Magnetic helicity signature and its role in regulating magnetic energy spectra and proton temperatures in the solar wind
G. Q. Zhao, Y. Lin, X. Y. Wang, H. Q. Feng, D. J. Wu, H. B. Li, A. Zhao, Q. Liu
aa r X i v : . [ phy s i c s . s p ace - ph ] S e p Magnetic helicity signature and its role in regulating magneticenergy spectra and proton temperatures in the solar wind
G. Q. Zhao , , Y. Lin , X. Y. Wang , H. Q. Feng , D. J. Wu , H. B. Li , A. Zhao , and Q.Liu 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, ChinaReceived ; accepted Correspondence should be sent to: [email protected] 2 –
ABSTRACT
In a previous paper, we found that proton temperatures are clearly associ-ated with the proton-scale turbulence in the solar wind, and magnetic helicitysignature appears to be an important indicator in the association. Based on 15years of in situ measurements, the present paper further investigates the mag-netic helicity of solar wind turbulence and its role in regulating magnetic energyspectra and proton temperatures. Results show that the presence of the helicitysignature is very common in solar wind turbulence at scales 0 . . kρ p .
1, with k being the wavenumber and ρ p the proton gyroradius. The sign of the helic-ity is mostly positive, indicating the dominance of right-handed polarization ofthe turbulence. The helicity magnitude usually increases with k and β k p (theproton parallel beta) when kρ p and β k p are less than unity. As helicity magni-tude increases, the power index of the energy spectrum becomes more negative,and the proton temperatures T ⊥ p and T k p rise significantly, where T ⊥ p and T k p are the perpendicular and parallel temperatures with respect to the backgroundmagnetic field. In particular, the rise of T ⊥ p is faster than T k p when β k p < T ⊥ p with the helicity magnitude may be interpretedas the result of the preferentially perpendicular heating of solar wind protons bykinetic Alfv´en wave (KAW) turbulence. Subject headings:
Solar wind (1534); Interplanetary turbulence (830); Solar coronalheating (1989); Space plasmas (1544); Plasma physics (2089)
1. Introduction
It has long been known that the solar wind undergos nonadiabatic expansion, withproton temperatures usually much higher than theoretical prediction (e.g., Marsch et al.1982b; Gazis & Lazarus 1982). Early researches revealed that the average dependence ofproton temperatures on the heliocentric radial distance follows a power law r n T , where n T & − − /
3. Moreover, considering theweakly collisional solar wind with proton distributions far from thermodynamic equilibrium,the double-adiabatic theory predicts the perpendicular proton core temperature decreasesas r − (Chew et al. 1956; Matteini et al. 2012), whereas observations showed that thistemperature decreases significantly more slowly. An index around − .
2. Data and Analysis Methods
The data used in the present paper are over a long time period from 2004 June to 2019May. They are from the
Wind spacecraft, a comprehensive solar wind laboratory in a haloorbit around the L1 Lagrange point. The magnetic fields are sampled at a cadence of 0.092s (Lepping et al. 1995), and the plasma data are at a cadence of 92 s (Ogilvie et al. 1995).The proton temperatures are yielded through a nonlinear-least-squares bi-Maxwellian fit ofion spectrum from the Faraday cup (Kasper et al. 2006). The survey is through dividingthe long time series into consecutive and overlapping time segments. Each time segmenthas a span of 200 s, and the overlap time is set to be 100 s. In each segment with dataavailable, the magnetic energy spectrum is produced by standard fast Fourier transformtechnique. The plasma parameters are obtained as average values over the time segment.They are composed of the proton density N p , perpendicular and parallel thermal velocities w ⊥ p and w k p , and bulk velocity V p , where ⊥ and k are with respect to the backgroundmagnetic field B . Following the paper (Zhao et al. 2020b), segments with A c < . A c is the Coulomb collisional age (Livi et al. 1986). The angle between B and V p is alsorequired in the range from 60 ◦ to 120 ◦ , which could reduce the possible heating/coolingeffects due to the alpha − proton differential flow (Zhao et al. 2019b, 2020a). In total about3 . × time segments satisfying these constraints are selected.Magnetic helicity has been widely used to indicate the presence of circularly/ellipticallypolarized waves in the solar wind (He et al. 2011; Zhao et al. 2018, 2019a; Woodham et al.2019). The helicity is reduced for the magnetic field measured by a single spacecraft. Itcan be expressed as kH rm ( k ) /P ( k ), where k is the reduced wavenumber, and H rm and P are the reduced fluctuating magnetic helicity and the magnetic energy at wavenumber k , respectively (Matthaeus & Goldstein 1982). It can be further expressed as a functionof frequency associated with the spacecraft time series when Taylor frozen-in-flowhypothesis holds, where the frequency is related to wavenumber (Taylor 1938; He et al.2011). The equation for the magnetic helicity used in this paper is finally written as(Matthaeus & Goldstein 1982; Zhao et al. 2020b) σ f = 2Im[ B y ( f ) · B ∗ z ( f )] P i = j [ B i ( f ) · B ∗ j ( f )] , (1)where Im means the imaginary part, B i · B ∗ j are the elements of the energy spectral tensorcoming from Fourier spectra of time series of magnetic fields, and the subscripts i and j indicate the components of the magnetic field vector in the GSE coordinate system.Equation (1) also means that the helicity is normalized, with values in the range from − x direction that points to the Sun. In order to obtain the polarization sense withrespective to the background magnetic field, the helicity will be multiplied by − − P f ) and the helicity ( σ f ) of the fluctuationin the frequency domain. The energy spectrum is characterize by two power laws. Oneis with an index nearly − / − . kρ p = 0 .
1, where ρ p = w ⊥ p / Ω p is the proton thermal gyroradius and Ω p is the proton cyclotron frequency,and the right line is chosen accordingly, with the spectral energy P f > − nT /Hz sothat signal level is much higher than the instrument noise level (Lepping et al. 1995). Theconversion from the frequency domain to the wavenumber domain is conducted accordingto the Taylor frozen-in-flow hypothesis, 2 πf = kV p (Taylor 1938). This step should bemeaningful for an analysis of various observations with different plasma parameters. Thenthe helicity ( σ k ) in the wavenumber domain can be presented, where an averaging operationover f e − . ≤ f ≤ f e . is conducted to produce a smoothed helicity spectrum, as shownin panel (c). Panel (d) displays the local spectral index ( α k ) that is yielded by fitting theenergy spectrum over the same frequency range for the averaging. The blue horizontaldashed line in this panel marks the constant − /
3. Comparing panel (d) with panel (c),one may note that the more negative α k occurs with the larger σ k . 8 –Fig. 1.— An example to illustrate (a) magnetic energy spectrum P f in frequency domain,(b) magnetic helicity σ f in frequency domain, (c) local average magnetic helicity σ k inwavenumber domain, (d) local magnetic spectral index α k in wavenumber domain. Twovertical dotted lines in panels (a) and (b) indicate the range plotted in panels (c) and (d). 9 –
3. Statistical Results
Based on the data set described in Section 2, statistical investigations on themagnetic helicity, magnetic energy spectra, and proton temperatures are conducted. Inthe investigations the sign of the helicity is taken into account, which was ignored inour previous study (Zhao et al. 2020b). Subsection 3.1 presents the statistical results formagnetic helicity distributions. Subsections 3.2 and 3.3 display the results for magneticspectral indices and proton temperatures regulated by the magnetic helicity, respectively. σ k The distributions of σ k depend on the wavenumber and the proton beta. Figure 2 plotsthe distributions at a fixed wavenumber kρ p = 0 .
8, but for different beta ranges, where thebin of 0.02 for σ k has been used. Panel (a) in Figure 2 is for all β k p observed, while panels(b) and (c) are for the data subsets with β k p < . . < β k p <
1, respectively, where β k p = w k p /v A is the proton parallel beta. One may first find that the distributions areasymmetrical with respect to σ k = 0, since the data with positive σ k are much more thanthose with negative σ k . There are two peaks arising at σ k ≃ ± .
22, although they are weakin the case of lower beta (pane (b)). This result implies that the majority of the data arecharacterized by a considerable helicity with the magnitude greater than 0.1. On the otherhand, the data size in a bin drops dramatically when the helicity magnitude approaches to0.4, which happens nearly irrespective of β k p .Figure 3 displays the color plot of distributions of σ k for various k . Three panelscorrespond to three β k p ranges as those in Figure 2, and the red color means the peak ofthe distribution. In order to share a common color bar, the data numbers in panels (b)and (c) have been amplified by 10 times and 6 times, respectively. One can see that the 10 –Fig. 2.— Distributions of σ k at kρ p = 0 . β k p , (b) β k p < .
3, (c)0 . < β k p < σ k at various k for cases of (a) all β k p , (b) β k p < .
3, (c) 0 . < β k p <
1. 11 –helicity distribution significantly depends on the wavenumber in each panel. For a smallwavenumber kρ p = 0 .
1, the peak of the distribution occurs at σ k ∼
0, while two peaks withfinite σ k arise when kρ p & .
3. One can also see that the peak with positive σ k dominatesthe distribution and the other peak is much weak. As kρ p increases up to 0.7 ∼ β k p ), the magnitude of σ k begins to decrease rapidly. The cause of the decrease mightbe complex, possibly due to the instrument noise, aliasing, and/or great balance of waveturbulence (He et al. 2012b; Markovskii & Vasquez 2013; Klein et al. 2014). In addition,the data number decreases significantly when kρ p exceeds some value, 0.7 ∼
1. This isbecause a lot of time segments described in Section 2 have spectral energies lower than thethreshold 10 − nT /Hz at the larger k , which are discarded in the statistics to reduce theeffect of noise on the results.To explore the role of β k p in determining the helicity, Figure 4 presents medians of | σ k | with respect to ( β k p , k ). Here only the magnitude of σ k is used, since the sign of σ k justmeans the polarization sense. Result shows that statistically | σ k | increases with k when kρ p . kρ p >
1, which is consistent with Figure 3. For a given k with0 . . kρ p . σ k increases with β k p when β k p . β k p > σ k decreasesconsiderably. The increase of | σ k | with β k p is in agreement with the results obtainedby Markovskii & Vasquez (2013, 2016), who calculated the magnetic helicity via hybridnumerical simulations of two-dimensional turbulence for three beta values, i.e., 0.15, 0.5,and 0.65. σ k Existing literatures show that spectral indices of the proton-scale magnetic fluctuationsin the solar wind take vales usually between − − σ k , especially inthe case of low beta. Figure 5 displays the distributions of ( σ k , α k ), where α k is the spectralindex. In the figure panels (a) − (c) correspond to three cases of all β k p , β k p < .
3, and0 . < β k p <
1, respectively, and the wavenumber kρ p = 1 has been fixed. One can see that α k decreases with | σ k | overall, which implies that a larger | σ k | account for a steeper energyspectrum. We fit the data by α k = aσ k + b , distinguished between σ k > σ k <
0. Thefitted parameters are presented in Table 1. It shows that α k , relative to the situation for σ k <
0, tends to have stronger dependence on σ k when σ k >
0. It also appears that thedependence is stronger if β k p is lower, and the strongest dependence of α k on σ k occurs inthe case of β k p < .
3, with α k = ( − . ± . σ k − . ± .
01 for σ k > σ k This subsection is presented to show how the helicity regulates proton temperatures.Figure 6 plots medians of proton perpendicular and parallel temperatures ( T ⊥ p and T k p )against σ k , respectively, with a given wavenumber kρ p = 0 . β k p . From panel(a), one may find the regulations of σ k on the temperatures. They are (1) both T ⊥ p and T k p increase with | σ k | when | σ k | is larger than some threshold, 0.15 ∼ σ k ; (2) T ⊥ p tends to increase faster relative to T k p ; (3) the temperature curvesTable 1: Fitted parameters for the expression of α k = aσ k + b at kρ p = 1. σ k > σ k < a b a b All β k p − . ± . − . ± .
002 2 . ± . − . ± . β k p < . − . ± . − . ± .
007 3 . ± . − . ± . . < β k p < − . ± . − . ± .
005 2 . ± . − . ± .
007 13 –Fig. 4.— Color scale plot of medians of | σ k | in the ( β k p , k ) space.Fig. 5.— Distributions of ( σ k , α k ) at kρ p = 1 for cases of (a) all β k p , (b) β k p < .
3, (c)0 . < β k p <
1. 14 –Fig. 6.— Medians of proton perpendicular temperature T ⊥ p (solid lines) and parallel tem-perature T k p (dashed lines) against σ k at kρ p = 0 . β k p , (b) β k p < .
3, (c)0 . < β k p < T ⊥ p /T k p against σ k at kρ p = 0 . β k p < . . < β k p < . . < β k p < β k p > β k p (black line). The orange and black lines have been up-shifted by adding0.2 and 0.1, respectively. 15 –Fig. 8.— Color scale plots of medians of T ⊥ p (upper panels) and T ⊥ p /T k p (lower panels) inthe ( σ k , k ) space. Panels (a) − (c), as well as panels (d) − (f), are for cases of all β k p , β k p < . . < β k p <
1, respectively. 16 –are asymmetrical with respect to σ k = 0 and positive σ k tends to correspond to highertemperatures. According to panel (b), the result appears to be very clear for the low betacase, i.e., β k p < .
3. In this case T ⊥ p and T k p show significantly different dependences on σ k . T ⊥ p rapidly rises with | σ k | while T k p is nearly irrespective of | σ k | , and the temperaturecurves in this case are less asymmetrical. In contrast, the asymmetry for T k p becomesevident when β k p is large, as shown in panel (c) for the case of 0 . < β k p <
1. Consequentlypositive σ k results in distinctly higher T k p , and the minimum of T k p happens with σ k around − .
2. In addition, it is also interesting that the fastest increase of T ⊥ p with | σ k | occursin panel (b) with σ k >
0, in which the spectral index shows the fastest decrease with | σ k | (panel (b) of Figure 5).For the sake of discussion, we adopt the following perspectives: (1) the increase oftemperatures means the occurrence of heating; (2) the faster increase of T ⊥ p ( T k p ) than T k p ( T ⊥ p ) implies that the heating occurs preferentially in the direction perpendicular (parallel)to the background magnetic filed. With these perspectives, further investigation showsthat the preferentially perpendicular heating occurs when β k p .
1, while the preferentiallyparallel heating tends to happen if β k p >
2. To illustrate this point, Figure 7 plots mediansof the temperature ratio T ⊥ p /T k p against σ k at kρ p = 0 .
8, where the lines with differentcolors correspond to different ranges of β k p . To avoid the overlapping of these lines, theorange and black lines have been up-shifted by adding 0.2 and 0.1, respectively. One cansee that T ⊥ p /T k p almost always rises as | σ k | increases when β k p <
1. On the other hand, T ⊥ p /T k p shows somewhat reduction with increasing positive σ k when β k p >
2. Furtherinvestigation shows that the dependence of T ⊥ p on σ k becomes much weak if β k p >
2, while T k p moderately increases with σ k for σ k & − . T ⊥ p (upper panels) and T ⊥ p /T k p (lower panels) against ( σ k , k ) for the three 17 –cases of β k p range. To highlight the color comparison in panels (d) and (f), T ⊥ p /T k p hasbeen multiplied by 1.5 in both panels. Strong dependences can be found for 0 . . kρ p . T ⊥ p and T ⊥ p /T k p in principle increase with | σ k | . For larger wavenumber with kρ p >
1, the dependences tend to remain but are not very clear, where | σ k | significantlydrops according to Figure 3. Note that T ⊥ p appears to be higher for a larger k , whichshould be attributed to the selection criteria that just allow the time segments withgreater turbulent energy to survive (due to the effect of noise at the larger k ). The highertemperature resulting from the greater magnetic energy at proton scales has been found inthe previous research (Zhao et al. 2020b). In addition, consistent with the result in Figure6, the approximate symmetry with respect to σ k = 0 occurs in the case of β k p < .
3, i.e.,panels (b) and (e), though the larger β k p tends to break the symmetry according to panels(c) and (f).
4. Summary and Discussion
Based on 15 years of in situ measurements, this paper performs a statistical researchon the magnetic helicity, magnetic energy spectra, and proton temperatures in the solarwind. Results from the magnetic helicity distributions show that the helicity signature withmoderately high magnitude (0 . < | σ k | < .
4) frequently arises in solar wind turbulence atscales 0 . . kρ p .
1. The distributions are generally asymmetrical, with the helicity mostlypositive. There are two peaks in the distributions, occurring at σ k ≃ ± .
22 for kρ p = 0 . β k p < .
3, while they can be strong for larger β k p . The magnitude of the helicity depends on β k p as well as k . For a given k , the helicitymagnitude increases with β k p when β k p < β k p >
1. Thisincrease with β k p is consistent with the prediction by hybrid simulations of two-dimensionalturbulence with beta less than unity (Markovskii & Vasquez 2013, 2016). 18 –The magnetic helicity appears to play an important role in regulating magnetic spectralindices at proton scales. The spectral indices will become significantly more negative if thehelicity magnitudes are larger. This correlation between the spectral indices and helicitymagnitudes is particularly clear for the case of β k p < .
3. By fitting the data at kρ p = 1,we obtain an expression of the correlation as α k = − . σ k − . β k p is larger, as shown in Table 1 for details.The magnetic helicity, on the other hand, also play a considerable role in regulatingproton temperatures. Overall, proton temperatures ( T ⊥ p and T k p ) usually increase withhelicity magnitudes at 0 . . kρ p .
1. The temperature increases show different behaviorsin different cases of beta ranges. In the case of β k p < .
3, it is clear that T ⊥ p fastly increasesas | σ k | increases, while this trend is very weak for T k p . The increase of T ⊥ p faster than T k p also occurs in the case of 0 . < β k p <
1. (An opposite result happens if β k p > T ⊥ p and T k p curves against σ k are asymmetrical, with positive σ k contributing to higher T ⊥ p and T k p . The asymmetry is more obvious for T k p when β k p is large. The investigation on thetemperature ratio T ⊥ p /T k p reveals that T ⊥ p /T k p almost always increases as | σ k | increaseswhen β k p < T ⊥ p than T k p (Figure 6).The magnetic helicity signature discussed in this paper should mainly result fromproton-scale KAW turbulence. A lot of researches on the nature of solar wind turbulenceat proton scales support the KAW turbulence model (Bale et al. 2005; Howes et al. 2008;Sahraoui et al. 2009, 2010; Salem et al. 2012; Chen et al. 2013; Groˇselj et al. 2018). Ourstatistical examination in terms of the long-axis direction of magnetic fluctuations at protonscales also favors the model of KAW turbulence (Zhao et al. 2020b). Note that KAWturbulence can naturally raise the non-zero magnetic helicity (Howes & Quataert 2010; 19 –He et al. 2012a; Podesta 2013). Hence it can be expected that the majority of the solarwind turbulence at proton scales is characterized by the considerable magnetic helicitysignature, as shown in this paper.Further, one may conclude that the majority of KAWs in solar wind turbulence appearto be outward propagating with respect to the Sun. Note that the magnetic helicity in thispaper is measured in the spacecraft reference frame, whose sign marks the polarization inthe spacecraft frame. KAWs are inherently right-handed polarized waves (positive helicity)in the solar wind reference frame (Gary 1986; Zhao et al. 2014), but they may appear asleft-handed polarized waves (negative helicity) in the spacecraft frame when they propagatetoward the Sun, in which the large Doppler-shift effect could result in the polarizationreversal. Our results in Figures 2 and 3 show that the measured helicity is mostly with thepositive sign, implying the dominance of the right-handed polarization in the spacecraftframe. This mostly positive helicity should imply that the KAWs are usually outwardpropagating (without polarization reversal).We interpret the elevation of proton temperatures with enhanced magnetic helicity asthe occurrence of proton heating in the solar wind. The heating may be attributed to thedissipation of proton-scale KAW turbulence that comes from the fluctuations in the inertialrange by turbulent cascade. In this idea, the inertial-range fluctuations would determine theability of the heating; the inertial-range fluctuations with higher energy would contributeto larger-amplitude KAW turbulence at proton scales, and therefore have greater ability toheat protons. Consequently, higher proton temperatures could be expected if the turbulenceamplitude is larger. Existing researches revealed positive correlation between protontemperatures and magnetic fluctuation level in the inertial range (Smith et al. 2006b;Vech et al. 2018). Our study particularly demonstrated that higher proton temperaturescorrelate with the larger turbulent amplitudes at proton scales (Zhao et al. 2020b). 20 –Existing literature also documented that higher proton temperatures are associatedwith steeper proton-scale turbulent spectra based on 33 event study, and concluded “Thissuggests that steeper dissipation range spectra imply greater heating rates” (Leamon et al.1998). The present study is in line with this literature. Our results first agree with thefinding of higher proton temperatures associated with steeper proton-scale spectra, sincewe have showed that the magnetic helicity enhancements can simultaneously correlatewith higher proton temperatures and steeper spectra at proton scales. We also speculatea specific process as follows. Some dissipation mechanism efficiently operates and quicklyremoves energy from the proton-scale turbulence, which results in a steeper proton-scalespectrum (with an index less than − / T ⊥ p /T k p with | σ k | for β k p < T ⊥ p than T k p ; both T ⊥ p and T k p in principle increase with | σ k | . We interpret this phenomenonas the heating that occurs preferentially in the perpendicular direction with respect to thebackground magnetic field. In the context of KAW turbulence, two mechanisms couldcontribute to the perpendicular heating, i.e., cyclotron resonance and stochastic heating.Theoretical researches show that cyclotron resonance is possible between KAWs andprotons, causing the perpendicular heating of protons (Gary & Borovsky 2004; Smith et al.2012; Isenberg & Vasquez 2019). Simultaneous observations of wave fluctuations andparticle kinetics reveal that KAWs seem to be responsible for the anomalous cyclotronresonance of proton beams, causing the perpendicular heating of proton beams (He et al. 21 –2015b). In the presence of large-amplitude electromagnetic fluctuations at the protongyroradius scale, the perpendicular heating by turbulent KAWs can also be expected dueto stochastic heating (Johnson & Cheng 2001; Chandran et al. 2010; Hoppock et al. 2018).The occurrence of the stochastic heating in the solar wind has been studied in recent years,and results support the stochastic heating as an effect mechanism to heating the solar wind(Xia et al. 2013; Vech et al. 2017; Arzamasskiy et al. 2019; Martinovi´c et al. 2019, 2020).Before concluding, we make a final remark as follows. Figures 6 and 8 show that thetemperature distributions against the helicity are asymmetrical, especially for the protonparallel temperature. The cause of this asymmetry is not clear. It seems to imply thatdifferent helicity signs, and therefore different wave propagation directions with respectto the Sun, correspond to different heating efficiencies. It has been well known thatthe relative directions between ion cyclotron waves and alpha − proton differential flowsignificantly affect the cyclotron resonance efficiency (Podesta & Gary 2011; Zhao et al.2019b, 2020a). It is unclear whether the propagation directions of KAWs relative to thedifferential flow, which usually points away from the Sun in the fast solar wind, also affectthe (cyclotron/Landau) resonance efficiency and result in the asymmetry of the temperaturedistributions. Further research on this issue is desirable.In summary, based on the long period in situ measurements, this paper investigates themagnetic helicity and its role in regulating magnetic energy spectra and proton temperaturesin the solar wind. This study should be helpful to discuss the solar wind turbulence and thenonadiabatic behavior of the solar wind. Note that the present discussion is preliminaryand further researches are needed.The authors acknowledge the SWE team and MFI team on Wind for pro-viding the data, which are available via the Coordinated Data Analysis Web(http://cdaweb.gsfc.nasa.gov/cdaweb/istp − public/). G.Q.Z. is grateful to the hospi- 22 –tality by Auburn University in USA as a visiting scholar. This research was supportedby NSFC under grant Nos. 41874204, 41974197, 41674170, 41531071, 11873018,41804163, 11903016, and supported partly by scientific projects from Henan Province(19HASTIT020,16B140003). 23 – REFERENCES