). Clearly, there are both technical and concep-tual aspects to address regarding this issue and we defer fullerconsideration to future work.While we have attempted to avoid adopting bias towardsor against any specific theoretical approach, it seems clearthat the present results favor a somewhat simplified descrip-tion of anisotropy in MHD associated with a mean magneticfield. The idea inherent in the derivation of RMHD, namelythat the 2D or quasi-2D modes represent the core nonlinear-ities of a turbulent system, appears to be largely consistentwith this single detailed numerical experiment. Both resonantand nonresonant estimated transfer rates point toward main-tenance of anisotropic spectra with strong perpendicular real-space gradients relative to the (suppressed) parallel gradients.The critical-balance condition, χ =
1, appears to be better in-terpreted as an order-of-magnitude estimate of the extent ofthe spectra in the parallel direction, at least in this case wherethe measurements are made relative to a global magnetic field.In this way most of the realizable implications of critical bal-ance point to a dynamical description in terms of quasi-2Dor Reduced MHD. Such a description of anisotropic MHDturbulence is a simplification relative to the chain of reason-ing leading to critical balance models, and we await furtheranalyses that provide support for the present viewpoint, or al-ternatives. A more complete discussion of the role of criticalbalance ideas in MHD turbulence is under review (Oughton &Matthaeus 2020). Related future work could examine similardistributions of the timescales in high cross helicity simula-tions as well as in spacecraft observations.
ACKNOWLEDGMENTS
We thank R. Bandyopadhyay for useful discussions. Thisresearch was supported in part by NASA Heliospheric Sup-porting Research grants NNX17AB79G, 80NSSC18K1210,and 80NSSC18K1648. The data that support the findings ofthis study are available from the corresponding author uponreasonable request.
Appendix A: Spectral Transfer and Scale Locality
Here we develop some background to motivate our ap-proach to estimating semi-local contributions in wavevectorspace to spectral transfer in anisotropic MHD turbulence. Thespectral density of energy S ( k ) evolves in incompressibleMHD according to an equation of the same formal structure asthe analogous equation for the spectral density in incompress-ible hydrodynamics. Note that S ( k ) is defined as the trace ofthe energy spectrum tensor, which in turn is defined in termsof the Fourier transform of the two-point correlation functionof the turbulent fields (see, e.g., Refs. 1,92). Specifically, theenergy spectrum evolves as (e.g., Chapter 6 of Ref. 93) ∂ S ( k ) ∂ t = N ( k ) − D ( k ) , (A1)where N ( k ) = (cid:82) d p d q T ( k , p , q ) is the nonlinear term repre-senting the net effect of all triadic interactions on the energydensity at wavevector k , T is the Fourier-space triple correla-tion, and D is the dissipation term. Each term in Equation (A1)is time dependent.Energy conservation in the absence of dissipation corre-sponds to the property that (cid:90) d k N ( k ) = (cid:90) d k d p d q T ( k , p , q ) = . (A2)We have neglected possible sources at very low | k | ∼ / L ,while in the usual way the dissipation is assumed to be ef-fective only for very large kL . Then there exists an inertialrange in which energy is conserved by nonlinearities, even asit is transferred from scale to scale. The net dissipation is ε = (cid:82) d k D ( k ) and in steady state this is equal to the transferrate across any sphere of radius k lying in the inertial range.We gain insight by integrating the equation for the spectraldensity Equation (A1) over all | k | < k ∗ , and defining E < ( k ∗ ) = (cid:90) k ∗ d q (cid:90) q d Ω q S ( q ) and E > ( k ∗ ) = (cid:90) ∞ k ∗ d q (cid:90) q d Ω q S ( q ) , (A3)where the integral is expressed in spherical polar coordinatesand d Ω = sin θ d θ d φ is the differential solid angle. Ignoringdissipation, it is clear thatd E < d t + d E > d t = . (A4)In solving Equation (A1) we are confronted with the classi-cal closure problem of turbulence. The second-order quantity S depends on the third-order correlation T . Further develop-ment would show that the time evolution of T depends onfourth-order correlations, and so on. To proceed, various ap-proximations can be made to solve for, or bring closure to, thishierarchy (e.g., Ref. 7).The principal complication lies in the nonlinear term de-scribed in Equation (A2), which, through the definition ofthe triple correlation T , depends on convolutions of the(schematic) form N ( k ) ∼ (cid:90) d p d q δ ( p + q − k ) v ( k ) v ( p ) v ( q ) . (A5)nisotropy and Timescales in 3D MHD Turbulence 13 FIG. 13. Diagram of a 2D cut through k -space, indicating locality oftriadic interactions on (near) a shell of radius k . Wavevectors for thequasi-2D modes lying on this shell terminate in the region shadedin red. Here δ ( p + q − k ) is a Dirac delta function, and so only triadswith p + q = k contribute to the integral. The associated tri-adic interactions have been studied and classified in a numberof studies. The development of models for the third-order correlations T is a principal goal of turbulence theory. Rigorous treatmentsare difficult, and usually remain inexact (e.g., Refs. 94,98). Auseful approach, adopted here, is to justify and adopt phe-nomenological models for the nonlinear effects, which in-cludes the physics of the cascade, and appropriate approxi-mations.To guide our reasoning, we refer to Figure 13, whichschematically shows a 2D cut (in the k (cid:107) - k ⊥ plane) through3D wavevector space. A set of wavevectors corresponding toa triadic interaction in the inertial range is illustrated. In eachtriadic interaction in incompressible MHD, there are specta-tor modes, and exchange modes. The spectator modes (say, q in Figure 13) induce energy transfer between the exchangemodes, while the spectator mode energy remains unchanged.The spectator wavevector therefore indicates the (unsigned)direction in which a particular triad induces transfer. Thisproperty is general and is useful for classifying differenttypes of triads. Note that because the net spectral transfer into S ( k ) is an integral over two sets of wavevectors, q and p , thetriad depicted in Figure 13 actually corresponds to two triads:one where q is the spectator mode and another where p is thespectator mode.The idea of locality in scale is a powerful assumption inKolmogorov theory that is supported, even if inexactly, in nu-merical experiment and theory. The basic idea isthat the net transfer from wavenumbers < k to wavenumbers > k is dominated by triadic interactions involving wavevectorsthat do not differ (in magnitude) greatly from k itself. Thiscan be visualized in Figure 13 as consisting of cases in whichthe interactions occur among k , p , and q , all having approx- imately the same magnitude, k ∼ p ∼ q . That is, all threeparticipating wavevectors, or perhaps two of these (“Modifiedlocality”; see Ref. 35), lie on or near the shell with radius k .The locality property stands in contrast to the general caseof triadic interactions (Equations (A2) and (A5)) that may de-pend on spectral amplitudes that are greatly distant both fromthe direction of k or from the shell with | k | = k . Those thatare distant are nonlocal interactions. Ignoring the nonlocaleffects and adopting the approximation of scale locality, onemay interpret the inertial range energy-balance Equation (A4)as the conservative exchange of energy across the boundary atwavenumber k . In that case, inertial range conservation maybe expressed as N ( k ) = ∇ k · F ( k ) . (A6)With this approximation and definition, both locality and con-servation are guaranteed in the inertial range. Note that the vector flux F admits functional dependence on correlationsat wavevectors other than k , but by locality the dependenceis dominated by wavevectors of similar magnitude. Then, interms of the flux, the time rate of change of energy within asphere of radius k isd E < d t ( k ) = (cid:90) q < k d q ∇ q · F ( q ) = k (cid:90) d Ω k F k ( θ k , φ k ) (A7)where the spherical polar representations of the wavevectors k = ( k , θ k , φ k ) and q = ( q , θ q , φ q ) are employed.The approximations leading to the expression Equation(A7) did not include the assumption of isotropy. Neverthelesswe see that the rate of transfer of energy to all wavevectors q having | q | > k is determined by the conditions on the shell | q | = k . This is crucial for motivating the phenomenologicalestimates of spectral transfer in the main text. In particular,a model of local spectral transfer will specify the surface flux F k = k k · F k and requires consideration only of the physicalparameters in the shell with radius k . However, it is clear thatphysical insight is required to understand how F k will dependon contributions from different parts of the shell.In this regard it is also possible to examine types of triads,allowing several classes of interaction to be identified. Additional considerations include the vector polarization ofthe fluctuations, and the comparison of the scale-local turbu-lence amplitude to the mean field strength. An important fac-tor that enters the modeling of triadic interactions is whetherinteractions are resonant interactions or nonresonant inter-actions, when turbulence occurs in the presence of a suffi-ciently strong externally supported uniform or very large scalemagnetic field. Most interacting triads are affected by this(effectively) DC magnetic field, in that the strength of thetransfer they induce is suppressed, as described by the factor σ = χ / ( + χ ) in Equation (10). However, there is a partic-ular class of triads that are unaffected by the large-scale fieldand are resonant interactions; that is the set of triads that in-clude one (or all three) participating wavevector that lies in theplane perpendicular to the field B . Such modes have “zerofrequency” and the triads that involve these “2D” modes ex-change energy but they do so without changing the associatednisotropy and Timescales in 3D MHD Turbulence 14Alfvénic frequency of the affected fluctuations. These corre-spond to resonant interactions in an iterative weak turbulenceapproach. In designing a phenomenological descriptionof scale-local transfer it is necessary to properly distinguishthe frequency changing non-resonant interactions from thefrequency-preserving resonant interactions. Appendix B: Results from an Initially Isotropic Run
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