Kinetic features for the identification of Kelvin-Helmholtz vortices in \textit{in situ} observations
A. Settino, D. Perrone, Yu. V. Khotyaintsev, D. B. Graham, F. Valentini
aa r X i v : . [ phy s i c s . s p ace - ph ] F e b Draft version February 9, 2021
Typeset using L A TEX twocolumn style in AASTeX62
Kinetic features for the identification of Kelvin-Helmholtz vortices in in situ observations
A. Settino,
D. Perrone, Yu. V. Khotyaintsev, D. B. Graham, and F. Valentini Dipartimento di Fisica, Universit`a della Calabria, 87036 Rende (CS), Italy Swedish Institute of Space Physics, Box 537 SE–751 21 Uppsala, Uppsala, Sweden ASI – Italian Space Agency, via del Politecnico snc, 00133 Rome, Italy (Received February 9, 2021; Revised February 9, 2021; Accepted February 9, 2021)
Submitted to ApJABSTRACTThe boundaries identification of Kelvin-Helmholtz vortices in observational data has been addressedby searching for single-spacecraft small-scale signatures. A recent hybrid Vlasov-Maxwell simulationof Kelvin-Helmholtz instability has pointed out clear kinetic features which uniquely characterize thevortex during both the nonlinear and turbulent stage of the instability. We compare the simulationresults with in situ observations of Kelvin-Helmholtz vortices by the Magnetospheric MultiScale satel-lites. We find good agreement between simulation and observations. In particular, the edges of thevortex are associated with strong current sheets, while the center is characterized by a low value forthe magnitude of the total current density and strong deviation of the ion distribution function froma Maxwellian distribution. We also find a significant temperature anisotropy parallel to the magneticfield inside the vortex region and strong agyrotropies near the edges. We suggest that these kineticfeatures can be useful for the identification of Kelvin-Helmholtz vortices in in situ data.
Keywords:
Space Plasmas — Space probes — Planetary magnetosphere — Interplanetary discontinu-ities INTRODUCTIONCoherent structures, such as flux tubes and vortices,are some of the main features in both fluids and plasmas,whose presence is strongly associated with the develop-ment of turbulence. Turbulent phenomena play an im-portant role in plasma transport, energy transfer acrossdifferent scales and dissipation mechanisms beyond theinertial range where they can eventually take place.Studies on both solar-wind and near-Earth plasma haveshown that ion scales are characterized by strong mag-netic discontinuities (Retin`o et al. 2007; Greco et al.2016; Perrone et al. 2016, 2017, 2020; Wang et al. 2019),which are connected through spatial scales from ion toelectron scales. Moreover, vortices have also been iden-tified at larger scales, mainly associated to the Kelvin-Helmholtz (KH) instability, even if the identification ofsuch vortices in in situ observations is not straightfor-ward.
Corresponding author: Adriana [email protected]
The KH instability is an ubiquitous phenomenon thatcan develop, in both ordinary fluids and plasmas, whena shear flow exists. For instance, an unstable config-uration is found when the velocity jump ∆ u is largerthan the component of the Alfv´en velocity parallel tothe bulk flow (Chandrasekhar 1961). Shear flows havebeen observed, for instance, at the interaction regionbetween fast and slow solar wind (Bruno & Carbone2013) and both teories and simulations have shown thatsuch sites are good candidates for the KH instabilityto grow (Korzhov et al. 1985; Ismayilli et al. 2018). Inaddition, in such regions, the wave-particle interactionwith the non-uniform velocity field can produce small-scale fluctuations leading to the dissipation of the waves(Pezzi et al. 2017; Valentini et al. 2012, 2017). KH in-stability has been observed in different plasma environ-ments, such as in the solar corona at the edge of coronalmass ejections (Foullon et al. 2011) and at the planetarymagnetospheres (Kivelson & Chen 1995; Fairfield et al.2003; Hasegawa et al. 2004, 2006; Foullon et al. 2010).In particular, at the Earth’s magnetopause the interac-tion with the shocked solar wind frequently leads to the Settino et al. formation of a so-called low latitude boundary layer andto the generation of KH vortices which propagate alongthe flanks of the magnetosphere and further towards thetail.The KH instability, during its nonlinear and turbu-lent phases, can lead to the formation of thin currentsheets which are possible sites for magnetic reconnec-tion. Such coupling between KH instability and mag-netic reconnection has been investigated by means ofnumerical simulations in both magnetohydrodynamics(MHD) and kinetic frameworks (see i.e. Nakamura et al.2013; Faganello & Califano 2017; Franci et al. 2020), aswell as in observational data. Indeed, the Magneto-spheric MultiScale (MMS) satellites are providing adeeper knowledge of the kinetic dynamics for KH in-stability (see i.e. Eriksson et al. 2016; Li et al. 2016;Sorriso-Valvo et al. 2019). Nonetheless, several KHevents has been observed during previous satellites mis-sions, namely Cluster and THEMIS, at both flanksof the magnetosphere (see Hwang et al. 2012, and ref-erences there in), providing also a statistical analy-sis of the dawn-dusk asymmetry (Henry et al. 2017;Kavosi & Raeder 2015).Since KH instability plays a central role in several phe-nomena in space plasmas and, especially, in the contextof near-Earth environment, it is crucial to identify KHvortices in order to better understand small-scale plasmadynamics. However, if the detailed study of KH vorticesin numerical simulations is straightforward, due to theknowledge of both temporal evolution and spatial be-havior, the identification of KH vortices in real spacedata, as collected by spacecraft, is very hard since onlyone point in space-time is provided and no informationabout the trajectory inside the vortex are available apriori .The main guidelines for the identification of KHvortices in observational data have been provided byHasegawa et al. (2004). Besides the observation ofquasi-periodic fluctuations, a rotating pattern can bedisplayed by the hodograms of the velocity and/or mag-netic perturbations. However, this vortical motion ofthe flux tubes can be clearly captured when the dis-tances among the four satellites is large, as in the caseof Cluster, whose average distance, during the detec-tion of KH instability, was about 2000 km. In such acase, it is impossible to study turbulent dynamics andall the related phenomena which characterize kineticscales. Moreover, MHD simulations of KH instability inthe Earth’s magnetospheric–like environment suggesteda specific pattern to use for the identification of highlyrolled-up vortices in in situ measurement: the presenceof lower density and faster than magnetosheath plasma regions (Takagi et al. 2006). Nonetheless, this featuredoes not uniquely identify KH rolled-up vortices, butcan also be a signature of different phenomena as forexample magnetosheath jets (Plaschke et al. 2014).Among the main quantities used for the identificationof KH vortices, there are also the vorticity vector andthe pressure minimum which are, however, inevitably af-fected by the choice of a threshold (Hussain & Hayakawa1987; Hunt et al. 1988). In order to overcome suchissues, mathematical techniques have also been devel-oped for the identification of vortices in both fluids(see Jeong & Hussain 1995; Kida & Miura 1998, fora review) and magnetized plasmas (Cai et al. 2018).Nonetheless, it is worth noting that these criteria havealso some important limitions: i) the necessity of amulti-spacecraft analysis because they are based on theestimation of eigenvalues for the gradient of velocity (ormagnetic) field vector; ii) the strong dependence of thevortex dimension on the relative distance between thespacecrafts.The aim of our work is to provide new quantities thatcan be used as guidelines for the identification of theboundaries of KH vortices using only single-spacecraftmeasurements. In this paper, we present a compari-son of small-scale signatures observed in both simulationand in situ data for a KH event. Recent hybrid Vlasov-Maxwell simulations have shown the presence of clearkinetic features in the vortices during both the nonlin-ear and turbulent stage of KH instability (Settino et al.2020). Here, we investigate the presence of those fea-tures in a KH event observed by MMS during a pe-riod of northward interplanetary magnetic field. Lead-ing and trailing edges of KH vortices have already beenidentified for this event by Hwang et al. (2020) andKieokaew et al. (2020). The paper is organized as fol-lows: in Section 2 we discuss hybrid Vlasov-Maxwellsimulation of KH instability and we present the fea-tures that identify the KH vortices during the nonlinearregime; in Section 3 we describe these same quantitiesas observed in the MMS data; in Section 4 we discussand compare the results from both simulation and ob-servational data, pointing out kinetic signatures in thedistribution functions and, finally, in Section 5 we pro-vide conclusions. KH VORTICES IN VLASOV SIMULATIONSWhen KH instability develops at ion scales, mean-ing that the thickness of the shear layer is of theorder of the characteristic ion scales, such as at theEarth’s magnetopause, kinetic effects come into playand a kinetic numerical approach can better describethe dynamics of the instability compared with a fluid dentification of Kelvin-Helmholtz vortices y / d i (a) Ion Pressure (b) Ion Vorticity (c) Total current density x/d i y / d i (d) Ion non-Maxwellianity x/d i (e) Ion Temperature Anisotropy x/d i (f) Ion Agyrotropy Figure 1.
Two-dimensional contour plots for the HVM simulation at a fixed time during the turbulent stage of the KHinstability. (a) Ion kinetic pressure, P i = n i T i (in normalized code units); (b) magnitude of the ion vorticity, | ω i | , and (c) totalcurrent density, | j | ; (d) ion non-Maxwellianity, ǫ M ; (e) ion temperature anisotropy, T ⊥ /T k ; and (f) ion agyrotropy, √ Q . Thevertical dashed lines indicate a one-dimensional path in the two-dimensional box domain which crosses the vortex in the center.The spatial behavior of these quantities, along the 1D cut, will be shown in Figure 3. The white star and magenta circle in eachiso-contour indicate the two spatial positions selected for investigating the ion distribution function (see Figure 4). approach (Nakamura et al. 2013; Henri et al. 2013;Karimabadi et al. 2013; Rossi et al. 2015). An impor-tant step for a correct description of the KH instabilityis to set up the equilibrium unperturbed state. How-ever, the choice is not trivial since this setting has strongconsequences for the onset of the instability. The sim-plest choice is to use a shifted Maxwellian distributionfunction, with the associated moments varying in space(Umeda et al. 2014). However, a shifted Maxwellian isnot an equilibrium distribution in a kinetic frameworkand can give rise to spurious oscillations of the orderof the ion gyroperiod, which can affect the dynamics ofthe instability itself.An exact stationary solution for the hybrid Vlasov-Maxwell system of equations, which describes a magne-tized shear flow, has recently been found by Malara et al.(2018) and it has been used to study in detail the dy-namics of the KH instability during its nonlinear andturbulent stages, focusing on the kinetic effects at ionscales (Settino et al. 2020). In the present paper, weuse the results described in Settino et al. (2020) for the exact kinetic equilibrium to select quantities able toidentify KH vortices in in situ data.The KH instability has numerically been studiedby means of the Hybrid Vlasov-Maxwell (HVM) code(Valentini et al. 2007) in a 2D-3V phase space configura-tion (two dimension in physical space and three dimen-sions in velocity space), with a uniform magnetic field, B , perpendicular to the velocity shear. The Vlasovequation is integrated for the ions, while electrons aretreated as a massless fluid, whose response is takeninto account through a generalized Ohm’s law for theelectric field. The quasi neutrality condition is adopted( n i = n e ) and the electron pressure is considered as afurther independent quantity (see Malara et al. 2018,for details). The system is perturbed with a broad-band spectrum of velocity fluctuations, generated inform of random noise. Finally, the ion plasma beta is β i = 2 v th /v A = 2, being v th and v A the ion thermaland the Alfv´en speed, respectively; while the electronto ion temperature ratio is T e /T i = 1. The followingnormalization for the HVM set of equations has beenused: density is normalized by n (the density far from Settino et al. the shears); time by the inverse proton cyclotron fre-quency Ω cp = eB /m i (where e is the electron chargeand m i is the ion mass); velocity by the Alfv´en speed v A = B / √ πn m i ; lengths by the ion skin depth, d i = v A / Ω cp ; magnetic field by B , the electric fieldby v A B /c (being c the speed of light) and pressure by n m i v A The unperturbed configuration is characterized by avelocity shear along the y -direction and varying along x ,imposed as a hyperbolic tangent profile which has beenduplicated to satisfy periodic boundary conditions in thephysical space. The rotational motion, induced by thevelocity shear, leads to the generation of a centrifugalforce and, as a consequence, the formation of vorticesalong the two shear layers. During the nonlinear stagethe vortices start merging, the ones at each of the shearlayer and then the two shear layers interact. Moreover,an enhancement of the space averaged total current den-sity occurs during the evolution of the instability, dueto both the mixing of vortices at large scales and thenonlinear coupling of modes at short wavelengths.The centrifugal force becomes stronger toward thecenter of the vortex, so that a pressure gradient is gen-erated to balance it. Therefore, a local ion pressureminimum inside the vortex is recovered, as shown in thecontour plot of P i in panel (a) of Figure 1. Moreover,the strong swirling motion enhances the magnitude ofthe ion vorticity, namely | ω i | = |∇ × u i | , which peaks atthe edges of the vortices, but, inside them, still reachesvalues higher than the background vorticity (see panel(b)). The spatial variation of the local ion plasma beta(not shown) indicates that the ion pressure is every-where higher than the magnetic pressure ( β i >
1) and,in particular, it peaks within the KH vortices. There-fore, the magnetic field is carried by the vortical flowsand the field lines are twisted and rolled-up according tothe whirling motion of the plasma. Moreover, the mag-netic field lines are highly distorted in correspondence ofthese structures and the generation of strong turbulentactivity is also found. Indeed, in panel (c) of Figure 1,an enhancement in the magnitude of the total currentdensity, namely | j | , is observed at the edges of the KHvortices, while a minimum is found at the center.During the nonlinear phase of the KH instability, thenonlinear coupling of the modes generates an energy cas-cade towards small scales and kinetic effects come intoplace, through complicated non-Maxwellian deforma-tions. Indeed, the ion distribution functions are foundto be far from thermodynamical equilibrium. Deviationsfrom the Maxwellian shape have been quantified via thenon-Maxwellian parameter, defined for a fixed time as (Greco et al. 2012) ǫ M = 1 n i sZ [ f i − g M ] d v (1)where f i is the actual distribution function and g M isthe associated Maxwellian built with the moments of f i .The contour plot of ǫ M , in panel (d) of Figure 1, showsan opposite behavior with respect to | j | . Indeed, whilethe total current density provides information about theboundaries of the vortices, the non-Maxwellianity peaksinside the structure, allowing the identification of thevortex core. Departures from the thermodynamic equi-librium are also observed in the iso-contours of both theion temperature anisotropy ( T ⊥ /T k , where T k and T ⊥ are the temperatures in the direction parallel and per-pendicular to the local magnetic field, respectively) andthe ion agyrotropy ( √ Q ), shown in panels (e) and (f)of Figure 1, respectively. The agyrotropy is defined as(Swisdak 2016) Q = P xy + P xz + P yz P ⊥ + 2 P ⊥ P k ; (2)where P ij are the components of the ion pressure tensorin the reference frame in which one of the axes is alongthe local magnetic field and the two perpendicular pres-sures are equal. √ Q ranges from 0 (fully gyrotropic con-figuration) to 1 (maximum agyrotropy). We found that T ⊥ /T k has a pattern similar to ǫ M , while √ Q exhibitsa behavior similar to those visible in the contour plotsof | j | . Indeed, the strongest anisotropies are observed inthe center of each vortex (same as ǫ M ), while √ Q peaksat the edges of the vortices (similar to | j | ). KH VORTICES IN MMS DATAOn the May 5 2017, between 19:30:00 and 21:00:00UT, MMS was located at the dawn flank of the Earthduring a period of mostly northward interplanetarymagnetic field. In particular, MMS was at [ − , − , ∼
156 km. Considering thatthe ion inertial length, d i , is ∼
100 km in the mag-netosheath side and ∼
230 km in the magnetosphericside, the spacecraft separation allows detailed studies ofkinetic scales. During this interval, MMS observed sev-eral fluctuations with a period between 2 . dentification of Kelvin-Helmholtz vortices MMS4 E i [ e V ] (a) D E F k e V / ( c m s s r k e V ) n i [ c m - ] (b) -1001020 B G SE [ n T ] | B |B x B y B z (c) P i [ n P a ] (d) | i | [ H z ] (e) | j | [ n A / m ] | j | plasma| j | curlometer (f) M (g) T / T || (h) (i) Figure 2.
MMS data in the interval 20:01–20:12 UTC on May 5 2017. Measurements are averaged on ∼
1s and all coordinatesare in GSE. From top to bottom: (a) ion energy spectrogram; (b) ion density; (c) magnetic field; (d) ion pressure; (e) magnitudeof the ion vorticity;(f) magnitude of the total current density evaluated from particle moments (black) and with the Curlometertechnique (red); (g) dimensionless ion non-Maxwellianity defined as in Equation 1 (black), and Equation 3 (red); (h) iontemperature anisotropy, where the horizontal red line indicates isotropic distribution function; and (i) ion agyrotropy. Coloredshaded areas highlight two KH vortices and the vertical solid (dashed) lines indicate the leading (trailing) edge of the vortex.
In the present analysis, we focus on the time in-terval 20:01:00–20:12:00 UTC to investigate the be-havior of the significant quantities for the identifica-tion of KH vortices highlighted by the HVM simula-tion. We use magnetic field data from fluxgate mag-netometer (Russell et al. 2016) sampled at 128Hz, andthe ion data from Fast Plasma Investigation instrument(Pollock et al. 2016) with a resolution of 150 ms . Anoverview of the interval is shown in Figure 2, where thecrossing of the two vortices by MMS4 has been markedby the green and yellow shaded areas. Moreover, lead-ing and trailing edges are indicated by vertical solid anddashed lines, respectively, corresponding to the timesprovided by Hwang et al. (2020). The energy spectrogram of ions (panel a) displaysthe crossing of a mixing layer by the spacecraft. In-side the vortices (shaded areas) a less mixed region ofmore magnetospheric-like ions can be recognized. Thisfeature is in agreement with the crossing of a lead-ing edge of the vortex because of the centrifugal forcewhich tends to confine the low density and high en-ergy plasma (magnetosphere-like) close to the vortexcore (Hasegawa et al. 2004; Hwang et al. 2012, 2020).The same behavior can be recognized in the ion density, n i , shown in panel (b). Indeed, two clear minima areobserved in correspondence with each vortex and closeto the trailing edge (vertical dashed lines). It is worthpointing out that the presence of such low density re- Settino et al. gions indicates a low number of particle counts insidethe vortices and, thus, a possible increase of uncertaintyin the particle measurements. Therefore, in order to im-prove the counting statistics and to have reliable obser-vations, we averaged all the quantities shown in Figure2 over seven time steps (roughly 1 s), which correspondsin terms of spatial scale to ∼ d i .Panel (c) shows the magnetic field in the GSE coor-dinate system. It can be easily seen that B is directedmostly northward along the whole interval (black andred lines), while an inversion in the sign of B y and B x arefound, in agreement with the crossing of vortex bound-aries. Panel (d) shows the ion pressure, P i = k B n i T i ,where k B is the Boltzmann constant and T i is the ionscalar temperature. The pressure inside the first vortex(green shaded area) is lower than the surrounding re-gions, while for the second vortex (yellow shaded area)we do not see this decrease in pressure. This differentfeature can be connected to the presence, before the sec-ond vortex, of a Flux Transfer Event (FTE) as reportedby Kieokaew et al. (2020), and/or to a high value of thebackground pressure. Indeed, the FTE determines anincrease of the magnetic pressure, playing the main rolein balancing the centrifugal force.In panels (e) and (f) we show the ion vorticity, ω i ,and the total current density, j , respectively, estimatedthrough multi-spacecraft techniques. In particular inpanel (f) we compare the magnitude of the current den-sity, by using both the single-spacecraft plasma mea-surements j = en i ( V i − V e ) (black line), where thequasi-neutrality assumption has been used, and the Cur-lometer technique (red line) (Dunlop et al. 1988, 2002).Since the behavior of these two quantities is similar, weare confident that the multi-spacecraft technique workswell. Thus, we estimate the ion vorticity in the samemanner as in Perri et al. (2020). We find that | ω i | peaksat the boundaries of the vortices, although strong spikesare also observed inside. It is worth pointing out thatstrong spikes are observed along the whole interval sug-gesting that the level of the background vorticity (due tothe presence of the velocity shear) is comparable to thevorticity enhancement connected to the rotational mo-tion. Thus, it is difficult to distinguish vortex bound-aries by using only this quantity. On the contrary, aclearer behavior is found for the magnitude of the totalcurrent density in panel (f). We find strong spikes at theedges of both the vortices, while low values are generallyobserved inside them.Figure 2g shows the departure from a Maxwellian dis-tribution, ǫ M . We use the same definition as in Equa-tion 1, multiplied by v / A (being v A the Alfv´en speedin the magnetosheath) in order to have a dimensionless quantity. Moreover, for completeness, we use a seconddefinition for the non-Maxwellianity, recently introducedin the framework of MMS observations (Graham et al.,2021, in prep. )˜ ǫ M = 12 n i Z | f i − g M | d v, (3)a dimensionless quantity that ranges from 0 (Maxwelliandistribution) to 1 (highest deviation from a Maxwellian).To reduce the artificial increase of non-Maxwellianitydue to the noise associated with the lowest energy chan-nels, we have integrated, both ǫ M and ˜ ǫ M , from 15 eVto 30 keV (see horizontal lines in panel (a) of Figure 2).Furthermore, to reduce the impact of the low countsstatistics inside the vortices, we have estimated both˜ ǫ M and ǫ M by averaging the ion distribution function on ∼ ǫ M (red line) and ǫ M (black line) display a sim-ilar trend. Indeed, the correlation between the two non-Maxwellianity parameters is very high, with a correla-tion coefficient of ∼ . T ⊥ /T k , and the ion agy-rotropy, √ Q , respectively. The definition for these quan-tities is the same used in Section 2. Anisotropic distri-bution functions are observed inside the vortices, whilestrong agyrotropy is found near the edges, particularlyevident for the second vortex (yellow shade). Moreover,the change of the anisotropy direction in panel (h) de-fines the boundary of the vortices. Indeed, it passes froma parallel ( T ⊥ /T k <
1) to a perpendicular ( T ⊥ /T k > DISCUSSIONIn this section we compare the results from the hy-brid KH simulation (Section 2) with MMS observations(Section 3). It is worth pointing out that, although amulti-spacecraft mission has been used for the present dentification of Kelvin-Helmholtz vortices HVM simulation (a)E C0.020.040.060.080.10.120.14 C (b)E0.020.040.060.080.1 CE (c)0.010.020.030.040.05 CE (d)0.811.2 CE (e)30 40 50 60 70 80 90 y/d i -3 CE (f)
MMS4 P i [ n P a ] C E (a) | i | [ H z ] C E (b) | j | [ n A / m ] C E| j | plasma| j | curlometer (c) M C E (d) T / T || C E (e)
C E (f)
Figure 3. (Left) simulation: one-dimensional cuts of the quantities shown in Figure 1, along vertical dashed lines at x = 53 . d p .(Right) MMS: zoom of the data in Figure 2. From top to bottom: (a) ion kinetic pressure; (b) magnitude of the ion vorticity;(c) magnitude of the total current density; (d) ion non-Maxwellianity; (e) ion temperature anisotropy; and (f) ion agyrotropy.The green shaded area highlights the vortex, while the magenta circles and the white stars indicate the points selected for theinvestigation of the ion distribution function at the edge (E) and inside (C) the vortex. work, the quantities we suggest for the identification ofthe vortices can be easily estimated with high-resolvedsingle-spacecraft measurements. Moreover, since in thesimulation we focus on a vortex at a fixed time duringthe nonlinear phase of the instability, we assume thatthe vortex observed by MMS does not evolve duringthe spacecraft crossing. Indeed, the large scale vorticesare propagating faster than the spacecraft and the Tay-lor hypothesis is therefore valid. To be more quantita-tive, the spacecraft speed, V sc , is of the order of 2 km/s,while the vortex speed can be evaluated as defined byOtto & Fairfield (2000): V vort = V i n MSH − n MSP n MSH + n MSP , (4)where V i is the ion bulk velocity averaged in the timeinterval analyzed and n MSH(MSP) is the ion density av-eraged in the magnetosheath (magnetospheric) side. Inour case V vort = 120 km/s ( ≫ V sc ). Since the spacecraftcan be considered at rest, we can assume that vorticesare observed in the same evolutionary phase. Moreover,the evolution of the KH vortices generally occurs whilethey are propagating towards the tail.Since in situ measurements only provide informationon a one-dimensional trajectory in time/space, we havedecided to choose a 1D path of the numerical quantitieswhich crosses the center of the vortex, i.e. x = 53 . d i in Figure 1 (vertical dashed lines) compared with theMMS4 observations (right panels). Moreover, for a one-to-one comparison, we have decided to consider, for the MMS observations, only the crossing of the first vortex(green shading in Figure 2). The results for the simula-tion (left) and MMS (right) data are shown in Figure 3,where the vortex regions are highlighted by the greenshaded areas.Both HVM simulation and in situ data show a localminimum for the ion kinetic pressure, P i , within thevortex (see panels a), in agreement with the physicalmechanism that produces this structure. Indeed, therotational force leads to the generation of a pressure gra-dient which balances the force itself. However, while thepressure minimum is clearly visible in the HVM simula-tion, much more fluctuations are observed in the MMSdata. In addition, as discussed in the previous section,no pressure minimum is observed in the second vortex(yellow shade in Figure 2), suggesting that P i is not agood parameter to identify KH vortices in space obser-vations. A similar argument can be used for the vorticityshown in panels (b). Indeed, for the HVM simulation,evident peaks at both the edges and inside the vortexare observed. In the MMS case, instead, strong spikesare not limited to the vortex region but are observed allalong the interval.Panels (c) show the magnitude of j . We found thatthe edges of the vortex are not a thin structure, but in-stead display a multiple filamentary structure (alreadyobserved in Figure 1c). High values of | j | are observedat the edges of the vortex and even if few small spikescan be recognized inside the shaded region, they still re-main lower than the outside region. Moreover, really low Settino et al. -2 0 2 v ExB /v th -3-2-10123 v B / v t h (a) -2 0 2 v Bx(ExB) /v th -3-2-10123 v B / v t h HVM simulationEDGE (b) -2 0 2 v Bx(ExB) /v th -3-2-10123 v E x B / v t h (c) -2 0 2 v ExB /v th -3-2-10123 v B / v t h (d) -2 0 2 v Bx(ExB) /v th -3-2-10123 v B / v t h CENTER (e) -2 0 2 v Bx(ExB) /v th -3-2-10123 v E x B / v t h (f) Figure 4.
Two-dimensional contour plots of the reduced ion distribution function at E = (53 . , . d i (top row) andat I = (53 . , . d i (bottom row). From left to right, the ion velocity distribution is shown in the planes ( v E × B , v B ),( v B × ( E × B ) , v B ), and ( v B × ( E × B ) , v E × B ), respectively. values are reached close to the trailing edge of the vor-tex (vertical dashed line) where the non-Maxwellianityis high (panel d on the right). Indeed, ǫ M displays anopposite behavior respect to | j | , since it is enhanced in-side the vortex, while decreases in correspondence of theboundaries.At the edge of the vortex, the transition from the out-side region to the vortex structure is well representedby a change in the direction of the ion temperatureanisotropy (see panels e). However, while in the HVMcase we pass from T ⊥ > T k to T ⊥ < T k (crossing the redhorizontal line, which identifies isotropy), in the MMScrossing the inverse is observed. Moreover, while numer-ical results show T k > T ⊥ in the center of the vortex,MMS observe either parallel and perpendicular temper-ature anisotropies inside the green shade. In Figure 3f,we show the ion agyrotropy which is characterized by asimilar behavior with respect to | j | . Indeed, the highestvalues are observed at the edges, while a minimum isfound inside the vortex region. The comparison between hybrid simulation and MMSmeasurements suggests that the combined use of thenon-Maxwellianity parameter and of the magnitude oftotal current density can preliminary identify the KHvortex region and provides information about the edgesand the center of the vortices. Moreover, the peaks inthe agyrotropy and the change of direction in the tem-perature anisotropy can be used for the identification ofthe outer boundaries of the vortex. At this stage, it isinteresting to consider the effects of the KH instabilityon the ion velocity distribution function (VDF), focus-ing on two distinct locations in physical space, namelythe outer edge of the vortex, where | j | has a peak while ǫ M is low, and the center of the vortex, in correspon-dence of an enhancement of ǫ M and the minimum of thetotal current density. These two points, E and C , havealso been indicated in both Figures 1 and 3 as a magentacircle and a white star, respectively.In Figure 4, we show the two-dimensional contourplots of the numerical ion VDF at E (top) and C (bot- dentification of Kelvin-Helmholtz vortices -2 0 2 v E x B /v th -3-2-10123 v B / v t h (a) -2 0 2 v B x( E x B ) /v th -3-2-10123 v B / v t h MMS4 - 2017-05-05T20:05:08.10 + 1.05 sEDGE (b) -2 0 2 v B x( E x B ) /v th -3-2-10123 v E x B / v t h (c) -2 0 2 v E x B /v th -3-2-10123 v B / v t h (a) -2 0 2 v B x( E x B ) /v th -3-2-10123 v B / v t h MMS4 - 2017-05-05T20:04:12.00 + 1.05 sCENTER (b) -2 0 2 v B x( E x B ) /v th -3-2-10123 v E x B / v t h (c) Figure 5.
Two-dimensional reduced ion velocity distributions at the edge (top row) and inside (bottom row) the vortexhighlighted in Figure 2 by the green shadow. The same reference frame used for the simulation results in Figure 4 has beenchosen. tom), integrated along the out-plane direction, wherethe velocity grid has been normalized to the ion thermalspeed. From left to right, the VDF is shown in the planes( v E × B , v B ), ( v B × ( B × E ) , v B ), and ( v B × ( E × B ) , v E × B ), re-spectively. The ion VDF appears to be strongly dis-torted both at the edge and center of the vortex, butdifferent features can be recognized. In particular, atthe edge of the vortex, a strong non-gyrotropic VDF isobserved (panel c), with a significant elongation in theoblique direction. Moreover, while in panel (b) the VDFis mostly isotropic, in panel (a) a small beam around v th can be observed in the direction perpendicular tothe magnetic field. The presence of these acceleratedparticles could be generated by the sharp changes in themagnetic field that characterize the vortex boundaries.The field lines are highly distorted by the rotational mo-tion of the plasma and, as a consequence, strong currentsare generated. On the other hand, at the center of thevortex, the VDF is almost gyrotropic (see panel f) and is significantly elongated in the direction parallel to themagnetic field (see both panels d and e).In Figure 5 we show the reduced ion VDF observedby MMS4 at both the edge of (top) and inside (bottom)the vortex. The time at which the two VDFs have beenpicked is also indicated in the right panels of Figure 3with a magenta circle (edge of the vortex) and a whitestar (center of the vortex). We averaged the VDFs overseven time steps to improve the counting statistics. Inthe chosen averaging time interval, all the VDFs havethe same behavior, since they lie in regions with thesame characteristics, i.e. high | j | and low ǫ M at theedge of the vortex (and the opposite in the center).At the edge of the vortex (top) some common fea-tures with the numerical simulation can be observed(see Figure 4). The ion distribution function is stronglyagyrotropic, with an elongation in the oblique direc-tion (panel c). It is worth pointing out that high non-gyrotropic signatures have also been observed in theelectron distribution function in correspondence of the0 Settino et al. boundaries of KH vortices in a fully kinetic KH simu-lation, while a mostly gyrotropic distribution has alsobeen found inside the vortex (Nakamura et al. 2020).The presence of the same trend found in KH vortices atelectron scales and in different simulations suggests thatsuch feature is independent of both scalelength and ini-tial conditions. Moreover, in panel (a) a clear beam canbe observed in the plane perpendicular to the magneticfield and along the positive direction. Here, we arguethat in our simulation the beam at the thermal speedseems less evident because of the high value of the ionplasma beta ( β i = 2) which means a more spread for thecore of the ion distribution, compared with the average β i = 0 . v B × ( E × B ) (panel b), in contrast with the mostly isotropic distribu-tion function observed in the numerical simulation. Thisis probably due to the transition from magnetospheric-like to magnetosheath-like particle populations in corre-spondence of the edges of the vortices, not described inour simulation.Finally, at the center of the vortex (bottom), we find agood agreement with the HVM simulation. We observedan almost gyrotropic ion distribution and a significantelongation is found in the direction parallel to the mag-netic field. However, unlike what is found in the simula-tion, where a parallel temperature anisotropy is presentin the whole vortex region, inside the vortex crossed byMMS also distributions with significant elongation inthe direction perpendicular to the local magnetic fieldcan be observed. CONCLUSIONSIn this paper, we have compared in situ data of a KHevent and results from a numerical simulation, suggest-ing new quantities to help the detection of KH vorticesin single-spacecraft measurements. A very recent hy-brid Vlasov simulation of KH instability (Settino et al.2020) has shown some interesting features which ariseat kinetic scales. Motivated by these clear small-scalesignatures, we have decided to test for their presencein a KH event observed by MMS for which the trail-ing and leading edges of the vortices had already beenrecognized (Hwang et al. 2020). We point out that oursimulation has been performed in different initial con-ditions respect to the MMS event, but nonetheless theresults are in reasonable agreement, being, the identifiedpatterns, an intrinsic characteristic of the vortices.We found, in both synthetic and in situ data, a clearenhancement of the magnitude of the total current den- sity at the edges of the vortex, followed by a minimumnear the center of the vortex itself. A opposite be-havior is observed for the ion non-Maxwellianity, whichpeaks inside the vortex, denoting a high distortion ofthe distribution function, and is low at the edges. More-over, in correspondence of these strong non-thermal fea-tures, also temperature anisotropy and agyrotropy areobserved. In particular, we found that the ion agy-rotropy behaves similarly to the magnitude of the totalcurrent density, since it reaches a local minimum insidethe vortex, while increases at the boundaries. A changein the direction of the ion temperature anisotropy is alsoobserved at the edges of the vortex.The main techniques to detect KH vortices inplasmas are based on multi-spacecraft approaches(Hasegawa et al. 2004; Cai et al. 2018), while singlespacecraft techniques have strong limitations (Jeong & Hussain1995; Kida & Miura 1998; Plaschke et al. 2014). For in-stance, techniques based on local pressure minima andenhancement in the vorticity magnitude fail when thebackground fluctuations are comparable to the valueinside the vortex or when the dynamics of the KH in-stability is affected by other ongoing phenomena. Theinefficiency of such quantities, namely the kinetic pres-sure and the vorticity magnitude, is also recovered inour analysis, since a well defined minimum of the ionkinetic pressure is observed inside the KH vortex of thehybrid simulation (Figure 3a), in agreement with the3D fully kinetic PIC simulation in southward interplan-etary magnetic field condition (Nakamura et al. 2020),but not in MMS data (Figure 2a). Indeed, a flux trans-fer event has been observed in the MMS interval weconsidered (Hwang et al. 2020; Kieokaew et al. 2020),suggesting that the absence of a pressure minimum isconnected to the increase of the magnetic pressure.Our analysis suggests that the investigation of the to-tal current density, non-Maxwellianity and both tem-perature anisotropies and agyrotropy enables the iden-tification of KH vortices in space measurements, requir-ing only a single satellite and a good resolution for par-ticle instruments. Therefore, we suggested that thesequantities could be also used in the framework of thenew ESA’s Solar Orbiter mission (M¨uller et al. 2020),launched in February 2020, to investigate the presenceof KH vortex structures at the interaction region be-tween fast and slow solar wind, providing a significantinsight on the instability and more generally on kineticeffects in the near-Sun solar wind.ACKNOWLEDGMENTSNumerical simulations have been run on Marconi su-percomputer at CINECA (Italy) within the ISCRA dentification of Kelvin-Helmholtz vortices