Interplay of Turbulence and Proton-Microinstability Growth in Space Plasmas
Riddhi Bandyopadhyay, Ramiz A. Qudsi, William H. Matthaeus, Tulasi N. Parashar, Bennett A. Maruca, S. Peter Gary, Vadim Roytershteyn, Alexandros Chasapis, Barbara L. Giles, Daniel J. Gershman, Craig J. Pollock, Christopher T. Russell, Robert J. Strangeway, Roy B. Torbert, Thomas E. Moore, James L. Burch
IInterplay of Turbulence and Proton-Microinstability Growth in Space Plasmas
Riddhi Bandyopadhyay, Ramiz A. Qudsi, William H. Matthaeus, Tulasi N. Parashar, and Bennett A. Maruca
Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA andBartol Research Institute, Newark, DE 19716, USA
S. Peter Gary and Vadim Roytershteyn
Space Science Institute, Boulder, Colorado 80301, USA
Alexandros Chasapis
Laboratory for Atmospheric and Space Physics, University of Colorado Boulder, Boulder, Colorado, USA
Barbara L. Giles and Daniel J. Gershman
NASA Goddard Space Flight Center, Greenbelt, Maryland 20771, USA
Craig J. Pollock
Denali Scientific, Fairbanks, Alaska 99709, USA
Christopher T. Russell and Robert J. Strangeway
University of California, Los Angeles, California 90095-1567, USA
Roy B. Torbert
University of New Hampshire, Durham, New Hampshire 03824, USA
Thomas E. Moore
NASA Goddard Space Flight Center, Greenbelt, Maryland, USA
James L. Burch
Southwest Research Institute, San Antonio, Texas 78238-5166, USA (Dated: February 21, 2020)Both kinetic instabilities and strong turbulence have potential to impact the behavior of spaceplasmas. To assess effects of these two processes we compare results from a 3 dimensional particle-in-cell (PIC) simulation of collisionless plasma turbulence against observations by the MMS spacecraftin the terrestrial magnetosheath and by the Wind spacecraft in the solar wind. The simulation devel-ops coherent structures and anisotropic ion velocity distributions that can drive micro-instabilities.Temperature-anisotropy driven instability growth rates are compared with inverse nonlinear tur-bulence time scales. Large growth rates occur near coherent structures; nevertheless linear growthrates are, on average, substantially less than the corresponding nonlinear rates. This result castssome doubt on the usual basis for employing linear instability theory, and raises questions as to whythe linear theory appears to work in limiting plasma excursions in anisotropy and plasma beta.
The interplanetary plasma typically exhibits weak col-lisionality and strong turbulence [1, 2]. Similar condi-tions exist in many astrophysical systems. In such high-temperature, low-density magnetized plasmas, Coulombcollisions between particles are rare, which allows thevelocity distribution fuction (VDF) of a given particlespecies to persist in a state far from local thermody-namic equilibrium. Consequently the VDFs are gener-ally non-Maxwellian, and the distortions of the VDFs aremanifested through substantial anisotropy in the pres-sure (or equivalently, temperature) tensor. Approximat-ing the VDF as a bi-Maxwellian, the anisotropy in par-ticle species j can be quantified as R j = T j ⊥ T j (cid:107) . (1) Here, T j ⊥ and T j (cid:107) are respectively the temperature ofspecies j parallel and perpendicular to the local magneticfield ( B ).Although deviations from equilibrium are observedin all charged plasma species [3–5], here we focus onprotons. The extreme values of proton-temperatureanisotropy in the solar wind exhibit a strong dependenceon the parallel-proton beta [6–8] β (cid:107) p = n p k B T p (cid:107) B / (2 µ ) , (2)where, n p is the proton number density, k B is the Boltz-mann constant, and µ is the permeability of vacuum.For progressively larger β (cid:107) p values, the range of observedtemperature-anisotropy values narrows in the solar wind[7] and the terrestrial magnetosheath [9]. a r X i v : . [ phy s i c s . s p ace - ph ] J un Kinetic microinstabilities [10] offer an appealing theo-retical explanation for the observed correlation betweentemperature anisotropy and plasma beta. Lineariza-tion of the Vlasov-Maxwell system about an assumedanisotropic equilibrium predicts that for extreme valuesof R p and β (cid:107) p , the distribution function becomes un-stable, triggering the growth of waves. It is typicallyassumed that upon reaching finite amplitude, these fluc-tuations drive the plasma toward (temperature) isotropy.The initial growth rate of the unstable waves is derivablevia linear theory from the values of β (cid:107) p and R p .An important question is whether the unstable wavesproduced in this way are merely a passive “side effect”,or if they actively modify the dynamics. Some authorsadopt the interpretation that the ion-driven microinsta-bilities may “feed” strong fluctuations [11] in regions ofinstability, materially impacting the plasma dynamics.A different point of view is that turbulence-cascade gen-erated localized inhomogeneities, i.e, coherent structuressuch as current sheets [12, 13], drive the temperature-anisotropies to extreme values, setting the stage for linearinstabilities that might occur in regions of strong nonlin-ear effects.The dissipation of turbulent fluctuations in weakly-collisional space plasmas involves the transfer of fluc-tuation energy from field and flow energies to ther-mal energies. The processes that contribute to thisdissipation generally fall into one of two categories:strongly nonlinear intermittent processes, and quasilin-ear processes. Here, we use the term “nonlinear” todenote the former category which includes the cascadeof turbulent energy from longer to shorter wavelengthswhere weak collisions and collisionless wave-particle in-teractions heat and isotropize the plasma species, and“linear” or “quasilinear” to denote the latter categorywhereby plasma anisotropies lead to the growth of short-wavelength plasma microinstabilities which also scatter,heat, and isotropize the plasma species. Within the lim-ited scope of hybrid simulations, turbulence and microin-stabilitites have been shown to coexist [14–16].Indeed, strong fluctuations are found near the sameextreme regions of the β (cid:107) p , R p -plane where the instabil-ity growth rates are large, causing the plasma to remain(marginally) unstable to temperature-anisotropy insta-bilities [11, 17, 18]. Similarly, computations of shear-driven turbulence [19] have shown that local instabilitiescan sporadically arise due to kinetic effects that are in-evitably found near current sheets and vortices [20, 21].From these studies, it is evident that regions contribut-ing to strong intermittency are also regions of strongkinetic activity, and furthermore these are often juxta-posed. It remains unclear which type of process – lin-ear or nonlinear– dominates on average and determinesthe dynamics of large-scale phenomena. One may studythis relationship by comparing the relative time scales ofnonlinear and linear dynamical processes [22, 23]. There is some subtlety in this comparison when the mediumis inhomogeneous, in that intermittency enters into thiscomparison in a significant way, while the standard in-stability calculation that we employ assumes extendedplane wave solutions.Recent studies of turbulence-driven cascade andtemperature-anisotropy driven microinstability [16, 22,23] find that the majority of solar-wind intervals, in anidealized situation, would support the proton-driven mi-croinstabilities. However, the associated growth ratesare rarely faster than all the other relevant time scales.Quantitatively, the non-linear time scales, estimatedfrom the spectral amplitude near the ion-inertial scale,are faster than the growth rates for most of the analyzedsamples. This comparison suggests that the turbulentcascade quickly destroys the ideal situation for harbor-ing micro-instabilities which would, otherwise, grow tomacroscopic values as unstable modes.As suggested above, the important physics of inter-mittency [24] motivates modification of results obtainedfrom globally based estimates such as average non-lineartime or average spectral amplitude near a given scale.Intermittent structures occupy a small fraction of thevolume, but are likely responsible for a large fraction ofthe plasma heating and particle energization [25]. Keep-ing this in mind, we propose that, instead of comparingtimescales based on average fluctuation amplitude withgrowth rates, it is reasonable to compare the two basedon the corresponding local values of plasma and turbu-lence properties.To address the above issues, here we carry out a localanalysis of both the instability growth rates and the non-linear time scales. We analyze three datasets:1. A three-dimensional, kinetic, particle-in-cell (PIC)simulation,2. In situ observations of Earth’s magnetosheath bythe MMS spacecraft, and3. In situ observation of the interplanetary solar windby the
Wind spacecraft.For all three cases we will show that both instabil-ity growth rates and non-linear rates are intermittentwith enhanced values near coherent structures, and that,pointwise, the nonlinear processes are faster than the in-stabilities for a majority of cases.
Linear Vlasov Theory – Solving the dispersion relationfor the linearized Vlasov and Maxwell’s equations in ahomogeneous plasma, one obtains the angular frequen-cies, ω , associated with a given wavevector k . The imag-inary component of ω is the growth or decay rate of the k mode. The dominant growth rate of a particular in-stability, expected in linear theory to trigger macroscopiceffects, is: γ max ≡ max k (cid:61) ( ω ) , (3)where the maximum operation is taken over all wave-vectors k associated with that instability. The plasma isconsidered unstable to a given instability if γ max > β (cid:107) p , R p )-values, the value of γ max is determined for eachof the four instabilities by computing the maximum valueof (cid:61) ( ω ) over a range of k -values. For every point with γ max >
0, we select the maximum growth rate from the 4types of instabilities, associated with proton-temperatureanisotropy: γ = max { γ cyclotronmax , γ mirrormax , γ (cid:107)− firehosemax , γ ∦ − firehosemax } . (4)Values of γ less than 10 − Ω p are taken to be 0 (i.e., effec-tively stable). Note that in strong turbulence the plasmaparameters vary significantly in space, so a separate cal-culation of γ is required at each point r . Nonlinear Timescales – The local nonlinear timescale,at a position r , for a lengthscale (cid:96) can be estimated as τ nl ( r ) ∼ (cid:96)/δb (cid:96) , (5)where the longitudinal magnetic field increment is δb (cid:96) = (cid:12)(cid:12)(cid:12) ˆ (cid:96) · [ b ( r + (cid:96) ) − b ( r )] (cid:12)(cid:12)(cid:12) , (6)and b is the total magnetic field expressed in Alfv´enspeed units. The vector lag (cid:96) has a magnitude (cid:96) anddirection ˆ (cid:96) . The timescale τ nl ( r ) is a strongly varyingfunction of position, and may take on large values nearcoherent structures. Accordingly, we compare the localvalues of γ and τ nl .For comparison with instability growth rates, it is con-venient to compute an equivalent frequency from the non-linear timescales as ω nl = 2 π/τ nl . We focus on a spatiallag of (cid:96) = 1 d i , the ion-inertial length, a scale at which amajority of highly unstable modes are found. PIC simulation – We analyze data obtained from athree-dimensional, fully kinetic, particle-in-cell (PIC)simulation [27]. The simulation has 2048 grid points,with L = 41 . d i , β p = β e = 0 . m p /m e = 50, δB/B = 1. The analysis is performed on a snapshotlate in time evolution of the simulation. For more details,refer to [27]. We emphasize that no attempt is made toclosely align the simulation parameters with those of themagnetosheath or the solar wind.Figure 1 shows the estimated values of probability den-sity of ( β (cid:107) p , R p )-values in the 3D PIC data , along withthe contours of constant instability growth rate, indi-cating involvement of β (cid:107) p -dependent constraints on R p ,in the simulation data. Although, for any given β (cid:107) p -value, a distribution of R p -values is observed, the distri-bution’s mode occurs near R p ≈
1, and its width becomesprogressively narrower with increased β (cid:107) p . Thus, theplasma likely hosts processes that favor isotropic proton-temperatures (limiting both R p > R p <
1) and
FIG. 1. Two plots of the estimated probability density, ˜ p , of( β (cid:107) p , R p )-values for the 3D PIC data. The two panels areidentical except for the overlaid curves, which show contoursof constant growth rate for different instabilities. The curvesin the top panel show the parallel instabilities: the proton-cyclotron ( R p >
1) and parallel-firehose ( R p > R p >
1) and oblique-firehose ( R p < γ , in units of the proton cyclotronfrequency, Ω p . these processes likely become more active at higher val-ues of β (cid:107) p . We believe these are the first reports of such β (cid:107) p -dependent constraints on R p in a three-dimensional,fully kinetic PIC simulation. Similar plots are obtainedfor the solar wind [26] and magnetosheath [9].The left panel of Fig. 2 shows the distribution of max-imum growth rate, γ , (Eq. 4) for a plane perpendicularto the mean magnetic field, at z ≈ . d i . The centerpanel illustrates the nonlinear frequencies at each point,averaged over lags of 1 d i along the x, y , and z directions.From the first two panels of Fig. 2, it is evident thatboth kinds of frequencies are distributed intermittentlyin space, with clusters of large values in similar regions.However, from the right panel, the ratio of these frequen-cies rarely exceeds unity. Even if both kind of processes FIG. 2. Plots (from left to right) of maximum growth rate γ , nonlinear frequency ω nl at 1 d i , and the ratio γ/ω nl at z ≈ . d i from PIC simulation. are enhanced near the same regions of physical space, thenon-linear processes are typically faster. Although Fig. 2plots only one plane, later we show an analysis from thefull 3D simulation domain. In situ Observation – Though our analysis in the pre-ceding section has important implications, the PIC sim-ulation carries several limitations, e.g., artificial protonto electron mass ratio, small system size. Therefore, wenext perform similar analyses, for two naturally occur-ring turbulent plasma systems: Earth’s magnetosheathand the interplanetary solar wind.We use burst-mode
MMS [28] data sampled in theEarth’s magnetosheath for several burst-mode periodsin both quasi-parallel and quasi-perpendicular shockedplasmas, including the ones reported in [9]. The
MMS /Fast Plasma Investigation [29] moments provide β (cid:107) p , R p -values and magnetic-field measurements fromthe Flux Gate Magnetometer [30] are used to computethe longitudinal increment (Eq. 5) at a spatial separa-tion of 1 d i . We select the magnetosheath intervals wherethe flow speed is greater than the Alfv´en speed and usethe Taylor hypothesis to convert the temporal separa-tion to spatial separation ( (cid:96) = −(cid:104)| V |(cid:105) τ ). The non-linearfrequencies were computed from the magnetic-field in-crements and interpolated to the ion cadence of 150 ms.The instability growth rates are calculated at ion cadencefrom the β (cid:107) p , R p values.The final statistics, shown later, are accumulated fromall the intervals. However, in Fig. 3, we show, as anexample, a 40 min burst-mode sample from 06:12:43 -06:52:23 UTC on 26 December 2017. Note that this in-terval is typical and not chosen for any special properties,other that the preliminary observation that it is turbu-lent and contains current sheets [31, 32]. The bottompanel on the top plot of Fig. 3 clearly shows that theratio γ/ω nl for this interval rarely exceeds unity.In the bottom plot of Fig. 3, we show a similar analy-sis for 1 au solar wind. We use measurements from Wind satellite, accumulated over a period of about 10 years.We use 11 Hz magnetic field measurements from
Wind ’s FIG. 3. Time series of the nonlinear frequency ω nl at 1 d i (top), the maximum instability growth rates γ (middle), andthe ratio γ/ω nl (bottom) for a burst-mode magnetosheathsample observed by the MMS spacecraft (top) and an inter-planetary solar wind interval sampled by the
Wind spacecraft(bottom). Note that due to the large difference in the mea-surement resolution of the
MMS and
Wind spacecraft, thetime scales in the two figures are vastly different ( ∼
40 minversus ∼ Magnetic Field Investigation [33] to calculate ω nl for aTaylor-shifted separation of 1 d i . The two Faraday cupsin the Solar Wind Experiment [34] return one ion spec-trum every ≈
90 s and the ω nl values are interpolated tothis cadence. A bi-Maxwellian distribution is fit to eachion spectrum to compute proton moments [35] and thusinfer values of R p and β (cid:107) p . The Wind data used here areidentical to those reported in [26]. In the small sampleof ≈ Wind data, shown in the bottom plotof Fig. 3, the exhibited behavior closely resembles themagnetosheath results (Fig. 3, top), apart from the dif-ferences in time scales. Again, the nonlinear frequency, ω nl , is greater than the instability growth rate, γ , for themajority, and the regions in which the growth rate is ofrelative significance are sporadic.The main result of this paper is shown in Fig. 4. Here,we plot joint probability distribution functions of the in-stability growth rates ( γ ) and the non-linear frequencies FIG. 4. Joint probability distribution functions of the max-imum instability growth rate γ and the nonlinear frequency ω nl from PIC simulation, MMS data in the magnetosheath,and
Wind data in the interplanetary solar wind. ( ω nl ) for all three datasets. In all three cases, the coreof the distribution resides well below the γ = ω nl line.From this result, we can conclude that for most data sam-ples, the non-linear processes are faster than the linear-instability growth. Discussion.
Temperature-anisotropy driven microin-stabilities are often considered to constrain the tem-perature anisotropy values in weakly-collisional plas-mas [7, 36, 37]. Recall that the linear Vlasov theoryof instabilities assumes a homogeneous background, inwhich background a small perturbation grows exponen-tially. The established success of linear-microinstabilitytheories suggests that the conditions near the extremelyanisotropic temperature may be uniform enough to jus-tify an application of linear theories. Turbulence, on theother hand, is an intrinsically nonlinear process. Thincurrent sheets, and other coherent structures generatedby the energy cascade, are sites of extreme temperatureanisotropy [20] and therefore, the high growth rates dueto the microinstabilities also reside in the same vicinity.It is therefore not a priori obvious whether the presenceof intermittency and coherent structures favors or dis-favors instabilities in comparison with nonlinear effects.This question has motivated the present study.To address this question, we have examined the statis-tical distribution of growth rates associated with protontemperature-anisotropy driven microinstabilities and thelocal nonlinear time scales, in three distinct systems. Thethree systems cover different ranges of ( R p , β (cid:107) p )-valuesamong other parameters. However, both simulation andobservation results show that, when the comparison isperformed in this way, locally in space, a negligible frac-tion of the samples support long-lived linear instabilities.For the majority of cases, it appears that the nonlineareffects do not allow sufficient time for the instabilitiesto grow large enough to affect the global dynamics toany significant degree. In this regard it is interestingthat the instabilities appear to delimit the anisotropieseven though the theory assumes homogeneous perturba-tions. One possibility is if the initial fluctuations arelarge amplitude to begin with, in which case application of a linear theory becomes questionable. In either case,clearly, a substantial revision in the present theoreticalunderstanding is in order.In this study, we have used a basic homogeneousplasma model to estimate the growth rate of pressure-driven instabilities. A more realistic extension will con-sider other ion species, as well as modifications to the lin-ear theory introduced by the presence of strong spatialinhomogeneity near coherent structures. Finally, con-sideration of more general equilibrium VDFs may giverise to more rapid instability. Each of these refinementswould represent a significant subsequent study.TNP was supported by NSF SHINE Grant AGS-1460130 and NASA HGI Grant 80NSSC19K0284. WHMis a member of the MMS
Theory and Modeling Teamand was supported by NASA Grant NNX14AC39G.The research of SPG was supported by NASA grantNNX17AH87G.This study used Level 2 FPI and FIELDS data accord-ing to the guidelines set forth by the
MMS instrumen-tation team. All data are freely available at https://lasp.colorado.edu/MMS/sdc/ . We thank the
MMS
SDC, FPI, and FIELDS teams for their assistance withthis study.
Wind
SWE and MFI data are available from CDAWeb( https://cdaweb.gsfc.nasa.gov/ ). The authorsthank the
Wind team for the
Wind magnetic field andplasma data.The bi-Maxwellian analysis code is summarized in [35].The simulation described in this paper was performedas a part of the Blue Waters sustained-petascale comput-ing project, which was supported by the National ScienceFoundation (awards OCI-0725070 and ACI-1238993)and the State of Illinois. Blue Waters allocation wasprovided by NSF PRAC award 1614664.The authors would like to thank Rohit Chhiber foruseful discussions and Yan Yang for helping to preparethe PIC dataset.
SUPPLEMENTAL MATERIAL
The first three panels of Fig. 5 show the three param-eters — R p , β (cid:107) p , and J z — across a plane at z ≈ . d i of the simulation box. The system is strongly turbulentand exhibits structures of various scales. The extremevalues of each parameter occur in distinct regions thatoccupy only small fractions of the total volume. Thatis, these quantities are intermittent, which is correlatedwith the existence of sharp gradients and coherent struc-tures. Further, the extreme values of R p and β (cid:107) p residenear (but not necessarily exactly coincident with) theextreme values of J z . These concentrations of currentdensities frequently correspond to current sheets. x ( d i )5152535 y ( d i ) β k p x ( d i ) R p = T ⊥ p /T k p x ( d i ) -0.3 -0.1 0.1 0.3 J z x ( d i ) − − − γ ion − cyclotron γ k− firehose x ( d i ) − − − γ mirror γ firehose FIG. 5. Colorplot of (left to right) β (cid:107) p , R p and J z , from a fully kinetic 3D PIC simulation at z ≈ . d i . The fourth andfifth panels show the spatial distribution of growth rate in units of proton-cyclotron frequency, γ/ Ω p , for parallel and obliquepropagation corresponding to the first two panels. Using the method described in the main text, we com-pute γ for the ( β (cid:107) p , R p )-pair at each grid point of the sim-ulation, where γ is the maximum value of growth rate forall possible values of propagation vector ( k ), for a giveninstability. The fourth panel of Fig. 5 shows the spatialdistribution of growth rates for the solutions with posi-tive growth rates, corresponding to the first two panels ofthe same figure. As described in main article, for γ max ,we imposed a cut-off at 10 − Ω p ; thus growth rates lessthan 10 − Ω p are considered to be 0. 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