Sub-ion scale Compressive Turbulence in the Solar wind: MMS spacecraft potential observations
Owen Wyn Roberts, Rumi Nakamura, Klaus Torkar, Yasuhito Narita, Justin C. Holmes, Zoltan Voros, Christoph Lhotka, C. Philippe Escoubet, Daniel B. Graham, Daniel J. Gershman, Yuri Khotyaintsev, Per-Arne Lindqvist
aa r X i v : . [ phy s i c s . s p ace - ph ] S e p Draft version September 2, 2020
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Sub-ion scale Compressive Turbulence in the Solar wind: MMS spacecraft potential observations
Owen Wyn Roberts, Rumi Nakamura, Klaus Torkar, Yasuhito Narita, Justin C. Holmes, Zolt´an V¨or¨os,
1, 2
Christoph Lhotka, C. Philippe Escoubet, Daniel B. Graham, Daniel J. Gershman, Yuri Khotyaintsev, andPer-Arne Lindqvist Space Research Institute, Austrian Academy of Sciences, Graz, Austria Geodetic and Geophysical Institute, Research Centre for Astronomy and Earth Sciences (RCAES), Sopron, Hungary ESA, European Space Research and Technology Centre, Noordwijk, Netherlands Swedish Institute of Space Physics, Uppsala, Sweden Goddard Space Flight Center, National Aeronautics and Space Administration, Greenbelt, MD, United States Department of Space and Plasma Physics, KTH Royal Institute of Technology, Stockholm, Sweden (Received; Revised; Accepted)
Submitted to ApJABSTRACTCompressive plasma turbulence is investigated at sub-ion scales in the solar wind using both the FastPlasma Investigation (FPI) instrument on the Magnetospheric MultiScale mission (MMS), as well asusing calibrated spacecraft potential data from the Spin Plane Double Probe (SDP) instrument. Thedata from FPI allow the sub-ion scale region ( f sc & f sc ≈ f sc ≈ Keywords: plasma turbulence, spacecraft charging, solar wind INTRODUCTIONThe solar wind is an excellent example of a turbulent plasma, which can be easily accessed for in situ plasma measure-ments. Plasma turbulence is characterized by disordered fluctuations in the electromagnetic fields (Bale et al. 2005;Salem et al. 2012) as well as flow velocity and temperature (Podesta et al. 2006; Safrankova et al. 2013; Roberts et al.2019). Moreover, the solar wind plasma is weakly compressive with fluctuations present in the magnitude of themagnetic field and the density (Hnat et al. 2005; Roberts et al. 2017c, 2018). Fluctuations are seen from the scale ofthe largest eddies at around 10 km (Matthaeus et al. 2005) down to the scales of the electron gyroradius at ∼ Corresponding author: Owen Wyn [email protected]
Roberts et al. function. An alternative approach to derive the electron density which can be used in the solar wind is to use thespacecraft potential. Using the lower time resolution electron density data from FPI, a calibrated measurement of theelectron density from the spacecraft potential can be obtained with much higher time resolution than is possible usingthe direct measurement.To investigate the compressive fluctuations in the magnetic field either a mean magnetic field direction needs tobe defined or for small fluctuations ( δB/B ≪
1) the magnitude of the magnetic field can be used. At large scales,a fluid description is appropriate and magnetic and density spectra often show a Kolmogorov like spectral index of − /
3. This region is often termed the inertial range (Tu & Marsch 1995; Smith et al. 2006; Bruno & Carbone 2013).When fluctuations approach ion scales, kinetic or Hall effects become important, a break is seen in the magnetic fieldpower spectra (Leamon et al. 1998; Markovskii et al. 2008; Bourouaine et al. 2012; Chen et al. 2014a; Bruno & Trenchi2014) which is followed by a steepening in the spectra. At the sub-ion scales the fluctuations become more compressive(Kiyani et al. 2013; Roberts et al. 2017a) and the morphology of the magnetic spectrum is unclear (Alexandrova et al.2009, 2012; Sahraoui et al. 2013). The magnetic field spectrum is typically observed to have a spectral index closeto -8/3 but can be variable due to coherent events such as parallel whistler waves or coherent structures (Lion et al.2016; Roberts et al. 2017a).The density spectrum has a slightly different morphology, where a flattening is often seen between the ion inertialand the ion kinetic ranges (Unti et al. 1973; Neugebauer 1975, 1976; Celnikier et al. 1983). This flattening in thespectra has been observed to have a variable range in frequency (Celnikier et al. 1983, 1987; Kellogg & Horbury 2005;Chen et al. 2014b). Furthermore, spectra in electric fields are also variable and may differ when a monopole or dipoleantenna is used. This may be due to density fluctuations affecting the monopolar measurement (Kellogg et al. 2003).The observed flattening in the density spectra has been modeled as being due to slow waves in the inertial scaleswhich are passively cascaded in the inertial range before being heavily damped at the ion scales (Harmon & Coles2005; Chandran et al. 2009; Schekochihin et al. 2009). The smaller scales in the spectrum can be modeled as an activecascade of kinetic Alfv´en waves. In this model, the flattening and its frequency range are related to the plasma β (the ratio of the thermal to magnetic pressures) and are explained due to the competition between the large scaleslow waves and the small scale kinetic Alfv´en waves (KAWs). This interpretation is supported at large scales by theanti-correlation of density and magnetic field magnitude (e.g. Howes et al. 2011; Verscharen et al. 2017; Roberts et al.2018), which is a characteristic of the MHD slow-wave. Furthermore, the dispersion relations of density fluctuationsfound in the solar wind which show a broad range of plasma frame frequencies when compared to the trace magneticfluctuations (Roberts et al. 2017c) possibly due to wave-wave interactions between Alfv´en waves and slow waves.In the sub-ion range, the magnetic helicity (e.g. He et al. 2011; Podesta & Gary 2011) signature is positive whenthe radial component of the field is positive. The normalized magnetic and density fluctuations are also of the sameorder of magnitude (Chen et al. 2013; Roberts et al. 2018). Both of these observations are consistent with the KAWinterpretation. Alternatively, the flattening in the density spectrum is also predicted more generally by the increasinginfluence of the Hall effect at small scales (Narita et al. 2019; Treumann et al. 2019). The Hall effect occurs due tothe demagnetization of protons as the fluctuations in the magnetic field vary on faster timescales than the protonscan follow, while electrons remain magnetized (Huba 2003; Kiyani et al. 2013; Narita et al. 2019; Treumann et al.2019; Bandyopadhyay et al. 2020). Another alternative is that this region in scale could be populated by compressiblecoherent structures (e.g. the structures discussed by Perrone et al. 2016).One of the limitations in the study of the electron density spectra is that the range of scales is very limited. Oftenonly the inertial range can be probed due to the low time resolution. To overcome this limitation remote sensingtechniques have often been used (e.g. Woo & Armstrong 1979; Harmon & Coles 2005) or for in-situ study specialplasma instruments have been developed such as FPI (Pollock et al. 2016). Alternatively, the spacecraft potential canbe calibrated to give a time resolution for electron density which is equal to the electric field time resolutions (Pedersen1995; Escoubet et al. 1997; Nakagawa et al. 2000; Pedersen et al. 2001, 2008). In this study, we will use data from boththe FPI instrument and the spacecraft potential from the Spin Plane double probes (SDP) instrument (Lindqvist et al.2016) to perform a new study of the electron density spectrum in the solar wind which allows frequencies deep in thesub-ion range to be investigated. Previous studies have been limited due to instrumental noise, limiting the maximumphysical frequency to near 10Hz on THEMIS (e.g. Chen et al. 2013) or due to a time resolution of 5Hz on Cluster(Yao et al. 2011; Roberts et al. 2017c).There are several challenges in using the spacecraft potential such as strong spin tones in the data (e.g. Roberts et al.2017c) and the influence of nanodust (e.g. Vaverka et al. 2019; Escoubet et al. 2020). The goal of this paper is to de- ompressive turbulence at sub-ion scales
30 20 10 0 −10 −2020100−10−2030 20 10 0 −10 −20x [GSE] (Re)20100−10−20 y [ G SE ] ( R e ) B0 MMS
Figure 1.
Location of the MMS spacecraft in GSE coordinates. The magnetopause of Earth (black dashed lines), the Bowshock (blue dot dashed lines) and the orientation of the magnetic field vector (red arrow) are also displayed. DATA/METHODOLOGY2.1.
Event Overview
The Magnetospheirc MultiScale Mission (MMS) mission (Burch et al. 2016) consists of four identical spacecraft in atetrahedral configuration optimized for studying magnetic reconnection in the Earth’s Magnetosphere. The main focusof this study is on a particular one-hour burst mode interval of slow solar wind on the 24th of November 2017 between01:10:03-02:10:03. During this time the MMS spacecraft were located at [ x GSE , y
GSE , z
GSE ] = [16 . , . , . R E . Thesubscript GSE denotes the Geocentric Solar Ecliptic coordinate system, where the x component points from Earthtowards the Sun, z points to the North solar ecliptic. A plot of the location of the MMS spacecraft is given in figure 1.The magnetopause is estimated from the model of Shue et al. (1998) and the bow shock model of Je´ab et al. (2005) isalso displayed. The mean magnetic field makes a large angle with the x-direction. The location of the spacecraft andthe large angle of the magnetic field with the x-direction indicate that connection with the foreshock is unlikely. Thiswill also be validated later from the electric field spectrogram and the ion energy spectra.The magnetic field is measured by the fluxgate magnetometers (FGM) (Russell et al. 2016) which have a samplingrate of 128 Hz in burst mode and sensitivity which allows the study of the magnetic fluctuations at inertial (fluid scales)and the start of the ion kinetic range before noise becomes significant at about 5Hz. The particle measurements areprovided by the FPI’s Dual Electron Spectrometers (DES) and Dual Ion Spectrometers (DIS) and have a sampling rateof 33Hz and 6.6Hz respectively. The spacecraft potential and the spin plane components of the electric field are obtainedfrom the SDP (Lindqvist et al. 2016) and have a sampling rate of 8.192kHz in burst mode. The third component of theelectric field comes from the axial double probe (ADP) instrument (Ergun et al. 2016). A figure showing an overviewof the event is shown in Fig 2 and the mean plasma parameters and their respective standard deviations are presentedin Tab 1. As there are problems for determining the ion temperature from FPI (Bandyopadhyay et al. 2018) (andconsequently β i ) the temperature from OMNI (King & Papitashvili 2005) which is measured at the L1 Lagrange pointand propagated to the bow shock nose is also quoted. Roberts et al.
Figure 2.
Measured data from the MMS3 spacecraft. The panels (from top to bottom) show the the magnetic field, the ionvelocity components, the electric field, the electric field spectra, the spacecraft potential, the ion omnidirectional spectra, theion temperature, the electron omnidirectional spectra and the electron temperatures.
Figure 2 shows that there are no high energy particles, and the magnetic field makes a large angle with the radialdirection with the y component dominating. Furthermore, the electric field spectrogram shows no signs of high-frequency electric field waves associated with the foreshock. This is important for the study of the solar wind as amore radially pointing field may result in a connection to the foreshock, which may pollute the solar wind plasmawith backstreaming particles and large amplitude wave activity (e.g. Turc et al. 2019). Additionally, the electrontemperature is low, and the electric fields have small amplitudes. These conditions are necessary as they can affectthe determination of the electron density from the spacecraft potential (Pedersen et al. 2001). Hot electrons in themagnetosheath can cause secondary emission of electrons from the surface (e.g. Lai 2011) with a maximum yield, whichdepends on the material but is typically in the range of 300-800eV. Strong electric fields or those which change abruptlycan enhance the photoelectron emission from the surface (e.g. Torkar et al. 2017; Graham et al. 2018a; Roberts et al.2020) causing the spacecraft potential to follow electric field fluctuations rather than the density fluctuations. In thisinterval the magnitude of the electric fields is small. The relevant equations and the details of the various effects onthe spacecraft potential to estimate electron density will now be discussed.2.2. Spacecraft Potential Calibration
In the following sections, the methodology to obtain an electron density estimation from the spacecraft potentialwill be discussed. These methods will be applied to derive the electron density in 16 burst mode intervals and 96 fastsurvey mode intervals in total (details of which can be found in the Appendix), including the interval presented in theprevious section. ompressive turbulence at sub-ion scales Table 1.
Table of means and standard deviations for several plasma parameters during the one hour burst mode intervalbetween 01:10:03-02:10:03 on 24/11/2017 taken from MMS3. The magnetic field data are from the fluxgate magnetometer andthe particle data are from the Fast Plasma Investigation. Ion temperatures and β i are also shown from OMNI B [nT] V i [km/s] n e [cm − ] β i (OMNI) β e T i [eV] (OMNI) T e [eV]6 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
1) (4 . ± . A spacecraft embedded in plasma becomes charged, and is affected by several different processes, with currentsflowing to and from the spacecraft. Typically, the two dominant processes in a plasma such as the solar wind are theelectron thermal current I e flowing to the spacecraft and the electron photocurrent flowing from the spacecraft I ph .There can be other sources of current such as in a dense plasma the ion thermal current, or currents from the secondaryemission of electrons when the electron temperature is high. There are also instruments on MMS that emit currentssuch as the Active Spacecraft Potential Control (ASPOC) (Torkar et al. 2017) or the Electron Drift Instrument (EDI)(Torbert et al. 2016) both of which are not operating in this interval. A bias current is also sent to the electric fieldprobes, from the body of the spacecraft but is much smaller than the photoelectron or the electron thermal currents.The balance of these various currents determines the spacecraft potential.The spacecraft potential is measured on MMS from the SDP instrument which consists of four biased probes inthe spacecraft spin plane which are mounted on the end of 60-meter wire booms and measure the potential differencebetween the probe and the spacecraft. The probe to plasma potential is stabilized by the use of a bias current whichhas the goal of keeping the probe to plasma potential positive and close to zero. In reality, the probes are not at zeropotential with respect to the plasma, however, this would only introduce a small systematic error (e.g. Pedersen 1995;Torkar et al. 2015). The spacecraft potential is calculated using the four probes of the SDP instrument. If one probeis not operational due to a failure then two opposing probes are used in the calculation. The mean of the individualprobe potentials give the probe to spacecraft potential, which is converted into a spacecraft to plasma potential (whichwe term spacecraft potential) by correcting for a boom shortening factor of 1.2 and a correction for each spacecraft onethrough four as [ c MMS1 , c
MMS2 , c
MMS3 , c
MMS4 ] = [1 . , . , . , . V psp tospacecraft potential V sc is, V sc,i = 1 . × ( − V psp ) + c MMSi (1)where, V psp = 1 n n X i =1 V i (2)where n is the number of probes used either 2 or 4.In solar wind typically the electron photocurrent and the electron thermal current dominate the current balanceequation such that the two can be approximately equated I ph ≈ I e . The electron thermal current can be approximated,assuming a Maxwellian velocity distribution function (Mott-Smith & Langmuir 1926) as: I e = − A spac qn e r k B T e m e π (cid:18) qV sc k B T e (cid:19) (3)Where A spac is the MMS spacecraft’s approximate surface area ∼ .An expression for the photocurrent is found by fitting the photoelectron current (which is approximately equalthermal current) to the spacecraft potential giving the photocurve using Eq. 4. I ph = I phot0 exp (cid:18) − V sc V phot0 (cid:19) (4)The parameters I phot0 , V phot0 , are the photoelectron current and potential obtained by fitting I e to the spacecraftpotential V sc . The photocurve consists of several different ranges (Lybekk et al. 2012; Andriopoulou et al. 2015), Roberts et al.
MMS1 sc (V)2628303234363840 I e ( µ A ) I e =I ph =301.54exp(−V sc /2.04)(a) MMS2 sc (V)2628303234363840 I e ( µ A ) I e =I ph =474.22exp(−V sc /1.71)(b) MMS3 sc (V)2628303234363840 I e ( µ A ) (c) I e =I ph =375.93exp(−V sc /1.95) MMS4 sc (V)2628303234363840 I e ( µ A ) I e =I ph =219.99exp(−V sc /2.09)(d) Figure 3.
The photocurves and the exponential fits for all four spacecraft on the 24th of November 2017 where a different exponential is fitted based on the spacecraft potential. The spacecraft potential in the intervalsanalyzed here is always less than 5V, meaning fitting with a single exponential function is sufficient (Grard 1973;Andriopoulou et al. 2015). Typically a longer time interval than the one studied is required so that a large enoughrange of potentials can be sampled and the relationship between the spacecraft potential and the photoelectron currentcan be determined effectively. Here we use the entire day of the 24th of November 2017 for calibration (the one hourof burst mode shown in Fig 2 occurs on the same day) and only select times when the ion and electron densitymeasurements from FPI are within 10% of one another. This is to ensure the density measurement is accurate, as in acommon plasma quasi-neutrality ( n e = n i ) is expected to be valid, and strong deviations from this are likely to indicatea low-quality measurement of the density. Additionally, only times when the magnitude of the DC electric field isbelow 10 mV/m are used for calibration. This is because strong electric fields can alter the photoelectron emissionfrom the spacecraft causing the spacecraft potential to follow the electric field (e.g. Torkar et al. 2017; Graham et al.2018b; Roberts et al. 2020) rather than the density. The photoelectron current is modeled by performing a fit of thethermal current (Eq. 3) to the spacecraft potential. Figure 3 show the photocurves for the different spacecraft andthe parameters I phot0 , V phot0 used for calibration are indicated. MMS1-3 shows similar potentials, however, MMS4has slightly different photoemission properties possibly due to top ring attached to MMS4. Furthermore, there wasa probe failure on MMS4 which affects the potential measurement. However, the differences in the photoemission ofMMS4 were observed before the probe failure (e.g. Andriopoulou et al. 2018).Providing that all other current sources are small, and an assumption (or a direct measurement if possible) is madeabout the ambient electron temperature, the spacecraft potential can be calibrated to give a measurement of theelectron density (Pedersen 1995,Escoubet et al. 1997,Nakagawa et al. 2000,Pedersen et al. 2001,Pedersen et al. 2008)given in Eq 5. n e,SC = 1 qA spac s(cid:18) πm e k B T e (cid:19) (cid:18) qV sc k B T e (cid:19) − (cid:18) I ph exp (cid:18) − V sc V ph (cid:19)(cid:19) (5) ompressive turbulence at sub-ion scales Spin Tone Removal
The spacecraft potential is subject to a strong spin effect due to the sunlit area of the spacecraft changing throughoutthe spin, which affects the amount of photoelectron emission from the surface. This results in high power spikes inthe Fourier power spectra of the spacecraft potential and the derived electron density (e.g. Kellogg & Horbury 2005;Yao et al. 2011; Chen et al. 2013; Roberts et al. 2017c). Furthermore, the spacecraft potential is measured by fourprobes in the spin plane of the spacecraft. As one of the probes passes the rear of the spacecraft (the front pointing atthe Sun) they pass a plasma wake (e.g. Engwall et al. 2009). These effects need to be removed before further analysis.This can be done by notch filtering (Yao et al. 2011), removing parts of the spectra (Chen et al. 2013) subtractingharmonics, or developing an empirical model of the spacecraft charging and subtracting it (Roberts et al. 2017c). TheFPI instrument also suffers from some spin effects in the solar wind, in this section, we will present a method to removesuch fluctuations from both the FPI density measurement and the SDP potential measurement.We will use the same approach as Roberts et al. (2017c), where an empirical model of the spacecraft chargingthroughout a spin is obtained. The spacecraft potential is converted to a potential fluctuation by subtracting asuitable average. If the plasma is stable with no large changes then an average over the entire interval is sufficient.However, if there are long term trends then a moving average based on the spacecraft spin period of 20 seconds canbe used. As these are low-frequency fluctuations, the fast survey mode spacecraft potential data sampled at 32Hzcan be used for obtaining the empirical model rather than the full-resolution burst mode. The spacecraft potentialfluctuation is shown in Fig 4 as a function of the spacecraft spin phase angle, and a clear dependence can be seen. Thepotential fluctuation in Fig 4a is binned into angle bins of 0.5 degrees and a model is derived by fitting the medianvalues of each bin which is shown in Fig 4b. For the MMS spacecraft, this model is based on a superposition of 20sine waves, the model is more complicated than for Cluster studied in Roberts et al. (2017c) owing to the angularoctagonal shape of MMS as compared to the smoother cylindrical shape of Cluster. When the model is derived themodel fluctuation is subtracted from the data removing the fluctuation due to spin leaving the other fluctuations weare interested in undisturbed. The spin removed potential data can then be used in place of the measured potentialdata to obtain the electron density without spin effects. As we subtract a fluctuation, there may be uncertainties inthe mean value that is subtracted. For example, if there is a long term trend the potential fluctuation in 4a may notbe centered near zero at the edges, and a mean based on a moving average should be used. Additionally, if there is alarge change in the plasma conditions that perturb the potential quickly a global or even a 20-second average may notbe suitable, furthermore, the wake is sensitive to the plasma conditions. Therefore this method should be applied topotentials and plasma conditions that are fairly stable. Therefore this method may appropriate when crossing largeboundaries for example.The FPI data are also subject to some spin effects as the instruments are not optimized for the solar wind (e.g.Bandyopadhyay et al. 2018). However, the same approach can be used to remove the spin effects from the directmeasurement. Figure 5 corresponds to Fig4b but for the FPI electron density measurement. There is also a dependenceon the spacecraft spin phase angle but it is more complex. In this case, the fluctuations are fitted with a superpositionof Gaussian functions and are then removed in the same way as for the spacecraft potential.The Fourier Spectra of the measured electron density from FPI-DES and the spacecraft potential are shown later inFigures 8a and c and the method is successful in removing the spikes. Figure 6 shows a comparison between the FPIburst mode data and the corrected and calibrated electron density derived from the spacecraft potential on MMS3. Itcan be seen that there is very good agreement between both measurements.2.2.2.
Dust impacts
As previously mentioned dust/micrometeorites can strongly affect the spacecraft potential giving abrupt changes inthe potential and the derived density. The typical profile of these is a sharp decrease in the potential followed by anincrease and then an exponential tail where the spacecraft recovers to its initial state. One such example in the datainterval is shown in Figure 7. The signatures shown here can also be seen in Fig2 as the large negative spike in thedata near the beginning of the interval and a positive spike after 0130UT.The profile is seen in the potential in Figure 7a can be explained as follows; a dust impact initially causes someelectrons to be lost, it then vaporizes causing a sharp increase in the plasma density near the spacecraft before the
Roberts et al.
Figure 4. (a) shows the spacecraft potential data as a function of the spacecraft spin phase angle. (b) shows the median valuesand the errors for the median value in each bin, the red curve is the model which is fitted to the data.
Median of Density Fluctuation θ [degrees]−0.3−0.2−0.10.00.10.20.3 δ n e [ c m − ] Figure 5.
The median of density fluctuations in a bin of 0.5 degrees from the FPI DES measurement of electron density. spacecraft recovers to its initial state as the spacecraft gathers electrons again (e.g. Meyer-Vernet et al. 2014; Zaslavsky2015; Vaverka et al. 2017; Ye et al. 2019). The effect on the density estimate as shown in Fig 7 is extreme. It isimportant to remove these signatures, as it can have a large effect on some typical techniques for analyzing turbulencee.g. calculating kurtosis or other higher-order moments which are heavily influenced by outliers (e.g. Dudok de Wit2004; Kiyani et al. 2006).Dust strike events have been studied on the STEREO spacecraft (Malaspina et al. 2015; O’Shea et al. 2017), andwere also estimated to occur on close to an hourly basis on MMS (Vaverka et al. 2019). Curiously the interval studiedhere is close to the peak of the Leonid meteor shower which may cause an increase in dust impacts, although definitivelydemonstrating a link between the meteor shower and the dust impacts seen here is not feasible. Figure 7b shows an ompressive turbulence at sub-ion scales Figure 6.
Comparison of the direct measurement of electron density from FPI-DES (black) and from the calibrated spacecraftpotential (red).
Estimation of the Noise floor
To ensure that our results are physically significant, knowledge of the noise floor is required for FPI and the SDPderived density. In figure 8, the Fourier power spectra are presented. In figures 8a the measured data from FPI isshown without spin correction. A few spikes can be seen in the spectra which correspond to harmonics of the spinfrequency. When the spin removal method discussed previously is applied to the FPI electron density measurementthe spectra in Fig8b are obtained. The spikes in Fig 8a have predominantly been removed. The estimated noise floor(Gershman et al. 2018) is given in grey and noise becomes significant near 3Hz in the FPI data.In Figure 8c a 17.30-minute sub-interval between 01:10:03-01:28:33 is shown. The shorter interval is shown for directcomparison with an interval of quiet solar wind where ASPOC is on so the potential is regulated and fluctuations inthe potential are smaller. The quiet interval is shown in grey. This interval on 2019/02/24 between 16:39:53-16:57:13the electric field measurement is near the preamplifier noise at f sc > Roberts et al. V sc [ V ] (a) V sc [ V ] (b) n e ( c m − ) (c) n e ( c m − ) (d) Figure 7. (a) shows an example of a dust strike event beginning at 01:11:52.729UT on MMS3. (c) shows the correspondingdensity estimation. (b) and (d) shows a different event starting at 01:32:41.638UT where the potential increases, which hascurrently no satisfactory explanation. the largest frequency where the power remains above three times the noise (e.g. Alexandrova et al. 2010; Roberts et al.2017a). For this signal, we see that this occurs at 47Hz which motivates the upper limit of 40Hz used here.The spin removed and calibrated electron density derived from spacecraft potential is shown in Fig 8d, where the spintones prominent in Fig 8c, have also been satisfactorily removed. There are some spikes at 1Hz and at higher frequenciesnear 16 and 32Hz and above 40Hz. These are likely to be instrumental in origin. To summarize this section; we havepresented a methodology for obtaining the electron density from the spacecraft potential. This includes methodsfor calibration, spin removal, and an approximate estimate for the noise floor of the potential measurement. Someexamples of dust strikes and inverted dust signatures have been shown. Following the methodology here we have beenable to obtain a measurement of the electron density which allows the sub-ion range to be investigated. RESULTSIn Figures 8b the FPI electron density is shown and the spectral breaks are found by fitting a straight line fromeither side of the break to determine the break frequency (Bruno & Trenchi 2014). The first break in the densityspectra doesn’t correspond to any of the Taylor shifted ion scales, however, the second break scale is near the shiftedinertial length. It is not surprising that the combined scale ( ρ i + d i ) associated with cyclotron resonance does not ompressive turbulence at sub-ion scales Figure 8. (a) black traces show the power spectral density of the electron density measured from FPI without spin removal. Thegrey trace denotes the estimated noise floor (Gershman et al. 2018) the solid coloured lines denote the different characteristicscales the ion cyclotron frequency f ci ion Larmor radius ρ i , the inertial length d i , and the combined scale f ρ i + d i . The dashedlines show the same for the electrons. The grey dot-dashed line at 3Hz denotes the scale where the noise becomes significant. (b)shows the spin removed electron density power spectral density and the black dot-dashed lines denote the spectral breaks. (c)shows the spectra of the measured potential data in black while the grey denotes a different interval when ASPOC is operating.(d) shows the electron density spectra obtained from the spacecraft potential. seem to link to the density spectrum here as ion cyclotron waves are not compressible. The electron density estimationfrom the spacecraft potential is shown in Fig 8d which allows the fitting of the sub-ion range to be performed over alarger range of scales [0.983,40]Hz than for the FPI measurement [0.65,3]Hz. It is interesting to note is that there isvery good agreement between both measurement methods at large scales, but at smaller scales, the spectral indicesare different with the FPI measurement being significantly flatter. This is likely due to the smaller range of scalesavailable before instrumental noise becomes significant at 3-5Hz.For comparison, the trace magnetic and magnitude fluctuations are also calculated from the data measured by theFluxgate magnetometer. The magnetic spectra are shown in Fig 10. It can be seen here that there is a flattening near5Hz in the trace spectra and near 3Hz in the magnitude spectra. Unfortunately, the MMS search coil does not havethe required sensitivity necessary for solar wind turbulence studies at frequencies higher than 5Hz therefore we willonly use the magnetic field measurement from FGM.The spectral break locations are found in the same manner as for the density spectra. For the trace magnetic field,the break is closest to the combined scale (Bruno & Trenchi 2014). The error on the break in all cases is near 0.06Hzwhich is calculated by propagating the errors of the two linear fits. The standard deviations of the Taylor shiftedscales are at most 0.07Hz. The results suggest that the magnetic field spectral break is closest to the combined scalewhile the density and compressible magnetic fluctuations ion scale break is closest to the ion inertial scale which is thelarger of the two scales. One interpretation is that this early density break is due to kinetic slow waves which begin tobe damped at lower frequencies causing the flattening seen in the density spectra. Slow waves are not the dominantsource of power in the trace magnetic spectra. In the trace spectra, cyclotron resonance becomes important causing the2 Roberts et al.
Figure 9. (a) shows the Fourier spectra from 01:10:03-01:28:33 (black) and the quiet interval where ASPOC is operating whichoccurs on 2019/02/24 between 16:39:53-16:57:13 (grey). The dark grey spectra denote the quiet spectra in grey which has beenmultiplied by 3. To better understand the noise properties the signal is separated into 64 windows and averaged. (b) shows thecorresponding wavelet spectra. The vertical grey line denotes the region where the wavelet spectra are approximately equal totimes the quiet signal multiplied by 3. The black line denotes where we limit our analysis to in frequency. spectral break before kinetic Alfv´en waves become important at smaller scales leading to similar morphologies for bothspectra at sub ion scales. However, this interpretation has the limitation that ion cyclotron waves are observed moreoften in the fast solar wind when the magnetic field is predominantly in the -x GSE (radial) direction (He et al. 2011;Podesta & Gary 2011) which is not the case for this interval. An alternative interpretation is that the region where theflattening is seen in the density spectra corresponds to a region where strongly compressible coherent structures aremore abundant such as pressure-balanced structures. At smaller scales, the structures may have comparable powersin both the compressible and incompressible components. There is some evidence that the scale-dependent kurtosisincreases with decreasing scale up to ion scales before becoming smaller again (Chhiber et al. 2018; Chasapis et al.2018). A detailed analysis of the scale-dependent kurtosis is planned but is outside the scope of this work.The flattening seen in the density spectrum could also be an indication of Hall effects becoming important(Narita et al. 2019; Treumann et al. 2019). Unfortunately, the electron velocity power spectral density becomesnoisy near 0.1 Hz, at lower frequencies both ion and electron velocity have a similar power spectral density whichis slightly shallower than -5/3 (Bandyopadhyay et al. 2018). Multi-spacecraft increments shown in the study ofBandyopadhyay et al. (2018) do suggest at smaller scales the Hall effect is present as the velocity increments ofelectrons have larger power than the ions at scales of 15km. However, this is at scales far smaller than the flatteningof the density spectrum observed here. An additional way to test whether it is Hall effects or the transition between ompressive turbulence at sub-ion scales Figure 10. (a) trace magnetic power spectral density (b) magnitude power spectral density. the Alfv´en and slow wave-dominated inertial range and the kinetic Alfv´en wave kinetic range is to investigate theelectric field. The ratio of the electric field to magnetic field fluctuations has been calculated previously using Clustermeasurements (Salem et al. 2012). In tandem with the results of the compressibility of the fluctuations, this wasinterpreted as being due to kinetic Alfv´en wave-like fluctuations.To investigate the electric field in this interval we calculate the second-order structure functions of the electricand magnetic field. A fluctuation is defined as the difference between two time-lagged measurements δ B ( t, τ ) = B ( t + τ ) − B ( t ) and the second-order structure-function is defined as D ( τ ) = h| δ B ( t, τ ) | i . The mean magnetic field ispredominantly in the y GSE direction B = ( − . , . , − .
39) nT, while the mean electric field direction is primarilyin the z GSE direction E = ( − . , . , .
24) mV/m. It is important to note that the three orthogonal componentsof the electric field come from two different instruments, i.e. the x and y GSE components are calculated from the spinplane booms and the z component comes from the Axial Double Probe instrument which has lengths of approximately60m and 12m respectively. This interval has a favorable magnetic field direction as the largest component in theelectric field is measured by the instrument with the largest baseline. The SDP instrument will sample the parallel andmost of one perpendicular component while the remaining perpendicular component will be measured by the ADP.However, effects due to the spacecraft wake (e.g. Engwall et al. 2009) will affect the SDP measurement. Effects ofthe wake and shadowing from the ADP make the measurement of the E x component difficult. In the L2 data of theelectric field, these effects have been removed but some residual effects may be present. Additionally, there are spikesin the E z data in between two burst mode files, these are removed by linear interpolation before analysis.4 Roberts et al.
Rather than use a global magnetic field direction a local magnetic field direction will be used. This will be used forthe analysis as there is increasing evidence that fluctuations in solar wind turbulence are aligned with a local magneticfield based on the size of the fluctuation rather than a global field defined by the mean over the entire time interval(e.g Horbury et al. 2008; Podesta 2009; Chen et al. 2012; Kiyani et al. 2013).The coordinate system for each structure-function pair is defined as the local mean-field direction B loc ( t, τ ) =[ B ( t + τ ) + B ( t )] /
2. The other directions are defined with respect to the unit vector of the local mean-field e k = ˆ B loc where the cross product of this direction and the bulk velocity direction makes the first perpendicular direction e ⊥ = e k × V sw | V sw | and the second perpendicular direction is orthogonal to both e ⊥ = e k × e ⊥ . As the e ⊥ is alongthe projection of the bulk velocity direction it can be compared to the x GSE component which is the more difficultto measure. This is further complicated by the fluctuations being very small in this component. Figure 11 shows thesecond-order structure functions expressed as an equivalent spectrum, for both the electric and magnetic fields. As theelectric field is not frame invariant (Kellogg et al. 2006; Chen et al. 2011; Mozer & Chen 2013) the equivalent spectraare presented in the spacecraft frame (denoted subscript SC) and the solar wind frame denoted subscript (SW) whichare related by the Lorentz transformation; E SW = E SC + V sw × B (6)This transformation is done for each structure-function pair ( E ( t ) and E ( t + τ )) after they have been put in the localmagnetic field direction, and the mean ion velocity over the entire period is used for V sw , as there are some challengesusing the FPI ions point to point in the solar wind (Bandyopadhyay et al. 2018). For comparison of the amplitudesof the different spectra a normalization needs to be performed as follows; B → B /B , (7) E → E /v A B , (8) n e → n e /n e , (9)where the subscript zeroes denote the mean over the interval.The analysis shows that the magnetic power is dominated by the perpendicular power while the electric power isdominated by the parallel component. At large scales an MHD Alfv´en wave does not have any associated electric field,therefore the strong parallel electric field fluctuations here are likely due to kinetic slow waves which have a large parallelcomponent. At ion scales, the parallel electric field may be due to the KAW (Narita & Marsch 2015). The flatteningof the electric field spectra was observed by Bale et al. (2005) and was interpreted to be due to KAW turbulence.However numerical simulations of gyrokinetic turbulence (Howes et al. 2008) and Hall turbulence (Matthaeus et al.2008) both show this enhancement. The presence of large parallel electric field fluctuations suggests that LandauDamping is an important mechanism for turbulent heating. e.g. (TenBarge & Howes 2013; Chen et al. 2019). Themagnetic field also becomes more compressible at the start of the sub-ion range. However, noise becomes significantnear 5Hz (Fig 10). The electric field exhibits some flattening f sc > e ⊥ component as it is perpendicular to boththe magnetic field and approximately perpendicular to the bulk flow direction. The Lorentz transformation has theeffect of flattening the e spectra as has been observed in the statistical study in Chen et al. (2011).At large scales the parallel electric field shows a similar Kolmogorov like power law as in the magnetic field, while theperpendicular components are flatter with indices closer to -3/2 rather than -5/3 which has often been measured forthe velocity fluctuations in the solar wind (Podesta et al. 2006; Bandyopadhyay et al. 2018) and in the magnetosheathRoberts et al. (2019). At sub-ion scales, it is difficult to make any firm conclusions as noise becomes significant. Asthe electric field measurement is based on taking the difference in the potential between two probes, the noise is likelyto be a larger problem for the electric field than the spacecraft potential. To contrast with the potential measurement,the measurement is based on an average of the four probes meaning that the signal to noise ratio will be larger for thepotential measurement allowing frequencies up to 40Hz to be resolved when compared to a few Hz for the electric fieldmeasurement. With the data available it is difficult to discriminate between the KAW scenario and the Hall scenario.However, in this region, the spectrum does steepen slightly for all components especially in the solar wind frame. This ompressive turbulence at sub-ion scales −2 −1 −8 −6 −4 −2 Magnetic field −2 −1 f [Hz]10 −8 −6 −4 −2 D /f [ s ] f −1.53 f −1.68 f −1.64 f −1.30 f −1.77 f −2.33 f −2.27 B B ⊥ ⊥ Electric field −2 −1 f [Hz]10 −8 −6 −4 −2 D /f [ s ] f −1.69 f −1.41 f −1.45 f −2.08 f −1.42 f −1.46 E E ⊥ ⊥ Electric field Solar Wind Frame −2 −1 f [Hz]10 −8 −6 −4 −2 D /f [ s ] f −1.68 f −1.30 f −1.40 E sw E sw ⊥ sw ⊥ −2.08 f −1.55 f −1.55 (c) Electron Density −2 −1 f [Hz]10 −8 −6 −4 −2 D /f [ s ] f −1.42 f −1.25 f −2.06 n e (d) Figure 11.
Second order structure functions expressed as equivalent spectra for (a) the magnetic field, (b) Spacecraft frameelectric field (c) solar wind frame electric field and the (d) the electron density. steepening is predicted for KAW turbulence before the spectrum flattens at higher wavenubmers (Narita et al. 2020),while Hall Turbulence would exhibit only a flattening (Narita et al. 2019). Although it should be noted that thereis only a short range of scales where the steepening is seen and it is difficult to determine conclusively between bothscenarios. Furthermore, the flattening predicted at kinetic scales in the electric field is difficult to distinguish fromnoise (e.g. Alexandrova et al. 2013). Perhaps more subtly both effects are present and contribute to the observations.The strong parallel electric field fluctuations do however support the dissipation of energy through Landau damping.At sub-ion scales, the relationship between the spectral indices of the magnetic and the density spectra is investigatedby performing by fitting the slopes of the power spectral density between the frequencies of 1Hz and 3Hz. The resultsare presented as histograms in Figure 12. As it is the ion kinetic scales that concern us rather than the spectral breaklocations we can split the time interval into smaller intervals of 64 seconds (2 data points for the spacecraft potentialand 2 for the magnetic field). The power spectral density is estimated using a Hanning window with no overlap, andare averaged over 7 windows to reduce the variance. From the one hour interval, there is a total of 55 spectra whichwe use to investigate the spectral index in the sub-ion range.The mean and median values of the spectral indices for the density are close to -2.6 which compare well with magneticfield measurements of this range given in Figure 8b and are consistent with previous studies of the magnetic spectra(Smith et al. 2006; Alexandrova et al. 2012; Sahraoui et al. 2013). However there is a large spread in the values, ashas been seen in Smith et al. (2006) for the dissipation range in a statistical study of the magnetic field spectra using6 Roberts et al. the ACE spacecraft containing various plasma conditions. For example, a change in the orientation of the magneticfield could cause the spectral index to change Horbury et al. (2008); Wicks et al. (2011); Roberts et al. (2017c, 2019).As the interval is only one hour and the dissipation range indices show high variability we interpret this as being aresult of the limited frequency range where we can fit both magnetic and density spectra.To test this hypothesis we perform the same fittings over the frequency range 1-40Hz for the density spectra whichis shown in the blue values in figure 12a and the spread in the values is significantly limited, however, the mean valuesare consistent over both frequency ranges. Therefore it is important to fit spectra over the largest range possible,otherwise one might conclude that there is a large variability in the spectrum where a fitting over a larger frequencyrange reveals that this is not correct. It is also noted that the spectra of the magnitude are flatter than the trace andthe density spectra. This is due to the noise becoming an issue at higher frequencies causing an artificial flattening inthe spectra similar to what is seen in the FPI density spectra in Figure 8. −4 −3 −2 −1 α den H i s t og r a m D en s i t y Mean=−2.63+/−0.59Median=−2.71+/−0.50Mean=−2.62+/−0.10Median=−2.63+/−0.10 (a) −4.0 −3.5 −3.0 −2.5 −2.0 −1.5 α trace mag H i s t og r a m D en s i t y Mean=−2.60+/−0.42Median=−2.61+/−0.45 (b) −3 −2 −1 α magnitude mag H i s t og r a m D en s i t y Mean=−2.22+/−0.56Median=−2.29+/−0.55 (c)
Figure 12. (a) shows the histogram of the spectral indices of the density power spectral density fitted between 1 and 3Hz (red)and between 1 and 40Hz in blue. (b) shows the fitting of the trace magnetic field spectra fitted between 1 and 3 Hz (c) showsthe same for the power spectral density of the magnitude. The solid and dashed lines denote the mean and the median valuesof the spectral indices
In Figure 13 we compare our values with the spectral indices which are obtained from the magnetic field spectra forthe same intervals in Figures 12. For direct comparison, we fit them over the shorter frequency range between 1-3 Hz. ompressive turbulence at sub-ion scales p < . −5 −4 −3 −2 −1−5−4−3−2−1−5 −4 −3 −2 −1 α trace mag −5−4−3−2−1 α den ρ =0.41p<0.01 α den =(1.58+/−0.18) α trace mag +(1.44+/−0.46) Figure 13.
Comparison of the spectral indices of the density and the trace magnetic field spectra in the range [1,3]Hz. Theerror bars are from the residuals of fitting a straight line to the log of frequency to the log of power.4.
Roberts et al. A. SOLAR WIND INTERVALS ANALYZED
Table 2.
Table of the mean and standard deviations of solar wind burst mode intervals where the density calibration has beenperformed. Plasma β is calculated by reasmpling the magnetic field data onto the FPI time tags, to avoid large values near theedges the first and last seconds are removed from the calculation. Consequently should there be magnetic holes the standarddeviation will be very large. The one minute resolution OMNI data are used for the calculation of β . This is only calculatedfor times when T i , B , and n i are all available. Date Time B [nT] V i [km/s] n e [cm − ] β i (OMNI) β e T i [eV] (OMNI) T e [eV]2016-12-06 11:37:34-11:44:03 7 . ± . . ± . ± . ± . . ± . ± . ± . . ± .
1) (2 . ± . . ± . . ± . . ± . . ± . . ± . ± . ± . . ± .
1) (1 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
1) (4 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
1) (3 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
1) (2 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
1) (3 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . (1 . ± .
4) (18 ± . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
1) (1 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
2) (5 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
2) (4 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . ±
1) (1 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
2) (1 . ± . . ± . . ± . . ± . . ± . . ± . ±
12 9 . ± . . ± .
7) (18 . ± . . ± . . ± . . ± . . ± . . ± . ± . . ± . . ± .
1) (3 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
1) (2 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
1) (12 . ± . ) ompressive turbulence at sub-ion scales Roberts et al.
Table 3.
Table of the mean and standard deviations of solar wind fast survey mode intervals where the density calibration hasbeen performed 2016-2017.
Date Time B [nT] V i [km/s] n e [cm − ] β i (OMNI) β e T i [eV] (OMNI) T e [eV]2016-12-06 11:20:01-11:59:55 7 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
2) (2 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . (0 . ± .
2) (1 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
3) (4 . ± . )2017-11-15 12:00:00-13:55:21 9 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
01) (2 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . (0 . ± .
2) (5 . ± . . ± . . ± . . ± . . ± . . ± . ± . . ± . (0 . ± .
3) (4 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
04) (1 . ± . . ± . . ± . . ± . ± . . ± . . ± . . ± . . ± .
6) (28 . ± . )2017-11-23 22:36:36-23:59:56 6 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
06) (3 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
06) (4 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
5) (3 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
2) (3 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
2) (6 . ± . )2017-12-01 16:40:00-19:21:59 4 . ± . . ± . . ± . . ± . . ± . ± . . ± . . ± . ) (8 . ± . )2017-12-02 14:10:00-17:09:59 4 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
2) (2 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . ) (2 . ± . . ± . . ± . . ± . . ± . . ± . . ±
20 13 . ± . . ± .
7) (19 . ± . )2017-12-08 03:05:00-04:04:59 3 . ± . . ± . . ± . . ± . . ± . . ± . . ± . (0 . ± .
3) (7 . ± . )2017-12-11 00:00:03-00:39:59 3 . ± . . ± . . ± . . ± . . ± . . ± . . ± . (0 . ± .
06) (1 . ± . . ± . . ± . . ± . . ± . . ± . ± . . ± . (0 . ± .
2) (6 . ± . )2017-12-15 18:40:00-19:39:59 2 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
4) (1 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
2) (2 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
3) (8 . ± . )2017-12-30 13:30:00-18:59:59 4 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
8) (4 ± . ) ompressive turbulence at sub-ion scales Table 4.
Table of the mean and standard deviations of solar wind fast survey mode intervals where the density calibration hasbeen performed 2018.
Date Time B [nT] V i [km/s] n e [cm − ] β i (OMNI) β e T i [eV] (OMNI) T e [eV]2018-01-02 08:30:00-14:09:59 5 . ± . . ± . . ± . . ± . . ± . . ± . . ± . (2 . ± . ) (7 . ± . )2018-01-05 09:00:00-10:39:59 2 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
6) (2 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . (1 . ± .
5) (4 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . (2 . ± . ) (32 . ± . )2018-01-19 06:00:00-11:29:59 8 . ± . . ± . . ± . . ± . . ± . . ± . . ± . (0 . ± .
04) (2 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . (0 . ± .
6) (11 . ± . )2018-01-25 02:50:00-08:19:59 7 . ± . . ± . . ± . . ± . . ± . . ± . . ± . (0 . ± .
6) (10 . ± . )2018-01-30 21:50:00-23:09:59 2 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . ) (1 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
2) (1 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . (0 . ± .
4) (2 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
06) (17 . ± . )2018-02-19 16:00:00-17:29:59 6 . ± . . ± . . ± . . ± . . ± . . ± . . ± . (1 . ± . ) (13 . ± . )2018-02-22 10:50:00-13:59:59 6 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
6) (5 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . (1 . ± .
6) (3 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
06) (3 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . (0 . ± .
3) (14 . ± . )2018-03-25 12:00:00-15:59:59 4 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . ) (7 . ± . )2018-04-(07-08) 21:30:00-01:59:59 3 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
2) (3 . ± . Roberts et al.
Table 5.
Table of the mean and standard deviations of solar wind fast survey mode intervals where the density calibration hasbeen performed 2019.
Date Time B [nT] V i [km/s] n e [cm − ] β i (OMNI) β e T i [eV] (OMNI) T e [eV]2019-03-(06-07) 20:30:00-05:59:59 5 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
5) (5 . ± . )2019-03-10 09:30:00-15:59:59 3 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
1) (6 . ± . )2019-03-14 04:45:00-06:39:59 4 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
05) (2 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
02) (2 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . (0 . ± .
2) (7 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
4) (7 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
1) (2 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
08) (3 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
2) (2 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
04) (1 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . (0 . ± .
3) (2 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
08) (4 . ± . . ± . . ± . . ± . . ± . . ± . . ± . ± . (2 . ± . ) (25 . ± . )2019-11-23 17:30:00-18:09:59 7 . ± . . ± . . ± . . ± . . ± . . ± . . ± . (0 . ± .
1) (8 . ± . )2019-11-30 13:30:00-19:09:59 4 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
7) (5 . ± . )2019-12-04 00:00:00-06:59:59 4 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
9) (2 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . (1 . ± . ) (1 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
08) (3 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . ) (4 . ± . )2019-12-22 19:05:00-19:54:59 5 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
5) (3 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
1) (1 . ± . ompressive turbulence at sub-ion scales Table 6.
Table of the mean and standard deviations of solar wind fast survey mode intervals where the density calibration hasbeen performed 2020.
Date Time B [nT] V i [km/s] n e [cm − ] β i (OMNI) β e T i [eV] (OMNI) T e [eV]2020-01-02 05:20:00-07:19:59 3 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
3) (2 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . (0 . ± .
05) (2 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . (0 . ± .
2) (9 . ± . )2020-01-12 17:20:00-20:09:59 2 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
3) (5 . ± . )2020-01-19 20:00:00-21:59:59 1 . ± . . ± . . ± . . ± . ± . . ± . . ± . . ± . ) (1 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
09) (2 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . ) (2 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
9) (2 . ± . . ± . . ± . . ± . . ±
50 1 . ± . . ± . . ± . (1 . ± . ) (9 . ± . )2020-02-02 17:00:00-18:59:59 4 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
2) (6 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
2) (2 . ± . . ± . . ± . . ± . . ±
200 1 . ± . . ± . . ± . (0 . ± .
3) (6 . ± . )2020-02-(06-07) 14:00:00-01:59:59 7 . ± . . ± . . ± . . ± . . ± . . ± . . ± . (0 . ± .
5) (21 . ± . )2020-02-08 13:30:00-16:29:59 2 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
5) (6 . ± . )2020-02-09 21:10:00-22:29:59 2 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
2) (4 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . (1 . ± . ) (2 . ± . . ± . ± . . ± . . ± . . ± . . ± . . ± . (2 . ± . ) (13 . ± . )2020-02-20 10:00:00-10:39:59 4 . ± . . ± . . ± . . ± . . ± .
10 31 . ± . . ± . . ± .
06) (5 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . (1 . ± .
9) (15 . ± . )2020-02-23 17:40:00-19:09:59 4 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
05) (3 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
1) (1 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . (1 . ± . ) (9 . ± . )2020-03-01 20:50:00-21:29:59 4 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
2) (5 . ± . )2020-03-06 14:50:00-15:29:59 6 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
3) (7 . ± . )2020-03-07 15:00:00-19:29:59 5 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
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5) (6 . ± . Roberts et al.
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