SPECTRAL PROPERTIES OF LARGE GRADUAL SOLAR ENERGETIC PARTICLE EVENTS. II. SYSTEMATIC Q/M DEPENDENCE OF HEAVY ION SPECTRAL BREAKS

In Figure 3(a) we show statistical properties of the SEP spectra by plotting histograms of the SEP-band parameters γ_a (red) and γ_b (blue) for all species in all 46 SEP events. Only those values with relative uncertainties <100% and finite values for γ_a and γ_b are included; 398 spectra were fitted. Consistent with the statistical properties of the O-band parameters in Paper 1, we note that γ_a for all species has a mean μ ∼1.23 and median m ∼1.19, which are both significantly smaller compared to the corresponding values for the high-energy Band parameter γ_b: μ ∼3.63 and m ∼3.57. We also note that γ_a has a 1σ standard deviation of ∼0.58 and varies over a smaller range of values between ∼0–3.5. In contrast, γ_b has a 1σ value of ∼1.12 and varies between ∼0.7–9. In summary, based on 398 fitted spectra, the average power-law spectral slope changes by ∼2.4 units from an average value of ∼1.23 at low energies to ∼3.63 above the break.

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Figure 3(b) shows a scatter plot of γ_b versus γ_a obtained for each individual species in all SEP events. As seen for the O spectral slopes in Paper 1, we note that most SEP spectra are flatter at lower energies and steepen above the break energy. The energy spectra in 10 cases flatten at higher energies (see Table 2); these are for H: events 30 and 40; He: event 44; O: event 31; Si: events 2 and 34; Ca: events 2 and 32; and Fe: event 34 and 45. Event 31 was discussed in Paper 1. Since γ_b < γ_a occurs for different species in different events, including these outliers in our analyses does not affect the overall results and conclusions of this paper. Note the many cases where γ_a is small in Figure 3(b); this is due to the previously mentioned coupling with E_B and is not due to low-energy spectral slopes γ₁ being close to 0.

Figure 4 investigates the relationships and differences between the SEP Band parameters for different species: (a) γ_a for H and Fe versus γ_a of O; (b) γ_b for H and Fe versus γ_b of O; (c) γ_b–γ_a for H and Fe versus γ_b–γ_a of O; and (d) E_B for H and Fe versus E_B of O. In general, the SEP Band parameters for H and Fe track those of O reasonably well over the corresponding range of values. We note the following. (1) In many events, γ_a for H and Fe are different compared to that of O; the differences between H and O are somewhat larger. (2) γ_b and γ_b–γ_a for H and Fe show tighter correlations with the corresponding values for O. (3) E_B for H and Fe in most SEP events show significant differences compared to corresponding values for O; the H E_B in 35 out of 40 events (∼87%) is larger than that of O, while the Fe E_B in 32 out of 40 events (∼80%) is smaller than that of O (also see Desai et al. 2016).

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Figure 5 examines the event-to-event variations of the SEP spectral parameters for all species: (a) γ₁; (b) γ_b; and (c) E_B. Red dots show the values for each species in each event, the solid black curves in (a) and (b) show the mean value for γ₁ and γ_b, respectively, and the solid black curve in (c) shows the proton spectral break energy E_H for each event. Dashed lines show the species-averaged mean values obtained by averaging for all species in all 46 events; yellow shaded regions depict the 1σ standard deviation (also see Figure 3(a)). γ₁ for events 20 and 34, and γ_b and E_B for event 20 are not plotted (see Section 3.5). We note the following: (1) γ₁ has an event-averaged mean of ∼1.64 ± 0.03, with 1σ standard deviation of ∼0.6; large deviations of ≫1σ from the event-averaged mean value are seen in events 12, 28, 32, 43 and 45, where γ₁ ≥ 2.5 for most species; (2) in most events, γ_b < 6 for most species; exceptions are events 17 and 18, where γ_b > 6 for most species; and (3) In most events, the H E_B is greater than those for the heavier species in the same event. Note that in events 30 and 40, H E_B is greater than ∼100 MeV. Overall, within an event, the proton spectral parameters γ₁ and γ_b do not stand out from the corresponding heavy ion spectral parameters but rather lie in the midst of the others; in contrast, the proton spectral break energy is almost always greater than the spectral break energies of the heavier ions.

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We now investigate the species-associated variations in the (a) low-energy spectral slope γ₁ and Band parameters γ_a and (b) γ_b within an event by plotting the distributions of their corresponding mean deviations in Figure 6. The mean deviation for each parameter in each event is calculated from the data shown in Figure 5. Figure 6(c) shows the actual distribution of the spectral break energy E_B for all species in all SEP events compared to that of the transition energy E_T. The following features are striking. (1) The mean deviations of γ₁, γ_a, and γ_b have narrow distributions that result in well-behaved Gaussian-like distributions (black curves) with small 1σ values. (2) Typically, E_T > E_B, with the mean of E_T almost a factor of 2 greater than that of E_B. More importantly, E_B and E_T vary over more than three orders of magnitude, which results in broad distributions with large 1σ standard deviations. These results indicate that: (1) within a given SEP event, the three parameters γ₁, γ_a, and γ_b, have remarkably similar values for all species, that is, for each event, both the low-energy and high-energy spectral slopes are similar within ∼10%–15%; (2) for a given event, most of the species-associated, spectral variations are driven by differences in the break energy or the transition energy E_T at which the spectra steepen; and (3) examples of such species-associated differences in E_B are shown in Figures 1(b), (d), and (f).

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To investigate the species-dependence of E_B within an SEP event, as seen in Figure 1(b), 1(d), 1(f), 4(d), and 6(c), as well as the event-to-event variations in E_B shown in Figure 5(c), we examine the E_X/E_H versus the ion's Q/M ratio for 38 SEP events in Figure 7. This figure, along with the three cases shown in Figure 1, demonstrates that in most of the events the break energies are well ordered by Q/M ratio. In the five events not shown in this paper, either the ULEIS and SIS spectra for many species did not match near overlapping energies (event 20; also see Section 3.3), or only the proton spectra showed a spectral break (events 29, 30 and 41), or the spectra for most species are well-described by a single power-law across the combined ULEIS and SIS energy range (event 34).

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As in Figure 1, we obtain the power-law exponent α for each event; values of α are provided in Table 2 and in each panel. In four events, 4, 9, 19 and 31, we found that the relative uncertainties in α were >100%. In four other events, 2, 3, 7, and 45, α < 0. From Table 2, we note that in event 7, the Fe E_B ∼81 MeV nucleon⁻¹ compared with ∼11 MeV nucleon⁻¹ for both H and O (and for many other species), which results in the Fe E_B being an outlier in Figure 7, and thus yielding α < 0. For events 2, 3, and 45 we only used the >0.2 MeV nucleon⁻¹ He–Fe spectra (>0.5 MeV nucleon⁻¹ for event 45), because below these energies, the ULEIS spectra for most species exhibited an upturn or downturn such that the Band function could not be fitted to the entire spectrum. When we force-fitted the spectra by including these lower energy data, the resulting Band fits were poor and the Q_X/M_X versus E_X/E_H plots were similar to that shown for event 31, with relative uncertainties in α > 100%. We therefore exclude these eight outlier events along with the five events not shown in Figure 7 from Table 2 and the subsequent discussion concerning α.

The figure clearly shows that the fit (solid red line) and the slope α well characterizes the systematic Q/M-dependence of the heavy ion spectral break energies in most of the remaining 33 SEP events in our survey. Other noteworthy features are: (1) in these 33 SEP events, including the three cases shown in Figure 1, α > 0.2 (dashed black line); and (2) in events 16 (shown in Figure 1(f)), 25, and 32, α > 2 (dotted black line, also see Section 4.1). Note that in event 32, the values of E_X/E_H for Ca and Fe are significantly larger than the fit, which yields α ∼2.16. Likewise, larger values of E_X/E_H for Fe and/or Ca are also seen for events 21, 28, 37, and 42, resulting in large deviations from the fits. Such deviations are probably affected by our assumption of the average SEP charge states for these species. For instance, if both the Ca and Fe were highly ionized in these events, as has been observed in some large SEP events with significant enhancements of Fe/O and Ca/O (e.g., Klecker et al. 2007), then their corresponding Q/M values would be larger. This would shift the data points to the right and closer to the fitted line. Indeed, Table 3 shows that four out of these five events have enrichments in the ³He/⁴He ratio and in the Fe/O ratio when compared to the corresponding abundances measured in the solar wind, indicating the presence of impulsive event suparthermals in the seed population (see Desai et al. 2006).

We now investigate the properties of α and its relationship with the solar source properties given in Table 1 of Paper 1. Figure 8(a) shows the histogram of α along with the mean, standard deviation, and median value of the distribution, while Figures 8(b), (c), and (d) plot α versus flare longitude, CME speed, and the peak proton flux (PFU) obtained by NOAA GOES/EPS, respectively. The peak proton flux is the integral flux measured by GOES/EPS for energies >10 MeV and has units of cm⁻² sr⁻¹ s⁻¹. The flare longitude and CME speed for events 1–29 are obtained from http://umbra.nascom.nasa.gov/SEP/, ftp://ftp.ngdc.noaa.gov/STP/swpc_products/daily_reports/solar_event_reports/, Cane et al. (2010), and Kahler (2013). CME and flare properties for events 30–46 are from Richardson et al. (2014), cdaw.gsfc.nasa.gov/CME_list/UNIVERSAL/, and Ding et al. (2013).

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Figure 8 shows the following: (1) α has a mean value of 1.27, median value of 1.16, and is confined between ∼0.2–3, with values for 3 SEP events greater than 2; two of these events were also accompanied by GLEs (see Table 2, Figures 1(f) and 7); (2) α exhibits no clear trend with the flare longitude but SEP events with α > 2 are associated with source longitudes west of W45; this is probably a result of the selection criteria, which are biased toward western hemisphere events; and (3) α exhibits statistically significant, positive trends with the peak proton flux and CME speed, with values for correlation coefficients r ∼0.48 and r ∼0.41, which have probabilities p < 1% and p < 2%, respectively, of being exceeded by uncorrelated pairs of parameters. It is evident from the figure that the correlations with CME speed and peak proton flux are largely due to the presence of events associated with higher CME speeds and/or GLEs (see Section 5.3).

Using the α values for the 33 events provided in Table 2, Figure 9 examines the relationships between α and (a) the ∼0.16–0.23 MeV nucleon⁻¹ Fe/O, (b) the ∼15–21 MeV nucleon⁻¹ Fe/O, and (c) the ∼0.5–2.0 MeV nucleon^{−1 3}He/⁴He ratios for those SEP events for which these key heavy ion abundances are finite (taken from Table 2 in Paper 1). The ∼0.16–0.23 MeV nucleon⁻¹ Fe/O exhibits a statistically significant, positive trend with α; correlation coefficient r ∼0.39 for 33 events has <2.5% chance of being exceeded by an uncorrelated pair of parameters. In contrast, α is not well correlated either with the ∼15–21 MeV nucleon⁻¹ Fe/O or the ∼0.5–2.0 MeV nucleon^{−1 3}He/⁴He ratios (see Section 5.3 for a discussion of these abundances in the extreme SEP events).

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Figure 10(a) investigates the relationship between the O SEP Band parameters γ_a and γ_b and the power-law exponent α, while Figure 10(b) plots the difference, γ_b–γ_a, versus α. We note the following: (1) α is not correlated with γ_a, γ_b, or γ_b–γ_a and (2) SEP events with α between ∼0.6–1.4 have a larger range of values for γ_b and γ_b–γ_a, i.e., events for which the spectra steepen significantly at higher energies occur for α values between ∼0.6–1.4.

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Figure 11 examines the relationship between the O break energy E_B and (a) the difference between the spectral slopes γ_b–γ_a and (b) α. Overall, the O E_B is not correlated with γ_b–γ_a or α. However, the O E_B in SEP events with γ_b–γ_a < 3 and γ_b–γ_a > 3 appear to exhibit negative and positive trends, respectively.

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While there is a general consensus that large gradual SEP events and their observed properties at 1 au result from DSA at CME-driven coronal or interplanetary shocks followed by transport through the interplanetary medium (e.g., Desai & Giacalone 2016), Cane et al. (2006) have alternatively proposed that the initial increase and subsequent decrease in the ∼25–80 MeV nucleon⁻¹ Fe/O time profiles in large SEP events is due to direct contributions of Fe-rich material accelerated in concomitant flares during the onset phase followed by contributions of material accelerated at CME-driven interplanetary shocks. Cane et al. (2006) also proposed that large SEP events in which the event-averaged ∼25–80 MeV nucleon⁻¹ Fe/O is enhanced by more than a factor of ∼2 compared with the average SEP value of 0.134 likely comprise significant direct contributions from flares.

We first note that enhanced Fe/O ratios in SEP events reflecting the presence of impulsive material are also typically associated with high charge states of Fe (e.g., Klecker et al. 2007), and so the suggestion that high Fe/O values in the early phases of SEP events are due to the presence of such material predicts that the Fe ionization states during these periods should also be high; however, this is not observed (e.g., Guo et al. 2014). Next, we point out that the increases in the Fe/O ratio observed during the early phases of large SEP events has been successfully and quantitatively modeled in many studies as a transport effect (e.g., Tylka et al. 1999, 2005, 2012; Mason et al. 2012, 2014; Reames et al. 2013), and that in particular, the following three observational results in conjunction with the new results presented in this paper provide strong support for the counterargument that ∼0.1–500 MeV nucleon⁻¹ H–Fe nuclei observed during large gradual SEPs are indeed accelerated by near-Sun CME-driven shocks. Thus, the bulk of the observations show that the suggestion that hypothesized "flare" contributions dominate the early phases of SEP event composition or the event-integrated SEP Fe/O abundances is not supported.

• 1.  
  First, the Cane et al. (2006) scenario makes no specific predictions about the magnitude of ³He enrichments below ∼2 MeV nucleon⁻¹ in large SEP events. Previously, we found that the ∼0.5–2 MeV nucleon^{−1 3}He/⁴He ratio is enhanced between factors of ∼2–150 in ∼46% of large SEP events (Desai et al. 2006), and in Figure 9 we show that the ∼0.5–2 MeV nucleon^{−1 3}He/⁴He ratio is enhanced between factors of ∼2–100 in ∼58% of the events. The magnitudes of these ³He-enrichments are substantially smaller than those found in the flare-related, impulsive SEP events (e.g., Mason et al. 2004) but are similar to those found in local interplanetary shock-associated events (Desai et al. 2003; Allegrini et al. 2008) and in particle events associated with corotating interaction regions or CIRs (Mason et al. 2008). This supports the notion that CME- and CIR-shocks routinely re-accelerate pre-existing suprathermal material already enriched in ³He from prior flare activity (Mason et al. 1999, 2005).
• 2.  
  Second, many studies (e.g., Tylka et al. 1999; Mason et al. 2006, 2012; Reames 2015) have shown that temporal variations in the <100 MeV nucleon⁻¹ Fe/O abundances in large SEP events are diminished or even eliminated if the Fe and O time-intensity profiles are compared at a scaled energy for the different species, indicating that these variations are due to differences in the Q/M ratios of the different species, and therefore better explained by interplanetary transport models that include the effects of focusing, diffusion, convection, adiabatic deceleration, and pitch-angle scattering.
• 3.  
  Finally, Desai et al. (2006) surveyed the ∼0.1–60 MeV nucleon⁻¹ event-integrated heavy ion abundances in 64 large SEP events, which included 61 of the 97 events studied by Cane et al. (2006). Twenty-seven of the events studied by Desai et al. (2006) and 21 of the first 30 events studied here were classified by Cane et al. (2006) as prompt and/or Fe-rich events, and therefore presumed to comprise significant direct flare contributions above ∼25 MeV nucleon⁻¹. However, Desai et al. (2006) found that, regardless of the absolute values of the ∼0.11–0.14 and the ∼12–60 MeV nucleon⁻¹ Fe/O ratios (which were indeed enhanced over the corresponding average solar wind value in 55 and 23 of the 64 events, respectively), the Fe/O ratio either decreased or remained constant with increasing energy in all but three events (see Figure 5(b) in Desai et al. 2006). This earlier result pointed to the significant influence of Q/M-dependent CME shock acceleration mechanisms at least up to ∼60 MeV nucleon⁻¹ in the vast majority of the large SEP events classified by Cane et al. (2006) as being flare-dominated. In fact, this energy-dependence of the Fe/O ratio is fully consistent with the decrease in the spectral break energy with decreasing Q/M ratio, as observed in the case of 33 events studied here, consistent with the dominance of the effects of CME shock acceleration mechanisms up to the highest energies surveyed in this paper, i.e., up to ∼500 MeV nucleon⁻¹ for H and up to ∼170 MeV nucleon⁻¹ for He–Fe.

Comparing our survey to prior studies, we note that some of the five events studied by Mewaldt et al. (2005a) and Cohen et al. (2005) also included the local shock-accelerated ESP component that accompanied the larger SEP event. In the present study, we eliminated all events with possible contributions from local IP shock-associated populations (see Paper 1) in order to highlight SEP acceleration processes. Further, we use event-integrated fluences, rather than time-intensity profiles (see Mason et al. 2012), to study the SEP spectral properties. In particular, Mason et al. (2012) used a detailed model of interplanetary propagation and showed that transport from the inner solar system can lower the break energy systematically for all species as well as lower the slopes by ∼10%–20% but that the basic spectral form remained intact (their Figure 14). Alternatively, we note that Li & Lee (2015) fitted the double power-law proton spectra in 9 of the 16 GLE events studied by Mewaldt et al. (2005a) with an analytical model that included interplanetary transport effects, and found that single power-law spectra injected by CME shocks near the Sun can exhibit spectral breaks at 1 au due to scatter-dominated transport through the interplanetary medium. However, this model predicts that α for the GLE-associated SEP events should lie in the range ∼0.18–0.75, which is clearly inconsistent with the α > 1.58 observed in five of the seven GLEs in our survey (see Table 2). On this basis, we contend that the formation of the double power-law SEP spectral forms, their associated properties, and the observed Q/M-dependence of E_B primarily reflect conditions near the distant CME-driven shocks where the acceleration takes place, and are not significantly affected by contributions from local interplanetary shock-accelerated populations nor by Q/M-dependent transport and scattering in the interplanetary turbulence en route to 1 au (e.g., see Zank et al. 2000; Cohen et al. 2005; Mason et al. 2012).

A fundamental prediction of early one-dimensional (1D) steady state as well as the more recent time-dependent DSA-based SEP models is that, in a given event, the differential energy spectrum of the accelerated particles below the break energy is characterized by a low-energy power-law spectral slope γ given by dj/dE ∝ E^−γ (e.g., Drury 1983; Lee 2005; Schwadron et al. 2015b). These models also predict that γ is independent of ion species and determined solely by γ ≈ (H + 2)/(2H − 2), where H is the strength of the CME-driven shock. Our results show that both the SEP Band parameter γ_a and the low-energy spectral slope γ₁ in a given SEP event are remarkably similar for all species, and that such species-independent spectral slopes are observed at both low and high energies for most of the events in our survey (Results 1 and 2). We therefore suggest that, to first order, the formation of double power-law spectra in large SEP events is consistent with DSA at near-Sun CME shocks (e.g., Schwadron et al. 2015b).

We now use the DSA-predicted relationship between γ (here we use the species-averaged γ₁ for each event) and H to infer compression ratios of the near-Sun CME shocks. Figure 12 compares these inferred values to three key properties of CMEs and SEPs: (a) CME speed from SOHO/LASCO, (b) peak proton flux from GOES/EPS, and (c) α from Table 2. The main features of this figure are: (1) the inferred shock compression ratios for the 37 events shown here lie between ∼1–5.5, with H > 4 for 4 events. Note that all cases of H > 4 have sizeable uncertainties. Based on our low-energy spectral slopes, we find that the average inferred shock compression ratio for 33 near-Sun CME shocks is 2.49 ± 0.08. These inferred values are remarkably consistent with the predicted range of values for CME shock compression ratios (see Schwadron et al. 2015b), and are well within the constraints of the Rankine–Hugoniot discontinuity conditions for the allowable range of ∼1–4, with an upper limit of <4 for shocks in non-relativistic space plasmas (e.g., Viñas & Scudder 1986). (2) The compression ratio H exhibits weak but positive correlations with all three parameters: (a) CME speed: for 37 events, r ∼0.38 has <2% chance; (b) peak proton flux: for 29 events, r ∼0.35 has <6% chance; and (c) α: for 31 events, excluding events with H > 4, r ∼0.47 has <0.57% chance of being exceeded by uncorrelated pairs of parameters.

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The heavy ion fluence spectra in most SEP events have an average value of ∼1.23 at energies below ∼1 MeV nucleon⁻¹ and steepen by ∼2.4 units above a roll-over or break energy to an average value of ∼3.63; the break energy decreases systematically with the ion's Q/M ratio (results 3 and 4). The Q/M-dependence of E_B in a given SEP event is well represented by the function $\mathrm{log}({E}_{X}/{E}_{H})=\mathrm{log}({n}_{0})+\alpha \,\mathrm{log}({Q}_{X}/{M}_{X})$ and characterized by the power-law exponent α (Result 4). The values of α for 33 SEP events lie in the range ∼0.2–3 (result 5), which encompasses the range of α values reported by Mewaldt et al. (2005a) and Cohen et al. (2005). Thus, with the exception of three events with α > 2 (see Figures 1(f) and 7, and Section 5.3), the range of values for α in our survey is consistent with the corresponding range of ∼0.2–2 predicted by Li et al. (2009). In this model, the Q/M-dependence of the spectral break energies in a given SEP event occurs due to the "equal diffusion coefficient" or the "equal acceleration time" condition, and the event-to-event variations in the power-law exponent α are driven by differences in the slopes of the turbulence spectra present near shocks with different obliquity.

Assuming that the spectral break energies for different species in a given SEP event occur at the same value of the diffusion coefficient ${\kappa }_{//}$ ⁸ , which scales as (M/Q)^a with observed values of a between ∼0.8–2.7, Cohen et al. (2005) followed Dröge (1994) and inferred that the power-law index η of the turbulence or wave intensity spectrum, given by $I(k)\propto {k}^{-\eta }$, near the CME shock acceleration region ranged between 1.2 to −0.7. Here $\eta =2-a$, and a is related to the exponent α in our survey by $a=\alpha /(2-\alpha )$. We now follow the approach of Mewaldt et al. (2005a) and Cohen et al. (2005) to infer the power-law exponent a, which determines the scaling between the particle diffusion coefficient and the ion's M/Q ratio, and the power-law index η of the wave intensity spectrum for 27 SEP events in our survey. In this analysis, we only include events that satisfied the following criteria: (1) fitted values of α and the inferred values of a and η have relative uncertainties <100% and (2) −4 < η < +4 (see Section 5.3).

Figure 13 plots histograms of (a) a and (b) η; the red histograms represent extreme SEP events discussed in Section 5.3. Within the estimated uncertainties, the power-law exponent a in 15 out of 27 SEP events lies between ∼0.75–2.75 and is comparable to the range of values obtained by Cohen et al. (2005), as shown by the yellow shaded region. Also, a varies between ∼−0.33–3.9, which is roughly consistent with the typical range of ∼0.5–7 proposed recently by Schwadron et al. (2015b); in this model a < 1 implies a weak dependence of ${\lambda }_{//}$ on the ion's Q/M ratio, while a > 1 indicates that ${\lambda }_{//}$ depends strongly on Q/M (also see Li et al. 2009; Battarbee et al. 2011, 2013; Vainio et al. 2014). We note that η in 27 events lies between −1.87 and 2.33; η in 15 events lies within the range ∼0.7–1.2 reported by Cohen et al. (2005). In nine events η > 1.2–the largest value reported by Cohen et al. (2005). Overall, in four events η > 5/3, where the turbulence intensity spectra near the distant CME shocks may be significantly steeper than the typical interplanetary Kolmogorov k^−5/3 turbulence spectrum. In contrast, η ≤ 5/3 in 23 events, which implies that the turbulence spectra near the corresponding CME shocks are probably significantly flatter than the Kolmogorov spectrum. Finally, result 8 indicates that when E_B is plotted versus γ_b–γ_a, the SEP events separate into two groups (γ_b–γ_a < 3 and >3), perhaps indicating that two competing mechanisms occur simultaneously in all SEP events: (1) Q/M-dependent processes that produce modest values for α (<1.4), and steeper spectra at higher energies that steepen significantly as the break energy increases, and (2) much stronger Q/M-dependent processes that produce higher values of α, relatively flatter spectra at high and low energies, and higher break energies (see Section 5.3).

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Nine events in our survey can be considered "extreme" since they produced GLEs (see Mewaldt et al. 2012) or had CME speeds >2000 km s⁻¹. These events are shown with solid color-coded symbols in Figures 8–12. Comparing these events as a group to the remaining events in our survey, the extreme events had:

• a.  
  stronger dependence of the break energy on the Q/M ratio, resulting in α ≥ 1.4 versus α < 1.4;
• b.  
  higher peak proton fluxes between ∼4 × 10¹–2 × 10³ versus ∼10¹–5 × 10² cm⁻² sr⁻¹ s⁻¹;
• c.  
  source locations from the "well-connected" region of the western hemisphere (longitude locations between W45–W90 versus E90–W120);
• d.  
  a stronger positive correlation between the low-energy Fe/O and α;
• e.  
  low-energy spectral parameters γ_a and γ₁ similar to other events, and to the mean and median values of the overall distributions (e.g., γ_a ∼1.2);
• f.  
  flatter spectra at higher energies compared to other events (γ_b ∼2.5–4 versus ∼2.5–7); and
• g.  
  higher average O break energy compared with other events (∼1–12 MeV nucleon⁻¹ versus 0.1–10 MeV nucleon⁻¹).

While some of these features are due to the bias in our selection criteria (e.g., faster CME speeds, higher proton flux, flatter high-energy spectra), the others are not, and so may provide clues to the properties of extreme SEP events. Figures 10 and 11 show that all of these extreme events with α ≥ 1.4 have low values for γ_a, γ_b, and γ_b–γ_a, and that, collectively, they exhibit a negative trend between E_B and γ_b–γ_a (Results 7 and 8). This indicates that the corresponding spectra are relatively flat at low and high energies, and that the break energy increases as the difference γ_b–γ_a between the SEP O Band parameters decreases. The fact that α > 2 in two of the seven GLE-associated SEP events in our study, taken together with the general result that higher values of α (≥1.4) are typically observed in SEP events associated with higher >10 MeV proton fluxes and faster (>2000 km s⁻¹) western hemisphere CMEs and GLEs (see Figures 8 and 10), indicates that spectral properties in these extreme SEP events are most likely governed by highly efficient trapping and stronger Q/M-dependent scattering due to substantially enhanced wave power near the distant CME-driven shocks (see also Cohen et al. 2005; Li et al. 2009).

The above scenario is consistent with the following inferred results shown in this paper:

• 1.  
  Figure 12 shows that the inferred values for the shock compression ratio H in these extreme events tend to be somewhat larger than the event average of 2.49 ± 0.08 for 33 events, and that the correlations between H versus peak proton flux and α are more significant for, and therefore likely to be driven by, these events.
• 2.  
  Figure 13 (red histograms: see Section 5.2 for the selection criteria used to infer a and η) shows that ${\lambda }_{//}$ has a strong Q/M-dependence with a ≥ 2.8 in 3 and a < −0.2 in two extreme SEP events. Also, η > 2 or η < −1.1 in these five events, which corresponds to substantially enhanced wave power.
• 3.  
  Four of the five SEP events (not shown in Figure 13) with the strongest observed Q/M-dependence in E_B, i.e., with α > 1.6 and $| \eta | \gt 4$, are also extreme events, as defined above.

In summary, the extreme SEP events in our survey exhibit strong Q/M-dependencies in the E_B, and therefore have larger α values, which correspond to extreme or above average values for a, η, and H. In contrast, most events in our survey (see Section 5.2) exhibit weaker Q/M-dependencies in the E_B and are associated with steeper spectra at higher energies probably because the SEPs are accelerated at much slower and relatively weaker CME shocks, where the somewhat weaker turbulence allows the accelerated particles to escape more easily. Note that this interpretation is in contrast to the model of Schwadron et al. (2015a, 2015b), where the stronger Q/M-dependence of the diffusion coefficient facilitates particle escape and therefore produces steeper spectra at higher energies.

The question is, what special conditions or physical processes are responsible for causing the significantly stronger Q/M-dependence in the spectral break energies in extreme SEP events? We note that in many SEP acceleration models, strong Q/M-dependent scattering occurs primarily at quasi-parallel shocks where turbulence levels are expected to be higher and self-generated Alfvén waves may also be present (Ng et al. 2003; Li et al. 2009). We rule out the possibility that the five SEP events with α > 1.6 and $| \eta | \gt 4$ are due to strong scattering at the Bohm diffusion limit, i.e., when ${\lambda }_{//}\sim {\rho }_{g};$ here ρ_g is the ion gyroradius, because the Bohm approximation represents the case for η = 1 and α = 1 (for details see Li et al. 2009), which is significantly smaller than the α values obtained for these events. Another possible source of enhanced turbulence are the Alfvén waves excited by protons, accelerated at earlier times, that escape and propagate upstream of the CME shock (e.g., Zank et al. 2007). Such self-generated Alfvén waves can trap and scatter the particles that are accelerated at later times much more efficiently (e.g., Ng et al. 2003) or stochastically re-accelerate the shock-accelerated particles downstream of the CME shocks (e.g., Afanasiev et al. 2014). Indeed, Ng et al. (2003) modeled the excitation of Alfvén waves by protons streaming away from CME shocks and found that the wave spectra could exhibit flat spectra with η ≈ 0, which corresponds to a = 2 and α = 4/3. In contrast, Battarbee et al. (2011) modeled SEP acceleration through self-generated turbulence near CME shocks with speeds of 1250, 1500, and 1750 km s⁻¹, and predicted that the maximum value for α is ∼1.5–1.6. Comparing these predictions with α in the extreme SEP events suggests that such models, even though they include the nonlinear effects of self-generated Alfvén waves, still cannot account for the significantly stronger Q/M-dependence in the heavy ion spectral break energies reported here.

Enhanced turbulence conditions could also occur when the so-called "equal resonance condition" is met, as discussed by Li et al. (2003) and Rice et al. (2003). According to Zank et al. (2007) and Li et al. (2009), this condition occurs when α = 2 and $\eta =\pm \infty$ at parallel shocks in the limit of strong turbulence and scattering, i.e., when the wave power or intensity spectrum I(k) approaches a discontinuity. The equal resonance condition is a special case of the more general condition—the equal acceleration time or equal diffusion coefficient—discussed in Section 5.2 (see Li et al. 2009). However, we note that the values of α in two GLEs (events 16 and 25) are more than 1σ greater than the maximum value of α ≈ 2, predicted by Li et al. (2005, 2009). Thus, for these events, α > 2 corresponds to a < 0 or a > 8 and $| \eta | \gt 4$, i.e., where the wave power I(k) becomes essentially discontinuous. This implies that scattering and trapping of particles near distant CME shocks in such events is so strong that the Q/M-dependence of the spectral break energies exceeds the limit of the equal resonance condition. We therefore suggest that the larger than predicted values for α in some of the extreme events in our survey indicate that the underlying mechanisms have not yet been fully incorporated in current theoretical models.

In most existing theoretical models, the strongest Q/M-dependence occurs at quasi-parallel shocks (e.g., Li et al. 2009). Such shocks, with low-injection thresholds, are expected to primarily inject and accelerate the low-energy solar wind or ambient coronal ions (e.g., Tylka & Lee 2006; however, see Giacalone 2005 for an alternative viewpoint). In contrast, and consistent with the results reported in Paper 1, we find that many of the extreme events that exhibit strong Q/M-dependent spectral break energies are also Fe-rich and ³He-rich (see Figure 9 and Paper 1). Specifically, we note that the ∼0.16–23 MeV nucleon⁻¹ Fe/O ratio is enhanced between factors of ∼2–10 compared to the average SEP value of ∼0.134 in seven of the nine extreme SEP events, while the ∼0.5–2.0 MeV nucleon^{−1 3}He/⁴He ratio is enhanced by more than an order of magnitude over the corresponding solar wind value in five extreme SEPs. This points to the importance of contributions of suprathermal flare-origin material to the seed populations even in cases where turbulence levels are significantly enhanced and the fast CME shocks are quasi-parallel. We suggest that in such events, the enhanced turbulence traps, injects, and accelerates the higher energy suprathermals much more efficiently than the co-existing lower energy solar wind or ambient coronal ions. Simultaneously, the equal diffusion coefficient condition causes the spectral break energies to exhibit stronger Q/M-dependence, occasionally exceeding the equal resonance condition limit, as in the case of two SEP events that produced GLEs. We therefore suggest that our results for extreme events can be reconciled with SEP models provided that they include suprathermal flare-origin material as an important component of the seed population that is available for acceleration at near-Sun CME shocks.