The Role of Proton Cyclotron Resonance as a Dissipation Mechanism in Solar Wind Turbulence: A Statistical Study at Ion-kinetic Scales

Several different characteristic ion plasma scales have been suggested to correspond to the observed spectral steepening, and each one is associated with a different plasma heating process. Two  scales that are commonly proposed to correspond to the spectral break are the ion inertial length, d_i = v_A/Ω_i, and the ion gyroscale, ρ_i = v_th,⊥/Ω_i. Here, ${v}_{{\rm{A}}}={B}_{0}/\sqrt{{\mu }_{0}{n}_{i}{m}_{i}}$ is the Alfvén speed, n_i is the ion number density, ${v}_{\mathrm{th},\perp }=\sqrt{2{k}_{B}{T}_{i,\perp }/{m}_{i}}$ is the ion thermal speed perpendicular to the background magnetic field, ${{\boldsymbol{B}}}_{0}$, and T_i,⊥ is the ion perpendicular temperature. The inertial length is associated with the onset of dispersive effects due to the Hall current term, as well as reconnection of small-scale current sheets (Dmitruk et al. 2004; Galtier 2006; Galtier & Buchlin 2007), whereas the transition from Alfvén wave to KAW-dominated turbulence occurs at scales comparable to the gyroscale (Howes et al. 2008a; Schekochihin et al. 2009; Boldyrev & Perez 2012).

Another explanation for the observed spectral steepening is cyclotron resonance of Alfvén waves with solar wind ions (e.g., Coleman 1968; Marsch et al. 1982a, 2003; Denskat et al. 1983; Goldstein et al. 1994; Gary & Borovsky 2004; Smith et al. 2012). Here, the only ions we consider are protons, and throughout this paper we use the subscript i to refer exclusively to protons. Leamon et al. (1998b) proposed a wavenumber for the onset of cyclotron damping of Alfvén waves (see also Gary 1999). The cyclotron resonance condition for protons is given by equating the Doppler-shifted wave frequency in the plasma frame, ω, to the proton gyrofrequency, Ω_i (e.g., see Stix 1992),

$$\begin{eqnarray}&&\omega ({k}_{\parallel })-{k}_{\parallel }{v}_{\parallel }=\pm {{\rm{\Omega }}}_{i},\end{eqnarray} \tag{ 1 }$$

where v_∥ is the parallel velocity of the resonant protons and k_∥ is the parallel component of the wavenumber with respect to ${{\boldsymbol{B}}}_{0}$. The ± sign takes into account the sense of polarization of the wave. The wave electric field vector of left-hand circularly polarized Alfvén/ion-cyclotron waves (hereafter, AICs) propagating parallel to ${{\boldsymbol{B}}}_{0}$ rotates in the same direction as proton gyration, so we use the positive sign. This interaction is most effective if ${k}_{\parallel }{v}_{\parallel }\lt 0$, reducing the resonance condition to

$$\begin{eqnarray}&&\omega ({k}_{\parallel })+{k}_{\parallel }\,| {v}_{\parallel }| ={{\rm{\Omega }}}_{i}.\end{eqnarray} \tag{ 2 }$$

To obtain the minimum wavenumber, ${k}_{\parallel }={k}_{c}$, at which dissipation of the waves by cyclotron resonance with the background solar wind proton distribution occurs, we take ${v}_{\parallel }={v}_{\mathrm{th},\parallel }$, where ${v}_{\mathrm{th},\parallel }$ is the parallel thermal speed of the proton velocity distribution, and for simplicity we substitute for $\omega ({k}_{\parallel })$ using the wave dispersion relation of Alfvén waves (e.g., Gary 1993): $\omega ({k}_{\parallel })={k}_{\parallel }{v}_{{\rm{A}}}$,

$$\begin{eqnarray}&&{k}_{c}=\displaystyle \frac{{{\rm{\Omega }}}_{i}}{{v}_{{\rm{A}}}+{v}_{\mathrm{th},\parallel }}\equiv \displaystyle \frac{1}{{d}_{i}+{\sigma }_{i}}.\end{eqnarray} \tag{ 3 }$$

Here, σ_i is the pseudo-gyroscale, defined as ${v}_{\mathrm{th},\parallel }/{{\rm{\Omega }}}_{i}$ using the parallel proton temperature, ${T}_{i,\parallel }$, which we distinguish from the typical definition of the ion gyroscale, ρ_i. The waves do not necessarily need to be parallel-propagating for resonance to occur; as long as there is a large enough ${k}_{\parallel }$ component, a wave can resonate with the proton population, even if it also has a significant k_⊥ component. If there is a substantial population of AICs in the solar wind, then we may expect the spectral break to occur at the scale 1/k_c.

Several past studies have explored the physical processes behind the observed spectral steepening by comparing characteristic ion scales with the spectral break measured from in situ data (Leamon et al. 1998b, 2000; Smith et al. 2001; Markovskii et al. 2008; Perri et al. 2010; Bourouaine et al. 2012; Bruno & Trenchi 2014; Chen et al. 2014; Roberts et al. 2017; Wang et al. 2018) or through simulations (e.g., Ghosh et al. 1996; Howes et al. 2008b; Cerri et al. 2016; Franci et al. 2016, 2017). However, these studies have produced various different conclusions, and there is currently no consensus on the dominant dissipation mechanism. The difficulty in determining the break scale arises from the fact that the measured scales d_i and ρ_i at 1 au are linked by the proton perpendicular plasma beta, ${\beta }_{i,\perp }={n}_{i}{k}_{B}{T}_{i,\perp }/({B}_{0}^{2}/2{\mu }_{0})$,

$$\begin{eqnarray}&&\displaystyle \frac{{\rho }_{i}}{{d}_{i}}=\sqrt{{\beta }_{i,\perp }},\end{eqnarray} \tag{ 4 }$$

and typically β_i,⊥ ∼ 1, so that these scales are inseparable, except in cases where β_i,⊥ ≪ 1 or β_i,⊥ ≫ 1 (for example, see Chen et al. 2014). Therefore, the spectral break may be associated with different scales, depending on changing solar wind conditions.

We can gain a better understanding of the possible dissipation mechanisms by looking at the nature of the fluctuations at these frequencies. The presence of fluctuations with different properties such as polarization will limit the role of certain mechanisms under different conditions. A useful quantity that can be used to diagnose certain types of fluctuations is the magnetic helicity, which characterizes the solenoidal structure of the magnetic field and twistedness of field lines (Moffat 1978; Woltjer 1958, see also Smith 2003; Telloni et al. 2013). For solar wind turbulence, the quantity of interest is the fluctuating magnetic helicity density (Matthaeus & Goldstein 1982a). A reduced form, H_m(k), can be computed from single-spacecraft measurements; based on several assumptions (Batchelor 1970; Montgomery & Turner 1981; Matthaeus et al. 1982), this is

$$\begin{eqnarray}&&{H}_{m}(k)=\displaystyle \frac{2\mathrm{Im}\{{{\boldsymbol{P}}}_{{yz}}(k)\}}{k},\end{eqnarray} \tag{ 5 }$$

where ${{\boldsymbol{P}}}_{{yz}}$ is the y–z component of the reduced power spectral tensor of the magnetic field fluctuations in geocentric solar ecliptic (GSE) coordinates (for details on reduced spectra, see Wicks et al. 2012). We define the reduced normalized magnetic helicity, σ_m(k), as

$$\begin{eqnarray}&&{\sigma }_{m}(k)=\displaystyle \frac{{{kH}}_{m}(k)}{{E}_{b}(k)}\equiv \displaystyle \frac{2\mathrm{Im}\{{{\boldsymbol{P}}}_{{yz}}(k)\}}{\mathrm{Tr}\{{{\boldsymbol{P}}}_{{ij}}(k)\}}.\end{eqnarray} \tag{ 6 }$$

Here, E_b(k) is the reduced magnetic spectral energy,  which is given by the trace of the reduced power spectral tensor: $\mathrm{Tr}\{{{\boldsymbol{P}}}_{{ij}}\}={{\boldsymbol{P}}}_{{xx}}+{{\boldsymbol{P}}}_{{yy}}+{{\boldsymbol{P}}}_{{zz}}$. The normalized magnetic helicity gives a dimensionless measure of the polarization of magnetic fluctuations to identify wave modes at a particular frequency in the turbulent spectrum; σ_m is zero for linearly polarized waves and ±1 for right- or left-hand circularly polarized fluctuations, respectively.

Past studies using a global mean magnetic field have found a lack of coherent helicity at low frequencies in the inertial range, i.e., fluctuating almost randomly between negative and positive values (Matthaeus & Goldstein 1982a). However, at ion-kinetic frequencies there is a dominant coherent signature that suggests right-hand polarization for outward propagating fluctuations (Goldstein et al. 1994; Leamon et al. 1998b; Hamilton et al. 2008; Brandenburg et al. 2011; Markovskii et al. 2015). More recently, wavelet-based studies (Telloni et al. 2012 and references therein) using the technique first developed by Horbury et al. (2008) for local mean field analysis have been employed (see also Podesta 2009; Wicks et al. 2010; Forman et al. 2011; Podesta & Gary 2011b, for more details). These studies attribute the right-handed signature to the presence of KAWs propagating at large angles to the local mean field and have also revealed the presence of a weaker left-hand polarized component likely due to quasi-(anti)parallel-propagating AICs (He et al. 2011, 2012a, 2012b; Podesta & Gary 2011b; Klein et al. 2014; Bruno & Telloni 2015; Telloni et al. 2015).

From these results, we may interpret the coherent helicity signature first observed by Goldstein et al. (1994) as arising from the dominance of the right-handed component over the left-handed component, implying dissipation of AICs at these frequencies that may be due to proton cyclotron resonance. In fact, Bruno & Telloni (2015) showed that on transitioning from fast to slow wind in the trailing edge of a fast wind stream (i.e., for decreasing Alfvénicity), both signatures weaken and eventually disappear, although the left-handed component is first to fade completely. However, Howes & Quataert (2010) showed that KAWs alone can also reproduce the observed helicity signature without the need for cyclotron resonance.

In this paper, we present a rigorous analysis of solar wind turbulence at ion-kinetic frequencies using a combined identification of the frequency of the spectral break and onset of the magnetic helicity signature. We compare these spectral properties of the fluctuating magnetic field with the characteristic plasma scales, d_i, ρ_i, and 1/k_c, and attempt to link the coherent helicity signature with the spectral steepening to help identify possible dissipation mechanisms at these frequencies. We use magnetic field spectra at a much higher resolution than undertaken previously so that plasma scales do not vary considerably over the time series of data used to compute the spectra. Our use of a large data set over the course of a year also enables us to identify how changing solar wind conditions affect possible dissipation mechanisms. We find evidence of proton cyclotron resonance that occurs at least half the time in our studied interval, particularly in the more Alfvénic fast wind, and discuss the possible implications for plasma heating at ion-kinetic scales.

To compute solar wind spectra, we employ a continuous wavelet transform (CWT) with a Morlet wavelet of frequency-width ω₀ = 6, using the method described by Torrence & Compo (1998). We obtain wavelet coefficients, W(s,t), as functions of the scale, s, at which the wavelets are evaluated, and time. We then convert these scales into equivalent Fourier frequencies using f ≈ ω₀/2πs and calculate components of the reduced power spectral tensor,

$$\begin{eqnarray}&&{{\boldsymbol{P}}}_{{ij}}(\,{f,t})={W}_{i}(\,{f,t}){W}_{j}^{* }(\,{f,t}),\end{eqnarray} \tag{ 7 }$$

where the asterisk indicates complex conjugate and the indices describe the three GSE coordinates, i, j = x, y, z. The power spectral density (PSD) is then

$$\begin{eqnarray}&&\mathrm{PSD}(\,{f,t})=\displaystyle \frac{2}{{f}_{s}}\mathrm{Tr}\{{{\boldsymbol{P}}}_{{ij}}(\,{f,t})\},\end{eqnarray} \tag{ 8 }$$

where f_s = 10.87 Hz is the sampling frequency of the MFI instrument. We first pad the signal to account for any border effects arising from the finite width of the Morlet wavelet, and then calculate an estimate of the PSD at each frequency for every measurement of the original time series. After removal of padding, we average the PSD over every 1000 MFI measurements. This averaging improves the accuracy of the amplitude of the power spectrum at each scale and results in one spectrum for every 92 s of data, which is the cadence of the SWE instrument. We take the time stamp of each 92 s spectrum as the middle of the time series used to produce that spectrum. Finally, we interpolate the time series of plasma measurements onto the time series of averaged PSD estimates to associate one measurement of the ion moments with each 92 s power spectrum.

We note that the length of time over which we average the spectra is shorter than the correlation time of solar wind turbulence, and so the assumptions of stationarity and ergodicity do not hold at low frequencies (Matthaeus & Goldstein 1982b; Perri & Balogh 2010). As such, many of the usual results of turbulence are not recovered; for example, the spectra do not converge to the typical f^−5/3 power law expected in the inertial range. However, we are attempting to measure turbulent behavior at ion-kinetic frequencies (0.1–5.4 Hz) and not in the inertial range. At these high frequencies there are a larger number of wavelengths sampled at these smaller scales during the advection of the turbulence past the spacecraft, and therefore the stationarity and ergodicity conditions are satisfied in our data set for our frequency range of interest.

To estimate the break frequency of each spectrum, f_b, we fit the PSD to the following linear function:

$$\begin{eqnarray}&&{\mathrm{log}}_{10}(\mathrm{PSD})=m\,{\mathrm{log}}_{10}(f)+c,\end{eqnarray} \tag{ 9 }$$

where m is the gradient of the line or spectral exponent. To accommodate greater uncertainty in the spectra at low frequencies, we fit this function to the power spectra using windows in frequency that increase in width in logarithmic space toward lower frequencies, giving us a value for m for each window. The frequencies, f, for fitting Equation (9) to the spectrum included in each window are given by

$$\begin{eqnarray}&&{\mathrm{log}}_{10}({f}_{m})-0.1j\leqslant {\mathrm{log}}_{10}(f)\leqslant {\mathrm{log}}_{10}({f}_{m}),\end{eqnarray} \tag{ 10 }$$

where f_m is the maximum frequency within the window and the index, j = 1, 2, 3, ..., increases by an integer factor for each successive window so that the term 0.1j widens the window as j increases. For each successive window, we set

$$\begin{eqnarray}&&{\mathrm{log}}_{10}({f}_{m,j+1})={\mathrm{log}}_{10}({f}_{m,j})-0.1j/20,\end{eqnarray} \tag{ 11 }$$

shifting the windows to lower frequencies as j increases. The division by 20 in the last term allows us to overlap the windows and provide a sufficient number of fits for m over the frequencies at which we evaluate the power spectra. We continue our windowing process along the spectrum as long as ${\mathrm{log}}_{10}({f}_{m})\gt -1$, giving us a total of 26 windows. The central frequency of each window, which we associate with a value of m, is taken as the median of the frequencies in that window.

We show an example 92 s spectrum from 2012 July in the top panel of Figure 1 and our results for m from our fitting process in the bottom panel.  We see a change in m from about −1.2 to −3.8 from low to high frequencies, resulting from the transition between the power laws for the inertial range and the ion-kinetic range. This transition is not a simple step function because of the finite width of the fitting window at the frequency of the spectral break.  To determine the width of this transition, we calculate two frequencies that bound either side of it. We identify the first, f₁, when the difference between two successive values for m exceeds the threshold $| {m}_{j+1}-{m}_{j}| \geqslant 0.05$, and the second, f₂, as the frequency with the minimum value for m. We then estimate the break frequency f_b for each spectrum as f_b = (f₁ + f₂)/2, in a similar fashion to Chen et al. (2014). The black dashed line in Figure 1 is our estimate of f_b for the example spectrum using this method, which we see agrees well with the break in the spectrum.

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Toward higher frequencies in Figure 1, the spectrum flattens and m increases. This flattening is most likely due to the increasing contribution of instrumental noise to the signal at these frequencies. To ensure that our estimated f_b is physical, we determine a cut-off frequency, f_noise, where the spectrum is equal to a signal-to-noise ratio (S/N) of 10 times our noise-floor estimate (see the Appendix), indicated by the vertical red dashed line in Figure 1. We neglect an estimate of the break frequency if f_b ≥ f_noise. Close to the Nyquist frequency, there is a second decrease in the spectral exponent, which we attribute to artifacts of the CWT.

To test the robustness of our automated fitting procedure and method to calculate f_b, we first apply it to consecutive 92 s power spectra over the course of one month, using data from 2012 July. Panels (a)–(d) in Figure 2 show time series of the components of the magnetic field, ${\boldsymbol{B}}$, the solar wind speed, v_sw, proton density, n_i, and proton perpendicular beta, β_i,⊥, respectively. We smooth ${\boldsymbol{B}}$ using a 51-point median filter here to emphasize the sectoral structure of the interplanetary magnetic field from the numerous crossings of the heliospheric current sheet, highlighted by the changing sign of the B_x and B_y components. We see that v_sw varies between 300 and 700 km s⁻¹ and n_i from less than 1 cm⁻³ to almost 35 cm⁻³. There are several periods, often during fast wind intervals, where β_i,⊥ ∼ 1. At other times β_i,⊥ typically does not exceed unity and reaches a minimum value of almost 1 × 10⁻³. The spacecraft sampled periods of both slow and fast wind, as well as shocks, density enhancements, and transient ejecta, illustrating the variability of the solar wind during this interval.

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Panel (e) in Figure 2 shows a contour plot of consecutive 92 s power spectra over 2012 July, i.e., a time series of spectra over the course of a month. In comparison with panels (a)–(d), we see that the spectra and therefore, the turbulent processes in the solar wind, depend on the overall plasma conditions, particularly at high frequencies. Here, white areas indicate data that we have removed, either due to the presence of a large data gap or because the frequencies exceed the defined noise-floor cut-off, f_noise. We also show as solid lines the three characteristic plasma scales, 1/k_c, d_i, and ρ_i, in green, red, and black, respectively. We plot these three scales as frequencies assuming Taylor's hypothesis: f_L = v_sw/2πl, where l is the appropriate length scale. According to panel (e), there are several periods during which f_noise < f_L, emphasizing the importance of our noise-floor treatment.

In Figure 2(f) we show a contour plot of the spectral exponent, m, versus frequency for each corresponding spectrum in panel (e), along with our estimated f_b and f_kc in black and red, respectively, for comparison. We note that the break frequencies are discretized by the scales of the wavelets and hence the windowing process in our fitting procedure. We discard values where f_b ≥ f_noise, and also f_b ≤ 0.1 Hz. This second condition allows us to avoid times when the amplitude of fluctuations is so low that a physical break between two power laws is obscured by noise, and therefore an estimate for f_b by our automated method is unreliable. We smooth f_b here only for this figure using a 21-point median filter. We find that our fitting procedure performs an accurate estimate of f_b for the ∼29,000 spectra from 2012 July, since f_b agrees well with the break in the spectrum from visual inspection of panel (f).

We now compare the three plasma scales as frequencies, f_L, with f_b, where L = 1/k_c, d_i, and ρ_i, and extend our analysis to a year of data from 2012. We calculate ∼344,000 spectra, estimating f_b for each spectrum and comparing it to the corresponding values for the characteristic plasma scales, f_L. Figure 3 shows two-dimensional histograms for f_b with f_kc, f_di, and f_ρi in the top, middle, and bottom rows, respectively. We show the results for all data and then separate according to slow (v_sw < 400 km s⁻¹) and fast wind (v_sw > 500 km s⁻¹) in the left, middle, and right columns, respectively. Separating by wind speed allows us to test for systematic effects due to the large-scale structure of different solar wind streams. We normalize each column of the binned data in each plot by the maximum number of spectra in a bin for that column, highlighting the most probable f_b measured as a function of f_L. We neglect values for f_b when f_b ≥ f_noise and f_b ≤ 0.1 Hz, as discussed before. We also omit bins with ≤10 spectra to avoid undersampling. In each panel, the black dashed lines give the line f_b = f_L, and similarly the red dashed lines are ${f}_{b}={f}_{L}\sqrt{2}$ and ${f}_{b}={f}_{L}/\sqrt{2}$, which indicate the resolution of the wavelet transform about the line f_b = f_L due to the finite width of the Morlet wavelet in frequency space (i.e., the e-folding frequency, see Torrence & Compo 1998).

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To quantify the relationship between f_b and f_L, we conduct a statistical analysis using this year of data. We first calculate the Pearson correlation coefficient,

$$\begin{eqnarray}&&R({f}_{b},{f}_{L})=\displaystyle \frac{1}{N-1}\displaystyle \sum _{i=1}^{N}\left(\displaystyle \frac{{f}_{b,i}-{\mu }_{b}}{{\sigma }_{b}}\right)\left(\displaystyle \frac{{f}_{L,i}-{\mu }_{L}}{{\sigma }_{L}}\right),\end{eqnarray} \tag{ 12 }$$

where μ is the mean and σ is the standard deviation. The coefficient $R\in [-1,+1]$ measures the linear correlation between f_b and f_L. A value of R = ±1 indicates a positive or negative linear correlation, respectively, whereas zero indicates no linear correlation. If $| R| =1$, a linear equation describes the relationship between the variables f_b and f_L. We also define a residual, ρ, which in analogy to the standard deviation is

$$\begin{eqnarray}&&\rho ({f}_{b},{f}_{L})=\sqrt{\displaystyle \frac{1}{N-1}\displaystyle \sum _{i=1}^{N}{| {f}_{b,i}-{f}_{L,i}| }^{2}},\end{eqnarray} \tag{ 13 }$$

where ρ ≥ 0. The residual gives the difference between our measured f_L and estimated f_b; in other words, a value of ρ closer to zero indicates there is less spread of data about the line f_b = f_L. In our calculations of R and ρ, we take the logarithms of both f_b and f_L. We place more weight here on the statistical significance of ρ over R since a linear relationship does not necessarily imply that f_b = f_L. The correlation coefficients are also intrinsically linked because of the definition of 1/k_c and the fact that ρ_i and d_i differ only by a factor of $\sqrt{{\beta }_{i,\perp }}$. As a final test, we count the number of spectra that lie between the two red dashed lines in each panel of Figure 3 to determine a percentage of the total number of spectra that satisfy f_b ≃ f_L, within the e-folding frequency. For the total number of spectra, we do not include instances where there are large data gaps, and where we have discarded values for f_b. For instances where we filter data according to slow or fast wind, the total number of spectra we use is only for that filtered data set. We give the results of our statistical analysis in Table 1.

For all 2012 data, regardless of wind speed, both f_kc and f_di have moderate correlations with f_b, with values of R = 0.56 and R = 0.52, respectively, whereas the correlation for f_ρi is weaker at R = 0.34. The lowest residual is ρ = 0.17 for f_kc, while for f_di it is ρ = 0.25 and for f_ρi it is even higher at ρ = 0.43. These values show that the cyclotron resonance scale, 1/k_c, is most closely associated with the spectral break (i.e., closest to f_b ≃ f_L) during the interval we study. This finding is supported by 52.08% of the total number of spectra in our data set falling between the two red dashed lines for f_kc in Figure 3(a). In contrast, we find only 30.19% for f_di in panel (b) and 9.14% for f_ρi in panel (c). The gyroscale, ρ_i, therefore has a poor relationship with the break frequency, suggesting that it is overall least likely to be associated with the spectral steepening during the interval we study.

These findings hold when we separate the data according to wind speed. For slow wind, f_kc has the highest correlation coefficient at R = 0.54 and lowest residual at ρ = 0.12. During periods of fast wind streams, the residual for f_kc is about half that of slow wind at ρ = 0.06, the smallest for all three scales. From panels (d) and (g) in Figure 3, we are unable to visually differentiate between the cases of fast and slow wind without statistical analysis. Also, comparing panels (f) and (g), we find that 57.01% of spectra fall within the resolution limit of our wavelet transform for f_kc in fast wind, which is higher than the 49.42% for slow wind. The correlation coefficients for both f_kc and f_di in fast wind are equal, at R = 0.58, which is only slightly larger than their slow wind values. Transitioning from slow to fast wind in panels (e) and (h), the percentage of spectra where f_b ≃ f_di increases from 27.74% to 36.08%. From these values, we see that the relationship between f_kc and f_b is better than for f_di and maintained even when the large-scale stream structure of the wind varies, although it is strongest in fast wind streams.

According to Equations (3) and (4), 1/k_c will coincide with the larger of the two scales, d_i or ρ_i, when β_i,⊥ ≪ 1 or β_i,⊥ ≫ 1, respectively, assuming an isotropic temperature (i.e., ρ_i ≃ σ_i). This expectation is consistent with observations by Chen et al. (2014) showing that the spectral break occurs at d_i for β_i,⊥ ≪ 1 and at ρ_i for β_i,⊥ ≫ 1, which they note is consistent with a break at 1/k_c in both cases. However, by definition, when β_i,⊥ ∼ 1, ρ_i ≃ d_i and therefore 1/k_c ≃ρ_i + d_i ≃ 2ρ_i ≃ 2d_i. Indeed, for periods with β_i,⊥ ∼ 1 we see in Figures 2(d)–(f) that f_di and f_ρi coincide and f_kc is shifted to lower frequencies by about a factor of 2. During these periods, there is a good agreement between f_kc and f_b. To address what happens when β_i,⊥ ∼ 1 quantitatively and clearly show the difference between 1/k_c and d_i or ρ_i, we filter our year of data to include only periods where 0.95 ≤ β_i,⊥ ≤ 1.05 and show the corresponding 2D histograms in Figure 4. In addition, the results from our statistical analysis are shown in the bottom panel of Table 1. We note that we do not remove bins with ≤10 spectra here because of the smaller amount of data available for these periods, but this only affects bins furthest from the black dashed line.

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Comparing panels (b) and (c) to (a) in Figure 4, we see that our measured f_b is consistently shifted to frequencies lower than f_di and f_ρi, i.e., the yellow enhancement in panels (b) and (c) is below the black dashed line, but in panel (a) we see that it is closer to the dashed line. These plots show that 1/k_c is a more likely candidate for the break scale than d_i or ρ_i, and we quantify this result by calculating R and ρ for this data set. The correlation coefficients are the same for f_di and f_ρi at R = 0.61, and almost the same at R = 0.60 for f_kc; however, the latter has the lowest residual at ρ = 0.02, compared to ρ = 0.04 for f_di and f_ρi. We note that the statistics for f_ρi improve considerably when considering only periods of β_i,⊥ ∼ 1, and are the same in this case for f_di.  Again, we find a high percentage of the number of spectra where f_b ≃ f_kc at 51.81%, almost double that for the other two scales.

When we consider all data in our interval, the results from our statistical analysis for f_kc and f_di do not differ significantly, particularly in fast wind, as we see from similar values for R and ρ in Table 1. From our analysis of periods where β_i,⊥ ∼ 1, we explain this result as being due to the ratio of spectra where β_i,⊥ < 1 to those where β_i,⊥ > 1, which is almost 8 in our data set. Finally, we conclude that f_b is best associated with f_kc and so the spectral break is most likely related to proton cyclotron resonance. We then explain the correlations with d_i and ρ_i as being due to the dependence of 1/k_c on both variables, and the fact that d_i and ρ_i are separated only by a factor of $\sqrt{{\beta }_{i,\perp }}$.

To further explore the possible role of proton cyclotron resonance at ion-kinetic frequencies, we now investigate the nature of the fluctuations at these frequencies. We calculate helicity spectra from successive periods of 92 s using the normalized magnetic helicity, σ_m, from Equation (6). Again, we use Taylor's hypothesis to obtain σ_m as a function of frequency instead of wavenumber, giving us one helicity spectrum for each corresponding power spectrum from the previous section. To quantify the relationship between 1/k_c and the coherent helicity signature at high frequencies, we devise a method to calculate the onset frequency, f_h, of the helicity signature, defined as the threshold frequency at which we see an enhancement in the helicity at ion-kinetic scales. We first fit a Gaussian function to the helicity spectra,

$$\begin{eqnarray}&&{\sigma }_{m}=\displaystyle \frac{1}{\sqrt{2\pi }{\sigma }_{D}}\exp \left\{-\displaystyle \frac{{(f-{f}_{p})}^{2}}{2{\sigma }_{D}^{2}}\right\},\end{eqnarray} \tag{ 14 }$$

where the fitting parameters are the standard deviation, σ_D, and the mean, f_p, which corresponds to the frequency of the peak in the helicity signature. We perform the Gaussian fitting in linear space so that the method is biased toward the peak in helicity at the highest frequencies, i.e., the coherent helicity signature. We show in the top panel of Figure 5 the example power spectrum from Figure 1, along with its corresponding helicity spectrum in the bottom panel, both in black. In the bottom panel, we also plot in red the Gaussian fit to the helicity spectrum using Equation (14). The red dashed line gives f_p from the fitting, whereas the black and gray dashed lines are f_b and f_noise from before, respectively. To estimate the onset frequency f_h, we calculate the FWHM of the Gaussian peak using ${\rm{\Delta }}f\,={\sigma }_{D}\sqrt{8\mathrm{ln}(2)}$ and then

$$\begin{eqnarray}&&{f}_{h}={f}_{p}-{\rm{\Delta }}f/2.\end{eqnarray} \tag{ 15 }$$

The minus sign is used to determine the onset frequency bounded toward lower frequencies. This method is independent of whether the peak in helicity is negative or positive and allows for an automated process estimating both f_h and f_p for ∼344,000 helicity spectra. In Figure 5, f_h is given by the blue dashed line. We see that f_b and f_h are separated by only about 0.1 Hz. From the results of Section 2.2, this result implies that f_h may also be associated with f_kc and suggests that the presence of the helicity signature is related to cyclotron resonance. We further investigate this relationship between f_b and f_h using our statistical analysis from the previous section. We also follow up the work of Markovskii et al. (2015) to confirm a relationship between f_p and f_ρi using our data set.

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We first use our method described above to estimate both f_h and f_p for 2012 July in order to check that it accurately reproduces the features of the helicity spectra. In Figures 6(a) and (b) we show again time series of ${\boldsymbol{B}}$ and v_sw for 2012 July. In addition, panels (c) and (d) show contour plots of σ_m for consecutive 92 s spectra over the course of 2012 July, where we normalize the frequency of each spectrum by 1/f_kc in panel (c), and similarly by 1/f_ρi in panel (d). We also plot the lines f_kc = 1 and f_ρi = 1 for reference in red, as well as our estimated f_h and f_p in black, in panels (c) and (d), respectively. We normalize the spectra here only for the figure, and not in any future analysis. From panel (c), we can see that f_h bounds at lower frequencies the enhanced red and blue signature as we expect. Also, from panel (d) we see that the peak of the helicity signature is located close to the middle of the helicity signature before the enhancement disappears completely (it should not be located directly in the middle due to the logarithmic scale in frequency). Therefore, we conclude that our method works and quantifies the onset and peak of the helicity signature accurately. We will now discuss our findings from Figure 6 in more detail.

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The persistent band of enhancement in σ_m at higher frequencies varies between about −0.4 and +0.2. Figure 6(a) suggests that the sectoral structure of the solar wind is likely responsible for this changing sign in helicity over the course of the month, which is consistent also with the findings of He et al. (2011). From Figure 6(c), we can see that the positive helicity signal is typically weaker in amplitude than the negative signal by a factor of 2. We currently have no explanation for this finding, but it is an interesting observation that should be explored in another study. We see that when an enhancement in helicity signature is present at high frequencies, f_h is well correlated with f_kc. By comparing panels (b) and (c), we find that the helicity enhancement weakens or almost completely disappears during periods of slow solar wind. Both these results show that the findings of Bruno & Telloni (2015) and Telloni et al. (2015) apply to large volumes of the solar wind. Also, from Figure 6(d), the peak of this coherent helicity signature is correlated with f_ρi, especially in fast wind where the helicity signature is strongest, which is consistent with Markovskii et al. (2015) and Telloni et al. (2015). While we have not shown a similar plot for f_di = 1, we find that it is not closely associated with either f_h or f_p, and confirm this in our subsequent analysis.

At lower frequencies than f_kc, the helicity fluctuates about zero, as expected for the inertial range of solar wind turbulence (Matthaeus & Goldstein 1982a), showing either a lack of or no dominant coherent circular polarization of fluctuations. There is an enhanced signature in the helicity that significantly deviates from the characteristic plasma scales between July 12 and July 16, peaking at around 0.1 Hz (from Figure 6(c), about 0.5 in normalized frequency units). We associate this signature with AICs produced by instabilities from unstable particle distributions. These waves are often Doppler-shifted toward lower frequencies than the spectral break since they typically propagate toward the Sun, in the opposite direction to the turbulent magnetic fluctuations we consider here (e.g., see Tsurutani et al. 1994; Jian et al. 2009, 2010, 2014; Gary 2015; Roberts & Li 2015; Roberts et al. 2015; Wicks et al. 2016). To exclude these events from our analysis, we discard data with f_h ≤ 0.2 Hz and f_p ≤ 0.2 Hz.

At frequencies f > f_p, the enhancement disappears, and the helicity returns to a value close to zero. If the trace of the power spectral tensor is increased artificially by instrumental noise, then the helicity will also reduce to zero, by definition from Equation (6). In fact, the increasing contribution from noise should not affect the phase contribution to the helicity, but rather just its amplitude. Therefore, despite seeing a return to zero, we do not see the return of the signature to an incoherent one similar to that at low frequencies seen in Figure 6, where σ_m oscillates in color between red and blue for opposite polarizations. Instead, the signal remains coherent with a value close to zero for f > f_p. Therefore, we cannot determine whether this effect is physical or due to the MFI noise-floor. An alternative explanation is aliasing (see the Appendix). Despite this, we find that typically f_p < f_noise and therefore we take the peak in the helicity signature, and hence f_p, as physical.

We now extend our analysis to include an entire year of data from 2012 in the same way as in Section 2.2. In Figure 7 we show histograms in the same format as Figure 3 for f_L against f_h for all data and then separated into periods of slow and fast wind in panels (a)–(c), (d)–(f), and (g)–(i), respectively. In Figure 8 we show, in a similar fashion to Figure 4, f_L against f_h for periods where β_i,⊥ ∼ 1. Finally, in Figure 9 we plot histograms for f_L against f_p. Here, we discard data with f_h ≥ f_noise and f_p ≥ f_noise to ensure that instrumental noise does not affect our results, and data where f_h ≤ 0.2 and f_p ≤ 0.2 Hz, as discussed previously. We provide the results from our statistical analysis for f_L and f_h in Table 2 and for f_L and f_p in Table 3.

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We find that f_ρi has the lowest correlations and highest residuals with f_h regardless of wind speed, which is consistent with Figure 7, where the distribution of data deviates significantly from the black dashed line. We conclude that f_ρi is not directly comparable to f_h within our studied interval. When we consider all data, f_kc has the highest correlation coefficient of R = 0.48 and lowest residual of ρ = 0.09, compared to R = 0.40 and ρ = 0.15 for f_di. We find that 64.29% of the total number of spectra fall between the two red dashed lines for f_kc in Figure 7(a), compared to 33.37% for f_di in panel (b). These percentages are similar regardless of wind speed. Besides comparable correlation coefficients of about R = 0.42–0.45 in fast wind streams, f_kc is closer to the relationship f_L ≃ f_h than f_di, from visual comparison of panels (g) and (h). In particular, we see that f_kc has the lowest residual of ρ = 0.04 during fast wind streams, compared to ρ = 0.06 in slow wind. Comparing panels (d) and (g), the percentage of spectra between the red dashed lines for f_kc is lower in the fast wind at 61.38% than in the slow wind where it is 64.72%, despite a lower residual in the former.

As in the previous section, we also filter the data to include only periods where 0.95 ≤ β_i,⊥ ≤ 1.05 and show the corresponding 2D histograms in Figure 8, as well as the results from our statistical analysis in the bottom panel of Table 2. Our results are similar to those for f_b and f_L, where we see clearly that f_kc best corresponds to f_h. From our statistical analysis, the correlation coefficients are the same for both f_di and f_ρi at R = 0.47, and almost the same at R = 0.46 for f_kc; however, the latter again has the lowest residual at ρ = 0.01, compared to ρ = 0.03 for the other two scales. Again, we find a high percentage of the number of spectra where f_b ≃ f_kc at 62.29%, almost triple that of the other two scales.

Following Section 2.2, we conclude that the onset of the helicity signature is also related to the cyclotron resonant scale, and therefore both the spectral steepening and coherent helicity signature are likely linked to the same physical process: proton cyclotron resonance. This signature is most prevalent when the spacecraft measures fast wind streams, and therefore we conclude that there is a stronger relationship between f_kc and f_h during these periods, as we also see for f_kc and f_b. The lower percentage for f_kc in fast wind is likely due to the smaller number of measurements available than for slow wind periods, as we see in plots in the right column of Figure 7, or because of the limited applicability of wind speed as the only criterion to categorize wind streams.

Moving now to the peak frequency of the helicity signature and our results presented in Figure 9 and Table 3, we find that f_kc and f_di have similar correlation coefficients with f_p of R = 0.49–0.56, regardless of wind speed. We can also see little difference when comparing visually the three columns in the figure for these two scales. However, the lowest residuals are seen for f_di, giving ρ = 0.07 and ρ = 0.05 during slow and fast wind streams, respectively. We find that f_ρi has lower correlation coefficients at R = 0.28–0.41 than f_kc and f_di. However, its residuals are comparable to that of f_di, at ρ = 0.06 for fast wind and ρ = 0.11 for slow wind. Figure 9 shows that f_p correlates with both f_di and f_ρi, as expected since they differ only by a factor of $\sqrt{{\beta }_{i,\perp }}$, but there is a constant offset in frequency for f_di that is not present for f_ρi. When we consider all data, 46.62% of spectra satisfy f_p ≃ f_ρi compared to 44.79% for f_p ≃ f_di, within the e-folding frequency.

For frequencies f_ρi > 1 Hz, the most likely value for f_p diverges from the black dashed line in panel (c) of Figure 9, which results in the low values for R with f_ρi, and higher correlation with f_di. However, we find that 39.82% of spectra in slow wind and 51.74% in fast wind satisfy f_p ≃ f_ρi within the e-folding frequency. The higher percentage in fast wind is likely due to the presence of a stronger helicity signature than for slow wind, making detection easier. The divergence at high frequencies may be caused by undersampling, because f_p can exceed f_noise at these frequencies, but we try to account for this by discarding bins with ≤10 spectra. We also see a similar feature in panel (i) of Figure 7 as in panel (i) in Figure 3.  Due to the noise-floor, it is difficult to distinguish whether this feature is physical or an artifact. Despite this divergence at high frequencies, we conclude that ρ_i better corresponds with the peak in the helicity signature at ion-kinetic frequencies than the other two scales.