Abstract
The main sources of charged particles in the Earth’s inner magnetosphere are the Sun and the Earth’s ionosphere. Furthermore, the Galactic cosmic radiation is an important source of protons in the inner radiation belt, and roughly every 13 years, when the Earth and Jupiter are connected via the interplanetary magnetic field, a small number of electrons originating from the magnetosphere of Jupiter are observed in the near-Earth space. The energies of solar wind and ionospheric plasma particles are much smaller than the particle energies in radiation belts. A major scientific task is to understand the transport and acceleration processes leading to the observed populations up to relativistic energies. Equally important is to understand the losses of the charged particles. The great variability of the outer electron belt is a manifestation of the continuously changing balance between source and loss mechanisms, whereas the inner belt is much more stable.
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The main sources of charged particles in the Earth’s inner magnetosphere are the Sun and the Earth’s ionosphere. Furthermore, the Galactic cosmic radiation is an important source of protons in the inner radiation belt, and roughly every 13 years, when the Earth and Jupiter are connected via the interplanetary magnetic field, a small number of electrons originating from the magnetosphere of Jupiter are observed in the near-Earth space. The energies of solar wind and ionospheric plasma particles are much smaller than the particle energies in radiation belts. A major scientific task is to understand the transport and acceleration processes leading to the observed populations up to relativistic energies. Equally important is to understand the losses of the charged particles. The great variability of the outer electron belt is a manifestation of the continuously changing balance between source and loss mechanisms, whereas the inner belt is much more stable.
In the preceding chapters we already have encountered various aspects of acceleration and loss of charged particles, e.g., betatron and Fermi acceleration (Sect. 2.4.4), bounce and drift loss cones (Sect. 2.6.1), magnetopause shadowing (Sect. 2.6.2), as well as the basics of growth and damping of waves in Chaps. 4 and 5. In this chapter we present the general framework of quasi-linear theory of diffusion and transport, and discuss the sources and losses of different particle species in more detail. At the end of the chapter (Sect. 6.7) we point out that the different mechanisms do not only affect the radiation belt particles additively but also synergistically, e.g., through nonlinear modulation of whistler-mode or EMIC waves by large-amplitude ULF waves.
6.1 Particle Scattering and Diffusion
The response of charged particles to temporally and spatially variable electric and magnetic fields is deterministic and, according to Liouville’s theorem of statistical physics, in absence of external sources and losses the phase space density is conserved along the dynamical trajectories of the particles. However, the empirical determination of temporal evolution of the phase space density ∂f∕∂t in any given location is limited by imperfect observations of electromagnetic fields and waves and the particle populations. The finite angular, energy, and temporal resolution of particle instruments makes them insensitive to phase mixing .Footnote 1 Consequently, we are not able to observationally distinguish individual particles with different phases in their gyro, bounce, or drift motions and the empirical information is in most cases limited to phase-averaged description of the radiation belts. On the other hand, the phase mixing makes the theoretical description much more tractable allowing the use of diffusion formalism to describe the time evolution of particle distributions. As stated by Schulz and Lanzerotti (1974): “Thus, the ultimate inability to distinguish particle phases by observations is a simplifying virtue”.
Diffusion is a statistical concept to describe the evolution of the phase space density. It was already encountered in Sect. 5.1.3 as random walk of particles along single-wave characteristics. Although it is customary to talk about particle diffusion, individual particles actually do not diffuse. They are scattered in the phase space by spatial and temporal inhomogeneities, wave–particle interactions and collisions. In wave–particle interactions the resonant scattering is the most efficient, but not the only, cause of diffusion.
Wave–particle interactions can act both as sources and losses of particles in the belts. For example, acceleration of lower-energy electrons can be considered as a source of higher-energy electrons. The losses through wave–particle interactions are due to lowering the particles’ pitch angles small enough to precipitate into the atmosphere.
In Chap. 5 we have discussed the growth and decay of waves in Landau and gyro resonances ω − k ∥ v ∥ = nω cα∕γ with the particles. The Landau resonance (n = 0) either increases or decreases the parallel energy of the particles depending on the shape of the particle distribution function close to the resonant velocity v ∥,res = ω∕k ∥, which leads to energy and pitch-angle diffusion of the phase space density. The perpendicular momentum does not change in Landau resonance and thus the first adiabatic invariant \(\mu = p_\perp ^2/(2mB)\) is conserved but not the second \(J = \oint p_\parallel \,ds\).
The gyro resonance (n ≠ 0) breaks the invariance of μ and, consequently, the invariance of J and Φ, and leads again to pitch-angle and energy diffusion. Since the gyro resonance takes place in much smaller temporal and spatial scales than the azimuthal drift, its effect on Φ (or on L ∗) of the particle remains practically negligible, and thus does not need to be taken into account in pitch-angle diffusion calculations.
In radiation belts gyro-resonant interactions with whistler-mode chorus waves and plasmaspheric hiss at kHz frequencies and with EMIC waves around 1 Hz are the most efficient mechanisms to scatter charged particles toward the atmospheric loss cone. A single wave–particle interaction does not change much the pitch angle and energy, unless the wave amplitude grows to nonlinear regime (see the discussion in Sect. 6.4.4, Fig. 6.4). As the width of the equatorial loss cone in radiation belts is only a few degrees (Fig. 2.6), particles at large equatorial pitch angles must scatter numerous times before they approach the edge of the loss cone and can precipitate to the atmosphere. Consequently, the pitch-angle scattering often is a slow process depleting the radiation belts in timescales of days to hundreds of days, depending on the wave mode and particle energy. Furthermore, for the whistler-mode waves with ω < ω ce the resonance vanishes when v ∥→ 0. Thus, in order to limit a larger than observed excess of close-to-equator mirroring electrons additional mechanisms are needed to scatter the electrons to smaller pitch angles. The same is true for scattering of radiation belt and ring current ions with EMIC waves.
In theoretical investigations it is common to consider particles mirroring exactly at the equator (α = 90∘), as is frequently done also in this book. For such particles there is no bounce motion and J = 0.Footnote 2 This is a bit of a singular special case because the inner magnetosphere never is so symmetric that the motion of the particles would remain strictly perpendicular. Due to finite temperature there are thermal velocities and fluctuations in the parallel direction, and once a particle gets parallel momentum, it will be affected by the mirror force. However, this does not solve the problem of finding efficient enough scattering mechanism for almost equatorially mirroring particles.
A mechanism that has been invoked to scatter electrons from nearly 90∘ pitch angles toward larger parallel velocity, where the whistler-mode scattering can take over, is resonance between the electron bounce motion and the equatorial magnetosonic noise introduced in Sect. 5.3.2. The bounce motion requires, by definition, that μ is conserved and the bounce frequency ω b must be lower than ω ce. If the wave frequency matches with a multiple of ω b, the resonance can break the invariance of J and lead to scattering in pitch angle and energy.
The bounce-resonant scattering can be investigated by supplementing the parallel equation of motion with a time-dependent force field F ∥(s, t), where s is the coordinate along the magnetic field line
where μ is the relativistic adiabatic invariant \(p_\perp ^2/(2mB)\) (e.g., Shprits 2016, and references therein). The parallel force F ∥ can be due to an electrostatic wave or, in the case of equatorial magnetosonic noise, due to the parallel electric field component of an oblique (WNA ≈ 89∘) X-mode wave.
Another mechanism to break the invariance of J may arise from compressional ULF fluctuations that affect the length of the bounce path. The net parallel acceleration can in this case be described as Fermi acceleration due to the mirror force oscillating in resonance with the bounce motion. As pointed out by Dungey (1965), the bounce motion is associated with the azimuthal drift. Expanding the azimuthal fluctuation as \(\exp (-\mathrm{i} \omega t + \mathrm{i} m\phi )\) the resonance condition can be expressed in terms of bounce and drift frequencies as
where l is the longitudinal mode number and m the azimuthal mode number of the wave.
We can consider the bounce-resonance associated with a ULF wave mode of a given m. Let 𝜗 be the co-latitude of spherical coordinates, which at the equator is by definition 90∘. Defining the azimuthal wave number as \(k_\phi = m/(r\sin \vartheta )\) and bounce-averaged drift velocity as \(v_\phi = \omega _d\,r\sin \vartheta \) at given radial distance r and co-latitude, the resonance condition can be rewritten analogous to the gyro resonance as
where k ϕ v ϕ takes the role of the Doppler shift k ∥ v ∥ in gyro-resonant interactions. As co-latitude cancels from the equation, the resonance condition is independent of latitude.
Finally, the third invariant Φ, which is inversely proportional to L ∗, is violated by the bounce-averaged drift resonance
The drift resonance is associated with cross-field motion of trapped particles when the first two adiabatic invariants are conserved. This leads to changes in particle energy and to spreading of the particle distribution to different drift shells, which is commonly referred to as radial diffusion (for a review, see Lejosne and Kollmann 2020, and references therein). The concept of radial diffusion was introduced already during the early days of radiation belt research to explain the existence of the outer radiation belt (e.g., Parker 1960). The attribute “radial” is a bit misleading because L ∗ is not a spatial coordinate but inversely proportional to the magnetic flux enclosed by the particle’s drift shell, corresponding to the equatorial radial distance LR E in a purely dipolar field only. The drift resonance needs low enough wave frequency to match with the timescale of electron’s drift motion. Comparison of Tables 2.2 and 4.1 indicates that ULF Pc4–Pc5 waves can interact resonantly with electrons from about 1 MeV upward. Resonance with lower energy electron populations is also possible, but requires high azimuthal wave numbers. This is the case usually with externally excited poloidal waves (Sect. 5.4.1). We discuss the radial diffusion further in Sect. 6.4, including other ways to transport particles across the drift shells, e.g., inductive electric fields related to shock driven compressions of the dayside magnetopause or to substorm expansive phases.
Table 6.1 summarizes the main features how different waves discussed in Chap. 5, can be in resonance with radiation belt charged particles. These waves often co-exist in the inner magnetosphere and affect the particle populations in multiple ways. For example, some of the resonances increase/decrease the perpendicular energy of the particle (gyro and drift resonances), while others (Landau and bounce resonances) increase/decrease the parallel energy and velocity of electrons. To have higher order (|n| > 1) gyro resonances the Doppler-shifted wave frequency ω − k ∥ v ∥ must not become so large that the dispersion equation would no more be fulfilled in the plasma conditions of the inner magnetosphere.
6.2 Quasi-Linear Theory of Wave–Particle Interactions
The time evolution of the phase space density is commonly described using the diffusion equation that can also include effects beyond wave–particle interactions or Coulomb collisions. In this Section we introduce the diffusion equation in the framework of quasi-linear theory , which is a standard approach in numerical simulation and modeling studies addressing the wave–particle interactions in the inner magnetosphere. An important element is to find the properties of the waves related to wave–particle interactions (e.g. amplitude, wave normal angle, intensity and MLT distribution). These must often be estimated empirically from various observations.
Quasi-linear theory is a theoretical framework in the domain between the linearized Vlasov theory (Sect. 4.2) and nonlinear plasma physics of shocks, large-amplitude waves, wave–wave couplings, strong plasma turbulence, etc. In quasi-linear theory the wave modes are those of the linear plasma theory, but the slow temporal evolution of particle distribution functions is taken into account. The restriction to linear waves is an evident limitation and cannot rigorously address the problem of plasma perturbations growing to large amplitudes. The practical limits of the validity of quasi-linear computations are difficult to assess. For example, it is not clear what is the effect of small-scale nonlinear chorus elements of whistler-mode waves on the larger-scale diffusion process.
As always, there are no general methods to deal with nonlinear plasma equations and nonlinear processes must in practice be considered on a case-by-case basis. Often the best that can be done is to compute the orbits of a large number of randomly launched charged particles in the presence of prescribed nonlinear fluctuations. If it is possible to determine the diffusion coefficients from, e.g., numerical simulations or empirically from observations, they can be inserted in the diffusion equation and used in computation of the temporal evolution of the phase space density, even if the underlying particle scattering would be due to nonlinear interactions.
6.2.1 Elements of Fokker–Planck Theory
The fundamental task is to find a description for the temporal evolution of the charged particle distribution function ∂f∕∂t at a given location in the phase space in the presence of plasma waves, including inter-particle collisions when needed. While the inner magnetospheric plasma is almost collisionless, in addition to various wave–particle interactions, Coulomb and charge-exchange collisions often need to be included in computations of the ring current and radiation belt dynamics. This can formally be done by introducing a collision term (∂f∕∂t)c on the right-hand side of the Vlasov equation and rewriting it as the Boltzmann equation (3.17). The Fokker–Planck approach is a common, although not the only, method to determine the frictional and diffusive effects arising from the RHS of the Boltzmann equation and it can also be applied to “collisions” between plasma waves and charged particles.
To formally introduce the Fokker–Planck approach let us consider the function ψ(v, △v) that gives the probability that a particle’s velocity v is deflected, or scattered, by a small increment △v due to a collision or to an interaction with a wave electric field. Integrating over all possible deflections that may occur during a period △t before the time t gives the distribution function
In the Fokker–Planck approach ψ is assumed to be independent of t. Thus, the scattering process has no memory of earlier deflections and the process can be characterized as a Markovian random walk in the phase space.
Next Taylor expand the integral in (6.5) in powers of the small velocity changes △v
where : indicates scalar product of two dyadic tensors aa : bb =∑ij a i a j b i b j. The total probability of all deflections is unity \(\int \psi (\triangle \mathbf {v})\,d(\triangle \mathbf {v})=1\) and the rate of change of f due to collisions is
where the averages 〈△v〉 and 〈△v△v〉 are defined as
and the terms of the second and higher orders in △t have been dropped. Note that the denominator in both terms on the RHS of (6.7) is △t. In random walk the mean square displacements increase linearly with time.
By inserting (6.7) as the collision term to the Boltzmann equation we have arrived to the Fokker–Planck equation. The first term on the RHS of (6.7) describes the acceleration/deceleration (∝〈△v〉∕△t) of particles due to collisions, which in classical resistive media corresponds to dynamical friction. The second term is the diffusion term, containing the diffusion coefficient D vv ∝〈△v△v〉∕△t. Note that diffusion can change both the absolute value and the direction of the velocity of the particles. The former corresponds to energy diffusion , the latter, in magnetized plasma, to pitch-angle diffusion .
The SI units of D vv are m2 s−3 because the diffusion takes place in the velocity space. In radiation belt diffusion studies the mostly used coordinates are the drift shell, momentum and pitch angle, which have different units. The diffusion equations are commonly normalized so that all diffusion coefficients are given in same units, e.g., momentum2 s−1 or s−1.
Thus far we have nothing more than a formal equation and the hard task is to determine the correct form of the probability function ψ. The diffusion through Coulomb collisions is treated in several advanced plasma physics textbooks (e.g., Boyd and Sanderson 2003) and we skip the technical details. Our focus is on the diffusion resulting from wave–particle interactions and large-scale inhomogeneities of the magnetic field.
6.2.2 Vlasov Equation in Quasi-Linear Theory
Although the Fokker–Planck theory is fundamentally a collisional theory, also wave–particle interactions can be cast to the same formulation within the framework of the quasi-linear approach. The method is to consider the slowly evolving and fluctuating parts of the distribution function separately.
6.2.2.1 Diffusion Equation in Electrostatic Approximation
The critical assumption of quasi-linear theory is that the temporal evolution of the distribution function f(r, v, t) takes place much more slowly than the oscillations of the waves interacting with the particles. The separation is most transparent for electrostatic waves in non-magnetized plasma familiar from the Landau solution of the Vlasov equation in Sect. 4.2.
Let us consider f as a sum of a slowly varying part f 0, which is the average of f over the fluctuations, and of a fluctuating part f 1. For simplicity, we further assume that f 0 is spatially uniform and write
Now the Vlasov equation is
where the charge density fluctuations are related to the fluctuating electric field through the Maxwell equation
Assuming that the fluctuations in f 1 and E are nearly sinusoidal waves, the averages of functions linear in f 1, including E, over the fluctuation period vanish. The average of (6.10) denoted by 〈… 〉 is thus
This equation describes the temporal evolution of f 0.
By subtracting (6.12) from (6.10) we get an equation for the rapid variations of f 1
In the quasi-linear approximation we neglect the second order nonlinear terms on the RHS as smaller than the linear terms on the LHS, which leads to
This is formally the same as the linearized Vlasov equation (4.1) with the exception that now, according to (6.12), f 0 is time-dependent.
From here on we continue in the same way as in the derivation of the Landau solution. Assuming, for simplicity, that there is only one pole in the complex Laplace-transformed time domain, corresponding to the complex frequency ω 0, we find the fluctuating part of the distribution function in the k-space
where
In this expression K(k, ω) is the dielectric function of Vlasov theory (4.4).
Finally, by substituting (6.15) and (6.16) to (6.12) and making the inverse Fourier transformation back to the r-space the temporal evolution of f 0 is found to be given by the diffusion equation
where the subscripts {i, j} refer to the cartesian components of the velocity vector and to the elements of the diffusion tensor D. Here summing over repeated indices is assumed. The tensor elements D ij are the diffusion coefficients
where \(\langle E_i(-\mathbf {k},t)E_j(\mathbf {k},t)\rangle /\mathcal {V}\) is the spectral energy density of the electrostatic field and \(\mathcal {V}\) denotes the volume of the plasma.
Note that the components of the electric field in (6.18) are given in Fourier-transformed configuration space. In the following we often express the phase space density in other than cartesian velocity coordinates, e.g., as f(p, α) or f(μ, K, L ∗). In practice the diffusion coefficients must be calculated from the observed or modeled amplitude and polarization of the electric field in the r-space and transform thereafter the diffusion equation into the appropriate coordinate system.
Now we have a recipe to calculate the diffusion of the distribution function f 0 in the velocity space if we can determine the spectrum of electric field fluctuations for a given wave mode (ω 0, k).
6.2.2.2 Diffusion Equation for Magnetized Plasma
The inner magnetospheric plasma is embedded in a magnetic field and the fluctuations are electromagnetic, which makes the treatment of the diffusion equation technically more complicated than in the electrostatic case. The fundamental quasi-linear theory of velocity space diffusion due to small-amplitude waves in a magnetized plasma was presented by Kennel and Engelmann (1966) and has been discussed thoroughly in the monographs by Schulz and Lanzerotti (1974) and Lyons and Williams (1984).
Kennel and Engelmann (1966) derived the diffusion equation for f 0 due to electromagnetic waves into the form
where the diffusion tensor D is defined as
Here \(\mathcal {V}\) is the volume of the plasma, the sum is over all harmonic numbers, the vectors a n,k contain information on the amplitude and polarization of the wave electric field, the asterisk indicates the complex conjugate, ω k is the complex frequency corresponding to the wave vector k, and ∥ refers to the direction of the background magnetic field. It is evident that accurate empirical determination of the amplitude and polarization is essential for successful numerical computation of the components of the diffusion tensor.
Kennel and Engelmann (1966) further showed that diffusion brings the plasma to a marginally stable state for all wave modes. In the proof no assumption of a small growth rate was made. Thus the conclusion applies to both non-resonant adiabatic diffusion, e.g., large-scale fluctuations of the magnetospheric magnetic field, and to resonant diffusion at the limit where the imaginary part of the frequency ω ki → 0. At the limit of resonant diffusion the singularity in the denominator of (6.20) is replaced by Dirac’s delta that picks up the waves for which
for an integer n. This is the resonance condition familiar from Chap. 5. The theory describes the diffusion resulting from both Landau (n = 0) and gyro-harmonic (n≠0) resonances assuming that the conditions of quasi-linear approach are met.
In radiation belts the particle distribution functions are safe to assume gyrotropic, which motivates formulation of the quasi-linear theory in two-dimensional (v ⊥, v ∥) velocity space. It is a straightforward exercise in coordinate transformations (e.g., Chap. 5 of Lyons and Williams 1984) to write the diffusion equation in the (v, α)-space as
where the subscript 0 has been dropped and the slowlyFootnote 3 evolving velocity distribution function is denoted by f. Note that here the diffusion equation is written in a form where all diffusion coefficients are given in units of velocity2 s−1. The non-relativistic equation (6.22) can be formulated relativistically by replacing v with p = |p| = γmv. Formally the Lorentz factor only appears as a relativistic correction to the gyro frequency in the calculation of the diffusion coefficients. The relativistic calculations are, however, more complicated because the resonant lines become resonant ellipses as discussed in Sect. 5.1.3.
The diffusion equation (6.22) expresses the already familiar fact that wave–particle interactions can cause diffusion both in the absolute value of the velocity (or kinetic energy W = mv 2∕2) and in pitch angle. Kennel and Engelmann (1966) pointed out that the particles scatter primarily in pitch angle. Only for particles whose velocities are of the order of or slightly below the wave phase velocity is the energy scattering rate comparable to the rate of pitch-angle scattering.
The direction of the diffusion depends on the shape of the particle distribution function close to the velocity of the particle. For example, when the anisotropy of suprathermal electron distribution amplifies whistler-mode waves in the outer radiation belt (Sect. 5.2.1), the suprathermal electrons scatter toward smaller pitch angles and lower energy. On the other hand the velocity distribution of radiation belt electrons (100 keV and above) is more isotropic with ∂f∕∂W < 0 and the scattering in energy leads to electron acceleration at the expense of wave power.
6.2.3 Diffusion Equation in Different Coordinates
In radiation belt physics the phase space density in six-dimensional phase space f(r, p, t) is often given as a function of the action integrals {J i} = {μ, J, Φ} and the corresponding gyro-, bounce- and drift-phase angles {φ i}. If an action integral is an adiabatic invariant, the corresponding phase angle is a cyclic coordinate and the phase space density is independent of that angle. In a fully adiabatic case, where all action integrals are conserved, the phase space is three-dimensional and f(μ, J, Φ).
When the adiabatic invariance of one or several action integrals is broken, particles with different phase angles respond differently to the perturbation. For example, gyro-resonant electrons in the same phase as the electric field of a whistler-mode wave are scattered most efficiently leading to gyro-phase bunching of scattered electrons. However, within the quasi-linear approximation, the random walk of the particles in the phase space leads to phase mixing and after a few oscillation periods the individual phases are no more possible to distinguish in observational data. In the case of whistler-mode waves the phase mixing randomizes the phase angles within a few milliseconds, which is well below the temporal resolution of most particle instruments.
The phase mixing is much slower in the drift motion around the Earth. For example, substorm related particle injections from the magnetotail into the inner magnetosphere and abrupt energization due to interplanetary shocks hitting the dayside magnetopause take place much faster than the drifts around the Earth and break the third adiabatic invariant. The drift periods are from a few minutes to several hours (Table 2.2) and the bunches of energetic particles are readily observable in particle spectra as drift echoes. Figures 7.6 and 7.8 in Chap. 7 are two illustrative examples of drift echoes after shock-driven acceleration.
The phase mixing facilitates the use of phase-averaged phase space density \(\overline {f}(\{J_i\},t)\) in diffusion studies. As the phase information is lost in the averaging, \(\overline {f}\) is not consistent with the Liouville theorem in case of broken adiabatic invariance. However, the Fokker–Planck equation can still be applied in the quasi-linear approximation. Now the kinetic equation, supplemented with external sources and losses, can be written as
where 〈dJ i∕dt〉ν are the frictional transport coefficients and D ij the elements of the diffusion tensor. The loss and source terms (\(\overline {f}/\tau _q\) and \(\overline {S}\)) represent the average lifetime of immediate loss processes (e.g., magnetopause shadowing or charge exchange) and the drift-averaged external sources of \(\overline {f}\). From here on we simplify the notation by dropping the bars above f and S.
It is customary to write the kinetic equation in some other coordinates {Q i} than the basic action integrals {J i}. In radiation belt studies J is frequently replaced by K and Φ by L (or L ∗). The general coordinate transformation of the kinetic equation is
where \(\mathcal {J}= \mathrm {det}\{\partial J_k/\partial Q_l\}\) is the Jacobian determinant of the transformation from coordinates {J k} to coordinates {Q l} and \(\tilde {D}_{ij}\) denotes the transformed diffusion coefficients. For example, the Jacobian for the transformation from {J i} = {μ, J, Φ} to {Q i} = {μ, K, L} is \(\mathcal {J} = (8m\mu )^{1/2}(2\pi B_E R_E^2/L^2)\) , where B E is the equatorial magnetic field on the surface of the Earth.
Let us neglect the frictional term. Assuming that μ and K are constant, \(\mathcal {J} \propto L^{-2}\). This way we obtain the important radial diffusion equation
Radial diffusion refers in this context to the statistical effect of the motion of radiation belt particles across the drift shells while conserving the first two adiabatic invariants.
In radiation belt studies the phase space density is often considered as a function of pitch angle, momentum and drift shell. In this case the detailed formulation of the diffusion equation is a bit more complicated (e.g., Schulz and Lanzerotti 1974)
Here α is the pitch angle at equator, \(G = p^2T(\alpha )\sin \alpha \cos \alpha \), and T(α) the bounce function (Eq. (2.76) or (2.77)). D LL is the diffusion coefficient in L ∗ (the asterisk has been dropped for clarity). D αα, D αp and D pp are the diffusion coefficients in pitch angle, mixed pitch angle–momentum, and momentum. In (6.26) all diffusion coefficients are given in units of s−1. Due to vastly different temporal and spatial scales the cross diffusion between L and (α, p) has been neglected. As the gyro- and Landau-resonant processes scatter particles both in momentum and pitch angle, there is no reason to diagonalize the diffusion coefficients in the (α, p)-space.
The determination of the diffusion coefficients is the most critical part in studies of transport, acceleration, and loss of radiation belt particles. We discuss the procedures in the context of radiation belt electrons in Sect. 6.4.
6.3 Ring Current and Radiation Belt Ions
There are two partially spatially overlapping energetic ion populations in the inner magnetosphere. The strongly time-variable ring current is carried primarily by westward drifting ions in the energy range 10–200 keV, peaking at geocentric distances 3–4 R E and reaching roughly to 8 R E. The much less variable proton population of the inner radiation belt is located earthward of 3 R E. It consists mainly of 0.1–40-MeV protons with a high-energy tail up to relativistic energies of 1–2 GeV.
Although the ring current is not the main focus of our book, the basic dynamics of the current-carrying ions is similar to the dynamics of radiation belt particles. Thus we start this Section with a brief review of the characteristics of the ring current. As noted in Chap. 1, the ring current is the main, but not the only, cause of temporal perturbations in the north component of the equatorial magnetic field on ground. These perturbations are used to calculate the Dst and SYM-H indices, which, in turn, are commonly used measures of the strength of the storms in the magnetosphere. As the energy density of ions is much larger than that of electrons, the net current is carried mostly by the westward drifting ions. The variability of Dst and SYM-H during magnetospheric storms is a signature of the variability in the energy density of the current carriers.
6.3.1 Sources of Ring Current Ions
The ultimate sources of the ring current are the ionosphere and the solar wind. The main carriers of the current are energetic protons and O+ ions. While singly-charged oxygen must be of ionospheric origin, the protons may come from both sources. Table 6.2 summarizes the relative abundances of ring current ions during quiet and storm-time conditions based on AMPTE/CCE and CRRES satellite observations. The data were gathered during a relatively small number of storm-time observations and the numbers shall be taken as indicative. As always, individual storms exhibit large deviations from typical values.
The ion energies in the ionosphere and the solar wind are smaller than the energies of the ring current carriers. While the solar wind proton population already is in the keV-range, the ionospheric plasma has to be accelerated all the way from a few eV. The main ion outflow from the ionosphere takes place in auroral and polar cap latitudes. The ions are first transported to the magnetospheric tail and only thereafter to the inner magnetosphere, being meanwhile gradually energized.
The acceleration and heating of the outflowing ionospheric plasma takes place in several steps (for a review, see Chap. 2 of Hultqvist et al. 1999). Enhanced O+ ion outflows are observed during substorm growth and expansion phases in the ionosphere by ground-based radars and by satellites traversing the auroral field lines. Thus the observed large storm-time fluxes of O+ in Table 6.2 are not surprising. Some amount of heating by fluctuating electric fields already takes place in the ionosphere. The more energy the ions gain, the more efficiently the mirror force pushes them up. Further acceleration is provided by the magnetic field-aligned electric potential structures of the order of 1–10 kV, which accelerate auroral electrons downward and ions upward. As different particle species move up and down along the magnetic field and drift across the field, the regions above the auroral zones host a large variety of plasma waves, many of which can contribute to the energization of the ionospheric plasma to the keV-range.
In the magnetotail current sheet J ⋅E > 0, which according to Poynting’s theorem of electrodynamics implies energy transfer from the electromagnetic field to the charged particles. Ions crossing the current sheet with a finite but small magnetic field component normal to the sheet (B n) are transported for a short while in the direction of the electric field and gain energy (see, e.g., Lyons and Speiser 1982). This is an example of non-resonant diffusion in pitch angle and energy, which is due to the breaking of the first adiabatic invariant and results in chaotization of particle motion (e.g., Chen and Palmadesso 1986; Büchner and Zelenyi 1989).
Figure 6.1 illustrates how a low-energy ion entering the nightside magnetosphere from the high-latitude mantle is transported first to the distant tail and from there earthward with the large-scale convection while bouncing between the magnetic mirrors. The closer to the Earth the particle comes, the more frequently it crosses the current sheet. Because the stretching is strongest in the distant tail, the particle motion is most chaotic there and, consequently, the acceleration is most efficient for particles entering the plasma sheet far in the tail. Numerical test-particle simulations by Ashour-Abdalla et al. (1993) indicated that a particle encountering the current sheet for the first time beyond 80 R E in the tail with the energy of 0.3 keV can gain energy by a factor of 50 through this process alone.
Schematic picture of transport of two solar wind particles with slightly different initial conditions entering to the magnetosphere through the high-latitude mantle to the inner plasma sheet. The difference in the particle trajectories illustrates the sensitivity to the initial conditions, which is characteristic to chaotic motion. The X line is the distant reconnection line of the Dungey cycle (Sect. 1.4.1). The abbreviations CPS and PSBL refer to central plasma sheet and plasma sheet boundary layer (From Ashour-Abdalla et al. 1993, reprinted by permission from American Geophysical Union)
The north component of the magnetic field in the current sheet increases toward the Earth and the current sheet heating becomes less efficient in the near-Earth space. The particles advecting adiabatically to the inner magnetosphere also gain energy through the drift betatron mechanism (Sect. 2.4.4). However, this is not sufficient to account for ion energies above 100 keV, and wave–particle interactions are called for. Advanced diffusion codes must deal with both resonant and non-resonant source and loss processes (e.g., Jordanova et al. 2010, and references therein).
Substorm dipolarizations can also contribute to ion acceleration through transient inductive electric fields, whose role in reaching 100-keV energies may be important (e.g., Pellinen and Heikkila 1984; Ganushkina et al. 2005). The inductive electric fields can lead to preferential acceleration of O+ over H+ because all adiabatic invariants of O+ can be violated while the magnetic moment of H+ remains conserved as has been demonstrated in test-particle simulations (e.g., Delcourt et al. 1990).
6.3.2 Loss of Ring Current Ions
The intensity of the ring current is determined by the balance between the sources and losses of the current carriers. The losses are taking place all the time, but during the storm main phase they are overshadowed by the injection of new current carriers. The enhancement of the ring current is a relatively fast process, whereas the losses take more time. This is evident in the rapid negative evolution of the Dst index during the storm main phase and much slower decay of the current during the recovery phase (Fig. 1.7).
The main loss of energetic ions, originally suggested by Dessler and Parker (1959), is due to charge-exchange collisions between the ring current ions and the neutral hydrogen atoms in the extension of the Earth’s collisionless exosphere known as geocorona. A typical charge-exchange process is a collision between a positively charged ion and a neutral atom, in which the ion captures an electron from the atom. After the process the charge state of the ion is reduced by one and the neutral particle becomes positively charged
At ring current altitudes the geocorona consists almost purely of hydrogen atoms, but for ions mirroring at low altitudes also the charge exchange with heavier atoms needs to be included in detailed calculations.
The temperature of the neutral geocorona is of the order of 0.1 eV. Thus after a charge exchange with a ring current ion, the emerging particles are an ion of very low energy and an energetic neutral atom (ENA) . The ENA moves to the direction of the incident ion at the time of the collision and leaves the ring current region. The charge exchange does not directly decrease the number of current carriers, but transfers the charge from fast to very slowly drifting ions. These ions are no more efficient current carriers, instead they become a part of thermal background plasma.
The efficiency of charge exchange as a loss mechanism depends on the lifetimes of the current carriers, which are inversely proportional to the charge-exchange cross sections. The cross sections cannot be calculated theoretically and their empirical determination is difficult because the exosphere is a much better vacuum than can be created in laboratories. Furthermore, the density profile of the geocorona as well as the L-shells and pitch angles of the incident ions need to be taken into account because ions mirroring at different altitudes encounter different exospheric densities.
Also Coulomb collisions and wave–particle interactions have a role in removing ring current carriers. The Coulomb collisions are most efficient at lower energies (<10 keV). However, charge exchange and Coulomb collisions jointly do not remove enough ions with energies larger than a few tens of keV, and above 100 keV they lead to flatter pitch-angle distributions (smaller loss cones) than observed (Fok et al. 1996). On the other hand the ring current is embedded in a domain populated by EMIC waves, plasmaspheric hiss and equatorial magnetosonic waves, which can scatter the higher-energy ring current ions to the atmospheric loss cone.
A challenge in inclusion of wave–particle interactions in numerical ring current models is that both the growth and decay of the waves must be modeled self-consistently with the evolution of the particle populations. For example, the growth rate of EMIC waves needs to be calculated solving the hot plasma dispersion equation simultaneously with the kinetic equation. From the growth rate the wave amplitudes are estimated using empirical relations. The effect of wave–particle interactions on the ions is thereafter treated as a diffusion process where the diffusion coefficients are determined using the calculated wave amplitudes (e.g., Jordanova et al. 2010, and references therein).
6.3.3 Sources and Losses of Radiation Belt Ions
The inner radiation belt is relatively stable against short timescale perturbations. The energetic particle content of the inner belt is dominated by protons at MeV energies extending up to a few GeV. For higher energies the gyro radii become comparable to the curvature radius of the background magnetic field and particles cannot any more be trapped in the magnetic bottle. The residence times of protons are long, from years close to the atmospheric loss cone (large adiabatic index K) to thousands of years for equatorially mirroring particles (K ≈ 0). The particle spectra display variations in decadal (solar cycle) to centennial (secular variation of the geomagnetic field) timescales (e.g., Selesnick et al. 2007).
While the spectrum of trapped ions at energies larger than 100 keV appears to turn quite smoothly from ring current to radiation belt energies, the histories of the ions are different. Ring current carriers up to energies of 100–200 keV originate from the much lower-energy ionosphere and the solar wind being accelerated and transported by various magnetospheric processes. Different mechanisms are needed to produce the radiation belt ions up to tens or hundreds of MeV.
The two main sources of inner radiation belt protons are solar energetic particle (SEP) events and the cosmic ray albedo neutron decay (CRAND) mechanism. Below energies of 100 MeV and for L ≳ 1.3 the solar source dominates, whereas below altitudes of 2000 km and at higher energies CRAND is the dominant source.
The solar flares and CMEs produce large fluxes of energetic protons, of which most are shielded beyond L ≈ 4 by the geomagnetic field. SEPs arriving to the magnetosphere with pitch angles already within the atmospheric loss cone are lost directly, whereas most ions are just deflected by the magnetic field and escape from the near-Earth space. During strong solar particle events solar protons and heavier ions have been found to be injected to L-shells 2–2.5 (e.g. Hudson et al. 2004b, and references therein), from where they are transported inward through radial diffusion. However, the trapped orbits in the innermost magnetosphere are equally difficult to enter into as to escape from, and only a small fraction of incoming protons become trapped. At L = 2 the trapping efficiency of 10-MeV protons has been estimated to be of the order of 10−4 and of 100-MeV protons only 10−7 (Selesnick et al. 2007).
Solar storms have a twofold role in the radiation belt ion dynamics. They provide intermittent source populations and drive perturbations in the magnetosphere that are necessary for particle trapping. However, the energetic solar particles from solar eruptions arrive to the Earth faster than the associated ICME. Thus the trapping is more efficient if the magnetosphere at the time of SEP arrival is perturbed, e.g., by a former ICME or a fast solar wind stream.
Decadal to centennial time series of fresh solar proton injections have been derived from enhancements of the NO3-rich layers in Arctic and Antarctic ice. Protons with energies larger than 30 MeV penetrate to the Earth’s atmosphere and enhance the production of odd nitrates (including NO3) in the troposphere. The molecules thereafter precipitate to ground and become archived in the polar ice (McCracken et al. 2001, and references therein).
The CRAND mechanism as a source of inner belt protons was suggested by Singer (1958) soon after the early observations of trapped radiation. It results from Galactic cosmic ray bombardment in the atmosphere, which produces neutrons that move to all directions. While the average neutron lifetime is 14 min 38 s, during which a multi-MeV neutron hits the ground or escapes far from the Earth, a small fraction of the neutrons decay to protons while still inside the magnetosphere. Because the Galactic cosmic ray spectrum is hard and temporally constant, the CRAND mechanism produces a hard and stable spectrum.
At energies below 50 MeV the observed proton spectra are too intense and variable to be explained by the CRAND mechanism. Figure 6.2 illustrates results of a model study by Selesnick et al. (2007), in which the main source and loss terms were integrated over 1000-year timescales. The proton fluxes at L = 1.2 and L = 1.7 are computed for several values of the adiabatic invariant K. The fluxes are highest for the protons mirroring closest to the equator (smallest K), where the trapping times are longest. At L = 1.2 the fluxes are several orders of magnitude smaller at all energies than at L = 1.7 and the spectrum is completely dominated by CRAND-produced protons. At L = 1.7 the softer solar proton spectrum becomes visible below 100 MeV.
Model calculations of energetic proton spectra at L = 1.2 (top) and L = 1.7 (bottom). The colors correspond to different values of the adiabatic index K. The uppermost curves are for equatorially mirroring particles (K ≈ 0). The largest K in the upper picture is 0.09 G1∕2 R E, in the lower 0.58 G1∕2 R E, each corresponding to the value for which the mirror points of the entire drift shell are above the Earth’s atmosphere. Note different scales on vertical axes in the panels (From Selesnick et al. 2007, reprinted by permission from American Geophysical Union)
The main energy loss mechanisms of the proton belt are the charge exchange and inelastic nuclear reactions with neutral exospheric atoms as well as Coulomb collisions with ionospheric and plasmaspheric charged particles. Furthermore, although small, the adiabatic compression or expansion of the drift shells related to the solar cycle and the secular variation of the geomagnetic field affect the proton energies during their long residence in the inner belt.
The charge exchange cross sections decrease rapidly at energies above 100 keV and the mechanism is a much slower loss process in the inner proton belt than in the ring current. Coulomb collisions and nuclear reactions slowly decrease the ion energy from hundreds of MeV to levels where the ENA production finally can take over the role as a loss process.
6.4 Transport and Acceleration of Electrons
During and after the Van Allen Probes mission the dynamical evolution of the energetic electron populations, in particular of the relativistic and ultra-relativistic electrons, has been in the focus of radiation belt research. This has a strong practical motivation because some of the most serious spacecraft anomalies have been addressed to large fluxes of relativistic “killer electrons” . In this section we discuss the physical mechanisms of electron acceleration and transport. Electron losses are the topic of Sect. 6.5.
6.4.1 Radial Diffusion by ULF Waves
The traditional theory of the electron belt formation, introduced in the 1960s, is based on inward radial diffusion due to low-frequency electromagnetic fluctuations in asymmetric quasi-dipolar magnetic field. The fluctuations are assumed to conserve the first and second adiabatic invariants but break the third, which in radiation belt studies is usually represented by L ∗. In the following discussion we drop the asterisk for clarity and write the diffusion equation (6.25) without external source and loss terms
where electromagnetic fluctuations determine the radial diffusion coefficient D LL. When the seed population is transported from the tail toward larger magnetic field, the particles gain energy due to the conservation of the magnetic moment \(\mu =p^2_\perp /(2m_eB)\) and, in the presence of ULF waves, by resonant interactions between the waves and the azimuthal drift motion of the electrons.
The practical challenge is to determine the diffusion coefficient D LL. In theoretical analysis one has to make quite a few simplifying assumptions and approximations. Already a slightly distorted dipole field geometry together with standard convection electric field models leads to complications. Furthermore, the intensity of the electromagnetic fluctuations is different at different magnetic local times and a function of magnetospheric activity. On the other hand, the empirical determination of the diffusion coefficients is severely constrained by available observations and different studies have led to different, sometimes contradictory, results (e.g., Ali et al. 2016, and references therein).
Based on purely theoretical arguments Fälthammar (1965) demonstrated that the diffusion coefficient from magnetic field perturbations for equatorially mirroring particles \(D^{em}_{LL,eq}\) is proportional to L 10. He considered small time-dependent perturbations of the magnetic field assuming that the spatial asymmetry of the perturbation was a stationary stochastic process. Lejosne (2019) re-derived the coefficient in the form
where B E is the terrestrial magnetic field at equator, P A(ω d) the power spectral density of the asymmetric compressional magnetic fluctuation and ω d the angular drift frequency. D LL decreases with decreasing equatorial pitch angle and is at the edge of the atmospheric loss cone reduced to about 10% of the coefficient for equatorially mirroring particles.
For compressional perturbations the diffusion coefficient is determined by the azimuthal component of the inductive electric field (∇×E = −∂ B∕∂t). Fälthammar (1965) considered electrostatic (∇×E = 0) perturbations separately, for which he found the diffusion coefficient
where P E(nω d) is the power spectral density of the nth harmonic of the electric field fluctuation at the drift resonant frequency ω = nω d. In the electrostatic approximation the magnetic field lines are electric equipotentials and the expression is valid for all pitch angles. The SI unit of the ratio of power spectral densities P E∕P A is that of velocity squared (m2 s−2). Thus both expressions (6.29) and (6.30) have the same physical dimension (SI unit s−1).
The division of electromagnetic fluctuations to inductive and electrostatic disturbances can be theoretically justified due to their different sources. However, these are difficult to distinguish in satellite observations. Another approach is to calculate a “pure” magnetic diffusion coefficient \(D^b_{LL}\) and combine the electrostatic and inductive electric fields into an electric diffusion coefficient \(D^e_{LL}\). This approach was taken by Fei et al. (2006), who developed further the earlier calculations of Elkington et al. (2003). They assumed the electric and magnetic perturbations to be those of compressional Pc5 ULF waves in an asymmetric quasi-dipolar magnetic field in the equatorial plane
where b 1 describes the global compression of the dipole field and b 2 is the azimuthal perturbation.Footnote 4
The calculation of Fei et al. (2006) was relativistic, which is important because radial diffusion is often applied to relativistic electrons. They found the diffusion coefficients
where m is the azimuthal mode number and P B,m and P E,m are the power spectral densities of the compressional component of the magnetic field and the azimuthal component of the electric field. \(D^e_{LL}\) has the same form as Fälthammar’s \(D^{es}_{LL}\) (6.30) but the power spectral densities are different. Here the P E,m includes the spectral power of the entire electric field, whereas in (6.30) P E,n represents the electrostatic fluctuations only.
Fei et al. (2006) claimed that their coefficients reduce to those of Fälthammar (1965) in the nonrelativistic limit and taking the different treatment of the electric field into account. However, as pointed out by Ali et al. (2016) and Lejosne (2019) the sum of \(D_{LL}^e\) and \(D_{LL}^b\) is about a factor of 2 smaller than \(D^{em}_{LL}\). The reason is the assumption of Fei et al. (2006) that the electric and magnetic field perturbations are independent of each other. As demonstrated by Perry et al. (2005), the azimuthal component of the electric field E ϕ and the time derivative of the poloidal component of the magnetic field ∂B θ∕∂t are anticorrelated in the model magnetic field (6.31), as they should be according to Faraday’s law (∇×E = −∂ B∕∂t).
The factor of 2 difference in Fei’s and Fälthammar’s diffusion coefficients may be a somewhat academic problem in practical diffusion studies, which often involve magnetospheric storms. In such cases the magnetic field model (6.31) is too simple and very different empirically determined diffusion coefficients have been found in different studies. In addition to the compression and stretching of the magnetic field, the time evolving ring current affects the electromagnetic field. For application of different empirically derived diffusion coefficients we refer to the investigation by Ozeke et al. (2020) of the two Saint Patrick’s Day (March 17) storms in 2013 and 2015. An example of the practical difficulties is that while most of the time \(D^b_{LL}\ll D^e_{LL}\), during the storm main phase this relationship can be the reverse.
Due to the great variability of inner magnetospheric conditions the empirical determination of the diffusion coefficients on the case by case basis may be the only way of finding diffusion rates consistent with particle observations. For a given D LL the diffusion equation (6.28) is fast to compute numerically, which makes it possible to look for an optimal coefficient as a function of appropriate magnetospheric parameters. A widely-used parameterization is that of Brautigam and Albert (2000) based on observations of the magnetospheric storm on 9 October 1990. They used the Kp-index as a parameter and found the diffusion coefficient
with coefficients a = 4.73 × 10−10, b = 10, and c = 0.506. More empirical event-based parameterizations can be found, e.g., in the above cited publications by Elkington et al. (2003), Ali et al. (2016) and Ozeke et al. (2020), and in articles cited therein. It is clear that event-based derivations using different parameters lead to different results but the critical L-dependence is in most cases close to Fälthammar’s original L 10.
6.4.2 Electron Acceleration by ULF Waves
We illustrate the drift resonant acceleration of electrons by discrete ULF wave modes following Elkington et al. (2003). They considered equatorial electrons in the model magnetic field of Eq. (6.31). The drift contours (2D drift shells) are determined by the constant magnetic field strength, where the L-parameter is replaced by
For small perturbations (b 1 ≪ B 0), \(\mathcal {L}\approx L\) within the radiation belts.
The electric field of the ULF waves in the equatorial plane was given in Eq. (4.107) as
Here E 0(r, ϕ) is the time-independent convection electric field. δE rm are the electric field amplitudes of the toroidal modes and δE ϕm of the poloidal modes, and ξ rm and ξ ϕm represent their phase lags.
For radial diffusion to be efficient the fluctuations should be global and resonate with a multiple of the angular drift frequency of the electrons
In the outer radiation belt the m = 2 mode fulfils the resonance condition at the fundamental drift frequency (ω = ω d) of relativistic electrons. At L = 6 the drift periods of 1–5-MeV electrons are 2.7–12.3 min (Table 2.2), matching with the period range of Pc5 waves. Resonance with the higher-frequency Pc4 oscillations is possible for larger azimuthal mode numbers. For a few tens to a few hundred keV electrons (drifting around the Earth at L = 6 in timescales of an hour to about ten hours) to be in drift resonance with Pc4–Pc5 waves the azimuthal mode numbers have to be very large (∼10–100). In summary, global large-amplitude ULF waves in the Pc4–Pc5 frequency band are natural agents of radial diffusion.
According to (2.68) the adiabatic (μ conserving) acceleration of equatorial electrons is given by
where v d is the electron drift velocity around the Earth. The magnetic perturbation of the toroidal mode δB ϕ and the dominant magnetic field component of the poloidal mode δB r both have a node at the equator. In both cases the electric field of the ULF wave has a component along the direction of the of the electron’s GC drift velocity in certain parts of drift path around the Earth resulting in drift-betatron acceleration. The compressional component δB ∥ of the poloidal mode is assumed to be so small, that the gyro-betatron acceleration μ ∂B∕∂t can be neglected and the energization is due to the drift-betatron effect only. In the toroidal mode δB ∥ = 0 by definition. Figure 6.3 illustrates how a drift resonant (ω = ω d) electron is accelerated by the toroidal (left) and poloidal (right) m = 2 ULF wave in a distorted dipole.
Illustration of drift resonance of an electron with m = 2 toroidal (left) and poloidal (right) mode wave electric field. Noon is to the right
Test-particle simulations of electron interaction with a whistler-mode wave packet. Small/large A refers to the wave amplitude and λ is the magnetic latitude. The spreading in the equatorial pitch angle (α eq) and energy (E) is caused by the different phases of the particles when they interact with the wave packet. The oscillatory behavior of pitch angle and energy is due to their η-dependence close to the resonance dη∕dt ≈ 0 and decays when the particle moves further away from the site of the resonance (from Bortnik et al. 2008a, reprinted by permission from American Geophysical Union)
Let us first consider an electron interaction with the toroidal mode (Fig. 6.3, left). Start from the point in the dusk sector where the radial component of the electron velocity v r reaches a maximum in inward direction. Here the electron encounters an outward electric field δE r. Thus dW∕dt = qδE r v r > 0 , and the electron is accelerated. Half a drift period later the electron is in the dawn sector, where it has a maximal outward velocity component and encounters an inward electric field, and is accelerated again. The dawn and dusk sectors are in this case regions of maximal energy gain. The electron actually gains energy throughout of the orbit around the Earth, except at noon and midnight where v r = 0. The asymmetric compression of the magnetic field is an important factor in the process. Increasing distortion increases v r in the dawn and dusk sectors and thus increases the energy gain.
Efficient drift acceleration can also occur from the resonance with the poloidal mode δE ϕ, where the electric field perturbation δE ϕ is in the azimuthal direction (Fig. 6.3, right). The electron on the nightside encounters an electric field that is in the direction opposite to its velocity and is accelerated. On the other hand, if the electron is in drift resonance with the wave, it encounters on the dayside an electric field that is in the direction of the drift motion and loses energy. In the compressed dipole configuration |δE ϕ v ϕ| is, however, smaller on the dayside than on the nightside. Thus the electron gains net energy over the drift period around the Earth. Adding a static convection electric field E 0 weakens the net acceleration by the poloidal mode of this particular electron, because E 0 is in the same direction as δE ϕ when the electron moves on the nightside and on the nightside dayside.
It is important to realize that the two examples in Fig. 6.3 describe only one electron in drift resonance with a discrete single-frequency wave. Considering electrons in different drift phases with respect to the phase of the wave, some electrons gain, some others lose energy. Some of them are pushed closer to the Earth, others further away from the Earth. The net result is both radial and energy diffusion (see also the discussion by Lejosne and Kollmann 2020).
If the poloidal modes are distributed over a range of frequencies, or a non-static convection field is acting on the electron, the dominant component of the electron’s drift velocity in the azimuthal direction may permit even more efficient acceleration than the interaction with purely toroidal modes of the same amplitude. Based on numerical calculations with a continuum of frequencies Elkington et al. (2003) concluded that the resonant mechanism can lead to very efficient inward diffusive radial transport of electrons and their acceleration from 100 keV to MeV energies.
It is evident that the radial transport and acceleration by ULF Pc4–Pc5 waves are closely related to each other. While the diffusive transport is generally considered as a relatively slow process (of the order of days) the ULF waves may also lead to fast radial diffusion. Jaynes et al. (2018) studied radiation belt electron response during the magnetic storm on 17 March 2015. The electron fluxes from a few hundred keV to relativistic energies recovered soon after the peak of the storm, while ultra-relativistic electron fluxes stayed low for a few days. While the energization up to relativistic energies could have been related to enhanced chorus wave activity to be discussed in Sect. 6.4.5, the reappearance and inward transport of ultra-relativistic electrons (up to 8 MeV) occurred when the observed chorus activity had already subsided, whereas empirical estimates of radial diffusion coefficients suggested fast diffusion. Because empirical diffusion coefficients are event-specific, also the amounts of radial diffusion and acceleration are event-specific.
6.4.3 Diffusion Coefficients in the (α, p)-Space
The challenges in the determination of diffusion coefficients in the pitch-angle–momentum space are different from those of D LL. The diffusion tensor is given by Eq. (6.20) but the calculation of its elements requires the use of an appropriate approximation of the dispersion equation and knowledge of the amplitude and polarization of the waves interacting with particles. In practical computations one needs to use realistic models of the spatial distribution and properties of the waves.
We consider the relativistic formulation of the diffusion equation (6.22) without external sources and losses following Lyons and Williams (1984)
Here D αα, D αp = D pα and D pp are the drift- and bounce-averaged diffusion coefficients
The multiplicative factors p 2∕2, p∕2 and 1∕2 normalize the units of all coefficients to momentum2 s−1. These coefficients can be computed from coefficients for given harmonic number n and perpendicular wave number k ⊥ as integrals over all wave numbers and sums over all harmonics as
The integrals are calculated over perpendicular wave vectors only, as the resonance condition yields Dirac’s delta in the parallel direction.
The expressions (6.40) are the components of the diffusion tensor (6.20) originally derived by Kennel and Engelmann (1966) in the non-relativistic approximation and generalized to relativistic particles by Lerche (1968). The diffusion coefficients for given n and k ⊥ are related to the pure pitch-angle diffusion coefficients, which after a lengthy calculation turn out to be
for a given particle species j. Here \({\mathcal V}\) is the plasma volume, and the derivative ∂ω∕∂k ∥ is to be evaluated at the resonant parallel wave number
The function Θ nk contains the information of the amplitude and polarization of the wave electric field
Here L, R, and ∥ refer to the left-hand, right-hand and parallel polarized components of the wave electric field for a given wave vector. The argument of the Bessel functions J n is (k ⊥ v ⊥ γ∕ω cj) and s j is the sign of the particle species j. Finally, \(D_{\alpha p}^{nk_\perp }\) and \(D_{pp}^{nk_\perp }\) are
The terms multiplying \(D_{\alpha \alpha }^{nk_\perp }\) in these equations are smaller than 1, which is consistent with the conclusion of Kennel and Engelmann (1966) that pitch-angle diffusion dominates over diffusion in energy (or in the absolute value of momentum) as noted in Sect. 6.2.2.
In practical computations the distribution of the wave power as a function of frequency is often approximated by a Gaussian as
Here A is a normalization constant and ω m and δω are the frequency and bandwidth of the maximum wave power. However, the determination of the diffusion coefficients for waves fulfilling the electromagnetic dispersion equation is still a formidable technical task requiring heavy numerical computations (e.g., Glauert and Horne 2005). Restricting the analysis to parallel propagating (k ⊥ = 0) whistler-mode and EMIC waves, the integrals assuming Gaussian distribution in frequency can be expressed in closed form (Summers 2005). This speeds up the computations significantly but means a neglection of effects of obliquely propagating waves, which are critical to the dynamics of radiation belts.
As will be discussed in Sect. 6.5, wave–particle resonances with the electron bounce motion also result in pitch-angle diffusion, which is found to be important for nearly equatorially mirroring particles. In that case the diffusion coefficients must be calculated without bounce-averaging. Detailed calculations of such diffusion coefficients have been presented by Tao and Li (2016) for equatorial magnetosonic waves, by Cao et al. (2017a) for EMIC waves and by Cao et al. (2017b) for the low-frequency plasmaspheric hiss.
6.4.4 Diffusion due to Large-Amplitude Whistler-Mode and EMIC Waves
When whistler-mode or EMIC waves grow to large amplitudes, the quasi-linear approach to calculate diffusion coefficients becomes invalid. Different schemes to estimate the diffusion by nonlinear wave–particle interactions have been introduced in the literature, e.g., the formation of electron phase-space holes discussed in Sect. 5.2.4 (Omura et al. 2013) and the dynamical systems approach (Osmane et al. 2016, and references therein). Here we present a straightforward approach to numerically integrate the equation of the electron motion in a wave field (E w(r, t), B w(r, t)) determined from observations or theoretical arguments. The relativistic equation of motion in a latitude-dependent background magnetic field B 0(λ) is
By launching a large number of electrons with different initial conditions representing the original f(α, p), it is possible to estimate the diffusion coefficients (6.39) from △α and △p averaged over a time period △t.
In practical computations (6.47) is convenient to transform to coupled differential equations for momentum parallel (p ∥) and perpendicular (p ⊥) to B 0 and for the phase angle η between the perpendicular velocity of the electron v ⊥ and the perpendicular component of the wave magnetic field B w⊥. After gyro-averaging, neglecting second order terms, and assuming parallel propagating (k = k ∥) waves the relativistic equations are (Albert and Bortnik 2009)
Here s is the coordinate along the magnetic field, the gradient (∂B 0∕∂s) represents the mirror force, ω ce is the electron gyro frequency in the background field B 0 and n the harmonic number. The velocity along the magnetic field is given by ds∕dt = p ∥∕(γm e) (For corresponding non-relativistic equations, see Dysthe 1971; Bell 1984). This formulation is applicable to both whistler-mode waves (n ≥ 1) (e.g., Bortnik et al. 2008a) and EMIC waves, in which case it is sufficient to consider the first order resonance (n = −1) only (e.g., Albert and Bortnik 2009).
Because the electron gyro frequency is higher than the wave frequency, in case of EMIC waves much higher, the terms relative to \(\sin \eta \) and \(\cos \eta \) average to zero for small-amplitude waves after a few gyro periods and dη∕dt can be approximated as
In resonant interaction η is practically constant over a short period △t and (6.49) reduces to the familiar resonance condition. As long as △v ∥ remains small compared to the adiabatic motion, the interaction can be described using resonant ellipses (Sect. 5.1.3).
However, if B w grows large enough, the nonlinear terms (\(\propto \sin \eta \)) are no more negligible and the scattering can be quite different. To understand the nonlinear resonant interaction in case of large-amplitude wave field, take the second time derivative of the last equation of (6.48) and insert dv ∥∕dt from the first. This gives an equation for a nonlinear driven oscillator, assuming here, for simplicity, non-relativistic motion (γ = 1)
where the terms of smaller orders have been neglected (Bortnik et al. 2016).
The type of interaction depends on the relative effect of the nonlinear term on the LHS and the driving term on the RHS of (6.50). In addition to the amplitude of the wave the result depends on the particle’s pitch angle (α) and on the latitude where the interaction takes place through the latitude-dependence of the mirror force ∝ ∂ω ce(λ)∕∂s. If the driving term dominates, the interaction remains linear. Note that at equator ∂B 0∕∂s → 0 and the nonlinear interaction can become important also for small B w (see also discussion at the end of Sect. 6.5.4).
Bortnik et al. (2008a) applied this method to whistler-mode chorus waves using both typical and very large amplitudes. They launched 24 test particles with initial phases η 0 distributed uniformly between 0 and 2π to move through a whistler-mode wave packet. The frequency of the wave packet representing the whistler-mode chorus elements was 2 kHz and the wave propagated away from the equator at L = 5 in the geomagnetic dipole field (Fig. 6.4).
In cases A and B of Fig. 6.4 the initial energy of the particles was 168 keV and the equatorial pitch angle α eq = 70∘. The parameters of the particles were selected so that they started at the latitude λ = −9∘ and were in resonance with the wave at λ ≈−5∘ (case A) and λ ≈−6.5∘ (case B). The interaction time △t with a single wave packet was about 10–20 ms.
In case A the wave amplitude was 1.4 pT. The scattering from a single wave packet remained small as expected: in equatorial pitch angle about 0.03∘–0.04∘ and in energy 30–40 eV. After several encounters with similar wave packages the result would be similar to quasi-linear diffusion.
In case B the interaction of the same particles with a large-amplitude wave B w = 1.4 nT, corresponding to observations by Cattell et al. (2008), was quite different. The equatorial pitch angles of all particles dropped about 5∘ and the energies by 5 keV. In this case the initially uniform phases η became bunched by the wave at the time of interaction. Such a non-linear behavior is known as phase bunching.
In case C the initial energy was 500 keV and the initial equatorial pitch angle 20∘. The particles were launched from λ = −30∘ and they resonated at λ ≈−23∘ with a large-amplitude (B w = 1.4 nT) wave with WNA 50∘. In this case a large fraction of the particles were again phase bunched with decrease in pitch angle and energy, whereas some of them scattered to larger pitch angles and energy. One of the particles (the red track in Fig. 6.4) became trapped in the wave potential and the particle remained trapped in the constant phase of the wave electric field for a longer time period. Consequently, the particle wandered to a much larger pitch-angle and the total energy gain was 300 keV at the time the particle met the boundary of the simulation. This behavior is known as phase trapping .
Trapping of electrons in the nonlinear wave potential is an essential result also in the above mentioned theories by Omura et al. (2013) and Osmane et al. (2016). As discussed at the end of the next section, efficient acceleration of relativistic electrons by nonlinear whistler-mode wave packets is consistent with wave and particle observations of Van Allen Probes (Foster et al. 2017).
6.4.5 Acceleration by Whistler-Mode Chorus Waves
Electron acceleration by chorus waves is different from the drift-resonant acceleration by ULF waves. It takes place through the gyro resonance between the waves and the electrons breaking the first adiabatic invariant. The right-hand polarized chorus waves interact with a fraction of energetic electrons through the Doppler-shifted gyro resonance ω − k ∥ v ∥ = nω ce∕γ . Variables ω, v ∥ and, ω ce can often be measured but k ∥ must, in practice, be determined by solving the dispersion equation, which in turn depends on plasma density and composition. As discussed in Sect. 5.1.3, the resonance with a wave of a particular ω and k ∥ defines a resonant ellipse in the (v ⊥, v ∥)-plane, which reduces to a resonant line in nonrelativistic case when the resonant condition depends only on electron’s parallel velocity.
The chorus is a wide-band emission, so there is a continuum of resonant ellipses and, consequently, there is a finite volume in the velocity space intersected by single-wave characteristics (Eq. (5.9)). Thus a large number of electrons are affected as long as they fulfil the resonance condition. The resonant diffusion curves illustrate that gyro-resonant interactions with whistler-mode chorus waves can efficiently energize electrons from a few hundred keV to MeV energies (Summers et al. 1998). The energization takes place near the equator where chorus waves propagate almost parallel to the background magnetic field (small WNAs). This is important to the Doppler shift term k ∥ v ∥ that must be large enough for the wave frequency and particle’s (relativistic) gyro frequency to match.
An example of strong and rapid local acceleration of electrons in the heart of the outer radiation belt via interaction with whistler-mode waves was presented by Thorne et al. (2013b). They studied the geomagnetic storm on 9 October 2012 in the early phase of the Van Allen Probes mission. Intense chorus activity was observed from the dawn to dayside sector. The electron diffusion was calculated solving the Fokker–Planck equation (6.26) in the pitch-angle–momentum (α, p) space. The results of the diffusion calculations were consistent with Van Allen Probes data. The authors, however, noted that a definitive conclusion about the relative importance of chorus waves vs. other processes was not possible to give due to observational limitations.
Similarly to diffusion caused by ULF waves the critical issue is the determination of the diffusion coefficients, now in pitch angle and momentum. The procedure has to be based on empirical or modelled plasma and wave properties. The drift averaging is needed in practice although the wave distribution in local time is inhomogeneous and varies from one event to another. Thus the estimation of net acceleration is a challenge.
An example of determining the diffusion coefficients from several sources as the input to the analysis of the electron acceleration during the so-called Halloween storm in autumn 2003 was published by Horne et al. (2005). The interaction with chorus waves is most efficient when ω pe∕ω ce is relatively small (\({\lesssim }4\)), which is the case outside the plasmapause. On 31 October 2003, this condition was met, as the high-density plasmasphere was confined inside L = 2 and remained inside L = 2.5 in the pre-noon sector (06–12 MLT) until November 4. In their numerical calculations the authors used relativistic electron data from the SAMPEX satellite, Kp and Dst indices, ground-based ULF observations and kHz-range wave observations from the Cluster spacecraft. They argued that the radial diffusion due to the ULF waves could not explain the strong increase of 2–6-MeV electron fluxes between L shells from 2 to 3 in the late phase of the storm after 1 November 2003. Instead, the Fokker–Planck calculations, based on diffusion rates calculated for chorus wave amplitudes measured by Cluster at somewhat higher drift shell (L = 4.3), suggested that the gyro-resonant interaction was sufficient to explain the establishment of very high electron fluxes in the slot region during this exceptionally strong storm period. We will discuss the slot region and the Halloween storm more thoroughly in Sect. 7.4.
While quasi-linear diffusion computations assuming linear whistler-mode waves seem to be able to produce observed acceleration of MeV electrons, the role of the large-amplitude rising-tone whistler elements (Sect. 5.2.4) raises interesting questions. For example, how well does a quasi-linear diffusion model represent the collective effect of nonlinear wave–particle interactions?
Foster et al. (2017) investigated the recovery of 1–5-MeV electrons after they had become depleted during the main phase of the storm on 17–18 March 2013. The recovery of MeV electrons was preceded by the occurrence of electrons in the energy range from a few tens to hundreds of keV that were injected by substorms in the storm recovery phase. Foster et al. (2017) used Van Allen Probes observations of the rising-tone whistler-mode wave packets to compute electron energization in the theory of Omura et al. (2015, and references therein, see Sect. 5.2.4). Figure 6.5 summarizes their results, according to which the nonlinear interaction turned out to be very efficient. For example, resonant 1-MeV electrons were found to be able to gain 100 keV in a single interaction with a wave packet of 10–20 ms duration. The observed wave packets had oblique WNAs from 5∘ to 20∘, which was taken into account in the analysis, and both gyro and Landau resonances were accounted for. The Landau resonance was effective at energies below 1 MeV and comparable to the gyro resonance below 100 keV.
The energy gain at different initial energies due to observed rising-tone whistler-mode wave packets. The upper panel shows the acceleration by gyro resonance and the lower panel by Landau resonance for different initial energies. The black lines indicate acceleration summed over all wave packets, the red lines the maximum energization by a single wave packet. The blue curve shows the most probable acceleration at pitch angles >87∘ (From Foster et al. 2017, reprinted by permission from American Geophysical Union)
6.5 Electron Losses
In this section we discuss the basic features of electron loss processes from the outer radiation belt. The complexity of the dynamics of the belt is further discussed in Chap. 7 in the light of recent observations.
The main loss mechanisms of radiation belt electrons are magnetopause shadowing and pitch-angle scattering to the atmospheric loss cone via wave–particle interactions. Observations indicate that the electron flux in the belts can be strongly depleted in timescales of days or hours, sometimes even minutes. Also Coulomb collisions cause pitch-angle scattering, but they are much less efficient. For example, the lifetimes against Coulomb collisions of 100-keV electrons exceed one year beyond L = 1.8 and is about 30 years at L = 5 (Abel and Thorne 1998).
Practically all wave modes discussed in Chap. 5 can contribute to the losses of the outer belt electrons. What is the dominant scattering mechanism, depends on the electron energy and equatorial pitch angle. To remove an electron away from the belt it needs to be close to the loss cone, which is at the equator only a few degrees. Furthermore, gyro-resonant interactions are inefficient at equatorial pitch angles close to 90∘, where Landau- and bounce-resonant processes turn out to be important in scattering the electrons to smaller pitch angles where the gyro-resonant interactions can take over. Because a single interaction in the quasi-linear domain changes the pitch angle only by a very small amount (Sect. 6.4.4), a large number of interactions are needed to change the electron’s pitch angle so much that it moves to the loss cone. Nonlinear interactions with large-amplitude waves can, however, lead to significant changes of pitch angle even in one interaction. If the electron interacts with a wave at higher latitude, where the loss cone width is much wider, it is easier to nudge it out of the belt.
6.5.1 Magnetopause Shadowing
Losses through magnetopause shadowing occur when the drift paths of electrons touch the magnetopause. Due to their large gyro radii the high-energy radiation belt electrons can cross the magnetopause even if the background plasma remains frozen-in the magnetospheric magnetic field. Figure 6.6 illustrates different factors contributing to the shadowing.
Schematics of magnetopause shadowing during nominal (left) and strongly compressed (middle) magnetosphere as well as the inflated drift shells during the main phase of magnetospheric storms (right). The picture on the left reminds that also local perturbations of the magnetopause, such as Kelvin–Helmholtz instabilities and flux transfer events, can let radiation belt electrons to escape from the magnetosphere. The blue trace indicates the drift shell of a particle that crosses the magnetopause. The figure is a simplification of a similar picture in Turner and Ukhorskiy (2020)
The nominal distance to the subsolar magnetopause during magnetospheric quiescence is about 10 R E, which is well beyond the typical radiation belt electron drift shells (Fig. 6.6, left). Local inward ripples and excursions in the magnetopause can, however, allow electrons’ drift paths to cross the magnetopause although the nominal distance would be beyond the drift shell. For example, during relatively quiet conditions Kelvin–Helmholtz vortices and/or flux transfer events at the magnetopause can cause such local inward excursions leading to losses.
During periods of large solar wind dynamic pressure the subsolar magnetopause can be compressed inside the geostationary distance (6.6 R E) as illustrated in the middle of Fig. 6.6. The compression enhances the shadowing losses as does the erosion of the magnetic field due to dayside reconnection. During the main phase of geomagnetic storms the ring current is enhanced, which leads to decrease of the equatorial magnetic field earthward of the current and on the surface of the Earth. Outside of the peak current on the equatorial plane the magnetic field is inflated. To conserve the third adiabatic invariant electrons move outward so that their drift shells enclose the same flux inside their drift paths illustrated on the right of Fig. 6.6.
As discussed in Sect. 2.6.2, the drift shell splitting due to the dayside compression of the magnetosphere shifts the electrons with large pitch angles furthest out. Consequently, such particles are lost most efficiently, which leads to butterfly-type of electron distribution function at large L. The electron energy also affects how effective shadowing losses are MeV electrons drift around the Earth in minutes (Table 2.2) and can get lost even in the case of a short time inward magnetopause excursion. The drift periods of lower-energy electrons can be hours and if the disturbance is short-lived, it can remove only a small fraction of the population.
6.5.2 Losses Caused by Whistler-Mode Waves in Plasmasphere
Interaction of electrons with whistler-mode waves scatters electrons both in energy and pitch angle. Whether this leads to acceleration or loss of radiation belt electrons depends on the shape of the particle distribution function close to the resonant velocity. It is important to keep in mind that in numerical and theoretical studies the detailed results depend on the chosen models of frequencies and WNA distributions of the wave amplitudes and on the properties of background plasma and magnetic field. For example, the frequency of the plasmaspheric hiss is less than 0.1 times the local electron gyro frequency, whereas outside the plasmapause, where the gyrofrequency is smaller, the frequencies of chorus waves are in the range 0.1 ω ce < ω < 1.0 ω ce. Thus different approximations of the dispersion equation need to be used in wave–particle interaction calculations if the complete dispersion equation is numerically too demanding to apply.
Plasmaspheric hiss plays a central role in the loss of electrons from the inner parts of the outer radiation belt and in the formation of the slot region between the inner and outer belts. Lyons et al. (1972) calculated diffusion coefficients based on, at time still relatively limited, observations. They were able to demonstrate that the core of the inner belt is not affected much by hiss-induced diffusion but its outer edge, i.e., the inner edge of the slot, is energy dependent, being closest to the Earth at highest energies, which is consistent with modern observations to be discussed more thoroughly in in Sect. 7.2 (see, Fig. 7.2).
The diffusion coefficients are proportional to the wave power, i.e., the square of the wave amplitude (6.41), and the estimated lifetimes depend on the distribution of wave power along the orbits of the particles. Lyons et al. (1972) used the amplitude of B w = 35 pT and found electron lifetimes within the slot to be 1–10 days, increasing with increasing energy up to 2 MeV. Using a smaller hiss amplitude of B w = 10 pT Abel and Thorne (1998) found electron lifetimes in the energy range 100 keV–1.5 MeV to be of the order 100 days, which they found to be consistent with several satellite observations in the outer electron belt but yet inside the plasmasphere. As shown in Fig. 5.11 the hiss amplitudes vary from a few to a few to a few tens of pT during quiet magnetospheric conditions to 100–300 pT during storms, resulting in large variations of radiation belt electron lifetimes in the plasmasphere.
Lyons et al. (1972) pointed out that to obtain correct electron lifetimes, in addition to the sum over a sufficient number of harmonic gyro-resonant (n ≠ 0) terms, it is necessary to include the Landau resonance (n = 0) in the calculation of the diffusion coefficients. This is because for large WNA, where the whistler mode turns to the magnetosonic/X-mode (Fig. 4.3), the minimum gyro-resonant velocity v ∥,res = nω ce∕k ∥ becomes larger than the velocity of the particles and the Landau resonance starts to dominate the scattering process. These conclusions have been confirmed and refined in several later investigations taking advantage of much more detailed and extensive modern observations (e.g., Ni et al. 2013; Thorne et al. 2013b, and references therein).
Ni et al. (2013) investigated the effects of gyro- and Landau-resonant terms on electron lifetimes up to ultra-relativistic energies for plasmaspheric hiss and oblique magnetosonic/X-mode waves. Figure 6.7 illustrates the diffusion coefficients. The green curves were calculated assuming quasi-parallel propagating whistler mode, the red curves with a model including a latitude-dependent WNA, to represent the observations that the WNA of the whistler mode is more oblique at higher latitudes (Chap. 5), and the blue curves indicate the diffusion caused by the magnetosonic/X-mode waves.
Examples of pitch-angle diffusion coefficients at L = 3.2. The colors correspond to different propagation directions of plasmaspheric hiss: The green curves represent a model of quasi-parallel propagation of the whistler mode, the blue highly oblique propagation in the magnetosonic/X mode. The red curves are calculated using a model where the WNA of the whistler mode increases with increasing latitude. The “bottleneck” discussed in the text is the drop in the diffusion rates between gyro-resonant interaction at smaller equatorial pitch angles and Landau interaction close to 90∘ (From Ni et al. 2013, reprinted by permission from American Geophysical Union)
A striking feature in Fig. 6.7 is the so-called “bottleneck” of very small pitch-angle diffusion coefficients between nearly-perpendicular and smaller equatorial pitch angles. It is due to that gyro-resonant scattering, which dominates at small and intermediate pitch angles, is not efficient at α eq ≈ 90∘, where the Landau resonance takes over. Thus, the bottleneck slows down their transport from very large pitch angles toward the atmospheric loss cone. This problem was already recognized by Lyons et al. (1972) who pointed out that the combined gyro- and Landau-resonant interactions are not efficient enough to scatter electrons from large to the intermediate pitch angles. This results in larger non-relativistic electron fluxes at nearly-equatorial pitch angles than observed.
According Ni et al. (2013) the drop between the gyro-resonant and Landau-resonant diffusion is energy-dependent and extends also to relativistic energies. They found that below 2 MeV (top three panels of Fig. 6.7) the inclusion of the first-order gyro and Landau resonances and quasi-parallel propagation is an equally good approximation as calculations including higher-order terms. At ultra-relativistic energies realistic latitude-dependent WNAs and higher harmonics need to be taken into account. Ni et al. (2013) noted that above 3 MeV the higher harmonics even become dominant at intermediate pitch angles. The diffusion due to the nearly-perpendicular propagating magnetosonic/X-mode waves (blue curves in Fig. 6.7) was found to be weaker at all pitch angles, being most notable for lower-energy (<1 MeV) electrons at both small (\({\lesssim }40^\circ \)) and large (≳80∘) pitch angles.
Ni et al. (2013) estimated the lifetimes of equatorially mirroring electrons to be days at 500 keV, a few tens of days at 2 MeV and more than 100 days at 5 MeV. Thus, in those rare cases where ultra-relativistic electrons get access to the slot region as a result of strong magnetospheric perturbations, they can remain trapped for weeks or months as will be discussed in Sect. 7.4.
A possible way to overcome the bottleneck is the pitch-angle scattering due to resonance with the bounce periods of the electrons (Sect. 6.1). Because the electron bounce frequencies in the plasmasphere are of the order of a few Hz (Table 2.2), which are smaller than the lowest observed hiss frequencies of a few tens of Hz, the interaction can only take place at high multiples of the (angular) bounce frequency ω = lω b. Cao et al. (2017b) calculated the diffusion coefficients including bounce terms up to l = 50 in the L-range 4–5. They found the diffusion rates to be comparable with the Landau resonance at energies below 0.5 MeV. At energies above 1 MeV the bounce resonance exceeds the Landau resonance in particular at intermediate pitch-angles ≳50∘. They concluded that the Landau and bounce resonances are critical to move electrons from α eq ≈ 90∘ to smaller pitch angles.
The different electron lifetimes due to scattering by plasmaspheric hiss are demonstrated in high-resolution observations by Van Allen Probes. Zhao et al. (2019b) studied the high-energy electron spectra beyond L ≈ 2.6 and found that instead of decreasing monotonically as a function of energy they tend to peak around 2 MeV. They called these reverse or bump-on-tail spectra similar to the familiar gentle-bump of the elementary Vlasov theory (Fig. 5.1). In the plasmasphere the energy density of relativistic particles is, however, much smaller than that of the dense and massive background plasma and the bump is too gentle to drive an instability, being a consequence rather than a driver of plasma waves. The reverse high-energy spectra form during a few days after the storm main phase when the plasmaspheric hiss scatters electrons of a few hundred keV to 1 MeV to the atmospheric loss cone followed by slow inward transport of >1-MeV electrons. Hiss scatters also >1-MeV electrons, but very slowly. Numerical simulation of the spectral evolution solving the diffusion equation (6.39) following the big (Dst min = −222 nT) Saint Patrick’s day storm on 17 March 2015 was found to reproduce the observations very well (Fig. 6.8).
Comparison of the evolution of observed and simulated spectra during 20–29 March 2015. The upper panels show the Van Allen Probe A MagEIS and REPT spectra and the lower panels the Fokker-Planck simulation of equatorially trapped electrons using a time-varying plasmaspheric hiss model. The colors indicate the time from 20 March (blue) to 29 March (red) (From Zhao et al. 2019b, reprinted by permission from Springer Nature)
Other whistler-mode waves potentially leading to pitch-angle diffusion in the plasmasphere are lightning-generated whistlers and emissions from ground-based VLF transmitters. The lightning-generated whistlers have maximum amplitudes at frequencies 3–5 kHz, which are higher than the typical plasmaspheric hiss frequencies. Meredith et al. (2009) added a model of lightning-generated whistler spectra to their diffusion calculations and found that they introduce a possible way to overcome the bottleneck by resonating with high pitch-angle ultra-relativistic electrons (2–6 MeV) and scattering them to lower pitch angles.
The strongest artificial signals propagating in the whistler mode to the inner magnetosphere arise from U.S. Naval communication transmitters at frequencies around 25 kHz. As to be discussed in Sect. 7.4.2, the emissions form a radio bubble around the Earth, which has been proposed as an explanation why ultra-relativistic electrons only seldom penetrate to the L-shells below 2.8.
6.5.3 Losses due to Chorus Waves and Electron Microbursts
Outside the plasmapause the whistler mode appears as chorus emissions. The basic wave–particle interactions with radiation belt electrons are similar to those in the plasmasphere but here the background plasma is both hotter and much more tenuous, and the frequencies of chorus waves are closer to the local electron gyro frequency than in the plasmasphere. All these factors affect the propagation characteristics of the waves and how they interact with electrons of different energies and pitch angles. When suprathermal electrons advect from the magnetospheric tail, their distribution function becomes anisotropic in the velocity space, which leads to instability and energy transfer from the electrons to the whistler-mode chorus waves (Sect. 5.2). At higher energies chorus waves are considered as efficient accelerators of electrons up to relativistic energies (Sect. 6.4.5).
The chorus emissions are also important agents of pitch-angle diffusion toward the atmospheric loss cone, because the pitch-angle diffusion coefficients are larger than the energy diffusion coefficients (Eq. (6.40)). While the scattering of \({\lesssim }100\) keV electrons is efficient near the equator, the chorus-wave losses of MeV electrons is most efficient at higher latitudes (λ ≳ 15∘), where the waves propagate increasingly obliquely (e.g., Thorne et al. 2005, and Fig. 5.5).
A specific feature of chorus waves, in particular of the lower-band chorus below 0.5 f ce, is that they are composed of short nonlinear rising-tone emissions in frequency (Sect. 5.2.4). These large-amplitude wave packets may lead to the brief electron microbursts . The microbursts were originally identified in balloon-borne observations as X-ray bremsstrahlung of ≳200-keV electrons precipitating into the atmosphere (Anderson and Milton 1964). The microbursts have later been observed with instruments on several high-altitude balloons and sounding rockets and using fast-sampling electron detectors looking upward into the atmospheric loss cone onboard spacecraft traversing the high-latitude ionosphere at low altitudes where the loss cone is wide.
The microbursts occur in timescales of milliseconds and they are observed in the energy range from a few tens of keV to several MeV. At lowest energies they are related to the generation of chorus waves, while at higher energies microbursts have been shown to be able to empty the outer radiation belt in a timescale of one day. Both quasi-linear gyro-resonant interaction with small-amplitude chorus waves and nonlinear interaction with large-amplitude chorus wave packets have been suggested to cause the very rapid pitch-angle scattering of electrons, even from a brief interaction with a single chorus wave packet (Bortnik et al. 2008a). In addition, the nonlinear Landau trapping by large-amplitude oblique whistler-mode chorus at high geomagnetic latitudes has been suggested to play a significant role in losses as they increase efficiently the parallel energy of electrons in a region where the loss cone is relatively wide (e.g., Osmane et al. 2016, and references therein).
The number of events with simultaneous high-resolution observations of large-amplitude chorus emissions and microburst precipitation in close magnetic field conjunction is limited. Mozer et al. (2018) investigated an event on 11 December 2016 when high-resolution wave data from Van Allen Probes B was available. The observed wave amplitude exceeded occasionally 1 nT, being in the nonlinear regime. The cross-correlation between 1-s averaged precipitating electron flux observed with the low-altitude AeroCube 6B microsatellite and the Van Allen Probes wave magnetic field was close to 0.9, which is an exceptionally high correlation in this context.
Mozer et al. (2018) calculated the standard bounce-averaged quasi-linear pitch-angle diffusion coefficient for an average amplitude of 100 pT. They found that the observed precipitating electron flux corresponded remarkably well to the estimated flux from quasi-linear diffusion once the data was averaged over 1 s, which extends over several periods of both chorus elements and microbursts. This result suggests that, on the average, the Fokker–Planck approach may describe quite well the pitch-angle scattering although the underlying scattering process may be nonlinear interaction with high-amplitude elements of whistler-mode chorus waves.
Another interesting conjugate event occurred on 20 January 2016. The CubeSat FIREBIRD II observed microbursts of 200-keV to 1-MeV electrons, and Van Allen Probe A detected lower band chorus of similar cadence and duration (Breneman et al. 2017). As microbursts were dispersionless, the scattering was considered to be a nonlinear first-order gyro resonance. AeroCube and FIREBIRD observations illustrate that even CubeSat-class satellites can have great scientific value.
The Japanese Arase satellite, launched in December 2016, made it possible to conduct. together with Van Allen Probes, high-resolution magnetically conjugate wave and particle observations simultaneously close to the equator and at higher magnetic latitudes. An example of conjugate observations between Arase and Van Allen Probes A on 21 August 2017 was published by Colpitts et al. (2020) (Fig. 6.9). This was the first time when the propagation of individual whistler-mode wave packets from the lower (12∘) to higher (21∘) magnetic latitude was directly observed.
Observations of the propagation of whistler-mode wave packets from magnetic latitude λ = 12∘ (Van Allen Probes A) to λ = 21∘ (Arase) over a period of 10 s. The top four panels show the Van Allen Probes A power spectral density (here PSD does not mean phase space density!) and wave normal angle (WNA), Arase PSD and WNA, all these in the frequency range 5.8–7.3 kHz, i.e., in the lower-band whistler mode (f ce∕2 ≈ 7.9 kHz). The lowest panel shows the magnetic field-aligned component of the Poynting vector calculated from Van Allen Probes A electric and magnetic field data (From Colpitts et al. 2020, reprinted by permission from American Geophysical Union)
Figure 6.9 shows a few chorus elements of the lower-band whistler mode observed by both satellites. The Poynting vector in the bottom panel indicates that wave energy was propagating at the location of Van Allen Probes A (λ = 12∘) toward the higher latitude. The dashed vertical line indicates the time when the first chorus element arrived at Van Allen Probes A. The same element arrived 0.2 s later at Arase at λ = 21∘, consistent with a ray-tracing study presented by Colpitts et al. (2020). The wave normal angle became increasingly oblique while the wave propagated to the higher latitude, thus making the wave more efficient to scatter the relativistic electrons toward the loss cone.
Due to observational limitations it is difficult to answer the question how large fraction of the total electron precipitation losses beyond the plasmapause are in the microbursts. Greeley et al. (2019) investigated their role during storm recovery phases using SAMPEX observations from 1996 to 2007. They found that the microburst losses had a high correlation with the global loss of 1–2 MeV electrons, in particular during storms driven by interplanetary coronal mass ejections (ICME), when the microbursts may even be the main loss process. The correlation was weaker for stream interaction region (SIR) driven storms. (For further discussion of the different storm drivers, see Sect. 7.3.)
6.5.4 Losses Caused by EMIC Waves
The quasi-linear pitch-angle diffusion of relativistic electrons due to whistler-mode chorus waves is a relatively slow process. On the other hand, electromagnetic ion cyclotron waves have been found to lead to enhanced electron losses at L-shells close to the plasmapause where the waves are frequently observed, in particular during storm-time conditions. Summers et al. (1998) demonstrated that EMIC waves lead to almost pure pitch-angle diffusion. Contrary to chorus, the EMIC waves are not efficient electron accelerators.
According to the resonance condition ω − k ∥ v ∥ = nω ce∕γ, in which n can be both a positive and negative integer (or zero), both right-hand and left-hand polarized waves can be in resonance with right-hand gyrating electrons. The whistler-mode resonances correspond to n ≥ 1. For the resonance with left-hand polarized EMIC waves it is, in practice, sufficient to consider the lowest order term (n = −1) only, due to much smaller wave frequency compared to the electron gyro frequency. The relative direction of the wave propagation and the parallel electron velocity must, of course, be such that in the electron’s guiding center frame the wave rotates in the same sense as the electron. Furthermore, the resonance condition requires that the energy of the electron is sufficiently high. Not only the Lorentz factor γ has to be large but also the parallel velocity must be large enough to Doppler shift the wave frequency close to ω ce∕γ. The minimum resonant energies are of the order of 1 MeV or larger, assuming that the plasma frequency is considerably higher than the electron gyrofrequency (ω pe∕ω ce ≥ 10) (Summers and Thorne 2003). This condition is met close to the plasmapause in the afternoon sector where the EMIC waves are frequently observed.
In numerical diffusion studies it is essential to apply an appropriate background plasma model. An example is the investigation by Jordanova et al. (2008) of the intense storm on 21 October 2001. The model included all major loss processes and was coupled with a dynamic plasmasphere model with 77% H+, 20% He+ and 3% O+. The EMIC wave amplitudes were calculated self-consistently with evolving plasma populations, resulting to He+ band amplitudes B w ≈ 5 nT at L = 4.5 and B w ≈ 10 nT at L = 6.25. The analysis was performed considering separately EMIC scattering alone, all processes except EMIC waves, and all scattering processes including EMIC waves. The highest pitch-angle diffusion coefficients for relativistic electrons were found to be in the range 0.1–5 s−1 and limited to equatorial pitch angles \({\lesssim }60^\circ \). Considering that the applied He+ band frequencies were below 1 Hz, so strong diffusion is at the limit of the quasi-linear approach. Jordanova et al. (2008) concluded that scattering by EMIC waves enhances the loss of >1-MeV electrons and can cause significant electron precipitation during the storm main phase. This conclusion has been verified observationally during the Van Allen Probes era when phase space densities have been possible to calculate with better accuracy than before (e.g., Shprits et al. 2017, and references therein).
In theoretical calculation of the resonant energy for interaction of EMIC waves with electrons both electron and ion terms in (4.63) must be retained and the derivation is a bit more complicated than the derivation of the ion resonant energy (5.19) (e.g., Summers and Thorne 2003; Meredith et al. 2003). Considering nearly parallel propagating hydrogen band EMIC waves and assuming small He+ (<10%) and O+ (<20%) concentrations Mourenas et al. (2016) derived a simplified equation for the minimum resonant energy of the electrons
where
Here the electron pitch angle and electron and proton gyro and plasma frequencies are given at equator and η p is the proton concentration (typically η p > 0.7). The main message of these equations is that the minimum resonant energy is approximately proportional to \(B/\sqrt {n_e}\) and inversely proportional to \(\cos \alpha _{eq}\) and to the wave frequency ω.
Based on CRRES observations Meredith et al. (2003) concluded that minimum energy conditions of 1–2 MeV electrons was met during about 1% of the electron drift motion around the Earth. They noted that while 1% may sound small, it actually is enough to keep diffusion significant and, at the same time, loss timescales in the range of hours to one day. If the interaction would take place within a much wider part of the drift motion, the electrons would disappear too fast compared to the observations.
During favorable conditions the electron loss may also be faster. Kurita et al. (2018) analyzed Van Allen Probes and Arase observations following each other during moderate substorm activity on 21 March 2017. They concluded that the relativistic electrons where lost in the L-shell range 4–5 in a timescale of 10 min or even faster, which corresponds to only a couple of drift periods. From the satellite and ground-based observations the EMIC wave activity was estimated to occur within a few-hour period around the magnetic midnight. It is possible that in this particular case the interaction took place during a much longer fraction of the drift path than estimated by Meredith et al. (2003).
Similar to the whistler-mode waves, the gyro-resonant scattering due to EMIC waves is limited to small and intermediate pitch angles because the minimum resonant energy increases beyond the electron energies when α eq → 90∘. The WNAs of the waves are \({\lesssim }30^\circ \), but if the wave amplitude is large enough, the parallel component of the wave electric field may be sufficient to lead to pitch-angle scattering of the electrons through the bounce resonance. The H+ band waves can fulfil the resonance condition ω = lω be at low resonant numbers in the outer radiation belt up to \(L \lesssim 6\). Cao et al. (2017a) calculated pitch-angle diffusion coefficients at energies >100 keV using oblique EMIC waves with the amplitude of 1 nT. They found that at equatorial pitch angles >80∘, where the gyro-resonant diffusion became weak, the bounce-resonant diffusion took over and exceeded 10−3 s−1 close to 90∘. At L = 3 the dominant bounce harmonic number was l = 2, whereas l = 1 dominated at L = 4 − 5.
Using typical plasma, wave and particle observations from the Van Allen Probes Blum et al. (2019) demonstrated that 50–100 keV electrons can be scattered efficiently by bounce-resonant interaction with both He+ and H+ band EMIC waves. They found pitch-angle diffusion coefficients exceeding 10−3 s−1 for electrons with equatorial pitch angles approaching 90∘.
The nonlinear interaction between EMIC waves and electrons can result in resonant pitch-angle scattering of α eq = 90∘ electrons (with v ∥ = 0) at the equator even in case of an exactly parallel propagating wave. To see this, write the electron’s equation of motion in the wave field (E w, B w) in the form (6.47). The Lorentz force due to the wave magnetic field ∝p ×B w has a component parallel to B 0. According to Eq. (6.48) the acceleration due to the nonlinear term is proportional to \(B_w\sin \eta \), where η is the phase angle between the perpendicular velocity of the electron v ⊥ and B w⊥. Because the gyro frequency is much higher than the frequency of the wave, η is highly oscillatory. For a small-amplitude wave the nonlinear term averages out rapidly and its effect is negligible causing just a small oscillation of α around 90∘. But for larger amplitudes the small oscillation may hit the bounce resonance and scatter the electron to off-equatorial motion.
Lee et al. (2020) performed test-particle simulations of 5-MeV electrons with an initial α eq = 90∘ and different phase angles η integrating Eq. (6.47) in a dipolar B 0(λ). In case of B w∕B 0 = 0.05 and wave normal angle 0∘ the diffusive effect on the pitch angle remained small (about 5∘) and constant during the length of the simulation (1600 gyro periods). Increasing the wave amplitude to B w∕B 0 = 0.1 increased the diffusive △α to about 20∘. Increasing the wave normal angle, when also the wave electric field causes a parallel force, led to rapid growth of the pitch-angle scattering. Consequently, the interaction with large-amplitude EMIC waves can contribute to the pitch-angle diffusion of equatorial and nearly equatorial ultra-relativistic electrons.
6.6 Different Acceleration and Loss Processes Displayed in Phase Space Density
It is likely that both local acceleration by whistler-mode chorus waves and ULF wave driven inward radial transport contribute to electron energization. Both wave modes can be significantly enhanced during geomagnetically active periods, but there is no one-to-one correlation between them making individual events different from each other. An important aspect is also the energy-dependence of acceleration to the highest energies. A plausible scenario is that electrons are first accelerated to MeV energies by chorus waves and then further to ultra-relativistic energies through inward transport by ULF waves (e.g., Jaynes et al. 2018; Zhao et al. 2019a).
Which one of these mechanisms is more important, and under which conditions, has, however, remained a highly controversial subject where new observations and refined computer simulations have been found to be in favor of one or the other. In cases when the phase space density (PSD, Sect. 3.5) as a function of adiabatic integrals f(μ, K, L ∗) can be determined from multisatellite observations with sufficient accuracy and wide enough coverage, its temporal evolution can be used to investigate the relative roles of the mechanisms that are fully adiabatic (e.g., the Dst effect, Sect. 2.7) and processes that break one or more adiabatic invariants.
The method is illustrated schematically in Fig. 6.10, in which the temporal evolution of PSD as a result of different processes is sketched as a function of L ∗ for a given μ. In a fully adiabatic process conserving all adiabatic invariants the PSD does not change. Radial transport and local acceleration/losses show different time evolution of the PSD.
In the case of inward radial transport alone the source is typically at large radial distances, and a wide range of energies over a wide domain in L ∗ is affected (Fig. 6.10, top left). The PSD increases with time at all drift shells and maintains its monotonous gradient ∂f∕∂L ∗ > 0 during the transport toward the Earth. Since the inward transport brings new electrons into the radiation belt region from larger distances, the PSD increases in absolute sense, as indicated by the upward arrow.
The local acceleration through wave–particle interactions, in turn, enhances the PSD within a limited radial distance leading to a temporally growing peak as a function of L ∗ (Fig. 6.10, top middle). The subsequent radial diffusion spreads the peak to both directions. Note, however, that a local peak (Fig. 6.10, top right) may also appear after the radial transport has first enhanced the PSD (time t 0 to t 1) and then magnetopause shadowing (Sect. 6.5.1) removes electrons from the outer parts of the belts (time t 1 to t 2).
Similarly, it is possible to distinguish between different loss mechanisms in the the temporal evolution of the PSD. The sketch in the bottom left of Fig. 6.10 illustrates the gradual loss due to pitch-angle scattering to the atmospheric loss cone through interaction with plasmaspheric hiss and whistler-mode chorus waves at a wide range of L-shells. The picture in the bottom middle describes outward radial diffusion and subsequent loss to the magnetopause. The time evolution of the PSD on the bottom right illustrates fast local loss by the EMIC waves.
It is important to understand that the PSD is not a magic wand. The method is constrained by the resolution and spatial coverage of the observations and, in particular, by the accuracy of the applied magnetic field model in the process to convert particle fluxes to PSD (Sect. 3.5).
The importance of wide enough L ∗ coverage was emphasized by Boyd et al. (2018) who combined Van Allen Probes and THEMIS observations. Of the 80 events they investigated only 24 featured a clear peak in the PSD as a function of L ∗ when the PSD was calculated from Van Allen Probes data alone. However, when THEMIS data from larger distances were included in the analysis, 70 of the 80 events indicated local acceleration.
Figure 6.11 shows two examples of the Boyd et al. (2018) study. The event on 13–14 January 2013 was one where the gradient was clearly positive as a function of L ∗ in the Van Allen Probes data but turned negative at larger distances and there was a clear growing peak from L ∗≈ 4.3 to at least L ∗≈ 7.5, when THEMIS data were included. The PSD from Van Allen Probes observations alone thus suggested radial transport but the wider radial coverage supported the interpretation that local acceleration was the dominant one.
Evolution of the phase space densities combined from Van Allen Probes and THEMIS observations on 13–14 January 2013 (top) and on 6–8 December 2014 (bottom). The different times are given with different colors increasing from blue to red (From Boyd et al. 2018, Creative Commons Attribution-NonCommercial-NoDerivs License)
The event on 6–8 December 2014 was different. In that case the Van Allen Probes data hint a local peak slightly earthward of the apogee of the spacecraft. However, there was no clear negative gradient in the PSD calculated from THEMIS observations.
6.7 Synergistic Effects of Different Wave Modes
In the previous sections we have mostly considered the source and loss effects of various wave modes one at a time. In reality the picture is more complicated. During its drift motion an individual electron encounters different wave environments at different MLT sectors. For example, whistler-mode chorus can accelerate the electron in the dawn side and the same electron may be scattered toward the loss cone by EMIC waves in the afternoon sector. Note also that these emissions are not strictly limited to these sectors and, as discussed in Chap. 5, may occasionally be observed at all local times, also simultaneously in the same location.
As discussed earlier, the combined effect of different wave modes on charged particles can be additive. Examples of this are processes in which a bounce resonance may first move electrons from large equatorial pitch angles to smaller pitch angles, where gyro resonance can take over, or in which the electrons are first accelerated to MeV energies by chorus waves, whereafter ULF waves may take care of the acceleration to ultra-relativistic energies.
The effect of wave modes can also be synergistic, where nonlinear interaction between the waves modifies the properties of the wave that scatters the particles either in energy or pitch angle. In particular, large-amplitude ULF waves have been found to modulate the key parameters of particle interactions with EMIC, chorus and plasmaspheric hiss emissions.
An early suggestion that ULF oscillations might modulate the electron scattering by whistler-mode waves was presented by Coroniti and Kennel (1970). They found that such modulations ought to be found in a wide range of periods 3–300 s. This corresponds to observed precipitation pulsations with periods 5–300 s in X-ray emissions and riometer absorptions caused by >30-keV electrons.
Modern observations of poloidal mode ULF oscillations with mirror-like magnetic and density oscillations were illustrated in Fig. 5.19. Xia et al. (2016) found that the ULF modulation strengthened both upper- and lower-band chorus emissions in the troughs of the magnetic fluctuation, whereas chorus waves weakened at the crests of the fluctuation. Careful analysis of electron and proton pitch-angle distributions suggested that the chorus emissions below 0.3 f ce were consistent with linear growth by enhanced low-energy electrons, whereas some, likely nonlinear, mechanism may be required to excite the chorus at higher frequencies, perhaps similar to the formation of the chirping emissions discussed in Sect. 5.2.4.
The mirror-type appearance can also affect the minimum resonant energy of electrons with EMIC waves. According to (6.51) the minimum resonant energy depends on the background magnetic field and plasma density as
Consequently, during the half period of the ULF wave, when the oscillation reduces the magnetic field and enhances the plasma density, W res,min is reduced from the constant background level. In order this to be effective requires, of course, that the amplitude of the modulating wave is large enough.
Using THEMIS (in 2007–2011) and Van Allen Probes (in 2012–2015) observations Zhang et al. (2019) investigated in total 167 large-amplitude ULF wave events colocated with hydrogen-band EMIC waves in the L-shell range 4–7. The events were found at all MLTs, mostly in the evening sector. The magnetic fluctuations were in the range 0.01 < △B∕B < 2 and the density fluctuation was in most cases even larger. Under the background conditions 5 < ω pe0∕ω ce0 < 25 the average fluctuation ratio (△n e∕n e)∕(△B 2∕B 2) was in the range 1–3.
Theoretically these levels of ULF fluctuations could reduce W res,min up to about 30% from the constant background level. Thus assuming that the background plasma conditions without the ULF fluctuation would suggest minimum resonant energy of 1 MeV, the presence of such fluctuations would reduce it to 0.7 MeV. Zhang et al. (2019) also noted that their event selection criteria limited the frequency range of the ULF waves to higher than 5 mHz in THEMIS observations and higher than 10 mHz in Van Allen Probes observations, thus missing the lowest-frequency end of Pc5 waves where △B∕B can be expected to be larger facilitating scattering of electrons of even smaller energy.
How important this mechanism is to the loss of sub-MeV electrons remains unclear. The occurrence rate of the events with simultaneous EMIC and ULF oscillations studied by Zhang et al. (2019) was not very large. It peaked at L-shells 5.5–6, where it was (3 ± 1) × 10−3.
Also plasmaspheric hiss has been found to be modulated by ULF fluctuations. Breneman et al. (2015) investigated global-scale coherent modulation of the electron loss from the plasmasphere using Van Allen Probes hiss observations, balloon-borne X-ray counts due to precipitating electrons in the energy range 10–200 keV, and ground-based ULF observations. The intensity of the hiss emission was modulated by the ULF oscillation and there was an excellent correspondence between hiss intensity and electron precipitation during the two in detail analyzed events on 3 and 6 January 2014. Thus the global-scale forcing of the plasmasphere by ULF waves can lead to enhanced hiss emission and consequent scattering of plasmaspheric electrons to the atmospheric loss cone.
Simms et al. (2018) performed an extensive statistical analysis of the effects of ULF Pc5, lower-band VLF chorus, and EMIC waves on relativistic and ultra-relativistic electron fluxes in four energy bands (0.7–1.8 MeV, 1.8–4.5 MeV, 3.5–6.0 MeV and 6.0–7.8 MeV) observed at geostationary orbit during 2005–2009. They used autoregressive models where the daily averaged fluxes were correlated with the fluxes and wave proxies observed in the previous day. The models were constructed separately for different pairs of the wave modes including both linear and quadratic terms of each wave, and a cross-term of the waves to represent the synergistic effects.
The regression coefficients contain a lot of information about linear and nonlinear influences of the individual modes and of their mutual interactions, thorough discussion of which is beyond the scope of this book. Here we focus on the synergetic effects suggested by the analysis.
The influence of Pc5 waves was found to be largest at midrange power and decreased due to the negative effect of the nonlinear term, and this was more pronounced when Pc5 waves were paired with VLF chorus. The synergistic interaction of Pc5 and chorus emissions was found to mutually increase their effects being statistically significant at higher energies. This is consistent with the idea that both Pc5 and chorus waves contribute to the electron acceleration to relativistic and ultra-relativistic energies and that the combined effect is not only additive but also synergistic. Simms et al. (2018) suggested that the nonlinearity of the Pc5 influence may be responsible for different conclusions found in different studies of its effectiveness relative to VLF chorus.
The EMIC waves had a negative effect on electron fluxes, in particular, at the highest energy ranges. This is consistent with the fact that the electron energy must exceed the minimum resonant energy. The negative effect was enhanced when both the EMIC waves and either Pc5 or chorus waves were at high levels, and again there was clear indication of synergistic interaction.
6.8 Summary of Wave-Driven Sources and Losses
Table 6.3 summarizes the sources and primary regions of occurrence (MLT, L-shell and latitude) of the wave modes presented in the previous sections as well as the resonances of radiation belt electrons with different wave modes.
Because radiation belt electron gyro frequencies are higher than the wave frequencies, the electron energies and the Doppler-shift of the wave frequency k ∥ v ∥ must be large enough to fulfil the resonance condition. Near the equator most of the relevant wave modes (hiss, chorus, EMIC) propagate indeed quasi-parallel to the background magnetic field with small wave normal angles (Table 4.2). The requirement for high v ∥ implies that resonance occurs mostly from small to intermediate pitch angles (up to 60–70∘).
Chorus waves can resonate with a wide range of equatorial pitch angles of the electrons (up to nearly 90∘) and over wide range of energies due to their wide range of frequencies. The gyro resonances with lower-frequency hiss and EMIC waves are, in turn, limited to lower/intermediate pitch angles and to the highest energies. Landau and bounce resonances may work from intermediate to 90∘ pitch angles.
Additionally, nonlinear interactions with large-amplitude waves can lead to rapid acceleration and scattering losses. The waves often propagate obliquely, typically at high-latitudes where they can most efficiently scatter electrons close to the loss cone.
Recall that the relation of wave frequency and parallel wave number depends on the dispersion equation. For example, Eq. (5.11) indicates that the resonant energy of whistler-mode waves is inversely proportional to the wave frequency.
It is, in general, a highly important but complex question on what timescales electrons are lost from the belts due to wave–particle interactions. But on the other hand, observed loss timescales can give insight to the scattering wave mode. Particularly interesting are the cases where the whole high-energy belt population is lost as fast as in ten minutes at low L-shells where the magnetopause shadowing is unlikely to occur. Hiss, chorus and magnetosonic waves all scatter relativistic electrons, but in timescales from a day to months. Even in case of nonlinear and strong interaction with large-amplitude waves the effect to the whole population is expected to take time. The most plausible cause for the fast wave–particle scattering are EMIC waves. Another possible cause for fast depletions is magnetopause shadowing and drift shell splitting.
Notes
- 1.
Phase mixing is analogous to the hiding of the initial perturbation in the ballistic term in the Landau damping process (Sect. 4.2.3).
- 2.
Recall that the bounce time is well-defined and finite also for “equatorially mirroring” particles, as discussed in Sect. 2.4.2.
- 3.
Recall that “slowly” refers to slow compared to the wave oscillation.
- 4.
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Koskinen, H.E.J., Kilpua, E.K.J. (2022). Particle Source and Loss Processes. In: Physics of Earth’s Radiation Belts. Astronomy and Astrophysics Library. Springer, Cham. https://doi.org/10.1007/978-3-030-82167-8_6
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