Solar Wind Magnetic Field Background Spectrum from Fluid to Kinetic Scales

arXiv:1708.01145v1 [astro-ph.SR] 3 Aug 2017 MNRAS t.TeT ste olwdb idwihprogressively ⋆ which wind a by followed then Alfvé is of TE level temperature, is The highest and ity. which the speed and (TE), compressibility, wind lowest edge the highest trailing the the by by characterized followed is Alfvénicit low region and This temperature high magnetic, and kinetic © 2016 oet Bruno Roberto fluid from scales spectrum kinetic background to field magnetic wind Solar uy2018 July 8 ih hr h yaia neato ewe h w flows two ( the stronger the between is stream interaction is slow dynamical and the region fast where first right between located The (SI), character. interface and different stream field a fluc- have regions. scales, and values tuations daily average different different on assume by parameters extending plasma characterized regions, po- are these of Within extension holes, equatorial coronal the lar from i. coming hereafter), (HSS’ streams streams those speed high corotating Typical INTRODUCTION 1 3 2 1 nvriyo ’qia eateto hscladChemica and Physical of Department L’Aquila, Obser of Astrophysical A University Astrophysics, Space for for Institute Institute National Astrophysics, for Institute National otc e-mail: Contact 07TeAuthors The 2017 .TeS scaatrzdb ihlvlo rsue both pressure, of level high by characterized is SI The ). 000 cwn n ash1990 Marsch and Schwenn , 1 – 7 [email protected] 21)Pern uy21 oplduigMRSL MNRAS using Compiled 2018 July 8 Preprint (2017) 1 ⋆ ail Telloni Daniele , rnsae,teseta ne ean rud-. n,bsdon based and, -2.7 frequ around higher remains for spe at index available c) the spectral -2.7; the break, around scales, value frequency tron a high towards the flatter and beyond flatter w right b fluctuations decade slower scaling; magnetic one Kolmogorov to typical of about the level faster keeping expans power decreases, from trailing largely the range regions, the a) behavior: two by following formed these in the region between stream rarefaction a Moving by slower stream. c characterized a mainly is and Alfvénic stream fluctuations, region fast less corotating and typical amplitude The win smaller Alfvénic, slow Fast by mainly transport. fluctuations, they populated an fluctuations incompressible field of amplitude, magnetic type of large the values for dy average also two the These but flows. for eters slow only and not fast in differ structured remarkably highly is wind solar The ABSTRACT ed plasmas – inter fields observe the superpo the into words: be wind time Key would Alfvénic faster any spectrum amplitude the but, large channeling wind and fluxtube Th turbulent slower a a spectrum. of and background part faster field inner s both magnetic the high to of of common sort characteristic a be features, of spectral existence these the All speed. flow ; rn n Carbone and Bruno 4 ooaigsras h oe ee osntcag,irrespect change, not does level power the streams, corotating Sn)slrwn ublne–wvs–Sn eishr magnetic – heliosphere Sun: – waves – turbulence – wind solar (Sun:) 2 aioDeIure Danilo , aoyo oio i sevtro2,105Pn Torinese, Pino 10025 20, Osservatorio Via Torino, of vatory tohsc n lntlg,VadlFsodlCvlee10 Cavaliere del Fosso del Via Planetology, and strophysics nic- e. y. cecs ’qia Italy L’Aquila, Sciences, l , fafwH,byn h ihfeunybekseparating break ( index variabilit high-frequency remarkable spectral frequencies a the the recorded to scales, beyond up kinetic from Hz, scales, fluid proton few at a slope of spectral the of ior ( low Alfvénicity changes which to across high separated region are from stream narrow abruptly, the very of a portions by two these enough, ing h udrgm,teeaeas la ieecsa proton at differences clear also are withi scales. there differences kinetic regime, above-cited the fluid besides the sca Kolmogorov-like addition, persistent In a ing. shows slope clearly spectral scales the fluid although Alfvénic a character its level observes power to one its also to wind relatively slow only not to compress- turbulence, fast different plasma from and moving field Thus, helio- higher ibility. the but by Alfvé temperature low followed low by ity, is characterized crossing region sheet rarefaction current spheric the time, the of firstly was region This by cooler. region becomes rarefaction named and down slows 3 n ran Pietropaolo Ermanno and rn tal. et Bruno mt ta.2006 al. et Smith ( 2014 Hundhausen rn eln 2015 Telloni & Bruno ,ivsiaigtebehav- the investigating ), idi hrceie by characterized is wind edsras suggest streams, peed nis eoeteelec- the before encies, tpoo cls for scales, proton at ) utbeobservations suitable ; ta ne becomes index ctral efc,afs wind fast a terface, aroie l 2010 al. et Sahraoui ssetu would spectrum is id eobserve we wind, lsaparam- plasma d ti h inertial the ithin A lntr space, planetary aia regimes namical e oit. to sed T o deo the of edge ion ( E 1972 ,013Rm,Italy Roma, 00133 0, ol cross would r sgenerally is d tl l v3.0 file style X Italy ompressive. .Interest- ). v fthe of ive .Most ). 3 nic- of y but at l- n ) 2 R. Bruno et al. within the following frequency decade or so. The steepest Table 1. Time intervals referring to the 4 high-speed streams spectra corresponded to the trailing edges of fast streams observed with CLUSTER and THEMIS, along with the corre- while the flattest ones were found within the subsequent sponding average wind speed. slow wind regions. The same authors found an empirical re- lationship between the power associated to the inertial range and the spectral slope observed within this narrow region # Year Time interval s/c hVi −1 which allowed to estimate that this slope approaches the DoY:hh km s Kolmogorov scaling within the slowest wind and reaches a 1 2007 72:06 - 73:09 CLUSTER-1 671 limiting value of roughly 4.4 within the fast wind. In addi- 74:15 - 75:18 CLUSTER-1 612 tion, Bruno et al. (2014) suggested the possible role played 79:09 - 80:12 CLUSTER-1 296 also by Alfvénicity in this spectral dependence. Recent the- 2 2011 36:18 - 37:01 THEMIS-C 580 oretical results seem to move in this direction attributing a 37:18 - 38:01 THEMIS-C 527 relevant role to Alfvénic imbalance within the inertial range 39:08 - 39:21 THEMIS-C 422 (Voitenko & De Keyser 2016). 40:18 - 41:01 THEMIS-C 323 The location of the frequency break and the nature of the fluctuations within this restricted frequency range 3 2011 149:15 - 150:00 THEMIS-C 680 was addressed by several authors within the recent lit- 152:15 - 153:00 THEMIS-C 514 153:15 - 154:00 THEMIS-C 447 erature on this topic (see reviews by Alexandrova et al. 154:15 - 155:00 THEMIS-C 399 (2013); Bruno and Carbone (2016)). Although turbulence phenomenology would limit the role played by parallel 4 2011 174:15 - 175:00 THEMIS-B 605 wavevectors k k, Bruno & Trenchi (2014) firstly found that 178:15 - 179:02 THEMIS-C 408 the cyclotron resonant wavenumber showed the best agree- 179:15 - 180:00 THEMIS-B 389 ment with the location of the break compared to results obtained for the ion inertial length and the Larmor radius. Successively, Chen et al. (2014) found similar results of the spectral power density adopted in order to collapse all k but did not consider a possible role of k because it was the spectra on top of each other to highlight a remarkable not consistent with the observed anisotropy of turbulence similarity in the spectra. The main difference with the (Horbury et al. 2008). Thus, this problem is still open and analysis by Bruno et al. (2014) is that these authors did these observations expect to have a theoretical explanation. not apply any normalization and the collapsing of the Telloni et al. (2015) analysed the nature of the magnetic spectra within the high frequency range came out naturally. fluctuations within this limited frequency range beyond the Moreover, another major difference was in the choice of the spectral break and found that their character was com- intervals. In Alexandrova et al. (2009) most of the intervals patible with left-hand, outward propagating ion cyclotron belonged to compressive regions while, in our analysis, they waves and right-hand kinetic Alfvén waves, confirming pre- were avoided since we aimed to show the spectral evolution vious findings by authors like He et al. (2011, 2012a,b); from fast to slow wind within typical corotating HSS. Podesta and Gary (2011). Thus, the main goal of the present work is that of In particular, Bruno & Telloni (2015) found that these checking whether the results highlighted in Figure 4 by two populations of fluctuations experienced a different fate Bruno et al. (2014) are fortuitous or represent the canon- within different regions of the velocity stream. Both fluctu- ical situation within corotating HSS. ations become indeed weaker and weaker when analysing slower and slower wind, and the left-handed population would be the first one to disappear. Another interesting fea- 2 DATA ANALYSIS AND RESULTS ture of the analysed spectra, which stimulated the present work, was represented by the results shown in Figure 4 of In order to perform this kind of analysis it is necessary to Bruno et al. (2014). Those spectra refer to different time in- have observations covering a wide range of time scales. Since tervals chosen along the speed profile, from fast to slow wind. we aim to study the magnetic field power density spectrum The low frequency range and the high frequency range were between fluid and proton kinetic scales we need to cover obtained by Fourier transforming WIND data from the flux- time scales ranging from several tens of minutes to tens of gate magnetometer and THEMIS-C data from the searchcoil hertz. Such a wide dynamical range is generally covered by magnetometer, respectively. While within the inertial range two different kind of magnetometers, one for the low fre- the power density level greatly differs from fast to slow wind, quency or DC field and another one for the high frequency as expected, no evident differences can be observed within or AC field. Two common kinds of magnetometers adopted the high frequency range above 1Hz, all the spectra col- onboard magnetospheric and heliospheric s/c are flux-gate lapse on top to each other showing a spectral slope that and the search-coil instruments for DC and AC field, re- Bruno et al. (2014) estimated to be roughly −2.5. spectively (Pfaff et al. 1998). For our purposes, it is neces- A similar study was already produced by sary that both instruments operate within the same time Alexandrova et al. (2009) who showed that spectra interval although we are aware that the search-coil data are measured under different plasma conditions have a similar available only for a much shorter time with respect to the shape concluding that there is a sort of universality of fluxgate. Moreover, the present study limits the observations solar wind turbulent spectrum from MHD to electron to the solely corotating high-speed streams, not influenced scales. However, this universal spectrum obtained by by magnetospheric up-stream waves or solar wind transients, Alexandrova et al. (2009) was the result of a normalization and the following low-speed incompressive region. We paid

MNRAS 000, 1–7 (2017) magnetic background spectrum 3

Wilcox Solar Observatory Source Surface Synoptic Charts repository provides also directly power density spectra com- time puted from SCM (Search Coil Magnetometer) data sampled between 0.1 Hz to 4 kHz. The THEMIS data repository provides 4 and 128 Hz resolution measurements from ﬂuxgate and search-coil magnetometers, respectively.

Earth trajectory Particular attention has to be paid when analyzing FGM data at frequencies larger than a few Hz since this frequency range can be aﬀected by noise due to the digitization process (Bennett 1948; Russell 1972). As reported by

800 Bruno et al. (2014), the spectral level due to the digitization 700 C 700 vb CLUSTER 700 noise tends to ﬂatten out the power spectrum of FGM data 600 600 ] ] ] ] -1 -1 -1 600 -1 March 2007 2 3 500 500 for frequencies above roughly − Hz. Hence, in the present 500 [km/s] (descending

sw V V[km s V[km V[km s V[km V [km s [kmV s [kmV 400 400 400 phase of solar analysis, the study of the power spectra obtained from both cycle 23) 300 300 300 CLUSTER and THEMIS fluxgate measurements is limited 200 200 200 3/9 10/03/2007 3/13 13/03/2007 3/17 16/03/2007 3/21 19/03/2007 22/03/2007 3/17 3/15 3/13 3/11 3/9 3/7 3/5 date at the observerdate in situ [UT] on 2007 date at the source [UT] on 2007 to frequencies lower than 2 Hz. time As discussed in Alexandrova et al. (2013), CLUSTER search-coil measurements are severely affected by instru- Figure 1. Top panel: source surface synoptic chart from Wilcox mental noise above 100 Hz. Hence, in the present work, the Observatory, evaluated at 3.25 solar radii, for solar rotation #2054. power spectra obtained from SCM data are displayed up to Blue contour lines and light grey shading show positive magnetic 64 polarity regions while dark grey area and dotted red lines refer to Hz, to avoid noise problems and to match the Nyquist the opposite polarity. The black solid line shows the location of frequency of THEMIS search-coil measurements. Moreover, the neutral line. The dashed red line shows the Earth’s trajectory CLUSTER search-coil data are provided already filtered at as projected onto the Sun. Time runs from right to left. Lower 0.6 Hz and data below this value are not relevant and should left-hand-side panel: speed profile (black colour line) of the stream not be used. In our case, we adopted a conservative choice of interest between March 11 and 23 and Alfvénic correlation (red showing spectra only above 2 Hz which, on the other hand, colour line). The three coloured vertical bars highlight the selected is the high frequency limit of spectra built from FGM data. three time intervals. Lower right-hand-side panel: profile of the It is worth reminding that the THEMIS probes spin 1 same stream projected back to the Sun using hr solar wind with a frequency of 1/3 Hz. This leads to an artificial large speed averages. amplitude periodic signal at the spin frequency in the SCM measurements (see Le Contel et al. (2008) for a complete particular attention in avoiding slow wind compressive re- treatment of spurious signals and noise in THEMIS data). gions generated by the dynamical interaction between fast To remove the spin modulation effects from the spectral and slow wind, i.e. stream-stream interface, or belonging to analysis, THEMIS search-coil data are provided already the heliospheric current sheet crossing rather than to the high-pass filtered at a frequency slightly higher than the slow wind tail of the corotating high-speed stream. spin frequency. However, in the present analysis, the power Among all the past and current space missions probing spectra obtained from THEMIS search-coil measurements the solar wind, the only two allowing this kind of study, with are shown only for frequencies above 2 Hz, which, on the available data freely accessible on the web, are THEMIS and other hand, is the highest frequency of the spectra we show CLUSTER, two constellations of 5 and 4 flight-formation from the fluxgate data. s/c, respectively. However, both missions were primarily de- THEMIS search-coil data are furthermore affected by signed for studying the Earth’s magnetosphere and only for two different sources of spurious noise: large amplitude relatively short periods of time CLUSTER and THEMIS spikes at twice the spin frequency (1/3 Hz), and at 8 and are/were immersed in the solar wind. Unfortunately, it was 32 Hz, (Le Contel et al. 2008). Their harmonics are indeed possible to use THEMIS observations only during the rela- found to dominate the power spectrum at higher frequen- tively short period of time when THEMIS B and C s/c were cies. However, since both types of spurious noises are phase transferring from terrestrial to lunar orbit. On the other locked (with the spin frequency and with the onboard 1 sec- hand, Cluster did not offer a much better situation rela- ond instrument clock, respectively) and are quite constant tively to the choice of appropriate time intervals. Indeed, the in amplitude with time, it is possible to reduce their level by average Cluster orbital period of about 2 days and the aver- means of the Superposed Epoch Analysis (SEA) as described age short duration of a high-speed stream of about 1 week in Le Contel et al. (2008). makes quite low the probability that the s/c probes both the We applied this SEA technique to THEMIS search-coil trailing edge and the rarefaction region of the stream during data before the spectral analysis for all the time intervals consecutive passages at the apogee. Moreover, the apogee is we chose. However, as reported in Le Contel et al. (2008), in the solar wind only for a limited period of time during the this technique, very effective in reducing phase-locked noise, year. As a consequence, all the data analysis requirements is not equally effective in reducing noise that is not phase- on the choice of the time intervals and the orbital features locked and whose effects are still present in the spectra after of the above two missions made the data selection extremely SEA filtering, as we will comment in the following. difficult and we were able to select only 4 HSS’ from Cluster We compute Power Spectral Densities (PSD) of mag- and THEMIS B and C s/c, listed in Table 1. netic field data from fluxgate and search-coil magnetome- The CLUSTER data repository provides magnetic field ters onboard CLUSTER and THEMIS s/c from the trace vector measurements from the FGM (Flux Gate Magne- of the spectral matrix using the Morlet Wavelet Transform. tometer) instrument with a cadence of 0.2 sec. The same In particular, the wavelet PSD shown in this paper corre-

MNRAS 000, 1–7 (2017) 4 R. Bruno et al.

Wilcox Solar Observatory Source Surface Synoptic Charts time 800

700 ]

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500 V[km s V[km 400 Earth trajectory 300

200 3/9 3/13 3/17 3/21 date at the observer [UT] on 2007

800 800 Cvb Themis-C 700 700 May 2011 ]

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Figure 2. Spectral analysis of the selected time intervals as in- Figure 3. HSS observed by THEMIS-C in May 2011 is shown dicated in the inset plot by different coloured bars. Vertical lines in the same format of Fig. 2 refer to the position of the wavenumbers corresponding to the the proton inertial length κi , the proton Larmor radius κL and the smallest wavenumber corresponding to the resonance condition for parallel propagating Alfvén waves κr . The highest speed is recorded when the s/c is farther in latitude with respect to the local heliomagnetic equator and the lowest speed is recorded when the s/c is approach- sponds to the global wavelet spectrum (that is, the time- ing the smallest heliolatitude (Bruno et al. 1986), i.e. the averaged wavelet spectrum over the whole time interval), border of the coronal hole. The presence of spikes visible be- which provides an unbiased and consistent estimation of the tween the beginning of March 13 and the end of March 16 true power spectrum of a time series (Percival 1995). (left-hand-side panel) reveals the Alfvénic nature of those The first HSS we analysed was observed by Cluster in fluctuations (Matteini et al. 2014). This consideration is in- mid March 2007, during the descending phase of solar ac- deed strengthened by the corresponding values assumed by tivity cycle 23. The top panel of Fig. 1 is a source surface the red curve that runs through the panel. This curve shows synoptic chart from Wilcox Observatory. This chart, which the value of RV B that represents the Alfvénic correlation co- covers solar rotation #2054, shows contour levels of magnetic efficient. This parameter takes into account not only the de- field intensity evaluated at 3.25 solar radii (Hoeksema et al. gree of correlation between magnetic and kinetic fluctuations 1983). Blue contour lines and light grey shading show pos- but also the degree of equipartition between the two since 2 itive magnetic polarity regions while dark grey area and RV B = σC /p1 − σR where σC/ and σR are the normalized dotted red lines refer to the opposite polarity. The black cross-helicity and residual energy (Bavassano et al. 1998). solid line shows the location of the neutral line, or helio- In particular, the Alfvénicity curve shows a rapid increase magnetic equator, that smoothly runs between the two op- through the velocity enhancement around March 13 and an posite unipolar magnetic regions, characteristic of low solar abrupt decrease at the beginning of day 17 when we enter the activity. The dashed red line shows the Earth’s trajectory rarefaction region of the stream (see also Telloni and Bruno as projected onto the Sun. In this kind of map the time (2016) for similar events). This is a common characteristic runs from right to left. The lower left-hand-side panel shows of corotating HSS’ and the abrupt decrease in Alfvénicity the speed profile of the stream of interest between March might be related to particular physical conditions develop- 11 and 23. The data shown in this panel are 1 min plasma ing during the expansion or inherent of the peripheral re- averages taken from Wind because the vantage location of gions of the coronal hole. Moreover, the Alfvénic character this s/c at the Lagrangian point L1 allows to monitor the of the fluctuations continues to decrease as we get closer to solar wind continuously. This is a typical high-speed stream the typical slow wind around day 22 of March. with a bulk velocity ranging roughly between 750 and 300 In Fig. 2 we show the results of our spectral analysis km/s. The three coloured vertical bars highlight the three relative to the selected time intervals within the stream ob- time intervals that we could select due to the restrictions served by Cluster. To make it easier for the reader to follow imposed by the orbit of Cluster s/c and the availability of our results, we show again the relative location of the se- the data. In the lower right-hand-side panel we report the lected time intervals within the stream in the inset plot. profile of the same stream projected back to the Sun using Different colours refer to the different spectra shown in the 1 hr averages of solar wind speed. Although relatively low main panel. For sake of completeness, we show also the po- time resolution, 1 hr is largely appropriate to back project sition of the wavenumbers corresponding to the proton iner- onto the solar surface in-situ measurements. In this panel, tial length κi = ωp/c = Ωp/vA and the proton Larmor radius accordingly with the top panel, time is running from right to κL = Ωp/vth, where c is the speed of light, ωp is the proton left. Due to the different values of the wind speed at the be- plasma frequency and Ωp is the ion-cyclotron frequency. ginning and at the end of each timer interval, the temporal Moreover, we also show the position of the minimum width of the regions mapped onto the Sun differs from the wavenumber corresponding to the resonance condition for original duration of the time intervals at 1 AU (see left-hand- parallel propagating Alfvén waves κr = Ωp/(vA + vth) since, side panel). This back-projection shows that Cluster, during as already shown by Bruno & Trenchi (2014), this wavenum- the selected time intervals, was immersed in the equatorial ber gives the best match with the observed location of the extension of a southern polar coronal hole. spectral break.

MNRAS 000, 1–7 (2017) magnetic background spectrum 5

all spectra together [Mar. 2007, Feb., May, June 2011]

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Figure 5. All the spectra from all the 14 available time inter- Figure 4. Power density spectra of the time intervals indicated vals listed in Table 1. The red coloured segments highlight the in Fig. 3 in the same format adopted for Fig. 2. approximate position of the magnetic background spectrum.

Moving from red to blue colour is equivalent to move also in this case, the break location is in better agreement from fast to slow wind. All the three spectra show the with the ion-cyclotron resonance prediction than with the two different scalings characteristic of the fluid and kinetic wavenumbers corresponding to the inertial length or Lar- range, as we will discuss in the following. In addition, as mor radius. expected, the transition region between these two regimes Specific results from the other two time intervals anal- (Bruno et al. 2014) is particularly clear for the fast wind ysed in this work will be omitted for sake of brevity, as al- samples and it is well indicated by the location of the es- ready stated before, but all the spectra from all the 14 avail- timated value of κr . The different spectral index is not the able time intervals are reported in Fig. 5. Although Cluster only feature that characterizes these two regimes. As a mat- and THEMIS intervals belong to different phases of the solar ter of fact, while the inertial range shows large differences cycle, the level of the kinetic part of their respective spectra, in the power associated to magnetic fluctuations of fast and encircled by the green dashed line, remains remarkably con- slow wind, the level of the fluctuations within the kinetic stant in time while the level of the inertial range varies from regime does not change appreciably, confirming previous re- fast to slow wind as expected. The spectral slopes marked by sults obtained by Bruno et al. (2014). the red lines are shown just to drive the eye since the main As representative of the remaining three HSS’ observed goal of this paper is to show the different spectral variabil- by THEMIS we will show and discuss in detail only one ity between fluid and kinetic scales. As matter of fact, while of them since the results are very similar. In any case, we the scaling at kinetic scales estimated from CLUSTER time will show all the spectra from all the analysed time intervals intervals is around −2.6 the scaling estimated from all the together in summary plots that will be discussed later in the THEMIS intervals is around −2.3 due to the unavoidable paper. effect of the residual noise not phase locked and thus not This corotating HSS is shown in Fig. 3, in the same for- filtered by the SEA technique. mat of Fig. 2. These observations were taken by THEMIS-C The above results introduce some constraints on the about four years after the corotating stream observed by spectrum. We have to consider the following points: Cluster. This time interval falls within the ascending phase a) the border between fluid and kinetic scales is rea- of solar cycle 24 while the previous one was recorded during sonably well identified by the wavenumber associated with the descending phase of the previous cycle. The magnetic the ion-cyclotron resonance mechanism (Bruno & Trenchi configuration at the Sun has changed from the time inter- 2014); val previously discussed, and the magnetic equator is much b) the spectral slope in the inertial range is well approx- more warped than before, anticipating the general magnetic imated by the Kolmogorov’s scaling; field inversion that would take place around the maximum c) fluid and kinetic spectral branches must be con- of the same cycle, a few years later. For this stream there nected. were four possible time intervals to analyse between fast It follows that there should be a lower threshold for the and slow wind, as highlighted by the coloured stripes across spectrum of magnetic fluctuations and it can be graphically the lower left-hand-side panel. The projected location of the identified by the red segments in Fig. 5; a sort of background s/c onto the Sun (lower right-hand-side panel) shows that spectrum characteristic of corotating HSS. THEMIS-C orbit was immersed in the equatorial extension A further property of interplanetary magnetic field of a northern polar coronal hole. spectra can be used to infer what should be the highest Power density spectra related to these intervals are possible level of spectral density associated with turbulence shown in Fig. 4 in the same format adopted for Fig. 2. As fluctuations characterizing the inertial range. As a matter of already observed for the latter, spectra of fluid and kinetic fact, if we consider the following points: scales behave differently when moving from fast to slow wind a) the location of the break is between 3 and 4 × 3 1 samples. To the large variability in the inertial range corre- 10− km− (Bruno & Trenchi 2014); sponds a rather stable spectrum at proton scales although b) the maximum spectral slope within the transition the noise that could not be filtered, as previously discussed, region is roughly −4.4 (Bruno et al. 2014); causes the bumps that we notice at kinetic scales. Moreover, c) the width of this transition region is less than

MNRAS 000, 1–7 (2017) 6 R. Bruno et al.

all spectra together [Mar. 2007, Feb., May, June 2011]

~10 2 { -5/3 -4.4

-5/3 ?

? -2.5

~3.4E-3

Figure 6. Geometrical construction to estimate the upper bound of the magnetic power spectral density within HSS. See text for details. Figure 7. Main plot: 2D histogram of the total magnetic PSD, for a scale of 2 × 10−5km−1, as a function of wind speed. Right- hand-side graph: histograms built for slow (< 400 km/s) and fast > 600 1 decade, as inferred from previous studies (Bruno et al. ( km/s) wind, respectively. The difference between points ”A” and ”B” is an estimate of the range of variability of the PSD 2014); from slow to fast wind. we can draw the upper bound for magnetic fluctuations within the inertial range as shown by the dashed red line in Fig. 6. At this point, we can estimate the maximum width cade experienced by these fluctuations, this limiting value of the interval of variability for the power spectral density should depend on the power associated with the injection between slow and fast wind. This interval results to be of scales, which, in turn, should not depend on the wind speed the order of 2 decades, as indicated in the plot. Obviously, either. In particular, it is possible, although a dedicated there are uncertainties associated with this rough estimate analysis would be needed before any conclusion, that the but the expected effect cannot produce differences by orders larger variability within the slow wind is probably due to of magnitude. the presence of those slow wind intervals characterized by In order to make this prediction more robust from a large amplitude Alfvénic fluctuations recently studied by statistical point of view, we analyzed 12 years of 1-min D’Amicis and Bruno (2015). As a matter of fact, these au- magnetic field measurements of the WIND s/c from 2005 thors found that the level of Alfvénic fluctuations within to 2016. Based on the Morlet wavelet transform, we com- the Alfvénic slow wind are of the same order of amplitude puted the PSD of magnetic field fluctuations at a single time of those characteristic of the fast Alfvénic wind. 5 1 scale well within the inertial range, namely 2×10− km− . To A final consideration would suggest that, using the ra- make this statistical analysis the most compatible with the dial dependencies found by Bavassano et al. (1982) it would spectral analysis discussed so far, we eliminated from the be possible to predict the upper level of the inertial range WIND dataset all the intervals corresponding to stream in- spectrum that future space missions like Parker Solar Probe terfaces and to interplanetary coronal mass ejections, as cat- and Solar Orbiter might observe in the inner heliosphere. 4.2 alogued by Ian Richardson and Hilary Cane since January Using the radial dependence R− found by these authors 1996 (Richardson and Cane 2010). In this way, we focused for the inertial range, we would expect to find an upper limit on corotating high-speed streams, mostly. The 2D histogram at 0.29 AU about 1.8×102 times higher than the upper limit of Fig. 7 shows the distribution of the PSD as a function of found at 1 AU shown in Fig. 6. wind speed, from the slowest to the fastest wind. This distribution appears to be asymmetric being char- acterised by a wider range of variability within the slow 3 SUMMARY AND DISCUSSION wind. As expected, larger PSD is preferentially associated with faster wind. The 2 distributions shown in the right- Power spectral density of interplanetary magnetic field fluc- hand-side graph are two histograms built for slow (i.e. < 400 tuations shows an unpredicted behavior at kinetic scales. km/s) and fast (i.e. > 600 km/s) wind, respectively. They Irrespective of what happens at fluid scales, the power spec- show the range of variability for the PSD on a scale of tral density level of kinetic scales remains largely unchanged 2 × 10−5km−1 within these velocity intervals. The peaks of when analysing either fast or slow wind data taken within the distributions are well separated by slightly less than 1 corotating HSS’. decade. However, if we take into account the variability of Because of this and because of the roughly fixed location the PSD we have to consider the difference between points of the break point separating fluid from kinetic scales, we can ”A” and ”B” which have been set at half maximum of their establish, for the slow wind, a sort of background spectrum respective distributions. In doing so we end up with a total extending from fluid to proton scales. The power spectral variability of the order of 2 decades, similar to the estimate level of this spectrum remains frozen when we look at dif- obtained from Fig. 6. ferent streams even if they are largely separated in time, It is interesting to notice the rather flat top of the dis- i.e. coming from different source regions. Slower solar wind tribution extending from slow to fast wind. This represents comes from the peripheral areas of the equatorial extension a sort of limiting value for the PSD for the scale stud- of a polar coronal hole and is relatively poor in Alfvénic ied here. In the framework of a non-linear turbulent cas- fluctuations. The present literature suggests that this spec-

MNRAS 000, 1–7 (2017) magnetic background spectrum 7 trum is likely due to the non-propagating background mag- Bavassano, B., Dobrowolny, M., Fanfoni, G., Mariani, F., & Ness, netic structure advected by the wind (Tu and Marsch 1995; N. F. 1982, Sol. Phys., 78, 373 Bruno and Carbone 2016). The Alfvénic spectrum, which Bavassano, B., Pietropaolo, E., & Bruno, R. 1998, J. Geo- characterizes the fastest wind found in the trailing edge of phys. Res., 103, 6521 ˘ the streams, is then superimposed onto the background spec- Bennett, W. 1948. Bell System Technical Journal, 27, 446âA¸S472. Bruno, R., Villante, U., Bavassano, B., Schwenn, R., Mariani, F. trum and can vary in power from one stream to the other. 1986. Solar Physics 104, 431-445. Alfvénic magnetic fluctuations, being much larger than those Bruno, R., & Trenchi, L. 2014, ApJ, 787, L24 due to the advected magnetic structure, completely cover Bruno, R., Carbone, V. 2016, Lecture Notes in Physics, Berlin the background spectrum. However, it is still unclear what Springer Verlag, 928 determines the spectral differences between these two con- Bruno, R., Trenchi, L., & Telloni, D. 2014, ApJ, 793, L15 tiguous regions of the HSS. Is it due to differences inher- Bruno, R., & Telloni, D. 2015, ApJ, 811, L17 ent the nature of the source regions or is it due to some Chen, C. H. K., Leung, L., Boldyrev, S., Maruca, B. A., & Bale, mechanism that acts differently during the expansion? A S. D. 2014, Geophys. Res. Lett., 41, 8081 dedicated study to this topic is highly needed and future D’Amicis, R., Bruno, R. 2015. The Astrophysical Journal 805, 84. observations by Parker Solar Probe and Solar Orbiter will He, J., Marsch, E., Tu, C., Yao, S., & Tian, H. 2011, ApJ, 731, 85 test these different possibilities. In any case, also for the He, J., Tu, C., Marsch, E., & Yao, S. 2012, ApJ, 749, 86 Alfvénic spectrum there seems to be a threshold which, in He, J., Tu, C., Marsch, E., & Yao, S. 2012, ApJ, 745, L8 this case, limits the maximum amplitude of the fluctuations, Hoeksema, J. T., Wilcox, J. M., Scherrer, P. H. 1983. Journal of as expected for any physical phenomenon. The fact that Geophysical Research 88, 9910-9918. the level of the kinetic spectrum does not depend on the Horbury, T. S., Forman, M., & Oughton, S. 2008, Phys. level of the inertial spectrum suggests that the magnetic en- Rev. Lett., 101, 175005 ergy associated with the Alfvénic fluctuations is somehow Hundhausen, A. J. 1972. Coronal Expansion and Solar Wind. transferred to particles, within the spectral transition re- Physics and Chemistry in Space 5, . gion, to modify their plasma kinetics. Several mechanisms, Le Contel, O., and 17 colleagues 2008. Space Science Reviews 141, invoking wave particle interaction to replace collisions in a 509-534. Marsch, E. 2006, Living Rev. Solar Phys., 3, 1 non-collisional fluid like the solar wind, or invoking the role Matteini, L., Horbury, T. S., Neugebauer, M., & Goldstein, B. E. of strong field fluctuations within current sheets generated 2014, Geophys. Res. Lett., 41, 259 by turbulence reconnection processes able to energize parti- Percival, D. B., 1995, Biometrika, 82, 619âA¸S31.˘ cles, have been proposed (see reviews by Alexandrova et al. Pfaff, R. F., Borovsky, J. E., Young, D. T. 1998. Washington DC (2013), Marsch (2006) and Bruno and Carbone (2016) and American Geophysical Union Geophysical Monograph Series references therein). Unfortunately, we do not yet have ob- 103, . servations of the particles velocity distribution function at Podesta, J. J., Gary, S. P. 2011. The Astrophysical Journal 734, kinetic scales in the solar wind to better understand which 15. are the main physical mechanisms at work. However, we Richardson, I. G., Cane, H. V. 2010. Solar Physics 264, 189-237. are confident that ESA-Solar Orbiter, NASA-Parker Solar Russell, C. T., 1972. Solar Wind, NASA sp-308, 365-374 Sahraoui, F., Goldstein, M. L., Belmont, G., Canu, P., Rezeau, Probe and possibly ESA-THOR, will provide all the neces- L. 2010. Phys. Rev. Lett. 105, 131101. sary observations to shed light on this fundamental question. Schwenn, R., Marsch, E. 1990. Physics of the inner heliosphere. 1. Large-scale phenomena.. Physics and Chemistry in Space 20, . Smith, C. W., Hamilton, K., Vasquez, B. J., Leamon, R. J. 2006. ACKNOWLEDGEMENTS ApJ, 645, L85-L88. One of the Authors (DT) is financially supported Telloni, D., Bruno, R., & Trenchi, L. 2015, ApJ, 805, 46 Telloni, D., Bruno, R. 2016. Monthly Notices of the Royal Astro- by the Italian Space Agency (ASI) under con- nomical Society 463, L79-L83. tract I/013/12/1. WIND and THEMIS-B/C data Tu, C.-Y., Marsch, E. 1995. Space Science Reviews 73, 1-210. were obtained from the NASA-CDAWeb website Voitenko, Y., De Keyser, J. 2016. The Astrophysical Journal 832, https://cdaweb.sci.gsfc.nasa.gov/index.html. Cluster L20. data were obtained from Cluster data repository website https://www.cosmos.esa.int/web/csa. The list of inter- This paper has been typeset from a TEX/LATEX file prepared by planetary coronal mass ejections, compiled by Ian Richard- the author. son and Hilary Cane, can be found at the following website: www.srl.caltech.edu/ACE/ASC/DATA/level3/icmetable2.htm. Finally, we like to thank our anonymous Referee for her/his valuable comments.

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