A study of density modulation index in the inner heliospheric solar wind during solar cycle 23
Susanta Kumar Bisoi, P. Janardhan, M. Ingale, P. Subramanian, S. Ananthakrishnan, M. Tokumaru, K. Fujiki
aa r X i v : . [ a s t r o - ph . S R ] A ug A study of density modulation index in the inner heliosphericsolar wind during solar cycle 23
Susanta Kumar Bisoi and P. Janardhan
Astronomy & Astrophysics Division, Physical Research Laboratory, Ahmedabad 380 009, India. [email protected],[email protected] and
M. Ingale and P. Subramanian
Indian Institute of Science Education and Research, Dr. Homi Bhabha Road, Pashan, Pune 411 021, India. [email protected],[email protected] and
S. Ananthakrishnan
Department of Electronic Science, University of Pune, Pune 411 007, India. [email protected] and
M. Tokumaru, and K. Fujiki
Solar-Terrestrial Environment Laboratory, Nagoya University, Nagoya 464-8601, Japan. [email protected], [email protected]
ABSTRACT
The ratio of the rms electron density fluctuations to the background density in the solar wind(density modulation index, ǫ N ≡ ∆ N/N ) is of vital importance in understanding several problemsin heliospheric physics related to solar wind turbulence. In this paper, we have investigated thebehavior of ǫ N in the inner-heliosphere from 0.26 to 0.82 AU. The density fluctuations ∆ N havebeen deduced using extensive ground-based observations of interplanetary scintillation (IPS) at327 MHz, which probe spatial scales of a few hundred km. The background densities ( N ) havebeen derived using near-Earth observations from the Advanced Composition Explorer ( ACE ).Our analysis reveals that 0 . . ǫ N . .
02 and does not vary appreciably with heliocentricdistance. We also find that ǫ N declines by 8% from 1998 to 2008. We discuss the impact of thesefindings on problems ranging from our understanding of Forbush decreases to the behavior of thesolar wind dynamic pressure over the recent peculiar solar minimum at the end of cycle 23. Subject headings: turbulence — solar wind — interplanetary medium
1. Introduction
The solar wind is an unparalleled natural lab-oratory for the study of magneto-hydrodynamicturbulence e.g., (Tu & Marsch 1995; Goldstein et al. 1995; Bruno & Carbone 2005; Marsch 2006; Spangler2009). It involves fluctuations in magnetic field,density and velocity over a wide range of spa-tial and temporal scales. Turbulent densityfluctuations in the solar wind have been ob-1erved over heliocentric distances ranging from ∼ R ⊙ to 1 AU or 215 R ⊙ fromthe Sun, where R ⊙ is the solar radius (Coles1978; Marsch & Tu 1990; Bavassano & Bruno1995; Janardhan et al. 1996; Efimov et al. 2000;Spangler 2002; Bird et al. 2003; Spangler 2009;Tokumaru et al. 2012). Moreover, density fluc-tuations are often believed to be better tracersof solar wind flows as compared to solar winddensity (Ananthakrishnan et al. 1980; Woo et al.1995; Huddleston et al. 1995). Detailed measure-ments of solar wind density fluctuations near theEarth have been made using in-situ data fromspacecraft, such as Helios 1 , Helios 2 , Wind , and
Ulysses .MHD turbulence theory generally assumes in-compressibility, and density fluctuations do notfit into the narrative. Furthermore, the scalinglaw in (spatial) wavenumber space exhibited bydensity turbulence observations is generally con-sistent with the Kolmogorov theory, which in factholds for incompressible fluid turbulence in theabsence of magnetic fields. The implications ofcompressibility (as evidenced by observations ofturbulent density fluctuations) vis-a-vis theoriesof MHD turbulence is a subject of considerablediscussion (Tu & Marsch 1994; Hnat et al. 2005;Shaikh & Zank 2010). In particular, knowing themanner in which the density modulation index ǫ N ≡ ∆ NN (1)varies with distance from the Sun is of vital im-portance for a variety of applications.In the expression for ǫ N (Eq 1), the quan-tity ∆ N represents the turbulent density fluc-tuation while N is the background density. Anunderstanding of ǫ N is important for under-standing turbulent dissipation and consequentlocal heating of the solar wind (Carbone et al.2009). It is also an important ingredient in con-structing models for the quantity C N , which isthe amplitude of the density turbulence spec-trum (Thejappa & MacDowall 2008). In turn, C N is crucial in understanding angular broad-ening of radio sources due to solar wind turbu-lence (Janardhan & Alurkar 1993; Bastian 1994;Subramanian & Cairns 2011) and in explainingthe rather low brightness temperatures of thesolar corona at meter to decameter wavelengths (Thejappa & MacDowall 2008). A crucial role isalso played by ǫ N in influencing the propagationof energetic electrons, produced by solar flares andother explosive solar surface phenomena, throughthe heliosphere (Reid & Kontar 2010).Recently, using IPS measurements of scintil-lation index from 1983 to 2009, the solar windmicro-turbulence levels in the inner heliospherewere shown to be steadily declining since ≈ ≈ ACE and
Wind ) (Jian et al. 2011) andout-of-ecliptic (
Ulysses ) (McComas et al. 2008)solar wind measurements, during the recent min-imum of solar cycle 23, in 2008 – 2009, haveshown a reduction in solar wind dynamic pres-sure of about 20%. Under these very unusualand unique circumstances of declining solar po-lar field strengths and density turbulence levels( ∝ ∆N) (Janardhan et al. 2010, 2011; Bisoi et al.2014), studies of the temporal changes of ǫ N in theinner-heliosphere are both important and crucialin understanding the relation between magneticfield fluctuations and density fluctuations. Sucha study also impinges on the important questionof the role of the dynamic pressure exerted by thesolar wind on the Earth’s magnetosphere duringthis unusual phase.The first measurements of ǫ N were made atheliocentric distances . R ⊙ , by Woo et al.(1995) using Ulysses measurements obtained in1991. Subsequently, density fluctuations in differ-ent types of solar wind flows have been reportedat 1 AU (Huddleston et al. 1995) and also in theregion from 0.3 to 1 AU using the
Helios 2 space-craft, interplanetary plasma data, obtained witha time cadence of 45 mins (Bavassano & Bruno1995). These authors reported a ǫ N of ≈ . . ǫ N . .
15 in the heliocentric dis-tance range 16 – 26 R ⊙ . Using Wind spacecraftdata at 1 AU, Spangler & Spitler (2004) have es-timated ǫ N of the order of 0.03 – 0.08 and proposedboth a linear and quadratic relationship between2he ǫ N and the magnetic field index ( ǫ B ) in regionsof the near-sun solar wind. The data used in pre-vious papers have been sparse, with either the ob-servations being confined to a small region of theheliosphere or covering periods from a few days toyears. However, in this paper, we have made use ofobservations spanning the whole inner-heliospherecovering the heliocentric distance range of 0.26 –0.82 AU corresponding to 55 – 175 R ⊙ . In addi-tion, our data set of eleven years covers the wholeof solar cycle 23, thereby enabling a study of thelong term temporal variation in ǫ N as well.In this study, we have made use of exten-sive and systematic IPS measurements to in-vestigate the radial evolution of ǫ N defined inEq (1). While electron density fluctuationshave been estimated at 327 MHz using mea-surements from the multi-station IPS observatoryof the Solar-Terrestrial Environment Laboratory(STEL), Japan, the solar wind densities used werederived from in-situ observations from the ACE spacecraft (Stone et al. 1998) with ǫ N being es-timated for the period 1998 – 2008, covering thewhole of solar cycle 23.The rest of the paper is organized as follows:section 2 briefly discusses interplanetary scintilla-tion as well as phase modulation of plane wavesby the solar wind. In section 3, the use of IPSand ACE data and their analyses are discussed.Subsequently, in section 4 we verify the long termtemporal and spatial behavior of ǫ N . Finally, sec-tion 5 summarizes our results.
2. Interplanetary scintillation
IPS is a diffraction phenomenon in which coher-ent electromagnetic radiation from a distant radiosource passes through the turbulent and refract-ing solar wind and suffers scattering. This resultsin random temporal variations of the signal inten-sity (scintillation) at the Earth. A schematic ofthe typical IPS observing geometry is shown inFigure 1. The broken lines in Figure 1 lie in theecliptic plane, while the solid lines lie out of theecliptic plane. The long-dashed line is the orbitof the Earth around the Sun. The line-of-sight(LOS) to a distant compact radio source with re-spect to the Sun (‘S’) and the Earth (‘E’) is shownby a solid line from E passing through the point‘P’, the point of closest approach of the LOS to the Sun. The angles ǫ and γ are respectively, the solarFig. 1.— A schematic of the IPS observing ge-ometry. The Earth, the Sun, the point of closestapproach of the LOS to the Sun, and the foot pointof a perpendicular from P to the ecliptic plane areshown by points E, S, P and A while the angles ǫ and γ are the solar elongation and heliographiclatitude of the observed source.elongation and heliographic latitude of the sourcewhile ‘A’ is the foot point of a perpendicular fromP to the ecliptic plane. The heliocentric distance‘r’ of the radio source, in AU, is given by r = sin( ǫ ).It must be noted that the scintillations observed atthe Earth are modulated by the Fresnel filter func-tion Sin ( q λ z4 π ) where, q is the wave number of theirregularities, z is the distance from E to P, and λ isthe observing wavelength. Due to the action of theFresnel filter, IPS observations at 327 MHz enableone to probe solar wind electron density fluctua-tions of scale sizes ≤ ≤ km, there are largerscale solar winddensity fluctuations caused by structures such ascoronal mass ejections (CMEs) and solar flares,which originate on the solar surface. The typicalscale sizes of these structures range from 10 to 10 km. The action of the Fresnel filter for scale sizes ≥ km is such that it will give rise to scintil-lation at distances > m = ∆S < S > , where ∆S is the scintillating fluxand < S > is the mean flux of the radio sourcebeing observed. For a given IPS observation, mis simply the root mean-square deviation of thesignal intensity to the mean signal intensity andcan be easily determined from the observed inten-sity fluctuations of compact extragalactic radiosources.Though IPS measures only small scale fluctu-ations in density and not the bulk density itself,it has been shown (Hewish et al. 1985) that therewere no variations in IPS measurements of ∆ N that were not associated with corresponding vari-ations in density N . These authors used a nor-malized scintillation index ‘g’ (a good proxy forthe density) to derive a relation between ‘g’, andthe density given by g = ( N cm − /9) . ± . .For an ideal point-like radio source and at anobserving wavelength λ , m will steadily increasewith decreasing distance ‘r’ from the Sun untilit reaches a value of unity at some distance fromthe Sun. As r continues to decrease beyond thispoint, m will again drop off to values below unity.This turnover distance is a function of observingfrequency and at 327 MHz ( λ = 92 cm) occursat ≈ ≈ ⊙ . The region beyond theturn-over distance is known as the weak scatteringregime. In addition to the dependence on heliocen-tric distance, m will also reduce with an increasein the angular diameter of the radio source beingobserved. The assumption that the solar wind is consid-ered to be a confined to a thin slab as depicted inFigure 1 is due to the fact that the solar wind scat-tering function β (r) ∝ r − . Hence, most of thecontribution to the scintillation will come from the point ‘P’ on the LOS that is closest to the sun.Plane waves from distant, compact extragalacticradio sources on passing through the thin slab ofdensity irregularities will have an rms phase de-viation ( φ rms ) imposed across their wave fronts.The expression for φ rms is φ rms = ( π ) λ r e ( aL ) [ < ∆ N > ] (2)where, r e is the classical electron radius, λ is theobserving wavelength, and a is the typical scalesize in the thin screen of thickness L (see Fig.1).In the weak scattering regime, m is given by m ≈ √ φ rms (3)Equations 2 and 3 can be rewritten as ∆ N = m ( ) ( π ) λ r e ( aL ) (4)Equation (4) gives us a prescription for deter-mining the quantity ∆ N from observations of m .
3. Data analysis
Regular IPS observations on a set of about200 chosen extragalactic radio sources have beencarried out to determine solar wind velocities andscintillation indices at 327 MHz (Kojima & Kakinuma1990; Asai et al. 1998) since 1983 at the multi-station IPS observatory of STEL, Japan. Prior to1994, these observations were carried out by thethree-station IPS facility at Toyokawa, Fuji, andSugadaira. In 1994, one more antenna was com-missioned at Kiso forming a four – station dedi-cated IPS network that has been making system-atic and reliable estimates of solar wind velocitiesand scintillation indices (Tokumaru et al. 2012)except for a data gap of one year in 1994. System-atic observations have been carried out on abouta dozen selected radio sources each day such thateach source would have been observed over thewhole range of heliocentric distances between 0.2and 0.8 AU in a period of about 1 year. We haveemployed the daily measurements of m, spanningthe period from 1998 to 2008, covering solar cycle23.Very compact radio sources are extremelyrare and it has been established at a num-ber of frequencies, using both IPS (Bourgois1969; Bourgois & Creynet 1972; Milne 1976) and4 mas150 mas300 mas450 mas0.17 0.34 0.76Solar Distance (AU)
Fig. 2.— shows curves of theoretically values of mas a function of solar elongation for various sourcesizes corresponding to sizes of 0 mas, 150 mas, 300mas, and 450 mas. These theoretical values of mare computed using Marians (1975) model.long baseline interferometry (Clark et al. 1968;Clarke et al. 1969) that the radio source 1148-001has an angular diameter of ≈
10 milli arcsecond(mas) at meter wavelengths. Thus, the source1148-001 can be treated as a nearly ideal pointsource at 327 MHz, with almost all of its flux con-tained in a compact scintillating component withvery little flux outside this compact component(Swarup 1977; Venugopal et al. 1985). As statedearlier, for such ideal point sources m will be unityat the turn-over distance, and will drop as the dis-tance of the LOS to the source moves further awayfrom the sun. For sources with larger angular di-ameters, m will be less than unity at the turn-overdistance.Marians (1975) computed values of m for radiosources of a given source size as a function of r byobtaining theoretical temporal power spectra us-ing a standard solar wind model assuming weakscattering and a power law distribution of densityirregularities in the IP medium. Figure 2 showscurves of theoretical m, computed using the Mari-ans model (Marians 1975), as a function of ǫ (in de-grees) for source sizes of 0 mas, 150 mas, 300 mas,and 450 mas, respectively. All the curves are plot-ted for ǫ ranging from 15 ◦ to 55 ◦ correspondingto the weak scattering regime at 327 MHz whichcovers heliocentric distances between 0.26 and 0.82AU.For the present analysis and in order to obtain S c i n t ill a t i on I nde x Solar Elongation (deg.)Solar Distance (AU)
Fig. 3.— The upper panel shows by filled bluedots, the actual measurements of normalized scin-tillation indices for the source 0003-003. The the-oretically computed curve for m using Marian’smodel Marians (1975) for both 0003-003 (dottedblack) and 1148-001 (red line) are overplotted.The middle panel shows the same two theoreti-cal curves for sources, 1148-001 and 0003-003 af-ter the data of 0003-003 has been multiplied by afactor, determined from ratio of theoretical curvesof 1148-001 and 0003-003 at each ǫ , to remove theeffects of source size. The lower panel shows thedata for all 27 sources after being normalized toremove the source size effect. It can be seen thatthe data is well fitted to the theoretical curves ofthe source 1148-001.a uniform data set, it would be necessary to ei-ther choose sources of the same angular size orremove the effect of the finite source size by ap-propriately normalizing the data. The normaliza-tion was carried out using a least squares mini-mization to determine which of the Marians curvesbest fits the data for a given source. Since it isknown that 1148-001 is a good approximation to apoint source, the observed values of m of all othersources were multiplied by a factor equal to thedifference between the best fit Marians curve forthe given source and the best fit Marians curvefor 1148-001, at the corresponding ǫ . The bestfit Marians curve for 1148-001 corresponds to that5ig. 4.— Shows the coordinates (RA and Dec.) of the 27 selected radio sources by numbered open circles.The solid curve represents the path (RA and Dec.) of the Sun. Each numbered source name is indicated atthe bottom left of the figure.obtained for a source size of 10 mas.The upper panel of Figure 3 shows, by filledblue dots, one example of the actual observationsof m as a function of heliocentric distance for thesource 0003-003. The dashed red line is the Mari-ans curve corresponding to a source size of 10 mas,while the dashed black line is the Marians curvewhich best fits the the data for the source 0003-003. The middle panel of Figure 3 shows the samedata after it has been normalized, as describedabove to remove the effect of the finite sourcesize. After normalizing all the observations inthe above manner, we shortlist only those sourceswhich had at least 400 observations distributeduniformly over the entire range of heliocentric dis-tances without any significant data gaps. Usingthis criteria we finally shortlisted 27 sources forfurther analysis. The normalized points for all27 sources are shown in the lowermost panel ofFigure 3 and they fit the theoretical curve of thesource 1148-001 very well. The Right Ascensionand Declination (J2000 epoch) of the 27 short-listed radio sources are shown in Figure 4 by num-bered open circles with the corresponding names of the sources (B1950 epoch) listed at the bottomof Figure 4. The ecliptic radio sources in Figure 4are those in the declination range ± ◦ , while thenon-ecliptic or high latitude sources lie above thisrange of declinations.Using equation 4, ∆ N has been obtained atheliocentric distances in the range 0.26 – 0.82 AU(55 – 175 R ⊙ ) from 1998 to 2008, using daily IPSmeasurements of m. In order to estimate thebackground solar wind density, we use values ofthe daily average solar wind density ( N ) obtainedfrom the Solar Wind Electron, Proton, and Al-pha Monitor (SWEPAM) onboard the ACE space-craft, covering the period from 1998 to 2008. How-ever,
ACE density measurements are effectivelyat a distance of 1 AU. Thus, for estimation ofdensity at the locations, spread over distances of0.26 – 0.82 AU, the measured
ACE densities at 1AU were extrapolated in the sunward direction us-ing a background density model by Leblanc et al.(1998). According to this model, the background6ensity, N at r (in units of AU) is given by N = . − + . × − r − + . × − r − cm − (5)This equation assumes a density of 7.2 cm − at 1 AU. In order to derive the background den-sity at a given r , we use equation 5 multiplied by N (1 AU)/7.2, where N (1 AU) denotes the value ofthe density from the ACE data. As discussed ear-lier, the ∆ N is deduced from IPS measurementsof m using Eq. 4. We compute N by using near-Earth ACE measurements that are contemporane-ous with the measurement of m and extrapolate itsunwards to the heliocentric distance where m ismeasured. For instance, let us consider the obser-vation of the source 1148-001 in 1999 at an ǫ (he-liocentric distance) of 15 ◦ (0.26 AU). We use ACE data at 1 AU from year 1999 and extrapolate itsunwards to a heliocentric distance of 0.26 AU todetermine the appropriate N to be used in Eq 1.The ratio of ∆ N to N gives the ǫ N (Eq 1). Asstated earlier, the m of a given source is a func-tion of the both the distance of the LOS from theSun and the source size, with ideal point-like ra-dio sources giving an m of ≈ R ⊙ ).
4. Temporal and Spatial Behavior of ǫ N The upper panel of Figure 5 shows the ǫ N asfunction of r in the range 0.26 to 0.82 AU andspanning the period 1998 – 2008. The solid blueand red dots represent the ǫ N derived for eclipticand non-ecliptic source observations respectively,while their running averages at heliocentric dis-tance intervals of 0.1 AU are shown by large opencircles with 1 σ error bars. The decline in the ǫ N is only 0.22%. So it is quite apparent that ǫ N isalmost independent of heliocentric distance. Thesolid black line is a fit to the running averages of ǫ N , which emphasizes this trend. The Mariansmodel, by assuming a spherically symmetric dis-tribution of density fluctuations ignores any lati-tudinal structure in the density fluctuations. IPSdata of non-ecliptic sources are therefore likely tobe affected by the latitudinal structure caused forexample by polar coronal holes. So, the differencebetween ecliptic and non-ecliptic sources may be -4 -3 -2 -1 D en s i t y m odu l a t i on i nde x (1998 - 2008)0.26 ≤ r ≤ N o . o f M ea s u r e m en t s ≤ r ≤ Fig. 5.— shows, in the upper panel, spatial vari-ation of the density modulation index, ǫ N , of allthe 27 selected sources, in the period from 1998 to2008. While the blue and red solid dots are theactual measurements of normalized modulation in-dices for ecliptic sources and non-ecliptic sourcesrespectively, the large open circles in black repre-sent averages of all observation at intervals of 0.1AU. The solid line is a fit to these average values.The lower panel shows a histogram of the ǫ N , witha median and mean of 0.006 and 0.01 respectively.attributed to a bias caused by the effect of thesolar wind latitudinal structure.Histograms of ǫ N for the 27 selected sourcesused in the present analysis are shown in thelower panel of Figure 5. The total number ofmeasurements are mentioned on the top rightcorner of Figure 5. An inspection of the his-togram of ǫ N shows that 0 . . ǫ N . . . . ǫ N . .
08 reported using
Wind spacecraft measurements of density fluctua-tions at 1 AU (Spangler & Spitler 2004). A mod-ulation index ǫ N . . Helios 2 spacecraft between 0.03 – 1 AU.7owever, in both these papers, the data used cov-ered only a limited time interval (albeit with a highsampling frequency of 45 min), whereas this studyuses data for eleven years, covering almost the en-tire solar cycle 23 (with a sampling frequency ofone day). -4 -3 -2 -1 (1998 - 2008)0.26 ≤ r ≤ -4 -3 -2 -1 (1998 - 2008)0.26 ≤ r ≤ D en s i t y m odu l a t i on i nde x Heliocentric distance (AU)
Fig. 6.— shows, in the upper panel, spatial vari-ation of the ǫ N for ecliptic sources, in the periodfrom 1998 to 2008. While the lower panel showsthe spatial variation of the ǫ N of non-eclipticsources.Figure 6 shows the spatial variation of ǫ N forIPS measurements of ecliptic (upper panel) andnon-ecliptic sources (lower panel). The mean val-ues of ǫ N for ecliptic and non-ecliptic sources are0.03 ± .
03 and 0.01 ± .
02 respectively, showing aslightly higher ǫ N for the ecliptic sources. The de-cline in ǫ N with heliocentric distance for the eclip-tic and non-ecliptic sources are 0.7% and 0.25%respectively. So it is again clearly evident that ǫ N is independent of heliocentric distance for bothecliptic and non-ecliptic sources. ǫ N A study of the long-term changes in IPS mea-surements of m, a good proxy for solar wind mi-croturbulence levels, has shown a systematic and -4 -3 -2 -1 D en s i t y m odu l a t i on i nde x ≤ r ≤ Fig. 7.— shows the ǫ N as function of time forthe selected 27 sources, at heliocentric distances of0.26 − ǫ N , the large open circlesin red represent annual means. The solid curve isa linear fit to annual means of ǫ N .steady decline in m since ≈ N would also exhibit a sim-ilar decrease. In fact, a consistent decrease in elec-tron density turbulence, in regions of the inner-heliosphere has been reported (Tokumaru et al.2012) using IPS measurements from STEL. UsingIPS measurements from the Ooty Radio TelescopeManoharan (2012) also reported a declining trendof the density turbulence from the year 2004 to2009 (see Figure 3 in Manoharan (2012)). It istherefore of interest to see how the ǫ N during theperiod 1998 – 2008 vary in time.Figure 7 shows the temporal variation of ǫ N ,covering the period 1998 – 2008, at heliocentric dis-tances ranging from 0.26 to 0.82 AU. The bluesolid dots are the derived density modulation in-dices while annual means of the modulation in-dices are shown by large red open circles with 1sigma error bars. The annual means of ǫ N show adecline of 8% in ǫ N . This finding impacts our un-derstanding of the steady temporal decline in solarwind dynamic pressure; we discuss this further inthe next section.
5. Summary5.1. Conclusions
We have carried out an extensive survey ofthe density modulation index ( ǫ N ) in the inner-8eliosphere using IPS observations at 327 MHz.We have used observations of 27 sources span-ning the heliocentric distance range 0.26 – 0.82AU for the period 1998 – 2008. One of the broadconclusions of our study is that ǫ N ≈ . ǫ N fromHelios data at heliocentric distances between 0.3and 0.5 AU have found 5% . ǫ N . ǫ N being independent of helio-centric distance agrees with those proposed byWoo et al. (1995) for the slow solar wind. Us-ing Ulysses time delay measurements, Woo et al.(1995) have shown that the relative density fluc-tuations obtained over a period of 5 hours for theslow solar wind ( ≤
250 km s − ) in the distancerange from 0.03 to 1 AU is independent of helio-centric distance.The long-term temporal variation of the rel-ative density fluctuations over heliocentric dis-tances of 0.26 – 0.82 AU, have shown a decline of8% during the period 1998 – 2008. We now comment on the implications of ourresults on some of the problems we have outlinedin the introduction: • The scintillation levels in the inner-heliosphere(which are ∝ ∆N) have been shown tobe declining monotonically since ≈ N ∝ the back-ground density N, this has prompted spec-ulations about a steady temporal declinein the pressure exerted by the bulk so-lar wind on the Earth’s magnetosphere.McComas et al. (2013) have calculated thecanonical standoff distance of bow shocknose of the Earth’s magnetosphere whichis about 11 Earth radii ( R E ) for the period 2009 – 2013 compared to about 10 R E for theperiod 1974 – 1994. According to these au-thors, this change is in view of the observeddecline in solar wind dynamic pressure from ∼ ∼ ǫ N ≡ ∆ N/N with time.Furthermore, if there is a linear relation-ship between the relative density fluctua-tions and the magnetic field fluctuations(Spangler & Spitler 2004), it would implythat the magnetic field fluctuations also de-cline steadily over period 1998 – 2008. Soit appears reasonable to conclude that thedecrease in density fluctuations is connectedto the unusual solar magnetic activity dur-ing the long deep solar minimum at the endof the solar cycle 23. It has been shownthat both solar polar fields and the level ofturbulent density fluctuations (∆ N ) havedecreased monotonically since around 1995(Janardhan et al. 2010, 2011; Bisoi et al.2014). • We note that the IPS technique used in thiswork to infer density fluctuations is sen-sitive to spatial scales of 50 to 1000 km(Pramesh Rao et al. 1974; Coles & Filice1985; Fallows et al. 2008). It is worth ex-amining how these scales relate to the dissi-pation scale of the turbulent cascade (oftenreferred to as the inner scale). If the lengthscales probed by the IPS technique are in theinertial range, it is reasonable to presumethat the magnetic field is frozen-in, and thedensity fluctuations can then be taken as aproxy for magnetic field fluctuations (e.g.,Spangler (2002)). We note, however, thatthe flux-freezing concept might not hold forturbulent fluids (e.g., Lazarian & Vishniac(1999)). In order to investigate this issue,we consider three popular inner scale pre-scriptions. One prescription for the innerscale assumes that the turbulent wave spec-trum is dissipated due to ion cyclotron reso-nance, and the inner scale is the ion inertialscale (Coles & Harmon 1989). In this case,the inner scale ( l i ) is given as a function of9
50 100 150 200r s ion cyclotron resonance (4 × Newkirk density)ion cyclotron resonance (Leblanc density)Electron gyroradiusProton gyroradius
Fig. 8.— The inner scale l i in km as a function of heliocentric distance in units of solar radii ( r s ). The dashedlines show the proton gyroradius using a proton temperature of 10 K. The solid and dotted lines showsthe inner scale governed by ion cyclotron resonance using the Leblanc et al density model and the fourfoldNewkirk density model respectively. The dot-dashed line shows the electron gyroradius using an electrontemperature of 10 K. The light gray region denotes the range of spatial scales for which IPS observationsare sensitive.heliocentric distance r by l i = 684 n e ( r ) − / km (6)where n e is the number density in cm − . Asecond prescription identifies the inner scalewith the proton gyroradius (Bale et al. 2005;Alexandrova et al. 2012). In this case theinner scale is given by l i ( r ) = 1 . × µ / T / i B ( r ) − cm (7)where µ ( ≡ m p /m e ) is the proton to electronmass ratio, T i is the proton temperature ineV and B is the Parker spiral magnetic fieldin the ecliptic plane (Williams 1995). How-ever, recent work seems to suggest that thedissipation could occur at scales as small asthe electron gyroradius (Alexandrova et al.2012; Sahraoui et al. 2013). The third pre-scription we therefore consider is one wherethe inner scale is taken to be equal to theelectron gyroradius and is given by l i ( r ) = 2 . × T / e B ( r ) − cm (8) where T e is the electron temperature in eV.The inner scales using these three prescrip-tions (Eqs 6, 7, 8) are shown in Figure 8 asa function of heliocentric distance. The greyband denotes the range of length scales ( ≈
50- 1000 km) to which the IPS technique is sen-sitive. As explained in the caption of Figure8, we use electron and proton temperaturesof 10 K in order to compute the proton andelectron gyro radii respectively. The mag-netic field is taken to be a standard Parkerspiral (Williams 1995). In order to computethe inner scale using Eq (6), we need a den-sity model. We have used two representa-tive density models – the Leblanc densitymodels (Leblanc et al. 1998) and the four-fold Newkirk density model (Newkirk 1961).If the length scales probed by the IPS tech-nique (denoted by the grey band in Figure 8)are larger than the inner scale, we can con-clude that the density fluctuations discussedin this paper lie in the inertial range of theturbulent spectrum. From Figure 8, it is ev-10dent that this is the case all the way fromthe Sun to the Earth only if the inner scaleis the electron gyroradius, or if it is due toproton cyclotron resonance, and the densityis given by the fourfold Newkirk model. Onthe other hand, if the inner scale is given bythe proton gyroradius, or if the inner scaleis due to proton cyclotron resonance and thedensity model is given by the Leblanc et al.(1998) prescription, the density fluctuationsprobed by the IPS technique are probablysmaller than the dissipation scale for helio-centric distances beyond 30–40 R ⊙ . • In order to account for the magnitude ofcosmic ray Forbush decreases observed atthe Earth, Subramanian et al. (2009) andArunbabu et al. (2013) deduce that the levelof magnetic field turbulence in the sheath re-gion ahead of Earth-directed CMEs rangesfrom a few to a few 10’s of percent. Themagnetic field turbulence level is often takento be a proxy for ǫ N (Spangler 2002). Gen-erally, the turbulence level in the sheathregion would be expected to be somewhathigher than (but not very different from)its value in the quiescent solar wind. Theresults of this paper regarding the mag-nitude of ǫ N in the quiescent solar windare thus broadly consistent with the de-ductions of Subramanian et al. (2009) andArunbabu et al. (2013) regarding the mag-netic field turbulence level. • Reid & Kontar (2010) have argued that themodulation index ǫ N needs to be around10% near the Earth and be proportional to R . (where R is the heliocentric distance)in order to account for the Earthward trans-port of electron beams produced in solarflares. However, we find that the modula-tion index shows no change with increasingheliocentric distance, and that its value nearthe Earth is considerably smaller than 10%.
6. Acknowledgments
IPS observations were carried out under the so-lar wind program of STEL, Japan. We thank theACE SWEPAM instrument team and the ACEScience Center for providing the ACE data avail-able in the public domain via World Wide Web.
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