As a matter of tension -- kinetic energy spectra in MHD turbulence
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As a matter of tension – kinetic energy spectra in MHD turbulence
Philipp Grete, Brian W. O’Shea,
1, 2, 3 and Kris Beckwith Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA Department of Computational Mathematics, Science and Engineering, Michigan State University, East Lansing, MI 48824, USA National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, MI 48824, USA Sandia National Laboratories, Albuquerque, NM 87185-1189, USA
ABSTRACTMagnetized turbulence is ubiquitous in many astrophysical and terrestrial systems but no complete,uncontested theory even in the simplest form, magnetohydrodynamics (MHD), exists. Many theoriesand phenomenologies focus on the joint (kinetic and magnetic) energy fluxes and spectra. We highlightthe importance of treating kinetic and magnetic energies separately to shed light on MHD turbulencedynamics. We conduct an implicit large eddy simulation of subsonic, super-Alfv´enic MHD turbulenceand analyze the scale-wise energy transfer over time. Our key finding is that the kinetic energyspectrum develops a scaling of approximately k − / in the stationary regime as the kinetic energycascade is suppressed by magnetic tension. This motivates a reevaluation of existing MHD turbulencetheories with respect to a more differentiated modeling of the energy fluxes. Keywords:
MHD — methods: numerical — turbulence INTRODUCTIONWhile our understanding of incompressible hydrody-namic turbulence has significantly advanced over thepast decades, many critical questions in the realm ofcompressible magnetohydrodynamic (MHD) turbulenceremain unanswered. This regime is of particular inter-est in both astrophysics and in terrestrial systems whereprocesses on a huge variety of scales are either governedor at least influenced by MHD turbulence. Astrophysicalexamples include energy transport in the solar convec-tion zone (Canuto & Christensen-Dalsgaard 1998; Mi-esch 2005), angular momentum transport and energyrelease in accretion disks (Balbus & Hawley 1998), thecore collapse supernova mechanism (Couch & Ott 2015;M¨osta et al. 2015), the interstellar medium with its star-forming molecular clouds (V´azquez-Semadeni 2015; Fal-garone et al. 2015; Klessen & Glover 2016), and clustersof galaxies that can be used to determine cosmologicalparameters (Brunetti & Jones 2015; Br¨uggen & Vazza2015). In the terrestrial context, this is of interest for arange of plasma experiments, such as laser-produced col-liding plasma flows, Z-pinches, and tokamaks (see, e.g.,
Corresponding author: Philipp [email protected].
Tzeferacos et al. 2018; Haines 2011; Mazzucato et al.2009; Ren et al. 2013).At the same time, MHD turbulence theory and phe-nomenology also made significant progress from earlyisotropic models (Iroshnikov 1964; Kraichnan 1965),to critically balanced turbulence (Sridhar & Goldreich1994; Goldreich & Sridhar 1995), to dynamic alignment(Boldyrev et al. 2009), but it is still a highly debatedtopic – see, e.g., Galtier (2016); Beresnyak (2019) forrecent reviews. Different theories make a variety of pre-dictions for the scaling of the energy spectra dependingon the strength of the mean magnetic magnetic field(either external or local), on the cross helicity (bal-anced versus unbalanced turbulence), and on the mag-netic helicity (encoding the topology of the magneticfield configuration). In the majority of cases scalingpredictions are only concerned with the total energyspectrum ( E ( k ) = E kin ( k ) + E mag ( k ) with wavenum-ber k ) and assume a moderate or strong backgroundfield so that dynamics are differentiated between par-allel and perpendicular to the mean field. Thus, thereis no differentiation between the kinetic ( E kin ( k )) andmagnetic ( E mag ( k )) energy spectra. A complemen-tary theoretical approach to modeling magnetohydro-dynamic turbulence is the use of shell models, which area computationally-inexpensive semi-analytical means ofmodeling turbulence. Notable examples of this include a r X i v : . [ phy s i c s . p l a s m - ph ] S e p Grete, O’Shea & Beckwith
Biskamp (1994); Frick & Sokoloff (1998); Plunian &Stepanov (2007) who also observe, for example, flatterspectra, spectral breaks, and different scaling behaviorof the kinetic and magnetic energy spectra. However,the behavior strongly depends on the characteristics ofthe system being modeled (with, e.g., properties of thesystem such as a mean magnetic field, helicity and crosshelicity contributing significantly to the observed out-comes similarly to the locality of the interactions con-sidered). By contrast to predictions from analytic andsemi-analytic modeling efforts, numerous computationalstudies of magnetized turbulence have reported different scaling behavior of kinetic and magnetic energy spectra(Haugen et al. 2004; Moll et al. 2011; Teaca et al. 2011;Eyink et al. 2013; Porter et al. 2015; Grete et al. 2017;Bian & Aluie 2019) and, perhaps even more importantly,different scaling behavior of kinetic and magnetic energyspectra has been reported in observations of the solarwind (Boldyrev et al. 2011).In order to gain a deeper insight into this discrep-ancy, we present and analyze the evolution and station-ary state of the kinetic and magnetic energy spectra andfluxes separately in the context of an implicit large eddysimulation of ideal MHD turbulence in its simplest con-figuration (vanishing mean field, cross-helicity, and mag-netic helicity). We confirm prior results (Haugen et al.2004; Moll et al. 2011; Teaca et al. 2011; Eyink et al.2013; Porter et al. 2015; Grete et al. 2017; Bian & Aluie2019) that the kinetic and magnetic energy spectra ex-hibit different scaling behavior. In particular, we findthat the kinetic energy spectrum exhibits a scaling closeto k − / – i.e., it is shallower than the spectra predictedin the theories above, which mostly range between k − / and k − / . We further demonstrate, using a shell-to-shell energy transfer analysis, that this “shallow” ki-netic energy spectrum is associated with magnetic ten-sion, which acts to suppress the kinetic energy cascadeand provides the major contribution in the energy fluxfrom large to small scales. This result is in markedcontrast with incompressible hydrodynamic turbulence,where the kinetic energy cascade is the only means oftransferring energy between scales in a self-similar fash-ion (which in turn leads to the emergence of the k − / scaling) and departures from this expected scaling inhydrodynamic turbulence simulations and experimentshave been associated with the existence of “bottlenecks”(Falkovich 1994; Schmidt et al. 2006; Frisch et al. 2008;Donzis & Sreenivasan 2010; K¨uchler et al. 2019; Agrawalet al. 2020). As such, the results presented in this workdemonstrate the rich physics phenomenology that canoperate even in the simplest scenarios (vanishing meanfield, cross-helicity, and magnetic helicity) where mag- netic tension is dynamically important and further serveto highlight the necessary ingredients that MHD turbu-lence theory and phenomenology should incorporate inorder to explain scalings of kinetic and magnetic energyobserved in both simulation and observation of magne-tized turbulence.The rest of this paper is structured as follows. In Sec-tion 2 we introduce the simulation setup and summarizethe energy transfer analysis. In Section 3 we present thekinetic and magnetic energy spectra, their temporal evo-lution, and scale dependent energy dynamics. Finally, inSection 4, we summarize our results, the limitations ofthe simulations upon which they are based and discussthe implications for both modeling of magnetohydrody-namic turbulence and astrophysical systems. METHOD2.1.
Simulation setup
We use the open source code,
K-Athena (Greteet al. 2021), a performance portable implementationof
Athena++ (Stone et al. 2020) based on
Kokkos (Ed-wards et al. 2014), to solve the ideal MHD equations .The second-order finite volume scheme employed is com-prised of a Van-Leer integrator, constrained transportMHD algorithm, piecewise-linear reconstruction, andRoe Riemann solver, (see Stone & Gardiner 2009, formore details on the numerical method). Given that noexplicit physical dissipative terms are present dissipationis purely numerical; as such the simulations presentedhere utilize the implicit large eddy simulation (ILES)technique (Grinstein et al. 2007). Turbulent driving isaccomplished through a stochastic forcing approach de-scribed by (Schmidt et al. 2009), implemented within K-Athena using a communication-avoiding algorithm forefficient large scale parallel simulations on GPUs.We conduct a single simulation of a cubic domainwith side length of 1 (if not noted otherwise all unitsare in code units) and periodic boundary conditions ona 2 , grid. The plasma is initially at rest (veloc-ity u = 0) with uniform density ( ρ = 1) and thermalpressure ( p th = 1). The initial magnetic field configura-tion ( B = ∇ × A with A = (0 , , r − r ) for r < r with r = (cid:113) ( x − . + ( y − . ) is a cylinder in thez-direction with radius r = 0 . (cid:104) E mag (cid:105) = 0 . K-Athena is available at https://gitlab.com/pgrete/kathena.Commit e5faee49 was used to run the simulation and the pa-rameter file ( athinput.fmturb ) is contained in the supplementalmaterial for this paper. inetic energy spectra in MHD turbulence β p = 800. The plasma is kept approximatelyisothermal using an adiabatic equation of state with adi-abatic index of γ = 1 . k = 2, where k isthe normalized wavenumber). The driving field is purelysolenoidal and has an autocorrelation time of 1.0 so thatno artificial compressible modes are injected (Grete et al.2018). In the stationary regime the integral length is L = (cid:82) E kin ( k ) /k d k/ (cid:82) E kin ( k )d k = 0 .
32 (i.e., slightlysmaller than the forcing scale at 0.5), the root meansquare (RMS) sonic Mach number is M s = 0 .
54, theresulting large eddy turnover time is T = L/ (M s c s ) =0 .
59, the RMS Alfv`enic Mach number is M a = 2 .
8, andthe mean plasma beta is β p = 54.2.2. Energy transfer analysis
For a detailed analysis of the energy dynamics we ap-ply the shell-to-shell energy transfer analysis presentedin Grete et al. (2017), which is an extension of Alexakiset al. (2005) to the compressible regime. The key ideais to separate energy transfers by their source (some en-ergy budget at some spatial scale Q ), sink (some budgetat some scale K ), and a mediator. Given the isother-mal nature of the simulation, we focus on the kineticand magnetic energy budget only and neglect a detailedanalysis of the internal energy budget (cf. Schmidt &Grete 2019) or non-isothermal statistics (Grete et al.2020).In general, the energy transfers are given by T XY ( Q, K ) with X , Y ∈ { U , B } (1)expressing energy transfer (for T >
0) from shell Q ofenergy budget X to shell K of energy budget Y. U andB represent the kinetic and magnetic energy budgets,respectively. More specifically, the energy transfers are T UU ( Q, K ) = − (cid:90) w K · ( u · ∇ ) w Q + 12 w K · w Q ∇ · u d x (2) T BB ( Q, K ) = − (cid:90) B K · ( u · ∇ ) B Q + 12 B K · B Q ∇ · u d x (3)for kinetic-to-kinetic (and magnetic-to-magnetic) trans-fers via advection and compression, T BUT ( Q, K ) = (cid:90) w K · ( v A · ∇ ) B Q d x (4) T UBT ( Q, K ) = (cid:90) B K · ∇ · (cid:0) v A ⊗ w Q (cid:1) d x (5) for magnetic-to-kinetic (and kinetic-to-magnetic) energytransfer via magnetic tension, and T BUP ( Q, K ) = − (cid:90) w K √ ρ · ∇ (cid:0) B · B Q (cid:1) d x (6) T UBP ( Q, K ) = − (cid:90) B K · B ∇ · (cid:18) w Q √ ρ (cid:19) d x (7)for magnetic-to-kinetic (and kinetic-to-magnetic) energytransfer via magnetic pressure. Here, w = √ ρ u is amass-weighted velocity chosen so that the spectral ki-netic energy density based on w is a positive definitequantity (Kida & Orszag 1990) and v A is the Alfv´envelocity.The velocity w K and magnetic field B K in a shell K(or Q) are obtained by a sharp spectral filter in Fourierspace with logarithmic spacing. The bounds are givenby 1 and 2 n/ for n ∈ {− , , , . . . , } . Shells (up-percase, e.g., K) and wavenumbers (lowercase, e.g., k )obey a direct mapping, i.e., K = 24 corresponds to k ∈ (22 . , . RESULTS3.1.
Emergence of a power law in E kin ( k )In MHD turbulence simulations (independent of nu-merical method such as pseudospectral DNS, higher-order finite difference, or finite volume ILES) without astrong mean field ( B (cid:28) (cid:104) u (cid:105) RMS ) and magnetic Prandtlnumber Pm ≈ E mag ( k ) > E kin ( k ) onall scales smaller than the forcing scales. Second, thekinetic energy spectrum develops a power law regime onthe magnetically dominated scales with a slope close to-4/3, i.e., shallower than the Kolmogorov slope of -5/3.In order to understand the emergence of a flatter-than-Kolmogorov slope, indicative of a less efficient energycascade, we present a single simulation in more detail inthe following sections.3.2. Time evolution of the energy power spectra
Figure 2 illustrates the evolution of the mean magneticand kinetic energies and their ratio over time. First, the
Grete, O’Shea & Beckwith wavenumber k c o m p . k / E k i n ( k ) , k / E m a g ( k ) Bian & Aluie 2019this workPorter et al. 2015Eyink et al. 2013Haugen et al. 2004
Figure 1.
Kinetic (solid) and magnetic (dashed) energyspectra reported in literature from simulations with variousnumerical schemes, compensated by k / : pseudo-spectralDNS of incompressible MHD with hyperdissipation(Bian &Aluie 2019, Fig. 10), ILES of compressible, ideal MHD(Porter et al. 2015, Fig. 3) similar to this work, pseudo-spectral DNS of incompressible MHD (Eyink et al. 2013,Fig. 2), and higher-order finite difference DNS of compress-ible MHD with hyperdissipation (Haugen et al. 2004, Fig. 7).All spectra have in common that magnetic energy dominatesall scales smaller than the forcing scale and that the kineticenergy spectrum exhibits a region with scaling close to k / .(Lines are vertically offset for increased readability.) time t [T] t A t B t SE kin E mag E mag / E kin Figure 2.
Temporal evolution of mean magnetic energy(orange dash-dotted line), mean kinetic energy (blue solid),and their ratio (green dashed). The shaded gray area indi-cates the stationary regime. Specific times t A (peak kineticenergy), t B (nonlinear dynamo), and t S (stationary) corre-spond to snapshots that are analyzed in more detail. mean kinetic energy reaches its peak value at time t A . E k i n ( k ) , E m a g ( k ) t = t A E kin ( k ) E mag ( k ) E k i n ( k ) , E m a g ( k ) t = t B E kin ( k ) E mag ( k ) wavenumber k E k i n ( k ) , E m a g ( k ) t = t S E kin ( k ) E mag ( k ) Figure 3.
Kinetic (blue solid) and magnetic (orange dash-dotted) energy spectra at different times. The inset shows8 < k <
64 compensated by k / and illustrates the flatten-ing of the kinetic energy spectrum. The thin lines in eachpanel illustrate the stationary state (bottom panel) for refer-ence. The gray area at 22 . < k ≤ . (cid:98) = K = 24) indicatesthe scale that is used in the more detailed energy transferanalysis in Section 3.3. The corresponding spectra (top panel in Figure 3) showthat the kinetic energy on small scales ( k (cid:38)
32) is lowerthan the stationary value (indicated by the thin blacklines), whereas the kinetic energy on large scales is abovethe stationary value. The magnetic energy spectrumcrosses the kinetic energy spectrum at k eq ≈
24 (where E kin ( k eq ) ≈ E mag ( k eq )) so that the magnetic field be-come dynamically relevant on small scales.At time t B , which corresponds to the nonlinear phaseof the dynamo, the kinetic energy on small scales k (cid:38) A movie of the temporal evolution of the energy spectra(
Grete et al-spectra evol.mp4 ) is available in the supplemen-tal material. inetic energy spectra in MHD turbulence k ≈
24 withsteeper slope on large scales and a shallower slope onsmall scales. The crossover of magnetic and kinetic en-ergy has shifted towards larger scales and now occursaround k eq ≈ ≈ T with E mag saturating at ≈ . E kin . In the sta-tionary regime (represented by t = t S ) the crossover hasshifted to the largest scales k eq ≈ (cid:46) k (cid:46)
64 witha shallower-than-Kolmogorov slope close to ≈ − / Energy dynamics
In absence of explicit dissipation (and, thus, the ex-plicit mean dissipation rate), all energy transfer ratesare normalized using the mean total cross-scale flux at k = 26 . Magnetic tension
The role of magnetic tension in shaping the kinetic en-ergy spectrum becomes apparent in Figure 4. It showsthe net rate of change in kinetic energy (top row) andmagnetic energy (bottom) row for the different media-tors over time and for the reference shell K = 24, i.e., ∂ t E XYkin ( K ) = (cid:88) Q T XY ( Q, K ) and (8) ∂ t E XYmag ( K ) = (cid:88) Q T XY ( Q, K ) (9)with XY ∈ {
UU,BUT,BUP } for the kinetic energy andXY ∈ { BB,UBT,UBP } for the magnetic energy. In otherwords, this is the net rate of change in energy at somescale K from all other scales Q.While at time t A the kinetic cascade ( T UU ) is stillcontributing to a net increase of kinetic energy on thosescale, the rate of change by magnetic tension T BUT isnegative, i.e., removing kinetic energy from K = 24.The net effect remains positive. At t B the dynamicshave changed. The kinetic cascade still contributes witha growth in kinetic energy, but magnetic tension nowdominates so that the net effect is a removal of kinetic t E k i n ( K = ) t E UUkin t E BUTkin t E BUPkin
Sum0 2 4 6 8 10 12 time t [T] t E m a g ( K = ) t A t B t S t E BBmag t E UBTmag t E UBPmag
Sum
Figure 4.
Net rate of change in kinetic (top) and magnetic(bottom) energy at k = 24 over time. Blue lines indicateenergy transfer through advection, orange through magnetictension, and green through magnetic pressures. The pressuredilatation and forcing term are not shown as their contribu-tion is negligible. energy from those scales. This transfer of energy fromthe kinetic to the magnetic budget through magnetictensions causes the flattening of the kinetic energy spec-trum.In the stationary regime the net rate of change in bothkinetic and magnetic energy fluctuates around 0 (other-wise the regime should not be considered stationary).This balance is only maintained through energy trans-fers between kinetic and magnetic energy budgets. Onaverage, the kinetic and magnetic cascades remove en-ergy from intermediate scales of their respective budgets(blue lines are negative) and this deficit is filled throughtransfers mediated by magnetic tension between budgets(orange lines).The importance of magnetic tension is similarly ob-served in the cross-scale energy fluxes. These fluxes are Grete, O’Shea & Beckwith C r o ss - s c a l e f l u x a t k = . U < U > U < ,TB > U < ,PB > Sum0 2 4 6 8 10 12 time t [T] C r o ss - s c a l e f l u x a t k = . t A t B t S t A t B t S B < B > B < ,TU > B < ,PU > Sum
Figure 5.
Cross-scale energy transfer across k = 26 . k < .
9) to the small scale ( k > .
9) kinetic budget (top)and magnetic budget (bottom), respectively. obtained from the individual transport terms viaΠ U < U > ( k ) = (cid:88) Q ≤ k (cid:88) K>k T UU ( Q, K ) , (10)Π U < , TU > ( k ) = (cid:88) Q ≤ k (cid:88) K>k T UBT ( Q, K ) , (11)Π U < , PU > ( k ) = (cid:88) Q ≤ k (cid:88) K>k T UBP ( Q, K ) , (12)(13)for energy being transferred from the kinetic energy onall scales ≤ k to to the kinetic and magnetic energies onscales smaller than k by advection, magnetic tension,and magnetic pressure, respectively. The same notationapplies to transfers from the large scale magnetic energywith U and B indices exchanged.Figure 5 illustrates the energy flux across k = 26 . B < B > at t B and then remains at aconstant value, it already peaks for Π U < U > at t = t A andafterwards declines again to 0. Transfers via magnetic tension (orange lines) from both large kinetic and mag-netic scales steadily growth till t = t B . Similar to the ad-vection transfers, Π B < , TU > remains constant after the peakwhereas Π U < , TB > declines with the key difference that thedecline is not to 0 but to a non-zero value. Moreover, itis the only remaining contribution for the kinetic energycross-scale flux (at that scale) and, overall, the domi-nating cross-scale flux is marginally ( ≈ U < U > , which is the only cross-scale flux in incompressiblehydrodynamics, is completely suppressed here and thecross-scale energy transfer transfer from the kinetic en-ergy budget is solely mediated by magnetic tension.3.3.2. Large scale energy conversion wavenumber k e n e r g y t r a n s f e r s Total cross-scale fluxCross-scale flux from E kin : U < U > + U < ,TB > + U < ,PB > Cross-scale flux from E mag : B < B > + B < ,TU > + B < ,PU > Cum. large scale E kin -to- E mag conversion: UB ( k ) Figure 6.
Cross-scale energy transfer across k from the ki-netic budget (orange) and magnetic budget (green), and cu-mulative energy conversion from kinetic to magnetic energyon scales larger than k in the stationary regime. While cross-scale fluxes allow for intra- (via advec-tion) and inter-budget (via magnetic tension and pres-sure) transfers, only the latter contributes to a conver-sion of energy. Figure 6 illustrates the net cross-scalefluxes versus scale in the stationary regime along withthe cumulative large scale kinetic to magnetic energyconversion. The cumulative large scale conversion refersto the net energy transfer between those two budgets onall scales larger than the reference scale k , C UB ( k ) = (cid:88) Q,K ≤ k T UBT ( Q, K ) + T UBP ( Q, K ) , (14)where the magnetic pressure contribution is negligible inthe simulation presented here. The cumulative energyconversion tightly follows the the cross-scale flux from inetic energy spectra in MHD turbulence k ≈ k ≈
30) the contributioncontinuously grows while the kinetic energy cross-scaleflux contribution decreases. Eventually, the kinetic andmagnetic cross-scale fluxes become approximately thesame strength. Similarly, the cumulative energy con-version reaches a constant value. This implies that nosignificant net energy conversion occurs on intermediateand small scales and is in agreement with Bian & Aluie(2019), who show that mean field line stretching is apredominantly large-scale process. SUMMARY, DISCUSSION & CONCLUSIONSMotivated by an apparent discrepancy between kineticand magnetic energy spectra scalings measured in simu-lations (Haugen et al. 2004; Moll et al. 2011; Teaca et al.2011; Eyink et al. 2013; Porter et al. 2015; Grete et al.2017; Bian & Aluie 2019) and observations of the solarwind (Boldyrev et al. 2011) with expectations derivedfrom analytic theory (Galtier 2016; Beresnyak 2019), wepresented shell-to-shell energy transfer analysis of an im-plicit large eddy simulation of approximately isothermal,subsonic, super-Alfv´enic MHD turbulence with vanish-ing background magnetic field, cross-helicity, and mag-netic helicity. In the context of this analysis, we findthat magnetic tension suppresses the kinetic energy cas-cade resulting in a spectrum that is shallower than pre-dicted in various theories, e.g., k − / (Iroshnikov 1964;Kraichnan 1965). Overall, the results presented heredemonstrate that the energy flux across scales is domi-nated by magnetic tension, and similarly the scale localenergy balance in the stationary turbulence regime ismaintained by a constant energy transfer between thekinetic and magnetic reservoir mediated by magnetictension.The simulations on which the results are based, arenecessarily limited. While a clear signature of an ex-tended range with a scaling close to k − / is observedin the kinetic energy power spectrum, no such range isobserved in the magnetic energy power spectrum (seeFigure 1). We attribute this to a combination of thesimulation setup as well as a limited dynamical range.More specifically, the mechanical energy injection on thelargest scales provides a barrier for the large scale mag-netic field growth in the absence of a significant (ex-ternal) mean field. As a result, the magnetic field isstrongest on intermediate scales and gets weaker towardslarger scales. Similarly, in the limit of large Reynoldsnumbers we expect the ratio of E mag ( k ) /E kin ( k ) to grow from the smallest (non-dissipative) scales towards largerscales until the growth is inhibited by the forcing actingon the largest scales. This also explains why extendedscaling ranges are regularly observed in reduced MHDsimulations or in simulation with a significant mean field(potentially stronger than the velocity field on the forc-ing scales) where it, figuratively, provides a large scaleanchor, see (Beresnyak 2019).In this study, we focus on simulations with magneticPrandtl numbers of Pm (cid:39) ≈ > Re (i.e., Pm > < (cid:29) (cid:28) Grete, O’Shea & Beckwith the MHD approximation (e.g., adding anisotropic termsas per the Braginskii approximation, Braginskii 1965),it is likely that a kinetic or hybrid fluid/kinetic approx-imation will be required for some physical regimes.The key finding of this work is that magnetic ten-sion acts to suppress cross-scale kinetic energy transfer,resulting in a kinetic spectrum with a slope k − / , incontrast with theoretical expectations regarding incom-pressible hydrodynamic turbulence. Such a suppressionof cross-scale kinetic energy transfer is also observedin simulations of hydrodynamics turbulence, where the“bottleneck effect” (a pileup of energy on the small-est scales) results in shallower than k − / scaling inthe kinetic energy spectrum in hydrodynamic turbulence(Falkovich 1994; Schmidt et al. 2006; Frisch et al. 2008;Donzis & Sreenivasan 2010; K¨uchler et al. 2019; Agrawalet al. 2020). Recently, Gong et al. (2020) also attributedthe hydrodynamic bottleneck effect to the shallow ki-netic energy spectra they observe in their MHD simula-tions.The results presented here demonstrate that, contraryto Gong et al. (2020), the physical mechanism for theshallow slope of the kinetic energy spectrum is fun-damentally different between hydrodynamics and mag-netohydrodynamics due to magnetic tension (which isnaturally absent in hydrodynamics) causing a suppres-sion of the kinetic cascade cross-scale flux. In addition,the results presented here suggest that the kinetic cas-cade is practically absent instead of being decoupled(from a magnetic cascade), as was recently suggestedby Bian & Aluie (2019). While we still observe a sig-nificant energy flux in the magnetic energy cascade, thebalance in the kinetic energy budget is maintained bymagnetic tension. Thus, both energy budgets remaincoupled through dynamically significant energy fluxes.Note, compared to the vastly extended dynamical rangein Bian & Aluie (2019) (which comes from the useof higher-order hyperdisspative terms), the dynamicalrange in the simulation presented here is rather limited.Future simulations with a larger dynamical range willhelp to address this question.With these caveats in mind, the results presented herehave a number of implications. First, they motivatea reevaluation of MHD turbulence theories that com-monly are only concerned with the total (kinetic andmagnetic) energy spectrum and energy flux. In partic-ular, the results presented here suggest that flux-basedmodels should differentiate between the intra- and inter-budget cross-scale fluxes, and consider energy budgetsseparately. We note that the scaling of the total en-ergy will be dominated by the magnetic energy scalingon intermediate scales, which is important in the light of MHD turbulence theory on scaling relations. Second,in the interpretation of observations and their derivedspectra special care is required in inferring propertiesfrom one spectra to the other, as we see no indicationthat that kinetic and magnetic energy spectra follow thesame scaling laws, (cf., also Boldyrev et al. 2011). Third,subgrid-scale modeling in the context of large eddy sim-ulations (Miesch et al. 2015; Grete et al. 2016) may be-come simpler as, for example, one can neglect a purelykinetic cross-scale flux. Finally, we note that in natu-ral systems the effective large scale driving mechanisms,e.g., a galaxy cluster merger (Subramanian et al. 2006),provides an outer scale and limit for the amplificationof magnetic fields by the fluctuation dynamo.Finally, we note that our results should also be inter-preted with care and not be overgeneralized. As men-tioned in the Introduction, the configuration space ofMHD turbulence is vast and the results presented herecover only a single point. Additional data from (evenlarger-scale) simulations, observations, and experimentsis required in order to get a complete picture of MHDturbulence. ACKNOWLEDGMENTSThe authors thank Jim Stone and Ellen Zweibelfor useful discussions. PG and BWO acknowledgefunding by NASA Astrophysics Theory Program grant inetic energy spectra in MHD turbulence Software:
K-Athena (Grete et al. 2021), a perfor-mance portable version of
Athena++ (Stone et al. 2020) using
Kokkos (Edwards et al. 2014).
Matplotlib (Hunter2007).
NumPy (van der Walt et al. 2011). mpi4py (Dalcnet al. 2005). mpi4py-fft (Dalcin et al. 2019).REFERENCES