Observational evidence for solar wind proton heating by ion-scale turbulence
G. Q. Zhao, Y. Lin, X. Y. Wang, D. J. Wu, H. Q. Feng, Q. Liu, A. Zhao, H. B. Li
aa r X i v : . [ phy s i c s . s p ace - ph ] A ug manuscript submitted to Geophysical Research Letters
Observational evidence for solar wind proton heating byion-scale turbulence
G. Q. Zhao , , Y. Lin , X. Y. Wang , D. J. Wu , H. Q. Feng , Q. Liu , A.Zhao , and H. B. Li Institute of Space Physics, Luoyang Normal University, Luoyang, China Henan Key Laboratory of Electromagnetic Transformation and Detection, Luoyang, China Physics Department, Auburn University, Auburn, USA Purple Mountain Observatory, CAS, Nanjing, China
Key Points: • Nearly collisionless solar wind turbulence at ion scales is investigated with 11 yearsof in-situ data • Correlations between the spectral index and magnetic helicity, and between theproton temperature and turbulent energy are revealed • A scenario for the solar wind turbulence and heating at ion scales is proposed
Corresponding author: G. Q. Zhao, [email protected] –1–anuscript submitted to
Geophysical Research Letters
Abstract
Based on in-situ measurements by Wind spacecraft from 2005 to 2015, this letter reportsfor the first time a clearly scale-dependent connection between proton temperatures andthe turbulence in the solar wind. A statistical analysis of proton-scale turbulence showsthat increasing helicity magnitudes correspond to steeper magnetic energy spectra. Inparticular, there exists a positive power-law correlation (with a slope ∼ . ) betweenthe proton perpendicular temperature and the turbulent magnetic energy at scales . . kρ p . , with k being the wavenumber and ρ p being the proton gyroradius. These find-ings present evidence of solar wind heating by the proton-scale turbulence. They alsoprovide insight and observational constraint on the physics of turbulent dissipation inthe solar wind. Plain Language Summary
The solar wind is a tenuous magnetized plasma that serves as a natural laboratoryof nearly collisionless turbulence. It is streaming outward from the Sun and has temper-atures much higher than those from a spherically expanding ideal gas, implying a heat-ing process must occur. The heating has been extensively discussed in past decades, butstill not well understood. Based on in-situ measurements, we reveal a scale-dependentconnection between proton temperatures and the turbulence with enhanced magnetichelicity signature. The connection is particularly strong for the proton temperature inthe direction perpendicular to the background magnetic field. These observations pro-vide implications for the physics of turbulent dissipation and heating in a collisionlessplasma.
It is well known that the solar wind is pervaded with magnetic fluctuations thatare inherently turbulent. The fluctuation energy appears as a power-law spectrum overa broad range of spacial scales (Alexandrova et al., 2013; Bruno & Carbone, 2013). Theenergy-containing range indicates very large scales with frequencies f < − Hz in thespacecraft reference frame. The energy spectrum in this range goes as ∼ f − , and is in-terpreted as uncorrelated large-scale Alfvén waves that provide a source of energy. Theinertial range represents intermediate scales with spacecraft frame frequencies − . f . − Hz. The spectrum follows a f − / law that is dominated by the Kolmogorovcascade. At small scales comparable to the proton gyroradius or inertial length, the spec-trum becomes complicated with various spectral indices between − and − (e.g., Sahraouiet al., 2013; Bruno et al., 2014), frequently lower than the classic index − / predictedby strong-turbulence theory for dispersive cascade (Galtier, 2006; Wu & Chen, 2020).The spectral steepening at the small scales are often interpreted as the occurrence of tur-bulent dissipation due to cyclotron damping (Gary & Borovsky, 2004; Smith et al., 2012),Landau damping (He, Pei, et al., 2015; Howes et al., 2018), stochastic heating (Johnson& Cheng, 2001; Chandran et al., 2010), or plasma coherent structures such as currentsheets and magnetic vortices (Perri et al., 2012; Wang et al., 2019).Various studies were conducted to explore the nature of the proton-scale turbulencein the solar wind, which is fundamentally important to the understanding of the dissi-pation and heating. Two models were proposed theoretically, including kinetic Alfvénwave (KAW) and whistler wave turbulence models (Gary & Smith, 2009). KAW turbu-lence model was suggested by a large body of studies on the basis of observations (e.g.,Bale et al., 2005; Sahraoui et al., 2009; Howes & Quataert, 2010; He et al., 2012a; Podesta,2013) and simulations (Howes et al., 2008; Grošelj et al., 2018). Among them, severalstudies used non-zero magnetic helicity to diagnose the presence of KAWs in the turbu-lence (Howes & Quataert, 2010; He et al., 2012a; Podesta, 2013). The KAW turbulencewould heat the solar wind through cyclotron/Landau damping or stochastic heating, while –2–anuscript submitted to Geophysical Research Letters it is under debate which process is dominant (Parashar et al., 2015; Isenberg & Vasquez,2019).The issue on the heating of the solar wind is complicated because of the complexroles of double-adiabatic expansion, global heat flux, ion differential flows, Coulomb col-lisions, local wave action, microinstabilities, and turbulence (see a recent review by Ver-scharen et al., 2019). Various correlations in the solar wind have been reported. Protontemperatures are found to be correlated with the solar wind speed, and faster solar windusually corresponds to higher proton temperatures (Marsch et al., 1982; Newbury et al.,1998). Steeper spectra at proton scales are also linked to faster solar wind (Bruno et al.,2014). The faster solar wind tends to be more imbalanced with greater wave energy fluxanti-sunward than sunward (Tu & Marsch, 1995). These correlations with the solar windspeed are possibly a consequence of different source populations of the solar wind (Schwenn,2006). There is also a correlation between the temperature ratio of alpha particles to pro-tons and the alpha − proton differential flow (Kasper et al., 2008). This has been inter-preted as evidence of solar wind heating by field-aligned ion-cyclotron waves (Gary etal., 2006; Kasper et al., 2013). A link between inertial-range intermittency and protontemperature enhancement has also been reported, as evidence of nonuniform heating drivenby magnetohydrodynamic turbulence (Osman et al., 2011, 2012).In this letter, we report a new correlation between the spectral steepness and mag-netic helicity for nearly collisionless solar wind turbulence at proton scales. The corre-lation exists even within a very narrow range of the solar wind speed. Moreover, it is shownthat the proton perpendicular temperature is correlated with the turbulent magnetic en-ergy at proton scales. A higher magnetic helicity favors a better correlation of the tem-perature with the magnetic energy. These findings will provide improved understand-ing of the ion-scale turbulence and heating in the solar wind. The data used in this letter are from the Wind spacecraft. The magnetic field dataare from the Magnetic Field Investigation instrument and have a cadence of 0.092 s, (Leppinget al., 1995). The plasma data are from the Solar Wind Experiment instrument work-ing at a cadence of 92 s (Ogilvie et al., 1995). The proton temperatures are producedvia a nonlinear-least-squares bi-Maxwellian fit of ion spectrum from the Faraday cup (Kasperet al., 2006). The data set is surveyed from 2005 to 2015 through dividing the long timeseries into consecutive and overlapping time segments. In each segment with data avail-able, the magnetic energy spectrum is obtained via standard fast Fourier transform tech-nique. The plasma parameters, including the proton density N p , perpendicular and par-allel thermal velocities w ⊥ p and w k p , and bulk velocity V p , are given as average valuesover the time segment, where ⊥ and k are with respect to the ambient field B . Eachtime segment has a span of 200 s, a compromise time interval to reduce the averagingeffects for plasma parameters while to cover the magnetic spectrum of interest with anappropriate resolution simultaneously. An overlap is set to be 100 s, comparable to theplasma data cadence. The Coulomb collisional age A c is calculated, which is the ratioof the transit time of the solar wind to the collision timescale (Livi et al., 1986). Seg-ments with A c < . are selected for samples with negligible collision effects. This se-lection means that the faster solar wind would be chosen in principle. The selected seg-ments have a median of the solar wind speed being 519 km s − , while it is 388 km s − for all segments. It is also required that the angle between B and V p is in the range from ◦ to ◦ to reduce possible heating/cooling effects due to the alpha − proton differ-ential flow, which could be strong when B and V p are quasi-parallel (Zhao et al., 2020;Zhao, Li, et al., 2019). In total about . × time segments satisfying these constraintsare selected, and most ( ) of them have proton parallel beta in the range from 0.1 to3. –3–anuscript submitted to Geophysical Research Letters
The normalized reduced magnetic helicity is considered as an indicator of the pres-ence of polarized waves in this letter. The helicity is a measure of the spatial handed-ness of the magnetic field, and has been widely used to identify wave modes in the so-lar wind for nearly parallel propagating cyclotron waves (Zhao et al., 2018; Zhao, Feng,et al., 2019; Woodham et al., 2019) or oblique KAWs (Howes & Quataert, 2010; He etal., 2011). The helicity refers to the fluctuating helicity in spectral form and is definedas H m ( k ) ≡ A ( k ) · B ∗ ( k ) , where A is the fluctuating magnetic vector potential, B isthe fluctuating magnetic field, the asterisk represents the complex conjugate of the Fouriercoefficients, and k is the wave vector (Matthaeus & Goldstein, 1982; Woodham et al.,2019). For the magnetic field measured by a single spacecraft, only the reduced helic-ity spectrum is available, which can be written as H rm ( k ) = 2Im[ B ( k ) · B ∗ ( k )] /k (Matthaeus & Goldstein, 1982; Woodham et al., 2019), where k is the reduced wavenum-ber and the direction 1 is the direction in which the spectrum is reduced, Im means theimaginary part, and B ( k ) · B ∗ ( k ) is an element in the energy spectral tensor. The nor-malized reduced magnetic helicity is defined as σ k = k H rm ( k ) /P k , which takes val-ues between [ − , , where P k = | B ( k ) | is the total magnetic energy at wavenum-ber k . In practice we first calculate the magnetic helicity as a function of frequency in-stead of wavenumber, i.e., σ f = 2Im[ B y ( f ) · B ∗ z ( f )] /P f , where f is the frequency asso-ciated with the spacecraft time series of measured magnetic field in the GSE coordinatesystem. The calculation of σ f is allowed when Taylor frozen-in-flow hypothesis holds,by which the frequency is related to wavenumber (Taylor, 1938; Matthaeus & Goldstein,1982; He et al., 2011).A statistical examination is conducted for the nature of the solar wind turbulence.The examination is according to the direction of the long axis of magnetic fluctuationswith respect to local B . It is expected that the long axis is quasi-perpendicular to B for KAW turbulence but to be quasi-parallel for whistler wave turbulence based on thelinear kinetic theory (He et al., 2012a). To determine the long axis of the fluctuations,variance analysis technique is adopted to estimate the long-axis direction, which has max-imum variance of fluctuations (Sonnerup & Scheible, 1998). A large θ LB , the angle be-tween the long axis and local B , could be expected if it is KAW turbulence.Figure 1 presents an example to illustrate the parameters used in this letter. Fig-ure 1a plots the magnetic energy spectrum ( P f ) in the frequency domain (for data on5 June 2005, 01:33:23 − f − / law in the low fre-quency, and steepens with an index about − . when the frequency exceeds . Hz. Withthe frequency approaching 2 Hz, the spectrum flattens again possibly due to the instru-ment noise and/or aliasing. The two vertical dashed lines in Figure 1a indicate the rangeof interest in this letter. The left line corresponds to a wavenumber kρ p = 0 . , where ρ p = w ⊥ p / Ω p is the proton thermal gyroradius and Ω p is the proton cyclotron frequency.The right line is chosen accordingly with P f > − nT /Hz that is required for sig-nal level much higher than the instrument noise level (Lepping et al., 1995). The wavenum-ber k represents the reduced wavenumber and is calculated by the Taylor frozen-in-flowhypothesis (Taylor, 1938), πf = kV p , with which the frequency domain can be con-verted to the wavenumber domain. Figure 1a is then replotted as Figure 1b, with P k in-stead of P f for the range bounded by the vertical dashed lines in Figure 1a. In this stepan averaging operation is performed to show a smoothed spectrum; the running aver-age window runs over f e − . ≤ f ≤ f e . . A similar averaging is performed for mag-netic helicity spectrum, and the smoothed σ k is shown in Figure 1c. The helicity σ k isobviously non-zero at kρ p ∼ in this example. The drop of σ k at kρ p > could beattributed to the noise and/or aliasing (Klein et al., 2014). It also could be attributedto the greater balance of KAW turbulence or nonlinear effects at smaller scales that willweaken the helicity signature (He et al., 2012b; Markovskii & Vasquez, 2013).Figure 1d displays the local spectral index α k , which is determined by the energyspectrum over the same frequency range for the averaging. The index α k is around − / –4–anuscript submitted to Geophysical Research Letters within uncertainty when kρ p < . ; the large uncertainty is inherently due to the shorttime segment of 200 s that leads to an inefficient coverage for the inertial range. For kρ p > . , the index α k rapidly decreases to a more negative value, ∼ − . at . < kρ p < . . Note that the more negative α k occurs with helicity signature enhanced as shownin Figure 1c. The presence of the more negative α k with the higher helicity allows oneto speculate that the steep energy spectrum might be partly attributed to the dissipa-tion of the turbulence with enhanced helicity. An examination on the long-axis direc-tion of magnetic fluctuations, Figure 1e, gives θ LB > ◦ for . < kρ p < . , whichis in line with the model of KAW turbulence (He et al., 2012a). Statistical investigations are performed to explore the significance of the results inFigure 1. It is found that spectral indices are ordered by the magnetic helicity at pro-ton scales. As an example for kρ p = 0 . , Figure 2 displays, from top to bottom, dis-tribution of | σ k | , medians of α k and θ LB sorted by | σ k | , where | σ k | is the absolute valueof σ k that can be either positive or negative even for a specific wave mode, dependingon the directions of B and wave propagation with respect to the Sun. The distributionof | σ k | has a weak peak at | σ k | ∼ . , and about 61% of the data have | σ k | > . .As | σ k | increases from 0 to 0.4, it is clear that the median of α k decreases from about − . to − . (Figure 2b), implying a significant correlation between α k and | σ k | . FromFigure 2c, statistical examination concerning the nature of the related turbulence showsthat 91% (63%) of the data have θ LB > ◦ (80 ◦ ) , and there is a tendency of θ LB in-creasing with | σ k | . If we consider only those data with maximum variances of magneticfluctuations at least 10 times larger than other variances from orthogonal directions inthe examination, the corresponding percentage increases to 98% (83%). Such large per-centage supports KAW turbulence dominating the nearly collisionless solar wind at pro-ton scales (He et al., 2012a).One may speculate that the correlation between α k and | σ k | is possibly a conse-quence of the scaling with solar wind speed V p . Figure 2d is presented for this issue. InFigure 2d, different curves are for different sub-datasets bounded by V p , where the red,orange, green, blue, light blue curves correspond to the V p ranges of 400 − − − − >
600 km s − , respectively. One can see that the correlation existsfor each sub-dataset even when the V p range is narrow (50 km s − ). This should implythat the correlation of α k with | σ k | is common and does not result from the scaling withthe solar wind speed.Another important finding in this letter is that proton temperature T p is correlatedwith turbulent energy P k . The correlation is particularly strong for T ⊥ p and depends on k , | σ k | , and β k p , where T p and T ⊥ p are the total and perpendicular temperatures of pro-tons, respectively, β k p = w k p /v A is the proton parallel beta, v A is the Alfvén speed. Higher | σ k | contributes to a better correlation according to our tests. Figure 3 plots the distri-butions of ( P k , T ⊥ p ) at fixed scales of (a) kρ p = 0 . as well as (b) kρ p = 0 . . All datain the figure are bounded by | σ k | > . and by . ≤ β k p ≤ , producing the datawith a size N ≃ . × . The black dashed lines present linear fitting of the data inlogarithmic space, describing a positive power law for the correlation between T ⊥ p and P k at the proton scales. The slopes of the fitting lines are considerable, 0.38 at kρ i =0 . and 0.39 at kρ i = 0 . . The relative uncertainties are found to be very small, ∼ P k , V p ) at fixed scales of kρ p = 0 . and kρ p = 0 . , respectively. The dis-tributions are shown to be significantly dispersive, although there is a tend of positivecorrelation between V p and P k . Following the plot of Figures 3a and 3b, five sub-datasetswith different V p ranges, as displayed in Figure 2d, are checked. Results show that the –5–anuscript submitted to Geophysical Research Letters
Figure 1.
An example for a time segment of 200 s on 5 June 2005 to illustrate parametersused in this letter: (a) magnetic energy P f ; (b) local average magnetic energy P k ; (c) local aver-age magnetic helicity σ k ; (d) local spectral index of magnetic energy α k ; (e) the angle betweenthe long axis of magnetic fluctuations and B , θ LB . Two vertical dashed lines in (a) indicate therange shown in (b) − (e) in wavenumber domain.–6–anuscript submitted to Geophysical Research Letters
Figure 2.
Statistics at kρ p = 0 . for ∼ . × time segments between 2005 and 2015: (a)distribution of | σ k | , (b) median of spectral index α k , (c) median of the angle θ LB , (d) medians of α k , where different color curves are for different ranges of solar wind speed V p .–7–anuscript submitted to Geophysical Research Letters
Figure 3.
Top, middle, and bottom rows are distributions of ( P k , T ⊥ p ), ( P k , V p ), and ( P k , T ⊥ p /T k p ), respectively. Left panels are for the wavenumber kρ p = 0 . while right panels arefor kρ p = 0 . . The black dashed lines are the linear fitting of the data in logarithmic space,and n and N represent the data number in each pixel and the total data number in each panel( ∼ . × ), respectively. –8–anuscript submitted to Geophysical Research Letters
Figure 4.
Color scale plot for (a) slopes of fitting lines of ( P k , T ⊥ p ) and (b) medians of | σ k | inthe ( β k p , k ) space. correlation between T ⊥ p and P k remains for each sub-dataset. The slopes of fitting linesdecrease relative to those without the limitation of V p , but are still considerable. We there-fore believe that the correlation between T ⊥ p and P k is inherent and can not be fully at-tributed to the scaling with the solar wind speed.We have also investigated the possible correlation between the temperature ratio T ⊥ p /T k p and P k , where T k p is the proton parallel temperature. The temperature ratiois a critical parameter in the solar wind data analyses (Zhao, Feng, et al., 2019; Wood-ham et al., 2019), as well as in magnetosheath studies (Maruca et al., 2018). Figures 3eand 3f present the distributions of ( P k , T ⊥ p /T k p ) at scales of (e) kρ p = 0 . and (f) kρ p =0 . . One can find a trend that higher P k corresponds to larger T ⊥ p /T k p . This impliesa correlation between T ⊥ p /T k p and P k , whereas this correlation is weaker than that be-tween T ⊥ p and P k . The slopes of fitting lines of ( P k , T ⊥ p /T k p ) are around 0.15 in bothcases. If we use different datasets by changing the range of β k p and V p , the correlationremains. The slopes of fitting lines may rise when β k p decreases, but they are usuallyless than 0.25. A large T ⊥ p /T k p may excite proton cyclotron and mirror instabilities, whichdepends on the proton beta and plays a role in constraining the solar wind plasma (Hellingeret al., 2006).It is notable that the correlation between T ⊥ p and P k with considerable slopes ex-ists in particular at proton scales; at larger scales the correlation weakens greatly. Fig-ure 4a displays the slopes for cases with various k as well as β k p . In each case half of datawith higher | σ k | are used to fit the data since higher | σ k | contributes to a better corre-lation. Any case with a slope < . or relative uncertainty greater than 10% is markedby the white color. One can see that a highlight region arises at scales . . kρ p . for finite β k p ( . ), where the slopes are considerably large ( & β k p and have a peak value of 0.4 at β k p ∼ . . At smaller scales with kρ p > , the result is not clear because at these scales the instrument noise and/or aliasing maybe non-negligible (Klein et al., 2014).As somewhat expected, the large slopes in Figure 4a are accompanied by enhancedmagnetic helicity. Using the same data in Figure 4a, Figure 4b plots medians of | σ k | andproduces a comparable color region in the ( β k p , k ) space. The comparability betweenFigures 4a and 4b confirms that higher | σ k | contributes to a better correlation between T ⊥ p and P k . Further investigation on spectral index α k (not shown) reveals that energyspectra in the highlight regions are steeper and have indices usually lower than − / , –9–anuscript submitted to Geophysical Research Letters which could be interpreted as hint of the occurrence of turbulent dissipation in the high-light regions. According to Figure 4, on the other hand, there appears an increasing trendof kρ p with β k p for a given slope or | σ k | . Such a trend may be related to the increasingtrend of k b ρ p (dimensionless spectral break position) with β k p , as found by Duan et al.(2020). This letter performs a statistical research using in-situ measurements of the solarwind lasting for 11 years. For the nearly collisionless solar wind turbulence at proton scales,two new results are obtained as follows. Firstly, the turbulence with a higher magnetichelicity corresponds to a steeper energy spectrum statistically. Secondly, for the turbu-lence with high helicity ( | σ k | > . ), there exists a positive correlation between theproton perpendicular temperature and the turbulent magnetic energy at scales . . kρ p . . The correlation can usually be described by a power law and its slope is con-siderable ( & β k p . . In addition, our examination on the nature of the so-lar wind turbulence supports the model of KAW turbulence at proton scales.Based on our findings and existing literatures, we propose a scenario for solar windturbulence and heating as follows. The magnetic fluctuations in the energy-containingrange first undergo the Kolmogorov cascade in the inertial range, and predominantly be-come KAW turbulence at proton scales. The KAW turbulence raises magnetic helicitysignature on the one hand, and suffers from dissipation on the other hand. Consequently,part of the magnetic energy goes into protons and a steeper energy spectrum with anindex lower than − / emerges. During this process, the fluctuations with higher ini-tial energy in the energy-containing range, therefore higher energy at proton scales, willhave greater ability to heat protons and result in higher proton temperatures. In otherwords, the higher temperatures are attributed to higher turbulent energy at proton scalesthat comes from stronger large-scale magnetic fluctuations by turbulent cascade.As for the mechanisms of the heating, we propose cyclotron resonance and/or stochas-tic heating by the proton-scale KAW turbulence. Cyclotron resonance between KAWsand protons is possible, leading to the proton temperature increase in the direction per-pendicular to the background magnetic field (Leamon et al., 1999; Gary & Borovsky, 2004;He, Wang, et al., 2015; Isenberg & Vasquez, 2019). The heating by turbulent KAWs canalso be expected when stochastic heating arises due to the presence of large-amplitudeelectromagnetic fluctuations at the proton gyroscale (Johnson & Cheng, 2001; Chandranet al., 2010). Both mechanisms contribute to the perpendicular heating and are in agree-ment with our observations. These observations show that the power-law correlation re-lated to the proton perpendicular temperature ( T ⊥ p ) is the best correlation comparedwith the case for either the parallel temperature ( T k p ) or the total temperature ( T p ). Thecorrelation between T p and P k is secondary with a slope always < . , and becomesmuch weaker between T k p and P k . On the other hand, we remind that ion cyclotron waves(ICWs) in the solar wind may also contribute to the proton perpendicular heating. Ananalysis on the solar wind with time interval of 13.4 hours shows that the distributionof ( β k p , T ⊥ p /T k p ) associated with ICWs corresponds to higher T ⊥ p /T k p (Telloni & Bruno,2016). Direct measurements of the dissipation rate spectrum around ion scales in mag-netosheath turbulence indicate the perpendicular heating by ICWs (He et al., 2019, 2020).In this letter, we favor the scenario of the heating by KAW turbulence because the KAWturbulence appears to be ubiquitous and has energy much higher than ICWs in the so-lar wind near 1 AU (He et al., 2012b). We note that the present scenario is preliminaryand more investigations are required in future works.Finally, we emphasize that the present research is relevant to the frequent obser-vations of high ion perpendicular temperatures in the solar wind and potentially to thosein the solar corona (Hollweg & Isenberg, 2002; Stansby et al., 2019). The data used in –10–anuscript submitted to Geophysical Research Letters this letter are from measurements near 1 AU. Other data, especially produced by theSolar Probe Plus spacecraft in the inner heliosphere (Fox et al., 2016), should be help-ful for further studies.
Acknowledgments
The authors acknowledge the SWE team and MFI team on Wind for providing the data,which are publicly available via the Coordinated Data Analysis Web https://cdaweb.gsfc.nasa.gov/cdaweb/istp_public/ . G.Q.Z. is grateful to the hospitality by AuburnUniversity in USA as a visiting scholar. This research was supported by NSFC undergrant Nos. 41874204, 41974197, 41674170, 41531071, 11873018, 41804163, 11903016, andsupported partly by scientific projects from Henan Province (19HASTIT020,16B140003).The authors acknowledge two referees for useful suggestions that improved this paper.
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