Violation of the zeroth law of turbulence in space plasmas
Romain Meyrand, Jonathan Squire, Alexander A. Schekochihin, William Dorland
UUnder consideration for publication in J. Plasma Phys. Violation of the zeroth law of turbulence inspace plasmas
R. Meyrand , † , J. Squire , A. A. Schekochihin , , and W. Dorland Department of Physics, University of Otago, 730 Cumberland St., Dunedin 9016, NewZealand The Rudolf Peierls Centre for Theoretical Physics, University of Oxford, ClarendonLaboratory, Parks Road, Oxford, OX1 3PU, UK Merton College, Merton Street, Oxford OX1 4JD, UK Department of Physics, University of Maryland, College Park, MD 20742, USA(Received xx; revised xx; accepted xx)
The zeroth law of turbulence states that, for fixed energy input into large-scale motions,the statistical steady state of a turbulent system is independent of microphysical dissipa-tion properties. This behavior, which is fundamental to nearly all fluid-like systems fromindustrial processes to galaxies, occurs because nonlinear processes generate smaller andsmaller scales in the flow, until the dissipation—no matter how small—can thermalizethe energy input. Using direct numerical simulations and theoretical arguments, we showthat in strongly magnetized plasma turbulence such as that recently observed by theParker Solar Probe (PSP) spacecraft, the zeroth law is routinely violated. Namely, whensuch turbulence is “imbalanced”—when the large-scale energy input is dominated byAlfv´enic perturbations propagating in one direction (the most common situation in spaceplasmas)—nonlinear conservation laws imply the existence of a “barrier” at scales nearthe ion gyroradius. This causes energy to build up over time at large scales. The resultingmagnetic-energy spectra bear a strong resemblance to those observed in situ, exhibitinga sharp, steep kinetic transition range above and around the ion-Larmor scale, withflattening at yet smaller scales, thus resolving the decade-long puzzle of the position andvariability of ion-kinetic spectral breaks in plasma turbulence. The “barrier” effect alsosuggests that how a plasma is forced at large scales (the imbalance) may have a crucialinfluence on thermodynamic properties such as the ion-to-electron heating ratio.
1. Introduction
In his celebrated 1850 manuscript “The Mechanical Equivalent of Heat” (Joule 1850),James Prescott Joule described how a liquid stirred by a falling mass would heat upby a well-defined, fixed amount, thus demonstrating the equivalence of mechanical workand heat. Though less well appreciated, Joule’s study also revealed another, similarlyintriguing law of nature: his series of experiments, which measured the work done bydifferent masses on either water or mercury, showed that the damping rate of the liquid’skinetic energy must be proportional to its velocity, rather than its viscosity. This isdespite the fact that it is the viscosity that is ultimately responsible for the conversionof work to heat. The general principle, which has subsequently become known as the“Zeroth Law of Turbulence,” states that the dissipation rate of a turbulent flow underfixed large-scale conditions is independent of the value or mechanism of the microphysicalenergy dissipation (e.g., the viscosity). This distinctive property arises because turbulence † Email address for correspondence: [email protected] a r X i v : . [ phy s i c s . s p ace - ph ] D ec R. Meyrand and others nonlinearly generates motions at successively smaller scales, always reaching the scalewhere viscous effects become large, no matter how small the viscosity itself.Collisionless plasmas, although far more complex than water and mercury as used inJoule’s experiments, are generally assumed to satisfy the zeroth law. Energy injectedinto smooth, large-scale fluctuations in position and velocity space (phase space)—for example, Alfv´enic perturbations emitted from the Sun’s corona—must make itsway (linearly or nonlinearly) towards small scales before it can be converted to heat.If this is not possible—if the zeroth law is violated—the injected energy will not beefficiently thermalized, instead building up over time in large-scale motions and mag-netic fields. An inability of the system to transfer energy to small scales thus has adramatic impact on the large-scale behavior of the plasma. In this paper, we arguethat, counter to the assumptions of much previous work, the zeroth law can be violatedin magnetized (Alfv´enic) plasma turbulence such as that observed in the solar wind.The effect, which occurs when the turbulence is “imbalanced” (i.e., when the energiesof forward and backward propagating fluctuations differ), arises because both energyand a “generalized helicity” are nonlinearly conserved in strongly magnetized (low-beta)collisionless plasmas. At scales above the ion gyroradius ρ i , the generalized helicity is themagnetohydrodynamic (MHD) cross-helicity and naturally undergoes a forward cascade(nonlinear energy transfer to small scales); at scales below ρ i , the generalized helicitybecomes magnetic helicity and naturally undergoes an inverse cascade (nonlinear transferto larger scales). The collision of the two cascades creates a “helicity barrier”: it stopsthe system from dissipating injected energy through nonlinear transfer to smaller spatialscales.The resulting turbulence, which we illustrate in Figs. 1 and 2, bears a strong resem-blance to recent measurements from the Parker Solar Probe (PSP) spacecraft and others.While balanced turbulence shows the expected transition from Alfv´enic to kinetic-Alfv´en-wave (KAW) turbulence at ρ i scales (Howes et al. et al. ∼ k − / spectrum (Boldyrev 2006) above the transition. † At yet smaller scales, a ∼ k − . KAWspectrum (Schekochihin et al. et al. et al. et al. a ). In our theory, thespectral break occurs around the scale at which the helicity barrier halts the energyflux, and this barrier moves to larger scales as the outer-scale energy grows with time.Final saturation, which occurs only after many Alfv´en crossing times and depends onsimulation resolution, relies on fluctuations reaching large amplitudes and dissipatingthrough nonuniversal (and, in our simulations, artificial) means. This suggests thatobserved turbulent cascades in the solar wind are often not in a saturated state where † The range in which it is observed here is not wide enough to distinguish between k − / (Maron & Goldreich 2001; Boldyrev 2006; Perez et al. k − / (Goldreich & Sridhar1995; Beresnyak 2014), but this RMHD-range scaling is not the point of this work. We willcompare to k − / where necessary because it is well motivated in balanced turbulence andsupported by observations (Chen et al. ρ i ρ i imbalanced balanced ρ i ρ i imbalanced balanced Figure 1.
The spatial structure of the perpendicular electron flow u ⊥ , or equivalently, theperpendicular electric field E ⊥ = −∇ ⊥ ϕ . We compare imbalanced and balanced turbulence inthe left and right panels, respectively. Top panels show a parallel ( x, z ) slice ( B = B ˆ z left toright), bottom panels show a perpendicular ( x, y ) slice ( B out of the page). The dramaticdependence on imbalance arises because imbalanced turbulence is afflicted by the “helicitybarrier”: at a nonuniversal scale k ∗⊥ ρ i (cid:46) E + ) cannot proceed to smaller scales, violating the zeroth law of turbulence. The resulting sharpbreak in the spectrum is shown in Fig. 2, and is followed by the re-emergence of a cascade atyet smaller scales (see zoomed region of left-hand panel). These simulations have a resolution of2048 and are initialized by refining the 256 simulations of Figs. 4–6, starting at t ≈ τ A . energy input balances dissipation. It also explains the observed non-universality of thebreak scale and of the sub-break spectral scaling.
2. Theoretical Framework
Before continuing, we define the following symbols, with α signifying species (eitherions, α = i , or electrons, α = e ): n α is the background density; T α is the backgroundtemperature and τ = T i /T e ; B is the magnetic field, with B = B ˆ z the background; β α = 8 πn α T s /B is the ratio of thermal to magnetic energy; m α is the particle mass; q α is the particle charge with q e = − e and q i = Ze ; Ω α = | q α | B /m α c is the gyroradius; ρ α = c √ m α T α / | q α | B is the gyroradius; d α = ρ α / √ β α is the skin depth; c is the speedof light; and v A = B / √ πn i m i is the Alfv´en speed.In order to elucidate the key physical processes involved in this highly complex problem,our approach is to use the simplest plasma model that meets two important requirements:(i) it can be formally (asymptotically) derived in a physically relevant limit, whichallows us to evaluate critically the plasma regimes in which our results remain valid; R. Meyrand and others
Figure 2.
Energy spectra for the simulations pictured in Fig. 1. Purple and orange linesshow imbalanced and balanced turbulence, respectively, while solid and dashed lines show thedominant ( E + ) and subdominant ( E − ) energies, respectively [see Eq. (2.7)]. Thin lines showthe spectra of the 256 imbalanced simulation at the same time and parameters (see Fig. 5),emphasizing the re-emergence of a kinetic-Alfv´en-wave cascade ( ∼ k − . ) at small scales inimbalanced turbulence, if a sufficient range of scales is available. The resulting double-kinkedspectrum strongly resembles those observed in the solar wind (Sahraoui et al. et al. a ). As far as we know, this is the first time such spectra have been reproduced in a numericalsimulation. and (ii) it remains valid for perpendicular scales both above and below the ρ i scale,which is clearly a necessity for a study of the ion-kinetic transition. The minimal modelof Finite-Larmor-Radius MHD (FLR-MHD) described below meets these requirements(Passot et al. et al. β e (cid:28) d e and ρ e . Because ρ e (cid:28) d e at β e (cid:28) d e /ρ i = ( m e /m i ) / / √ β i in a neutral plasma, so long as β i > m e /m i and β i ∼ β e , FLR-MHD provides a valid description of the ion-kinetic transition. The low- β assumption is well satisfied in many astrophysical and space plasmas, including in thesolar corona and the near-Sun solar wind (Bruno & Carbone 2013).2.1. FLR-MHD Model
FLR-MHD can be self-consistently derived from the Vlasov equation, starting withthe assumptions that all fields (the magnetic field, flow velocity etc.) vary slowly in timecompared to the ion-cyclotron frequency, that there is a strong background magneticfield, and that the correlation length l (cid:107) of a perturbation in the field-parallel directionis much larger than its field-perpendicular correlation length l ⊥ (Schekochihin et al. β e ∼ β i (cid:28) et al. et al. l ⊥ (cid:29) ρ i ) cannot directly heat ions (within the low-frequencyapproximation), allowing the formulation of a simple closed set of fluid equations (i.e.,equations in 3-D space) to describe the Alfv´enic component of the turbulence both aboveand below the ρ i scale. † These are the FLR-MHD equations. We note that the assumption l ⊥ (cid:28) l (cid:107) , which is well tested in the solar wind (Chen 2016), is satisfied in standardmagnetized plasma turbulence phenomenologies (Goldreich & Sridhar 1995; Boldyrev2006; Schekochihin 2020). The key idea is that of a “critical balance” between linear andnonlinear times at all scales, which leads to the estimate l (cid:107) ∼ l / ⊥ (cid:29) l ⊥ (Boldyrev 2006;Mallet & Schekochihin 2017). At electron-skin-depth scales ( l ⊥ ∼ d e ) where the magneticfield is no longer frozen into the electron flow, FLR-MHD breaks down due to couplingto the electron distribution function. Although a model exists to capture this transitionaccurately (Zocco & Schekochihin 2011), its additional complexity is unnecessary fordescribing the ion-kinetic transition of interest here. We thus focus on scales above d e ,which also implies β i > m e /m i so that d e < ρ i .The FLR-MHD equations are (cid:18) ∂∂t + u ⊥ · ∇ ⊥ (cid:19) δn e n e = − c πen e (cid:18) ∂∂z + b ⊥ · ∇ ⊥ (cid:19) ∇ ⊥ A (cid:107) , (2.1) (cid:18) ∂∂t + u ⊥ · ∇ ⊥ (cid:19) A (cid:107) = − c ∂ϕ∂z + cT e e (cid:18) ∂∂z + b ⊥ · ∇ ⊥ (cid:19) δn e n e , (2.2) δn e n e = − Zτ (cid:16) − ˆ Γ (cid:17) eϕT e , (2.3)where δn e /n e = δn i /n i is the perturbed electron (and, by quasi-neutrality, ion) density, A (cid:107) is the ˆ z component of the vector potential, ϕ is the electrostatic potential, u ⊥ = c B − ˆ z × ∇ ⊥ ϕ is the perpendicular E × B (electron) flow, and b ⊥ = − B − ˆ z × ∇ ⊥ A (cid:107) is the perturbation of the magnetic field’s direction. The gyrokinetic Poisson operator1 − ˆ Γ = 1 − I ( α ) e − α , with α = − ρ i ∇ ⊥ / I the modified Bessel function, becomes1 − ˆ Γ ≈ − ρ i ∇ ⊥ / k ⊥ ρ i (cid:28)
1, and 1 − ˆ Γ ≈ k ⊥ ρ i (cid:29)
1. In the former limit, the FLR-MHD system becomes the well-known ReducedMHD (RMHD) model (Strauss 1976), in the latter it becomes the Electron RMHD model(Schekochihin et al. et al.
Imbalanced Alfv´enic Turbulence
A linearization of Eqs. (2.1)–(2.3), assuming a sinusoidal spatial dependence withwavenumber k = k ⊥ ˆ x + k z ˆ z , yields forward and backward propagating modes of fre-quency ω = ± k z v ph ( k ⊥ ) v A , where v ph ( k ⊥ ) = k ⊥ ρ i √ (cid:18) − ˆ Γ + Zτ (cid:19) / ≈ k ⊥ ρ i (cid:28) , (cid:18)
12 + Z τ (cid:19) / k ⊥ ρ i k ⊥ ρ i (cid:29) . (2.4)FLR-MHD thus recovers shear-Alfv´en waves when k ⊥ ρ i (cid:28) β ) kinetic Alfv´enwaves (KAWs) when k ⊥ ρ i (cid:29)
1. The eigenfunctions of these linear modes, known as thegeneralized Els¨asser potentials, will provide a useful basis for intuitive discussion of the † Compressive fluctuations, which cascade passively to ρ i scales, where they likely heat theions through nonlinear phase mixing (Meyrand et al. et al. ρ i scales by changing the relationship between δn e and ϕ (although theycannot exchange energy with Alfv´enic fluctuations; Schekochihin et al. R. Meyrand and others nonlinear problem and turbulence. At wavenumber k , these are Θ ± k = − Ω i v ph ( k ⊥ ) k ⊥ δn e n e ∓ A (cid:107) √ πm i n i . (2.5)At large scales k ⊥ ρ i (cid:28)
1, they have the property ˆ z ×∇ ⊥ Θ ± = Z ± = u ⊥ ± B ⊥ / √ πm i n i ,where Z ± are the Elsasser variables (Elsasser 1950).The utility of Θ ± arises from the fact that at large scales (i.e., in the RMHD limit),nonlinear interaction—and thus the turbulent cascade—requires the interaction between Z + and Z − (equivalently, Θ + and Θ − ). Thus, the difference in amplitude of Z + and Z − , which is known as the energy imbalance and is determined by the outer-scale forcingof the plasma, has a strong influence on the properties of the turbulent cascade. We willquantify it in the standard way with σ c = (cid:82) d x ( | Z + | − | Z − | ) (cid:82) d x ( | Z + | + | Z − | ) , (2.6)so σ c = ± Z − = 0 or Z + = 0. Although imbalanced RMHD turbulence remainspoorly understood (Perez & Boldyrev 2009; Chandran 2008; Beresnyak & Lazarian 2009;Lithwick et al. | σ c | (cid:38) . et al. k ⊥ ρ i (cid:29) Θ + with Θ + ). This implies thatthe two components can exchange energy and that a turbulent cascade is, in principle,possible with just one component Θ ± (Cho 2011; Kim & Cho 2015; Voitenko & De Keyser2016). 2.3. The “Helicity Barrier”
Here we argue that the conservation properties of FLR-MHD imply that a turbulentflux of energy cannot proceed in the usual way to small scales (where it needs to get to bedissipated). We term the barrier in the cascade at scales l ⊥ ∼ ρ i , the “helicity barrier.”2.3.1. Conservation laws of FLR-MHD
The FLR-MHD system has two nonlinearly conserved quadratic invariants, (free)energy and (generalized) helicity. These are most easily and clearly written in termsof the generalized Elsasser variables. The free energy is E = 14 (cid:88) k (cid:0) | k ⊥ Θ + k | + | k ⊥ Θ − k | (cid:1) , (2.7)which reduces to E ≈ (cid:90) d x V ( | Z + | + | Z − | ) = 12 (cid:90) d x V (cid:18) | u ⊥ | + | B ⊥ | πn i m i (cid:19) (2.8)at large scales. The generalized helicity is H = 14 (cid:88) k | k ⊥ Θ + k | − | k ⊥ Θ − k | v ph ( k ⊥ ) , (2.9)which reduces to the MHD cross-helicity at k ⊥ ρ i (cid:28) H ∝ (cid:82) d x u ⊥ · B ⊥ , and becomesmagnetic helicity at k ⊥ ρ i (cid:29) H ∝ (cid:82) d x δB (cid:107) A (cid:107) . † If the k ⊥ ρ i (cid:28) σ c ≈ H /E . We also define the Θ ± “energies,” E ± = (cid:80) k | k ⊥ Θ ± k | /
4, along with perpendicular spectra for E , H , and E ± ,denoted E ( k ⊥ ), E H ( k ⊥ ), and E ± ( k ⊥ ), respectively.2.3.2. The inevitability of the helicity barrier
Consider the case where energy and helicity are injected at large scales at the rates ε and ε H , respectively, with injection imbalance σ ε ≡ | ε H | /ε . The conservation laws abovetell us that in a statistical steady state, there must be a nonzero energy flux Π ( k ⊥ ) andhelicity flux Π H ( k ⊥ ) to small scales where they can be dissipated. If we further assumethat (i) energy transfer due to nonlinearity is significant only for modes with similarscales (locality), and (ii) parallel dissipation is small because eddies are highly elongatedalong the magnetic field, then Π ( k ⊥ ) and Π H ( k ⊥ ) must be constant between the forcingand dissipation scales. In the following argument, based on Alexakis & Biferale (2018), weassume such a constant-flux solution and find a contradiction, suggesting that this typeof solution is not possible in FLR-MHD when σ ε (cid:54) = 0. Fundamentally, the contradictionarises because at large scales H is the RMHD cross-helicity, which undergoes a forwardcascade, while at small scales H is magnetic helicity, which undergoes an inverse cascade(Schekochihin et al. et al. et al. Π ( k ⊥ ) (cid:39) ε (cid:39) ε diss ⊥ , Π H ( k ⊥ ) (cid:39) ε H (cid:39) ε diss H , ⊥ , where ε diss ⊥ = ν n (cid:80) k ⊥ k n ⊥ E ( k ⊥ ) and ε diss H , ⊥ = ν n (cid:80) k ⊥ k n ⊥ E H ( k ⊥ ) are the energyand helicity dissipation rates (we assume hyper-viscous dissipation of δn e and A (cid:107) of theform ν n k n ⊥ ). This solution satisfies the following inequalities: | Π H ( k ⊥ ) | (cid:39) ν n (cid:12)(cid:12)(cid:12) ∞ (cid:88) p ⊥ = k ⊥ p n ⊥ E H ( p ⊥ ) (cid:12)(cid:12)(cid:12) (cid:54) ν n v − ( k ⊥ ) (cid:12)(cid:12)(cid:12) ∞ (cid:88) p ⊥ = k ⊥ p n ⊥ v ph ( p ⊥ ) E H ( p ⊥ ) (cid:12)(cid:12)(cid:12) (cid:54) v − ( k ⊥ ) ν n ∞ (cid:88) p ⊥ = k ⊥ p n ⊥ E ( p ⊥ ) (cid:39) v − ( k ⊥ ) Π ( k ⊥ ) , (2.10)where we have used the fact that v ph ( k ⊥ ) is a monotonically increasing function of k ⊥ , aswell as the inequality v ph ( k ⊥ ) | E H ( k ⊥ ) | (cid:54) E ( k ⊥ ) from Eqs. (2.7)–(2.9). The ratio of fluxes | Π H ( k ⊥ ) | /Π ( k ⊥ ) (cid:39) σ ε must thus satisfy σ ε (cid:54) /v ph ( k ⊥ ) for all k ⊥ above the dissipationscales. But 1 /v ph ( k ⊥ ) decreases with k ⊥ to arbitrarily small values ( v ph ∝ k ⊥ at k ⊥ ρ i (cid:29) k ⊥ violate the inequality (2.10). ‡ In such a case, theconstant-flux solution fails, indicating that the system is unable to thermalize energy andhelicity input through small-scale dissipation. We further see that the failure occurs onlybelow the scale where 1 /v ph ( k ⊥ ) (cid:39) σ ε ; this is around k ⊥ ρ i (cid:39) σ ε ≈ . σ ε . This highlights an interesting difference comparedto the well-known inverse energy cascade of two-dimensional hydrodynamics (Fjørtoft † Here δB (cid:107) is the magnetic-field strength perturbation; δB (cid:107) ∝ δn e for k ⊥ ρ i (cid:29) et al. ‡ It is worth commenting briefly on the recent work of Milanese et al. (2020), which hasconsidered a similar system with a conserved energy and generalized helicity. In that system,the function 1 /v ph ( k ⊥ ) in the generalized helicity increases with k ⊥ at large k ⊥ , which isthe opposite of our Eq. (2.9). This leads to the phenomenon of “dynamic phase alignment”,whereby fluctuations become increasingly correlated at small scales, reducing the strength oftheir nonlinear interaction so as to maintain constant fluxes of energy and helicity. R. Meyrand and others
Figure 3.
Colored points show the saturation energy E sat versus parallel hyper-dissipation ν z for 5 FLR-MHD simulations with N ⊥ = 64, N z (cid:54) σ ε = 0 .
88, and ρ i = 0 . L . EquivalentRMHD simulations ( ρ i = 0) are shown with black points. The dependence of E sat on ν z demonstrates that the helicity barrier causes the violation of the zeroth law of turbulence. Theinset shows the time evolution of the energy in each case (colors match those of the points). ρ i ) scales beforeit hits the barrier. As a consequence, despite FLR effects not influencing directly thenonlinear interactions at MHD scales, they could strongly influence turbulence statisticsat those scales by insulating them from the dissipation scales.
3. Numerical Experiments
The argument above suggests that it is not possible to have a constant flux of bothenergy and helicity through the ion-kinetic transition scale. It does not, however, elucidatehow the system behaves in the presence of continuous imbalanced injection of energy atlarge scales. For this, we turn to numerical simulations.3.1.
Numerical setup
We solve Eqs. (2.1)–(2.3) using a modified version of the pseudospectral code TURBO(Teaca et al. L ⊥ = L z = L with N ⊥ × N z Fourier modes. Athird-order modified Williamson (1980) algorithm is used for time stepping. We addhyper-diffusion operators of the form ν ⊥ ∇ ⊥ + ν z ∇ z to the right-hand side of Eqs. (2.1)–(2.2); these are necessary to absorb energy at small spatial scales without significantlymodifying dynamics at larger scales. This form of the dissipation is not designed toapproximate a specific physical process, and the values of ν ⊥ and ν z are chosen based onthe numerical resolution, ensuring that the energy spectrum falls off sufficiently rapidlybefore the resolution cutoff. Fluctuations are stirred at large scales by added forcingterms ( f n e and f A (cid:107) ) in Eqs. (2.1)–(2.2). This forcing is confined to 0 < k ⊥ (cid:54) π/L and | k z | = 2 π/L and takes the form of negative damping ( f n e and f A (cid:107) proportionalto the large-scale modes of n e and A (cid:107) ); this method allows the level of energy andhelicity injection ( ε and ε H ) to be controlled exactly, while producing sufficiently chaoticmotions to generate turbulence. While σ ε = ε H /ε is thus fixed, the imbalance σ c ≈ H /E is determined by the turbulence and evolves in time. Initial conditions are random andlarge-scale with energy E = 10 ετ A , where τ A = L z /v A is the Alfv´en crossing time. Theperpendicular and parallel energy dissipation rates are ε diss ⊥ = ν ⊥ (cid:80) k ⊥ ,k z k ⊥ E ( k ⊥ , k z ) E (a) σ ε = 0 .
88 (RMHD) σ ε = 0 . σ ε = 0 . t/τ A . . . ε d i ss ⊥ / ε , ε d i ss z / ε ε diss = ε ε − (b) 10 k ⊥ L ⊥ / π k z L z / π (c) k ⊥ ρ i = k ∗⊥ ∝ k / ⊥ (c) − − ε diss /max ( ε diss ) Figure 4.
Energy and dissipation properties from a set of simulations at resolution N ⊥ = N z = 256. Panel (a) compares the time evolution of energy in imbalanced FLR-MHD( σ ε = 0 . ρ i = 0 . L ) to balanced FLR-MHD ( σ ε = 0, ρ i = 0 . L ) and imbalanced RMHD( σ ε = 0 . ρ i = 0). The stars indicate the time from which the higher-resolution simulations ofFigs. 1–2 were initialized. Panel (b) shows ε diss ⊥ (solid lines) and ε diss z (dotted lines) for each case,to show that saturation is reached through parallel dissipation (unlike in balanced turbulenceand in imbalanced RMHD). The black-dashed line shows 2 ε − = ε − ε H . Panel (c) shows the( k ⊥ , k z ) dissipation spectrum in the saturated state of imbalanced FLR-MHD, illustrating thatdissipation occurs primarily at the perpendicular break scale ( k ∗⊥ ρ i (cid:39) .
15) at high k z . and ε diss z = ν z (cid:80) k ⊥ ,k z k z E ( k ⊥ , k z ), where E ( k ⊥ , k z ) is the 2-D energy spectrum (insteady state, if it exists, we would have ε = ε diss = ε diss ⊥ + ε diss z ). Simulations are run acrossa range of resolutions up to N ⊥ = N z = 2048. For the highest-resolution cases, we use arecursive refinement procedure, restarting a lower-resolution case at twice the resolutionand running until ε diss ⊥ converges in time; this dramatically reduces the computationalcost to enable otherwise unaffordable simulations. All simulations use Z = 1 and τ = 0 . ρ i ).3.2. Results
Figure 3 illustrates the turbulent-energy saturation amplitude E sat in simulationsforced with nonzero injection imbalance, σ ε = 0 .
88. We compare FLR-MHD with ρ i = 0 . L ⊥ (colored points) to RMHD ( ρ i = 0; black points), varying the parallel hyper-dissipation ν z . The difference is obvious: FLR effects cause the turbulence to saturateat much larger amplitudes, which increase with decreasing dissipation; larger amplitudesare associated with longer saturation times (see inset; Miloshevich et al. N ⊥ = 64, 64 (cid:54) N z (cid:54)
256 chosen as appropriate foreach ν z ), ρ i lies only slightly above the scale where perpendicular dissipation becomesimportant; the presence or absence of FLR effects at small scales thus strongly impactsthe large scales. Second, the zeroth law is violated with respect to the parallel dissipation,0 R. Meyrand and others despite the fact that parallel dissipation is generally neglected in magnetized turbulencetheories because the increasing elongation of eddies at smaller scales usually implies ε diss ⊥ (cid:29) ε diss z . Consequently, in order to saturate, the system finds it must access small-scale parallel physics, thus escaping the ordering assumptions ( l (cid:107) (cid:29) l ⊥ ) used to derivethe FLR-MHD model. This suggests that detailed properties of this saturated state arenot relevant to real physical systems.These unusual characteristics, which are due to the formation of a helicity barrier inimbalanced FLR-MHD, motivate a more thorough exploration of the turbulence prop-erties. Below and in Figs. 1–2, we present detailed results from much-higher-resolutionsimulations to help explain the effect of the helicity barrier and its potential relevanceto space plasmas. We compare imbalanced FLR-MHD to an equally imbalanced RMHDsimulation and balanced FLR-MHD, all at the same ε . To aid discussion, we break thetime evolution into three phases: first, a transient phase during which small-scale motionsare produced from the initial conditions; next, a pseudo-stationary phase , which is thelong phase of slow energy growth (seen in the inset of Fig. 3) that occurs due to thehelicity barrier; and finally, saturation, when ε ≈ ε diss ⊥ + ε diss z and ∂ t E ≈
0. During thepseudo-stationary phase and saturation, the helicity barrier creates a sharp break in thespectrum at a wavenumber that we will denote k ∗⊥ .3.2.1. The effect of the helicity barrier
Figures 4–6 show the time evolution of the energy, dissipation ε diss ⊥ , (cid:107) , energy spectra E ± ( k ⊥ ), and total energy flux Π ( k ⊥ ), comparing imbalanced FLR-MHD at σ ε = 0 .
88 and ρ i = 0 . L ⊥ with balanced FLR-MHD ( σ ε = 0, ρ i = 0 . L ⊥ ) and imbalanced RMHD( σ ε = 0 . ρ i = 0). These simulations, which have a resolution of N ⊥ = N z = 256,are used as low-resolution seeds (starting at t ≈ τ A ) for the recursive resolutionrefinement allowing us to reach N ⊥ = N z = 2048 in Fig. 2 (the full time evolution isonly computationally accessible at modest resolution). Let us first describe the balancedFLR-MHD and imbalanced RMHD cases in order to highlight the effect of the helicitybarrier.The balanced FLR-MHD simulation reaches saturation after a transient phase lastingseveral τ A , exhibiting a ∼ k − / ⊥ spectrum at large scales (Fig. 5) and constant fluxof energy to small scales (Fig. 6) where it is dissipated with ε diss ⊥ (cid:29) ε diss z (Fig. 4b).While the transition to KAW turbulence ( ∼ k − . ⊥ ) at k ⊥ ρ i (cid:39) N ⊥ = 256, it is clearly visible in the N ⊥ = 2048 spectrum inFig. 2. The imbalanced RMHD simulation is similar, although it is slower to saturate,reaching steady state by τ A (cid:39)
40, with a ∼ k − / ⊥ spectrum in E + and E − (Fig. 5)and energy fluxes to small perpendicular scales (not shown). The larger saturated energyarises because the cascade time τ cas is larger in imbalanced turbulence due to its slowernonlinear interactions (Chandran 2008; Lithwick et al. E sat ∼ ετ cas islarger with fixed ε . As E grows, the parallel outer scale l (cid:107) decreases due to criticalbalance ( l (cid:107) ∼ L ⊥ v A /E / ), which causes a modest parallel dissipation ( ε diss z (cid:39) . ε diss )observed in RMHD for the chosen parameters (Fig. 4b). This disappears at either lower ε and/or higher resolution.Imbalanced FLR-MHD turbulence is markedly different from both its balanced coun-terpart and imbalanced RMHD turbulence. As noted above, the latter is especiallyremarkable because FLR-MHD is identical to RMHD at k ⊥ ρ i (cid:28)
1, and ρ i (vertical linein Fig. 5) lies only slightly above the resolution cutoff (where perpendicular dissipationdominates) at these parameters. After an initial transient phase ( t (cid:46) τ A ) when itsevolution is similar to RMHD, the system forms a sharp spectral break at k ∗⊥ , and the1 Figure 5.
Time evolution of the spectra, E ± ( k ⊥ ), for the simulations shown in Fig. 4, comparingimbalanced FLR-MHD (top panel), balanced FLR-MHD (middle panel), and imbalanced RMHD(bottom panel). Individual spectra are shown at times spaced by t = 0 . τ A , as indicated by thecolor. While the spectrum converges rapidly in balanced FLR-MHD and imbalanced RMHDturbulence, the spectra of imbalanced FLR-MHD turbulence continue to evolve until t (cid:39) τ A ,with the break continuously moving to larger scales. pseudo-stationary phase begins. During this phase, the outer-scale energy in E + ( k ⊥ )grows in time, while the spectral break, which lies near k ∗⊥ ρ i (cid:39) Π ( k ⊥ ) is confined to increasingly large scales(broadly matching k ∗⊥ , shown with colored lines), as well as fluctuating wildly comparedto balanced turbulence. Clearly, Π (cid:28) ε for k ⊥ > k ∗⊥ , which explains the continualincrease in E with time during this phase. The subdominant mode’s spectrum E − ( k ⊥ )behaves quite differently to E + ( k ⊥ ), undergoing a modest decrease at earlier times andsaturating well before E + . This implies that the energy imbalance σ c increases with timeduring the pseudo-stationary phase. Interestingly, the E − cascade appears agnostic tothe break in E + ( k ⊥ ) and proceeds to small perpendicular scales. This is consistent withthe observation (Fig. 4b) that the saturated perpendicular energy dissipation approaches ε diss ⊥ ≈ ε − = ε (1 − σ ε ) in the pseudo-stationary phase (a result that has been confirmedat higher resolution and at other σ ε ). This suggests a form of “flux pinning,” wherebythe energy flux to small scales is determined by the requirement of a near-balanced KAWcascade (as seen in Fig. 2), which thus avoids the problems associated with the inversecascade of helicity. The amplitude of this cascade appears to be limited by the availabilityof Θ − fluctuations arriving from the inertial range.The saturation mechanism in imbalanced FLR-MHD is fundamentally different to thebalanced case or to imbalanced RMHD turbulence, because Π ( k ⊥ ) at k ⊥ (cid:38) k ∗⊥ remains2 R. Meyrand and others
Figure 6.
Time evolution of the normalized energy flux Π ( k ⊥ ) /ε for the simulations of Figs. 4–5,comparing imbalanced FLR-MHD (top panel) and balanced FLR-MHD (bottom panel). Thecoloring is the same as in Fig. 5. While balanced FLR-MHD turbulence shows the expectednear-constant flux to small scales (where it is dissipated), imbalanced FLR-MHD turbulence ischaracterized by wild fluctuations in Π (note different ordinate scale and the position of the greyline at Π = ε ), which, with time, are increasingly confined to large scales. The time-dependentwavenumber of the break ( k ∗⊥ ) is shown with the colored vertical lines. The small flux Π (cid:28) ε at smaller scales provides direct evidence for the existence of the helicity barrier. limited to (cid:39) ε − , no matter what the turbulence amplitude. Saturation finally occurs—at t ≈ τ A with energy imbalance reaching σ c ≈ .
999 for the FLR-MHD simulation ofFigs. 4–6—once eddies of perpendicular scale k ∗⊥ reach sufficiently large amplitudes andsmall parallel scales to dissipate through parallel hyper-dissipation (Fig. 4). Our simula-tions indicate that this generation of small parallel scales occurs due to critical balancerather than through an independent parallel cascade to small l (cid:107) at fixed k ⊥ (which wouldimply a ν z -independent E sat ). We can thus estimate the saturation amplitude using l (cid:107) ( k ⊥ ) ∼ l (cid:107) ( L ⊥ k ⊥ ) − / and Z + k ⊥ ∼ E / ( L ⊥ k ⊥ ) − / , where Z + k ⊥ is the typical variationin Z + across scale k − ⊥ and l (cid:107) ( k ⊥ ) is the corresponding parallel correlation length (Mallet& Schekochihin 2017). Noting that saturation occurs at ν z l (cid:107) ( k ∗⊥ ) − ( Z + k ∗⊥ ) ∼ ε and l (cid:107) ∼ L ⊥ v A /E / , we find E sat ∼ ( ε/ν z ) − / ( k ∗⊥ L ⊥ ) − / ( v A L ⊥ ) / . (3.1)The ν − / z scaling is approximately satisfied by the simulations in Fig. 3 (which allsaturate with similar k ∗⊥ near the forcing scales), while the 2-D dissipation spectrumin Fig. 4c confirms directly that most dissipation occurs at small l (cid:107) on k ∗⊥ -scale eddies.Figure 4c also shows the critical-balance scaling l (cid:107) ∼ k − / ⊥ (although note that k z ratherthan k (cid:107) ∼ l − (cid:107) is plotted) and the finite ε diss ⊥ at larger k ⊥ from flux leakage throughthe barrier. As mentioned above, this saturation mechanism is unphysical: by growingto ε diss z > ε diss ⊥ , the system is trying to break the l (cid:107) (cid:28) l ⊥ ordering used to derive FLR-MHD. Nonetheless, only with this basic understanding of why the system saturates canwe evaluate how the helicity barrier might evolve in more realistic scenarios.3 − − − − − σ c . . . . . . . k ∗ ⊥ ρ i σ ε = 0 . σ ε = 0 . σ ε = 0 . σ ε = 0 . α Figure 7.
Position of the break k ∗⊥ ρ i versus the energy imbalance (1 − σ c ) for a number of N ⊥ = N z = 256 simulations with different injection imbalances σ ε . As σ c ( t ) grows in time dueto the helicity barrier, there is concurrent decrease in k ∗⊥ , with no obvious dependence on thehelicity injection σ ε = ε H /ε or other time dependence. The black line shows the empirical fit(3.2), the star shows the fit for Fig. 2, and the greyed out region indicates where k ∗⊥ gets withina factor of 2 of the forcing scale ( k ∗⊥ < × π/L ). The inset shows a histogram of the fittedspectral slope ∼ k − α ⊥ above and below the break for the σ ε = 0 .
88 simulation of Fig. 5 (theaverages are (cid:104) α (cid:105) ≈ .
67 above the break and (cid:104) α (cid:105) ≈ . The ion spectral break
For testing of the helicity-barrier hypothesis against observations, it is of interest tounderstand the position and spectral slope of the ion-kinetic transition region around ρ i scales. Given the unphysical saturation mechanism in our simulations, we hypothesizethat the pseudo-stationary phase is of more relevance to realistic space plasmas and thatthe “instantaneous” state of turbulence during that stage can be characterized by theenergy imbalance σ c ( t ) and the injection imbalance σ ε , i.e., that the time history of thegrowth is unimportant. Figure 7 shows k ∗⊥ ρ i versus imbalance (1 − σ c ) for simulationswith four different σ ε at N ⊥ = N z = 256, as well as the spectral slopes ( ∼ k − α ⊥ ) aboveand below k ∗⊥ for σ ε = 0 .
88 (inset; the values of α are obtained via a broken-power-lawfit; Astropy Collaboration 2013). We see good correlation of k ∗⊥ to σ c ( t ), approximately k ∗⊥ ρ i (cid:39) (1 − σ c ) / , (3.2)with little dependence on the injected flux. Spectral slopes in the ion-kinetic transitionrange ( k ⊥ > k ∗⊥ ) are seen to vary more than in the MHD range ( k ⊥ < k ∗⊥ ) and are verysteep, α (cid:39)
4, in good agreement with PSP observations (Bowen et al. a ).
4. Discussion
Our simulations show a dramatic difference in imbalanced Alfv´enic turbulence depend-ing on whether or not energy can be dissipated at spatial scales above the ion-gyroradiusscale. If it can, turbulence proceeds in a relatively conventional way, with energy reachingsmall perpendicular scales where it is thermalized by (hyper)-dissipation. If it cannot,the helicity barrier blocks the cascade at k ∗⊥ ρ i (cid:46)
1, only a small proportion (2 ε − ) of theenergy can reach the smallest perpendicular scales where it would heat electrons, while4 R. Meyrand and others E + grows with time until it becomes so large that modes at k ∗⊥ (which itself moves to largescales) dissipate on the parallel viscosity. The latter effect is unphysical—FLR-MHD isderived by assuming l (cid:107) (cid:29) l ⊥ —so the obvious question that arises is what would happenin a real plasma such as the solar wind. In order for the barrier to lose its importance forlarge-scale dynamics, some mechanism must either remove nearly all of the helicity inthe system, thus allowing the energy to be channelled into the small-scale KAW cascade,or significantly dissipate both energy and helicity at or above the scale of the barrier(as the parallel dissipation does in FLR-MHD). Presumably a real plasma will find away to accomplish one of these feats—the question is which, and what conditions (e.g.,fluctuation amplitudes) are required for it to do so. A definitive answer will have to waiteither for observations or for high-resolution six-dimensional kinetic simulations, but wecan nonetheless speculate on possibilities.The first possibility is that there exists another perpendicular dissipation mechanismthat stops the formation of a helicity barrier in the first place. Within the gyrokineticordering l ⊥ (cid:28) l (cid:107) , because the low ion-thermal speed at β i (cid:28) k ⊥ ρ i ∼ (cid:29) β (cid:29) m e /m i (e.g., normalized damping of (cid:39)
1% at β (cid:39) .
1; see Howes et al. k ⊥ d e ∼ et al. k ⊥ ρ i ∼
1; however,in order to have a significant effect, the compressive and Alfv´enic cascades must havesimilar energy contents, which is not generally observed in the solar wind (Bruno &Carbone 2013; Chen et al. et al. k ⊥ ρ i (cid:46) l (cid:107) (cid:28) ρ i . Stochastic-ion heating (Chandran et al. k ⊥ ρ i ∼ (cid:39) . β / —perhaps acting as a dissipation “switch”as the amplitude grows. It is worth noting, however, that if a k ∗⊥ ρ i (cid:46) ρ i -scale turbulenceamplitude substantially, which also reduces the heating efficiency.If the aforementioned perpendicular dissipation mechanisms fail to dissolve the barrier,it is also possible that—even in a real plasma—large-scale energy cannot dissipate, eithergrowing until significant power reaches small parallel scales (of order d i or ρ i ) or, instratified environments such as the solar wind (Chandran & Perez 2019), propagatingand growing without dissipation until wave reflection causes the imbalance to decreaseenough to allow the cascade to proceed. In the former case, various other, non-gyrokineticdissipation avenues are made available to the plasma; for example, other cascade channels(Saito et al. l (cid:107) ∼ ρ i ion-cycloton waves(Huang et al. et al. b ), which may be a signature of this mechanism.Finally, magnetic reconnection may play an important role by enabling nonlocal energyor helicity transfers (Mallet et al. et al. A critical imbalance?
Our theoretical arguments for helicity-barrier formation, which relied on conservationof energy and helicity in FLR-MHD, suggest that a barrier should form with any injectedimbalance σ ε , but with the constant-flux solution failing at smaller scales for smaller σ ε , viz., for 1 /v ph ( k ⊥ ) < σ ε . † Unfortunately, a robust test of this prediction for small σ ε is difficult: for σ ε (cid:46) . /v ph ( k ⊥ ) (cid:39) σ ε scale is below ρ i , and it is difficult tomaintain a reasonable MHD range while also ensuring that sub- ρ i scales are not affectedby perpendicular dissipation. In any case, it is clear that non-FLR-MHD effects will giverise to a critical σ ε , below which there is no barrier. FLR-MHD breaks down at d e scalesand helicity is no longer conserved, implying that the helicity barrier will likely not form if σ ε (cid:46) v ph ( d − e ) ∼ d e ρ i = (cid:114) m e m i β − / e (4.1)(the latter estimates assume β e (cid:29) m e /m i ). The discussion of the previous paragraphalso suggests that other effects (e.g., stochastic-ion heating) would further increasethe minimal σ ε for which the barrier forms, by dissipating some helicity or energy.Measurement of this critical imbalance, along with improved theoretical understandingof helicity-barrier formation and dissolution, is left to future studies.4.2. Implications for the solar wind
The qualitative agreement of our FLR-MHD energy spectra with those observed in thesolar wind is highly suggestive. As far as we are aware, no previous numerical simulationshave been able to produce similar double-kinked spectra. The position, shape, and causeof the ion-kinetic transition has been a decades-long puzzle with numerous proposedexplanations (Schekochihin et al. et al. et al. et al. et al. et al. et al. et al. a ). In addition, larger-scale transitions to steeper spectra correlate with higher-amplitude fluctuations, lower β , higher proton-scale magnetic helicity, and fast-windregions (Smith et al. et al. et al. et al. k ⊥ ρ i = 1, while steeper spectra and larger-scale breaks result from the energy growingin time (Fig. 5). It is also worth noting that direct measurement of the turbulentenergy flux in the solar wind has found the surprising result that the Z + flux seemsto reverse at large imbalance (Smith et al. et al. † Note that this is not incompatible with the observation of Fig. 7 that the break isindependent of σ ε : once the barrier forms, the system is under no obligation to have a constantflux above the break, meaning the flux can go to zero at scales above where σ ε v ph ( k ⊥ ) (cid:39) R. Meyrand and others for global heliospheric models (Verdini & Velli 2007; Chandran & Hollweg 2009). Itis also interesting to ask about the plausible relevance to the sudden large-scale fieldreversals, or “switchbacks,” observed ubiquitously by PSP (Kasper et al. et al. in-situ due to wave growth in the expanding plasma(Squire et al. T e /T i or β (Howes et al. et al. et al. et al. REFERENCESAlexakis, A. & Biferale, L.
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