Model of Electromagnetic Ion Cyclotron Waves in the Inner Magnetosphere

JournalofGeophysicalResearch: SpacePhysics

RESEARCH ARTICLE Model of electromagnetic ion cyclotron waves 10.1002/2014JA020032 in the inner magnetosphere

Key Points: K. V. Gamayunov1, M. J. Engebretson2, M. Zhang1, and H. K. Rassoul1 • Nonlinear cascade can provide seed energy for EMIC wave growth 1Department of Physics and Space Sciences, Florida Institute of Technology, Melbourne, Florida, USA, 2Department of + • RC O cause generation of EMIC Physics, Augsburg College, Minneapolis, Minnesota, USA waves at highly oblique wave normals • EMIC waves at low L shells may be important for the inner RB loss Abstract The evolution of He+-mode electromagnetic ion cyclotron (EMIC) waves is studied inside the geostationary orbit using our global model of ring current (RC) ions, electric field, plasmasphere, and EMIC Correspondence to: waves. In contrast to the approach previously used by Gamayunov et al. (2009), however, we do not use K. V. Gamayunov, kgamayunov@fit.edu the bounce-averaged wave kinetic equation but instead use a complete, nonbounce-averaged, equation to model the evolution of EMIC wave power spectral density, including off-equatorial wave dynamics. The major results of our study can be summarized as follows. (1) The thermal background level for EMIC waves Citation: Gamayunov, K. V., M. J. Engebretson, M. is too low to allow waves to grow up to the observable level during one pass between the “bi-ion latitudes” Zhang, and H. K. Rassoul (2014), Model (the latitudes where the given wave frequency is equal to the O+–He+ bi-ion frequency) in conjugate of electromagnetic ion cyclotron waves hemispheres. As a consequence, quasi-field-aligned EMIC waves are not typically produced in the model if in the inner magnetosphere, J. Geophys. Res. Space Physics, 119, 7541–7565, the thermal background level is used, but routinely observed in the Earth’s magnetosphere. To overcome doi:10.1002/2014JA020032. this model-observation discrepancy we suggest a nonlinear energy cascade from the lower frequency range of ultralow frequency waves into the frequency range of EMIC wave generation as a possible mechanism Received 21 APR 2014 supplying the needed level of seed fluctuations that guarantees growth of EMIC waves during one pass Accepted 3 SEP 2014 through the near equatorial region. The EMIC wave development from a suprathermal background level Accepted article online 8 SEP 2014 shows that EMIC waves are quasi field aligned near the equator, while they are oblique at high latitudes, and Published online 25 SEP 2014 the Poynting flux is predominantly directed away from the near equatorial source region in agreement with observations. (2) An abundance of O+ strongly controls the energy of oblique He+-mode EMIC waves that propagate to the equator after their reflection at bi-ion latitudes, and so it controls a fraction of wave energy in the oblique normals. (3) The RC O+ not only causes damping of the He+-mode EMIC waves but also causes wave generation in the region of highly oblique wave normal angles, typically for 𝜃>82◦, where a growth rate 𝛾>10−2 rad/s is frequently observed. The instability is driven by the loss cone feature in the RC O+ + + distribution function, where 𝜕F∕𝜕v⟂ > 0 for the resonating O . (4) The oblique and intense He -mode EMIC waves generated by RC O+ in the region L ≈ 2–3 may have an implication to the energetic particle loss in the inner radiation belt.

1. Introduction Electromagnetic ion cyclotron (EMIC) waves are commonly observed in the Earth’s magnetosphere [e.g., Anderson et al., 1992a, 1992b]. In many cases, the free energy for EMIC wave growth is provided by the pressure anisotropy of the ring current (RC) protons. EMIC waves are most easily generated in the vicinity of the magnetic equator with quasi-field-aligned wave normal angles, and they have frequencies below the proton gyrofrequency [Kennel and Petschek, 1966]. These waves strongly affect the RC protons [Gonzalez et al., 1989], thermal electrons [Cornwall et al., 1971], thermal/suprathermal ions [Anderson and Fuselier, 1994; Fuselier and Anderson, 1996; Horne and Thorne, 1997], hot heavy ions [Thorne and Horne, 1994, 1997], and the outer radiation belt (RB) relativistic electrons [Lorentzen et al., 2000; Sandanger et al., 2007], leading to nonadiabatic particle heating and/or pitch angle scattering in the eV-MeV energy range. In particular, EMIC waves in the outer RB cause relativistic electron loss on a time scale of several hours to aday[Summers et al., 2007], competing with adiabatic depletion caused by the Dst effect [Summers and Thorne, 2003], and currently, these waves are considered to be one of the most important causes for relativistic electron loss during the initial and main phases of storms. The rate of the electron loss is not simply controlled by the wave intensity but crucially depends on the wave spectral distribution [Glauert and Horne, 2005], and a properly specified global distribution of the EMIC wave power spectral density (PSD) is needed to accurately describe wave-particle interactions on the global magnetospheric scale throughout the

different storm phases. However, despite a number of statistical and case studies of EMIC waves [e.g., Anderson et al., 1992a, 1992b; Fraser et al., 2010; Halford et al., 2010; Clausen et al., 2011; Usanova et al., 2012; Minetal., 2012], there is still no reliable global and dynamic model of the EMIC wave PSD that could be incorporated in the model of relativistic electrons in the outer RB. Currently, the EMIC wave PSD is poorly specified in all models of the outer RB relativistic electrons. In this situation, a realistic model, verified and constrained by data, of the EMIC wave PSD on a global magnetospheric scale throughout the different storm phases can be very helpful. A hybrid simulation [e.g., Hu and Denton, 2009; Omidi et al., 2010] could be one of the possibilities to self-consistently simulate RC-EMIC wave electrodynamics. However, the hybrid approach treats electrons as a fluid and so is inherently unable to address the resonant wave Landau damping by thermal electrons, which is known to be very important for EMIC wave energetics. In addition, implementation of the hybrid simulation on global spatial and temporal magnetospheric scales requires enormous computational resources, and interpretation of the simulated results is not trivial. The other pos- sibility to obtain the EMIC wave PSD is a quasi-linear approach to the RC-EMIC wave system, which is based on a coupled system of RC ion and wave kinetic equations self-consistently treating the electrodynamic coupling between RC ions and EMIC waves. This approach allows one to model both the global dynamics of the RC ions, which drive the “local” EMIC wave generation on a microscale, and the large-scale feedback from EMIC waves to the RC ion phase space distribution function (PSDF). The global quasi-linear model of Gamayunov et al. [2009] is based on first principles and able to self-consistently simulate the temporal evolution of the spatial and spectral distributions of EMIC wave energy and RC ion PSDF in a multi-ion magnetospheric plasma. This model also includes a self-consistent treatment of the magnetospheric electric field along with a plasmaspheric density model dynamically linked with the magnetospheric electric field. The model of Gamayunov et al. [2009] was the first step in our attempt to develop a self-consistent physics-based model of the global EMIC wave evolution in the Earth’s magnetosphere. Because the He+-mode of EMIC waves is a dominant mode observed in the inner magnetosphere [e.g., Fraser and Nguyen, 2001], only this mode was considered in the paper by Gamayunov et al. [2009]. Our first attempt was based on the bounce-averaged treatment of the He+-mode EMIC wave evolution. The bounce-averaged approach, however, is a significant simplification to the real situation in the magnetosphere. The bounce-averaged approach implicitly assumes that the wave phase trajectories are quasiperiodic, and so waves return to nearly the same equatorial point with nearly the same wave normal angle after one bounce period. It also explicitly requires that the wave PSD does not change significantly during a bounce period. In general, both of these assumptions can be violated in the Earth’s magnetosphere. So a more complete approach based on the original, nonbounce-averaged, wave kinetic equation is needed to model the global EMIC wave evolution, including a latitudinal PSD evolution on a time scale less than a typical wave bounce period ∼1–2 min. The main purpose of this work is to present the first results for the He+-mode EMIC wave global evolution in the Earth’s inner magnetosphere (L < 6.6) that are based on a nonbounce-averaged description of the EMIC wave dynamics. We still neglect the contributions of H+- and O+-mode EMIC waves in this study because He+-mode of EMIC waves is a dominant mode observed in the inner magnetosphere. We also do not consider the so-called EMIC rising tone emissions although they are observed in a close association with the Pc1 waves [Pickett et al., 2010]. There are two reasons why we do not include the EMIC rising tone emissions in our model at this point. First, the rising tone emissions are extremely rare in Earth’s magnetosphere. For example, a focused survey of 8 years of Cluster data in order to increase the database of EMIC rising tone emissions has revealed unambiguously only three events [Grison et al., 2013]. A survey of the Time History of Events and Macroscale Interactions during Substorms (THEMIS) data over 5 years has obtained a few tens of rising tone emission events out of about 7000 events observed Nakamura et al. [2014]. For comparison, a study of Active Magnetospheric Particle Tracer Explorers (AMPTE)/CCE magnetic field data from 1984 day 239 through 1985 day 326 has found 9316 Pc1 and Pc2 events (the events occurred for a total of 347.3 h out of 7544 h of observations or 4.60% of the total observation time) [Anderson et al., 1992a]. Similarly, during the CRRES mission from 25 July 1990 through 11 October 1991, Fraser and Nguyen [2001] have found 830 EMIC events (over 96 h of the analyzed 7248 h or 1.33% of the total observation time). Second, there is an indication [e.g., Pickett et al., 2010; Nakamura et al., 2014] that EMIC rising tone emissions could result from a nonlinear wave-proton interaction, which is triggered when the quasi-linearly amplified Pc1 waves exceed some threshold. If this is the case, one first needs to have a reliable quasi-linear model of EMIC waves in the global magnetosphere in order to rigorously study the following nonlinear wave-proton interaction.

This article is organized as follows. In section 2, a set of governing equations is given along with the assumptions and initial and boundary conditions used in the simulations. In section 3, we consider the He+-mode EMIC wave development from the thermal background level. The distributions of wave energy, normal angle, and Poynting flux are shown and discussed. The presented wave characteristics reveal two model shortcomings: (1) a majority of the wave energy is concentrated in the region of oblique normals, while the observed He+-mode EMIC waves are generally quasi field aligned inside the geostationary orbit, and (2) the Poynting flux anisotropy does not demonstrate prevalence of any direction for the wave energy propagation, while observations show a predominant wave energy propagation away from the near equatorial source region. To find the reason(s) for these model-observation discrepancies, we then analyze in detail the He+-mode EMIC wave generation and damping. The analysis shows that in order for the quasi-field-aligned waves to be produced in our model, they should grow up to the observable level during one pass through the near equatorial unstable region of about ±10◦. The thermal background level for EMIC waves, however, is low, and as a consequence, waves typically cannot reach an observable level during one pass through the near equatorial region. Previously, to overcome this drawback, the mechanism of EMIC wave growth rate modulation by low-frequency compressional Pc5 waves was frequently invoked. However, this mechanism cannot account for all observations of EMIC waves. So we suggest a nonlinear energy cascade from the lower range of ultralow frequency (ULF) waves into the frequency range of EMIC wave generation as a possible mechanism supplying the needed level of seed fluctuations that guarantees growth of EMIC waves during one pass through the near equatorial region. The EMIC wave development resulting from a suprathermal background level shows that both model-observation discrepancies identified above are gone. In section 4, we provide a summary and conclusions. Finally, in Appendix A, we present the model distributions of the plasmaspheric density and RC ion parameters, which are among the most crucial characteristics that control EMIC wave generation and evolution, during the studied event.

2. Dynamic Coupling of RC Ions, Electric Field, Plasmasphere, and EMIC Waves 2.1. Governing Equations To model RC ions we use the same approach we used in our previous work [Khazanov et al., 2003, 2006; Gamayunov et al., 2009]. The RC ion module used in our code has been developed at the University of Michigan by Fok et al. [1993] and Jordanova et al. [1996]. This module is very well tested and widely used in RC studies. It is a core part of well-known RC models such as Comprehensive Ring Current Model (CRCM) at NASA/Goddard Space Flight Center [e.g., Fok et al., 2001], Ring current-Atmosphere interactions Model (RAM) at Los Alamos National Laboratory (LANL) [e.g., Jordanova et al., 2006], and Hot Electron and Ion Drift Integrator (HEIDI) at the University of Michigan [e.g., Liemohn and Katus, 2012]. The RC module solves the ,𝜑, ,𝜇 , + + + bounce-averaged kinetic equation for the PSDF F(r0 E 0 and t) of the major RC species H , O , and He . The PSDF depends on the radial distance in the magnetic equatorial plane r0, geomagnetic east longitude 𝜑 𝜇 , kinetic energy E, the equatorial pitch angle cosine 0, and time t. The resulting kinetic equation has the following form [e.g., Gamayunov et al., 2009]:

( ⟨ ⟩ ) (⟨ ⟩ ) (√ ⟨ ⟩ ) 𝜕F 1 𝜕 dr 𝜕 d𝜑 1 𝜕 dE + r2 0 F + F + √ E F 𝜕t 2 𝜕r 0 dt 𝜕𝜑 dt 𝜕E dt r0 0 E ( ⟨ ⟩ ) ⟨( ) ⟩ 𝜕 d𝜇 𝛿 1 𝜇 𝜇 0 F . + 𝜇 𝜇 𝜕𝜇 0h( 0) F = 𝛿 (1) 0h( 0) 0 dt t loss

The left-hand side (LHS) in equation (1) includes all the bounce-averaged drifts velocities, which are denoted as ⟨⋅⟩. The spatial drift velocities include the E × B drift and the magnetic gradient-curvature drift [e.g., Roederer, 1970]. The drift velocities for the kinetic energy and equatorial pitch angle cosine can be obtained from conservation of the ﬁrst and the second adiabatic invariants [e.g., Jordanova et al., 1994]. The energy drift velocity also includes energy degradation due to Coulomb collisions. The right-hand side (RHS) includes losses from charge-exchange, Coulomb collisions, RC-EMIC wave scattering, and RC ion precipitation into the atmosphere [Jordanova et al., 1996]. For EMIC waves, in contrast to the approach previously used by Gamayunov et al. [2009], we do not use the bounce-averaged kinetic equation but instead use the original equation to model the full evolution

3.0 of the EMIC wave PSD, including the nλ oﬀ-equatorial wave dynamics. This equation k has the following form [e.g., Stix, 1992]:

2.0 𝜕B2 (r, t, k) 𝜕𝜔(r, k) 𝜕B2 θ P nr k + k 𝜕t 𝜕k 𝜕r 𝜕 2 ) 𝜕𝜔 B E r − k = 2𝛾 (r, t, k) B2, (2) 1.0 𝜕r 𝜕k k where dr∕dt = 𝜕𝜔(r, k) ∕𝜕k, and dk∕dt = 2 λ r0 −𝜕𝜔(r, k) ∕𝜕r. In equation (2), B (r, t, k) 0.0 k

Distance Z (R is the EMIC wave PSD with normalization ∫ 2 , , ∫ 2 𝜋 2 𝜃 𝜃 Bk (r t k) dk = Bk2 k sin dkd = ∫ 2 𝜋 2 𝜃 |𝜕𝜔 𝜕 | 𝜔 𝜃 Bk2 k sin ∕ ∕ k d d = -1.0 ∫ 2 , ,𝜔,𝜃 𝜔 𝜃 2 , B (r t ) d d = Bw (r t), 2 , where Bw (r t) is the squared magnetic field of EMIC waves, B2 (r, t,𝜔,𝜃) = -2.0 2 𝜋 2 𝜃 |𝜕𝜔 𝜕 | 𝜔 𝜃 0 1 2 3 4 5 Bk2 k sin ∕ ∕ k , and are the angular wave frequency and wave normal Distance X (RE) angle, respectively, k is the wave vector, Figure 1. The coordinate systems used to integrate the ray tracing and 𝛾 (r, t, k) is the imaginary part of the equations. 𝜃 is the angle between the directions of the local mag- frequency which includes both the wave netic field and the wave normal vector. The dashed lines are two growth rate due to interaction with RC ions dipole magnetic field lines. and the damping rate due to absorption by the thermal plasma and RC ions. The wave PSD in equation (2) completely describes the EMIC wave energy evolution in r-k phase space. The characteristics of the LHS in equation (2) are solu- 2 , , tions of the ray tracing equations. So the general solution of equation (2) is an initial distribution Bk (r t k) that is just transported along characteristics with simultaneous growth and/or damping in accordance with 𝛾 (r, t, k) along these phase trajectories. To have characteristics of the LHS of equation (2), a set of ray tracing equations is also solved. These equations can be written for a plane geometry as follows [e.g., Haselgrove, 1954; Haselgrove and Haselgrove, 1960; Kimura, 1966; Gamayunov et al., 2009] dr (𝜕G∕𝜕k) =− r , (3) dt 𝜕G∕𝜕𝜔

d𝜆 (𝜕G∕𝜕k)𝜆 r =− , (4) dt 𝜕G∕𝜕𝜔 𝜕 𝜕 dkr d𝜆 ( G∕ r)r = k𝜆 + , (5) dt dt 𝜕G∕𝜕𝜔

dk𝜆 k𝜆 dr (𝜕G∕𝜕r)𝜆 =− + . (6) dt r dt 𝜕G∕𝜕𝜔 In equations (3)–(6), an Earth-centered polar coordinate system is used, which is shown in Figure 1. Any point P on the ray trajectory is characterized by its radius vector of length r and magnetic latitude 𝜆.The

components of the wave vector kr and k𝜆 are given in the local Cartesian coordinate system centered on the current point P with its axes oriented along the radius vector and the magnetic latitude directions, respectively. The function G (𝜔, k, r) has roots for the EMIC eigenmodes only, i.e., G (𝜔(k), k, r) = 0 at any point along the phase trajectory. An extensive verification of our ray tracing code against previously published results for different plasmaspheric density models and for all possible EMIC wave modes in a multi-ion magnetospheric plasma demonstrated a very good agreement between the results [Khazanov et al., 2006]. The quasi-linear pitch angle diffusion coefficient in the RHS of equation (1) is a functional form of the EMIC wave PSD, and 𝛾 (r, t, k) in equation (2) is a functional form of the RC PSDF. So there is a system of coupled equations, and the entire set of equations (1)–(6) self-consistently treats the interacting RC ions and EMIC waves.

20 2.2. Used Approaches and Initial and Boundary Conditions 0 The electric ﬁeld in our model is treated (nT)

Y self-consistently [Gamayunov et al., 2009], and

B -20 the total magnetospheric electric field is the -40 sum of a self-consistent convection field and a 20 corotation field. The module dealing with the

0 large-scale magnetosphere-ionosphere coupling and the model of the ionosphere used (nT) Z

B -20 in our code have been developed at the Uni- versity of Michigan [Liemohn et al., 2001; Ridley -40 and Liemohn, 2002; Liemohn et al., 2004], where 103 they are currently used in the RC HEIDI model and in the University of Michigan global MHD model of the magnetosphere. The ionospheric

V (km/s) boundary conditions for the electric potential

102 are taken from the Weimer [1996] model, which is driven by the interplanetary magnetic field 8 BY , BZ components and solar wind velocity. The 6 Hardy et al. [1987] ionospheric conductance Kp 4 model is used in this study, and this model is 2 driven by Kp. All of these driving parameters 0 are shown in Figure 2 during 4 May 1998. Inter- 0 planetary data are obtained from the Magnetic -100 Field Investigation [Lepping et al., 1995] and the Solar Wind Experiment [Ogilvie et al., 1995] -200 Dst (nT) instruments aboard the Wind satellite. The geo- measured Dst magnetic field in this study is taken to be a -300 100 dipole field. The equatorial plasmaspheric electron den-

10-1 + sity is calculated using the dynamic global core H plasma model of Ober et al. [1997]. This model O+ + is basically the same as the time-dependent GEO RC Frac. He 10-2 72 84 96 model of Rasmussen et al. [1993], which was Hours after 00 UT on May 1, 1998 used in our previous studies, except the Ober et al. model is linked with a self-consistent elec- Figure 2. The interplanetary and magnetospheric characteristics during 4 May 1998. (from top to bottom) The interplanetary tric field, while the Rasmussen et al. model is magnetic field GSM BY and BZ components, the solar wind driven by the Volland-Stern convection field velocity,the3hKp index, the measured Dst index, and the frac- [Volland, 1973; Stern, 1975] with Kp parameter- tions of major RC ions at geosynchronous distance according ization. The Ober et al. model is currently used, to the Kp and F parametrization given by Young et al. [1982]. 10.7 for example, in the HEIDI model of RC at the The hours shown are counted from 0000 UT on 1 May 1998. University of Michigan [e.g., Liemohn and Katus, 2012], and modification of the Rasmussen et al. model is used in RAM at LANL [e.g., Jordanova et al., 2006]. To obtain the meridional density distribution, we use the analytical density model of Angerami and Thomas [1964], which is adjusted to the Ober et al. model at the equator. So the resulting plasmaspheric model provides a 3-D spatial distribution of the electron density. The thermal ion content is currently externally specified, and nominally 77% H+, 21%He+, and 2% O+ are used. This ion content is in the observed range of 10%–30% for He+ and 1%–5% for O+ [e.g., Young et al., 1977; Horwitz et al., 1981]. The wave damping rates by the plasmaspheric particles are calculated assuming . a Maxwellian distribution for all species, where temperatures Te = 2 eV, TH+ = THe+ = 0 6 eV, and TO+ = 1 eV are assumed for the electrons, H+, He+, and O+, respectively. Only the RC H+ and O+ are modeled in this study. The initial RC ion distributions are constructed from statistical energy distributions of Sheldon and Hamilton [1993] and the pitch angle characteristics of Garcia and Spjeldvik [1985]. The energy distributions are derived from measurements by the charge-energy-mass

Table 1. Model Input Parameters for the Plasmaspheric and RC Speciesa

Species Plasmasphere RC Fraction at GEO RC Initial Distribution − e 100%, Te = 2 eV Not modeled Not modeled + . H 77%, TH+ = 0 6 eV F10.7 and Kp driven AMPTE/CCE-CHEM + . He 21%, THe+ = 0 6 eV Not modeled Not modeled + O 2%, TO+ = 1 eV F10.7 and Kp driven AMPTE/CCE-CHEM aThe listed plasmaspheric parameters are used in all the runs except sections 3.2.2 and 3.2.3, where the eﬀects of plasmaspheric electron temperature and O+ on wave energetics are analyzed. The fractions of RC ions at geosynchronous distance during 4 May 1998 are shown in Figure 2.

(CHEM) spectrometer on board the AMPTE/CCE satellite during quiet conditions with |Dst| < 11 nT and Kp < 2+. The pitch angle characteristics are derived from the quiet time radiation belt ion data obtained by the instruments flown on Explorer 45. The nightside boundary conditions for RC ions are imposed at the geostationary orbit. In this study we use flux measurements from the Magnetospheric Plasma Ana- lyzer (MPA) (up to 50 keV) [Bame et al., 1993] and the Synchronous Orbit Particle Analyzer (SOPA) (above 50 keV) [Belian et al., 1992] instruments on the geosynchronous LANL satellites during the modeled event. These instruments measure the total ion flux only, and we use a parametric formula derived by Young et al. [1982] to split the total measured flux between H+, O+, and He+. The Young et al. formula is based on the observations at geosynchronous orbit by the GEOS-1 and GEOS-2 satellites. It gives the ion composition

as a function of solar activity, as represented by F10.7 ﬂux, and magnetic activity, as represented by the Kp index. This formula is widely used in RC modeling [e.g., Kozyra et al., 2002; Jordanova et al., 2006; Liemohn and Katus, 2012]. The fractions of RC ions at geosynchronous distance during 4 May 1998 are shown in Figure 2, where the RC O+ content does not exceed 40%. However, we have to note that comparison with the measurements by the CHEM spectrometer on board AMPTE/CCE shows that the Young et al. formula underestimates the O+ fraction during geomagnetic storms, at least for the February 1986 storm [Hamilton et al., 1988]. The model input parameters used in this study for the plasmaspheric and RC species are summarized in Table 1. Finally, in all cases presented in this study, except section 3.4, the He+-mode EMIC waves grow from the thermal background level, which is taken from Akhiezer et al. [1975], and we prepare the entire system by running the model code during 3 days of 1–3 May 1998 before simulating the geomagnetic storm on 4 May.

Figure 3. The squared magnetic ﬁeld for the model He+-mode EMIC waves (ﬁrst row) in the equatorial plane and (second row) in the meridional plane averaged over all magnetic local times (MLTs).

3. Results The distributions of the plasmaspheric density and RC parameters are among the most crucial elements that control EMIC wave generation and evolution. For this reason, we show in Appendix A the model distributions of these controlling parameters during hours 74–82. All the distributions shown in Appendix A are in a very reasonable agreement with the observations. To have an overall global picture of He+-mode EMIC wave evolution and look at the model performance, we show in section 3.1 the wave magnetic field distributions on a large time scale during hours 74–82 (4 May 1998) and discuss the model distributions of wave normal angle and the Poynting flux. 3.1. EMIC Wave Evolution on Large Time Scale 3.1.1. Wave Magnetic Field The first row in Figure 3 shows the squared magnetic field of the He+-mode EMIC waves in the equatorial plane which is calculated as ( ) ( ) 2 ,𝜆 ,𝜑, 2 ,𝜆 ,𝜑, ,𝜔,𝜃 𝜔 𝜃. Bw r0 = 0 t = ∫ B r0 = 0 t d d (7)

The wave activity is mostly located in the noon-dusk MLT sector in agreement with the statistical studies of EMIC waves [e.g., Anderson et al., 1992a; Fraser and Nguyen, 2001; Halford et al., 2010; Usanova et al., 2012]. Persistent wave activity observed at the outer edge of the simulation domain is also in agreement with observations, which demonstrate an increase in the wave occurrence rate with L shell [Anderson et al., 1992a; Fraser and Nguyen, 2001; Halford et al., 2010; Usanova et al., 2012]. Comparison of the first row in Figure 3 with Figure 7 in the paper by Gamayunov et al. [2009] (hour 77) shows that a majority of the wave activity is located inside of the plasmaspheric plume, but the effect of the plasmapause on wave organization is not so obvious, similar to the CRRES observations reported by Fraser and Nguyen [2001]. The wave activity in the region L < 3 is due to RC O+, and we will discuss this feature later. In the second row of Figure 3 we show the meridional distribution of the He+-mode EMIC wave magnetic field averaged over all MLTs. To obtain the MLT average for any function C (r, t,𝜔,𝜃), the following integral is evaluated ( ) ∫ C (r, t,𝜔,𝜃) B2 (r, t,𝜔,𝜃) d𝜔d𝜃d𝜑 ⟨C r ,𝜆,t ⟩ = , (8) 0 ∫ B2 (r, t,𝜔,𝜃) d𝜔d𝜃d𝜑 ( ) 2 ,𝜆,𝜑, + where C = Bw r0 t in the case shown in Figure 3. In this study, we model all the He -mode EMIC wave trajectories that cross the equatorial plane at least once. However, for the further consideration we only keep the ray trajectories which get reflected at the O+–He+ “bi-ion latitudes” (the latitudes where the given He+-mode EMIC wave frequency is equal to the O+–He+ bi-ion frequency). A schematic of the reflected ray trajectories is similar to that shown in Figure 6 of the paper by Perraut et al. [1984], except we do not consider here the wave tunneling through the O+–He+ bi-ion latitudes. (Note that wave tunneling may be an important energy sink for the He+-mode EMIC waves when the O+ fraction in plasma is less than about 1% [e.g., Khazanov et al., 2006].) All the ray trajectories that pass the O+–He+ bi-ion latitudes propagating downward to the ionosphere eventually leave the simulation domain, and we neglect them. The fraction of the ray trajectories which leave our simulation domain, including those that pass the bi-ion latitudes and propagate downward to the ionosphere, usually does not exceed 20%. The waves that propagate and/or tunnel through the bi-ion latitudes can be detected at higher latitudes in the magnetosphere and ultimately on the ground, and we are planning to study this fraction of waves in the future. So Figure 3 shows energy distribution only for those waves that are bouncing between “bi–ion latitudes” in the conjugate hemispheres. In addition, only one hemisphere is shown in the second row because the plasma model used here assumes a north–south symmetry with the following change in the wave normal angle 𝜃 → 𝜃∕|𝜃|𝜋 − 𝜃. The most intense waves are seen in the region of low L shells, where they are generated by the RC O+. We will show the relevant wave growth rates and discuss this effect in more detail later. 3.1.2. Wave Normal Angle and Poynting Flux Distributions We also calculate the average meridional distributions of the wave normal angle (not shown). The averages are obtained from equation (8), where C = 𝜃 and integrations over wave normal angles 𝜃 = 0◦–90◦ and 𝜃 = 90◦–180◦ are performed for EMIC waves propagating from the equator and to the equator, respectively. A majority of the wave energy is concentrated in the region of oblique normals 𝜃 ≈ 70◦–110◦. To estimate

the wave normal angle from observations, the minimum variance direction is usually determined by apply- ing Fast Fourier Transform (FFT) decomposition to the data. The extracted 𝜃 is generally less than 30◦ [e.g., Fraser, 1985; Ishida et al., 1987]. While highly oblique EMIC waves are observed in the outer magnetosphere [e.g., Anderson et al., 1996a, 1996b; Minetal., 2012], they are generally quasi ﬁeld aligned inside the geostationary orbit. So the model results do not agree with observations of wave normals for the He+-mode EMIC waves in the inner magnetosphere. The average meridional distributions of the Poynting ﬂux for the He+-mode EMIC waves are also calculated

(not shown). The field-aligned component (vg) of the EMIC wave group velocity dominates the transverse component, and so we calculate only the field-aligned component of the Poynting vector. The averages ∫ 2 𝜔 𝜃 𝜋 𝜃 ◦ ◦ are again obtained from equation (8), where C = vgB d d ∕8 and integrations over = 0 –90 and 𝜃 = 90◦–180◦ are performed to obtain the Poynting flux from the equator and to the equator, respectively. The most comprehensive statistical study of the EMIC wave Poynting flux by Loto’aniu et al. [2005] is based on the CRRES data, and they reported the maximum Poynting flux of 25 W/km2 and the average value of 1.3 W/km2. Note, however, that a collection of all the observations reviewed by Loto’aniu et al. gives a wide range of 0.1–100 W/km2 for the EMIC wave Poynting flux in the inner magnetosphere. The model flux values are close to the statistical observations for EMIC waves. The largest Poynting fluxes of ∼ 25 W/km2 are seen in the region of low L shells because both the wave magnetic field (see Figure 3) and group velocity are large there. We also calculate the Poynting flux anisotropy in the meridional plane averaged over all , ,𝜔 (MLTs (not shown). The anisotropy) ( is calculated using equation) (8), where C =A (r t )= 𝜋 𝜋 𝜋 𝜋 ∫ ∕2 2 𝜃 ∫ 2 𝜃 ∫ ∕2 2 𝜃 ∫ 2 𝜃 0 vgB d + 𝜋∕2 vgB d ∕ 0 vgB d − 𝜋∕2 vgB d , i.e., A=1 if wave energy is propagating from the equator only, A=−1 in the opposite case, and A = 0 if the Poynting fluxes from and to the equator can- cel each other. On the whole, there is no clear prevalence of any direction for the wave energy propagation. This does not agree with the observations of EMIC wave Poynting flux that show predominant wave energy propagation away from the near equatorial source region [e.g., Loto’aniu et al., 2005]. 3.2. Detailed Analysis of He+-Mode EMIC Wave Energetics The wave characteristics presented in section 3.1 revealed two model shortcomings: (1) a majority of the wave energy is concentrated in the region of oblique normals, while the observed He+-mode EMIC waves are generally quasi field aligned inside the geostationary orbit, and (2) the Poynting flux anisotropy does not demonstrate the prevalence of any direction for the wave energy propagation, while observations show a predominant wave energy propagation away from the near equatorial source region. At the same time, however, the model distributions of the RC parameters and plasmaspheric density shown in Appendix A are all in very reasonable agreement with the observations. The results from our ray tracing module are also in very good agreement with the previously published results. So the reason(s) for the above model shortcomings is(are) likely related to the wave energetics. In this section we first consider in detail the He+-mode EMIC wave generation and damping in order to find the reason(s) for the above two model shortcomings. Then we consider the effects of plasmaspheric electron temperature and O+ fraction on wave energetics. 3.2.1. Effect of RC H+ and O+ on Wave Growth and Damping Both the RC H+ and O+, depending on the plasma and wave characteristics, are able to generate the He+-mode EMIC waves and cause their damping. Figure 4 shows the imaginary part of the He+-mode EMIC wave frequency for the case when only the resonant interaction with RC O+ is taken into account. The results shown are taken from our model at L = 3, MLT = 23, and hour 76. The growth/damping rates for four wave frequencies are shown. The RC O+ causes damping of the He+-mode EMIC waves for a majority of wave normals and for all latitudes because the wave frequency is higher than the frequency of marginal insta- + + bility, which is less than the O gyrofrequency ΩO+ , set by the RC O temperature anisotropy [e.g., Thorne + and Horne, 1994]. (Similar to the case of proton cyclotron instability considered( ) in Appendix A, the RC O + >𝜔 𝜔 + anisotropy can cause growth of the O -mode EMIC wave if AO+ ∕ ΩO+ − .) However, the RC O can also cause wave growth (𝛾>0) in the region of highly oblique wave normals (typically for 𝜃>82◦), where growth rate 𝛾>10−2 rad/s is frequently observed. The wave instability is driven by the loss cone feature + + [e.g., Denton et al., 1992] of the RC O PSDF, where 𝜕F∕𝜕v⟂ > 0 for the resonating O . The strongest EMIC wave generation takes place at the second O+ cyclotron resonance, while a weaker wave generation at the higher resonances is also seen in the model results. The largest growth rate is observed near midnight and for low L shells: for example, 𝛾 ∼ 10−1 rad/s around L ∼2. While the distribution of the growth rate in MLT

Figure 4. The imaginary part of the He+-mode EMIC wave frequency where only the resonant interaction with RC O+ is taken into account. The results are taken from our model at L = 3, MLT = 23, and hour 76. (a–d) The growth/damping rates for four wave frequencies are shown. The RC O+ can potentially cause wave growth (𝛾>0) only in the region of highly oblique wave normals, and we show the latitudinal extent where 𝛾>10−2 rad/s. (Note that to keep the validity of the equation used for the imaginary part of the wave frequency, we always hold 𝛾 = 𝛾∕|𝛾|10−1𝜔 when |𝛾| > 10−1𝜔,and the plateau in Figure 4a is due to this.)

becomes more homogeneous with L, the growth rates around noon are still smaller. The latitudinal extent of wave instability depends on wave frequency, but this extent is always large. We show in Figure 4 only the latitudinal extents where 𝛾>10−2 rad/s, but 𝛾<0 outside of the shown limits. (Note, however, that cyclotron resonance numbers contributing to the growth rate change with latitude, and 𝛾 for a specified frequency and oblique wave normals can be negative in a localized latitudinal extent inside the latitudes shown in Figure 4.) The highest latitudes shown in Figure 4 are only about 2◦–3◦ less than the respective bi-ion latitude for the specified frequency. So the RC O+ can generate EMIC waves with highly oblique wave normal angles over a wide latitudinal extent. Figure 5a shows the imaginary part of the He+-mode EMIC wave frequency where a resonant interaction with only the RC H+ is taken into account. The results for five frequencies are shown at L=4, MLT = 18, 𝜆=1.1◦, and hour 76. The RC H+ can cause both wave generation and damping depending on the wave frequency and normal angle. Wave generation inside the region from the field-aligned angles up to about 𝜃 =30◦–50◦ is due to the positive temperature anisotropy of the RC H+, where generation takes place at the first cyclotron resonance. This instability is well known [e.g., Kennel and Petschek, 1966], and it is usually considered as the cause of EMIC waves in the RC. Wave generation in the region of oblique wave normals is driven by the loss cone feature of the RC H+ distribution, similar to the case of RC O+ discussed above, and/or by the positive temperature anisotropy of the RC H+ at the second cyclotron resonance. Both these mechanisms were previously used for RC H+ and H+-mode EMIC waves to explain the linear polarization of waves observed by AMPTE/CCE in the dawn MLT sector of the outer magnetosphere (see Denton et al.

Figure 5. (a) The imaginary part of the He+-mode EMIC wave frequency where only the resonant interaction with RC H+ is taken into account. The results for ﬁve frequencies are shown at L = 4, MLT = 18, 𝜆 = 1.1◦, and hour 76. (b) Similar to Figure 4, except at the location L = 4 and MLT = 18.

[1992] for the former mechanism and see Anderson et al. [1996b] for the latter mechanism). Overall, significant wave generation by RC H+ usually takes place for L > 3 in the afternoon-midnight MLT sector, and the latitudinal extent where 𝛾>0 does not exceed 10◦–15◦ off the equator. Quasi-field-aligned EMIC wave generation can be only caused by the RC H+. However, these ions also support a generation of oblique waves, while the unstable wave normals are typically smaller compared to the case of highly oblique waves generated by the RC O+. 𝜔 . The most effective wave generation shown in Figure 5a takes place for the wave frequency ∕ΩO+,eq ≈ 2 8. For this frequency, the wave damping rate due to the RC O+ is shown in Figure 5b for six latitudes. It follows from comparison of Figures 5a and 5b that quasi-field-aligned EMIC waves can only grow over a latitudinal extent not larger than about ±10◦ off the equator but will eventually enter the region of damping by the RC O+ during wave propagation and refraction. The EMIC waves generated as quasi-field-aligned waves return to the equator with oblique normals, i.e., they cannot be re-amplified due to the quasi-field-aligned instability, except in the relatively rare cases where waves are guided by density gradients. So, for the quasi-field-aligned waves to be observed, they should grow up to the observable level during one pass through the near equatorial unstable region. However, the thermal background level for the He+-mode EMIC waves, which is used in all our simulations shown in section 3.1, is low (< 10−18 nT2), and wave intensity cannot be accumulated in the region of field-aligned normals because of wave refraction. So waves typically cannot reach an observable level (say ∼10−2 nT2) during one pass through the near equatorial region. To verify/confirm that, we did a simulation of the nominal case during hours 76–78, except that for any specified wave frequency 𝜔, the damping rate 𝛾 =−10−1𝜔 is prescribed inside the spatial cells where this particular frequency equals to the bi-ion frequency. The simulation results (not shown) indeed show that existing waves damp during less than 1 min, and there is no wave activity during the following 2 h of simulations. This explains why we do not see the quasi-field-aligned EMIC waves in the results shown in section 3.1. The situation is different for oblique EMIC waves. These waves are directly generated by the RC O+ and H+, and there is also a source of oblique waves from those waves that were initially generated by the RC H+ with the quasi-field-aligned normals but “survived” damping by the RC O+ during wave refraction into the region of highly oblique normals. Once waves get in the region of highly oblique normals, they will likely stay there for many bounce periods. The resulting energetics of highly oblique waves depends on the balance between wave growth due to the RC O+ and H+ from one side and wave energy dissipation by the thermal electrons, RC O+, and wave tunneling through the O+–He+ bi-ion latitudes from the other side. Figure 5 and the above analysis explain why a majority of the wave energy is concentrated in the region of oblique normals in the results discussed in section 3.1. In addition, when EMIC waves grow to an

Figure 6. The squared magnetic field for the model He+-mode EMIC waves (first to third rows) in the equatorial plane and (fourth to sixth rows) in the meridional plane averaged over all MLTs during hours 76–78 with a 20 min cadence. The first and fourth rows are for the case where both RC H+ and O+ are included in the imaginary part of the wave dispersion relation, while in the second/fifth and third/sixth rows only, the RC H+ or O+ are, respectively, included.

observable level during multiple oscillations between bi-ion latitudes in the conjugate hemispheres, the spatial localization of the near equatorial unstable region is masked. Effectively it appears as a wide wave source distributed between bi-ion latitudes in the conjugate hemispheres. This also explains why the Poynting flux anisotropy discussed in section 3.1 does not demonstrate the prevalence of any direction for the wave energy propagation. So the correct specification of the seed fluctuation level for EMIC wave growth is crucial to overcome the two model shortcomings revealed in section 3.1. Below, in sections 3.3 and 3.4, we will discuss in more detail the issue of seed fluctuation level for EMIC wave growth and show how this level affects the resulting wave normal angle and energy flux distributions. To illustrate how the RC H+ and O+ drive the global EMIC wave energetics together and independently, we modeled the end of the main and beginning of the recovery phases during hours 76–78 (see Figure 2). Despite the crucial effect of the wave background level on wave characteristics, we still keep here the thermal background level for the He+-mode EMIC waves. We show the wave energy distributions only, and the results shown are for comparative purposes only. While distributions were produced every minute, they are only shown with a 20 min cadence. Figure 6 shows the equatorial and meridional distributions of the squared wave magnetic field for three cases. The first and fourth rows are for the case where both the RC

H+ and O+ are included in the imaginary part of the He+-mode EMIC wave dispersion relation, while in the second/fifth and third/sixth rows, only the RC H+ or O+, respectively, are included. We also keep here the contribution in the imaginary part from Landau damping by thermal electrons. It is difficult to say whether the RC H+ or whether O+ contributes mostly in the wave development, except in the region around L ∼2 where the RC O+ determines wave energetics (compare the first/fourth and third/sixth rows in Figure 6). The RC H+ and O+ can reinforce each other making wave generation possible only when both species are included as, for example, at hour 76.67. These ions can also counteract each other leading to wave generation being suppressed as, for example, at hour 77. 3.2.2. Effect of Plasmaspheric Electron Temperature on Wave Energetics Landau damping by thermal plasmaspheric electrons is an important mechanism among those that control EMIC wave energetics [e.g., Cornwall et al., 1971; Thorne and Horne, 1992]. The most effective wave energy dissipation takes place where the wave phase speed along the background magnetic field is close to the electron thermal speed. The phase speed, in turn, is proportional to the Alfvén speed and so controlled by the models used for the magnetic field and plasmasphere. On the other hand, we do not model the plasmaspheric temperature, and the electron temperature is currently a prescribed parameter in our model. To look at the sensitivity of the EMIC wave energetics to the plasmaspheric electron temperature, we calculate the squared magnetic field for the model He+-mode EMIC waves for the nominal case . Te =2 eV and the case Te =5 eV during hours 76–78 (not shown). The equatorial Alfvén speed at L=6 5 is about the 1 eV electron thermal speed, and it is about 3 times smaller than the Alfvén speed at L=2. So the electron thermal speeds in both cases are between the minimum and maximum of the equatorial Alfvén speed in the simulation domain. As a result, an increase in the electron temperature from 2 eV to 5 eV leads to a more effective Landau damping in the region of low L shells and a decrease of damping at the large L shells. This tendency is clearly seen in the model results. The observed dependence on electron temperature is strong enough, suggesting that a dynamic model of the plasmaspheric electron temperature should be ultimately included in the global model to accurately describe EMIC wave evolution. In respect of the two model shortcomings pointed out above, it is important to emphasize that electron Landau damping maximizes in the region of highly oblique normals [e.g., Thorne and Horne, 1992]. So it controls, at least partly, the energy of oblique waves that propagate to the equator after their reflection at bi-ion latitudes. 3.2.3. Effect of Plasmaspheric O+ Density on Wave Energetics To illustrate the effect of the O+ density on EMIC wave development, we leave the RC O+ population as before but only change the fraction of cold plasmaspheric O+. (Note that despite including RC O+ only in the imaginary part of the EMIC wave dispersion relation, modification of the RC O+ not only causes a change of the growth/damping rate but the cold plasmaspheric distribution as well because the self-consistent electric field depends on RC O+. For this reason, we decided not to alter the RC O+ and vary only the thermal O+ fraction.) The change of the cold O+ density modifies the dispersive properties of waves that, in turn, may modify the wave propagation and growth/damping rate. In Figure 7 we show the equatorial and meridional distributions of the squared magnetic field for the model He+-mode EMIC waves during hours 76–78. We again use a thermal background level for waves. The first and fourth rows are obtained for 3% O+ in the plasmaspheric model, and the second/fifth and third/sixth rows are for 2% (nominal case) and 1.5%, respectively. A change of the O+ percentage modifies the ratio of the O+–He+ bi-ion frequency to 𝜔 . + ΩO+ . For example, bi∕ΩO+ ≈ 1 09 for the nominal case (2% O ), and this ratio approaches unity with + 𝜔 decrease in the O percentage. The bi∕ΩO+ ratio controls the EMIC wave damping rate in the vicinity of the bi-ion frequency that strongly grows with a decrease in the O+ fraction [e.g., Thorne and Horne, 1994]. A strong modification of the damping rate in the vicinity of the bi-ion frequency is the most pro- nounced effect of the cold O+ density change. The wave damping at the second cyclotron resonance is less sensitive to modification of wave dispersion caused by an increase/decrease in the cold O+ fraction, being mostly controlled by the RC O+ density. The change of the O+ fraction in the plasmaspheric model also modifies the pattern of wave propagation and Alfvén speed. The latter controls Landau damping by the thermal plasmaspheric electrons, and so an increase in the fraction of cold O+ (decrease in Alfvén speed) leads to more effective Landau damping in the region of low L shells and a decrease of damping at large L shells. It follows from Figure 7 that overall wave generation is more intense in the 3% O+ case compared to the nominal case shown in the second row. The latitudinal extent of the observable wave energy in the fourth

Figure 7. The squared magnetic field for the model He+-mode EMIC waves (first to third rows) in the equatorial plane and (fourth to sixth rows) in the meridional plane averaged over all MLTs during hours 76–78 with a 20 min cadence. The first and fourth rows are obtained for 3% O+ in the plasmaspheric model, while the second/fifth and third/sixth rows are for 2 % (nominal case) and 1.5%, respectively. (Note that we keep 77 % H+ in all the cases but adjust the He+ fraction accordingly to keep quasi-neutrality.)

row of Figure 7 is larger compared to the ﬁfth row indicating a suppressed wave damping in the vicinity of the bi-ion frequency [e.g., Thorne and Horne, 1994]. (Note that bi-ion latitudes themselves are closer to the equator in the 3% O+ case compared to the nominal case.) A decrease in the O+ fraction compared to the nominal case causes an increase in wave damping, and EMIC wave activity in the third and sixth rows is greatly reduced. So the abundance of O+ strongly controls the energy of oblique waves that propagate to the equator after their reﬂection at bi-ion latitudes, and so it controls the fraction of wave energy in the oblique normals.

3.3. Quasi-Field-Aligned EMIC Waves and Nonlinear Source of Seed Fluctuations It follows from section 3.2 that in order for quasi-ﬁeld-aligned waves to be observed, they should grow up to the observable level during one pass through the near equatorial unstable region. One of the

possibilities to do so is to increase the growth rate in Figure 5a by a factor of 2, at least, or probably even more. The larger growth rate could result from the modulation of plasma density and/or magnetic ﬁeld by compressional waves in the low-frequency part of the ULF wave band [e.g., Mursula et al., 2001; Mursula, 2007]. In order to “work”, this mechanism requires high intensity of the low-frequency compressional waves, about tens of nT or so [e.g., Demekhov, 2007]. However, EMIC waves are observed when the amplitudes of simultaneous compressional Pc5 are small. For example, Loto’aniu et al. [2005] observed not only EMIC waves but even repetitive structure of EMIC waves, but simultaneous Pc5 amplitudes were less than 0.3 nT. So the mechanism of growth rate modulation cannot account for all observations of EMIC waves. The other possi- bility to allow EMIC waves to grow up to an observable level during one pass between the bi-ion latitudes in conjugate hemispheres is to invoke some nonlinear mechanism that can supply a seed ﬂuctuating energy, which essentially exceeds the thermal background level, in the k-space of EMIC wave generation. Below we suggest and analyze such a nonlinear mechanism.

+ 2 −2 2 To estimate the needed fluctuation level for the He -mode EMIC wave growth, we adopt Bobs = 10 nT as an observable wave intensity, 𝛾 = 0.2 rad/s as a typical wave growth rate due to the RC H+ thermal anisotropy (see Figure 5a) , and 𝜏 = 20 s for the wave propagation time through the latitudinal region |𝜆| < ◦ 10 near the equator; half of the wave bounce period is Tw∕2 ≈ 60 s [e.g., Khazanov et al., 2006], and waves need approximately 20 s to propagate through the region |𝜆| < 10◦ if the bi-ion reflection latitude is estimated as 𝜆 ≈ 30◦. Then, we have the following estimate for the required seed fluctuation 2 2 𝛾𝜏 −6 2 level Bseed = Bobs exp (−2 ) ≈ 3 × 10 nT . This background level will allow EMIC waves to grow up to 2 the observable level during one pass through the near equatorial region if energy Bseed is supplied in the k-region of EMIC wave generation on a time scale smaller than 𝜏. Note that the estimates presented below are not so sensitive to the exact energy supply time scale, but physically this time scale cannot be larger than Tw∕2, which is a typical time scale of wave energy transfer from the field-aligned( wave) normal angles into the region of highly oblique normals. So there should be a source term like 𝛿B2∕𝛿t ∼ B2 ∕𝜏 in k seed seed the RHS of equation (2) to make the appearance of quasi-field-aligned EMIC waves possible in our model. A nonlinear wave-wave interaction generates a wave energy cascade in the region of larger wave numbers. So this nonlinear process may supply the needed seed energy in the k-region of the He+-mode EMIC wave generation. Let us estimate the effectiveness of this nonlinear source. The nonlinear wave energy transfer in the k-space can be described using a phenomenological diffusive approximation [e.g., Zhou and Matthaeus, 1990; Jiang et al., 2009; Gamayunov et al., 2012]. For the field-aligned waves, the simplest form of the governing equation is [e.g., Gamayunov et al., 2012] ( ) dB2 𝜕 𝜕B2 B2 k = D k , D = v k4 k , (9) dt 𝜕k kk 𝜕k kk A 2 2BE

where vA and BE are the Alfvén speed and the Earth’s magnetic ﬁeld, respectively. Integrating equation (9) 𝜔 + from the crossover frequency cr (EMIC waves are generated in the left-hand polarized mode) up to the He 2 𝜏 gyrofrequency, and then replacing the LHS by Bseed∕ we get

2 2 |∞ B 𝜕B | seed k | , = Dkk | (10) 𝜏 𝜕k | k1 ( ) 𝜔 𝜔 + where cr = k1 . The RHS in equation (10) vanishes at k =∞because the He -mode EMIC waves damp 𝜔 in the vicinity of ΩHe+ = (k =∞), and so the wave intensity is low around ΩHe+ . Equation (10) allows us to estimate the energy of low-frequency waves that would be suﬃcient to sustain the needed level of EMIC 2 −3∕2 wave seed ﬂuctuations. Taking the Kraichnan-Iroshnikov spectrum Bk ∼ k [Kraichnan, 1965; Iroshnikov, 1964], which is valid in the limit of weak turbulence [e.g., Zhou and Matthaeus, 1990; Gamayunov et al., 2012], for the low-frequency wave PSD we get: ( ) 1∕2 B2 2B B2 B2 = seed E , B2 = k . (11) k 𝜏 3∕2 𝜔 3vA k vA

2 𝜏 The dependencies on Bseed and in equation (11) are quite weak, and so the above estimate is not so sensitive to the exact values of the EMIC wave observable level used and/or time scale. Now, integrating

equation (11) from, for example, the typical frequency of Pc4 pulsa- 𝜔 𝜋 tions ( 0 = 2 ∕100 rad/s) up to, for example, the frequency of Pc2 𝜔 𝜋 ( 1 = 2 ∕10 rad/s), and adopting BE = 130 nT (for L ∼ 6), we get the following estimate for the energy density of low-frequency fluctuations 𝜔 1 2 𝜔 . 2. B𝜔d ≈ 0 3 nT (12) ∫𝜔 0 The estimate (12) has a weak depen- 𝜔 dence on 0 and an even weaker one 𝜔 on 1. So the above quite loose specification for the bounding frequencies is not crucial for the estimate. Actu- ally, we have to emphasize that an invoked energy cascade trans- fers wave energy from the lower frequency energy range into the higher-frequency range of EMIC wave generation, and so the typical frequencies of geomagnetic pulsations are used here for estimate purposes only. Given that long-period ULF waves in the Earth’s magnetosphere typically have amplitudes higher than a few nT [e.g., Anderson et al., 1990], the estimate (12) can be easily sat- isfied. This makes it quite possible that a nonlinear energy cascade from Figure 8. Ten-point smoothed FFT power spectra of 512 s of magnetic the Pc4 to Pc2 ULF range into the field data from the Polar satellite’s Magnetic Fields Experiment beginning frequency range of Pc1 and Pc2 pul- at 12:00 UT 24 February 2003. The eight samples/s data were rotated into sations is able to supply the needed mean field-aligned coordinates (X radial, Y azimuthal, and Z field aligned) level of seed fluctuations that guar- before processing. antees growth of EMIC waves during one pass through the near equatorial region |𝜆| < 10◦. As we already emphasized, the mechanism of growth rate modulation involves compressional low-frequency waves, while the nonlinear energy cascade mechanism operates with transverse waves. Another difference is a low-energy threshold given by equation (12) that makes the nonlinear energy cascade mechanism quite attractive. It is also obvious that both mechanisms can operate concurrently and complement each other. Careful analysis of satellite data can reveal the background wave power across the Pc1 to Pc4 wave bands. We have already begun such an analysis using data from the Van Allen Probes and Polar spacecraft. As an example, Figure 8 shows power spectra of magnetic field data in mean field-aligned coordinates from the Polar satellite’s Magnetic Fields Experiment for a 512 s interval on 24 February 2003. During this time, Polar was traveling near apogee very nearly along a magnetic field line in the outer dayside magnetosphere: it moved from 0.22◦ to 0.96◦ in GSM magnetic latitude, from L = 9.6 to L = 9.5, and at a constant MLT = 13:02. The total magnetic field during this interval varied from 56.6 to 59.1 nT. The power spectra were calculated using a 4096-point FFT algorithm applied to eight samples/s data beginning at 12:00 UT; a 10-point smoothing was applied to the spectra before plotting. The results in Figure 8 show on average the power law distributions of the magnetic fluctuating energy from about 10−2 Hz up to 1 Hz, which is a manifestation of the forward turbulent energy cascade, and EMIC waves around 0.3 Hz are sitting

Figure 9. The distributions of model wave characteristics for the nominal setup during hours 76–78 with a 20 min cadence. The background ﬂuctuation level B2 = 10−3 nT2 is adopted, and a damping rate 훾 =−10−1휔 is prescribed inside the spatial cells where a particular frequency equals the bi-ion frequency. seed

on top of this nonthermal background. (Note that the peak around 0.2 Hz is a satellite spin tone.) Figure 8 also shows that EMIC wave growth rate around 0.3 Hz is much larger than the nonlinear energy cascade rate because the nonlinear cascade is not able to wash out the EMIC wave energy from the region around 0.3 Hz. This supports our theoretical finding that a nonlinear energy cascade can supply wave energy from the lower frequency range into the higher-frequency range of EMIC wave generation, where EMIC waves grow from this nonthermal background level due to the proton cyclotron instability. It is also worth to note that a nonlinear energy cascade in Saturn’s magnetosphere has been recently reported after analysis of the magnetic field fluctuations measured by the Cassini spacecraft [von Papen et al., 2014].

3.4. Development of EMIC Waves From Nonthermal Background Level A full numerical implementation of either the mechanism of growth rate modulation or the nonlinear energy cascade mechanism outlined in section 3.3 is a very complex task and is beyond the scope of this study. However, we can simply increase the seed fluctuation level for EMIC wave growth and then analyze the tendencies in the resulting wave normal angle and energy flux distributions. It follows from section 3.2 that quasi-field-aligned EMIC waves can be routinely observed if the wave background level is high enough to guarantee wave amplification up to the observable level during one pass through the near equatorial unstable region. After wave reflection at bi-ion latitudes, the He+-mode EMIC waves will usually have oblique wave normals, and the fraction of wave energy in the oblique normals is controlled by the strength of wave energy dissipation around bi-ion latitudes and wave tunneling. Then, the equatorward propagating oblique waves can be further re-amplified due to interaction with the RC O+ and H+ as discussed above (see Figures 4 and 5a). However, the re-amplification of oblique waves requires many bounce periods just because their growth rate is much smaller compared to the growth rate of the quasi-field-aligned EMIC waves. The resulting wave PSD is formed by both EMIC wave components; the quasi-field-aligned, or “prereflection,” component and oblique, or “postreflection,” component.

2 −3 2 Figure 9 shows the distributions of model wave characteristics for the nominal setup where Bseed = 10 nT is adopted in the entire simulation domain as a wave seed fluctuation level. To eliminate a postreflection wave component, the damping rate 훾 =−10−1휔 is also prescribed inside the spatial cells where a particular frequency equals to the bi-ion frequency. It is clearly seen that both model shortcomings identified in section 3.1 are gone. The He+-mode EMIC waves are quasi field aligned near the equator, while they are oblique at high latitudes, and the Poynting flux anisotropy shows wave energy propagation away from the near equatorial source region. All the results in Figure 9 confirm the expected characteristics of the prereflection wave component. These results also indicate that our model correctly describes, at least qualitatively, the dynamics of He+-mode EMIC waves between the bi-ion latitudes in conjugate hemispheres. Figure 10 also shows the distributions of model wave characteristics for the nominal setup, except 1.5% O+ 2 −3 2 in the plasmaspheric model is used, and the background fluctuation level Bseed = 10 nT is adopted. Now both the prereflection and postreflection wave components contribute in the results shown in Figure 10. Comparing the second rows in Figures 9 and 10, we see that wave activity at L ≈ 2 is entirely due to the postreflection component generated by the RC O+ at the highly oblique normals. This is expected because the wave growth rate driven by the loss cone feature of the RC O+ PSDF is small compared to the growth rate driven by the positive temperature anisotropy of the RC H+. So many bounce periods are required for waves to reach an observable level. Currently, there are as yet no direct observations in the equatorial region that could support our model finding that intense He+-mode EMIC waves are generated in the region L ≈ 2–3.In addition, ground observations at the field line foot point indicate that waves originate in the higher-latitude > source region (L 3) and then propagate in the ionospheric F2 region waveguide to the lower latitudes. However, there are at least two circumstances which could prevent observations of EMIC waves at low L shells. First, when the magnetic field becomes large (for lower L shells), spacecraft must change their digiti- zation range; so their magnetometers become “blind” to the small-amplitude waves. This is why electric field data are often of more value than magnetic field data at low L shells. Second, the RC O+ generate He+-mode EMIC waves at highly oblique normals, and many bounce periods are required for waves to reach an observable level. So the fraction of O+ should be high enough to support wave bouncing and re-amplification. But, the large O+ fraction suppresses wave tunneling through the bi-ion latitudes that is only the way for waves to propagate downward to the ionosphere and ultimately to be detected on the ground. We have to emphasize, however, that ground observations cannot by themselves identify the band within which Pc1 waves are

Figure 10. The distributions of model wave characteristics for the nominal setup, except 1.5% O+ in the plasmaspheric model, are used. The background ﬂuctuation level B2 = 10−3 nT2 is also adopted. seed

Table 2. Number of Pc1 Events Versus L Shell as Observed by Van Allen Probe B During the Year of 2013a

2 < L < 33< L < 44< L < 55< L < 6 Total Events seen in B and E data (%) 9 (6%) 27 (19%) 44 (30%) 66 (45%) 146 (100%) Normalized relative occurrence 13% 30% 35% 22% 100% aEach event was observed in both magnetic and electric ﬁeld data. The second row shows the relative occurrence percentage after normalization by the fraction of orbit time spent in each given L range. Note that the table does not distinguish events by band.

observed, because of the presence of significant horizontal ducting. Only in situ spacecraft observations can simultaneously detect the ambient magnetic field and locally present waves. In our attempt to verify/support the above model finding, we have preliminarily analyzed the Van Allen Probes field data. Fourier spectrograms of all Electric and Magnetic Field Instrument Suite and Integrated Science (magnetic field instrument) and Electric Field and Waves Suite (electric field instrument) data from both Van Allen Probes spacecraft have been created and analyzed at Augsburg College. The number of Pc1 events versus L shell, as observed by the Van Allen Probe B during the year of 2013, is shown in Table 2. In total, 146 Pc1 events were observed (each in both electric and magnetic field data) during this period. Among these events, nine (6%) events were observed in the region 2 < L < 3, and 27 (18%) in the region 3 < L < 4. When the observations are normalized to account for the differing fractions of orbit time the spacecraft spends in a given L range, the relative occurrence rates increase to 13% for 2 < L < 3 and 30% for 3 < L < 4. Therefore, our preliminary data analysis reveals a considerable number of EMIC events observed at low L shells. This supports the model finding that EMIC waves could be generated in the region L ≈ 2–3. The wave normal angle distributions in Figure 10 still tend to be more field aligned near the equator and oblique at high latitudes, while a contribution of the oblique postreflection wave component is clearly seen in the third and fourth rows. The Poynting flux anisotropy is also “contaminated” by the postreflection component, while it still shows a predominance of wave energy propagation from the equator. Figures 9 and 10 show that the balance between prereflection and postreflection wave components is quite delicate. A departure from the “right balance” can cause both a concentration of wave energy in the region of oblique normals with no prevalence of any direction for the wave energy propagation as in section 3.1, and a quasi-field-aligned wave energy distribution near the equator with the wave energy propagating away from the near equatorial source region as in Figure 9. Currently, our model does not include the physics that could dynamically regulate this delicate balance. Inclusion of the needed physics is a very complex task, and it is beyond the scope of this study. In the future, to make our model more realistic, at least the following elements should be incorporated in the model: (1) a dynamic model of low-frequency ULF waves either as a means of EMIC wave growth rate modulation or as an energy source for the nonlinear energy cascade, (2) a dynamic model of the plasmaspheric ion composition and electron temperature, and (3) EMIC wave tunneling through the bi-ion latitudes.

4. Summary and Conclusions The scattering of outer RB relativistic electrons is highly sensitive to the EMIC wave energy distribution in spectral phase space, and a properly specified global distribution of the EMIC wave PSD is needed to accurately describe wave-particle interactions on the global magnetospheric scale throughout different storm phases. Currently, despite a number of statistical and case studies of EMIC waves, there is still no reliable model of EMIC wave PSD that could be incorporated in global models of relativistic electrons in the outer RB. In this situation, a dynamic physics-based model, verified and constrained by data, of the EMIC wave PSD will allow us to properly quantify the pitch angle diffusion coefficient for relativistic electrons in the outer RB, and so to quantify the electron loss. Then, this diffusion coefficient can be used by physical models of energetic electrons to study real events with the inclusion of a dynamic global EMIC wave distribution.

In this work we have presented initial wave results from our global magnetospheric model that provides a time-dependent, driven by real solar wind conditions, He+-mode EMIC wave PSD in the L-MLT-magnetic latitude-frequency-wave normal angle phase space. We have first considered EMIC wave development from the thermal background level. The wave characteristics presented in section 3.1 have revealed two model shortcomings: (1) a majority of the wave energy is concentrated in the region of oblique normals, while the observed He+-mode EMIC waves are generally quasi field aligned inside the geostationary orbit, and (2) the Poynting flux anisotropy does not demonstrate prevalence of any direction for the wave energy propagation, while observations show predominant wave energy propagation away from the near equatorial source region. Second, in order to identify the reason(s) for these two model-observation discrepancies, we have analyzed in detail the He+-mode EMIC wave generation and damping. The results of our analysis have shown that in order for quasi-field-aligned waves to be observed, they should grow up to the observable level during one pass through the near equatorial unstable region of about ±10◦. However, the thermal background level for the He+-mode EMIC waves is low (< 10−18 nT2), and as a consequence, waves typically could not reach an observable level during one pass through the near equatorial region. Third, to overcome this drawback, we have suggested a nonlinear energy cascade from the lower ULF range into the frequency range of EMIC wave generation as a possible mechanism supplying the needed level of seed fluctuations that guarantees growth of EMIC waves during one pass through the near equatorial region |휆| < 10◦. Finally, we have considered the EMIC wave development from a suprathermal background level, and the model results have shown that both model-observation discrepancies identified in section 3.1 are removed. The comparison of the model results with statistical studies of He+-mode EMIC waves, which is the dominant mode observed in the inner magnetosphere, shows that the spatial distribution of wave energy is in agreement with observations, being located in the noon-dusk MLT sector (see Figures 3, 6, 7, 9, and 10). The Pc1 and Pc2 wave amplitudes from AMPTE/CCE are in the range 1–3.5 nT [Anderson et al., 1992a], and the amplitudes are < 3 nT with the typical amplitude about 0.5 nT in the CRRES data [Fraser and Nguyen, 2001]. Our model results are in agreement with those observations as well. The data-model comparison of the EMIC wave normals is not so obvious and challenging for two reasons. First, FFT analysis usually gives wave normal angles less than 30◦ in the inner magnetosphere [e.g., Fraser, 1985; Ishida et al., 1987; Min et al., 2012]. At the same time, the CRRES statistics (and the AMPTE/CCE statistics as well) shows that EMIC events near the magnetic equator are evenly distributed from left-hand polarized to linearly polarized with some right-hand polarized admixture [Fraser and Nguyen, 2001; Meredith et al., 2003]. It is important to note that, as emphasized by Anderson et al. [1992b] and Fraser and Nguyen [2001], observation of a significant number of linearly polarized events near the equator cannot be explained by polarization reversal from left handed through linear to right handed at the crossover frequency, as suggested by Young et al. [1981]. (If the Young et al. mechanism takes place, then the quasi-field-aligned waves can have a linear polarization.) Therefore, the observed linear polarization including the near equatorial zone suggests that waves should be often highly oblique. In other words, the minimum variance direction (the direction of wave normal vector) extracted from data and wave ellipticity contradict each other. This suggests that new approaches to the data analysis should be applied. Second, as we showed above, the abundance of O+ strongly controls the energy of oblique He+-mode EMIC waves that propagate to the equator after their reflection at bi-ion latitudes. Currently, our model does not include a dynamic model of the plasmaspheric ion composition, so it cannot dynamically regulate the delicate balance between prereflection and postreflection wave components. Without this, we cannot make a meaningful prediction of the wave normal angle distributions because depending on the abundance of O+, our model can produce both a concentration of wave energy in the region of oblique normals with no prevalence of any direction for the wave energy propagation as in section 3.1, and a quasi-field-aligned wave energy distribution near the equator with the wave energy propagating away from the near equatorial source region as in Figures 9 and 10. Currently, we are working with the Van Allen Probes data and on the further model development in order to make a meaningful comparison of EMIC wave spectral characteristics with the model results. However, we are not able to do such kind of comparison in the present work. Finally, we would like to emphasize that we cannot use for the comparison purposes the EMIC event shown in Figure 8 because our model currently operates for L < 6.6 only. We have already begun working with the Van Allen Probes data, but the data analysis is more complicated in the inner magnetosphere because the averaged background magnetic field is changing during the observational time (at least ∼10 min) that is needed to produce power spectra of the wave magnetic field in the extended frequency domain.

Hours The particular major ﬁndings from our study and 103 74 conclusions can be summarized as follows. ) 75 -3 77 1. A nonlinear energy cascade from the lower ULF 79 + 80 range into the frequency range of He -mode 81 EMIC wave generation can supply the needed 82 102 level of seed ﬂuctuations that guarantees growth of EMIC waves during one pass through the near equatorial region |휆| < 10◦. This suggests that a dynamic model of low-frequency ULF waves needs to be linked with our global 1 10 model of EMIC waves.

Equatorial Thermal Density (cm 2. The abundance of O+ strongly controls the energy of oblique He+-mode EMIC waves that 2 3 4 5 6 propagate to the equator after their reflection L-shell at bi-ion latitudes, and so it controls the fraction of wave energy in the oblique normals. This Figure A1. The equatorial plasma density versus L shell for MLT = 16. suggests that a dynamic model of the plasmaspheric ion composition is needed, RC O+ should be included not only in the imaginary part of the wave dispersion relation but in the real part as well, and EMIC wave tunneling through the bi-ion latitudes should be also taken into account. 3. The RC O+ not only causes damping of He+-mode EMIC waves but also wave generation in the region of highly oblique wave normals, typically for 휃>82◦, where the growth rate 훾>10−2 rad/s is frequently observed (see Figure 4). The instability is driven by the loss cone feature of the RC O+ PSDF, where + 휕F∕휕v⟂ > 0 for the resonating O . 4. The oblique and intense He+-mode EMIC waves generated by RC O+ in the region L ≈ 2–3 mayhavean implication to the energetic particle loss in the inner RB. Finally, we would like to stress one more time that our preliminary analysis of the Van Allen Probes and Polar field data supports the theoretical findings of this study that a nonlinear energy cascade from the lower frequency range can supply energy into the frequency range of EMIC wave generation for further EMIC wave growth and that EMIC waves could be generated in the region L ≈ 2–3.

Appendix A: Distributions of Plasmaspheric Density and RC Parameters

An example of the model equatorial plasmaspheric density at hour 77 is shown in the paper by Gamayunov et al. [2009, Figure 7, middle row]. Figure 7 in the Gamayunov et al. paper shows the number density of the thermal plasma only. In the present study, we also included the RC H+ and O+ only in the imaginary part of the EMIC wave dispersion relation but neglected it in the real part. (The effect of RC H+ in the real part of wave dispersion relation was previously discussed, for example, by Gamayunov and Khazanov [2008].) As follows from Figure 3 in the paper by Gamayunov et al. [2009], the cross polar cap potential (CPCP) drop grows from about 50 kV up to about 275 kV during approximately2hfrom74to76.TheCPCP drop stays at its maximum up to hour 77, and then it goes down fluctuating around 50 kV after hour 79. As a consequence of the increased convection field, a plasmaspheric plume is seen at hour 77. Later, a decrease in the CPCP drop leads to an extension of the region of closed equipotential lines that causes erosion of the plume and the counterclockwise rotation of the plasmaspheric material. Figure 7 in the paper by Gamayunov et al. [2009] gives us a global view of the plasmaspheric density distribution. This picture, however, is rather quali- tative, and accurate values of density are difficult to estimate from this figure. The equatorial plasma density profiles versus L shell in Figure A1 give us the values of density during hours 74–82. Because the plasmaspheric plume is a favorable region for EMIC wave growth, we show in Figure A1 the plasma density profiles for MLT = 16. The model distributions of the RC number density and energy density are shown for H+ and O+ in Figures A2 and A3, respectively. A positive pressure anisotropy of RC H+ may cause He+-mode EMIC wave generation, and we show in Figure A2 the H+ anisotropy distribution as well. On the other hand, RC O+ cannot generate

+ Figure A2. The model equatorial parameters for RC H . The number density, energy density, and pressure anisotropy (A = P⊥∕P‖ − 1) are shown in the ﬁrst, second, and third rows, respectively.

the He+-mode EMIC waves due to their pressure anisotropy [e.g., Thorne and Horne, 1994]. The RC O+ pressure anisotropy can cause generation only of O+-mode EMIC waves, which are not considered in this study. For this reason, we do not show the RC O+ anisotropy in Figure A3. We also have to note that a RC ion density enhancement observed in the midnight-morning MLT sector for large L shells after hour 79 is due to the low-energy ions (< keV). This enhancement is not observed in the RC energy density distributions, and so it does not aﬀect the energetics of the RC. It is clearly seen from the energy density distributions that the strongest RC is developed by the end of storm main phase at hour 77. The energy density distributions in Figures A2 and A3 are very similar to those observed by the CHEM spectrometer on board of AMPTE/CCE [Hamilton et al., 1988] during the February 1986 storm. However, in our case, in contrast to the February 1986 storm, the RC H+ contains the major part of the RC energy during the entire modeled event, while the energy densities of RC H+ and O+ are comparable around hour 77.

+ The RC H pressure anisotropy, A = P⊥∕P‖ − 1, in Figure A2 increases toward lower L shells in accordance with observations, and during storm recovery phase, the model anisotropy distribution is quantita- tively very close to the anisotropy proﬁles observed during quiet magnetospheric conditions by the instruments on board AMPTE/CCE [Lui and Hamilton, 1992]. To estimate the anisotropy threshold for pro- + ton cyclotron( instability,) we may use the well-known result for a bi-Maxwellian distribution of RC H , >휔 휔 + A ∕ ΩH+ − , where ΩH+ is the H gyrofrequency [e.g., Kennel and Petschek, 1966]. (Note that the above inequality for the proton cyclotron instability threshold is still valid for a non-bi-Maxwellian dis- 휔 . tribution, but A is not a simple pressure anisotropy in that case.) Using ∕ΩH+ = 0 25 as the largest wave frequency, we estimate that RC H+ can cause growth of the He+-mode EMIC waves if A > 0.33. It follows from Figure A2 that the RC H+ pressure anisotropy exceeds the estimated threshold in the regions L < 4 and L < 6 in the 74 and 75 h snapshots and during the storm recovery phase,

Figure A3. The model equatorial parameters for RC O+. The number density and energy density are shown in the ﬁrst and second rows, respectively.

respectively. But, of course, we do not expect that EMIC waves will be at an observable level every- where inside these large regions. A high enough positive anisotropy is only a necessary condition in order to observe EMIC waves. The resulting EMIC wave energy will depend on many other magnetospheric characteristics such as the distribution and composition of cold plasma, ﬁne structure of the RC ion distributions, damping by RC O+ and thermal electrons, and the background energy level for EMIC wave growth.

It is worth noting that instead of the above inequality for the proton cyclotron( ) instability threshold, which > 훽 c . . . involves both the ion and wave characteristics, the inequality A a∕ ‖p , where a ≈ 0 8, c ≈ 0 4–0 5, 훽 휋 2 + and ‖p = 8 NpT‖p∕B is the RC H ﬁeld-aligned plasma beta, is also frequently used in the proton cyclotron instability analysis [e.g., Anderson et al., 1996b, and references therein]. The later inequality needs only the ion characteristics in order to be evaluated. Numerically, however, the above two thresholds are quite close. 휔 Indeed, using the resonance condition for the ﬁeld-aligned EMIC waves, − kv‖res −ΩH+ = 0, replacing 휔 v‖res by( the thermal) speed( v‖)T , and then substituting( ∼ kv) A, where vA is the Alfvén speed, we get >휔 휔 훼 훽 0.5 훼 0.5 A ∕ ΩH+ − ≈ ∕ ‖p where ≈ v‖T ∕v‖res Np∕Ne ∼1.

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