Whistler wave generation by halo electrons in the solar wind
Yuguang Tong, Ivan Y. Vasko, Marc Pulupa, Forrest S. Mozer, Stuart D. Bale, Anton V. Artemyev, Vladimir Krasnoselskikh
DDraft version May 23, 2019
Typeset using L A TEX twocolumn style in AASTeX62
Whistler wave generation by halo electrons in the solar wind
Yuguang Tong,
1, 2
Ivan Y. Vasko, Marc Pulupa, Forrest S. Mozer, Stuart D. Bale,
1, 2
Anton V. Artemyev, and Vladimir Krasnoselskikh Space Sciences Laboratory, University of California, Berkeley, CA 94720 Physics Department, University of California, Berkeley, CA 94720 nstitute of Geophysics and Planetary Sciences, University of California, Los Angeles, USA University of Orleans, Orlean, France
Submitted to ApJLABSTRACTWe present an analysis of simultaneous particle and field measurements from the ARTEMIS space-craft which demonstrate that quasi-parallel whistler waves in the solar wind can be generated locallyby a bulk flow of halo electrons (whistler heat flux instability). ARTEMIS observes quasi-parallelwhistler waves in the frequency range ∼ . − . f ce simultaneously with electron velocity distri-bution functions that are a combination of counter-streaming core and halo populations. A linearstability analysis shows that the plasma is stable when there are no whistler waves, and unstable inthe presence of whistler waves. In the latter case, the stability analysis shows that the whistler wavegrowth time is from a few to ten seconds at frequencies and wavenumbers that match the observations.The observations clearly demonstrate that the temperature anisotropy of halo electrons crucially af-fects the heat flux instability onset: a slight anisotropy T (cid:107) /T ⊥ > T (cid:107) /T ⊥ < Keywords: solar wind — plasmas — instabilities — waves INTRODUCTIONThe mechanisms controlling the heat flux in collision-less or weakly-collisional plasmas are of high interest inastrophysics (Cowie & McKee 1977; Pistinner & Eichler1998; Roberg-Clark et al. 2018a). In-situ measurementsin the solar wind indicate that the heat flux is gener-ally different from the classical Spitzer-H¨arm prediction(Feldman et al. 1975; Scime et al. 1994; Bale et al. 2013)and apparently constrained by a threshold dependent onlocal plasma parameters (Gary et al. 1999b; Gary & Li2000; Tong et al. 2018). Such observations have moti-vated many studies on the detailed physics of heat fluxinhibition in the solar wind.In the slow solar wind ( v sw (cid:46)
400 km/s) the elec-tron velocity distribution function can often be approx-imated by a bi-Maxwellian thermal dense core and a
Corresponding author: Yuguang [email protected] tenuous, suprathermal halo (Feldman et al. 1975; Mak-simovic et al. 1997). The heat flux is predominantlyparallel to the magnetic field and carried by suprather-mal electrons. Linear stability analysis shows that thecounter-streaming core and halo electrons are capable ofdriving whistler waves propagating quasi-parallel to thebulk flow of the halo population via the so-called heatflux instability (Gary et al. 1975, 1994; Gary & Li 2000).The quasi-linear theory (Gary & Feldman 1977; Pistin-ner & Eichler 1998) and numerical simulations (Roberg-Clark et al. 2018a; Komarov et al. 2018; Roberg-Clarket al. 2018b) suggest that the scattering of halo elec-trons by the whistler waves should suppress the heatflux below some threshold value that is in general agree-ment with the heat flux constraints observed in the solarwind (Gary et al. 1994, 1999b; Tong et al. 2018). How-ever, the aforementioned experimental studies did notprovide measurements of whistler waves accompanyingthe electron heat flux measurements, and are thereforeinsufficient to firmly establish the heat flux inhibition bywhistler waves in the solar wind. a r X i v : . [ phy s i c s . s p ace - ph ] M a y Tong et al.
It is not until recently that careful studies of whistlerwaves presumably generated by the heat flux instabil-ity in freely expanding solar wind have been reportedwith measurements on Cluster and ARTEMIS space-craft. Lacombe et al. (2014) reported whistler wavesobserved along with the heat flux values close to the the-oretical threshold given by Gary et al. (1999b). Stansbyet al. (2016) presented observations of similar whistlerwaves on ARTEMIS and determined the dependence ofthe whistler wave dispersion relation on β e . However,neither study showed that the whistler waves were in-deed generated by the heat flux instability in the lo-cal plasma, leaving the possibility that whistler waveswere generated in a very different plasma by an alter-native mechanism and propagated to the spacecraft lo-cation. We note that whistler waves in the solar windcan be associated with shocks and stream interactionregions (Lengyel-Frey et al. 1996; Lin et al. 1998; Bren-eman et al. 2010; Wilson et al. 2013), while we focus onwhistler waves in the freely expanding solar wind.In this study we analyze simultaneous particle andwave measurements for data intervals presented byStansby et al. (2016) and carry out linear stability anal-ysis on electron velocity distribution functions. We findthat the observed whistler waves are indeed generatedlocally by the heat flux instability on a time scale ofa few seconds. In this letter we present one of thoseevents, which also demonstrates crucial features of theheat flux instability. OBSERVATIONSWe consider observations of ARTEMIS (Angelopou-los 2011) on November 9, 2010 for ten minutes around10:17:00 UT as the spacecraft was in the pristine so-lar wind about 40 Earth radii upstream of the Earth’sbow shock. We use measurements of the following in-struments aboard ARTEMIS: the magnetic fields with3 second resolution provided by the Flux Gate Magne-tometer (Auster et al. 2008), the electron velocity distri-bution function (32 log-spaced energy bins from a few eVup to 25 keV and 88 angular bins) and particle moments(density, bulk velocity and temperatures) with 3 secondtime resolution provided by the Electrostatic Analyzer(McFadden et al. 2008), measurements of three magneticand electric field components at 128 Hz sampling rateprovided by the Search Coil Magnetometer (Le Contelet al. 2008) and Electric Field Instrument (Bonnell et al.2008).Figure 1 shows that the solar wind was streaming atabout v sw ∼
320 km/s, the quasi-static magnetic fieldwas gradually decreasing from B ∼ n ∼ − , and the electron tem- perature was T e ∼
15 eV. The ion temperature was T i ∼ f ce was varying from150 to 90 Hz, the Alfv´en speed v A = B / (4 πn m i ) / from 90 to 30 km/s, while β i,e = 8 πn T i,e /B ∼ . − f ce corresponds to whistler waves (Stansbyet al. 2016). Since the power spectra above 64 Hz can-not be obtained from the search coil magnetic field timeseries, we also checked the on board FFT power spec-tra of search coil magnetic fields (not shown) covering8 Hz-4 kHz and verified that there was no significantpower between 64 Hz and f ce . Panel (f) presents thespectral coherence between the two magnetic field com-ponents perpendicular to the quasi-static magnetic fieldand indicates a high coherence of the whistler waves.We carry out a spectral polarization analysis followingSantolik et al. (2003) to determine the obliqueness ofwhistler waves to the quasi-static magnetic field. Panel(g) presents the cosine of the propagation angle and con-firms that whistler waves propagate almost parallel oranti-parallel to the quasi-static magnetic field in accor-dance with the conclusions of Stansby et al. (2016). Theamplitude of whistler waves ranges from 0.05-2 nT, andis small compared to B .Figure 2 presents an example of the processed electronvelocity distribution function (VDF). The raw electronVDF measured around 10:17:49 UT is corrected for theeffect of spacecraft potential and transformed from thespacecraft frame into the solar wind frame using theion bulk velocity measurements. Panel (a) shows theprocessed VDF f ( v (cid:107) , v ⊥ ) averaged over the gyrophase,where v || and v ⊥ correspond to velocities parallel andperpendicular to the background magnetic field. TheVDF is asymmetric in the direction parallel to the mag-netic field with opposite asymmetries below and abovea few thousand km/s, indicating counter-streaming ofcold and hot electrons. Panel (b) shows VDF cuts f || , f ⊥ and f −|| corresponding to electrons streaming parallel(pitch angles α ∼ ◦ ), perpendicular ( α ∼ ◦ ) and anti-parallel ( α ∼ ◦ ) to the quasi-static magnetic field.Below ∼
30 eV, f −|| > f || , consistent with core electronsstreaming anti-parallel to the magnetic field. At higherenergies, f || > f −|| shows that the hotter electrons arestreaming in the opposite direction.The counter-streaming cold and hot electrons per-sist through the whole ten minutes in Figure 1. Is histlers in the solar wind (a) (f) (e) (d) (c) (b) (g) Figure 1.
ARTEMIS observations in the pristine solar wind on November 9, 2010 about 40 Earth radii upstream of theEarth’s bow shock: (a) quasi-static magnetic field; (b) ion bulk velocity in the GSM coordinate system; (c,d) electron and iondensities and temperatures; (e) wavelet power spectrum of one of the magnetic field components perpendicular to the quasi-staticmagnetic field; we use a Morlet wavelet with center frequency ω = 32 as the mother wavelet and normalize the wavelet power( W ) by the white noise power ( σ ); (f) the coherence coefficient between magnetic field components B x and B y perpendicularto the quasi-static magnetic field; (g) | cos θ kB | indicating obliqueness of the whistler waves ( k and B are the wave vector andthe quasi-static magnetic field). In panel (g) domains with coherence smaller than 0.6 have been masked out for clarity. 2Dmaps (e)-(g) are computed using the magnetic field measured at 128 Hz sampling rate. this plasma indeed capable of generating the observedwhistler waves? How fast is the instability? What con-trols the absence of whistler waves before 10:16:00 UTand their later appearance? To address these questionswe fit the processed electron VDFs and carry out a linearkinetic stability analysis using the previously developednumerical code (Tong et al. 2015). ANALYSISDuring this slow solar wind interval, the electronVDFs are well described by a combination of core andhalo populations f = f c + f h . The core and halo aremodelled respectively with drifting bi-Maxwellian and bi-kappa distributions f c = A c exp (cid:34) − m e (cid:0) v || − v c (cid:1) T || c − m e v ⊥ T ⊥ c (cid:35) ,f h = A h B κ (cid:20) m e ( v || − v h ) (2 κ − T || h + m e v ⊥ (2 κ − T ⊥ h (cid:21) − ( κ +1) , where A s = n s ( m e / πT / ⊥ s T / || s ) / , B κ = Γ( κ +1) / ( κ − / / Γ( κ − /
2) and n s , v s , T ⊥ s , T || s are densities,bulk velocities and temperatures (parallel and perpen-dicular to the quasi-static magnetic field B ) of the coreand halo populations ( s = c, h ). These parameters are Tong et al. (a) (b)
Figure 2.
Example of an electron VDF that has been transformed into the solar wind frame and calibrated for the spacecraftpotential: (a) gyrophase averaged f ( v || , v ⊥ ), where v || and v ⊥ are parallel and perpendicular to the magnetic field; (b) VDFcuts plotted vs. electron energy and corresponding to electrons streaming parallel ( f || = f ( v (cid:107) > , v ⊥ = 0)), perpendicular( f ⊥ = f ( v (cid:107) = 0 , v ⊥ )) and anti-parallel ( f −|| = f ( v (cid:107) < , v ⊥ = 0)) to the quasi-static magnetic field. estimated by fitting the model to VDF cuts f || , f ⊥ and f −|| using the standard χ minimization method. Fol-lowing Feldman et al. (1975) the electron current in thesolar wind frame is kept zero by restricting the param-eters to n c v c + n h v h = 0.Figures 3 (a) and (c) illustrate the fitting procedure,using an electron VDF measured in absence of whistlerwave activity at 10:12:11 UT and another VDF in pres-ence of whistler waves at 10:17:49 UT. Panels (a) and(c) present the VDF cuts, the model fits and the bestfit parameters. Only data points above the one countlevel have been used in the fitting procedure. Core elec-trons make up about 80-85% of the total electron den-sity, the bulk velocity is 100-200 km/s (anti-parallel to B in the solar wind frame), or about four times largerthan the local Alfv´en speed, the temperature is around9 eV, and the parallel and perpendicular temperaturesare slightly different, T ⊥ c /T || c ∼ .
06. The halo bulkvelocity is about 500-1000 km/s (parallel to B ) andthe temperature is about 30 eV. The halo populationis rather anisotropic in (a) with T ⊥ h /T (cid:107) ,h ∼ . T ⊥ h /T (cid:107) ,h ∼ . B in ourcase.Figures 3 (b) and (d) present growth rates and disper-sion curves of parallel propagating whistler waves com-puted for electron VDFs in (a) and (c). In agreementwith observations we find whistler waves to be stablefor VDF (a), but unstable for VDF (c). In the lat-ter case the linear stability analysis predicts the fastestgrowing whistler waves at the frequency of 0.05 f ce .Although it is in general agreement with the whistlerwave spectrum in Figure 1e, a careful comparison re-quires Doppler shifting the plasma frame frequency of0.05 f ce into the spacecraft frame (see below). Panel (d)shows that the maximum growth rate is about 10 − f ce or 0 . − in physical units, which corresponds to ane-folding time of about a second. During this time,whistler waves can only propagate a few hundred kilome-ters, because the phase velocity of the whistler waves isabout c ( f /f ce ) / f ce /f pe ∼
500 km/s, where f and f pe are whistler and plasma frequencies (see also Stansbyet al. (2016)). This indicates that the observed whistlerwaves were likely generated locally.In order to uniquely identify the free energy sourcedriving the whistler waves, we computed growth ratesfor electron VDFs (a) and (c), but with either 1) coreand halo bulk velocities set to zero or 2) temperature-isotropic core and halo. Panels (b) and (d) show thatthe electron VDFs with zero bulk velocities (blue curves) histlers in the solar wind T || h /T ⊥ h > T ⊥ c /T || c issteady and around 1.1, while the halo is temperature-anisotropic with T ⊥ h /T || h ∼ . v A . We perform the linearstability analysis on every electron VDF and determinethe growth rate γ m , frequency f m and wavenumber k m of the fastest growing whistler wave. In the spacecraftframe the whistler wave will be observed at a Doppler-shifted frequency f m + k m v sw , where k m is parallel tothe quasi-static magnetic field B .Panel (e) demonstrates that the Doppler-shifted fre-quency of the fastest growing whistler wave indeed tracesthe observed whistler waves. There are no whistlerwaves before about 10:16:00 UT, while the plasma isstable. Whistler waves suddenly appear around 10:16:00UT, when the plasma becomes unstable. Around10:21:00 UT the plasma is stable for a short time inter-val, and the coherent whistler waves disappear over thisinterval. The strong correlation between whistler wavesand the local plasma stability/instability indicates thatthe whistler waves are indeed generated locally. Panel(f) strengthens this conclusion by demonstrating thatthe e-folding time γ − m of the fastest growing whistlerwave is from 1 to 10 seconds.The abrupt transition from stable to unstable plasmaaround 10:16:00 UT coincides with the halo populationbecoming more isotropic. As we demonstrated in Fig-ure 3, the reason is that the halo temperature anisotropyquenches the whistler heat flux instability. The crucialrole of the temperature anisotropy is further demon-strated in Figure 5. Panel (a) presents the electronheat flux q e normalized to the free streaming heat flux q = 1 . n e T e ( T e /m e ) / versus β c || = 8 πn c T || c /B . Atany given β c || the heat flux is clearly below a threshold given by q e /q ∼ /β c || , that is similar to the marginallystable values in literature (Gary et al. 1999a; Pistinner &Eichler 1998; Roberg-Clark et al. 2018a; Komarov et al.2018; Roberg-Clark et al. 2018b). However, at a given q e /q both stable and unstable VDFs are observed, in-dicating thereby that some other parameter controls theonset of the whistler wave generation. Panel (b) showsthat the halo temperature anisotropy separates stableand unstable VDFs with a similar heat flux value. Thisre-emphasizes the crucial effect of the halo temperatureanisotropy on the heat flux constraints in the solar wind. DISCUSSION AND CONCLUSIONIn-situ observations indicated that whistler waves gen-erated by the heat flux instability highly likely constrainthe heat flux in the solar wind (Feldman et al. 1975;Gary et al. 1999b; Tong et al. 2018). However, therehave been no previous analyses that would prove thatwhistler waves in the solar wind are actually producedlocally by the whistler heat flux instability. In this let-ter we have presented a careful analysis of simultaneousparticle and wave measurements for one of the time in-tervals in Stansby et al. (2016). We have performedsimilar analysis for other Stansby et al. (2016) time in-tervals and confirmed that whistler waves are generatedlocally by the heat flux instability in those intervals aswell. The presented event has shown that the e-foldinggrowth time of whistler waves can be as short as one sec-ond and clearly demonstrated the crucial effect of thehalo temperature anisotropy T ⊥ h /T || h <
1. In someof the Stansby et al. (2016) events the halo popula-tion has T ⊥ h /T || h >
1. The linear stability analysishas shown that even a slight T ⊥ h /T || h > Tong et al. (a) (b) (c) (d)
Figure 3.
Illustration of the fitting procedure and linear stability analysis of VDFs associated with negligible and noticeablewhistler wave activity observed around 10:12:11 and 10:17:49 UT: (a,c) the VDF cuts f || , f ⊥ and f −|| corresponding to electronswith pitch angles around 0 ◦ , 90 ◦ and 180 ◦ are shown with dots; the VDF cuts are shifted vertically with respect to each otherfor visual clarity; only VDF values above one count level (dashed curves) have been used in the fitting procedure; the modelfits are presented with solid curves and the fitting parameters are indicated in the panels; (b,d) the growth rate and dispersioncurves of parallel propagating whistler waves; the growth rate computations are carried out for (red) the measured electronVDFs and for the measured VDF with either (blue) core and halo bulk velocities set to zero or (green) temperature-isotropiccore and halo. Up to this point we have been focused on the electronheat flux constrained by wave-particle interactions. Infact, Coulomb electron-electron collisions can also affectsolar wind electrons and constrain the electron heat flux(Salem et al. 2003; Bale et al. 2013; Pulupa et al. 2014;Landi et al. 2014). The Knudsen number for the ob-served solar wind K n ∼ − . histlers in the solar wind (a) (e) (d) (c) (b) (f) Figure 4.
The results of the fitting of 183 electron VDFs: (a) the total electron densities from the fitting and the electrondensity calibrated on the ground; (b,c) parallel temperatures and temperature anisotropies of the core and halo population; (d)the bulk velocity of core population v c with respect to the local Alfven speed v A . Panel (e) repeats Figure 1g that shows thecoherence between the two magnetic field components perpendicular to the quasi-static magnetic field (domains with coherencesmaller than 0.6 have been masked out for visual clarity). The spacecraft frame frequency of the fastest growing whistler modeis indicated in panel (e) with dots. Panel (f) presents the e-folding time (inverse of the growth rate) of the fastest growingwhistler wave. The absence of dots in some intervals implies that the plasma was stable. Tong et al. (a) (b)
Figure 5.
The demonstration of the crucial effect of the halo temperature anisotropy on the whistler heat flux instability. Panel(a) presents the electron heat flux q e normalized to the free-streaming heat flux q = 1 . n e T e ( T e /m e ) / versus core electronbeta parameter β c || computed for all 183 VDFs available over the ten-minute interval. q e /q = 1 /β c || is plotted in dashed linefor reference. Panel (b) presents the temperature anisotropy of the halo population versus β c || . Unstable (stable) VDFs arelabeled with red (blue) dots. REFERENCES
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