Magnetic field turbulence in the solar wind at sub-ion scales: in situ observations and numerical simulations
L. Matteini, L. Franci, O. Alexandrova, C. Lacombe, S. Landi, P. Hellinger, E. Papini, A. Verdini
MMagnetic field turbulence in the solar wind atsub-ion scales: in situ observations andnumerical simulations
L. Matteini , , ∗ , L. Franci , , O. Alexandrova , C. Lacombe , S. Landi , , P.Hellinger , E. Papini , and A. Verdini , Department of Physics, Imperial College London, London SW7 2AZ, UK LESIA, Observatoire de Paris, Universit ´e PSL, CNRS, Sorbonne Universit ´e, Univ.Paris Diderot, Sorbonne Paris Cit ´e, France School of Physics and Astronomy, Queen Mary University of London, UK Dipartimento di Fisica e Astronomia, Universit ´a di Firenze, Italy Astronomical Institute, CAS, Prague, Czech Rep. INAF, Osservatorio Astrofisico di Arcetri, Firenze, Italy
Correspondence*:Imperial College London, South Kensington Campus, London SW7 2AZ, [email protected]
ABSTRACT
We investigate the transition of the solar wind turbulent cascade from MHD to sub-ion rangeby means of a detail comparison between in situ observations and hybrid numerical simulations.In particular we focus on the properties of the magnetic field and its component anisotropy inCluster measurements and hybrid 2D simulations. First, we address the angular distributionof wave-vectors in the kinetic range between ion and electron scales by studying the varianceanisotropy of the magnetic field components. When taking into account the single-directionsampling performed by spacecraft in the solar wind, the main properties of the fluctuationsobserved in situ are also recovered in our numerical description. This result confirms that solarwind turbulence in the sub-ion range is characterized by a quasi-2D gyrotropic distribution ofk-vectors around the mean field.We then consider the magnetic compressibility associated with the turbulent cascade and itsevolution from large-MHD to sub-ion scales. The ratio of field-aligned to perpendicular fluctuations,typically low in the MHD inertial range, increases significantly when crossing ion scales and itsvalue in the sub-ion range is a function of the total plasma beta only, as expected from theoreticalpredictions, with higher magnetic compressibility for higher beta. Moreover, we observe that thisincrease has a gradual trend from low to high beta values in the in situ data; this behaviour iswell captured by the numerical simulations. The level of magnetic field compressibility that isobserved in situ and in the simulations is in fairly good agreement with theoretical predictions,especially at high beta, suggesting that in the kinetic range explored the turbulence is supportedby low-frequency and highly-oblique fluctuations in pressure balance, like kinetic Alfv ´en waves orother slowly evolving coherent structures. The resulting scaling properties as a function of theplasma beta and the main differences between numerical and theoretical expectations and in situobservations are also discussed. a r X i v : . [ phy s i c s . s p ace - ph ] A ug atteini et al. Properties of sub-ion turbulence
The solar wind constitutes a unique laboratory forplasma turbulence (Bruno and Carbone, 2013). Inthe last decade increasing interest has raised towardsthe small-scale behaviour of the turbulent cascade,i.e. beyond the breakdown of the fluid/MHD de-scription that takes place at ion scales. Spacecraftobservations of solar wind and near-Earth plasmasprovide unique measurements of the turbulent fluc-tuations at scales comparable and smaller than thetypical particle scales, the Larmor radius ρ (see Ap-pendix for definition of physical quantities used)and the inertial length d (e.g. Alexandrova et al.,2009; Sahraoui et al., 2010; Alexandrova et al.,2012; Chen et al., 2013a). However, the physicalprocesses governing the energy cascade at kineticscales and those responsible for its final dissipationare not well understood yet.What is well established is that in the transi-tion from MHD to kinetic regime, plasma turbu-lence modifies its characteristics. Observationaland numerical studies over the last few years havehighlighted the main differences between largeand small scale properties of solar wind fluctua-tions (e.g., Chen, 2016; Cerri et al., 2019). Themagnetic field spectrum typically steepens whenapproaching ion scales, leading at sub-ion scales(between ion and electron typical scales) to a powerlaw with spectral index close to − . (Alexan-drova et al., 2009, 2012; Kiyani et al., 2009; Chenet al., 2010; Sahraoui et al., 2013), steeper thanKolmogorov -5/3, but also than the theoretical pre-diction − / from EMHD (Biskamp et al., 1996)and KAW/whistler turbulence (Schekochihin et al.,2009; Boldyrev et al., 2013). The origin of such aspectral slope is still unknown and it has been pro-posed that it could be related to intermittency correc-tions (Boldyrev and Perez, 2012; Landi et al., 2019),magnetic reconnection (Loureiro and Boldyrev,2017; Mallet et al., 2017; Cerri et al., 2018), Landaudamping (Howes et al., 2008; Schreiner and Saur,2017) and/or the role of the non-linearity parameter(Passot and Sulem, 2015; Sulem et al., 2016). The change in the magnetic field spectrum is ac-companied by a rapid decrease in the power ofion velocity fluctuations ( ˇSafr´ankov´a et al., 2013;Stawarz et al., 2016) and the onset of the non-idealterms in the Ohm’s law which governs the electricfield associated to the turbulent fluctuations; as aconsequence the electric field spectrum becomesshallower at sub-ion scales (Franci et al., 2015a;Matteini et al., 2017). In this framework the electriccurrent (mostly carried by electrons) plays a majorrole, coupling directly with the magnetic field inthe cascade and likely affecting the energy cascaderate via the Hall term (Hellinger et al., 2018; Papiniet al., 2019). All these properties depend furtheron the plasma beta ( β = 8 πnk B T /B ), which con-trols, among other things, the scale at which themagnetic field spectrum breaks (Chen et al., 2014;Franci et al., 2016).One of the most significant differences with re-spect to the turbulent regime observed at large scaleshowever is the role of compressive effects. Whilein the inertial range fluctuations show a low levelof both plasma and magnetic field compressibility,hence can be reasonably well described by incom-pressible MHD, at sub-ion scales density and mag-netic field intensity fluctuations become significantand comparable to transverse ones (Alexandrovaet al., 2008; Sahraoui et al., 2010; Salem et al., 2012;Kiyani et al., 2013; Chen et al., 2012b; Perroneet al., 2017), in agreement with simulations (Franciet al., 2015b; Cerri et al., 2017). It is believed thatthis is related to a change in the properties of theturbulent fluctuations, which become intrinsicallycompressive at small scales. It is then by studyingin detail their properties that it is possible to shedlight on the nature of the fluctuations which supportthe cascade at kinetic scales (Chen et al., 2013b;Pitˇna et al., 2019; Groˇselj et al., 2019; Alexandrovaet al., 2020).Another important aspect of solar wind turbu-lence is its spectral anisotropy (Horbury et al., 2008;Wicks et al., 2010; Chen et al., 2010; Roberts et al.,2017). Studies about the shape of turbulent eddies,both at MHD (Chen et al., 2012a; Verdini et al.,2018, 2019) and kinetic scales (Wang et al., 2020), This is a provisional file, not the final typeset article atteini et al. Properties of sub-ion turbulence reveal the presence of a 3D anisotropy in the struc-tures when described in terms of a local frame. Onthe other hand, when the analysis is made in aglobal frame (without tracking the local orientationof the structures), the 3D anisotropy is not captured,and the k-vectors of the fluctuations show a sta-tistical quasi-2D distribution around the magneticfield (Matthaeus et al., 1990; Dasso et al., 2005; Os-man and Horbury, 2006). In this work we addressthis latter aspect and we investigate the distributionof the k-vectors with respect to the ambient mag-netic field at kinetic scales by using the magneticfield variance anisotropy (i.e. the ratio of magneticfield fluctuations in different components). Saurand Bieber (1999) have shown that, also in singlespacecraft observations, is possible to character-ize the 3D k-vector distribution by using varianceanisotropy. When the sampling occurs only alonga preferential direction, like in typical solar windobservations, their model predicts various possiblekinds of variance anisotropy as a function of theunderlying k-spectrum. In particular, assuming aquasi-2D gyrotropic distribution of k-vectors (ax-isymmetric with respect to the magnetic field), theratio of the power in the two perpendicular mag-netic field components is directly related to the localslope of the spectrum - which is assumed to have thesame form for all components and a slope indepen-dent of the scale within a given regime. Since bothquantities, spectral slope and perpendicular powerratio, can be easily measured in situ, the Saur &Bieber model constitutes an useful and simple toolto investigate underlying spectral anisotropies. De-spite the model was originally developed for MHDscale fluctuations, it basically corresponds to a ge-ometrical description built on the divergence-lesscondition for B , so it can be applied to any kindof regimes, including the low-frequency turbulenceexpected at sub-ion scales (Turner et al., 2011). Inthe work of Lacombe et al. (2017), we investigatedthe k-vector distribution at sub-ion scales using thetechnique by Saur and Bieber (1999). Based onthe comparison with the predictions, we concludedthat the distribution of the k-vectors in the sub-ionrange of solar wind turbulence is consistent with a quasi-2D gyrotropic spectrum, then approaching amore isotropic shape when reaching electron scales(Lacombe et al., 2017). However, such an applica-tion has not been benchmarked by kinetic numericalstudies yet.The aim of this work is then to focus on thespectral anisotropy properties and magnetic com-pressibility at small scales, by exploiting the de-tailed comparison of in situ observations and high-resolution kinetic numerical simulations. The paperis organised as follow: in Sec. 2 we introduce thespacecraft and numerical dataset used and in Sec. 3we describe their spectral properties. In Sec. 4 wediscuss the spectral anisotropy at sub-ion scales andtest, for the first time, the Saur and Bieber model innumerical kinetic simulations; in Sec. 5 we addressproperties of the magnetic compressibility and itsdependence on the plasma beta. Finally, in Sec. 6we discuss our conclusions and the implicationsof our finding for the interpretation of solar windobservations and simulations. In this study we compare properties of magneticfluctuations measured in situ by the Cluster space-craft with numerical results obtained by means of2D hybrid particle-in-cell (PIC) simulations.
For our analysis we use the dataset discussed byAlexandrova et al. (2012), when Cluster was in thefree solar wind, i.e., not magnetically connected tothe Earth’s bow shock. Details have been describedalso in Lacombe et al. (2017) and we recall herethe main aspects. Magnetic field fluctuations aremeasured by the STAFF (Spatio-Temporal Analysisof Field Fluctuation) instrument, composed by awave form unit (SC) and a Spectral Analizer (SA).Power spectra are computed on board in a magneticfield aligned system of coordinates (MFA), basedon the 4s magnetic field measured by the FGM(Fluxgate Magnetometer) experiment. A selectionof 112 spectra has been performed, retaining ineach spectrum only measurements above 3 times
Frontiers 3 atteini et al.
Properties of sub-ion turbulence
Figure 1.
Cluster STAFF spectra for different intervals with β = 0 . , . , from left to right. Colorsencode the magnetic field components B x (black dashed), B y (blue), and B z (red). The region highlightedin yellow corresponds to the the sub-ion range investigated in this study. Bottom panels show the ratio ofthe power in the perpendicular components P y ( k ) /P x ( k ) (black diamonds) and the value γ of the localslope of the total spectrum P ( k ) ∼ k − γ (red stars). The average values of P y ( k ) /P x ( k ) and γ in thesub-ion range are also shown as horizontal dashed lines. Figure 2.
Magnetic field spectra from hybrid simulations for different beta regimes ( β p = 0 . , , and β e = β p ). Components are encoded as in Figure 1 and the coloured region indicates the sub-ion range thatcan be directly compared with the analogous region in the observations.the noise level in every direction x, y and z (seeAppendix in Lacombe et al., 2017). Each sample isa 10 minute average of 150 individual 4s spectralmeasurements. This provides spectra above 1Hz upto typically 20-100Hz, depending on the amplitudeof the fluctuations in each interval. When converted into physical length scales, assuming the Taylorhypothesis ( k = 2 πf /V sw ), this leads to signals thatcover the range between ∼ d p and ∼ . d e (where d p and d e are the proton and electron inertial lengthsrespectively), enabling then a good description ofthe sub-ion regime from proton to electron scales. This is a provisional file, not the final typeset article atteini et al. Properties of sub-ion turbulence
The reference frame adopted (MFA) is such that B z is the component aligned with the mean mag-netic field B (relative to the 4s interval duringwhich an individual spectrum is calculated); B x is the component orthogonal to B z in the plane con-taining both the solar wind velocity V sw and themean magnetic field B , and B y is the third orthog-onal component. Note that a selection criterium isimposed on the angle θ BV , the angle between thelocal 4s magnetic field and the flow velocity, i.e.,that θ BV is large enough to avoid a connection withthe Earth bow shock during the sampled interval; θ BV in the dataset has an average value of ∼ de-grees. This implies that for each spectrum, the meanmagnetic field makes a big angle with respect to thesampling direction; moreover, we have checkedthat θ BV does not vary significantly during the 10minutes over which spectra are averaged.As a consequence, this procedure selects intervalsin which Cluster observed highly oblique k-vectorsand, to a good approximation, the component B x corresponds also to the sampling direction (radial)and is orthogonal to B ; B y corresponds to the otherperpendicular component and B z is identified as thecompressive component B (cid:107) . As already discussedin Lacombe et al. (2017), although the total tracepower measured in situ is an invariant observable,the fact that the sampling occurs only in a preferreddirection introduces a relative weight between B x and B y that is measurement dependent (Saur andBieber, 1999). To take this into account, we haveemployed an analogous approach in the analysisof the simulations data, as described in the nextsection. In situ observations are directly compared withnumerical simulations performed with the hybrid-PIC code CAMELIA (Matthews, 1994; Franciet al., 2018a). The hybrid model captures well thetransition from fluid to kinetic regime around ionscales. Moreover, it reproduces successfully manyof the main properties of solar wind turbulence ob-served by spacecraft at sub-ion scales (Franci et al.,2015b,a). It is then a suitable tool to investigate the turbulent regime probed by STAFF/Cluster data.We use here 2D simulations -computationally moreaffordable than 3D- in order to explore the param-eter space observed in situ; in particular we focuson the effects associated to variations in the protonand electron plasma beta β p and β e . Franci et al.(2016) have shown that 2D hybrid simulations areable to capture the ion-break scale behaviour indifferent beta regimes observed in solar wind tur-bulence (Chen et al., 2014). We then exploit thegood matching between the simulations and in situobservations to characterise further the propertiesof kinetic plasma turbulence in the sub-ion regime.On the other hand, 3D hybrid simulations (Franciet al., 2018b) have confirmed the solidity of the re-duced 2D results and the good agreement with insitu observations.In order to make a direct comparison with sub-ionspectra measured by Cluster, we have adopted asimilar approach in the computation of spectra inthe simulations. This means that numerical spectraare computed along the x direction only, to mimicthe radial sampling occurring in the solar wind. Thisis obtained by integrating along y the Fourier spec-trum P ( k x , y ) of each i magnetic field component: P i ( k x ) = (cid:90) P i ( k x , y ) dy (1)Therefore, also in the simulation B x correspondsto the sampling direction, orthogonal to the out-of-plane magnetic field B z , and B y is the mostenergetic fluctuating component, being orthogonalto both B and k = k x . With this approach andwithin the observational conditions previously de-scribed, we can perform a direct comparison ofsimulations and in situ data.The numerical dataset used was originally pre-sented in Franci et al. (2016) and is available online.It is constituted by a set of different β p = β e ; runs are initiated withrandom perpendicular Alfv´enic fluctuations with Frontiers 5 atteini et al.
Properties of sub-ion turbulence vanishing cross-helicity and equipartition in mag-netic and kinetic energies. Spectra are computed atthe maximum of the turbulent activity.
Figure 1 shows three examples of Cluster spectra(2003/02/18 04:45-04:55; 2004/02/22 05:40-05:50;2004/01/22 04:40-04:50), where frequencies havebeen converted into k-vectors and normalised to d p (original sampling frequencies are also shownfor reference). Observations cover ion and electronscales, with a transition accompanied by a slopechange around kd p ∼ . In this work, we focus onthe sub-ion regime highlighted in yellow in the pan-els, where electron physics effects can be neglected(at least for spectral properties) and a well-definedslope close to − . can be observed (Alexandrovaet al., 2012). The three cases, corresponding todifferent total beta β regimes [0 . , . , , show asimilar qualitative behaviour: as expected, the spec-trum P y of the perpendicular B y component (blue)is always the most energetic. The power in the otherperpendicular component P x (black dashed) is al-ways slightly smaller, however, its ratio with P y isroughly independent of beta and close to the localspectral slope (bottom panels); this is related to the3-D distribution of k-vectors (Lacombe et al., 2017)and will be discussed more in detail in Sec.4.On the other hand, the power P z of the fieldaligned component B z (red) is typically less ener-getic than P y , however, its weight is highly variablewith beta: P z is smaller than P x for β < , com-parable to P x for β ∼ , and larger the P x for β > . This obviously results in a variable mag-netic compressibility associated to the fluctuationsand its functional dependence on beta is the subjectof Sec.5Figure 2 shows an analogous selection from nu-merical simulations; note that in the simulations β e = β p . In this case the regime reproduced in thesimulation box includes the MHD inertial rangeand its transition to a sub-ion cascade at smallerscales. The yellow area highlights the region of the Figure 3.
Reduced spectra of the fluctuations ofthe magnetic field components B y and B x de-fined with respect to a fixed sampling direction k x for a simulation with β p = 0 . . The thick solidblack line corresponds to the total perpendicularpower P ⊥ ( k x ) ; the dashed line shows P ⊥ ( k x ) / ,also corresponding to the average power in anyperpendicular magnetic field component in theaxisymmetric case.spectra − roughly a decade between kd p ∼ and kd p ∼ − that can be directly compared withthe in situ data. In this region, the qualitative be-haviour of the spectra is similar to Figure 1: B y (blu) is always dominant, B x (black) contributesfor a constant fraction of it and is roughly the sameat all betas, while B z (red) varies significantly inthe panels and becomes comparable to B y for largebetas. This confirms that our method of comput-ing spectra in the simulations mimicking satelliteobservations really captures the main aspects ofin situ measurements and can then be exploited toinvestigate further the properties of the turbulentcascade. Saur and Bieber (1999) have investigated how dif-ferent types of k-vectors distributions can generatea variable anisotropy in the observed magnetic fieldcomponents, due to sampling effects. In the case ofa gyrotropic 2D distribution of k-vectors, the ratio P y /P x is expected to coincide with the local slope γ of the spectrum P ( k ) ∼ k − γ . This applies well This is a provisional file, not the final typeset article atteini et al. Properties of sub-ion turbulence to solar wind observations in the physical rangeof interest here, as it can be appreciated in Fig. 1,where the ratio P y /P x , shown in the bottom panels,is close to the spectral slope observed - typicallyin the range [-2.5,-3] - and appears roughly inde-pendent of the plasma beta. Interestingly, at smallerscales, when the magnetic spectrum steepens as ap-proaching electron scales (Alexandrova et al., 2009),this is not associated to an increase in the perpen-dicular power ratio P y /P x (which on the contraryhas a slight decrease); this doesn’t correspond tothe expectation for a quasi-2D spectrum accordingto the model and in fact Lacombe et al. (2017) hasinterpreted this signature as the result of a moreisotropic distribution of k-vectors close to electronscales.To validate further this observational conclusion,we verify here the applicability of the Saur andBieber model to sub-ion scale turbulence. In thesimulations the spectrum is two-dimensional byconstruction and, consistent with the axisymmet-ric initial conditions imposed in the x - y plane, itis also gyrotropic with respect to the out-of-planemagnetic field B z .First, it is instructive to discuss spectra shown inFigure 3. These are power spectra of the perpendic-ular components B x (purple) and B y (orange) asa function of k x , assuming then a fixed directionof sampling. As expected, P y ( k x ) > P x ( k x ) ; onthe other hand, their sum P ⊥ ( k x ) (solid black line)is statistically equivalent to the axisymmetric spec-trum P ⊥ ( k ) = P y ( k ) + P x ( k ) . The difference isthat when calculating the axisymmetric spectrum P ⊥ ( k ) all perpendicular magnetic field directionshave equal weight and one can assume that statisti-cally P y ( k ) ∼ P x ( k ) ; as a consequence the powerassociated to any individual perpendicular compo-nent corresponds to half of the total perpendicularpower P ⊥ ( k ) / ∼ P ⊥ ( k x ) / (thin dashed blackline). It is interesting to note that when samplingalong a fixed direction ( x ), as it happens with space-craft in the solar wind, none of the two measuredspectra P y ( k x ) and P x ( k x ) is really representativeof the power P ⊥ ( k ) / of the gyrotropic descrip-tion; instead the component along the sampling Figure 4.
Hybrid simulations: spectra of the ra-tio of the perpendicular magnetic components P y ( k x ) /P x ( k x ) and corresponding to the local spec-tral slope. Different colors encode different β p :0.125 (cyan), 1 (black) and 8 (red). The horizontaldashed lines show reference spectral slopes ob-served in the simulations at kd p < (-5/3) andat sub-ion scales kd p > .( B x ) is significantly reduced due to the solenoidal ∇ · B = 0 condition, while the orthogonal ( B y ) isamplified, in order to maintain the same total power P ⊥ ( k ) . This means that in solar wind spectra likein Fig. 1, neither P x nor P y are individually repre-sentative of the average power in a perpendicular Bcomponent: the individual measurements of P x or P y cannot be directly associated to it, but only theirsum.Bearing this in mind, Figure 4 shows the ratio ofthe power in the perpendicular components for thethree simulations shown in Figure 2. The P y /P x ratio well captures the transition from MHD to asteeper spectrum at smaller scales; in all cases theratio, close to 5/3 at large scales, starts increasing inthe vicinity of ion scales and reach a maximum inthe sub-ion regime, where the ratio saturates close to ∼ , in good agreement with the local spectral slopeobserved in the kinetic range, which is typicallyclose to − . At larger k the ratio then decreasesdue to the noise. In the framework of the Saur andBieber model for spectral anisotropy this indicatesa quasi-2D gyrotropic spectrum of the fluctuations,which corresponds well to the spectrum developedin these simulations. This confirms that the model is Frontiers 7 atteini et al.
Properties of sub-ion turbulence valid also at sub-ion scales, and reinforces the find-ing of Lacombe et al. (2017), where is found thatsolar wind spectra at kinetic scales are describedwell by a quasi-2D gyrotropic distribution.
There is another interesting indication suggestedby Figure 4, namely the fact that the P y /P x ra-tio in the sub-ion range seems to depend on beta:consistent with this, the sub-ion slope in Figure 2is slightly steeper for small β p and shallower forlarger β p . This behaviour is already discussed inFranci et al. (2016) and is found in all simulationsfor the spectrum of the transverse fluctuations B ⊥ ;conversely, the spectrum of the parallel component B (cid:107) is almost independent of β p (see Figure 4 inFranci et al., 2016). We have then looked for a sim-ilar trend also in the in situ data. Figure 5 showsthe histogram of the spectral slopes in the kineticrange for B ⊥ (top) and B (cid:107) (bottom), for larger (red)and smaller (black) total beta. Spectral slopes arecalculated between < kd p < for β p < andbetween < kρ p < for β p > , where a quitewell-defined power law scaling is observed. Theyare then separated in two groups defined by thetotal beta β < and β > . The mean of each his-togram is indicated by the small vertical line endedwith a diamond. For the parallel component (bottompanel), the distribution of the slopes is similar forboth beta regimes and centred around a value of ap-proximately − . ± . ; this is in good agreementwith the simulations. For the dominant perpendic-ular component (top panel), we observe averagevalues consistent with previous studies based onthe total power δB = δB (cid:107) + δB ⊥ of the fluctu-ations (Alexandrova et al., 2009, 2012; Sahraouiet al., 2013; Chen et al., 2013a). However, in thelower beta case (black), some slightly steeper slopesare observed for B ⊥ with respect to the high betacase, with an average of − . ± . with respectto − . ± . . This behaviour agrees qualitativelywith the simulations; however a more detailed in-vestigation is needed to fully identify the role of β p on the sub-ion spectral slope and is beyond thescope of the present study. Figure 5.
Spectral slope measured for differentbeta conditions in Cluster data; (black) β < and(red) β > . Top panel refers to the spectrum ofthe perpendicular magnetic component B ⊥ and thebottom panel to B (cid:107) . The short vertical lines end-ing with a diamonds indicate average values of thehistograms.What can be however pointed out is that a conse-quence of the behaviour in Figure 5 is that while athigh beta, δB (cid:107) and δB ⊥ are observed to have almostthe same scaling, so that their ratio remains approx-imately constant in the sub-ion range, at lower β their slightly different scaling is expected to resultin a slow increase of the δB (cid:107) /δB ⊥ ratio betweenion and electron scales. These properties are relatedto the evolution of the magnetic compressibility ofthe fluctuations in the sub-ion range, which is mainfocus of the next section. We now investigate the role of the third magneticfield component B z , which is aligned with the local(at 4s) magnetic field B . In particular, we focus onthe the magnetic compressibility C (cid:107) = δB (cid:107) /δB ,where δB = δB (cid:107) + δB ⊥ and its implication for thenature of the cascade at these scales. Note that inthis case, the measurement of B z is not affected by This is a provisional file, not the final typeset article atteini et al. Properties of sub-ion turbulence
Figure 6.
Examples of Cluster FGM (black) andSTAFF (red) spectra of magnetic compressibility C (cid:107) as a function of the frequency measured inspacecraft frame.the sampling direction (provided that this is orthog-onal to B to a good approximation) and since weuse the total perpendicular power P ⊥ , the cautiondiscussed in Sec. 4 is not needed here.Figure 6 shows C (cid:107) for three intervals of differenttotal β =1,3,4 ( β p = 0 . , . , . ) as measured fromSTAFF (red). For these three cases we also showthe spectrum of the magnetic field compressibilityas measured at lower frequencies (corresponding tophysical scales larger than d p ) by the fluxgate mag-netometer onboard Cluster (FGM, black). Note thatFGM spectra are linearly interpolated between . and . Hz to remove artefacts due to spacecraft spin( . Hz). There is a good matching between thetwo independent measurements at f ∼ Hz andwhere data points from both instruments are avail-able for a more extended range there is also a quitesatisfactory overlap between them. The overall be-haviour agrees well with the expected picture: atlarge scale, in the MHD inertial range, the level ofcompressibility is lower, typically C (cid:107) (cid:46) . (e.g.,Horbury and Balogh, 2001; Smith et al., 2006) andstarts to increase as approaching ion scales (Hamil-ton et al., 2008; Alexandrova et al., 2008; Salemet al., 2012; Kiyani et al., 2013), reaching some-times variance isotropy (indicated by the dashedhorizontal line) in the sub-ion range, where the com-pressibility seems to saturate. As already shown by Figure 7.
STAFF spectra of magnetic compress-ibility C (cid:107) as a function of the frequency measuredin spacecraft frame. Different colours and linesidentify different groups of intervals with given β .Lacombe et al. (2017), the level of magnetic com-pressibility developed at small scales is larger forhigh beta than for small beta. Since we focus onthe behaviour at sub-ion scales, in the following werestrict our analysis to STAFF measurements only.To highlight further the β -dependence of the mag-netic compressibility, Figure 7 shows C (cid:107) for aselection of spectra with different β , increasingfrom red to purple. There is a continuous transi-tion from lower to higher magnetic compressibilityas a function of beta, in agreement with linear the-ory expectations (e.g. Podesta and TenBarge, 2012).Moreover, at high beta it seems that the fluctuationsreach an asymptotic δB (cid:107) /δB ratio, leading to anextended plateau in the spectrum, while at the low-est beta a plateau cannot be clearly identified. Wenow want to identify more in detail what processand length scale control the level of C (cid:107) and in solarwind data. First it is useful to go again from frequency to k-vector spectra: in Figure 8 frequencies are convertedinto k-vectors and normalised with respect to theproton inertial length d p . Frontiers 9 atteini et al.
Properties of sub-ion turbulence
We first identify two big categories such that bothproton and electron betas are small, i.e. β p < and β e < , or both large, i.e. β p > and β e > . Weobtain an average total beta β ∼ in the former, and β ∼ in the latter. The average spectrum of mag-netic compressibility for each of the two familiesis shown in the top panel of Figure 8 as a functionof kd p ; the thin dotted lines identify the standarddeviation around the averages. In the high beta case(solid blue) the compressibility reaches a plateau af-ter kd p = 1 and saturates at an average level whichis very close to isotropy (same power in P x , P y and P z ), while in the low beta case (dashed red) C (cid:107) remains smaller and there is not a clear plateau at kd p > .The remaining spectra are further separated in twoother families: the first with β p < and β e > , thesecond β p > and β e < . In this case the averagetotal betas are very similar, β ∼ . ( β p ∼ . )and β ∼ . ( β p ∼ . ), respectively, and fall inbetween the other two groups (small and large β ).Consistent with this, the average spectrum of thesetwo families, shown in orange and green in the bot-tom panel, has a level of compressibility at sub-ionscales that is intermediate with respect to the othertwo curves. Moreover, they almost precisely fall ontop of each other. All this suggests that not only thetotal plasma beta is a good parameter for orderingthe level of compressibility generated at sub-ionscales, but also this level is roughly independent onthe individual weights of β p and β e , being their sum β = β p + β e the only relevant parameter.This observational finding is in very good agree-ment with the expectation from the relation below: C (cid:107) = β p / T e /T p )1 + β p (1 + T e /T p ) = β/
21 + β (2)where T e and T p are the electron and protontemperatures.Eq. (2) can be derived (Schekochihin et al., 2009;Boldyrev et al., 2013) under the assumption of low-frequency magnetic structures in pressure balanceat scales where the ion velocity becomes negligi-ble compared to the electron one, or equivalently Figure 8.
Cluster average spectra of magnetic com-pressibility for intervals with β e and β p < (red)and β e and β p > (blue); in the bottom panel, in-termediate values with similar average β but with β p < , β e > , and β p > , β e < are also shownin green and orange respectively. Thin dotted linesin the upper panel show the one-sigma dispersionof the data.the Hall term J × B becomes dominant over theideal MHD term − U × B . A special case is theregime of kinetic Alfv´en waves (KAW), howevereq. (2), which does not depend explicitly on k andthus on a specific dispersion relation, can be seen asa more general condition for highly oblique fluctua-tions in the sub-ion range (e.g. ion-scale Alfv´enicvortices, Jovanovic et al., 2020), under the assump-tions described above (see e.g. Appendix C2 ofSchekochihin et al., 2009). To improve our analysis we focus more in detailon the Cluster observations and compare them withnumerical results. Note that as in the simulations ofFranci et al. (2016) it is only considered the case β p = β e , we have made a selection of solar windspectra with similar properties ( β p ∼ β e ∼ β/ ).These have then been divided in 5 sub-groups asa function of β and averaged to obtain a mean C (cid:107) This is a provisional file, not the final typeset article atteini et al. Properties of sub-ion turbulence
Figure 9.
Top panels: (Left) Cluster spectra of magnetic compressibility for intervals binned on different β , encoded in different styles and colours. Only cases with β p ∼ β e have been retained. The horizontaldashed line denotes energy equipartition between components (i.e. isotropy). (Right) Spectra of magneticcompressibility for simulations with different β p = β e , shown with same style as left panel. The increase of C (cid:107) for kd i (cid:38) is due to numerical noise. Bottom: Same as top panels, but with k-vectors normalised withrespect to the ion gyro-radius ρ p . Horizontal dotted lines, coloured according to their β , are the theoreticalprediction of C (cid:107) from Eq.(2).profile for each β -family. The selection results in7, 13, 23, 9 and 1 spectra for β = 0 . , , , , respectively (only 1 spectrum fulfils the conditionfor high enough beta). Simulations with approxi-matively the same β p (and β ) are considered fora direct comparison. In the following analysis wewant to identify the physical scale associated to thechanges in the properties of the fluctuations and itspossible connection to either the ion Larmor radius ρ p or the inertial length d p , as they are related by: ρ p = (cid:112) β p d p . The results of this comparison are shown in Fig-ure 9, where scales are normalised to both d p (top)and ρ p (bottom). Left panels show spectra from insitu data and right panels results from simulations,where the colors encode the same range of β . Qual-itatively, the global trend seen in the simulationsmatches well that of the observations. First, thelevel of magnetic compressibility reached at sub-ionscales increases monotonically with β , as expected.Second, we can identify a plateau phase beyond ion Frontiers 11 atteini et al.
Properties of sub-ion turbulence scales whose extension is gradually reduced as β de-creases; for the smallest betas the plateau disappearsand is replaced by an almost monotonic increaseof C (cid:107) all along the sub-ion range - though with ashallower slope compared to that of the transitionfrom the MHD range.This seems to suggest a different behaviour of theturbulent fluctuations populating the sub-ion cas-cade as a function of the beta. To investigate furtherthis aspect, horizontal dotted lines in the right pan-els of Figure 9 show the theoretical prediction forthe asymptotic level of C (cid:107) between ion and electronscales predicted by Eq.(2), with same color scale.For simulations at large β , when a plateau is clearlyobserved, the level of magnetic compressibility alsoagrees well with the one predicted by the theory.In the low beta case there is a larger discrepancyand the observed level of magnetic compressibilityis larger than the constant level predicted by Eq.(2). The different behaviour of the compressibilityin low- and high-beta regimes found in our simu-lations, together with the larger discrepancy withrespect to the theoretical predictions observed atlow beta, are also consistent with results from previ-ous numerical studies (e.g. Cerri et al., 2016, 2017;Groˇselj et al., 2017).The situation is somewhat different when compar-ing predictions to the in situ data; in this case thereis a slight difference between the KAW level and theobserved one, and this is persistent at all β . In par-ticular, at high beta it is apparent that while Eq. (2)predicts a compressibility that goes beyond 1/3 (for β → ∞ we have C (cid:107) = 0 . , so δB (cid:107) = δB ⊥ ), a con-dition well recovered in the simulations, in Clusterdata C (cid:107) does not go beyond component isotropy( δB (cid:107) = δB ⊥ / , thus C (cid:107) = 1 / ). However, due tothe low statistics in the data (just 1 spectrum has β (cid:38) ) it’s hard to draw a firm conclusion here.Interestingly, from Figure 9 it seems that nor d p nor ρ p are able to fully capture and order the changein the spectrum of the magnetic compressibilityfor different betas; the saturation/plateau phase forlow β spectra results more shifted towards high k-vectors compared to the high β ones when nor-malising to d p , while the vice-versa is observedwhen normalising to ρ p . This suggests that the be-haviour can be better captured by an intermediatescale between the two. For this reason, in Figure 10we have normalised spectra to a mixed scale (cid:112) d p ρ p .Note that such a scale, proportional to d p β / p , wasfound to describe well the behaviour of ion-breakscale in magnetic field spectra in the range β p ∼ by Franci et al. (2016), and, although not shown, todescribes the variation of the break of the parallelmagnetic field spectrum at all betas; this then moti-vated our choice. When such a mixed scale is used(top right panel), all cases follow the same trend:they grow until they reach k (cid:112) d p ρ p ∼ and thenstart flattening, the saturation level depending onthe beta. In situ observations (top left panel) seemto follow the same trend, confirming that such anintermediate scale is a good candidate for control-ling the variation of the magnetic compressibilityspectrum at ion scales.It is then reasonable to use such a k-vector nor-malization to better evaluate the agreement withEq. (2). In the bottom panels of the same figure C ∗(cid:107) spectra are then normalised to the theoretical pre-diction for C (cid:107) . In simulations, as already pointedout, cases with β > display a good agreementwith the sub-ion compressibility level predicted bythe theory; as a consequence, when normalised to (cid:112) d p ρ p all spectra collapse on top of each other allalong ion and sub-ion scales. A worse agreementis observed at β ≤ when simulations display aslightly higher compressibility level than predicted.Quite differently, the ratio between the in situ ob-servations and the theoretical C (cid:107) is always below 1and around 0.7-0.8 for all β groups in the sub-ionrange (see also Figure 10 of Lacombe et al., 2017).This behaviour is consistent with the results of Pitˇnaet al. (2019) based on observations from the Windspacecraft, who find on average C (cid:107) ∼ . - with-out making a distinction among beta regimes - andwith most of the data displaying a slightly smallermagnetic compressibility than the prediction. Ourstudy confirms this scenario and suggests that the This is a provisional file, not the final typeset article atteini et al. Properties of sub-ion turbulence
Figure 10.
Same spectra as in Figure 9, but with k-vectors normalised to the mixed scale (cid:112) d p ρ p ; in thebottom panels C (cid:107) is normalised to the theoretical prediction by Eq.2.same trend is followed for all spectra, almost inde-pendently of the plasma beta. A ratio smaller than 1and close to ∼ . is also consistent with similarobservational results of the plasma compressibilityand based on the ratio between density and perpen-dicular magnetic fluctuations predicted by lineartheory (Chen et al., 2013a; Pitˇna et al., 2019). Thiswas interpreted by Chen et al. (2013a) as a conse-quence of the non-linear behaviour of the solar windfluctuations in the sub-ion range, in agreement withsimulations of strong KAW-turbulence (Boldyrevet al., 2013). On the other hand, for the magneticcompressibility, our fully non-linear simulations ofsub-ion turbulence do not recover the same effectseen in situ, as C ∗(cid:107) (cid:38) . Other reasons could explainsuch a discrepancy, e.g., the effect of some electronLandau damping on the fluctuations observed insitu (Howes et al., 2011; Passot and Sulem, 2015;Schreiner and Saur, 2017) and not captured by thehybrid model. In order to answer these questions, amore detailed study of the polarisation properties ofthe fluctuations in our simulations is in preparation. Finally, note that the increase in C (cid:107) observed athigher k in the in situ data could be related to afurther change in the properties of the fluctuationsas they approach electron scales; as discussed inLacombe et al. (2017) this also coincides with achange in the estimated spectral anisotropy. For ex-ample, Chen and Boldyrev (2017) have suggestedthat the increase in the magnetic compressibilitybeyond the sub-ion range could be related to elec-tron inertia corrections to Eq.2. This effect is thennot captured by the hybrid model and we cannotcompare any more the observations with the simula-tions in this range. It is however interesting to notethat while the further increase of compressibility atelectron scales is predicted for β e (cid:46) (Chen andBoldyrev, 2017; Passot et al., 2017), in the intervalsmeasured by Cluster it seems to be observed for allbeta ranges for kd p (cid:38) ( kd e (cid:38) / ). Moreover,it’s also interesting to note that spectra for all betasreach isotropy at roughly kρ p ∼ , correspondingon average to kρ e ∼ . . Frontiers 13 atteini et al.
Properties of sub-ion turbulence
In summary, we have discussed properties of mag-netic field spectra of turbulent fluctuations in thesub-ion regime and their main dependence on theplasma beta. We have carried out a detailed compar-ison between in situ Cluster magnetic field observa-tions in the frequency range f (Hz) = [1 , , corre-sponding to scales typically between d p < l < d e ,and high-resolution 2D hybrid simulations.First we investigated the spectral anisotropy ofmagnetic fluctuations at sub-ion scales. Our simu-lations confirm that the model of Saur and Bieber(1999), originally developed for MHD range fluc-tuations, is valid also at kinetic scales; by applyingthe model to the numerical spectra obtained mim-icking the sampling along a fixed direction madeby spacecraft, we were able to successfully captureoriginal spectral properties as well as their vari-ation with β . This then reinforces the finding ofLacombe et al. (2017) who applied the Saur andBieber model to kinetic-scale observations for thefirst time and concluded that fluctuations of the so-lar wind spectrum in the sub-ion range are quasi-2Dand gyrotropic. Moreover, we have shown that thecomponent anisotropy measured in situ - leading toan apparent non-gyrotropic spectrum from an orig-inal gyrotropic one (see also Turner et al., 2011) -is a direct consequence of the solenoidal conditionof the magnetic field and the sampling procedure.This is not an effect related to the Doppler-shift ofk-vectors swept through the spacecraft by the fastplasma flow and in fact, we were able to reproduceit in simulations just imposing a fixed samplingdirection.Note that our result about the global 2D-symmetryof the k-vectors around the magnetic field is not in-consistent with studies addressing the local shapeof the eddies and suggesting the presence of a 3Danisotropy (e.g. Chen et al., 2012a; Verdini andGrappin, 2015; Verdini et al., 2018, 2019; Wanget al., 2020). In our approach we do not considerthe specific orientation of the turbulent structures inthe plane perpendicular to B, and it is reasonable toexpect that their local 3D anisotropy is then lost. In other words, despite the possible presence of a 3Danisotropy of the turbulent eddies their k-vectorscan be oriented isotropically around B, leading then- in a frame like the one used here - to the 2D spec-trum found in the Cluster observations. This doesnot exclude that some aspects of the 3D anisotropycould be still captured using also a global approach,however, our study suggests that in this case onehas to take carefully into account the effects of theapparent component anisotropy introduced by thesampling (Saur and Bieber, 1999, see also Figure 3in this work).For the magnetic compressibility C (cid:107) , we have con-firmed that it has a strong dependence on the plasmabeta (e.g., Alexandrova et al., 2008; TenBarge et al.,2012; Lacombe et al., 2017). In particular we haveshown that in Cluster observations C (cid:107) depends onthe total beta β only (Fig. 8), as expected for low-frequency pressure-balanced fluctuations at highlyoblique propagation (e.g., KAW). In the β rangeexplored we find a good qualitative agreement be-tween the trend observed in the data and in thesimulations. The compressibility is observed to in-crease as a function of β , leading to a plateau atsub-ion scales for high betas and in good agreementwith the prediction by Eq. (2). At low beta, a fulldeveloped plateau is not observed beyond ion scalesand the compressibility continue to slowly increasealong sub-ion scales, both in observations and sim-ulations (see also Groˇselj et al., 2019). There is,however, a difference in the asymptotic level ofcompressibility reached at high β in data and oursimulations; in the former, fluctuations seem notto exceed component isotropy ( C (cid:107) = 1 / ), whilein the latter they approach C (cid:107) = 0 . , which is thelimiting value predicted by Eq. (2). This aspect de-serves to be explored in future studies, extendingthe range of β explored, to then establish if theasymptotic condition observed in simulations andpredicted by the theory, which implies same powerin the parallel component as in the sum of the per-pendicular ones, can be also observed in situ forhigh enough β intervals. As a consequence of thebehaviour just described, there is a different quan-titative agreement of the magnetic compressibility This is a provisional file, not the final typeset article atteini et al. Properties of sub-ion turbulence observed in situ and in simulations, with the theo-retical prediction by Eq. (2). In simulations thereis very good matching with the predicted level athigher beta, but an excess of C (cid:107) at low beta; thiseffect was already observed in Cerri et al. (2017)and is confirmed here on a large range of β . Onthe other hand, in solar wind observations, the ra-tio is always lower than 1 (smaller compressibilitythan predicted by the theory), and close to ∼ . for all β , in agreement with similar studies on theplasma compressibility (Chen et al., 2013a; Pitˇnaet al., 2019).Our analysis also suggests that the increase inthe compressibility at ion scales is controlled byan intermediate scale between the Larmor radius ρ p and the proton inertial length d p (Fig. 9). Forsimulations this was already anticipated in Franciet al. (2016), and we could identify it as relatedto (cid:112) d p ρ p , thus proportional to d p β / p (Fig. 10).Such a scaling with β p also corresponds to the scal-ing observed for the spectral ion-break in the range β p ∼ . However, it is worth to highlight that bothobservations (Chen et al., 2014) and our simulations(Franci et al., 2016) show that the spectral ion-breakscale follows the largest of ρ p and d p depending onthe beta, so that the correction term proportional to d p β / p identified in Franci et al. (2016) is impor-tant only around β p ∼ . On the other hand, thepresent study indicates that a scale proportional to (cid:112) d p ρ p orders well the spectra of compressibilityat all betas, for both in situ data and simulations,suggesting that such a mixed scale controls the tran-sition in the nature of the fluctuations from MHDto sub-ion range (see also the monotonic scalingwith β p of the ion-break in the parallel magneticfield spectrum shown in Fig.4 of Franci et al., 2016).This may imply that the two changes of regime -the steepening of the magnetic spectrum and the in-crease in the compressibility - can occur at differentscales for more extreme β values. In particular, weexpect the spectral break to occur at a larger scalewith respect to the plateau in the compressibilitywhen β p (cid:29) or β p (cid:28) , as in these cases (cid:112) d p ρ p is always smaller than the largest between ρ p and d p . A more detailed analysis on this aspect will be the subject of a future study, as well as the possi-ble implications of this behaviour for fluctuationsin the inner Heliosphere, where the plasma beta istypically lower than at 1AU, which can be observedby the Parker Solar Probe and Solar Obiter. APPENDIX: SYMBOL DEFINITIONSAND NORMALIZED UNITS
The subscripts ⊥ and (cid:107) refer to the direction withrespect to the ambient magnetic field B and p and e denote respectively protons and electrons. Allequations are expressed in the c.g.s. unity system. n and T denote the number density and the temper-ature of a species (we assume also n p = n e = n ). β e,p = 8 πnk B T e,p /B are the electron and pro-ton betas, and β = β p + β e is the total plasmabeta; here k B is the Boltzmann constant. For eachspecies of mass m and charge q , the inertial length d is defined a c/ω p , where ω p = (4 πnq /m ) / is the plasma frequency, and the Larmor radius ρ is defined as v th / Ω c where v th it the thermalspeed of each species and Ω c = q p B o /mc is thecyclotron frequency. V sw is the solar wind speedand f the frequency of the fluctuations measured bythe spacecraft; k denotes the module of the wavevector k . CONFLICT OF INTEREST STATEMENT
The authors declare that the research was conductedin the absence of any commercial or financial re-lationships that could be construed as a potentialconflict of interest.
AUTHOR CONTRIBUTIONS
LM anf LF performed the main analysis and pro-duced figures. OA and CL identified Cluster inter-vals, provided the in situ dataset, and contributedto the observational spectral analysis. PH providedthe hybrid code, LF performed the numerical simu-lations, and together with LM, SL, AV and EP, theydiscussed the use and interpretation of numericaldata. All authors contributed to the global interpreta-tion of the results, as well as to their discussion and
Frontiers 15 atteini et al.
Properties of sub-ion turbulence presentation in the manuscript. All authors revisedthe manuscript before submission.
FUNDING
This work was supported by the Programme Na-tional PNST of CNRS/INSU co-funded by CNES.It has also been funded by Fondazione Cassadi Risparmio di Firenze through the project HY-PERCRHEL. We acknowledge PRACE for award-ing us access to resource Cartesius based in theNetherlands at SURFsara through the DECI-13(Distributed European Computing Initiative) call(project “HybTurb3D”), and CINECA for the avail-ability of high performance computing resourcesand support under the ISCRA initiative (grantsHP10C877C4 and HP10BUUOJM) and the pro-gram Accordo Quadro INAF-CINECA 2017-2019(grants C4A26 and C3A22a). L.F. was supportedby Fondazione Cassa di Risparmio di Firenze,through the project “Giovani Ricercatori Protag-onisti”, and by the UK Science and TechnologyFacilities Council (STFC) grants ST/P000622/1 andST/T00018X/1. PH acknowledges grant 18-08861Sof the Czech Science Foundation. OA and CL aresupported by the French Centre National d’EtudeSpatiales (CNES).
ACKNOWLEDGMENTS
Authors acknowledge useful discussions with J.Stawarz, G. Howes, and A. Pitna.
DATA AVAILABILITY STATEMENT
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