Warm plasma effects on electromagnetic ion cyclotron wave MeV electron interactions in the magnetosphere

Authors


Abstract

[1] The full kinetic linear dispersion relation in a warm plasma with He+ and O+ ions is used to estimate the minimum resonant electron energies required for resonant scattering of relativistic electrons by electromagnetic ion cyclotron waves. We find two significant differences from the cold-plasma approximation: (1) waves can be excited inside the stop bands and at ion gyrofrequencies for relatively small wave numbers k < Ωp/vA and (2) short wavelengths with k > Ωp/vA experience strong cyclotron damping. We show that, in general, minimum resonant energy of electrons Emin depends only on the wave number k, magnetic field strength B, and plasma mass density ρ and depends on the wave frequency ω only implicitly, via the dispersion relation. Formulae for Emin as function of ω based on cold-plasma approximation predict the lowest energy loss where ωimage since in this approximation k → ∞ at these frequencies. We show this inference is incorrect and that kinetic effects mean that the ion gyrofrequencies are no longer necessarily preferential for low energy loss. The lowest values of Emin are obtained where the dispersion supports the largest wave numbers k and in the regions of the largest mass densities ρ and the lowest magnetic fields B. For realistic magnetospheric conditions Emin ∼ 2 MeV and can only drop to ∼500 keV inside dense plasmaspheric plumes, with plasma density of the order of 500 cm−3, or during plasmaspheric expansions to high L shells (L ∼ 7).

1. Introduction

[2] Electromagnetic ion cyclotron (EMIC) waves are regularly observed in the terrestrial magnetosphere [e.g., Troitskaya and Gulelmi, 1967; Gendrin, 1975; Mauk and McPherron, 1980; Anderson et al., 1992a, 1992b; Anderson and Hamilton, 1993; Usanova et al., 2008]. These waves are typically driven by the temperature anisotropy of an ion species (T > T) and are believed to play an important role in the loss of radiation belt particles by scattering them into the atmosphere [Summers and Thorne, 2003; Albert, 2003; Summers et al., 2007a]. Although the kinetic linear dispersion of EMIC waves in a pure proton-electron plasma, as well as in plasmas with heavier ion species, has long been known [e.g., Davidson and Gladd, 1975; Kozyra et al., 1984; Gendrin et al., 1984] most studies examining MeV electron loss by EMIC scattering use the cold plasma approximation [Summers and Thorne, 2003; Albert, 2003; Summers et al., 2007a, 2007b].

[3] There are, however, numerous experimental observations of hot ion species, in particular, hot protons which drive the EMIC waves, as well as singly ionized helium He+, which have been associated with EMIC wave activity [e.g., Young et al., 1981; Anderson and Fuselier, 1994; Lund et al., 1999]. Initially, Young et al. [1981] believed that He+ played some sort of catalytic role in the excitation of EMIC waves, since there was a clear correlation between the EMIC wave power and hot helium ion abundance. However, Anderson and Fuselier [1994] using data from the AMPTE/CCE spacecraft later showed that this was simply caused by strong heating of helium ions, so that more of them could penetrate the potential barrier of the spacecraft and reach the detector. These authors estimated that in the absence of wave activity the average helium ion temperature was approximately 0.85 of the proton temperature Tp, while during strong EMIC wave events it rose to about 2.3 Tp. By analyzing the particle data from the FAST mission, Lund et al. [1999] later discovered that during EMIC wave activity the thermal He+ ions have ∼3–30 times the energy of thermal protons or O+ ions. It has also been shown in numerical simulations, e.g., by Omura et al. [1985], that He+ ions are heated much more efficiently by EMIC waves than any other ion species. Initially cool He+ ions were found to reach temperatures approximately 10 times higher than the cool proton background.

[4] In a series of papers [e.g., Horne and Thorne, 1994], the full kinetic dispersion relation was used to study the thermal ion effects on the EMIC wave propagation in the inner magnetosphere. They showed that high plasma beta has a strong positive effect on the EMIC wave growth rates.

[5] In addition to the hot protons which excite the EMIC waves, He+ ions also appear to undergo strong secondary heating by EMIC waves. Consequently, the cold plasma approximation may no longer be appropriate to describe the waves dispersion in space plasma correctly, since observations usually record the nonlinear saturated state of an instability. In this paper we compare the results of linear dispersion theory in the cold fluid and warm kinetic approximations. Further, we examine the consequences of the warm kinetic dispersion relation for EMIC waves in relation to their potential role in causing the scattering loss of MeV radiation belt electrons.

2. Linear Dispersion Relation of EMIC Waves in a Kinetic Approximation

[6] Studies of the effect of EMIC waves on radiation belt electrons have typically assumed that the magnetospheric plasma is cold; that is, its plasma beta is very small, β = 2μ0ΣjnjκBTj/B2≪ 1. In this case the EMIC wave dispersion relation for propagation parallel to the ambient magnetic field is described in the cold approximation by the equation,

equation image

where Ωj =ejB/mj are signed gyro frequencies and ωpj = (njej2/mjε0)1/2 are plasma frequencies of various particle species j = e, p, He+, O+ [e.g., Stix, 1962].

[7] The consequence of using the cold approximation is that for certain ranges of wave frequencies equation (1) can only be satisfied if k is imaginary. These ranges of frequencies are called “stop bands” and wave propagation inside these bands is impossible. For all other frequencies, however, the wave numbers and frequencies are purely real; that is, the waves propagate without amplification or damping. As a result, in such an approximation it is impossible to obtain wave excitation or growth.

[8] The full kinetic linear dispersion relation for EMIC waves in a warm plasma with a finite plasma beta is given by

equation image

where Aj = Tj/Tj − 1 and vTj = (κBTj/mj)1/2 are the temperature anisotropies and thermal velocities parallel to the external magnetic field [e.g., Davidson and Gladd, 1975; Cuperman et al., 1975]. It should be kept in mind that the wave number k here is, in fact, k = ∣k∣. In this study we consider plasmas containing 3 ion species: protons, He+ and O+, but in order to approximate the magnetospheric plasma we use up to six ion populations, hot and cool populations of the same ion species, so there may be up to seven particle populations in the actual dispersion relations we solved. The wave frequency is now, in general, a complex number and consists of real and imaginary parts, Ω = ω + . Charge neutrality is ensured by the relation between the average particle species densities, Σjejnj = 0. Z(zj) is the plasma dispersion function,

equation image

and the arguments zj are given by

equation image

[9] Cuperman et al. [1975] and later Gomberoff and Neira [1983] proposed an approximate “quasi-cold” solution, assuming that the imaginary part of the frequency is small (i.e., damping or growth is weak) and the real part of the frequency satisfies the cold approximation relation (1). This approximate solution is often used in EMIC wave studies [e.g., Kozyra et al., 1984; Omura et al., 1985]. However, one should keep in mind that this approximation is achieved by asymptotic expansion of the plasma dispersion function assuming that its argument is large and real, or ∣ω − Ωj∣ ≫ kvTj and keeping only the lowest-order term. This solution becomes invalid either in the vicinity of each gyrofrequency Ωj if the plasma is cold and vTjvAj, or it becomes invalid for all frequencies if the plasma is sufficiently warm and vTjvAj, where vAj = B/(μ0mjnj)1/2 is the Alfvén velocity for ion species “j.” These assumptions, under which the cold approximation is applicable, are equivalent to assuming that plasma betas for each ion species βpj = 2μ0njκBTj/B2 are much smaller than unity. We will show that plasma betas in the magnetosphere are not necessarily small and the assumption that the real frequency closely follows the cold plasma dispersion may be invalid for realistic magnetospheric parameters.

[10] Examples of the solutions of the full linear kinetic dispersion relation (2) are given in Figures 1, 2, 3, and 4 for several sets of realistic magnetospheric parameters. For comparison, the solutions of the cold dispersion relation (1) are also shown as thin lines. All relevant plasma and magnetic field parameters are summarized in Table 1.

Figure 1.

Linear dispersion curves for EMIC waves in multi-ion plasma. (left) Real frequencies and (right) growth rates. Thin lines correspond to the cold approximation. Solid lines correspond to the full kinetic approach. Dashes are used to show real frequency and growth rates for different branches. Different rows correspond to different plasma parameters from Table 1: (a) case 1, (b) case 2, (c) case 3, (d) case 4, (e) case 5, and (f) case 6.

Figure 2.

Linear dispersion curves for EMIC waves with the same parameters as in (top) Figure 1a and (bottom) Figure 1b, cases 1 and 2 from Table 1, respectively, but using the cold approximation for the cool ion populations.

Figure 3.

Linear dispersion curves for EMIC waves in multi-ion plasma. (left) Real frequencies and (right) growth rates. Thin lines correspond to the cold approximation. Solid lines correspond to the full kinetic approach. Dashes are used to show real frequency and growth rates for different branches. Different rows correspond to plasma parameters from Table 1: (a) case 7, (b) case 8, (c) case 9, and (d) case 10.

Figure 4.

Linear dispersion curves for EMIC waves in multi-ion plasma. (left) Real frequencies and (right) growth rates. Thin lines correspond to the cold approximation. Solid lines correspond to the full kinetic approach. Dashes are used to show real frequency and growth rates for different branches. Different rows correspond to different L shell values. Plasma parameters are listed in Table 1; cases 11 to 15 from top to bottom.

Table 1. Plasma Parameters Used in the Dispersion Relationsa
 np, hotnp, coolimage hotimage coolimage hotimage coolTp,hot,∥image image hotLABβ
  • a

    All densities are given in cm−3, temperatures of hot species are in keV, and magnetic field B is given in nT. Temperature anisotropy A = T/T − 1 is the same for all hot ion species. The temperatures of all cool species, including electrons, were set to 1 eV. The last column gives the total parallel plasma beta β = 2μ0ΣjnjκTj,∥/B2.

Case 153001005250415900.15
Case 253001005250711104.2
Case 31520552.52.52525415900.66
Case 41520552.52.525257111019
Case 52510100502525415901.2
Case 625101005025257111034
Case 72551001005250425900.73
Case 82551001005250523002.8
Case 925510100502510425900.91
Case 1025510100502510523003.5
Case 1125510010052503313900.13
Case 1225510010052503.538700.33
Case 132551001005250435900.73
Case 1425510010052504.534100.97
Case 152551001005250533002.8

[11] In Figure 1 we take a plasma which consists of 70% protons, 20% He+ ions and 10% O+ ions, as in the work by Meredith et al. [2003]. Only a fraction of the protons is hot and anisotropic, there is always a cool proton population. The term “cool” is used in order to remind the reader that even though the temperature of the population is very low and the plasma beta may be very small, β ∼ 10−4, still the full kinetic description of that particle population is used. The term “cold” is reserved for the approximation, where particle temperature is assumed to be exactly zero. We now compare the dispersion relations varying the value of the L shell and the temperature of the heavy ions. The wave numbers are normalized to Ωp/vA, where vA = B/(μ0ρ)1/2 is the Alfvén velocity and ρ = Σjmjnj is the plasma mass density. One can see, that with this normalization the wave numbers are insensitive to the local magnetic field magnitude and only vary with plasma density and ion composition.

[12] Figures 1a and 1b show the dispersion curves at L = 4 and L = 7, respectively, when all heavy ions are cool at temperatures of only 1 eV and only 5 cm−3 protons are hot with parallel temperature of 25 keV and temperature anisotropy A =1 (cases 1 and 2 in Table 1). One can see that the unstable wave numbers are larger at L = 4 and the dispersion is in good agreement with the cold approximation. Notice that all branches are damped at large wave numbers. This happens only when the full kinetic approximation is used for the cool ion populations, even though their temperatures are only 1 eV and partial plasma betas are very small, βp,c ∼ 10−4.

[13] Figures 1c and 1d show the dispersion at L = 4 and L = 7, respectively, when half of the He+ and O+ ions (10 and 5 cm−3) have the same temperature and anisotropy as the hot protons (15 cm−3), while the other half is cool (cases 3 and 4 in Table 1). The results remain very similar if the temperatures of the heavy ions are reduced to 10 or 5 keV. One can see that the unstable waves can be excited inside the stop band, in particular at the He+ gyrofrequency (Figure 1d). In the warm kinetic regime, the resonant frequencies no longer coincide with the ion gyrofrequencies. The unstable wave numbers, however, remain fairly small. Again, waves with larger wave numbers can be excited at lower L shells.

[14] Figures 1e and 1f (cases 5 and 6 in Table 1) demonstrate wave dispersion at L = 4 and L = 7, respectively, when all of the He+ and O+ ions (20 and 10 cm−3) are at the same temperature as the hot protons (25 cm−3). The results remain qualitatively similar when the temperature of the heavy ions is decreased to 10, 5, 2 or even 1 keV and the density of the hot proton population is decreased. The hot ion species cause strong damping at low L shells, but may drive EMIC waves inside the stop band at higher L shells. The range of the unstable wave numbers remains, however, approximately the same as in the previous cases.

[15] One clear result is that the discrepancy between the warm and cold dispersion curves is largest in the regions of strong wave growth or damping, where the imaginary part of the frequency γ becomes significant. Thus, we would expect to see the strongest deviations from cold theory exactly in those regions where the waves are either generated or absorbed.

[16] Sometimes, when the plasma consists of a mixture of hot and cool particle populations, a mixed approach is used, where the hot populations are treated kinetically and the cool populations are treated in the cold approximation [e.g., Gomberoff and Neira, 1983, equation (3); Kozyra et al., 1984, equation (2)]. However, even if the temperature of the cool population is very low, 1 eV in our case, there is still a significant difference between describing it in the cold and kinetic approximations. Figure 2 (top) and Figure 2 (bottom) show the dispersion curves for exactly the same parameters as in Figures 1a and 1b (cases 1 and 2 in Table 1), respectively, but now the cool populations are described using the terms from the cold plasma dispersion relation (1). Apparently, the real frequencies remain unchanged. However, all wave branches which used to undergo strong cyclotron damping in the warm approximation when ω → Ωj are now undamped! This demonstrates that in the kinetic approximation even fairly cold plasmas with T ∼ 1 eV and β ≪ 1 will damp EMIC waves at relatively large wave numbers. We shall see why large numbers are of particular importance for interaction with relativistic electrons in section 3.

[17] Figure 3 shows a comparison of four different cases when a dense cool proton background (510 cm−3) which corresponds to the plasma density inside the plasmasphere in the vicinity of L ∼ 4 is added to the plasma considered in Figure 1. The result is that now the He+ and O+ abundances drop to approximately 2% and 1%, respectively. This makes the stop bands much narrower.

[18] Figures 3a and 3b (cases 7 and 8 in Table 1, respectively) show the EMIC waves dispersion in a plasma with cool heavy ions at L = 4 and L = 5. The temperature anisotropy of hot protons (25 cm−3) is now set to A = 2, which is still within the reasonable range of values, according to Cornwall et al. [1971]. One can see, that the growth rates strongly increase with increasing L shell value, but the unstable wave numbers decrease. All three branches are linearly unstable, but the growth rate of the lowest branch is approximately 2 orders of magnitude smaller than the growth of the upper two branches.

[19] Figures 3c and 3d (cases 9 and 10 in Table 1) show the EMIC waves dispersion in a plasma with helium and oxygen ion temperatures THe∥ = TO∥ set to 10 keV in the parallel direction and with the same anisotropy A = 2 as the hot protons at L = 4 and L = 5, respectively. As a result, the upper and lower branches are strongly damped for all wave numbers except very small ones. The lower branch undergoes a polarity reversal; that is, it becomes right-hand polarized at large wave numbers. However, it is strongly damped and probably can never be observed in situ. The middle branch merges together with the upper branch across the stop band and is strongly unstable.

[20] It is not necessary to have hot heavy ions in order to obtain EMIC waves at the heavy-ion gyrofrequencies, as we have obtained in Figures 1d, 1f, 3c and 3d. Figure 4 shows the sequence of dispersion curves for the same plasma parameters with cool heavy ions as considered for Figures 3a and 3b, but with temperature anisotropy A = 3, at several different L shells, between L = 3 and L = 5 (cases 11 to 15 in Table 1, respectively). One can see a clear transition of the unstable wave region from higher to lower wave numbers and lower frequencies. While at low L shells the upper branch grows fastest, at higher L shells, the middle branch becomes dominant. At the intermediate L shells the two branches merge together and breach the stop band completely.

3. Minimum Electron Resonant Energies

[21] In order to compare the kinetic description of EMIC wave dispersion in a warm plasma to the previous cold plasma results, we estimate the minimum resonant electron energies necessary for effective radiation belt electron scattering by EMIC waves. These energies are obtained from the consideration of the resonance condition in a relativistic regime (for more details see, e.g., Summers et al. [1998]),

equation image

where v = (v + v)1/2 is the electron speed, ω is the wave frequency, k is now a signed wave number parallel to the ambient magnetic field k = k and n = 0, ±1, ±2,… is an integer corresponding to the cyclotron harmonic. We continue to use signed gyrofrequencies, therefore Ωe < 0 and the minimum electron kinetic energy Emin for resonant electron interaction with EMIC waves is obtained by setting v = 0 and n = 1 in (5) and solving the resulting equation for v:

equation image

Emin is then given by

equation image

By definition of kinetic energy, Emin → 0 as v → 0. For efficient resonant scattering of relativistic electrons Emin must be as low as possible. Let us consider equation (6) to see how v can be minimized. First of all, we should realize which terms are small, so that the expression can be simplified. Obviously, for EMIC waves ω < Ωp ≪ ∣Ωe∣, so the term ω2/c2 can be safely omitted. Also, for EMIC waves ∣k∣ is typically of the order of ωpp/c. But then, if ∣k∣ ≫ ∣Ωe∣/c, the ωk term is negligibly small in comparison to the ∣Ωe∣(Ωe2/c + k2)1/2 ≈ ∣Ωek term, since ω ≪ ∣Ωe∣. And if ∣k∣ ≪ ∣Ωe∣/c, ωk becomes even more negligible in comparison to Ωe2/c. So, equation (6) with good accuracy can be reduced to

equation image

where we chose the sign in order to consider only the electrons moving in the same direction as the wave. This simple analysis shows that the minimum resonant energy, in fact, is practically independent of the wave frequency! The main wave parameter, which affects the minimum resonant energy is, indeed, the wave number k and Emin only depends on the frequency implicitly via the ω(k) dispersion relation. The lowest resonant energies are obtained for electron interaction with shortest possible wavelengths.

[22] Rewriting equation (8) as

equation image

and substituting it into equation (7) gives

equation image

We want to stress that equation (10) was obtained using the full relativistic resonance condition and the only assumption made was that for EMIC waves ω ≪ Ωe. No other assumptions about the wave spectrum were made thus far.

[23] Now, in order to estimate what realistic values of Emin can be obtained under realistic magnetospheric conditions, let us express all quantities in equation (10) in terms of the plasma parameters. It is convenient to normalize the wave numbers k in terms of Ωp/vA, since in this normalization the wave numbers are independent of the magnetic field strength and only depend on the plasma mass density ρ. We can then introduce a dimensionless wave number K

equation image

As we have seen in the dispersion curves in Figures 14, the unstable waves always lie in the range 0 < K ≤ 1 with the fastest growing waves in the range 0.2 ≤ K ≤ 0.4. Then, the minimum resonant energy can be expressed in terms of plasma parameters, physical constants, and the dimensionless wave number K as

equation image

[24] The most important question is whether EMIC waves are capable of resonant scattering of low-energy electrons, with energies of the order of 1 MeV or less. Let us see what kind of values of Emin can be obtained for the plasma parameters used in our Table 1. At L = 4 the magnetic field strength is approximately B = 590 nT and EMIC wave growth is fastest at K = 0.4. If we use the total ion number density n = 550 cm−3 and assume that the plasma is composed of 70% protons, 20% He+ and 10% O+ ions, the mass density ρ ≈ 2.6 × 10−18 kg/m3. With these parameters equation (12) gives Emin ≈ 2.1 MeV. Even if K = 1, the minimum resonant energy would still be relatively high, Emin ≈ 600 keV. However, cyclotron damping at such high wave numbers would be very strong and their excitation is highly unlikely!

[25] At higher L shells, say, L = 7, we used B = 110 nT and K = 0.2. If the plasma density is around 50 cm−3 and plasma composition is the same as above, then ρ ≈ 2.6 × 10−19 kg/m3. With these numbers, the expected minimum resonant energy is Emin ≈ 2.5 MeV. Only by using unrealistically high wave numbers can one obtain low resonant energies, e.g., for K = 1, Emin ≈ 280 keV.

[26] In order to see how much the increase of heavy ions abundances may lower the minimum resonant energy, let us consider a plasma with 30% He+ and 30% O+ ions. For the same total plasma density of 50 cm−3, the mass density now becomes ρ ≈ 5.3 × 10−19 kg/m3. An increase in heavy ion density cannot lead to excitation of shorter wavelengths. On the contrary, the unstable wave range will move toward lower k values. Hence, even if we use the same field intensity and normalized wave number K = 0.2 at L = 7 instead of 2.5 MeV, we will obtain 1.6 MeV. Using the same high abundances (30% of He+ and 30% of O+) for L = 4, ne = 550 cm−3 and K = 0.4 will give Emin ≈ 1.3 MeV (compared to 2.1 MeV obtained for 20% He+ and 10% O+ abundances). However, heavy ions produce strong cyclotron damping of all waves slightly above their gyrofrequency [see, e.g., Mace et al., 2011]. Thus, is seems unlikely that EMIC waves could have sufficient amplitudes in plasmas with high concentrations of heavy ions to produce any significant electron scattering.

[27] The only plausible scenario for obtaining low minimum resonant energies is when high-density plasma is ejected to high L shells. Then, for plasma density of the order of 500 cm−3 at L = 7 even for realistic wave number K = 0.2 Emin ≈ 560 keV. Thus, equation (12) demonstrates that the lowest minimum resonant energies should be expected at higher L shells where B is lowest and in the presence of high plasma density ρ, whereas K must be as high as possible. Such conditions may be expected during expansions of the plasmasphere to large L shells, beyond the geosynchronous orbit, e.g., to L ∼ 7 or higher, or in “plasmaspheric plumes” [see, e.g., Darrouzet et al., 2008].

[28] Meredith et al. [2003], Albert [2003], and Summers et al. [2007a] used the cold plasma dispersion relation to eliminate k from equations (5)(7), yielding a function Emin(ω). In cold plasma theory ∣k∣ → ∞ when ω → Ωj and so Emin drops to zero near the heavy ion gyrofrequencies Ωj. However, as one can see from the accurate solutions of the warm kinetic dispersion relation in Figures 14, the ion gyrofrequencies no longer necessarily correspond to resonances, changing this simple cold-plasma picture. To assess the impact of the warm plasma theory on the resonant Emin, we now replace the cold-fluid solutions for ω and k by the values obtained from the full kinetic dispersion relations, and we present the results in Figures 5 and 6.

Figure 5.

Minimum electron energies for resonant interaction with EMIC waves. (left) L = 4 and (right) L = 7. Dashed lines correspond to the cold fluid approximation. Solid lines correspond to the full kinetic approach. The color bar denotes growth/damping rates along each curve. Panels correspond to parameters from Table 1: (a) case 1, (b) case 2, (c) case 3, (d) case 4, (e) case 5, and (f) case 6.

Figure 6.

Minimum electron energies for resonant interaction with EMIC waves. Dashed lines correspond to the cold fluid approximation. Solid lines correspond to the full kinetic approach. The color bar denotes growth/damping rates along each curve. (left) L = 4 and (right) L = 5. (top) THe = TO = 1 eV and (bottom) THe∥ = TO∥ =10 keV. Panels correspond to plasma parameters in Table 1: (a) case 7, (b) case 8, (c) case 9, and (d) case 10.

[29] In Figure 5 we plot the values of Emin as a function of ω for the wave numbers 0 ≤ kvAp ≤ 1.5. For each branch of the linear dispersion, there is a corresponding branch in the Emin(ω) plot (three curves in total). The lowest values of Emin correspond to the largest wave numbers as predicted by equations (6) and (7). For k = 0 Emin = ∞; that is, there is no resonant interaction. For comparison, the dashed lines represent the values of Emin obtained in the cold plasma approximation equation (1). The solid lines are the corresponding values obtained from solutions of the full kinetic dispersion relation (2). The damping/growth rate is encoded in a color palette. In case of strong damping, we sometimes limited the lower boundary of the color palette so that the growth or damping variations would still be visible. Even though we use the full formula, equation (6), for the parallel electron velocities in the computation of Emin, when Emin is plotted versus the wave numbers k, all six curves (three curves obtained in kinetic and three curves obtained in cold approximation) fall on top of each other. This follows from our observation that all terms containing the wave frequency ω in equations (6) and (7) are negligibly small. Thus, for a given wave number k all three wave branches predicted by linear theory will produce electron scattering at the same minimum resonant energy regardless of their frequency!

[30] Each panel of Figure 5 corresponds to the same case as in Figure 1 (cases 1–6 in Table 1). Figures 5a, 5c, and 5e correspond to L = 4, while Figures 5b, 5d, and 5f correspond to L = 7. One can immediately notice an order of magnitude difference in typical Emin between the two columns due to lower values of B2/ρ at L = 7. The damping at k → 0 at the top of each curve is zero (γ = 0), so one can easily see the transition from growth to damping by the change of the color of the curves: redder colors correspond to growth, while bluer colors correspond to damping. The growth rates for these cases shown in Figure 1 show that at the higher k (lower Emin) boundary the waves are damped. The curves obtained in the kinetic approximation agree best with the cold plasma theory curves for lowest plasma beta (case a) and diverge further and further away as the plasma beta increases.

[31] Figure 6 shows the minimum resonant electron energies for the four cases considered in Figure 3 (cases 7–10 in Table 1). The left and right columns correspond to L shell values 4 and 5, respectively. Figures 6a and 6b describe plasma with cool heavy ions, while Figures 6c and 6d correspond to hot anisotropic heavy ions. We want to warn the reader that the color palette in Figures 6c and 6d is strongly shifted into the negative region, so the unstable region is limited to a narrow red part of the central curves (the detailed profiles of growth and damping rates for these cases are shown in Figures 3c and 3d, respectively). The most notable feature of Figure 6 is that the unstable waves are generated right across the stop band. However, since the wave numbers of these waves are quite small, typically kvAp < 0.5, contrary to the prediction of the cold-plasma theory, the minimum resonant energies remain fairly high. As we have demonstrated in Figure 4, one can obtain similar dispersion at certain L shells even with cool heavy ions, so this effect may be quite common under a wide variety of plasmaspheric conditions.

4. Discussion and Summary

[32] We have examined the EMIC wave interaction with MeV energy radiation belt electrons using the full kinetic plasma theory. Storm time enhancement of the ring current generates hot-ion pressure in the inner magnetosphere, and hence such warm plasma effects should be included in the analysis of EMIC wave-electron interactions. Our analysis not only quantifies the minimum resonant energies for MeV electron-EMIC wave scattering, but also estimates the linear growth/damping rates of EMIC waves at particular frequencies. The main features arising from the thermal effects in relation to the resonant interaction of EMIC waves with relativistic electrons can be summarized as follows.

[33] 1. In the warm kinetic approximation one should expect to observe EMIC waves inside the stop bands. It is possible to excite EMIC waves at the ion gyrofrequencies or anywhere in the stop band frequency range for a broad range of plasma betas, heavy ion abundances and temperatures.

[34] 2. The sharp decrease of Emin when ω → Ωj in the cold plasma approximation appears only because in that theory k → ∞ as ω → Ωj and there is no damping of the short wavelengths for large k. In the full kinetic approach EMIC waves can easily be excited at ω = Ωj with finite k values, so Emin does not asymptotically drop to zero at the ion gyrofrequencies. For most plausible plasma parameters the spectrum of unstable EMIC wave numbers is limited from above by approximately 0.5Ωp/vA.

[35] 3. Solving the full kinetic dispersion relation for the cool ion components instead of using the cold approximation leads to strong cyclotron damping of the EMIC waves at large wave numbers.

[36] 4. The lowest resonant electron energies are achieved for largest wave numbers, but there is no clear dependence of Emin on EMIC wave frequency. For unperturbed realistic plasma parameters, Emin is typically around 2 MeV. Lowest values of the minimum resonant electron energies may be expected during plasmasphere expansion beyond L ∼ 7, or inside plasmaspheric plumes, where Emin may drop to about 500 keV. This typically requires high plasma density (n ∼ 500 cm−3) in the weak magnetic fields (B ∼ 100 nT). Enhanced abundances of heavy ions may also lead to decrease in resonant energies due to higher plasma mass density. However, the wave numbers in this case will tend to decrease and the waves above the ion gyrofrequencies will experience strong cyclotron damping.

[37] We have not considered the effects of wave propagation in inhomogeneous plasma and nonuniform magnetic fields, although these effects are clearly important in the terrestrial magnetosphere and have been studied intensively in the past [see, e.g., Horne and Thorne, 1994]. Our work addresses an entirely different aspect of the EMIC waves, namely their excitation and interaction with relativistic electrons and the possibility of radiation belt losses [see, e.g., Gomberoff and Neira, 1983; Kozyra et al., 1984; Meredith et al., 2003; Albert, 2003; Summers and Thorne, 2003; Summers et al., 2007a, 2007b]. The use of linear dispersion relation limits our investigation to homogeneous plasma density and uniform magnetic field, just as in all of the studies cited above. However, contrary to those studies, we did not use any simplifications regarding plasma dispersion function. We conclude that inclusion of kinetic effects in the plasma dispersion makes the EMIC wave interaction with low-energy relativistic electrons somewhat less likely compared to the traditional cold-plasma approach. In order to resonantly scatter relativistic electrons with energies below 2 MeV, short wavelengths are needed which are in general not supported by the kinetic dispersion relation. Hence, other loss processes, such as scattering by whistler mode waves [see, e.g., Horne and Thorne, 2003], or other mechanisms [see, e.g., Friedel et al., 2002; Shklyar and Kliem, 2006] may be required to explain the loss of radiation belt electrons below ∼2 MeV.

[38] Our analysis shows that in order to estimate the minimum resonant electron energy from the in situ observation of EMIC waves, the wave magnetic field spectra alone are insufficient. One cannot reconstruct the realistic wave dispersion relation ω(k) to obtain the wave numbers k, and hence estimate Emin, without the knowledge of the plasma composition, ion temperatures and plasma beta. As we have demonstrated above, the wave frequency is useless for estimation of Emin, it is the wave number k which is essential. Moreover, wave propagation and refraction in the inhomogeneous plasma and magnetic fields of the magnetosphere will make the interpretation of wave spectra observed far away from the wave source region even more complicated. Thus, the only hope to obtain realistic experimental estimates of Emin is when the wave spectra are measured locally in the wave source region along with the particle distribution functions and DC magnetic field. Then, the kinetic dispersion relation can be used to obtain wave numbers from the measured frequency spectra. These wave numbers could then be substituted into equation (12) to obtain a reliable estimate of Emin.

[39] It should be also borne in mind that nonlinear evolution of the waves may significantly alter the results presented here, which were all obtained using linear stability analysis. As the EMIC waves grow and saturate, they are likely to heat the He+ ions increasing plasma beta and the importance of warm plasma effects, especially below imageAccording to observations, the temperature of He+ in the regions of active EMIC wave activity can exceed the proton temperature by a factor of 2.3 [Anderson and Fuselier, 1994] or even more [Lund et al., 1999]. However, accurate measurements of heavy ion abundances and temperatures are very difficult [see, e.g., Meredith et al., 2003], so we used the plasma densities and temperatures over some range of values which seemed reasonable for the inner magnetospheric conditions, as also suggested by Meredith et al. [2003]. As well, EMIC wave growth and damping rates are expected to change as the waves and plasma evolve nonlinearly. The authors are currently preparing a report on the nonlinear evolution of EMIC waves in thermal plasmas with heavy ion species.

Acknowledgments

[40] We acknowledge fruitful discussions with J. M. Albert. This work is supported by Discovery Grants of the Natural Sciences and Engineering Research Council of Canada to I.R.M., R.D.S., and D.S. and by the WCU grant R31-10016 funded by the Korean Ministry of Education, Science and Technology.

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