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Keywords:

  • Van Allen Probes;
  • radiation belts;
  • energetic ions;
  • ring current

Abstract

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Kennel-Petschek Theory
  5. 3 Measured Ion Spectra
  6. 4 Ion Kennel-Petschek Analyses
  7. 5 Discussion and Conclusions
  8. Acknowledgment
  9. References

[1] A differential Kennel-Petschek (KP) flux limit for magnetospheric energetic ions is devised taking into account multiple ion species effects on electromagnetic ion cyclotron (EMIC) waves that scatter the ions. The idea is that EMIC waves may limit the highest ion intensities during acceleration phases of storms and substorms (~ hour) while other mechanisms (e.g., charge exchange) may account for losses below those limits and over longer periods of time. This approach is applied to published Earth magnetosphere energetic ion spectra (~ keV to ~1 MeV) for radial positions (L) 3 to 6.7 RE. The flatness of the most intense spectral shapes for <100 keV indicate sculpting by just such a mechanism, but modifications of traditional KP parameters are needed to account for maximum fluxes up to 5.4 times greater than expected. Future work using the new capabilities of the Van Allen Probes mission will likely resolve outstanding uncertainties.

1 Introduction

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Kennel-Petschek Theory
  5. 3 Measured Ion Spectra
  6. 4 Ion Kennel-Petschek Analyses
  7. 5 Discussion and Conclusions
  8. Acknowledgment
  9. References

[2] Following the pioneering work of Kennel and Petschek [1966], Schulz and Davidson [1988], and Summers et al. [2009], Mauk and Fox [2010] developed a general approach to evaluating a relativistic differential Kennel-Petschek (KP) flux limit for energetic magnetospheric electrons, providing a verified standard for comparing the relativistic electron intensities and spectral shapes of all of the strongly magnetized planets of the solar system. Here a similar approach is developed for energetic magnetospheric ions that includes multispecies (H, He, O) composition of the plasmas that determine properties of the electromagnetic ion cyclotron (EMIC) waves thought to scatter the energetic ions. The idea behind the KP approach is not to include all possible complexities associated with the wave-particle interactions but to reduce those complexities to some minimal set that provides reasonable scaling for particle intensities for a diversity of conditions and environments. It will become clear that ion composition is a critical component of that minimal set.

[3] The effects of heavy ions on the generation of EMIC waves have been examined by many authors beginning with Cornwall and Schulz [1971], and more recently by, e.g., Mauk [1982], Roux et al. [1982], and Omidi et al. [2013]. The role of EMIC waves in the loss of energetic ions generally is uncertain. While EMIC waves have certainly been invoked for explaining various features of ring current populations [e.g., Williams and Lyons, 1974a, 1974b], it is known that charge exchange and coulomb-scattering losses are important and perhaps dominant for storm-like time scales (days) [Daglis et al., 1999]. However, it is well established that these collisional processes cannot explain the ion losses during the initial rapid phases of storm recovery (hours) [e.g., Keika et al., 2006]. Therefore, the question is does the EMIC wave-moderated KP process limit the ion intensities over relatively short (hour) time frames?

2 Kennel-Petschek Theory

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Kennel-Petschek Theory
  5. 3 Measured Ion Spectra
  6. 4 Ion Kennel-Petschek Analyses
  7. 5 Discussion and Conclusions
  8. Acknowledgment
  9. References

[4] Considering only ions, the KP limit idea is as follows: (1) magnetic flux tubes within Earth's inner magnetosphere are robustly populated by energetic ions (protons assumed here) by some acceleration processes during magnetically active conditions, (2) accelerated protons trapped in magnetic bottle configurations are unstable to the generation (net gain: G) of EMIC waves near the magnetic equator, (3) a small fraction of the generated waves propagating along the magnetic field lines are reflected (R) back toward the equatorial regions (the original concept uses ionospheric reflection but any mechanism of feedback would suffice), and (4) if G · R > 1, there occurs runaway growth of the waves causing ions to be lost at the fast strong pitch angle diffusion limit. It is predicted that particle distributions will adjust themselves so that G · R ≤ 1. Kennel and Petschek [1966] formulated this condition as a limit on an integral of the intensity distributions by selecting a single optimal minimum resonant energy for evaluation.

[5] The differential KP limit examines the maximum allowed intensities not just at a single minimum resonant energy but as a function of the minimum resonant energy; this process constrains not only integral intensities but spectral shapes as well [Schulz and Davidson, 1988]. We approximate the net EMIC wave gain (G) with the expression:

  • display math(1)

where G is the ratio of final to initial wave amplitude as the waves propagate through the equatorial regions, γ is wave growth rate (1/s), Vg is the wave group velocity (cm/s), D is the distance in planetary radii along the magnetic field line where wave growth rate remains positive and large, and Rp is the planetary radius in centimeter. Assuming as did Kennel and Petschek [1966] that R ~ 0.05 such that ln[1/R] ~ 3 and D ~ L, G · R ≤ 1 yields

  • display math(2)

[6] Here both γ and Vg are shown to be functions of ωr, the EMIC mode wave frequency that is in gyroresonance with a proton with a user-specified minimum resonant energy Er. Equation (2) is evaluated with analytic fits to measured energetic ion distributions using [Mauk and Fox, 2010]

  • display math(3)

containing five fitting parameters, C, kT, γ1, γ2a or γ2b, and E2, and with the anisotropy parameter “S” set to 1/6, suggested by Kennel and Petschek [1966], discussed in section 4.

[7] We define two different values of C: Cm originates from the spectral fitting process and CK is the value needed for the intensity to be exactly at the KP limit for a given Er. Because γ[ωr(Er)] is linearly proportional to C, we can change (2) into an equality by replacing γ[ωr(Er)] with (CK/Cm) · γ[ωr(Er)] and rearranging to

  • display math(4)

[8] When this ratio is greater than 1, equal to 1, or less than 1 for a given specified resonant energy, Er, that means that the proton intensity is greater than, equal to, or less than the differential KP limit for the specified value of Er.

[9] The spatial wave growth rates (γS = γT/Vg) are evaluated with [Kennel and Petschek, 1966; Mauk and McPherron, 1980]:

  • display math(5)

where

  • display math(6)
  • display math(7)

where mp is proton mass, e is unit charge, c is speed of light, Vϕ is the EMIC wave phase velocity with a wave assumed to be propagating parallel to B, ω is wave frequency (rad/s), Ωp is proton gyrofrequency (eB/(mp c), where B is magnetic field strength), fp(P) is the hot proton phase space density as a function of momentum P derived using fp(P) = I(P)/P2, P and P|| are proton momentum perpendicular and parallel to B, and PR is the parallel momentum that a proton must have to be in gyroresonance with the EMIC wave. A+(PR) reduces simply to the “S” parameter for (3).

[10] We assume for the evaluation of Vϕ in (5)(7) that the EMIC waves are dominated by cold, multispecies plasmas [Stix, 1992]. Several authors have recently considered the effects of warm plasmas on the dispersion properties of EMIC waves [e.g., Silin et al., 2011]. The results are that the dispersion properties can shift somewhat quantitatively and occasionally qualitatively, and energetic proton scattering rates can be enhanced [Yoon et al., 2011]. However, a long history of statistical analyses shows qualitative and semiquantitative agreement with the cold plasma results at geosynchronous orbit [e.g., Mauk, 1982, 1983; Roux et al., 1982], and as one addresses regions closer to the plasmasphere-dominant regions, as we are doing here, the multicomponent plasma species are even colder [Horwitz et al., 1986]. Some of the more sophisticated simulations being run today [e.g., Omidi et al., 2013] utilize the cold plasma approximation.

[11] Figure 1 shows how complex the EMIC wave properties are in multispecies plasmas. While we will assume that the waves propagate parallel to B, a different value (25°) was chosen for Figure 1 to better reveal the mode structure. We focus here on the left-handed “transverse” modes (designated with the letters “LT,” where “transverse” refers to the orientation of the major axis of the magnetic perturbation ellipse with respect to the k-B plane; the compressional modes “LC” amplify much less robustly [Mauk, 1982]). Only because we are selecting such a small value of S, the value recommended by Kennel and Petschek [1966], it is found that only LT0 and LT1 contribute, and, with the addition of even small contributions of oxygen, LT0 is the principal player.

image

Figure 1. Example of dispersion properties of electromagnetic ion cyclotron (EMIC) waves propagating in a cold, multi-ion species magnetized plasma derived from cold plasma theory [Stix, 1992]. (top) Phase speed normalized by Alfven speed. (bottom) Ellipticity.

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3 Measured Ion Spectra

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Kennel-Petschek Theory
  5. 3 Measured Ion Spectra
  6. 4 Ion Kennel-Petschek Analyses
  7. 5 Discussion and Conclusions
  8. Acknowledgment
  9. References

[12] Figure 2 shows fits (using (3)) of the more intense > 5 keV ion distributions in Earth's inner magnetosphere found in the literature (Table 1 provides the fit parameters). With the exception of spectrum 5, these are “total ion” spectra. However, it is known with spectra measured with solid-state detector systems that dead layer, pulse height defect, and energy to momentum conversion factors conspire to yield almost pure proton spectra even when heavy ions contribute equally to the total ring current particle pressures.

image

Figure 2. Parametric fits to energetic ion spectra (the >5 keV portions) from the literature for various radial positions (L). The fitting expression is (3) and parameters for each fit are provided in Table 1. The sources are Fritz et al. [1977], Williams and Lyons [1974a, 1974b], Spjeldvik [1977], Williams [1987], Krimigis et al. [1985], and Lyons and Williams [1980].

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Table 1. Fitting Parameters (Equation (3)) for Figure 2 Spectra
IDLCkTγ1Eoγ2aγ2bCRM/ErM
  1. a

    These have the greatest uncertainty in the data and fits.

16.74.20E50.080.43062.44.5010.3/27
252.67E6~00.26987.35.7413.1/45
34.52.00E60.140.5482614.5011.1/72
44.25.72E6~00.5451282.7812.5/46
547.27E4~0−0.94021710.4604.0/56
647.66E6~00.4284944141.253.6/45
744.00E6~00.24489.04.0114.5/48
83.52.41E7~00.7792233.1214.0/45
93.55.00E6~00.16055.82.6714.6/37
10a36.54E6~00.30877.611.692.5/38
11a34.47E6~0−0.04243.02.2815.4/35

4 Ion Kennel-Petschek Analyses

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Kennel-Petschek Theory
  5. 3 Measured Ion Spectra
  6. 4 Ion Kennel-Petschek Analyses
  7. 5 Discussion and Conclusions
  8. Acknowledgment
  9. References

[13] The characteristic differential KP analysis is shown in Figure 3A for spectrum 7 (Lyons1, L = 4) for two different composition assumptions. For each of the three different assumed total densities in each panel (labeled “a,” “b,” and “c” and shown with dotted, dashed, and solid lines, respectively), the profiles show the degree to which each portion of the spectrum is greater than (Cm/CK > 1), equal to (Cm/CK = 1), or less than (Cm/CK < 1) the differential KP limit using nominal parameters. For each L value (L = 4 for Figure 3A), the three density values used are the mean “trough” density as determined by Sheeley et al. [2001] (25 cm−3 for Figure 3A), the mean “plasmasphere” density [Sheeley et al., 2001] (350 cm−3 for Figure 3A), and the plasmasphere density for a fully formed plasmasphere [Chappell, 1972] representing an upper limit to the reasonable densities that might be assumed (1100 cm−3 for Figure 3A). For all panels, we assume the nominal plasmaspheric helium fraction (20%) [Horwitz et al., 1986], and when oxygen ions are introduced, a value common to the outer plasmasphere regions is also assumed (≥ 2%) [Horwitz et al., 1986].

image

Figure 3. (A) The differential Kennel-Petschek analysis of spectrum 7 in Figure 2 (L = 4) for two different assumptions about the ion composition of the cold plasmas and three different total density assumptions within each panel (a, b, and c; see main text). (B) Similar analyses for Figure 2 spectrum 2 (top: L5) and spectrum 10 (bottom: L3) for just one assumption about the ion composition of the cold plasmas. In all panels, the red curves represent the results for the LT0 EMIC mode, and the black curves are for the LT1 mode. Clearly, when oxygen ions are introduced, the LT0 mode controls the upper intensity limits. The thick horizontal black lines in all panels highlight the traditional KP limit value, while the thick horizontal blue lines are suggested alternatives discussed in the text.

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[14] The colors red and black in Figure 3 represent the range of wave frequencies of the waves involved; red represents frequencies less than the oxygen cyclotron frequency (LT0, with or without oxygen), and black represents frequencies between the oxygen and helium cyclotron frequencies (LT1). Even for a very small oxygen contribution (e.g., 1%), the two modes (LT0 and LT1) break apart in their effects, and LT0 (red lines) becomes the principal limiter of energetic proton intensities. We have analyzed all spectra with an array of assumed helium and oxygen contributions (1% to 50% for oxygen) and have found that the red traces stay close to the same levels as shown in Figure 3, with the one major difference being how far to the left the red traces extend.

[15] There are several features of note in Figure 3A. With a small amount of oxygen included, the KP limit stays active to much lower resonant energies than occurs without oxygen (we arbitrarily stop the calculation, leaving unconnected ends to the red lines on the left side, when the wave frequencies approach the oxygen resonant frequency to within about 3%). We see also that the peaks of the Cm/CK profiles rise above the traditional KP limit (the thick black line) by almost a factor of 5. This mismatch can mean that the KP limit is not in force (the concept is flawed) or that a different set of parameters for the limit should be used. The horizontal blue line shows what the KP limit would be if we assumed that the reflective feedback is 0.5%, rather than the canonical 5%, and that the distance along the field line where the growth remains high is L/2 rather than L. An additional modification that could bring observations more in line with expectations is with the anisotropy parameter S. We chose the original value used by Kennel and Petschek [1966] (S = 1/6). However, Williams and Lyons [1974b] observed even flatter ion pitch angle distributions during the most active periods when the loss cone is apparently populated with scattered ions. Thus, even smaller values of S might be appropriate (a change that would lower the Cm/CK values). Note that the appropriate value of S to use is not from the diversity of S values that are generally observed in the magnetosphere. Over storm time periods, the S parameter may be established by other processes, for example, charge exchange and coulomb collisions. However, the S values to use here are those that the distributions evolve to as the intensities rise to or above the KP limit. The observations of Williams and Lyons [1974b] certainly indicate that during the periods of the highest intensities, the anisotropy parameter is much flatter than would be created by the collisional processes. Nearly flat pitch angle distributions (which the paper argues are generated by ion cyclotron turbulence) were observed as close to Earth as the observations allowed: L = 4.2.

[16] None of the “new” parameter possibilities (R, D, S) for achieving quantitative concurrence with the KP expectations are unreasonable, but the almost factor of 5 mismatch and the need for manipulating the model parameters certainly weaken arguments in favor of the efficacy of the KP limit for ions. More in favor are the consistencies for different spectra and L values. Figure 3B depicts the analysis for L = 5 and L = 3 showing substantially similar results, and the last column of Table 1 (showing maximum Cm/CK designated “CRM” and Er where that peak occurs, designated “ErM”) shows substantial consistency for all of the most intense spectra in Figure 2 (CRM values between 2.5 and 5.4). Additionally, there is the issue of the spectral shapes. One expects that as the ion distributions are accelerated, portions of the distribution will start bumping up against the horizontal line that represents the KP limit. Because of strong losses for those parts of the distribution, Cm/CK will flatten itself along the limiting line. For the more intense spectra observed, the <100 keV portions of the Cm/CK profiles are relatively horizontal. Horizontal Cm/CK profiles correspond to intensity spectral indices that are very hard (between 0 and 1), a condition that is observed with many of the Figure 2 spectra for lower energies (γ1 in Table 1).

5 Discussion and Conclusions

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Kennel-Petschek Theory
  5. 3 Measured Ion Spectra
  6. 4 Ion Kennel-Petschek Analyses
  7. 5 Discussion and Conclusions
  8. Acknowledgment
  9. References

[17] The question of whether the LT0 mode is observationally available to play the role assigned to it here is not known. Recent statistical studies [Usanova et al., 2012, and references therein] have bypassed the characterization of this mode because the authors claim that the range of frequencies is mixed together with other phenomena. The fact that substantial power can exist within the LT0 frequency regime within the inner magnetosphere during storms is well demonstrated by Figure 2 in Ukhorskiy et al. [2010], and it is only during intense ion activity periods that we require the active participation of the LT0 mode.

[18] It is well understood that the generation and propagation of EMIC waves in Earth's magnetosphere are generally far more complicated than the simple model represented in this paper [e.g., Omidi et al., 2013; Silin et al., 2011]. The question is can a simplified recipe that provides an approximate scaling standard for comparing different space environments and conditions be found? Such a simple recipe appears to work for the electrons [Summers et al., 2009; Mauk and Fox, 2010] despite the well-known complexities of whistler-electron interactions. The proof resides only in the extent to which we can obtain consistent results over a broad range of events and environments.

[19] The evidence discussed here provides indications that an energetic ion, differential KP limit is active in helping to control the maximum intensities of ring current ion intensities within Earth's inner magnetosphere, but the evidence is not definitive. Definitive evidence may come in the future from detailed studies needed using the comprehensively instrumented Van Allen Storm Probes Mission launched 30 August 2012, once instrument responses are well understood and a diversity of storms is encountered. Future work will evaluate whether more sophisticated analyses, for example, the inclusion of hot plasma effects, are needed to better organize the diversity of spectra.

Acknowledgment

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Kennel-Petschek Theory
  5. 3 Measured Ion Spectra
  6. 4 Ion Kennel-Petschek Analyses
  7. 5 Discussion and Conclusions
  8. Acknowledgment
  9. References

[20] The Editor thanks Michael Schulz and an anonymous reviewer for their assistance in evaluating this paper.

References

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Kennel-Petschek Theory
  5. 3 Measured Ion Spectra
  6. 4 Ion Kennel-Petschek Analyses
  7. 5 Discussion and Conclusions
  8. Acknowledgment
  9. References