The ion-scale magnetic holes, localized depletions of the magnetic field magnitude, represent one of the quite stable and commonly observed nonlinear plasma structures in the pristine 55,61 and shocked 25,27,58 solar wind, planetary, [12][13][14]16,17,46,54,66,68 and cometary 19,43,44,51 magnetospheres. Formation of such structures is typically attributed to the nonlinear stage of the ion mirror mode, 8,24,31,40 although smaller (sub-ion) scale magnetic holes may be generated by plasma turbulence 21,41 and various types of acoustic instabilities. 26,32,45 One important property of magnetic holes is their stability: such holes may travel for large distances in the solar wind. 42 This stability allows magnetic holes generated around the Earth's magnetosphere boundary, magnetopause, to propagate into the inner magnetosphere and transport hot trapped ion and electron populations. 30,64 The stress balance in magnetic holes and their formation mechanism assume that the trapped ion and electron populations are transversely anisotropic and hotter than the plasma background. 37,60 Thus, the transport of these populations into the inner magnetosphere provides an additional energy source for the generation of electromagnetic waves that play an important role in the dynamics of energetic particles. 7,52,67 In fact, spacecraft observations of magnetic holes in the Earth's inner magnetosphere are usually associated with localized bursts of electromagnetic whistler-mode waves, 12,64 electron cyclotron harmonics, 13 and electromagnetic ion cyclotron (EMIC) waves. 30,59,65 The latter wave mode is generated by hot, transversely anisotropic ions 22 trapped within magnetic holes. 5,60 EMIC waves are quite effective in resonantly scattering hot ions, 9,10,15 and their generation within magnetic holes is expected to result in isotropization of the ion population and precipitation into the atmosphere. Such isotropization and ion losses due to precipitation will alter the hot trapped ion population that is responsible for the pressure balance in magnetic holes. Thus, EMIC waves may influence the magnetic hole structure by scattering these hot trapped ions. See the schematic diagram in Fig. 1 Such a feedback mechanism will modify the magnetic holes and finally, when a sufficiently large ion population is lost, will destroy the holes. This natural limitation of magnetic hole lifetime in the Earth's inner magnetosphere is the focus of our study.
To illustrate the typical characteristics of the system consisting of magnetic holes, hot transversely anisotropic trapped ions, and EMIC waves, we provide an illustrative Fig. 2 showing the example of THEMIS 2 observations in the inner Earth's magnetosphere. Panel (a) shows several bursts (maxima) of electromagnetic wave emission within the helium and proton cyclotron frequency ranges; the spectrum analysis shows that these are left-hand circularly polarized and field-aligned EMIC waves. 53 The peaks of EMIC wave amplitude [panel (b)] are within local depletion of the background magnetic field shown in panel (c) as negative compressional fluctuations dB k < 0 of a few nanotesla magnitude; these are magnetic holes. Hot transversely anisotropic ions fill these magnetic field minima: panels (d) and (e) show peaks of ion flux anisotropy (better seen for the >3 keV range) and peaks of ion temperature anisotropy; these peaks coincide with peaks of the EMIC amplitude. The stress balance within magnetic holes is controlled by the hot-ion component: panel (f) shows the variations of magnetic and hot-ion thermal pressures. The EMIC wave generation within magnetic holes is supported by enhanced ion fluxes [see panel (g) for peaks of ion beta] and ion anisotropy. Thus, Fig. 2 shows a typical example of a magnetic hole with trapped hot ions unstable to EMIC wave generation. In this study, we plan to provide estimates of how such EMIC waves interacting resonantly with hot ions may reduce their contribution to the stress balance within magnetic holes, which will eventually determine the hole lifetime.
In this study, we analyze the event shown in Fig. 2 to reveal the potential EMIC wave effect on scattering and losses of the hot ion population trapped within the magnetic hole. In Sec. II, we use the linear analysis of the observed ion distribution function to confirm the generation mechanism of observed EMIC waves. In Sec. III, we estimate the effect of ion isotropization due to scattering by EMIC waves. In Sec. IV we estimate the effect of ion losses from the magnetic hole due to scattering by EMIC waves. Then, in Sec. V we summarize the results obtained and discuss the role of the EMIC wave in the stability of the magnetic hole.
To confirm that the transversely anisotropic ion population trapped within magnetic holes is responsible for the generation of observed EMIC waves, we performed a linear stability analysis of this population. 18,39,48 We fit the ion energy and pitch-angle distribution by the combination of three anisotropic Maxwell functions: cold population with the temperature $12 eV, warm population with the temperature $300 eV, and hot population with the temperature $3:6 keV. The fitting shows that the warm population is field-aligned anisotropic 62 and likely represents local background plasma provided by ion outflow from the Earth's ionosphere. The hot population is transversely anisotropic 11,63 and should be the main free energy source for the generation of EMIC waves. The cold population does not resonate with EMIC waves and contributes a lot to the total plasma density.
We use analytical fitting of the observed ion distribution function from Fig. 3 to evaluate EMIC wave growth rate and dispersion relation using Eq. ( 1) from Ref. 6. Figure 4(a) shows that the observed EMIC wave dispersion is largely affected by the hot plasma contribution and differs well from the cold plasma dispersion from Ref. 49. The difference of wave numbers, k, for normalized wave frequency x=X cp $ 0:4 is about factor Â2, i.e., in hot plasma, EMIC waves have much smaller k and will resonate with smaller energy ions [the resonant energy / ðx À X cp Þ 2 =k 2 , where X cp is the proton cyclotron frequency]. This hot plasma effect can be important for consideration of the EMIC wave contribution to the scattering of ions trapped within a magnetic hole. Figure 4(b) compares linear wave growth rate, c=X cp (in blue), and the observed wave intensity spectrum. The linear theory reproduces well the observed frequency range of EMIC waves, which confirms that these waves are indeed generated by ion population trapped within a magnetic hole.
The most unstable frequency rate where the growth rate reaches the maximum value is x=X cp 2 ½0:3; 0:5. Spacecraft observations show that within this frequency range, EMIC waves also have the maximum wave power. Therefore, this comparison shows a good agreement between linear theory results and spacecraft observations.
Generation of EMIC waves and their following scattering of hot ions should result in ion anisotropy relaxation. 23,24,40 In this section, we estimate the maximum possible effect of such relaxation. The pressure balance across the magnetic hole takes the form of a perturbation equation B 0 dB k þ 4pT ? n ¼ 0 connecting the magnetic field depletion magnitude dB k and the enhancement of the ion thermal pressure T ? n. The same balance equation after full relaxation of the ion distribution will take a form B 0 dB new k þ 4pT new ? n ¼ 0, where T new ? is the ion perpendicular temperature after scattering by EMIC waves. To determine T new ? , we consider the threshold of EMIC wave generation:
where constants S and A are given by numerical simulation of wave saturation. 23 This equation provides the condition for the final ion anisotropy after ion distribution relaxation to the state when EMIC wave generation will be sufficiently weak (the growth rate will be almost zero). If we take into account that such relaxation is mostly provided by the pitch-angle ion scattering within strong energy redistribution, 24,40 we may add the energy conservation law
The combination of these equations and observations of initial dB k and ion anisotropy provides the final dB new k . Figure 5 shows the distribution of observations in ðb; dB new k =dB k Þ with color coding the initial ion anisotropy. The ion scattering and isotropization can decrease the nT ? pressure (and dB k / nT ? ) by $5% for strongly anisotropic and high b case. Such variation of magnetic hole magnitude is not essential, and thus isotropization due to pitch-angle scattering cannot affect the hole lifetime. However, this scattering also provides ion losses, because magnetic holes within the inner magnetosphere are connected along magnetic field lines with the dense ionosphere. Therefore, if EMIC waves are sufficiently effective in scattering small pitch-angle ions, this scattering can drive ion precipitations (see discussion in Refs. 29, 33, and 36). We examine this effect in Sec. IV. FIG. 2. An example event with EMIC wave emissions from the magnetic holes observed in the Earth's inner magnetosphere by THEMIS-E spacecraft: (a) EMIC wave power spectrum with proton cyclotron and helium cyclotron frequencies shown by white curves, (b) EMIC wave amplitude derived from power spectrum integration, (c) compressional component of the background magnetic field perturbations, (d) energy distribution of ion thermal anisotropy (ratio of perpendicular and parallel ion fluxes) (e) ion temperature anisotropy (ratio of perpendicular and parallel ion pressure components) (f) pressure balance between magnetic field pressure perturbations (shown in blue) and plasma pressure perturbations (shown in red) and (g) ion b (ratio of perpendicular ion pressure and magnetic field pressure). To plot these data, we use measurements of THEMIS fluxgate magnetometer with Fast Survey datatype (1/4s sampling), 4 spin averaged (3s) measurements of ion <25 keV spectra and moments by the Electrostatic Analyzer (ESA) 35 and from 25 keV to 6 MeV by Solid State Telescope (SST). THEMIS data are processed with the SPEDAS software. 3 In this event, THEMIS-E was near the dusk-flank magnetopause, with X GSE % 1:4R E ; Y GSE % 12:8R E ; Z GSE % À2:6R E ; R E % 6380 km.
EMIC waves generated by hot ion population within magnetic holes should provide effective pitch-angle scattering of ions. To quantify this scattering, we evaluate the pitch-angle diffusion rate: 6,50
where ¼ ffiffiffi p p erf ðrÞ with r ¼ 4/3 is obtained from observed wave spectrum, x ¼ x=X cp is the normalized wave frequency, y ¼ kc=X cp is the normalized wave number, E is the dimensionless particle kinetic energy given by
0 is the ratio of the energy density of the wave magnetic field to that of the background field, i.e., the relative wave power; x m ¼ w m =X cp and dx ¼ dx=X cp are the normalized maximum and bandwidth frequency respectively, obtained from the characteristics of wave spectrum, and Fðx; yÞ À1 ¼ dy=dx is the wave group speed determined from the plasma dispersion of EMIC waves. We used observed spectrum characteristics to set x m (mean frequency), dx (spectrum dispersion), and R parameters. Figure 6(a) shows D aa [Eq. ( 2)] for several typical ion energies. There is a fairly strong scattering, with D aa $ 10 À3 s -1 , of low pitch angle ions for the ½2; 10 keV range. This is the ion population that is largely responsible for the pressure balance within a magnetic hole, whereas scattering of these ions may result in their precipitation and loss into Earth's atmosphere. The typical losscone size, the pitch-angle range of ions precipitating into the atmosphere, for radial distances of magnetic hole observations is Da LC $ 18, whereas the bounce period of field-aligned ions of $5 keV energies is s b % 4 min. Therefore, D aa $ 10 À3 s -1 well exceeds the strong diffusion limit, D SD % ðDa LC Þ 2 =s b $ 10 À6 s -1 , and EMIC waves will always keep the loss-cone full. 28 This estimate of ion losses should be corrected by the fact that bouncing ions will resonate with EMIC waves only on such a range of off-equatorial distances, i.e., the bounce averaged diffusion rate should be smaller than the local (equatorial) rate. To estimate this bounce averaged hD aa i, we adopted a dipole field to approximate the Earth's magnetic field topology,
where k is the magnetic latitude. The local ion pitch-angle can then be recalculated to the equatorial pitch-angle as sin a ¼ sin a eq ð1 þ 3 sin 2 kÞ 1=4 =cos 3 k. The bounce average diffusion rate is given as follows: 20,34
where Tða eq Þ ¼ 1:30 À 0:56 sin a eq is the dimensionless ion bounce period. 38
where a max is determined by the equatorial pitch-angle range of a nonzero hD aa i.
Figure 7 shows the profile of ion lifetime s à as a function of energy [Eq. ( 4)]. Ions with small energies (<1 keV) are generally scattered slower than hot ions at the energy range, ½1; 10 keV. The latter ion population should be isotropized and moved along the phase space density gradient toward the loss-cone within $10-30 min. Such hot ion losses from the magnetic hole should quickly reduce the ion pressure, and thus decrease the magnetic field depletion, i.e., $20 min is the timescale of magnetic hole decay. This timescale is comparable to the timescale of hole train observations by spacecraft: holes are generally observed by groups moving (or oscillating) across the spacecraft. 5,30 Although intervals of hole observations can last for hours, the spacecraft motion and hole motion do not allow spacecraft to trace dynamics of a specific hole for more than $10 À 30 min (see examples in Refs. 5 and 64). Therefore, the $20 min estimate for hole decay time does not contradict the spacecraft observations.
In this study, we have investigated the dynamics of magnetic holes, localized magnetic depletions, which are commonly observed in the inner Earth's magnetosphere 5,12,14 and near-Earth solar wind. 55,61 These magnetic holes are pressure-balanced structures supported by the thermal pressure of a hot, transversely anisotropic ion population trapped within the hole. In the Earth's inner magnetosphere, magnetic holes are often observed as a train of quasiperiodic structures that further modulate electromagnetic ion cyclotron waves 5,30 and transport hot ion populations closer to the planet. In particular, large magnetic field perturbation of the most intense magnetic holes allows them to effectively scatter relativistic electrons. 56,57 All these characteristics make magnetic holes an important element for the ion kinetics of the inner magnetosphere. The formation of magnetic holes is associated with the nonlinear stage of the ion mirror instability, 8,24,31,40,47 and thus their lifetime should be controlled by the dynamics of the iontrapped population.
The transversely anisotropic ion population inside magnetic holes can generate electromagnetic ion cyclotron waves, which may further scatter ions and reduce their contribution to the hole pressure balance. We examined two main consequences of such scattering that can result in magnetic hole decay: (1) reduction of ion anisotropy due to ion isotropization and (2) ion losses due to precipitation into the atmosphere. Even complete isotropization of the ion population cannot significantly reduce the ion contribution to the pressure balance. Thus, this mechanism does not result in magnetic hole decay, but only slightly reduces the magnetic hole's magnitude. The observed intensities of the electromagnetic ion cyclotron waves and their spectral characteristics allow these waves to effectively scatter the main (hot) ion population. The typical lifetime of this population is tens of minutes, and this estimate is quite comparable to the timescale of magnetic hole observations. 5 Therefore, this (second) mechanism is quite a prospective and the most promising candidate for magnetic hole destruction. FIG. 7. Lifetime of ions scattered by EMIC waves.
Based on our analysis, we propose the following probable steps for magnetic hole dynamics in the Earth's inner magnetosphere: (1) hole formation at the magnetopause due to mirror instability of hot ions that are anisotropically heated by the solar wind compression of the Earth's magnetosphere, (2) propagation (drift) of magnetic holes to the smaller radial distances due to plasma convection, (3) generation of EMIC waves within magnetic holes and hot ion scattering, and (4) precipitation of a significant fraction of hot ion population into the Earth's atmosphere and hole decay. This hypothesis assumes that the magnetic hole decay in the inner Earth's magnetosphere is contributed by two factors: the presence of cold, dense background plasma that reduces ion resonant energies and intensifies the EMIC wave generation, and the presence of loss cone (connection of magnetic field lines with the collisional atmosphere where ions can be lost). These two factors are rather unique for the inner magnetosphere conditions, and their absence in the solar wind likely explains why solar wind magnetic holes can survive for a much longer time. 42
Phys. Plasmas 31, 072103 (2024); doi: 10.1063/5.0205942
V C Author(s) 2024