The interaction of ultra-low-frequency pc3-5 waves with charged particles in Earth’s magnetosphere

  • Qiugang Zong
  • Robert Rankin
  • Xuzhi Zhou
Review Paper


One of the most important issues in space physics is to identify the dominant processes that transfer energy from the solar wind to energetic particle populations in Earth’s inner magnetosphere. Ultra-low-frequency (ULF) waves are an important consideration as they propagate electromagnetic energy over vast distances with little dissipation and interact with charged particles via drift resonance and drift-bounce resonance. ULF waves also take part in magnetosphere-ionosphere coupling and thus play an essential role in regulating energy flow throughout the entire system. This review summarizes recent advances in the characterization of ULF Pc3-5 waves in different regions of the magnetosphere, including ion and electron acceleration associated with these waves.


ULF waves Poloidal ULF waves Toroidal ULF waves Compressional ULF waves Alfven waves Field line resonances Wave–particle interactions Drift-resonance Drift-bounce resonance Bounce-resonance Electrons and ions “Killer” electrons Magnetosphere Radiation belts Ring current Plasmasphere Plasmaspheric boundary layer Ionosphere Magnetotail Interplanetary shock Solar wind dynamic pressure Energetic particle acceleration Magnetic storms Substorm Substorm injection 

1 Introduction

The solar wind and interplanetary magnetic field exert control over energy input to Earth’s magnetosphere, part of which is in the form of ultra-low-frequency (ULF) waves. These waves, also known as geomagnetic pulsations, have frequencies in the range of about 1 mHz to 1 Hz and play a role in transporting mass, momentum, and energy throughout the magnetosphere. Modulations of energetic particle fluxes by ULF waves with periods of several minutes (Pc 5 range) were first reported by Brown et al. (1961), who attributed them to solar wind dynamic pressure pulses. Subsequently, Southwood and Kivelson (1981, 1982) developed theory that has proved useful for interpreting observational features of ULF waves and the modulations of ion and electron flux produced by them (Zong et al. 2007, 2009a, b). The Southwood–Kivelson theory has recently been revisited by Zhou et al. (2015, 2016), who accounted for finite growth and decay of ULF waves previously not included in the Southwood and Kivelson (1981) formalism.

As illustrated in Fig. 1, external solar wind perturbations and internal plasma instabilities act as sources of ULF waves in Earth’s magnetosphere. The Kelvin–Helmholtz (K–H) instability and solar wind dynamic pressure pulses are believed to be two primary external sources of wave excitation. The K–H instability produces large-scale vortices that transport solar wind plasma into the magnetosphere (Fairfield et al. 2000; Hasegawa et al. 2004). Surface waves generated by the K–H instability propagate earthward to excite field line resonances (FLRs) (Rae et al. 2005; Claudepierre et al. 2008). In contrast, when a solar wind dynamic pressure pulse impinges on the magnetopause, it produces fast mode waves that propagate into the inner magnetosphere, transferring energy from the solar wind in the process. The generation mechanisms for ULF waves depend on the local time region in which they operate. As the K–H instability requires shear flow, the generation region lies near the dawn and dusk sides of the low-latitude magnetopause. Solar wind dynamic pressure pulses that compress the magnetopause and launch fast mode waves can stimulate FLRs in the dayside magnetosphere around magnetic local noon (Zhang et al. 2010). Numerical studies of ULF waves excited by solar wind impulses have confirmed coupling between global cavity modes and FLRs (Allan and Poulter 1992; Zhu et al. 1991; Lee et al. 1989; Zhang et al. 2009).
Fig. 1

Schematic illustrating that Solar wind disturbances and Kelvin–Helmholtz surface waves can stimulate ULF waves and field line resonances (FLR’s) under certain conditions. The region where energetic electrons experience drift-resonances with ULF waves is indicated (after Allan and Poulter (1992))

Positive and negative impulses correlate with sudden changes in solar wind dynamic pressure that compress and inflate the magnetosphere, respectively (Zhang et al. 2009, 2010). Solar wind density enhancements and sudden enhancements of solar wind speed are known to produce positive impulses. Interplanetary shocks (IP’s) associated with large-scale transient solar wind phenomena such as coronal mass ejections (CMEs) and co-rotating interaction regions (CIRs) also correlate with positive impulses. IP shocks (positive impulses) are the primary cause of Storm Sudden Commencement (SSC) and enhancements of magnetopause currents. The interaction between Earth’s magnetic field and positive impulses has been shown to excite ULF waves in the inner magnetosphere (Yang et al. 2008; Zhang et al. 2009; Zong et al. 2009a, b). Positive impulses at the front edge of the Heliosphere Plasma Sheet (HPS) may also act as a source of ULF waves (Winterhalter et al. 1994). Statistical analysis of IP shocks and solar wind sources of negative impulses (Takeuchi et al. 2002) suggest they are mostly caused by the solar wind discontinuity embedded inside CIRs, the front edge of interplanetary magnetic clouds, and the trailing edge of the HPS. Small-scale plasma bubbles located within CIRs also produce negative and positive pressure impulses at the front and trailing edge of the bubbles, respectively.

ULF waves excited by solar wind dynamic pressure variations have much larger amplitudes than VLF waves (Claudepierre et al. 2008; Zong et al. 2009a, b; Zhang et al. 2010) and hence the former can accelerate energetic particles at a much faster rate. The comparable timescales associate with drift and bounce motion of energetic particles and ULF wave periods makes drift-bounce-resonance (e.g., Southwood et al. 1981; Hudson et al. 2008, Zong et al. 2008) possible. This process can adiabatically accelerate inner-magnetosphere charged particles and significantly enhance their radial diffusion (e.g., Elkington et al. 2003; Loto’aniu et al. 2006). A close correlation between rapid particle flux enhancements and ULF wave activity has been reported both for case studies (Zong et al. 2007; Tan et al. 2004) and statistical surveys (Mathie et al. 2001; O’Brien et al. 2003). The spectral power density of ULF waves varies roughly inversely with the wave frequency (Lanzerotti and Southwood 1979). A schematic showing the relation between different plasma waves and energetic electrons is illustrated in Fig. 2, with the overlap regions indicating where wave–particle resonance may occur. The Figure does not reveal how the efficiency of wave–particle interactions in the inner magnetosphere is affected by the wave frequency, polarization, particle species, pitch angle distribution, resonance harmonic number, and other parameters. The effect of these wave properties on particle dynamics is the subject of this review paper.
Fig. 2

Overview of frequency range of wave–particle interaction regions for energetic electrons (left panel) and ions (right panel) in Earth’s magnetosphere. The left-side vertical axes show the power flux density of waves and the right-side vertical axes show the corresponding energy range. Gyration, bounce, and drift frequencies of different L-shell ranges are marked. The ULF wave frequency range overlaps the bounce and drift frequencies of electrons and ions in the inner magnetosphere, indicating that bounce and drift resonance involving these waves is possible (Zong et al. 2008, 2011)

In the early 1940s, the magnetic signature of the passage of an IP shock was detected by ground-based magnetometers (Chapman and Bartels 1940) and referred to as Storm Sudden Commencement (SSC) (Chapman 1940). During such events, ions and electrons are accelerated during the shock passage (Brown et al. 1968, Matsushita et al. 1961; Ortner et al. 1962; Ullaland et al. 1970). Energetic (tens of keV) electron precipitation lasting from 3 to 10 min is enhanced and energetic particles are injected into the inner magnetosphere (e.g., Arnold et al. 1982; Blake et al. 1992; Li et al. 1993). IP shocks generate harmonic radiation at frequencies around a few hundred Hz (Hayashi et al. 1978), fast magnetosonic waves (Wilken et al. 1982; Kepko and Spence 2003; Tan et al. 2011), along with whistler waves (Parks 1975), EMIC waves (Anderson and Hamilton 1993), and chorus waves (Fu et al. 2012a, b). They also generate toroidal and transient Pc5-band standing Alfvèn waves (Cahill et al. 1990; Kim et al. 2002) and mixed toroidal-poloidal mode standing waves (Zong et al. 2009a, b; Zhang et al. 2010). Plasma waves produced by IP shocks drive particle acceleration (Li et al. 1993; Zong et al. 2009a, b; Tan et al. 2011; Halford et al. 2015) and/or particle loss in the magnetosphere (Matsushita 1961; Ortner et al. 1962; Brown et al. 1968; Ullaland et al. 1970). Sudden density or velocity increases in the solar wind produced by IP shocks lead to a step-like enhancement of the solar wind dynamic pressure and a sudden compression of Earth’s dayside magnetosphere. The resulting magnetosonic waves launched into the magnetosphere (Wilken et al. 1982; Kepko and Spence 2003; Hudson et al. 2004; Claudepierre et al. 2009) excite FLRs around magnetic local noon (Zong et al. 2009a, b; Zhang et al. 2010) and interact with charged particles.

In general, inner-magnetosphere wave–particle interactions are affected by VLF waves (Brice 1964; Kennel and Petschek 1966; Lyons and Thorne 1970; Lyons et al. 1972; Abel and Thorne 1998a, b) and ULF waves (Southwood and Kivelson 1981, 1982; Mathie and Mann 2001; O’Brien et al. 2003). Wave–particle interactions involving ULF waves are discussed in Sect. 4. Table 1 summarizes properties of toroidal and poloidal mode standing ULF waves that produce different types of interaction with energetic particles.
Table 1

Properties of toroidal and poloidal mode standing ULF wave

ULF mode

Magnetic field

Electric field

Wave number

Toroidal waves

B_azimuthal (B ϕ )

E_radial (Er)

Small wave number m

Poloidal waves

B_radial (Br)

E_azimuthal (E ϕ )

Large wave number m

Compressional poloidal waves

B_parallel (Bp)

E_azimuthal (E ϕ )

Large wave number m

It remains unclear how shock-related energetic particles are produced and accelerated in the magnetosphere (Baker et al. 2004; Friedel et al. 2002). It has become clear that ultra-relativistic electrons with energies greater than 6 MeV can be injected deep into the inner magnetosphere (L ~ 3) within a few minutes after IP shock impact (Kanekal et al. 2016). Observationally, there is no obvious immediate response to IP shocks for electrons with energies between 250 and 900 keV. The natural conclusion is that electric fields induced by the rapid change in the geomagnetic field are responsible for the particle energization. In situ observations of drift-bounce-resonance between energetic electrons and ULF waves following IP shocks have been reported by (Zong et al. 2007, 2009a, b; Tan et al. 2004, 2011). As mentioned earlier, this is possible because of the comparable periods of energetic particle drift motion and ULF oscillations (e.g., Southwood and Kivelson 1981; Takahashi et al. 1985, 1990, 1992; Hudson et al. 2008; Zong et al. 2007, 2009a, b). Drift-resonance with poloidal mode ULF waves (Zong et al. 2009a, b) and compressional waves (Tan et al. 2011) induced by IP shock impact is associated with fast electron acceleration and even the formation of new radiation belts (Zong et al. 2011).

There have been relatively few studies of the interaction between toroidal and poloidal ULF waves with energetic ions (H+, O+) in the ring current region. Ring current ions (mainly H+) with a positive gradient in velocity space (Gkioulidou et al. 2014) are a potential source of ULF wave generation through drift-bounce resonance (e.g., Hughes et al. 1978; Glassmeier et al. 1999; Wright et al. 2001; Ozeke and Mann 2008). Low-energy ring current O+ ions (several to tens of keV) are known to be accelerated and decelerated by ULF standing waves via drift-bounce-resonance during storm times (Yang et al. 2010a, b, 2011a, b). The presence of heavy ions in the ring current is a characteristic feature of powerful magnetic storms with |Dst| > 100 nT. During these storms, O+ can become the dominant ion species (up to 60–80%) in terms of energy density in the ring current region (Daglis et al. 1999; Zong et al. 2001; Fu et al. 2001). The mechanism by which ionospheric oxygen ions are injected into the ring current region during storm periods is not well understood (Daglis and Axford 1996), but may have an association with ULF waves.

Questions surrounding ULF waves, such as what controls their global distribution, interaction with energetic particles, and the role they play in energy transport throughout Earth’s magnetosphere, can be addressed using a combination of multi-satellite observations and numerical simulations. Recent theoretical extensions of drift-resonance theory are discussed in Sect. 5, while Sect. 6 presents new simulation results using models that are designed to reproduce wave–particle interactions. A deeper understanding of ULF waves will contribute not only to an improved understanding of mass, energy, and momentum transport processes in Earth’s magnetosphere, but also the influence of ULF waves on space weather in general.

2 ULF wave generation: toroidal, poloidal, and compressional modes

2.1 ULF waves in the inner magnetosphere

Figure 1 illustrates that solar wind disturbances and K–H surface waves can stimulate ULF waves and field line resonances (FLRs) in Earth’s magnetosphere under different conditions. (Allan and Poulter 1992). Observations from the four cluster spacecraft (cf. Fig. 1) reveal that after an IP shock, energetic electrons are accelerated promptly in the outer radiation belt. One of the most important acceleration mechanisms is considered to be drift-resonance with shock-induced ULF waves. The region of the magnetosphere where drift-resonance occurs is illustrated in Fig. 1, together with relevant parts of the system sampled by spacecraft such as cluster and double star.

ULF waves excited by IP shocks or solar wind dynamic pressure pulses (Zong et al. 2009a, b, Zhang et al. 2010) separate into toroidal and poloidal modes, see Table 1. The toroidal mode has periodic radial electric field and azimuthal magnetic field perturbations, whereas the poloidal mode has an azimuthal electric field and a radial magnetic field. Large-scale compressional waves may also contribute azimuthal electric field oscillations to some extent (Hudson et al. 1997). The response of poloidal mode ULF waves to both positive and negative solar wind dynamic pressure impulses in different local time sectors has been discussed by Zong et al. (2009a, b) and Zhang et al. (2010). A decrease in ULF wave frequency due to inflation of the magnetic cavity by a drop in solar wind dynamic pressure (negative impulse) was reported by Shen et al. (2017).

Figure 3 shows THEMIS-D measurements of ULF waves during 17:40–18:40 UT on 19th November 2007 (Fu et al. 2012a, b). An IP shock, characterized by a sudden increase in magnetic field strength from 45 to 50 nT, was observed at 18:10 UT (Fig. 3b). At that time, THEMIS-D was in the morning sector at L = 9.7 and saw no clear evidence of wave activity before the shock. A significant enhancement of ULF waves in the range of 3–8 mHz was observed after the IP shock impact. The frequency of observed ULF waves and modulated energetic electrons satisfy the drift-resonance condition (Fu et al. 2011).
Fig. 3

THEMIS observations of ULF waves triggered by an interplanetary shock on 19 November 2007, a cartoon illustrating compression of magnetosphere by the shock, b magnetic field strength, c magnetic field By component in the frequency range 3–8 mHz, d magnetic field Bz component in the frequency range 3–8 mHz. Data are shown in GSM coordinates. The vertical red line indicates the time when the interplanetary shock hits Earth’s magnetosphere

(modified from Fu et al. (2012a, b))

Generation of ULF waves by solar wind dynamic pressure pulses or interplanetary shocks is viable for the scenario illustrated in Fig. 3 from at least two vantage points: (1) unlike other triggering mechanisms the specific energy source is clear; (2) the response of the magnetospheric system to solar wind dynamic pressure pulses and IP shocks yields a significant and easily identified electromagnetic signal. Thus, there is no temporal ambiguity for IP shock-related ULF wave generation.

ULF waves that are correlated with positive and negative solar wind dynamic pressure impulses have been investigated statistically by Zhang et al. (2010) based on 270 positive and 254 negative pressure pulse events, respectively. The positive pressure pulse events are summarized in Fig. 4, which shows (a) the parallel magnetic field (Bp, compressional); (b) the radial magnetic field (Br, poloidal); (c) the azimuthal magnetic field (Ba, toroidal), all at geosynchronous orbit. The observations are binned according to epoch time (5 min resolution) and local time (4 h resolution) from 3 h before impulse onset to 3 h following impulse onset.
Fig. 4

Superposed epoch analysis of excited ULF waves—three components a parallel magnetic field (B p, compressional component); b the radial magnetic field (B r, poloidal component), and c the azimuthal magnetic field (B a, toroidal component) at geosynchronous orbit for 270 solar wind dynamic pressure impulse events

In Fig. 4, it is worth noting that the radial/azimuthal oscillating magnetic field (poloidal/toroidal waves) at 12:00 LT and 24:00 LT are in antiphase, and that wave periods are similar and amplitudes larger at 12:00 LT than at 24:00 LT. The amplitude of the poloidal ULF wave (B r) is also larger than the toroidal wave (B a). The poloidal ULF wave shows an enhancement in the first 20 min after zero epoch time at all local times that lasts at least 3 h in the noon-afternoon sector.

Figure 5 illustrates various features of poloidal and toroidal waves excited by positive and negative impulses at three different locations (~ 0:00 local time (LT), 6:00 LT, 12:00 LT) at geosynchronous orbit. The variations of the magnetic field \(B_{\text{r }}\) and electric field \(E_{\text{a}}\) for poloidal waves in the figure are the result of numerical calculations. The amplitude, duration, and period of positive and negative impulses are the same, while the phase is opposite. The blue solid lines represent the magnetospheric response to the positive impulse, and the red dashed lines represent perturbations due to the negative impulse. Figure 4 demonstrates that the amplitude of the poloidal wave is larger at 12:00 LT than at 0 LT and 06 LT, while the waves excited by positive and negative impulses have almost the same amplitude but are in anti-phase (180º phase difference). These results have been confirmed by statistical analysis of GOES magnetic field data at geosynchronous orbit (Zhang et al. 2010).
Fig. 5

Illustration of the magnetic field and electric field variations of poloidal waves excited by a solar wind positive/negative impulse at 0 LT, 6 LT and 11 LT at geosynchronous orbit. The arrows indicate the drift motion direction of ions (solid) and electrons (dashed) relative to midnight. The variations of magnetic field B r and electric field E φ (E ϕ ) for poloidal waves are the result of numerical simulation. The blue solid line and red dashed lines represent the poloidal ULF waves excited by positive and negative impulse, respectively (Zhang et al. 2010)

Although excitation of ULF Pc4-5 waves has been traditionally attributed to the K–H instability, waves generated by this instability show minimum activity around noon near the subsolar stagnation region compared with the dawn and dusk sector (e.g., Anderson et al. 1990; Zhu and Kivelson 1991; Lessard et al. 1999; Hudson et al. 2004; Rae et al. 2005; Takahashi and Ukhorsky 2007). The observations reported here, on the other hand, show that poloidal ULF waves associated with solar wind dynamic pressure pulses exhibit a maximum at local noon. This is consistent with a scenario in which poloidal ULF waves are excited when a solar wind dynamic pressure pulse or interplanetary shock impinges on the magnetopause, as confirmed by multi-spacecraft observations (Zong et al. 2009a, b) and MHD simulations (Kress et al. 2007).

It should be noted that the observations reported here are based on spacecraft measurements rather than ground-based measurements from magnetometers. Because poloidal mode ULF waves usually have a large azimuthal wavenumber on the order of m ~ 50 or greater, they can be strongly attenuated by the ionosphere (e.g., Hughes et al. 1976a, b) and may not be detectable on the ground.

Figure 6 shows the equatorial distribution of ULF wave power investigated by Liu et al. (2009) based on 7 years of electric field measurements from THEMIS.
Fig. 6

Distributions of Pc5 ULF wave power in the \(E_{a}\) component in the equatorial plane based on 7 years, Jan 2008 to Dec 2014, observations of THEMIS for Kp levels from 0 to 5 in panels (a) to (f), respectively

Adopted from Liu et al. (2016a, b)

The THEMIS data sets discussed here are categorized into six geomagnetic conditions defined by values of the Kp index and magnetic latitude within 15° of the magnetic equator. The data are also binned in spatial pixels for 24 MLT sectors, from L = 3 to 8 with a 1 R E step. Median values of azimuthal electric field (E a) power spectral densities integrated over the Pc5 frequency range are plotted in Fig. 6, which shows there is larger power at high-L. This L-dependence is consistent with an interpretation that Pc5 ULF wave power is mainly supplied from sources in the solar wind (Liu et al. 2009, 2010). Wave power is also stronger on the dayside compared to the nightside, where there is larger wave power in the pre-midnight sector compared to the post-midnight sector. As geomagnetic activity increases, the power spectral density of E ϕ also increases, as shown in Fig. 6a–f. It is also evident that ULF wave power penetrates deeper into the inner magnetosphere as the Kp index increases (Liu et al. 2016a, b). The geomagnetic activity dependence of the distribution has been analyzed using seven subsequent years of data from the THEMIS mission (Liu et al. 2016a, b). The distribution of the power spectral density of the azimuthal electric field component is shown in Fig. 6, adopted from Liu et al. (2016a, b). Larger power spectral density (PSD) is observed at high L, consistent with the hypothesis that the poloidal Pc5 ULF wave energy is mainly supplied from external sources (e.g., solar wind) (e.g., Zong et al. 2009a, b; Zhang et al. 2010; Liu et al. 2010).

2.2 ULF Waves in the plasmasphere boundary layer and related ions of ionospheric origin

Cold plasma mass density affects the Alfvén speed and thus the magnetospheric response to ULF waves, while the electron number density controls the high-frequency (VLF and radio wave frequency range) response (Denton et al. 2004a, b, 2006). It is unclear how plasma mass density structure near the plasmapause and plasmasphere boundary layer (PBL) affect other properties of ULF waves and field line resonances (Fraser et al. 2005). Recent exploration of the role of heavy ion populations such as oxygen is providing new insight into ULF wave properties and resonant particles within the PBL. Here, we discuss how changes in mass density and ion composition in the inner magnetosphere influence energetic particle populations, and the generation, quenching, and propagation of ULF waves. The plasmapause and PBL (Carpenter and Anderson 1992; Carpenter and Lemaire 2004) are important in this context.

The PBL is the terminology used when the plasmapause lacks isolated steep mass density gradients. The generation of ULF waves is affected by the evolution of cold plasma in the PBL, while particle acceleration and transport driven by ULF waves affect properties of the PBL itself. Elevated mass densities also affect the response time of the magnetosphere to solar wind and IMF dynamic changes and can be estimated using the “cross-phase spectral technique” developed by Waters et al. (1991). This involves using pairs of ground magnetometers (at a given magnetic longitude and different latitudes) to compute field line eigenfrequencies by comparing the wave phase and amplitude at each of two locations. Knowledge of cold plasma mass density also enables the Alfven speed to be determined and hence the magnetospheric response to low-frequency ULF disturbances (Denton et al. 2004a, b, 2006).

Cluster spacecraft observations summarized in Figs. 7 and 8 provide examples of the importance of ion composition and mass loading around the PBL during the 2003 “Halloween” storm period. Figure 7 gives an overview of Cluster measurements between 2130 and 2230 UT on October 31, 2003. During this time interval, the IMF B z was small and mostly positive, while the D st index had recovered from a major storm from a minimum D st around −383 nT at 2300 UT on October 30, 2003, to around −68 nT at the time of interest. Pc5 waves and quasi-periodic variations of particle fluxes can be seen in the figures. The largest modulations are found mainly in the electron fluxes, with a peak-to-valley modulation ratio of about 3, whereas the proton flux modulation is weaker with a peak-to-valley modulation ratio of about 1.5. The modulations for both electrons and protons diminish after about 2223 UT when Cluster enters the PBL, as indicated by the elevation of the plasma ion density in the bottom panel of Fig. 8. During the period when ULF waves are present, from 2130 to 2223 UT, the plasma density is ~ 1 cm−3. However, when the particle flux modulations disappear, the plasma density is ~ 10 cm−3. This is not large enough to be identified with the plasmasphere density of ~ 100 cm−3 (Moldwin 2002), but it is obviously higher than in the plasmatrough, plasmasheet, and ring current (~ 1 cm−3). The elevated density region most likely corresponds to the PBL, according to the definition in Carpenter and Lemaire (2004).
Fig. 7

Energetic electron (30–500 keV) fluxes observed by Cluster in the dayside magnetosphere on 31 October 2003. The top four panels show the modulated electron spectrum and fluxes. The bottom panels are the observed X and Y components and magnitude of the magnetic field in GSE coordinates subtracted by the T89 magnetic field model. The electron flux and toroidal ULF wave magnetic field show a clear correlation at ~ 3.9 mHz frequency, implying that ULF waves cause the observed energetic electron flux modulation (after Zong et al. (2007))

Fig. 8

Plasma density a for H+ and O+ and b energetic electron flux variations near Cluster perigee in the inner magnetosphere during the recovery of the “Halloween” storm on October 31, 2003. The red dashed line marks the termination of energetic electron flux fluctuations. The red line marks the Cluster trajectory. The wave fluctuations terminate when the Cluster fleet enters plasmasphere boundary layer

To better comprehend the properties of ULF waves and their effect on plasma ion populations, it is necessary to understand the structure of the plasmapause and PBL regarding ion composition, mass density, and the presence of density gradients. Detailed investigation of variations of the plasmapause and PBL reveal that energetic electron flux modulations are observed to terminate when the oxygen density exceeds the plasma proton density. Figure 8 shows plasma density variations for different ion species (H+, O+) associated with radiation belt energetic electron flux near Cluster perigee in the inner magnetosphere during a magnetic storm. A large number of ions (H+, O+) of ionospheric origin have been injected into the inner magnetosphere and are provided directly from the plasmasphere or PBL.

The quenching of ULF waves and modulations in the energetic particle fluxes within the PBL in Fig. 8 may be related to an increasing ratio of heavy ions (O+) to light ions (H+) and its effect on the ULF wave frequency and propagation speed. These heavy ions quench ULF waves by reducing wave growth in the inner plasmasphere. On the other hand, during periods of geomagnetic activity oxygen ions loaded onto magnetic flux tubes reduce local Alfven wave eigenfrequencies, resulting in deeper penetration of externally excited Pc5 wave power onto lower L-shells than normal. These low-frequency Pc5 standing waves deep within the magnetosphere can be efficient at accelerating outer radiation belt electrons and ring current ions (Ozeke and Mann 2008, Yang et al. 2010a, b).

The presence of heavy energetic ions in regions of the inner magnetosphere such as the PBL and plasmasphere is possibly the result of local acceleration of a cold population by ULF waves or recycling of ions of ionospheric origin through the magnetotail. To investigate this possibility, multiple signatures of energy dispersion in the spectrogram of energetic ions can be used to trace the origin of particles that have interacted with ULF wave electric fields. It is found that multiple energy dispersions appear very shortly after the first half cycle of wave electric field oscillations in the case of IP shocks. In Fig. 9, field line lengths between mirror point of ions with local pitch angles of 15°, 35°, and 45° from the equator are 3.97, 2.89, and 2.09 R E, respectively. This suggests these ions may be of plasmasphere origin rather than coming directly from Earth’s ionosphere 5 R E away from the equator (Zong et al. 2012). The dispersion signatures in the spectrogram in Fig. 9 and the calculated time delay profiles match quite well. The period of successive dispersion signatures is comparable to the ULF wave period of ~ 100 s, implying that the modulations in flux are the local response of ions to ULF waves rather than escaping ions from Earth’s ionosphere. Once these oxygen ions escape into the magnetosphere they will alter the plasma mass density and hence the properties of ULF waves (Denton et al. 2004a, b, 2006).
Fig. 9

Multiple energy dispersions associated with ULF wave carried electric field in the energetic ion spectrogram from 100 eV to 36 keV observed by Cluster CIS instrument. The ULF wave carried electric field Y-component is over-plotted in thick black. The horizontal white dashed line marks the energy of 10 keV. The beginning and end of three dispersions are marked with vertical dotted lines. The calculated time delay due to energy difference for the three dispersions with the black, blue, and red lines corresponding to the local pitch angle of 30°, 15°, and 45° respectively. The shock arrival time is marked by the vertical dashed line. The position in GSM, L value, invariant latitude (ILat), and magnetic local time (MLT) for C1 are given in the labels at the bottom

The density structure of the plasmasphere itself may be affected by ULF waves, as suggested first by Adrian et al. (2004). The observations summarized in Figs. 8 and 9 support this view as they demonstrate that following the appearance of IP shock-induced poloidal mode ULF waves, the oxygen ion density in the PBL is enhanced by a factor of 2–3.

Liu et al. (2013) investigated a poloidal ULF wave event observed by the THEMIS spacecraft on 15 September 2011, as shown in Fig. 10. The electron density derived from the spacecraft potential is plotted in the first panel. There are two density drops at L = 4.4 and L = 6.5, respectively, suggesting the presence of the PBL. The wavelet power spectra of B r, B ϕ , B p, E r, and E ϕ in the mean field aligned (MFA) coordinate observed by the THEMIS-D spacecraft are plotted in panels b through f, respectively. High narrowband wave power is clearly seen in both the B r and E ϕ components, suggesting the existence of a poloidal mode ULF wave. The frequency of this wave gradually decreases from 15 to 5 mHz as the spacecraft moves outward in L-shell. An interesting feature is that the ULF waves exist only within the PBL marked by the two red vertical lines. The particle measurements from the ESA instrument in the last panel show that the energy flux of several to tens of keV ions is enhanced after 15:45 UT, suggesting possibly that a ‘bump-on-tail’ ion distribution source provides the free energy necessary for ULF wave generation. Detailed analyses by Liu et al. (2013) suggests that the observed narrow-band ULF waves are second-harmonic poloidal mode oscillations generated by the drift-bounce-resonance ‘bump-on-tail’ hot ion distribution. This event enables the resonance condition to be investigated in detail, as discussed below.
Fig. 10

Observations by THEMIS-D on 15 September 2011: a electron density derived from the spacecraft potential measurement; bf wavelet power spectra of magnetic field (b radial, c azimuthal, d parallel) and electric field (e radial, f azimuthal) components in the MFA coordinate system; g spectra of ion energy flux between 1 and 30 keV

Adapted from Liu et al. (2013)

The plasma morphology of the event summarized by Figs. 8 and 9 is in agreement with the presence of a heavy ion ‘‘torus’’ or ‘‘shell’’ in the vicinity of the plasmapause (Horwitz et al. 1984; Singh et al. 1992; Fraser et al. 2005) as the oxygen ion density significantly exceeds the proton density. Exactly how this affects ULF wave generation is not yet fully understood (Fraser et al. 2005) but it is suggested to be possible at locations where the drift-bounce resonance frequency matches the eigenfrequency of local standing Alfvén waves. The presence of cold high-density plasma may, therefore, be a controlling factor for ULF wave generation in this event as it determines the L value of the field line on which ULF waves are excited, i.e., where the field line eigenfrequency and drift-bounce resonance frequency match. To understand this, first note that the Alfven velocity is defined by
$$V_{A} = \frac{B\left( s \right)}{{\sqrt {\mu_{0} \rho \left( s \right)} }},$$
where \(\mu_{0}\) is the vacuum permeability, ρ(s) is the plasma mass density, and B(s) is the geomagnetic field flux density along a field line with parallel coordinate s. The approximate eigenperiod of the magnetic field line can be obtained by an integration along the field line:
$$T = \frac{1}{f}\sim \int \frac{ds}{{V_{A} }}.$$

In the inner magnetosphere, a population of heavy ions (e.g., O+) leads to a decrease in the ULF wave frequency, thus enabling the possibility of wave generation within the PBL. The conclusion from this study is that exploration of the ion density composition and radial variation of the Alfven velocity provides new insight into ULF waves and resonant processes in the PBL. Before leaving this topic, it should be noted that although the presence of oxygen ions affects ULF wave periods greatly it will not by itself lead to wave generation. Free energy, e.g., a bump-on-tail ion distribution or external solar wind driver, is required for wave generation.

2.3 ULF waves driven by the transient foreshock phenomena

A substantial decrease of solar wind dynamic pressure caused by flow deflection and density depletion inside hot flow anomalies (HFAs) results in an imbalance of total pressure on either side of the bow shock. The bow shock and magnetopause in this situation can expand significantly by as much as 5 R E (Sibeck et al. 1999). Jacobsen et al. (2009) showed that field-aligned currents and traveling convection vortices (TCVs) can be generated by the significant magnetopause deformation caused by HFAs. Localized intensification of auroral emissions associated with TCVs generated by HFAs is considered to be the result of upward field-aligned currents (Sitar et al. 1998; Fillingim et al. 2011). Shields et al. (2003) showed statistically that TCV events occur simultaneously with Pc 3 micropulsations and might share the same source as these waves. THEMIS ground-based stations observed a propagating magnetic impulse event caused by an HFA (Eastwood et al. 2008), while Hartinger et al. (2013) reported a Pc 5 ULF wave event generated by transient ion foreshock phenomena (TIFP) such as HFAs. Zhao et al. (2017b) have shown that HFAs can generate global Pc 3 ULF waves inside the magnetosphere, and that the generated waves are standing Alfvén waves.

Hot flow anomalies (HFAs) form when interplanetary discontinuities (probably tangential discontinuities) interact with the bow shock (Schwartz 1995; Zong and Zhang 2011) and are often observed in the foreshock region of the bow shock within hot plasma and flow deflection (Schwartz et al. 1985; Zhao et al. 2015). Since HFAs cause significant expansion of the magnetosphere similar to the effect of negative solar wind dynamic pressure pulses, the mechanism of HFA generation of ULF waves in the magnetosphere can also be considered similar. The main difference between these two ULF wave generation mechanisms is that solar wind dynamic pressure pulses interact with the entire bow shock, while HFAs interact with the bow shock locally and move parallel to the bow shock. The MHD simulation and observational studies by Zhang et al. (2009, 2010b) have shown that ULF waves are generated inside the magnetosphere by negative solar wind dynamic pressure pulses. However, the possible ULF wave generation by HFAs has not been fully explored.

The Cluster 1 spacecraft observed a HFA event upstream of the dawnside (10 MLT) bow shock on April 27, 2008. The magnetosphere was quiet and stable outside the event region. The red dashed line in the middle panel of Fig. 11 represents the event center defined as the time when the magnitude of Vx reaches its minimum. The HFA is associated with an interplanetary discontinuity in which the direction of By reverses across the event center. The components of the magnetic field within the event region have large fluctuations due to the hot plasma. The reduced magnitude of the magnetic field and density shows strong compression regions on the leading side, while the increase in the magnitude of V y and V z corresponds to a deflection of the ion velocity. The dynamic pressure shows a large decrease inside the event center with a strong compression region on the leading side. The HFA event starts at 18:45:42 UT marked by the red dashed line on the leading side (defined as the time when the temperature increase is larger than the background value) and ends at 18:47:12 UT marked by the red dashed line on the trailing side (defined as the time when the temperature reduces to the background value from the event center).
Fig. 11

The HFA event observed by Cluster 1 on April 27, 2008 (from top to bottom panels show the dynamic pressure, the three components of the magnetic field in GSE coordinates, the magnetic field strength, the x component of the ion velocity in GSE coordinates, the y and z components of the ion velocity in GSE coordinates, the ion temperature, and the ion density); the red dashed vertical lines represent the start, center, and end of the event

Figure 12 shows that GOES 10 and 12 were in the afternoon sector at geosynchronous orbit at the time when Cluster 1 observed the HFA event and can thus be used to investigate the magnetosphere response to the HFA. Figure 12a and b shows the magnetic field components (in GSM coordinates) observed by GOES 10 and GOES 12 after subtracting a 5-min running average from their wavelet analysis. The red dashed lines represent the time when Cluster 1 observed the HFA event. In Fig. 12a, oscillations in the magnetic field component B x (other components are at noise level) observed by GOES 10 last for about 5 min and start about 98 s after Cluster 1 observes the HFA at 18:48:20 UT (marked by the blue dashed line). The observed Pc 3 ULF wave is stable and nearly monochromatic with a period 32 s. In Fig. 12b, GOES 12 observes the Bx component of the ULF wave about 71 s after Cluster 1 observes the HFA at 18:47:53 UT (marked by the blue dashed line). The Pc3 waves observed by GOES 12 also have a period around 32 s and last for about 4 min.
Fig. 12

Positions of Cluster, GOES, and THEMIS spacecraft on the xy plane of the GSM coordinates at the time when Cluster 1 observed the HFA event. The magnetic field components after subtracting the 5-min running average and their wavelet spectrogram of GOES 10, GOES12, THEMIS-A, and THEMIS-E in the GSM coordinates. The red dashed lines show the time when Cluster 1 observed the HFA event. The blue dashed lines show the time when the Pc 3 ULF wave starts

GOES 12, THEMIS-A, GOES 10 and THEMIS-E observed the ULF waves with the same frequency in turn, indicating the generated ULF waves propagated from the dawn side to the dusk side. The schematic in Fig. 13 summarizes features associated with a HFA and its relation to ULF wave generation. Reflected ions (white arrows) from Earth’s bow shock become trapped by a tangential discontinuity (purple dashed line) and drift along with it. The trapped, reflected ions interact with incident solar wind ions to form a hot plasma region (yellow region) known as a hot flow anomaly (HFA). Expansion of the HFA forms shocks (blue arrows) on either side of the structure and generates ULF waves that propagate globally inside the magnetosphere. The Cluster fleet indicated schematically in Fig. 13 has one of the spacecraft positioned to observe ULF waves produced by the HFA (red dashed lines represent magnetic field lines that are unperturbed, while solid red lines represent magnetic field lines that support ULF waves).
Fig. 13

Schematic graph of an HFA generating global ULF waves in the magnetosphere

Figure 13 provides context for interpreting observations of HFA-generated near-monochromatic Pc 3 waves by multiple spacecraft and ground stations. The waves have the characteristics of standing Alfvén waves with more power in the poloidal mode than in the toroidal mode. Pc 3 ULF waves were observed in the dawn, noon, and dusk sectors, indicating that the response of the magnetosphere to the HFA is global. The global aspect of Pc 3 waves produced by HFAs implies the impact of the latter on the magnetosphere is more significant than previously thought.

2.4 ULF waves in the nightside magnetosphere and magnetotail

Sibeck et al. (1990) proposed a model for how a pressure pulse interacts with the magnetosphere on the dayside magnetosphere. The formation of a single or double vortex near the magnetopause is predicted. The ionospheric counterpart has been observed by the SuperDARN radar and ground magnetometers (e.g., Sibeck et al. 2003). Global MHD simulations have reproduced the single vortex aspect of the interaction on the dawn and dusk side of the magnetosphere (Wang et al. 2010; Samsonov et al. 2010; Samsonov and Sibeck 2013). Samsonov and Sibeck (2013) explained vortex formation as the result of compressional wave penetration and reflection at magnetosphere boundaries.

Shi et al. (2014) studied the magnetospheric flow response to a sudden impulse using THEMIS spacecraft observations in the equatorial plane. During the event, THEMIS-B, C, and D, located near the duskside flank of the magnetopause, observed an anti-clockwise rotating flow vortex, as shown in Fig. 14. Equivalent ionospheric currents (EICs) derived from the THEMIS ground magnetometer array show a vortex with counterclockwise rotation around the magnetic foot-points of THEMIS-B, C, and D in the magnetosphere. This rotational sense indicates an upward FAC, which is consistent with the vortex observed in the magnetotail. After the impulse sweeps past the magnetosphere, THEMIS-D observes a ULF wave (Shi et al. 2013). The toroidal mode component of this standing Alfven wave is stronger than the poloidal component. The wave period of ~ 7.6 min is close to the FLR period as calculated from different models (e.g., Rankin et al. 2000, Lui and Cheng 2001). The generation of the ULF wave in this event is suggested to be closely related to the flow vortex in the magnetosphere.
Fig. 14

Comparison of vortex perturbations corresponding to rotation as observed by THEMIS B and C with Panel (a) Global MHD simulation results on the dusk side magnetosphere. The background color indicates plasma pressure, and the arrows represent plasma velocities (b) THEMIS probe positions for satellite B, C, and D (c, d) Plasma velocity vectors observed by THEMIS B and C at different moments

From Shi et al. (2014)

The observations discussed above were all made in the magnetotail. Tian et al. (2016) reported in situ observations of a clockwise flow vortex in the dayside magnetosphere with a radial scale larger than 3 R E and positive field-aligned current (FAC). Observations made by the THEMIS satellites in the outer dayside magnetosphere suggest that the flow vortex is produced by a sudden impulse. In the ionosphere, the preliminary impulse (PI) and main impulse (MI) geomagnetic signals are induced by twin traveling convection vortices due to a sudden increase of solar wind dynamic pressure. This is the first simultaneous observation of a dayside magnetospheric vortex and ionospheric traveling convection vortex or TCV.

Flow vortices and ULF waves associated with negative solar wind dynamic pressure have not been widely reported. Recently, Zhao et al. (2016) presented such an event, which was observed by the THEMIS satellites on the dawnside of the magnetosphere. The flow vortex, in this case, has an anti-clockwise rotation. This observation suggests the rotation sense of dawnside and duskside vortices are opposite, as global MHD simulation confirms. Thus, compared to the results of Shi et al. (2014), the conclusion is that the rotation sense of vortices induced by positive and negative sudden impulses are opposite.

In the nightside magnetosphere, a shock-induced substorm-like event on 13 April 2013 was observed by the newly launched Van Allen Probes (Hao et al. 2014). Substorm-like electron injections with energies in the range of 30–500 keV were observed immediately after the shock arrival between L ~ 5.2 and 5.5, and were strongly modulated at the 150-s wave period (Fig. 15). Toroidal and poloidal mode ULF waves appeared following a reconfiguration of the magnetotail magnetic field after the IP shock passage. The poloidal mode was more intense than the toroidal mode with a phase shift of 90° between field components B r and E ϕ that indicates a standing wave structure observed in the northern hemisphere. The energetic electron flux modulations from both Van Allen Probes provide an estimate of the azimuthal wave number m ∼ 14, implying drift resonance is possible for injected electrons with energies between 150 and 230 keV.
Fig. 15

Observations of electron differential flux and plasma parameters by Van Allen Probes A and B

(From Hao et al. (2014))

There are at least two possible IP shock or solar wind dynamic pressure pulse mechanisms to explain excited ULF waves in the magnetotail, flow-driven FLRs and injected particle-driven drift-bounce resonance instability. In this review, we consider only ULF waves driven by IP shock or solar wind dynamic pressure pulses. Other mechanisms generating ULF waves on the nightside, such as the substorm current wedge, reconnection, and the tail waveguide, are discussed in the recent review paper by Keiling and Takahashi (2011).

In the model of Sibeck et al. (1990) the shock interaction with the magnetosphere involves force balance between the transient solar wind and the magnetopause. In this case, an IP shock of high-dynamic pressure impinging on the magnetopause generates a single or double vortex around the magnetopause. Inside the magnetosphere, a fast-mode compressional wave is excited, which usually propagates tailward faster than the IP shock in the solar wind (Sibeck et al. 1990). The magnetopause bulges outward ahead of the IP shock and subsequently causes an inward compression of the magnetopause after the shock. These outward–inward bulges generate vortex-like structures inside the magnetosphere. The electromagnetic Poynting flux and plasma flow obtained from the frozen-in condition E = − V × B exhibit a bipolar signature with initially negative (positive) values turning into positive (negative) values immediately after the IP shock passage. This reversal indicates that a vortex has been formed.

IP shock-induced vortex-like flow has been observed by Huttunen et al. (2005), Keika et al. (2008), Tian et al. (2012), and Shi et al. (2013) in the tail plasma sheet. The vortex-like plasma flow structure can excite Alfvén waves at certain frequencies depending on its azimuthal phase velocity. The fast-mode compressional wave acts as an intermediary that transports magnetopause energy of motion to the location of Alfvén resonance (Wright and Rickard 1995). Figure 16 illustrates that as the fast-mode wave propagates within the magnetosphere FLRs are excited. These excited waves can interact with energetic electrons via drift-resonance, which is similar to the IP shock-excited dayside ULF wave scenario developed by Zong et al. (2009a, b).
Fig. 16

Schematic showing how a pressure enhancement associated with an IP shock drives reconfiguration in the nightside magnetosphere. At L ∼ 5, the shock breaks dynamic pressure balance and causes particle injection. In the deeper tail, a vortex-like flow is generated that acts as a broadband source of earthward-propagating compressional waves that drive field line resonances. The excited poloidal standing Alfvén waves in the Pc 4–5 band then interact with energetic particles via drift-bounce resonance

The other possible scenario whereby poloidal ULF waves can be excited is through drift-bounce resonance (Southwood and Kivelson 1981). Ozeke and Mann (2008) suggested that energetic ring current ions can excite poloidal ULF waves of moderate m-number in the second harmonic. Fundamental mode poloidal waves are possibly related to eastward-drifting injected electrons (James et al. 2013). As shown in Fig. 12, the observed dispersionless energetic electron injections may excite poloidal mode ULF waves (first harmonic) through the drift-bounce resonance mechanism. However, it is unclear how this scenario explains that (1) both poloidal and toroidal mode ULF waves are excited simultaneously and damped at the same rate; (2) the observed ULF waves have a rather long time delay (~ 14 min) with respect to the injected energetic electrons. Therefore, the first scenario is more favorable to explain the observations although the second scenario cannot be ruled out entirely.

2.5 Growth and damping of ULF waves

ULF waves excited by IP shock impact (Zong et al. 2009a, b) or solar wind positive and negative dynamic pressure pulses (Zhang et al. 2010) sometimes decay rapidly. In Fig. 14, a strong interplanetary shock with a maximum solar wind dynamic pressure of 70 nPa impinged on Earth’s magnetosphere at 18:27 UT on November 7, 2004. The Cluster spacecraft fleet was on the morning side of the magnetosphere in the PBL at L = 4.4. At that time, ULF waves induced by the IP shock were observed with electric field amplitudes as high as 60 mV/m (Zong et al. 2009a, b). The black, blue, green, and red lines in Fig. 14 represent the azimuthal component of the electric field observed by all four cluster spacecraft during 18:20–18:40 UT, respectively. The wavelet analysis spectrum of the azimuthal electric field from cluster-C3 is shown in the bottom panel of Fig. 17.
Fig. 17

Top: Azimuthal component of the electric field observed by the four Cluster spacecraft on Nov. 7, 2004; Bottom: Wavelet spectrum of the electric field azimuthal component observed by C3

As discussed by Zong et al. (2009a, b), in this event the four Cluster spacecraft observed strong Alfvén waves with a period of about 100–150 s near the PBL after the arrival of the IP shock. The observed ULF waves had a mainly poloidal component of electric field that accelerated ions effectively and decayed quickly within ~ 15 min. The ULF waves were observed to have a faster decay rate inside the PBL than in the plasmasphere. According to Southwood (1983), possible sinks of ULF wave energy include at least three mechanisms: damping through ionospheric Joule heating, generalized Landau damping, and mode coupling. By comparing the effects of Landau damping, Joule heating, and waveguide propagation, Wang et al. (2015) found that Joule heating and magnetospheric waveguide propagation are insufficient to account for the observed decay rate of ULF wave energy in this event. However, Landau damping of the wave due to drift-bounce resonance with energetic ions was estimated to be higher than that provided by Joule heating (Fig. 18). Wang et al. (2015) estimated that drift-bounce resonance between ULF waves and O+ ions with energies ranging from a few keV to tens of keV is sufficient to explain the rapid wave damping rates observed by Cluster on Nov 7, 2004. It was further confirmed that Landau damping is more efficient when heavy ions such as O+ are present, which explains why the ULF waves excited by IP shocks and solar wind pressure impulses can have higher decay rates in the PBL, although higher densities of H+ also contribute.
Fig. 18

The variations of ULF wave amplitude caused by different damping mechanism. Black, blue, and red lines are the calculated damping rates at C3, based on joule heating, Landau damping, and the combined effect, respectively

In cases where amplitudes of ULF waves excited by IP shocks and solar wind dynamic pressure pulses are observed to decay, the process begins within the first half to one complete wave cycle. However, six ULF wave events observed between April 1, 2007, and December 31, 2012, show growth in wave amplitude for dynamic pressure enhancements (Shen et al. 2015). The electric field amplitude carried by ULF waves in these cases increased continuously by a factor of ~ 1.3–4.4 over three or four wave cycles. Wave growth was mainly in the toroidal (radial) electric field component and likely the result of the superposition of two wave modes; a standing wave excited by a solar wind dynamic impulse of the type described by Zong et al. (2009a, b), and a propagating compressional wave produced by solar wind oscillations. When these two wave modes are superposed, wave amplitude growth in the radial electric field is observed, as illustrated in Fig. 19.
Fig. 19

The observed and simulated Er for Events A and B. a The radial component of the unfiltered electric field in the FAC system from 16:20 to 17:20 UT observed by THA and b the radial component of the electric field calculated by superposition of two wave sources. In Fig. 10b, the blue and green dashed lines indicate two wave sources and the black solid lines illustrate the superposed waves. The observed and simulated Er for Events A and B

From Shen et al. (2015)

The fast decay of the electric field can be fitted using the simple model of the electric field considered by (Zong et al. 2011),
$$E(t) = Ae^{{ - D_{0} t}} \cdot \sin (\omega_{0} t + \phi ).$$
Here, A is the wave amplitude, \(D_{0}\) is the damping rate, \(\omega = \omega_{0}\) is the wave angular frequency, and \(\phi\) is the initial phase of the signal. The mean lifetime of a large amplitude ULF wave that undergoes fast damping is given approximately by \(\tau_{0} = {1 \mathord{\left/ {\vphantom {1 {D_{0} }}} \right. \kern-0pt} {D_{0} }}\). It is worth noting that fast decay or growth of electric fields of ULF waves implies an additional particle acceleration mechanism, even for non-resonant particles. The damping stage of the ULF electric field contribution to such an acceleration process can be estimated as follows:
$$\varSigma E = \int_{0}^{\infty } {[A} e^{{ - D_{0} t}} \times \sin (\omega_{0} t + \phi )] \times dt$$
$$\varSigma E = A\frac{{\omega_{0} }}{{D_{0}^{2} + \omega_{0}^{2} }}$$

As discussed previously, the observed drift motion of charged particles is mainly in the azimuthal direction, i.e., the direction of poloidal wave electric fields. However, during successive wave periods of decaying or growing electric field oscillations, particles do not lose all of the energy gained in the preceding cycle. Interaction over multiple cycles is, therefore, important for all ULF wave energetic particle acceleration mechanisms in the inner magnetosphere and does not require any resonance condition to be satisfied (Zong et al. 2011).

3 ULF waves’ interaction with energetic particles in the magnetosphere

3.1 Interaction between ULF waves and energetic particles: resonant conditions

A widely used theory of energetic particle modulation by ULF transverse waves was developed by Southwood and Kivelson (1981, 1982). In their theory, particles experience the wave-carried electric field during their drift-bounce motion and their energy is accordingly changed. The drift-bounce resonance of energetic particles is determined from the following condition (Southwood et al. 1969):
$$\varOmega - m\omega_{d} = N\omega_{b} ,$$
where N is an integer (normally ± 1, ± 2 or 0), m represents the azimuthal mode number of the ULF wave, and Ω, ω d and ω b are the wave frequency and particle drift and bounce frequencies, respectively. Because of the known energy dependence of ω d and ω b , the resonance energy can be calculated if the wave properties (the wave frequency and the azimuthal wave number) are known.
The drift-bounce resonance energy for electrons interacting with poloidal mode ULF waves is shown in the top panel of Fig. 20 for different N. The wave frequency is 10 mHz at L = 5 and the equatorial pitch angle is 30°. The resonant energy of H+ (O+) ions versus azimuthal wave number is given in the middle (bottom) panel for different equatorial pitch angles. Here, m is positive for a ULF wave propagating eastward, negative for a wave propagating westward, and the bounce frequency of particles is considered positive. Knowledge of the azimuthal wave number (m) is critical for studying ULF wave interactions with the ambient plasma environment. Three methods for determining the azimuthal wave number (m) of waves based on ground-based magnetometer data and in situ measurements have been used. They are summarized below:
Fig. 20

Drift-bounce resonant energy versus azimuthal mode number for electrons (top), Hydrogen (middle) and O+ (bottom), calculated from \(m = \left( {\varOmega - N\omega_{b} } \right)/\omega_{d}\), where \(f = \varOmega /2\pi =\) 10.0 mHz and L = 5.0. Solid (dashed) lines correspond to positive and negative N, respectively

(1) Azimuthal wave number determined from multiple points of observation.

Multi-point observations can be used to determine the m-number of ULF waves that propagate azimuthally around Earth. In this approach, the measured phase difference between two signals is used to calculate m. By definition, m is the number of wave cycles around Earth in the azimuthal direction:
$$m = \, \Delta \varphi / \, \Delta \lambda ,$$
where Δφ is the wave phase difference between two observation points at the same latitude and Δλ is the azimuthal or longitude separation between these two points. Thus, phase differences and associated wave numbers that are found to be positive (negative) correspond to eastward (westward) propagation.
To account for differences in latitude between two ground-based stations or spacecraft we need to consider possible latitudinal effects when evaluating the azimuthal wave number. A simple method used by Takahashi et al. (1992) is based on the assumption that the phase of the wave varies linearly both in latitude (\(\lambda\)) and longitude (\(\varPhi\)), i.e., as exp [i(l \(\lambda\) + n \(\varPhi\))]. The phase variation between two stations is determined by solving the following equation.
$$\varPhi j - \varPhi i = k \left( {rj - ri} \right),$$
where \(\varPhi\) is the phase of the oscillation in a magnetic field component (H or D in the case of ground measurements), k = (l, n) is the dimensionless wave vector and r = (\(\lambda , \varPhi\)) gives the geomagnetic latitude and longitude of a station. The subscripts j and i designate a station pair. For each magnetic field component, data from three stations yields a unique value of k (Takahashi et al. 1992).

Magnetic field measurements by MMS can be used to determine azimuthal wavenumbers of ULF waves by computing phase differences between the poloidal magnetic field component (b r) measured by each spacecraft, taking account of their separation.

Figure 21 shows the wave magnetic field from the four MMS satellite from 1545 to 1630 UT on June 23, including the compressional (b p), poloidal (b r), and azimuthal (b ϕ ) components. For any two MMS satellites that are separated longitudinally, the azimuthal phase shift of the ULF wave can be determined. As shown in the bottom three panels of Fig. 21, four MMS satellites can provide six such measurements. The bottom panels show the observed ULF wave phase differences (time lags) as a function of longitudinal separation for the six pairs and for three selected waves intervals. The time lag is determined by cross-correlation analysis of the poloidal components (b r) observed by each pair. The obtained m numbers of the poloidal waves are −135, −62, and −119, respectively. The negative sign of the wavenumber indicates that the azimuthal wave vector k is directed westward, which is in the same direction as proton drift.
Fig. 21

MMS spacecraft wave magnetic field in field-aligned (p), poloidal (r), and toroidal (a) directions. The three horizontal bars indicate the time intervals selected for estimating the azimuthal wavenumbers. The red lines are the least square fits. The observed time lags in b r versus longitudinal separations between MMS satellites (Li et al. 2017a, b)

(2) Azimuthal wave number estimated by resonant particle (single spacecraft).

Figure 22 shows the electron fluxes in different energy channels obtained by the Cluster RAPID instrument during 2140–2200 UT on Oct 31, 2003. The azimuthal magnetic field of the wave is shown in Fig. 3b. The cross-wavelet analysis (Grinsted et al. 2004) technique has been used to obtain the phase differences between the electron fluxes and the azimuthal magnetic field. The observed phase differences can be explained by the resonance theory of Southwood and Kivelson (1981, 1982). At the resonance energy, the oscillations of the particle flux are in quadrature with the fundamental mode ULF wave magnetic field and in-phase at higher and lower energy. Therefore, the observed phase difference of 90 degrees at 127 keV is a signature of drift-resonance in this energy channel from 2143 to 2153 UT.
Fig. 22

a Energetic electron fluxes from different energy channels obtained by RAPID/CLUSTER. b The azimuthal component of the magnetic field (B ϕ ). c The phase differences (at the period of 256 s) between the azimuthal magnetic field and the electron fluxes in different energies obtained from cross wavelet analysis

The resonance energy can be used to estimate the azimuthal wave number m of observed ULF waves based on the drift resonance condition:
$$m = \, \varOmega /\omega_{\text{d}}$$

The drift frequency ωd for 127 keV electrons at L = 9 is about 0.4 mHz and the ULF wave frequency Ω is 3.9 mHz. Thus, the azimuthal wave number m is around 10. After 2153 UT the resonance energy is higher as the satellite is at lower L where the electron drift frequency is larger. To satisfy the drift resonance condition, the value of m must decrease, consistent with the low-m feature (|m| < 10) of the ULF wave observed at L = 6.6 (e.g., Hughes et al. 1978). For a monochromatic ULF wave, however, the phase difference between the electron fluxes and the magnetic signals should be either 90° or 0°. The gradual phase shift that is observed thus suggests a spread in the resonance energy that is either caused by the effect of coexisting poloidal and toroidal waves, or a range of Ω and/or m similar to the poloidal mode cases suggested by Takahashi et al. (1990).

(3) Azimuthal wave number estimated by finite larmor radius effect (single spacecraft).

It is also possible to determine the transverse propagation vector for a compressional wave by using the finite Larmor radius of energetic particles observed at a single spacecraft (Su et al. 1977; Kivelson and Southwood 1983; Lin et al. 1988). The basic idea is the following (see also Fig. 23): a spinning particle detector with spin axis parallel to the ambient magnetic field detects particles whose gyro-centers fall on a circle at a distance R L (the Larmor radius of particles) from the spacecraft and thus see at a different phase of the wave. Therefore, spin-phase dependence can occur in measured particle distributions when R L is comparable to the wavelength (Kivelson and Southwood 1983). Under the assumption that the velocity of gyromotion is much larger than the drift velocity of the particles, the spin modulation of flux j (Su et al. 1977; Lin et al. 1988) is given by
Fig. 23

Illustration of the spin phase dependence in measured particle distribution with a compressional wave. Red dashed circle shows the location of gyrocenters of particles continuously sampled by the spinning detector. The black circle arcs show the trajectories of the particles measured by the detector in two opposite look direction represented by the blue arrows. Point A and B, which are separated by 2RL along the dashed line, are the corresponding gyrocenters of the particles. Shaded and light areas show wave peaks and valleys, respectively

$$j \sim \exp (ik \cdot R_{L} - i\omega t).$$
The phase shift Δϕ between flux modulations observed in two opposite look directions is equal to the difference of the wave phases separated by 2R L along the orthogonal direction (see blue arrows and point A and B in the Fig. 23). There is a 2nπ ambiguity of the phase that can be removed by finding a consistent solution for different energy channels (Lin et al. 1988). The transverse propagation of the wave can be completely determined after velocity components in two orthogonal direction are obtained. Thus, the phase velocity of the wave along this direction can be calculated by
$$V_{p} = 2\pi R_{L} \left[ {\left( {2n\pi + \Delta \phi } \right)T} \right]$$
where T is the wave period, and the wavelength is λ = V p T. As long as the wavelength is known, the wave number m is unambiguously determined.
As seen in Tables 2 and 3, ULF waves cover a wide range of m from m = −110 (westward) to m = 250 (eastward). Considering that particles in the magnetosphere have a large energy range, their interaction with either eastward- and westward-propagating ULF waves warrants detailed study and is presented in Sects. 3.1.1, 3.1.2.
Table 2

Westward-propagating ULF Wave (m < 0)

Wave number (m)

L-shell value

Calculation method


− 110


Drift-bounce resonance (proton)

Takahashi et al. (1990)

− 70


Drift-bounce resonance (proton)

Dai et al. (2013)

− (67–29)


Ekaterinburg coherent decameter radar

Chelpanov et al. (2016)

− (38 ± 6)

5.2–8.1 (65°–73°N, 12°–22°E)


Yeoman et al. (2012)

− 35

(69.7°N, 18.9°E)

EISCAT HF coherent radars

Yeoman and Wright (2001)

− 25

6.5–8.3 (7.4)

Phase differences of geomagnetic data

Wright et al. (2001)

− (22–24)


Phase differences of geomagnetic data

Ren et al. (2015)

− (9–44)



James et al. (2016)

− 18


Phase differences of spacecraft measurement

Takahashi et al. (2015)

− (14 ± 2)

33 (~80°S)


De Lauretis et al. (2016)

− 13


Phase differences of geomagnetic data, Phase differences of spacecraft measurement

Sarris et al. (2013)

− (10 ± 3)


Phase differences of spacecraft measurement

Wang et al. (2015)

− (5–6)

5.27 (64°18′S, 248°21′E)


Ponomarenko and Waters (2013)

− 2.6 ± 0.28


Phase differences of geomagnetic data

Tian et al. (2012)

− 1–62



76 out of 118 cases

Mazzino (2015)

Table 3

Eastward-propagating ULF Wave (m > 0)

Wave number (m)

L-shell value

Calculation method




Phase differences of geomagnetic data

Tan et al. (2011)


3.56–14.93 (~58°–75°)

Phase differences of geomagnetic data

Pilipenko et al. (2014)



Phase differences of geomagnetic data

Shen et al. (2015)



Phase differences of spacecraft measurement

Shah et al. (2016)



Phase differences of spacecraft measurement

Sarris (2014)




42 out of 118 cases

Mazzino (2015)


7 (5.5–10)

Drift resonance (electron)

Zong et al. (2007)



Drift resonance (electron)

Ren et al. (2016)



Drift resonance (electron)

Hao et al. (2014)

17 ± 3


Drift resonance (electron)

Yang et al. (2011)



Theoretical assumption

Ozeke and Mann (2008)

20 ~ 55


Phase differences of geomagnetic data

Sarris et al. (2009)

22 ± 3


Drift resonance (electron)

Yang et al. (2010a, b)



phase relation between particle flux of different energy and ULF wave electric field

Zhou et al. (2015)



Drift resonance (electron)

Claudepierre et al. (2013)

50 (poloidal)

1 (toroidal)


Phase differences of spacecraft measurement

Zong et al. (2009a, b)


6.55-7.44 (~67°–68.5°, magnetic)

SuperDARN (Statistics of 83 events)

James et al. (2013)



Larmor radius sounding technique

Lin et al. (1988)


(69.63°N, 19.52°E)

Advanced Rio‐imaging experiment data

Beharrell et al., (2010)


4 ≤ L≤6

Phase differences of spacecraft measurement

Eriksson et al., (2005)

100 (40–250, no direction)


Frequency Doppler shift of spacecraft-observed ULF signals

Le et al. (2011)


L = [4.4, 4.6]

Phase differences of spacecraft measurement

Schafer et al. (2008)

3.1.1 Interaction of charged particles with eastward-propagating ULF waves (m > 0)

For a plasmasphere electron with an energy of a few eV in the inner magnetosphere, Fig. 24 (right-side top panel) shows that drift-bounce resonance with ULF waves is possible. Since the drift frequency of the electron is much less than the bounce frequency, ω d  ≪ ω b , and ω b is comparable with the ULF wave frequency Ω, the resonance condition degenerates to a bounce-resonance condition:
Fig. 24

ULF waves at odd (top row) and even (bottom row) harmonics have different parallel structures and satisfy different resonance conditions. For odd modes (e.g., fundamental mode), N in the drift-bounce resonant condition should be 0, ± 2,…; for even modes (e.g., second harmonic mode), N = ±1, ± 3,… (Southwood et al. 1981). The interaction between fundamental mode ULF waves and energetic particles has been investigated intensively (e.g., Zong et al. 2007, 2011, 2012; Yang et al. 2011a; Ren et al. 2016)

$$\varOmega \, = \, N \, \omega_{\text{b}}$$

Note that bounce-resonance is independent of azimuthal wave number, implying that plasmasphere electrons can resonate with both poloidal and toroidal mode ULF waves.

For an electron with energy around a few hundred keV in the inner magnetosphere, the drift-resonance condition can be satisfied because the drift direction of the electron (eastward) is in the direction of wave propagation. As the drift frequency of energetic electrons in this case is much less than the bounce frequency: ω d  ≪ ω b , and ω d is comparable to the ULF wave frequency Ω, only the N = 0 drift-resonance condition can be satisfied;
$$\varOmega \, = \, m \, \omega_{d}$$

The drift-resonance condition for energetic ions (~ 100 keV) cannot be satisfied since the drift direction of energetic ions is opposite the direction of wave propagation.

For electron energies in the range of a few keV to tens of keV, drift-resonance, bounce-resonance, and drift-bounce resonance are not possible because the drift frequency of electrons is much less than the ULF frequency Ω, ω d  ≪ Ω and Ω ≪ ω b . Figure 20 (right-side middle and bottom panels) shows resonant energies of H+ (right-side middle panel) and O+ (right-side bottom panel) versus azimuthal wave number and different N. For ions with energies in the range of a few keV to tens of keV, drift-bounce resonance is possible for N = +1, +2, +3. etc. Note that the drift-bounce resonance condition for ions of thermal energy is more easily satisfied for O+ than H+ when the ULF wave has a limited wave number. A summary of resonant interactions between electrons, ions, and ULF waves is provided in Table 4.
Table 4

The interactions between particles and ULF waves

ULF mode

East-propagating ULF waves (m > 0)

West-propagating ULF waves (m < 0)






Plasmasphere (~ 10 eV)





Plasma (~ 1 keV)





Energetic (~ 100 keV)






Plasmasphere (~ 10 eV)





Plasma (~ 1 keV)





Energetic (~ 100 keV)





3.1.2 Interaction of charged particles with westward-propagating ULF waves (m < 0)

When ULF waves propagate westward they have a different interaction with electrons and ions. Similar to the eastward-propagating ULF wave situation, a plasmasphere electron with an energy of a few eV (Fig. 24, left-side top panel) can satisfy the drift-bounce resonance condition. Also, the drift-bounce resonance condition degenerates to bounce resonance (see Eq. (4) for the resonant condition), which again is independent of azimuthal wavenumber.

From inspection of the top panel of Fig. 20 and the top-right panel of Fig. 25, in the case of westward ULF wave propagatiion, energetic electron drift-resonance cannot be satisfied as the eastward drift motion of energetic electrons is opposite the direction of wave propagation. However, drift resonance involving energetic ions (~ 100 keV) is possible as ions drift in the same direction as waves. For ions of energy in the range of a few keV to tens of keV, Fig. 25 also shows that drift-bounce-resonance and bounce-resonance is possible. As we can see from Fig. 20, the drift-bounce-resonance condition can be satisfied for hydrogen and oxygen ions with different energy when N is either positive or negative. For westward-propagating ULF waves with a limited wave number, the drift- bounce resonance in thermal energy range for O+ ions of is easier to satisfy than H+ ions.
Fig. 25

Different harmonic mode ULF waves have different parallel structures and satisfy different drift-bounce resonant conditions

3.2 Phase relationship between resonant ions and ULF waves

Figure 26 illustrates that drift-bounce resonant ions (e.g., O+) of equal energy and different equatorial pitch angle develop velocity dispersion as they move from the equator along magnetic field lines toward polar latitudes. The phase relationship between ions of fixed energy depends on the location of satellite observations. A negative slope in pitch angle dispersion would be observed by a spacecraft (e.g., Cluster) located in the southern hemisphere, as illustrated in Fig. 21.
Fig. 26

Energy spectrogram and pitch angle distributions observed by the CIS-CODIF instrument on the Cluster C4 satellite on October 29, 2003. The satellites traverses from the southern to northern hemisphere (Yang et al. 2011a). The panel excerpts show phase-relationships illustrated in Fig. 27

As shown in Fig. 27, due to the velocity dispersion the red particle on the field line in the figure will arrive at the spacecraft before the blue one, assuming both particles move southward from the equator with the same energy but different equatorial pitch angles, \(\alpha_{\text{eq1}}\) (red particle) and \(\alpha_{\text{eq2}}\) (blue particle), with \(90^{ \circ } < \alpha_{\text{eq2}} < \alpha_{\text{eq1}} < 180^{ \circ }\). The corresponding local pitch angle of the red particle (\(\alpha_{s1}\)) will be larger than the blue one (\(\alpha_{s2}\)), while for a local pitch angle of 90° the particle mirror point is at the location of the spacecraft. After half a wave cycle the red and blue resonant particles will be reflected at their mirror points and will reach the spacecraft again with local pitch angles of 180° − \(\alpha_{s1}\) (red one) and 180° − \(\alpha_{s2}\) (blue one), respectively. This implies that the phase relationship between the pitch angle distributions of resonant ions and their bounce motion may help to further diagnose the spatial structure of ULF wave modes.
Fig. 27

Drift-bounce resonant ions in the southern hemisphere, magnetic equator, and northern hemisphere. A schematic of the time of flight effect as ions propagate along a magnetic field line. The red and blue particles have the same energy but different equatorial pitch angles. The symbols \(\alpha_{eq}\) and \(\alpha_{s}\) denote the equatorial pitch angle and the local pitch angle detected by the virtual satellite, respectively. The variable \(\varphi_{m}\) represents the corresponding mirror point magnetic latitude. Pitch angle distributions are shown qualitatively at different latitudes

To explore the parallel structure of the ULF waves, the uncoupled poloidal and toroidal mode ideal MHD wave equations can be solved. Three possible scenarios for the parallel structure have been considered for different ionosphere conditions and plasma pressures:

  1. (1)

    The magnitude of the electric field peaks at a high magnetic latitude under the assumption of infinite ionospheric conductivity in a cold plasma (Cummings et al. 1969), or in plasma with finite pressure (Mager et al. 2009),

  2. (2)

    A uniform electric field structure in latitude under finite ionospheric conductivity equal in both hemispheres (Ozeke et al. 2004; Takahashi et al. 2011) or under finite ionospheric conductivity with some difference between both hemispheres (Ozeke et al. 2004; Dai et al. 2013),

  3. (3)

    A “Quarter-wave” electric field structure under the assumption of an ionospheric conductivity larger than a certain critical value in one ionosphere and less than it in the conjugate ionosphere (Allan 1983).

The phase relationship between wave electric field oscillations and the resonant particle flux at the magnetic equator can be used to diagnose the parallel structure of the electric field fundamental mode along magnetic field lines. Figure 28 shows schematics of three types of electric field morphology considered by Ren et al. (2016) for fundamental-mode poloidal ULF waves, and the expected behavior of accelerated ions satisfying the drift-bounce resonance condition N = 2 in the wave frame.
Fig. 28

Schematic of wave-particle resonance in a fundamental mode poloidal wave electric field a equator-concentrated case: electric field intensity peaks at the magnetic equator. b Equator-rarefied case: intensity peaks at high magnetic latitude and is lower at the magnetic equator. c Latitude-uniform case: intensity remains constant up to high magnetic latitude along the magnetic field line. Blue lines represent guiding center orbits of accelerated ions that produce a maximum in the flux. The black horizontal lines indicate the spacecraft trajectory in wave frame. Blue and solid circles represent positions where a maximum in the flux of accelerated particles will be observed and its location after being traced backward in time to the magnetic equator. The bottom panels show the phase relationship between the electric field and ion flux after being traced back to the magnetic equator. From Ren et al. (2017)

Figure 28a is a schematic in which the electric field maximizes at the magnetic equator (equator-concentrated case). Resonant ions following the blue-line trajectory produce a maximum in the flux because they experience the strongest westward-accelerating electric field (plus signs) and weakest eastward-decelerating electric field (minus signs) during a bounce period. These ions, when detected by a spacecraft (blue filled circle) moving through the wave electric field, can be traced to a position on the magnetic equator (red filled circle) where the maximum accelerating electric field exists (bottom panel of Fig. 28a). The ion flux traced in this manner is in antiphase with the eastward electric field in agreement with the observations shown in Fig. 29e–f. Figure 28b shows a case when the electric field maximum occurs at high magnetic latitude (equator-rarefied case). Resonant ions corresponding to the maximum flux experience the largest accelerating electric field at high magnetic latitude and weakest accelerating electric field at the magnetic equator. In the bottom panel of Fig. 28b, these ions traced back to the magnetic equator are in phase with the electric field. In the latitude-uniform case (Fig. 28c) the electric field is uniform up to high magnetic latitude. In this case, ions experience no net acceleration or deceleration over a bounce period. However, at locations where ions have experienced only the accelerating part of the electric field (marked by blue filled circles) a maximum in the flux will occur. By tracing these ions back to the equator it is found that the flux is 90° out of phase with the electric field.
Fig. 29

Accelerated O+ (10.5–35.1 keV) and H+ ions (0.3–12.3 keV) observed by Cluster and after they are traced back to the magnetic equator, indicated by the red segments. The blue line is the poloidal-mode electric field (E ϕ ). Red stars mark start and end points of the measured flux when it is higher than the average. Red segments indicate the duration of the higher than average spacecraft flux after it is traced back to the magnetic equator (Ren et al. 2016)

Figure 29 shows observations of accelerated O+ and H+ ions and their location (marked by red segments) after they are traced backward in time along guiding-center trajectories to the magnetic equator. The resulting time-series of ion fluxes have an anti-phase relationship with the westward-accelerating electric field. Ren et al. (2016) suggest this is an evidence that electric fields of fundamental mode poloidal waves peak near the magnetic equator.

The analysis presented above is based on the observed anti-phase relationship between bounce resonant ions and waves, and differs from theoretical predictions based on solutions to the ideal MHD wave equations. However, general morphology of the wave electric field structure under more general conditions can be further explored using the diagnostic approach proposed by Ren et al. (2016, 2017).

3.3 Particle trapping by compressional ULF waves

The mirror force produced by the magnetic field-aligned component of compressional mode ULF waves can also modulate energetic particle fluxes. Large energetic electron and proton particle flux modulations produced by compressional ULF waves were observed by the Van Allen Probes between 1200UT and 1400 UT on 19 February 2014 (Fig. 30) (Liu et al. 2016a, b). During this event, the Van Allen Probes were in the dayside magnetosphere near apogee.
Fig. 30

Van Allen Probe-A MagEIS and REPT omnidirectional particle fluxes between 1200 and 1400 UT on 19 February 2014. Electron fluxes from 79.80 keV to 2.85 MeV are plotted at top panel and proton fluxes from 82.85 to 636.18 keV are plotted below. Solid and dashed lines represent the time of Bp minima and maxima, respectively

The detrended omnidirectional differential fluxes from the MagEIS and REPT instruments onboard Van Allen Probe-A also exhibit oscillations, as illustrated in Fig. 30. As shown in Fig. 31, a clear correlation between particle fluxes and magnetic compressional mode oscillations can be seen between 1250 UT and 1350 UT. In Figs. 30 and 31, solid and dashed lines indicate the time of Bp minima and maxima, respectively. Inspection shows that the fluxes of both protons and electrons oscillate out-of-phase with Bp but show no noticeable phase differences across energy. The conclusion from this event study is that the magnetic field-aligned compressional component of fast mode waves can modulate energetic particles in this event across a wide range of particle energy, from 79.80 keV to 2.85 MeV for electrons, and from 82.85 to 636.18 keV for protons. The data also show that the amplitude of de-trended flux oscillations are larger at higher energies, indicating a more effective mechanism operates at higher energies. Note that peak-to-valley ratios have been used to quantify the strength of the particle flux modulations. For VAP-A there are five peaks in each channel in Fig. 30 near the five solid lines, while for VAP-B there are eight (Fig. 31).
Fig. 31

Van Allen Probe B compressional component and electron and proton pitch angle distributions. From top to bottom the following items are plotted: a magnetic compressional component, be electron pitch angle distributions at different energies, and fh proton pitch angle distributions at different energies

A possible scenario by which particles can become trapped by compressional ULF waves (mirror effect) is sketched in Fig. 31. Trapped particles peaking at 90° pitch angle will be confined near the magnetic field minimum, while particles with small pitch angle can reach the magnetic field maximum. A spacecraft traveling through the magnetic field minimum near the center of this magnetic bottle will detect a particle flux maximum (Kivelson and Southwood 1996). Consequently, pitch angle distributions at ‘a*’ or ‘b*’ in Fig. 32 should be similar to distributions at ‘o’, assuming the kinetic energy and magnetic moment are conserved. This similarity can be verified if the distributions at ‘o’ are also observed in situ by spacecraft. Based on Liouville’s theorem, the equations relating the particle distributions are,
$$\left\{ {\begin{array}{*{20}l} {F\left( \alpha \right) = F_{0} \left( {\alpha_{0} } \right)} \\ {{ \sin }\left( \alpha \right) = { \sin }\left( {\alpha_{0} } \right)\sqrt {B/B_{0} } } \\ \end{array} } \right.,$$
where F(α) and F 0(α) are distributions at ‘a*’ and ‘a’, respectively, and B and B 0 are the magnetic field strengths.
Fig. 32

Schematic of compressional wave mirror magnetic field in the wave frame. Lines with arrows represent spacecraft trajectories. For the event discussed in the paper, there are five magnetic minima for VAP-A in each energy channel. Liouvilles theorem is used to map the pitch angle distributions to locations where four magnetic maxima are observed. The 741.60-keV electron flux shown in Fig. 24 is used

Van Allen Probes observations demonstrate that the mirror effect discussed above can trap relativistic electrons with energies up to 2.85 MeV for an amplitude variation of B p around 20%, while for ions the upper limit is ~ 600 keV because of larger gyroradius. For B p ~ 200 nT the gyroradius is ~ 502 km for a 636.18-keV proton. This is comparable to the local radius of curvature of the wave magnetic field, which implies the proton response to the wave is non-adiabatic at these energies. The non-adiabatic parameter \(\kappa = \sqrt {R_{ \hbox{min} } /\rho_{ \hbox{max} } }\) in this case is ~ 2, where \(R_{ \hbox{min} }\) is the minimum radius of curvature of the magnetic field and \(\rho_{ \hbox{max} }\) is the maximum Larmor radius (Büchner and Zelenyi 1989). For ions with  ~ 2 the wave structure is not able to trap particles with energy higher than about 600 keV, which places a limit on the effectiveness of mirror force trapping by waves. The observations in this case indicate that relativistic electrons (ions) up to 2.85 MeV (0.6 MeV) energy can be strong modulated by ULF wave compressional mode structure along the field.

4 Fast acceleration of charged particles by ULF waves

The theory of drift-bounce resonance developed by Southwood and Kivelson (1981, 1982) is valid for axisymmetric magnetic fields. In this case, particles experience the wave electric field along their drift and bounce motion and suffer net energy gain or loss when the resonance condition \(\varOmega - m\omega_{d} = N\omega_{b}\) is satisfied, usually for N = 0, ± 1, ± 2. Parallel electric fields are neglected as they are usually small in collisionless plasma. The rate of energy change of a charged particle interacting with a ULF wave is expressed by
$$\frac{{{\text{d}}W}}{{{\text{d}}t}} = \mu \frac{{\partial B_{p} }}{\partial t} + q\varvec{E} \cdot \varvec{V}_{d} ,$$
where E, V d, and µ denote the wave electric field, particle drift velocity, and magnetic moment, respectively, and subscript p denotes the parallel component of the magnetic field. The first and second terms on the right-hand-side of the equation represent acceleration due to the wave magnetic and electric fields, respectively.
Elkington et al. (2004) showed that a particle drifting in an asymmetric compressed dipole field can interact resonantly with low-m global toroidal waves when the wave frequency satisfies \(\omega = \left( {m \pm 1} \right)\omega_{d}\). The drift–resonance interaction in this case is illustrated schematically in Fig. 33 for an m = 2 toroidal mode wave. It can be seen in this figure that an electron initially at dusk and moving inward radially gains energy due to an outward directed electric field if it reaches the dawn sector one wave period afterwards when the wave electric field is radially inward (also see Sect. 5.2). The resonance condition derived by Elkington et al. (2004) is valid for a specific magnetic field model, but in general it demonstrates that electrons can gain energy from toroidal mode ULF waves in a magnetic topology with noon-midnight asymmetry.
Fig. 33

Drift path of a relativistic electron interacting with a toroidal wave in a compressed dipole field, after Elkington et al. (1999). The electric field of an m = 2 toroidal wave are indicated by solid arrows at t = 0 for an electron starting from duskside, and the dashed arrows show the electric field direction one wave period later at which the electron has moved to the dawnside

The Elkington et al. (2004) theory suggests that toroidal mode ULF waves can accelerate energetic particles in the radiation belt region under strong solar wind pressure. However, in the inner magnetosphere noon-midnight asymmetry becomes insignificant on low L-shells. Acceleration of energetic electrons by toroidal mode ULF waves may, therefore, be limited to the outer magnetosphere, while in the inner magnetosphere the poloidal mode is more likely responsible. The exception is the dayside outer magnetosphere where the poloidal electric field of fast mode waves can accelerate energetic electrons, as shown by (Degeling et al. 2010, 2014) based on computer modeling and data-model comparisons.

Figure 34 is a schematic illustration on the behavior of resonant electrons and ions interacting with different harmonics of poloidal mode standing waves in a simple field line topology viewed in the wave frame. Westward and eastward electric fields are indicated by plus and minus, respectively, with magnitudes corresponding to the density of symbols. The blue and red dashed lines show guiding center orbits of resonant particles in the fundamental mode and second harmonic, respectively.
Fig. 34

Behavior of resonant electrons satisfying (a) the N = 0 drift resonance in an eastward-propagating fundamental mode standing wave, (b) the N = 1 drift-bounce resonance condition in a second harmonic westward-propagating standing wave. The westward and eastward electric fields are indicated by plus and minus, with their magnitude indicated by the density of symbols

The resonance condition defined in Sect. 3.1 can be used to identify optimum acceleration scenarios for electrons and ions interacting with poloidal mode waves. As illustrated in Fig. 34a, an electron in N = 0 drift-resonance with an eastward-propagating fundamental mode standing wave will experience continual acceleration because its gradient-B drift motion is synchronized with the speed and direction of propagation of the wave (see Sect. 4.1). Figure 34b illustrates that ions in N = 1 drift-bounce resonance with a second-harmonic westward-propagating standing wave will also experience continual acceleration (see Sect. 4.2).

4.1 Fast acceleration of “killer” electrons in the earth’s radiation belt by poloidal ULF waves

Energetic electrons can be accelerated efficiently by poloidal mode ULF waves that have a sufficiently strong electric field in the direction of electron drift motion. For energetic electrons, the bounce frequency is normally much larger than both the drift and ULF wave frequencies. Only the N = 0 resonance condition can be satisfied so that electrons do not move azimuthally in the wave frame. The guiding center orbit shown in Fig. 34a indicates that acceleration and deceleration of electrons cancel over a bounce period in a second harmonic wave mode. Therefore, only fundamental and odd harmonic waves can accelerate electrons over many drift periods. As discussed earlier, the N = 0 drift-resonance condition \(\omega = m\omega_{d}\) has no explicit relationship to bounce motion.

Figure 34a is a schematic but whether electrons experience net acceleration depends on the wave frequency and wave vector, m-number, electron pitch angle distribution, location in the magnetosphere, and so on. To explain quantitative details of wave–particle interactions in the inner magnetosphere all of these factors must be considered. Complicating the situation is that IP shocks impinging on Earth’s magnetosphere also manifest energetic particle acceleration and generation of ULF waves. The first effect of shock impact is compression of the geomagnetic field and passage of the shock through the magnetosphere in ~ 1 min based on a speed of shock propagation on the order of the Alfven speed. It is reasonable, therefore, to assume the first adiabatic invariant is conserved during shock passage (Wilken et al. 1986).

An IP shock interaction with the magnetosphere was observed in situ on November 7, 2004 (Zong et al. 2009a, b, 2012). Figure 35 shows that during the event large amplitude ULF waves are generated, along with fast electron acceleration. The observed shock-induced ULF waves can be separated into the toroidal mode with time-varying radial electric field and azimuthal magnetic field, and the poloidal mode with azimuthal electric field and radial magnetic field. Large-scale compressional waves driven directly by the shock may also contribute to the observed azimuthal electric field perturbations to some extent. The wavelet coherence (Grinsted et al. 2004) between the electric field component and the integrated energetic electron flux is shown in Fig. 36b and d for the poloidal and toroidal modes, respectively. Both wave modes have similar power, but the highest coherence of 0.9 is at a wave period of about 110 s in the poloidal mode just after shock arrival, which further confirms strong correlation between the poloidal wave electric field E ϕ and energetic electron fluxes.
Fig. 35

From top to bottom: energetic electron pitch angle distributions (68–94 keV) measured by Cluster C2, C3, and C4 over-plotted with the azimuthal electric field E ϕ (black line) in the mean-field-aligned (MFA) coordinate system, and magnetic field B z component from the four satellites (in colors) in the 7 November, 2004 event. Note that positive E ϕ field is in the eastward direction. The equatorial radial distance in R e, the L value, for each satellite is given in the labels at the bottom of the figure. The dashed vertical line marks the time of arrival of the shock-induced field disturbances

Fig. 36

Measurements from Cluster C3 in the 7 November 2004 event. The magnetic field and electric field are projected onto a local mean-field-aligned (MFA) coordinate system in which the parallel direction p is determined by 15-min sliding averaged magnetic field, the azimuthal direction a is parallel to the cross product of the p and the spacecraft position vector, and the radial direction r completes the triad. a Continuous wavelet power spectrum of the radial electric field, b the squared wavelet coherence between the radial electric field and the integrated energetic electron flux, c the azimuthal electric field, and d the squared wavelet coherence between the azimuthal electric field and the integrated energetic electron flux; e phase difference between electric field and magnetic field for the toroidal mode (red) and poloidal mode (blue)

Wavelet transform (Grinsted et al. 2004) performed on the electric field components of the wave energy density spectra for the toroidal and poloidal mode are shown in Fig. 36a and c, respectively. The periods of these modes are in the range 100–200 s, and the electric and magnetic fields of both modes have a phase difference of about 90° (Fig. 36e), which suggests they are standing waves (Singer et al. 1982) along the magnetic field. Using the box model approximation for the magnetosphere (Southwood and Hughes 1983), the observed toroidal and poloidal standing waves are determined to be fundamental modes.

As spacecraft measurements are a mixture of temporal and spatial variation (Fig. 37), analysis of particle acceleration due to the compressional component of the wave magnetic field must have the latter removed (Zong et al. 2012). This can be accomplished using the following equation:
Fig. 37

The 6-s period of the shock-induced B p variation. Time variation of ∂B p/∂t calculated from Cluster magnetic field observations during the period from 18:27:18 to 18:27:24 UT. The vertical arrows mark the maximum value of ∂Bp/∂t. The average and maximum of ∂B p/∂t are labeled

$$\frac{{\partial B_{P} }}{\partial t} = \mu \frac{{{\text{d}}B_{p} }}{{{\text{d}}t}} - (V \cdot \nabla )B_{p} .$$
While the partial differential of B p cannot be directly observed by spacecraft, the temporal variation can be obtained from the observed magnetic field using
$$\frac{{\partial B_{P} }}{\partial t} = \frac{{B_{p} (t_{2} ,L_{2}^{*} ) - B_{p} (t_{1} ,L_{2}^{*} )}}{{t_{2} - t_{1} }}\sim \frac{{B_{p} (t_{2} ,L_{2}^{*} ) - B_{p} (t_{1} ,L_{1}^{*} )\left(\frac{{L_{1}^{*} }}{{L_{2}^{*} }}\right)^{3} }}{{t_{2} - t_{1} }},$$
where subscripts 1 and 2 represent values before and after the IP shock compression, L* is the corrected L shell parameter (Roederer 1970), and B p is the parallel component of the magnetic field determined from a local mean-field-aligned (MFA) coordinate system (Takahashi et al. 1990). As defined in Roederer (1970),
$$L^{*} = - \frac{{2\pi k_{0} }}{{\varPhi R_{E} }},$$
where k 0 is the magnetic dipole moment of Earth, R E is Earth radius, and \(\varPhi\) is the third adiabatic invariant calculated using the ONERA- DESP library. The IGRF model is chosen as the internal geomagnetic field and the T96 model is selected as the external geomagnetic field. Thus, the energy change due to the interplanetary shock compression can be estimated assuming the magnetic moment is conserved and the energy change of particles is \(\mu\) dependent. The spectra at 1825 UT, Nov. 7, 2004 is used as the initial condition.
For each energy channel, the energy after compression at 1829 UT is computed according to
$$E_{1} = E_{0} \left( {1 + \frac{{\int_{{t_{0} }}^{{t_{1} }} {\frac{{\partial B_{p} }}{\partial t}{\text{d}}t} }}{{B_{0} }}} \right) = \left( {1 + \frac{{ \langle \frac{{\partial B_{p} }}{\partial t} \rangle (t_{1} - t_{0} )}}{{B_{0} }}} \right),$$
where subscripts 0 and 1 refer to 1825 UT and 1829 UT, B 0 is the magnetic field magnitude at 1825 UT, and 〈∂B p/∂t〉 denotes the average during 1825–1829 UT. Then the estimated flux at 1825 UT for each energy channel is used as the spectra at 1829 UT. The observed hydrogen and oxygen ion spectra at 1825, 1829, and 1840 UT, as well as the calculated spectra for 1829 UT are shown in Fig. 38.
Fig. 38

Energetic electron spectra observed by Cluster C3 before (tb, black), immediately after the interplanetary shock compression (at the first peak, t1, blue), and about 12 min after the appearance of the shock-excited ULF waves in the inner magnetosphere (t2, red) in the 7 November 2004 event (Zong et al. 2009a, b)

As seen in Fig. 38 (compression part labeled t1), the calculated enhancement of the spectrum due to shock compression is rather small; less than 10% of the observed energy enhancement of the energetic electron flux in the inner radiation belt. This implies that electric fields carried by the observed ULF waves contribute the major part of the acceleration of energetic electrons after interplanetary shock arrival. Based on the observed ULF wave electric field of 40 mV/m along the drift path, an electron with an energy of a few hundred keV can double its energy in less than a few wave periods. This is much faster than timescales of ULF wave-driven radial diffusion (Perry et al. 2005). Recently, prompt acceleration of relativistic electrons (~ MeV) observed by the Van Allen Probes have been observed in association with impulsive shock-induced electric fields and associated ULF wave processes (Foster et al. 2015). In the nightside of the magnetosphere, Hao et al. (2014) have also reported the response of ULF wave-modulated substorm-injected electrons to interplanetary shock impact. These observations suggest interplanetary shocks are intimately connected to ULF waves and acceleration of energetic particles in Earth’s magnetosphere.

4.2 Fast acceleration of ring current ions by poloidal ULF waves

Ions satisfying the N = 2 drift-bounce resonance condition for the fundamental mode electric field illustrated in the left panel of Fig. 34 experience equal and opposite electric fields during each bounce and have no net acceleration. In contrast, ions moving along the trajectory through the second harmonic ULF wave illustrated in the right panel of Fig. 34 experience westward-directed electric field during each bounce. Consequently, N = 1 drift-bounce resonance is associated with successive accelerations in the direction of westward-propagating second-harmonic waves. The conclusion is that N = 1 drift-bounce resonance involving second-harmonic poloidal mode standing waves is more efficient than for fundamental mode and higher harmonic ULF waves.

Figure 39 shows pitch angle distributions for oxygen ions (8.3–35.1 keV) measured by the Cluster C1 spacecraft on June 3, 2003. Pitch angle dispersions in the oxygen particle flux are strongest in the energy range 10.5–35.1 keV, which is suggestive of an acceleration process involving poloidal mode ULF waves. The dashed lines in Fig. 39 show estimated arrival times of oxygen ions of different energy and pitch angle that have traveled from the equator to the spacecraft location. The delay times are consistent with the spacecraft observations, which indicates that the electric field carried by the ULF wave in this case peaks at the magnetic equator consistent with a fundamental mode ULF wave. In this scenario, the acceleration process occurs predominantly at the magnetic equator region.
Fig. 39

Magnetic drift-bounce resonant O+ ions. PADs of the oxygen ions in different energy channels as measured by the CODIF instrument from C1; the black dashed lines in each channel indicate the calculated pitch angle dispersion signatures of the oxygen ions with the maximum flux, which originate from the magnetic equator

Figures 40a and b present the averaged phase space density (PSD) of oxygen ions (1–40 keV) and hydrogen ions (0.05–20 keV) between 2210 and 2225UT, respectively. The dashed lines (red) show the background spectra fitted with a power-law distribution. Figure 34a shows that the accelerating process mainly occurs in the energy range (gray shaded area) of oxygen ions in drift-bounce resonance with ULF waves within the wave bandwidth. The energy range of protons that satisfy the drift-bounce resonance condition for ULF waves within the wave-bandwidth is about 0.85–4 keV (gray shaded area in Fig. 40b), where the decelerating process mainly occurs. Therefore, ULF waves play a dominant role in acceleration and deceleration of ions. The relative PSD variations of oxygen ions are much larger than for protons, which indicates a more efficient energy exchange between oxygen ions and ULF waves (Zong et al. 2012). The resonant energy of oxygen ions is higher than hydrogen ions due to the mass difference, which results in a higher energy exchange rate for oxygen ions.
Fig. 40

The averaged PSD spectra of a O+ and b H+ ions over the time periods from 2210UT to 2225UT, June 3, 2003. The dashed red lines represent the power law fit of the PSD spectra. The gray shadow areas in a and b represent the resonant energy range of O+ and H+ ions satisfying the drift-bounce resonant condition with ULF waves (Ren et al. 2016)

Figure 20 shows that the drift-bounce resonance condition for oxygen ions is more easily satisfied than for hydrogen for azimuthal wavenumbers |m| < 100 since the bounce speed of the former is closer to the gradient drift speed due to its mass dependence. In the energy range of ring current ions, oxygen ions can satisfy the N = ± 1, ± 2 resonance conditions, which implies drift-bounce resonance is potentially important for ring current oxygen acceleration.

To establish the link between ULF waves and the acceleration of H+ and O+ ions, the coherence between the wave electric field and the integrated fluxes for hydrogen and oxygen for both the poloidal and toroidal modes has been calculated (Fig. 41). In Fig. 41, the first panel shows the continuous wavelet power spectrum of the azimuthal electric field (poloidal mode). The second and third panels show the squared wavelet coherence between the azimuthal electric field and integrated hydrogen and oxygen ion fluxes. The fourth panel is the wavelet power spectrum of the radial electric field (toroidal mode). The fifth and sixth panels show the squared wavelet coherence between the radial electric field and integrated hydrogen (0.2–20 keV) and oxygen (2.12–18.65 keV) ion fluxes.
Fig. 41

Continuous wavelet power spectrum of both ULF poloidal and toroidal electric field and the squared wavelet coherence between the toroidal mode and poloidal mode electric field and the integrated Hydrogen (0.2–20 keV) and oxygen (2.12–18.65 keV) ion fluxes. The over-plotted black arrows are the phase angles between toroidal mode and poloidal mode electric field and the integrated hydrogen and oxygen ion fluxes (with in-phase pointing right, anti-phase pointing left, and 90° pointing straight down)

Although the poloidal (the first panel of Fig. 41) and toroidal (the fourth panel of Fig. 41) modes have similar power intensity, the coherences between ULF waves and oxygen and hydrogen ion fluxes are different. High coherences (> 0.9) exist between hydrogen and oxygen ion fluxes and the poloidal mode (two bands between 64 and 256 s and between 256 and 512 s) rather than the toroidal mode. The duration of high coherence for oxygen ions is also longer than for hydrogen. The phase differences between the poloidal mode electric field and integrated fluxes of hydrogen and oxygen ions are nearly 180°, indicating anti-coherence These results show that the strongest coherence is between the electric field E ϕ carried by the poloidal wave and the integrated fluxes of both hydrogen and oxygen ions. Toroidal mode ULF waves have much less effect on the integrated fluxes of hydrogen and oxygen ions. The close coherence and the duration of high coherence between the wave-like E ϕ oscillations and the integrated fluxes of oxygen ions suggest that shock-induced ULF waves produce strong modulation/acceleration effects on oxygen.

The rapid acceleration of energetic oxygen ions via drift-bounce resonance with poloidal-mode ULF waves contributes to the development of the magnetic storm ring current since the affected ions are within the ring current energy range. The spectra given in Fig. 42 were obtained from the CIS (31 energy channels) and RAPID (5 energy channels) instruments onboard the Cluster C3 and C4 spacecraft in the energy range from 10 eV to 1000 keV. There are obvious double-peak structures in the spectra of both hydrogen and oxygen ions with one peak at a few keV and the other at ~ 80 keV. As seen in Fig. 30, the spectra for oxygen ions change dramatically after interaction with ULF waves induced by IP shock impact (Zong et al. 2012).
Fig. 42

Spectra evolution of plasma and energetic oxygen ions during the interaction with ULF waves (Zong et al. 2012)

The formation of the ring current during a geomagnetic storm is one of the most fundamental issues in space physics. The acceleration of energetic ions (H+ and O+) by poloidal mode ULF waves from 10 s of keV to 100 s of keV in the ring current region is, therefore, important for understanding the origin of magnetic storms and other dynamic processes in the inner magnetosphere.

4.3 Simultaneous resonance of ULF waves with energetic electrons and ions

As shown in the top panel of Fig. 20 and in Table 2, it is possible that ULF waves can interact simultaneously with energetic electrons via drift-resonance and thermal hydrogen and oxygen ions via drift-bounce resonance.

Figure 43 provides a schematic view of the behavior of resonant electron and ions interacting with a fundamental mode standing wave. The particle behavior is again illustrated using a box model magnetosphere viewed in the frame of the wave. Plus and minus symbols on the figure indicate westward and eastward electric fields, respectively, with magnitudes represented by the density of symbols. The blue and red dashed lines show guiding-center orbits of electrons and ions in a fundamental mode wave, respectively. Figure 43a shows that energetic electron drift-resonance (N = 0) is possible, while Fig. 31b indicates that for hydrogen and oxygen ions drift-bounce resonance can occur. Ions that follow the red-dashed or blue dot-dashed trajectories satisfy the N = 2 drift-bounce resonance condition for the fundamental mode. They experience a strong (weak) westward-accelerating electric field and a weak (strong) eastward-decelerating electric field within each bounce period and thus suffer net acceleration (deceleration). An ion moving along the black solid line satisfying the N = 2 drift-bounce resonance condition will experience acceleration in the first half period and equal deceleration in the second half period, and thus gain no energy during the interaction.
Fig. 43

A schematic of ULF waves interacting with energetic electrons (N = 0 drift resonance) and with ions (N = 2 drift-bounce resonance) in a fundamental mode standing wave. Left: schematic of the behavior of resonant electron satisfying N = 0 drift resonance condition in a fundamental standing wave. Right: schematic of the behavior of resonant particle satisfying N = 2 drift-bounce resonance condition in the same fundamental mode standing wave. The westward and eastward electric fields are indicated by plus and minus, and their magnitude corresponds to the density of the symbols

Figure 44 shows energetic electron fluxes recorded by Cluster that are modulated at the frequency of an observed 3.3 mHz poloidal mode standing wave. The peak-to-valley ratios measured by all four spacecraft indicate that the drift-resonance energy of energetic electrons in the event is around 94 keV and that the resonance energy width is about 60 keV (Yang et al. 2010a, b). In the same time interval, periodic bi-directional O+ and H+ ion flux oscillations at the same frequency are observed at energies near 4.5 keV and 280 eV, respectively. Figure 39 shows O+ pitch-angle distributions measured by the CODIF instrument onboard Cluster C1 in different energy channels ranging from ∼ 2 to ∼ 20 keV. The behavior of the pitch angle distributions indicates that in this event oxygen and hydrogen ions are participating in bounce-resonance at energies of ∼ 4.5 keV and ∼ 280 eV, respectively.
Fig. 44

The electron flux variations from Cluster/RAPID measurements from 2300 UT, 21 October 2001 to 0000 UT, 22 October 2001; the energetic electron flux oscillation with removed background levels. The L value variations derived from the IGRF model. The first black dashed line marks the arrival of the solar wind pressure pulse

The observations summarized in Figs. 44 and 45 are the first to show simultaneous energetic electron drift-resonance and drift-bounce resonance with thermal ions in a fundamental poloidal mode standing wave in the inner magnetosphere. Third-harmonic ULF waves can also interact simultaneously with substorm-injected hot ions from the magnetotail, and cold ion outflow from the ionosphere (Ren et al. 2015). This implies that fundamental mode poloidal ULF waves can also be excited by energetic electrons via drift-resonance. The excited poloidal ULF waves can further interact with thermal ion species via drift-bounce resonance and vice versa. The 3.3-mHz frequency of the observed fundamental mode poloidal wave between L = 4.2 and 5.0 is much lower than expected of this region (∼ 10 mHz) (e.g., Wild et al. 2005). A density of O+ four times larger than H+ can perhaps explain the observed low frequency (see Sect. 2). There is already evidence of enhanced oxygen ion densities in the inner magnetosphere (e.g., Yau et al. 1985; Hamilton et al. 1988; Fu et al. 2001) and significantly decreased eigenfrequencies of magnetic field lines (Fraser et al. 2005). The source of this enhanced mass density may be drift-bounce resonant oxygen ions that mass load magnetic field lines, allowing Pc5 waves to penetrate to significantly lower L than normal (e.g., Lee et al. 2007). These standing waves at relatively low frequency in the inner magnetosphere can accelerate radiation belt electrons efficiently in this region via drift-resonance.
Fig. 45

Pitch angle distributions of the O+ ions in different energy channels as measured by the CODIF instrument on the Cluster C1 satellite1. From top to bottom, the energies decline from around 20–2 keV. The various energy channels are labeled on each panel

5 The Southwood–Kivelson resonance theory and its recent extensions

5.1 Theory of ULF wave poloidal mode drift resonance

The observations of particle flux oscillations in the ultralow frequency band have long been interpreted in the context of drift and/or drift-bounce resonance between ULF waves and charged particles (Southwood and Kivelson 1981, 1982). These important resonant processes, as indicated by their names, are associated with the drift and bounce motion of charged particles in the inner magnetosphere since the ULF wave periods are usually comparable to the energetic particle drift and bounce time scales through the waves. Let us start from considering the behavior of equatorially mirroring particles in the transverse ULF wave field. In this case, any change in particle kinetic energy must be attributed to the particle acceleration or deceleration during their drift motion by the wave-associated electric field. Given that energetic particles in the inner magnetosphere drift mainly in the azimuthal direction, their energies can be modulated most significantly by poloidal ULF waves with azimuthal electric field oscillations. The so-called drift resonance happens at a certain energy (the resonant energy) with which particles drift at the same azimuthal speed as the waves. These resonant particles can thus experience a steady electric field, which leads to a net energy excursion. In the following, we briefly review the mathematical framework of the drift resonant process, which has been comprehensively developed in Southwood and Kivelson (1981).

For particles of charge q experiencing drift motion in a ULF wave, the kinetic energy W changes at the following rate:
$$\frac{{{\text{d}}W_{A} }}{{{\text{d}}t}} = qE \cdot v_{d} ,$$
where subscript A signifies an average over many gyration periods, E is the wave associated electric field, and v d is the magnetic gradient and curvature drift velocity (Northrop 1963). For an equatorially mirroring particle (vanishing curvature drift) in Earth’s magnetic dipole field, the drift velocity v d can be approximated in the nonrelativistic limit by
$$v_{d} = - \frac{{3L^{2} W}}{{qB_{E} R_{E} }}\hat{e}_{\phi } ,$$
where \(\hat{e}_{\phi }\) is defined eastward, R E is Earth’s radius, L is the L-shell parameter (radial distance in R E of the equatorial crossing point a field line), and B E is the equatorial magnetic field on Earth’s surface. In Southwood and Kivelson (1981), ULF waves propagate in the azimuthal direction with the wave-associated electric field given by
$$E = E_{\phi } \exp i(m\phi - \omega t)\hat{e}_{\phi } ,$$
where ϕ is the magnetic longitude (increasing eastward), m is the azimuthal wave number, and ω is the wave angular frequency. From Eqs. (12)–(14) the average rate of change of the particle energy is thus given by the following equation:
$$\frac{{{\text{d}}W_{A} }}{{{\text{d}}t}} = - \frac{{3L^{2} W}}{{B_{E} R_{E} }} \cdot E_{\phi } \exp i(m\phi - \omega t),$$
which should be integrated along the particle drift orbit to t = −∞ to achieve a energy gain δWA from waves. Here, the particle’s drift orbit is assumed to be unperturbed (despite the energy change of the particle from the waves) with the angular drift frequency ωd given by.
$$\omega_{d} = \frac{{{\text{d}}\phi }}{{{\text{d}}t}} = - 3LW/qB_{E} R_{E}^{2} .$$
An integration of Eq. (15) backward in time along the particle’s drift orbit (16) leaves a result that depends on the initial condition assuming the amplitude of the sinusoidal waves remains constant. To circumvent this problem, Southwood and Kivelson (1981) assumed the wave angular frequency ω is complex with a small, positive imaginary part Im(ω) that represents a gradually growing wave signal. This assumption enables the particle to see a sinusoid for a finite interaction time, which on integration of Eq. (15) yields an averaged particle energy gain
$$\delta W_{A} = - i \cdot \frac{{3L^{2} W}}{{B_{E} R_{E} }}\frac{{E_{\phi } \exp i(m\phi - \omega t)}}{{\omega - m\omega_{d} }}.$$

As defined in Southwood and Kivelson (1981), drift resonance happens when the particle drifts at the same azimuthal speed as the wave phase velocity, which indicates that d equals Re(ω) so that the denominator of Eq. (17) becomes Im(ω) × i. This small imaginary term suggests that for resonant particles, δWA must oscillate at a large amplitude in antiphase with the wave electric field. For particles with lower or higher energies (smaller or larger ωd) the denominator is dominated by its real part and the corresponding δW A oscillations have much smaller amplitudes and are ± 90° out of phase with the wave electric field. In other words the amplitude δWA of energy gain must shift in phase by 180° as the resonance is crossed from low energy to high energy.

An actual particle detector cannot measure δWA directly, and, therefore, Southwood and Kivelson (1981) discussed the associated variations of particle fluxes and phase space densities (PSDs) that theoretical results can be compared directly with observational data. By assuming a negligible gradient of preexisting particle PSDs in the azimuthal direction, the wave-produced PSD variations, δfA, can be written as
$$\delta f_{A} = \delta W_{A} \left[ {\frac{L}{3W}\frac{\partial f(W,L)}{\partial L} - \frac{\partial f(W,L)}{\partial W}} \right],$$
which shows that δf A is proportional to δW A provided there is a pre-existing finite PSD gradient in energy and/or space. Here, the importance of spatial gradients in producing PSD oscillations is emphasized, which is caused by back-and-forth convection of particles as a response to the wave electric field. The PSD variations can be alternatively written as
$$\delta f_{A} = - \delta W_{A} \frac{\partial f(W,\mu )}{\partial W} = \delta W_{A} \frac{L}{3W}\frac{\partial f(L,\mu )}{\partial L},$$
where μ is the particle’s magnetic moment, assumed constant for adiabatic behavior of particles interacting with ULF waves. The linear dependence of δf A on δW A suggests that the phase shift of particle PSDs across the resonant energy should also be 180°. Such a phase shift is thus treated as a characteristic signature of ULF wave-particle drift resonance (Claudepierre et al. 2013; Dai et al. 2013; Mann et al. 2013).

5.2 Theory of ULF wave toroidal mode drift resonance

The drift-resonance theory developed by Southwood and Kivelson (1981) concerns only particle interactions with poloidal mode ULF waves because in a dipole magnetic field toroidal waves do not cause net radial transport over a drift orbit. However, as solar wind dynamic pressure can significantly compress the dayside magnetosphere an equatorially mirroring particle drifting along contours of constant magnetic field strength will experience a variation of drift velocity along a drift orbit (see Fig. 33) that causes net acceleration or deceleration by the toroidal wave radial electric field (Elkington et al. 1999, 2003). Elkington et al. (1999) proposed that drift-resonance is possible and derived a mathematical framework for it (Elkington et al. 2003) that is very similar to the Southwood and Kivelson (1981) theory. In Elkington et al. (1999), the equatorial magnetic field strength is assumed to be
$$B(L,\phi ) = \frac{{B_{0} }}{{L^{3} }} + b_{1} (1 + b_{2} \cos \phi ),$$
where constants b 1 and b 2 are parameters that can be determined according to measurements and the second term on the right-hand-side represents the ϕ-dependent magnetic field resulting from solar wind compression. The radial component of the drift velocity for an equatorially mirroring particle becomes
$$v_{r} = \frac{{b_{1} b_{2} \sin \phi W}}{{qB^{2} LR_{E} }},$$
which is no longer zero due to the solar wind distortion of the dipole field. For toroidal mode wave–particle interactions, the average rate of change of particle energy defined by Eq. (15) is replaced by an equation that involves the radial component of the electric field
$$\frac{{{\text{d}}W_{A} }}{{{\text{d}}t}} = \frac{{b_{1} b_{2} \sin \phi W}}{{B^{2} LR_{E} }} \cdot E_{r} \exp i(m\phi - \omega t),$$
which represents the average rate of change of the particle energy from toroidal ULF waves.
A similar procedure to that used in Southwood and Kivelson (1981) can be adopted to calculate the particle energy gain from waves and the drift-resonance condition for toroidal wave–particle interactions
$${\text{Re}}(\omega ) = ({\text{m}} \pm 1)\omega_{\text{d}} ,$$
which differs from the poloidal wave-particle drift resonant condition (Re(ω) = m ω d ) in the extra ± ω d term. This term originates from sin ϕ in Eq. (22), or, in other words, from the noon-midnight asymmetry of the equatorial magnetic field strength. Therefore, there are two resonant energies for toroidal ULF wave–particle interactions. As the azimuthal wavenumber m for toroidal waves is usually smaller than that for poloidal waves; this indicates larger resonant energies. According to Zong et al. (2009a, b), the different drift-resonance conditions for toroidal and poloidal mode wave–particle interactions expand the energy range of particle acceleration in the inner magnetosphere and thus play an important role in the radiation belt formation.

5.3 Observational signatures of ULF wave drift-resonance

A characteristic signature of ULF wave drift-resonance is the 180° phase shift of particle PSDs (or fluxes) for multiple energy channels across the resonant energy. This signature has been widely used as a diagnostic to identify drift-resonant wave–particle interactions (Mann et al. 2013, Claudepierre et al. 2013; Dai et al. 2013; Hao et al. 2014). The diagnostic is derived from Eq. (19), which suggests a linear dependence of the change in PSD, δfA, on the energy change δWA, and from equation (17), which suggests there is a 180°-phase shift of δWA across the resonant energy. These two equations can lead to different outcomes under certain circumstances and thus result in more complicated observational signatures.

According to Eq. (8), δf A is indeed proportional to δW A although their ratio (i.e., the pre-existing PSD gradient) depends on μ (or equivalently, W) at any given location. Near geosynchronous orbit, the PSD radial gradient for equatorially-mirroring electrons is statistically shown to be positive (outward) at lower μ values and positive (earthward) at higher μ values (Turner et al. 2012). However, the PSD gradient can be highly dynamic during active geomagnetic conditions (Chen et al. 2007; Turner et al. 2010). Therefore, the phase difference between δf A and δW A could be either 0 or 180° depending on the sign of the PSD gradient at the corresponding energy. In other words, even if δWA exhibits a 180° phase shift across the resonant energy, the phase shift of δfA could differ from 180° if the PSD gradient changes sign within the energy range of interest. An observation supporting this hypothesis was made by the Van Allen Probes on 29 October 2013, in which a 360° phase shift in energetic electron PSD modulations is present in four consecutive energy channels (see Fig. 46). According to Chen et al. (2016), the phase difference of 360° is comprised of a 180° phase shift across the resonant energy and an additional 180° phase shift associated with a reversal of the electron PSD gradient at higher energy. In this event, the PSD gradient reversal is caused by the drift motion of substorm injected electrons (Chen et al. 2016). This is an excellent example showing that particle modulation signatures may contain important information not only on ULF wave–particle interactions but also on pre-existing particle distributions in the magnetosphere.
Fig. 46

Van Allen Probe A observations of energetic electron modulations on 29 Oct 2013, after Chen et al. (2016). a, b Electron residual fluxes at four consecutive energy channels; cf pitch angle spectrograms of the electron fluxes for each energy channel. The phase shift in electron residual fluxes from the lowest to the highest energy channels adds up to ∼360°

Note that equation (17) does not necessarily correspond to a 180°-phase shift of δW A across the resonant energy, especially during wave excitation and growth stages when the wave angular frequency ω has a large imaginary part (Zhou et al. 2015). A large Im(ω), if comparable to Re(ω), suggests a large imaginary part of the denominator in equation (17), which significantly affects the amplitude and phase of particle PSD oscillations at different energies. This effect is best illustrated in the complex plane, as shown in Fig. 47, in which the two panels have different Im(ω)/Re(ω) values. The blue arrows, with colors varying from dark to light blue, represent the denominator of equation (17) with increasing energies. They all end on the dashed line y = Im(ω), with a right-side cutoff at x = Re(ω) representing the zero-energy limit. The reciprocals of the blue arrows, after multiplication by −i, are proportional to the complex division of δW A over E ϕ as represented by red arrows. The red arrows end on the dashed circle tangent to the imaginary axis at the origin, with a diameter of 1/Im(ω) and a zero-energy cutoff located at (−Im(ω)/|ω|, −Re(ω)/|ω|) in the complex plane.
Fig. 47

Amplitude and phase relationships between ΔW A and E ϕ at different energies in the complex plane, for a quasi-steady (Im(ω Re(ω)) and b rapidly growing (Im(ω Re(ω)) waves

According to equation (17), arguments of the red arrows in Fig. 47 represent phase differences between δW A and E ϕ at any given spacecraft location. Therefore, we have,
$$\Delta \varPhi = \arg (\delta W_{A} /E_{\phi } ) = 180^{ \circ } + \arctan \frac{{{\text{Re}}(\omega ){\text{ - m}}\omega_{\text{d}} }}{{{\text{Im}}(\omega )}},$$
where ΔΦ is the phase difference between δW A and E ϕ . It can be seen that ΔΦ is 180° when Re(ω) equals mωd, which indicates an antiphase relationship between δW A and E ϕ at the resonant energy. At other energies, since ωd is proportional to energy (Schulz and Lanzerotti 1974), ΔΦ varies from 180° + arctan Re(ω)/Im(ω) (at zero energy) to 90° (at infinite energy). In other words, the total phase shift of δW A across energies from zero to infinity equals 180° − arctan Im(ω)/Re(ω)

Figure 47a shows a case with a small Im(ω), which suggests a large diameter for the dashed circle with the most intense red arrows (including the zero-energy cutoff) located near the imaginary axis. The only exception appears when the particle energy is very close to the resonant energy so the denominator of equation (17) is dominated by its imaginary part, and the corresponding red arrow approaches the real axis at a large value, x = −1/Im(ω). This picture, consistent with descriptions in Southwood and Kivelson (1981), suggests that particle PSDs at increasing energies should have a sharp, approximately 180° phase shift near the resonant energy. However, this does not apply for large Im(ω), as is appropriate for the wave growth stage, see Fig. 47b for the case with Im(ω) comparable to Re(ω). The zero-energy cutoff of red arrows, which is now far from the imaginary axis, suggests that the total phase shift of particle PSDs across energies is much less than 180°, as predicted by Eq. (24). From the red arrows in Fig. 47b, one can also find that the phase changes more gradually with increasing energies than in Fig. 47a, and that the peak amplitude of δW A /E ϕ at the resonant energy becomes weaker than its counterpart in Fig. 47a and closer to the amplitudes at non-resonant energies.

The newly predicted particle signatures for growing ULF waves are also found in Van Allen Probes observations. It is shown in Zhou et al. (2015) that the phase shift of particle PSDs across the resonant energy is only slightly above 90°, rather than 180° as might be expected during the early growth stage of an ULF wave event. Moreover, a best-fit procedure can be carried out based on Eq. (24) to determine the ULF wave characteristics (such as frequency, growth rate, and azimuthal wave number) from phase information of particle PSD oscillations at multiple energy channels (Zhou et al. 2015). In other words, the application of the Southwood and Kivelson (1981) framework provides not only an understanding of energy coupling in the inner magnetosphere, but also a new diagnostic for the evolution of magnetospheric ULF pulsations.

5.4 Generalized theory of drift-resonance for ULF waves with finite amplitude growth and decay times

There is an implicit assumption in the derivation of the Southwood and Kivelson (1981) theory: the real and imaginary frequencies of the ULF waves must remain constant. In reality, a large imaginary frequency during the wave growth stage should eventually decrease to negative values when the wave amplitude starts to decay. Therefore, it is important to revise the current framework by introducing a time-dependent imaginary part of the wave angular frequency to accommodate the growth and damping of the ULF waves. In Zhou et al. (2016), the imaginary frequency is assumed to be linearly decreasing with time:
$${\text{Im}}(\omega ) = - t/\tau^{ 2} ,$$
where τ > 0 represents the time scale of wave growth and decay. In this case, the wave electric field, Eq. (14), can be rewritten as
$$E = E_{\phi } \exp ( - t^{2} /\tau^{2} )\exp i(m\phi - \omega_{r} t)\hat{e}_{\phi } ,$$
where ω r is the real part of the wave angular frequency, Re(ω). This equation describes a Gaussian envelope of the electric field oscillation amplitude; the wave amplitude grows until t = 0 after which it starts to decay. Then, the same procedure used in Southwood and Kivelson (1981) to derive the particle’s average rate of energy gain from waves is applied by integrating the particle’s drift orbit backward in time to obtain its energy change
$$\delta W_{A} = - \frac{\sqrt \pi }{2}\frac{{3L^{2} W}}{{B_{E} R_{E} }} \cdot E_{\phi } k(\tau )g(t,\tau )\exp i(m\phi - m\omega_{d} t),$$
where \(k\left( \tau \right)\) is defined by
$$k(\tau ) = \tau \exp \left[\frac{{ - (m\omega_{d} - \omega_{r} )^{2} \tau^{2} }}{4}\right],$$
$${\text{and}}\,\,\,\,g(t,\tau ) = {\text{erf}}\left(\frac{t}{\tau } - i\frac{{m\omega_{d} \tau - \omega_{r} \tau }}{2}\right) + 1$$

The energy gain equation appears much more complicated than equation (17); however, drift resonance still occurs when d  = ω r . At the resonant energy, k(τ) reaches its maximum, τ, and the complex g(t, τ) function degenerates to a real function, erf(t/τ) + 1. Therefore, the δW A oscillations near the resonant energy have the largest amplitude and are in antiphase with the electric field as predicted by Southwood and Kivelson (1981).

The inclusion of wave growth and damping in the drift resonance theory also predicts signatures different from the Southwood and Kivelson (1981) framework. Figure 48 shows an example of energetic electron interactions with ULF waves during the wave growth and damping stages. The wave electric field (26) at a fixed ϕ location is shown in Fig. 48a, and the corresponding δWA values at the same location are shown in Fig. 48b as a function of time and electron energy. The manifestation of δW A oscillations observable in a particle detector with finite energy and time resolutions, in the format of electron residual PSD δfA oscillations, is computed according to Eq. (19) and shown in Fig. 48c. In this case, the resonant energy is 250 keV as represented by the dashed lines in Fig. 48b and c.
Fig. 48

Predicted and observed electron responses to ULF waves during growth and damping stages, (cf. Zhou et al. 2016). a Azimuthal electric field; b predicted electron energy gain as a function of time and energy with the dashed line denoting the resonance energy ~ 250 keV; c predicted spectrum of electron residual PSDs detected by a MagEIS-like particle detector with finite time and energy resolution; d Van Allen Probe-B observations on 11th April 2014; e observed electron residual PSD modulations

Figure 48 shows that in the wave growth stage, the amplitudes of δW A and δf A oscillations increase gradually and the phase difference across the resonant energy increases from very small values to 180° when the wave stops growing. After that, despite the decreasing wave amplitude, both the δWA oscillation amplitude and the total phase shift continue to increase. The electron PSD oscillation amplitude also increases, until the phase mixing effect (due to the limited energy resolution of particle detectors) attenuates the PSD oscillations (Degeling et al. 2008). The predicted signatures in Fig. 48c, manifested as increasingly inclined stripes in the electron energy spectrum, are consistent with observations from Van Allen Probes (see Fig. 48d and e). The consistency between theory and observations validates the generalized theory of ULF wave–particle resonant interactions and provides new insights into our understanding of particle dynamics within the entire ULF wave lifespan (Zhou et al. 2016).

5.5 Drift resonance for spatially localized ULF waves

The theory of wave-particle drift resonance can be further generalized by introducing a spatially localized nature of the ULF waves. In Eqs. (15) and (27), the amplitude of the wave electric field are both assumed to be independent of location, which is not typical in the inner magnetosphere as ULF waves are usually confined in a limited range of longitudes (Liu et al. 2009; Claudepierre et al. 2013; Liu et al. 2016a, b). The localized nature of ULF waves can be taken into account by assuming a von Mises distribution of the wave amplitude in the theoretical framework (Li et al. 2017b). In this case, the wave electric field (27) is replaced by
$$E\left( {t,\phi } \right) = \frac{{E_{\phi 0} \exp \left( { - t^{2} /\tau^{2} } \right)}}{{2\pi I_{0} \left( \xi \right)}}{ \exp }\left[ {\xi \cos \left( {\phi - \phi_{0} } \right)} \right] \cdot {\text{exp }}i\left( {m\phi - \omega t} \right)\hat{e}_{\phi } ,$$
where ϕ 0 is the magnetic longitude with maximum ULF wave amplitude, ζ is the concentration parameter that represents the characteristic angular width of the von Mises function, and I0(ζ) is a normalization factor defined by the zeroth-order modified Bessel function of the first kind. In this case, the particle’s energy gain from the waves can again be derived from an integration along the particle’s drift orbit, although the integration can only be done numerically due to the introduction of the von Mises function. An example of the integration results is presented in Fig. 49, which predicts the electron responses to longitudinally localized ULF waves during the wave growth and damping stages.
Fig. 49

Electron responses to longitudinally localized ULF waves during the growth and damping stages. a The modeled wave-associated electric field as a function of time and magnetic longitude, with the dashed lines delineating the characteristic angular width of the waves. b The normalized energy gain of particles from the waves observed by a virtual satellite located at the region of strong wave activity. c Same as panel b, except for a different spacecraft location outside the region of strong wave activity

Figure 49a presents the ULF wave electric field as a function of time and magnetic longitude, from which one can clearly see the eastward propagation and the amplitude variation of the longitudinally localized ULF waves. Two different locations (ϕ = 0 and ϕ = π) are then selected to place two virtual satellites, which are located within and outside the region of strong wave activity, respectively. The normalized energy gain δWA of particles at these locations are shown in Fig. 49b and c as functions of time and energy. Figure 49b suggests that a satellite within the region of strong wave activity would observe signatures very similar to those in Fig. 48, namely the presence of increasingly tilted stripes in the energy spectrum. The only difference is that after t = 400 s, a series of newly appeared, highly tilted stripes are superimposed over the preexisting stripes at higher energies. These are particles that have been previously modulated by the localized ULF waves; these particles then drift around the Earth to reach the wave active region again, which enables the presence of a strongly distorted pattern as drift echoes of the earlier stripes. The energy spectrum of δWA outside the wave active region, shown in Fig. 49c, differs from Fig. 49b in that the stripes are much more tilted so the phase shift across energies significantly exceeds 180° even for the very first stripes. These signatures can again be understood as a distortion of the original spectrum (like in Fig. 48) by the energy-dependent drift motion of electrons.

Such kinds of electron signatures have been recently observed (Li et al. 2017a) in energetic particle data newly available from the BD-IGSO spacecraft. Figure 50a shows the BD-IGSO observations of the electron fluxes at different energy channels, with the corresponding energy spectrum given in Fig. 50b. One can clearly see from Fig. 50b the presence of increasingly tilted stripes, with significant (larger than 180°) phase shift across energies even at the very early stage of the electron flux modulations. These signatures are attributed to the localized nature of ULF waves, which enables Li et al. (2017b) to carry out a best-fit procedure and extract the wave characteristics from the particle data. The resultant best-fit spectrum is shown in Fig. 50c, which clearly provides an excellent match with the observations in Fig. 50b. Figure 50d provides the best-fit results of the wave electric field as a function of time and magnetic local time, with the BD-IGSO spacecraft location indicated by the green line near the eastern edge of the wave active region. In this scenario, it is indeed the shifted location of the spacecraft from the region of strong wave activity that enables the energy-dependent electron drift motion to distort the conventional picture (as in Fig. 49b) with an enlarged phase shift during the early stage of the ULF wave evolution. For this specific event, the localized nature of the ULF waves are also confirmed by magnetic field data from a series of ground-based stations (see Li et al. 2017b), which validates the scenario discussed above and highlights the importance of localized ULF wave–particle interactions in particle dynamics of the inner magnetosphere.
Fig. 50

BD-IGSO observations on 11th November 2015 and the corresponding best-fit results. a IGSO observations of electron fluxes at different energy channels. b Energy spectrum of the electron residual fluxes. c Best-fit results of the residual fluxes based on the scenario of a localized wave-particle drift resonance. d Best-fit resultant of ULF wave electric field as functions of time and magnetic local time

Further evidence of localized ULF wave-particle drift resonance, given in Hao et al. (2017), focuses more on the pitch-angle distribution of energetic electrons within and outside the region of strong wave activity. Provided that the bounce-averaged drift speed of electrons depends not only on their energies but also on their pitch angles, the electron pitch-angle spectrum within the wave active region can be also distorted after the electrons drift outside the region. For electrons outside the wave active region, the larger drift velocities of 90°-pitch angle electrons can result in a phase leading than electrons with other pitch angles. Figure 51 shows the predicted energy-spectrum and pitch-angle spectrum of electrons within and outside the wave active region. The predicted energy spectrums clearly exhibit the characteristic increasingly tilted stripes as in Fig. 49, and the phase shift across energies for the very first stripe is larger outside the region of strong wave activity (compare Fig. 51b and f). More interestingly, the predicted pitch-angle spectrum of electrons show “boomerang”-shaped structures with the peak (valley) of the flux modulation arriving earlier at pitch angles closer to 90°. Drift echoes of these “boomerang”-shaped stripes are also clearly present in Fig. 51.
Fig. 51

Calculation on electron response to a localized poloidal ULF wave. a equatorial Eφ of ULF wave as a function of time and MLT. Time profile of Eφ at MLT = 1200 is overplotted with black line. Imaginary probe is set to be at MLT = 1630, as marked with blue horizontal line. b 90° pitch angle electron energy gain from ULF wave as function of time and energy. c energy gain of 349.8 keV electron as a function of pitch angle and time. d the same as c but for 466.8 keV electrons. eh the same as ad but with imaginary probe located at MLT = 0630

Hao et al. (2017) also use Van Allen Probes observations to show that the “boomerang”-shaped stripes are indeed present in the inner magnetosphere after an interplanetary shock arrival. The observed signatures, shown in Fig. 52, are highly consistent with the modeled features in Fig. 51, which provides smoking-gun evidence that longitudinally localized ULF waves are excited by interplanetary shocks.
Fig. 52

Residual flux profile in pitch-angle-versus-time plot for 599.6, 466.8, and 349.8 keV electrons from MagEIS, Van Allen Probe-B observations on 7th June, 2014

Finally, there remain several assumptions not necessarily valid in the generalized drift resonant theory. A most obvious assumption is that the theory deals with non-relativistic particles, which overestimates the particle drift velocity and, therefore, leads to an overestimation of δWA and underestimation of the resonant energy. Another important assumption in the Southwood and Kivelson (1981) framework is that the particles drift orbit remains unperturbed despite its energy change by waves. This assumption is appropriate only if the particle energy gain δW A is much smaller than its initial energy so that nonlinear effects are insignificant. Full orbit calculations relaxing this assumption are discussed in the next section.

6 Test-particle simulations of resonant wave–particle interactions

The generalization of the Southwood–Kivelson theory (1981, 1982) of wave–particle interactions presented in Sect. 5 successfully reproduces energetic electron flux modulations observed by the Van Allen Probes-B satellite on April 11, 2014. An important aspect of the theory (Zhou et al. 2016) is that it incorporates finite growth and decay of ULF waves, a feature that is needed for quantitative analysis of satellite observations. As in Sect. 3.2, the theory uses a simple model of the equatorial electric field, Eq. 15, and assumes that charged particles move along unperturbed trajectories with no change in L. In this section, full Lorentz-force test particle simulations without these limitations are used to investigate wave–particle interactions in self-consistent electric and magnetic fields. The new results that are presented illustrate how improved theory (Sect. 5) can be used in combination with numerical techniques to better characterize resonant wave–particle interactions.

6.1 Eigenmode solutions to the ideal MHD wave equations

Self-consistent electric and magnetic fields of standing mode ULF waves can be calculated using the approach described in Rankin et al. (1999, 2005). A monochromatic driver with harmonic dependence \(\exp \left[ {i\left( {\omega t - m\phi } \right)} \right]\) is added to the wave equations to excite poloidal mode ULF waves with a radial (azimuthal) magnetic (electric) field in the equatorial magnetosphere. The amplitude of the driver can be made time-dependent, but when constant leads to linear growth of the wave amplitude with time. As geomagnetic field lines evolve independently in ideal MHD, \(\omega\) is explicitly a function of L. This enables growth, damping, and phase mixing to be accounted for in the model under the assumption of infinite ionospheric conductivity. The absence of finite height-integrated Pedersen conductivity, and hence ionospheric damping, means the wave amplitude is a free parameter. The mode structure along field lines will change under finite conductivity, but as the differences are expected to be minor for conductivities on the order of a few to several mho, effects of finite conductivity are neglected to simplify the analysis.

Figure 53 shows eigenmode solutions to the MHD wave equations for a fundamental mode poloidal wave excited on a L = 4.5 dipole field line. The left and right columns display the in-phase and out-of-phase wave electric and magnetic field components, respectively. The azimuthal electric field has an amplitude of 2 m Vm−1 at the equator and a maximum of 4 m Vm−1 off the equator. The compressional component of the wave magnetic field in the top-left panel of the figure is less than 10nT within a latitude range of −20° and +20° about the equator. In the full Lorentz force test-particle simulations discussed below this component is included to avoid violation of the first adiabatic invariant \(\mu\). It is worth noting that a guiding-center formalism conserves \(\mu\) regardless of whether the compressional component is included, although this excludes potentially important physics.
Fig. 53

Wave fields for a 20.94 mHz poloidal mode excited at L = 4.5. The compressional magnetic field (top left panel) and azimuthal electric field (bottom left panel) are in phase. The radial magnetic field (top right panel) and azimuthal electric field (bottom right panel) are in anti-phase. The background density is specified to have an \(r^{ - 6}\) dependence along geomagnetic field lines

It is worth comparing the eigenmode solutions shown in Fig. 53 with the results of the data analysis in Sect. 3.2, which consider N = 2 drift-bounce ULF wave–particle interactions involving the fundamental mode (cf. Figure 28a with the bottom-left panel of Fig. 53). The profile of the poloidal mode along field lines is affected by the mass loading, the L-shell of the field line, finite conductivity effects, and possible distortion from a dipole field. The results presented below are valid at relatively low L-shells in the inner magnetosphere where a dipole field model is justified.

6.2 Full Lorentz force calculations of drift-resonant H+ ions

For the test-particle simulations a phase-space grid in energy and pitch angle is defined under the assumption that there is no gyro-phase dependence of the distribution function. To compute the distribution function at a given location, phase space points are traced backward in time onto the magnetic equator prior to ULF waves being present. Liouville’s theorem is then used to map the initial distribution (assumed Maxwellian) forward to the observation point at time t.

Figure 54 shows results for an input wave (top panel) similar to that in Fig. 48a. The left-side bottom two panels in the figure show changes in the energy and particle distribution of equatorially mirroring H+ ions interacting with a m = 8, 20.94 mHz frequency poloidal mode wave excited at L = 4.5. The right column of Fig. 54 shows various quantities computed along the trajectory of a drift-resonant ion of energy 244 keV. The polarization drift velocity shown in the bottom-right panel is dominated by spatial gradients in the wave electric field. The magnetic moment is normalized by its value at t = 0 to highlight possible regions of non-adiabatic behavior. In this case, the particle trajectory is not strongly perturbed (top-right panel) by the wave, which has as a maximum electric field of 2 m Vm−1 at the equator. There is an obvious similarity between Fig. 54 and the unperturbed electron trajectory calculation of Zhou et al. (2016) shown in Fig. 48a.
Fig. 54

H+ ions interacting with a m = 8 poloidal wave of frequency 20.94 mHz. The left panels show a the wave profile; b the change in particle energy; c the relative change in the distribution function. The right panels show L, W, \(\mu\), \(E_{\phi }\), and the radial components of the \(\varvec{E} \times \varvec{B}\) (black line) and polarization drift (blue line) velocities of a drift-resonant ion of initial energy 244 keV placed initially at L = 4.5. Wave parameters are the same as in Fig. 52

The pitch angle dependence of drift resonance is illustrated in Fig. 55, with panel (a) in the left column showing the time profile of the wave and panels (b) and (c) showing changes in particle energy for pitch angles of 90° and 35°, respectively. The drift-resonance mechanism is more efficient at 35° pitch angle because ions experience a larger electric field off the equator (see Fig. 53) and hence a larger net acceleration along their bounce trajectory. The differential particle flux in the right column of Fig. 55 is binned in energy similar to a MagEIS-like instrument on the Van Allen Probes. The modulations of the differential flux in Fig. 55 have a maximum amplitude in the ~ 240 keV energy channel, which is consistent with the drift-resonance condition N = 0. The differential flux also exhibits a 180° phase change with energy across the resonance, as discussed in Sect. 5.
Fig. 55

Test particle simulation of H+ ions for parameters defined in the caption of Fig. 48. Panel a in the left column shows the time profile of the wave. The energy changes of particles with pitch angles of 90° and 35° are shown in panels b and c, respectively. Panels b and c in the right column show corresponding modulations in differential particle flux for the two values of pitch angle

It is informative to consider a wave that acts over a longer timescale than in Figs. 54 and 55. This is illustrated in Fig. 56 for a 90° pitch angle H+ ion of drift-resonant energy 244 keV. The trajectory is overlaid with snapshots of the electric field as a function of L at the equator. Parameters are the same as in Figs. 54 and 55 except that the wave reaches a maximum amplitude of 2 m Vm−1 at the equator after 800 s and is held constant for 3400 s before decaying over a time interval of 800 s. The left and right panels of Fig. 56 show different parts of the orbit of the drift-resonant ion up to 4200 s and 5000 s, respectively. After the wave reaches constant amplitude resonant ions drift azimuthally with constant energy and L (right panel). An interesting aspect of Fig. 56 is that resonant ions are trapped in an effective potential well in the wave frame and are continually accelerated and decelerated as they move periodically back and forth across L along their drift path. This effect is not captured by the methodology used in Sect. 5.
Fig. 56

Test particle simulation of drift-resonant H+ ions of energy 244 keV and 90° pitch angle interacting with a m = 8 poloidal mode ULF wave at L = 4.5. In the left panel the electric field grows linearly with time for 800 s and is then held constant for 3400 s. The right panel shows the remaining part of the trajectory up to t = 5000 s after the wave has decayed. Other parameters are defined in Fig. 54

Figure 57 shows the time dependence of L, W, \(\mu\), \(E_{\phi }\), and the \(\varvec{E} \times \varvec{B}\) and polarization drift velocities along the ion trajectory marked in Fig. 56. For the wave parameters used in this example, the first adiabatic invariant normalized by its value at t = 0 is subject to small variations during successive aperiodic excursions between L ~ 4 and 5. The bottom panel of Fig. 57 shows that the magnitude of the \(\varvec{E} \times \varvec{B}\) drift velocity is much larger than the polarization drift velocity (blue line), which implies the former is more important in causing radial motion.
Fig. 57

Equatorially mirroring H+ ions of resonant energy 244 keV interacting with an initially growing and then decaying poloidal mode wave with azimuthal wavenumber m = 8 and frequency 20.94 mHz. The quantities L, W, \(\mu\), \(E_{\phi }\), and the radial components of the \(\varvec{E} \times \varvec{B}\) (black line) and polarization drift (blue line) velocities shown in each panel are computed along the ion trajectory. Other parameters are the same as in Fig. 56

6.3 ULF wave–particle interactions for different ion species

As the gyro radius of O+ ions is much larger than for H+ ions of the same energy, it is informative to consider the mass dependence of drift resonance. Figure 58 shows the trajectory of a drift-resonant O+ ion overlaid with the azimuthal electric field of a poloidal mode ULF wave of the same form used in Fig. 56. Comparison of Fig. 58 with Fig. 56 shows that the dynamics of O+ and H+ is similar even though the gyro radius of the former is larger by a factor of 16. A different set of wave parameters is considered below.
Fig. 58

Test particle simulation of a resonant O+ ion of energy 244 keV interacting with a poloidal mode ULF wave with azimuthal wavenumber m = 8 and frequency 20.94 mHz. Wave parameters are the same as those used in Fig. 56

An obvious question is what ULF wave parameters lead to violation of the 1st adiabatic invariant? To answer this question a wave carrying a larger electric field is considered. The amplitude of the wave at L = 5.7 increases linearly with time to 3.23 m Vm−1 at the equator (peaking at 6.97 m Vm−1 off the equator) and is then held constant. The background magnetic field on the resonant field line is smaller than in Fig. 58, which implies the wave acts as a larger perturbation on the background state than in the previous cases considered. The eigenfrequency is 10 mHz and the azimuthal wavenumber is increased to m = 35 to make the spatial scale more comparable to the ion gyro-radius. Figure 59 shows trajectories of equatorially mirroring H+ and O+ ions of initial energy 130 keV as they interact with the wave.
Fig. 59

Equatorially mirroring ions of energy ~ 130 keV interacting with a poloidal mode ULF wave with a maximum amplitude of 3.23 mVm−1 at the equator. The trajectories of H+ and O+ ions are overlaid with the azimuthal electric field at t = 2000 s in panels (a) and (b), respectively. Panels (c) and (d) display corresponding values of L, W, \(\mu\), \(E_{\phi }\), and the radial components of the \(\varvec{E} \times \varvec{B}\) (black line) and polarization drift (blue line) velocities along particle trajectories ending at t = 10,000 s

Panel (a) in Fig. 59 and the variations in L(t) and W(t) evident in the first two frames of panel (c) show that H+ ions are trapped (radially confined) by the wave similar to Fig. 56. The amplitude of the modulations in L and W gradually increases with time as does the period of the modulations. At around t = 8000 s ions experience a small electric field [fourth frame of panel (c)] and drift at nearly constant L and W until they are trapped once again by the wave. The third frame in panel (c) shows that variations in the 1st adiabatic invariant are less than ten percent over the 10,000 s timescale of the simulation. The last frame in panel (c) again shows that the \(\varvec{E} \times \varvec{B}\) drift velocity is much larger than the polarization drift velocity. Panels (b) and (d) of Fig. 59 show corresponding behavior for O+ ions. The trapping mechanism is evidently less effective for higher mass ions for the wave parameters used. There is, however, a substantial variation in the 1st adiabatic invariant visible in the third panel of Fig. 59d, which points to non-adiabatic behavior of O+. Reasons for this are left for future study but may be related to the ion gyroradius becoming comparable to the length scale over which wave perturbations vary. The polarization drift velocity may also affect μ-conservation as it is significantly larger for O+ than for H+. Rapid time variations of the electric field experienced by O+ over a gyro-orbit might also be a factor. In Fig. 59c, abrupt changes in the direction of H+ particle motion at the inner and outer edges of the potential well are correlated with jumps in the magnitude of the 1st adiabatic invariant. Note that for O+ ions, Fig. 59d includes significant variations in L and W due to the large cyclotron radius in addition to wave electric and magnetic fields.

The methodology and results presented in Sect. 6 demonstrate the utility of using simple ULF and test-particle models to investigate resonant wave–particle interactions. In future studies, the same models will be applied to the study of localized waves by superposing solutions to the wave equation with different amplitudes, phases, and azimuthal wave numbers. The effects of gradients in phase space density will also be investigated in order to better understand the complex behavior of resonant processes affecting ions and electrons in Earth’s inner magnetosphere.

7 Concluding remarks and outstanding questions

Improved characterization of ultra-low frequency (ULF) waves will advance understanding of wave–particle interactions that are a naturally occurring phenomenon in the plasma universe and uniquely accessible in Earth’s magnetosphere. The characterization of resonant wave–particle interactions involving energy transfer between ULF waves and magnetospheric particles will help explain particle acceleration associated with interplanetary shocks that disturb the near-Earth space environment. Poloidal mode ULF waves are produced by interplanetary shocks (positive impulses) that compress the magnetosphere, and solar wind negative and positive pressure pulses that inflate and compress the magnetosphere, respectively. Excited poloidal mode ULF waves are in general stronger around local noon than at dawn and dusk. They have a larger amplitude than toroidal waves generated during positive and negative impulses, although the wave amplitudes in negative pressure pulse events are lower than for positive impulse events.

The results presented in this paper show that drift-resonance of energetic electrons and drift-bounce resonance of thermal ion species can occur simultaneously in the case of fundamental-mode poloidal standing waves in the inner magnetosphere. Similarly, third-harmonic ULF waves can simultaneously interact with substorm-injected hot ions from the magnetotail and cold outflow ions from the ionosphere. This suggests that fundamental-mode poloidal ULF waves can be excited by energetic electrons via drift-resonance and that these waves can further interact with thermal ion species via drift-bounce-resonance. Conversely, fundamental-mode poloidal ULF waves excited through drift-bounce resonance with thermal ion species can interact with energetic electrons via drift-resonance. Scenarios for efficient particle acceleration by poloidal mode ULF waves exist for both electrons and ions. Electrons satisfying the N = 0 drift-resonance condition with eastward-propagating fundamental mode ULF waves experience continual acceleration as their gradient drift is in the direction of wave propagation. Ions satisfying the N = 1 drift-bounce resonance condition for second-harmonic westward-propagating standing waves also experience continual acceleration for the same reason.

Vast numbers of accelerated energetic particles are produced immediately after interplanetary shock arrival at the magnetosphere. The shock interaction proceeds in stages: initial particle acceleration occurs due to strong shock-related magnetic field compression. This is followed by excitation of poloidal and toroidal mode standing ULF waves during the shock passage. In the inner magnetosphere, these excited ULF waves accelerate electrons and ions via drift resonance and drift-bounce-resonance. However, the interaction with poloidal mode standing waves is much more efficient at accelerating electrons and ions in the radiation belt and in the ring current region, respectively. Drift-resonance acceleration of ions and electrons by poloidal and toroidal mode ULF waves proceeds differently depending on the L-shell location. The acceleration of energetic electrons by toroidal ULF waves is more important in the outer magnetosphere when the magnetosphere is strongly deformed. In general, poloidal mode ULF waves are much more important for the acceleration of energetic particles in the inner magnetosphere.

Non-resonant particles can be accelerated by ULF waves when fast damping or growth of waves occurs. Net acceleration takes place in this case because particles accelerated in the second half cycle of growing or damping waves do not lose all of the energy gained in the first half cycle. Damping rates of ULF waves are significantly faster in the plasmasphere boundary layer (PBL) to an extent that wave–particle interactions terminate. Fast damping in the PBL may be caused by a change of ion composition, e.g., an increasing ratio of heavy ions (O+) to lighter ions (H+). The damping rate due to the presence of heavy ions increases because of the mass dependence of the ULF wave frequency and wave propagation speed. During periods of geomagnetic activity, oxygen ions that are loaded onto magnetic field lines reduce the local Alfven eigenfrequency, resulting in the deeper penetration of Pc5 wave power onto lower L-shells than normal. These low-frequency Pc5 standing waves deep within the magnetosphere can be efficient at accelerating outer radiation belt electrons and ring current ions.

Although satellite observations from Cluster, THEMIS, and the Van Allen Probes have led to significant progress in our understanding of ULF wave–particle interactions in Earth’s magnetosphere, some outstanding questions remain unanswered:
  1. 1.

    Are high-m poloidal mode ULF waves generated principally by an exterior solar wind driver or do ring current ions predominantly excite these waves? How can these different generation mechanisms be distinguished and how what is their relationship to eastward- and westward-propagating poloidal ULF waves? In the case of ring current wave generation, are high-m poloidal mode ULF waves excited during quiet times without any ring current intensification?

  2. 2.

    How common are high-m poloidal mode waves at the plasmapause and are they a signature of the existence of the plasmapause? What is the role of the plasmaspheric ion constituency?

  3. 3.

    What is the significance of high-m poloidal mode ULF waves for the energization of storm-time ring current ions (see, e.g., Zong et al. 2012)? Is this a prerequisite for a super magnetospheric storm?

  4. 4.

    Do ULF waves mediate coupling between plasmasphere and ring current ion species and high-energy electrons in the radiation belts? If so, is energization of radiation belt electrons caused by ring current ion-excited second-harmonic poloidal ULF waves of moderate m-number (Ozeke and Mann 2008)? Alternatively, do fundamental mode poloidal ULF waves excited by eastward-drifting injected electrons (James et al. 2013) accelerate ring current ions?

  5. 5.

    In the inner magnetosphere, the ULF waves are usually localized both radially and azimuthally. The temporal evolution of the ULF waves also has finite duration with wave growth and damping stages. How do these wave characteristics affect wave-particle dynamics, and how are the conventional pictures of drift-resonance and drift-bounce-resonance affected? What are the corresponding observational signatures?

  6. 6.

    What is the relationship between MeV electron radial diffusion and broadband low-m and high-m Pc4-5 waves and compressional waves? Is the traditional Fokker–Planck treatment valid at large ULF wave amplitude, and is the wave–particle interaction process nonlinear?

  7. 7.

    What is the role of ULF waves in magnetosphere-ionosphere coupling? The energy dissipated into the ionosphere by ULF waves is estimated to be 1014–1015W, or 30% of the substorm budget. ULF wave-modulated ionospheric ion outflow ranging from to hundreds of eV has been observed by Yang et al. (2010a, b) and Ren et al. (2015). What are the magnetospheric consequences of these ionospheric outflow ions that have extremely high fluxes?

  8. 8.

    How common are high-m poloidal mode waves at the plasmapause and can they be used to identify the plasmapause? How are ULF waves affected by the plasmasphere ion constituency?


In conclusion, simultaneous wave and multiple ion species measurements from multiple spacecraft during conjunctions are the key to understanding and quantifying ULF wave interactions with ring current ion populations in the inner magnetosphere. It is also crucial to understand how ULF waves are affected by ion composition and mass density changes in the plasmasphere boundary layer and its possible link to ionospheric outflow. Questions concerning ULF waves, such as their generation, dissipation, interaction with energetic particles, and roles in energy transport in Earth’s magnetosphere can be potentially resolved using a combination of multi-satellite and ground-based observations, and numerical simulation.



We acknowledge Dr. Wenlong LIU, Dr. Quanqi SHI, Dr. Huishan FU, Dr. Yongfu WANG, Mr. Han LIU, Mr. Yin LIU and Jie REN for providing very useful materials. This work has been supported by National Natural Science Foundation of China (41421003 and 41627805). R. Rankin acknowledges financial support from the Canadian Space Agency and NSERC. We also acknowledge the ESA Cluster team for providing us data sets, and NASA CDAWeb ( for ACE, GOES, and OMNI data.


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© Division of Plasma Physics, Association of Asia Pacific Physical Societies 2017

Authors and Affiliations

  1. 1.Institute of Space Physics and Applied TechnologyPeking UniversityBeijingChina
  2. 2.Department of PhysicsUniversity of AlbertaEdmontonCanada

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