Radiation Belts and Their Environment

Abstract

The Van Allen radiation belts of high-energy electrons and ions, mostly protons, are embedded in the Earth’s inner magnetosphere where the geomagnetic field is close to that of a magnetic dipole. Understanding of the belts requires a thorough knowledge of the inner magnetosphere and its dynamics, the coupling of the solar wind to the magnetosphere, and wave–particle interactions in different temporal and spatial scales. In this introductory chapter we briefly describe the basic structure of the inner magnetosphere, its different plasma regions and the basics of magnetospheric activity.

The Van Allen radiation belts of high-energy electrons and ions, mostly protons, are embedded in the Earth’s inner magnetosphere where the geomagnetic field is close to that of a magnetic dipole. Understanding of the belts requires a thorough knowledge of the inner magnetosphere and its dynamics, the coupling of the solar wind to the magnetosphere, and wave–particle interactions in different temporal and spatial scales. In this introductory chapter we briefly describe the basic structure of the inner magnetosphere, its different plasma regions and the basics of magnetospheric activity.

1.1 The Overall View to the Belts

The discovery of radiation belts dates back to the dawn of the space age when the knowledge of the physical properties of the magnetosphere was still in its infancy. In February 1958 the first U.S. satellite Explorer I¹ carried a Geiger–Müller instrument that was designed to measure cosmic radiation. It indeed did so until the spacecraft reached the altitude of about 700 km when the instrument mysteriously fell silent. The observations from Explorer III confirmed Explorer I observations only a month later. In their seminal paper James Van Allen and his co-workers (Van Allen et al. 1958) suggested that the instrument was saturated due to high-intensity corpuscular radiation trapped in the Earth’s magnetic field.

In December 1958 Pioneer III ventured further into space and understanding of the basic structure of inner and outer radiation belts started to evolve. It soon became clear that a population of multi-MeV protons, up to 1–2 GeV, dominates the ion radiation at equatorial geocentric distances of about 1.1 − 3 R _E (R _E ≃ 6370 km is the radius of Earth ).² The high-energy electrons exhibit a two-belt structure with a slot region in between (Fig. 1.1). The inner electron belt is partially co-located with the proton belt at equatorial distances of about 1.1 − 2 R _E. The outer belt is beyond about 3 R _E extending to distances of 7 − 10 R _E with electron energies from tens of keV to several MeV. Sometimes the outer belt exhibits two or even three spatially distinct parts. As the proton mass is 931 MeV c⁻² and the electron mass 511 keV c⁻², the highest-energy inner belt protons and the outer belt electrons are relativistic moving at almost the speed of light.

Fig. 1.1

A sketch of showing the inner and outer electron belts and a slot region in between embedded in the dipolar magnetic field of the Earth. The inner belt is within 2 R _E form the center of the Earth. The figure illustrates the structure when the outer belt is split to two spatially distinct domains as observed by the Van Allen Probes. (Image credits: NASA’s Goddard Space Flight Center and Grant Stevens, Rob Barnes and Sasha Ukhorskiy of the Applied Physics Laboratory of the Johns Hopkins University)

Since the early space age, the radiation belts have been investigated using a large number of satellites.³ The observations now cover more than five solar cycles and have revealed the extremely complex and highly variable structure of the belts. Based on these observations and theoretical reasoning great number of different numerical models of radiation belts have been constructed not only for scientific purposes but also to meet the needs of spacecraft engineers and space mission planners. As our focus is on the physical processes, we will not go into the details of these models. An interested reader can find the models with their descriptions at several web-sites, e.g., the Community Coordinated Modeling Center (CCMC)⁴ and the Space Environment Information System (SPENVIS)⁵ It is evident that the observations during the Van Allen Probes era—many of which are discussed in this book—and the subsequent modeling efforts will lead to important revisions and refinements of the models.

Although the fluxes of the highest-energy particles and their energy densities are considerably lower than those of the background plasma in the inner magnetosphere, they are of a significant concern due to their space weather effects, both posing risks to spacecraft and humans in orbit and affecting the upper atmosphere through energetic electron and proton precipitation. The energization of radiation belt particles is an interesting fundamental plasma physical process and much emphasis has been placed on understanding the dynamics of relativistic and ultra-relativistic populations.

The inner belt, in particular the proton population, is relatively stable, whereas the outer electron belt is in continuous change. The high-energy electron fluxes can change several orders of magnitude within minutes: the outer belt may suddenly become almost completely depleted of, or get abruptly filled with, relativistic electrons. Most activity occurs in “the heart of the outer belt”, at equatorial distances of about 4 − 5 R _E. While the Van Allen Probes mission has shown that there is an almost impenetrable inner edge of the outer belt ultra-relativistic (≳ 4 MeV) electrons at an equatorial distance of 2.8 R _E, there have been a few observed events when the slot region was filled with ultra-relativistic electrons and the electrons remained trapped in the region up to several months.

The highly variable configuration and complex dynamics of the outer belt owe to the continuous changes in the plasma and geomagnetic field conditions driven by variable properties of the solar wind caused, in particular, by coronal mass ejections, stream interaction regions, and fast solar wind flows carrying Alfvénic fluctuations. Locally the kinetic response to particle injections from nightside magnetosphere affect the thermodynamic properties of the radiation belt electrons.

The radiation belts overlap with different plasma domains of the inner magnetosphere: the ring current, the plasmasphere and the plasma sheet, whose properties and locations vary in time. In particular the boundary of the plasmasphere, moving between equatorial distances of 3 and 5 R _E as a response to the solar wind driving, is a critical region to the dynamics of the outer radiation belt.

The inner magnetospheric plasma exhibits complex wave activity transferring energy and momentum between different plasma populations. The waves are known to scatter and energize the electrons depending on the particle energy, wave amplitude and the direction of wave propagation. While much of elementary space plasma theory has been developed under the approximation of linear perturbations, in the case of observed large-amplitude waves nonlinear effects need to be considered. Furthermore, the plasma and magnetic environment of the belts is not spatially symmetric, but varies as function of local time sector and geomagnetic latitude, and of course, temporally.

1.2 Earth’s Magnetic Environment

In the first approximation the Earth’s magnetic field is that of a magnetic dipole. The dipole axis is tilted 11^∘ from the direction of the Earth’s rotation axis. The current circuit giving rise to the magnetic field is located in the liquid core about 1200–3400 km from the center of the planet. The current system is asymmetric displacing the dipole moment from the center, which together with inhomogeneous distribution of magnetic matter above the core gives rise to large deviations from the dipole field on the surface. The pure dipole field on the surface would be 30 μT at the dipole equator and 60 μT at the poles. However, the actual surface field exceeds 66 μT in the region between Australia and Antarctica and is weakest, about 22 μT, in a region called South Atlantic Anomaly (SAA) . The magnetic poles migrate slowly, and the SAA has during the past decades moved slowly from Africa toward South America being presently deepest in Paraguay. The SAA has a specific practical interest, as the inner radiation belt reaches down to low Earth orbiting (LEO) satellites at altitudes of 700–800 km above the anomaly.

1.2.1 The Dipole Field

Knowledge of the charged particle motion in the dipole field is essential in studies of radiation belts. In the main radiation belt domain at geocentric distances 2–7 R _E the dipole field is a good first approximation for the quiet state of the magnetic field. In reality, the dipole field is an idealization where the source current is assumed to be confined into a point at the origin. The source of planetary and stellar dipoles is a finite, actually a large, current system within the celestial body. Such fields, including the Terrestrial magnetic field, are customarily represented as a multipole expansion: dipole, quadrupole, octupole, etc. When moving away from the source, the higher multipoles vanish faster than the dipole making the dipole field a good starting point to consider the motion of charged particles in radiation belts. In the dipole field charged particles behave adiabatically as long as their gyro radii are smaller than the gradient scale length of the field (Chap. 2) and their orbits are not disturbed by collisions or time-varying electromagnetic field.

For the geomagnetic field it is customary to define the spherical coordinates in a special way. The dipole moment (m _E) is in the origin and points approximately toward geographic south, tilted 11^∘ as mentioned above. Similar to the geographic coordinates the latitude (λ) is zero at the dipole equator and increases toward the north, whereas the latitudes in the southern hemisphere are negative. The longitude (ϕ) increases toward the east from a given reference longitude. In magnetospheric physics the longitude is often given as the magnetic local time (MLT) . In the dipole approximation MLT is determined by the flare angle between two planes: the dipole meridional plane containing the subsolar point on the Earth’s surface, and the dipole meridional plane which contains a given point on the surface, i.e., the local dipole meridian. Magnetic noon (MLT = 12 h) points toward the Sun, midnight (MLT = 24 h) anti-sunward. Magnetic dawn (MLT = 6 h) is approximately in the direction of the Earth’s orbit around the Sun.⁶ The abbreviation h (for hour) is often dropped and fractional MLTs are given by decimals instead of minutes and seconds.

The SI-unit of m _E is A m². In the radiation belt context it is convenient to replace m _E by k ₀ = μ ₀ m _E∕4π, which is also customarily called dipole moment. The strength of the terrestrial dipole moment varies slowly. For our discussion a sufficiently accurate approximation is

$$\displaystyle \begin{aligned} \begin{array}{lcll} m_E &=& 8\times 10^{22}\,\mathrm{A}\,\mathrm{m}^2 & \ \\ k_0 &=& 8\times 10^{15}\,\mathrm{Wb}\,\mathrm{m} & (\mathrm{SI}: \mathrm{Wb} = \mathrm{T}\,\mathrm{m}^2)\\ \ &=& 8\times 10^{25}\,\mathrm{G}\,\mathrm{cm}^3 & (\mbox{Gaussian units},\ \ 1\,\mathrm{G} =10^{-4}\,\mathrm{T}) \\ \ &=& 0.3\,\mathrm{G}\,R_E^3 & (R_E\simeq 6370\,\mathrm{km}) \end{array} \end{aligned}$$

The last expression is convenient in practice because the dipole field on the surface of the Earth (at 1 R _E) varies in the range 0.3–0.6 G.

Outside its source, the dipole field is a curl-free potential field B = −∇Ψ , where the scalar potential is given by

$$\displaystyle \begin{aligned} \varPsi=-{\mathbf{k}}_0\cdot\nabla{1\over r}=-k_0{\sin\lambda\over r^2}\,, \end{aligned} $$

(1.1)

yielding

$$\displaystyle \begin{aligned} \mathbf{B}={1\over r^3}\lbrack 3({\mathbf{k}}_0\cdot{\mathbf{e}}_r){\mathbf{e}}_r-{\mathbf{k}}_0\rbrack\,. \end{aligned} $$

(1.2)

The components of the magnetic field are

$$\displaystyle \begin{aligned} \begin{array}{rcl} B_ r & =&\displaystyle -{2k_0\over r^3}\sin\lambda \\ B_\lambda & =&\displaystyle {k_0\over r^3}\cos\lambda \\ B_\phi & =&\displaystyle 0 \end{array} \end{aligned} $$

(1.3)

and its magnitude is

$$\displaystyle \begin{aligned} B = {k_0\over r^3}(1 + 3\sin^2\lambda)^{1/2}\,. \end{aligned} $$

(1.4)

The equation of a magnetic field line is

$$\displaystyle \begin{aligned} r = r_0\cos^2\lambda\,, \end{aligned} $$

(1.5)

where r ₀ is the distance where the field line crosses the equator. The length element of the magnetic field line element is

$$\displaystyle \begin{aligned} \mathrm{d} s = (\mathrm{d} r^2 + r^2\mathrm{d}\lambda^2)^{1/2} = r_0\cos\lambda(1 + 3\sin^2\lambda)^{1/2} \mathrm{d}\lambda\,. \end{aligned} $$

(1.6)

This can be integrated in a closed form, yielding the length of the dipole field line S _d as a function of r ₀

$$\displaystyle \begin{aligned} S_d \approx 2.7603\,r_0\,. \end{aligned} $$

(1.7)

The curvature radius R _C = |d² r∕ds ²|⁻¹ of the magnetic field is an important parameter for the motion of charged particles. For the dipole field the radius of curvature is

$$\displaystyle \begin{aligned} R_C(\lambda) = {r_0\over 3}\cos\lambda{(1+3\sin^2\lambda)^{3/2}\over 2-\cos^2\lambda}\,. \end{aligned} $$

(1.8)

Any dipole field line is determined by its (constant) longitude ϕ ₀ and the distance where the field line crosses the dipole equator. This distance is often given in terms of the L-parameter

$$\displaystyle \begin{aligned} L = r_0/R_E\,. \end{aligned} $$

(1.9)

The parameter was introduced in the early days of Explorer data analysis by Carl E. McIlwain to organize the observations in magnetic field-related coordinates. Consequently, L is known as McIlwain’s L-parameter.

For a given L the corresponding field line reaches the surface of the Earth at the (dipole) latitude

$$\displaystyle \begin{aligned} \lambda_e = \arccos{1\over\sqrt L}\,. \end{aligned} $$

(1.10)

For example, L = 2 (the inner belt) intersects the surface at λ _e = 45^∘ , L = 4 (the heart of the outer belt) at λ _e = 60^∘ and L = 6.6 (the geostationary orbit)⁷ at λ _e = 67.1^∘.

The dipole field line length in (1.7) was calculated from the dipole itself. Now we can calculate also the dipole field line length from a point on the surface to the surface on the opposite hemisphere to be

$$\displaystyle \begin{aligned} S_e \approx (2.7755\times L - 2.1747)\,R_E\,, \end{aligned} $$

(1.11)

which is a good approximation when L ≳ 2.

The field magnitude along a given field line as a function of latitude is

$$\displaystyle \begin{aligned} B(\lambda) = \lbrack B_r(\lambda)^2 + B_\lambda(\lambda)^2\rbrack^{1/2} = {k_0\over r_0^3}{(1 + 3\sin^2\lambda)^{1/2}\over\cos^6\lambda}\,.{} \end{aligned} $$

(1.12)

For the Earth

$$\displaystyle \begin{aligned} {k_0\over r_0^3} = {0.3\over L^3}\,\mathrm{G} = {3\times 10^{-5}\over L^3}\,\mathrm{T}\,. \end{aligned} $$

(1.13)

At the magnetic equator on the surface of the Earth, the dipole field is 0.3 G (30 μT), at the poles 0.6 G (60 μT).

The actual geomagnetic field has considerable deviations from the dipolar field because the dipole is not quite in the center of the Earth, the source is not a point, and the electric conductivity of the Earth is not uniform. The geomagnetic field is described by the International Geomagnetic Reference Field (IGRF) model, which is regularly updated to reflect the slow secular variations of the field, i.e., changes in timescales of years or longer (Fig. 1.2).

Fig. 1.2

The magnetic field magnitude on the surface of the Earth according to the 13th generation IGRF model released in December 2019. The South Atlantic Anomaly is the deep blue region extending from the southern tip of Africa to South America. The model is available at National Centers for Environmental Information (NCEI, https://www.ncei.noaa.gov)

1.2.2 Deviations from the Dipole Field due to Magnetospheric Current Systems

The Earth’s magnetosphere is the region where the near-Earth magnetic field controls the motion of charged particles. It is formed by the interaction between the geodipole and the solar wind. The deformation of the field, caused by the variable solar wind pressure, sets up time-dependent magnetospheric current systems that dominate deviations from the dipole field in the outer radiation belt and beyond.

The solar wind plasma cannot easily penetrate to the Earth’s magnetic field and the outer magnetosphere is essentially a cavity around which the solar wind flows. The cavity is bounded by a flow discontinuity called the magnetopause. The shape and location of the magnetopause is determined by the balance between the solar wind dynamic plasma pressure and the magnetospheric magnetic field pressure. The nose, or apex, of the magnetopause is, under average solar wind conditions, at the distance of about 10 R _E from the center of the Earth but can be pushed to the vicinity of the geostationary distance (6.6 R _E) during periods of large solar wind pressure, which has important consequences to the dynamics of the outer radiation belt. In the dayside the dipole field is compressed toward the Earth, whereas in the nightside the field is stretched to form a long magnetotail. The deviations from the curl-free dipole field correspond to electric current systems according to Ampère’s law J = ∇×B∕μ ₀ .

In the frame of reference of the Earth the solar wind is supersonic, or actually super-magnetosonic, exceeding the local magnetosonic speed $v_{ms} = \sqrt {v_s + v_A}$, where v _s is the sound speed, $v_A = B/\sqrt {\mu _0 \rho _m}$ the Alfvén speed and ρ _m the mass density of the solar wind. Because fluid-scale perturbations cannot propagate faster than v _ms, this leads to a formation of a collisionless shock front, called the bow shock , upstream of the magnetosphere. Under typical solar wind conditions the apex of the shock in the solar direction is about 3 R _E upstream of the magnetopause. The shock converts a considerable fraction of solar wind kinetic energy to heat and electromagnetic energy. The irregular shocked flow region between the bow shock and the magnetopause is called the magnetosheath.

The current system on the dayside magnetopause shielding the Earth’s magnetic field from the solar wind is known as the Chapman–Ferraro current , recognizing the early attempt of Chapman and Ferraro (1931) to explain how magnetic storms would be driven by corpuscular radiation from the Sun. In the first approximation the Chapman–Ferraro current density J _CF can be expressed as

$$\displaystyle \begin{aligned} {\mathbf{J}}_{CF} = {{\mathbf{B}}_{MS}\over B_{MS}^2}\times\nabla P_{dyn}\,, \end{aligned} $$

(1.14)

where B _MS is the magnetospheric magnetic field and P _dyn the dynamic pressure of the solar wind. Because the interplanetary magnetic field (IMF) at the Earth’s orbit is only a few nanoteslas, the magnetopause current must shield the magnetospheric field to almost zero just outside the current layer. Consequently, the magnetic field immediately inside the magnetopause doubles: about one half comes from the Earth’s dipole and the second half from the magnetopause current.

The Chapman–Ferraro model describes a teardrop-like closed magnetosphere that is compressed in the dayside and stretched in the nightside, but not very far. Since the 1960s spacecraft observations have shown that the nightside magnetosphere, the magnetotail , is very long, extending far beyond the orbit of the Moon. This requires a mechanism to transfer energy from the solar wind into the magnetosphere to keep up the current system that sustains the tail-like configuration.

Figure 1.3 is a sketch of the magnetosphere with the main large-scale magnetospheric current systems. The overwhelming fraction of the magnetospheric volume consists of tail lobes , connected magnetically to the polar caps in the ionized upper atmosphere, known as the ionosphere. The polar caps are bounded by auroral ovals. Consequently, in the northern lobe the magnetic field points toward the Earth, in the southern away from the Earth. To maintain the lobe structure, there must be a current sheet between the lobes where the current points from dawn to dusk. This cross-tail current is embedded within the plasma sheet (Sect. 1.3.1) and closes around the tail lobes forming the nightside part of the the magnetopause current .

Fig. 1.3

The magnetosphere and the large scale magnetospheric current systems. (Figure courtesy T. Mäkinen, from Koskinen 2011, reprinted by permission from SpringerNature)

The cusp-like configurations of weak magnetic field above the polar regions known as polar cusps do not connect magnetically to magnetic poles, but instead to the southern and northern auroral ovals at noon, because the entire magnetic flux enclosed by the ovals is connected to the tail lobes. Tailward of the cusps the Chapman–Ferraro current and the tail magnetopause current smoothly merge with each other. Figure 1.3 also illustrates the westward flowing ring current (RC) and the magnetic field-aligned currents (FAC) connecting the magnetospheric currents to the horizontal ionospheric currents in auroral regions at an altitude of about 100 km.

The magnetospheric current systems can have significant temporal variations, which makes the mathematical description of the magnetic field complicated. A common approach is to apply some of the various models developed by Nikolai Tsyganenko (for a review, see Tsyganenko 2013).⁸ Particularly popular in radiation belt studies is the model known as TS04 (Tsyganenko and Sitnov 2005).

For illustrative purposes simpler models are sometimes useful. For example, the early time-independent model of Mead (1964) reduces in the magnetic equatorial (r, ϕ) plane to

$$\displaystyle \begin{aligned} B(r,\phi) = B_E\left(\frac{R_E}{r}\right)^3 \left[1+\frac{b_1}{B_E}\left(\frac{r}{R_E}\right)^3 - \frac{b_2}{B_E}\left(\frac{r}{R_E}\right)^4\cos\phi \right]\,, \end{aligned} $$

(1.15)

where we have adopted the notation of Roederer and Zhang (2014). Here B _E is the equatorial dipole field on the surface of the Earth (approximately 30.4 μT = 30, 400 nT) and ϕ is the longitude east of midnight. The $\cos \phi$ term describes the azimuthal asymmetry due to the dayside compression and nightside stretching of the field. The coefficients b ₁ and b ₂ depend on the distance of the subsolar point of the magnetopause R _s (in units of R _E), which, in turn, depends on the upstream solar wind pressure

$$\displaystyle \begin{aligned} \begin{array}{rcl} b_1 & = &\displaystyle 25\left(\frac{10}{R_s}\right)^3\,\mathrm{nT} \\ b_2 & = &\displaystyle 2.1\left(\frac{10}{R_s}\right)^4\,\mathrm{nT}\,. \end{array} \end{aligned} $$

(1.16)

This model is fairly accurate during quiet and moderately disturbed times at geocentric distances 1.5–7 R _E.

1.2.3 Geomagnetic Activity Indices

The intensity and variations of magnetospheric and ionospheric current systems are traditionally described in terms of geomagnetic activity indices (Mayaud 1980), which are available at the International Service of Geomagnetic Indices webpages maintained by the University of Strasbourg.⁹ The indices are calculated from ground-based magnetometer measurements. The large number of useful indices illustrates the great variability of geomagnetic activity; sometimes the effects are stronger at high latitudes, sometimes at low, sometimes the background current systems are strong already before the main perturbation, etc. As different indices describe different features of magnetospheric currents, there is no one-to-one correspondence between them. The choice of a particular index depends on physical processes being investigated. Here we briefly introduce the most widely used indices for global storm levels, Dst and Kp, and for the activity at auroral latitudes, AE, which will be used later when discussing the relation of radiation belt dynamics with evolving geomagnetic activity.

The Dst index aims at measuring the intensity of the ring current. It is calculated once an hour as a weighted average of the deviation from the quiet level of the horizontal magnetic field component (H) measured at four low-latitude stations distributed around the globe. Geomagnetic storms (also known as magnetospheric storms or magnetic storms) are defined as periods of strongly negative Dst index, signalling enhanced westward the ring current. The more negative the Dst index is, the stronger is the storm. There is no canonical lower threshold for the magnetic perturbation beyond which the state of the magnetosphere is to be called a storm and identification of weak storms is often ambiguous. In this book we call storms with Dst from –50 to –100 nT moderate, from –100 to –200 nT intense, and those with Dst < −200 nT big. A similar 1-min index derived from a partly different set of six low-latitude stations (SYM–H) is also in use.

A sensitive ground-based magnetometer reacts to all magnetospheric current systems and, thus, Dst has contributions from other currents in addition to the ring current. These include the magnetopause and cross-tail currents, as well as induced currents in the ground due to rapid temporal changes of ionospheric currents. Large solar wind pressure pushes the magnetopause closer to the Earth forcing the magnetopause current to increase to be able to shield a locally stronger geomagnetic field from the solar wind. The effect is strongest on the dayside where the magnetopause current flows in the direction opposite to the ring current. The pressure corrected Dst index can be defined as

$$\displaystyle \begin{aligned} Dst^* = Dst - b{\sqrt{P_{dyn}}} + c\;, {} \end{aligned} $$

(1.17)

where P _dyn is the solar wind dynamic pressure and b and c are empirical parameters, whose exact values depend on the used statistical analysis methods, e.g., b = 7.26 nT nPa^−1∕2 and c = 11 nT as determined by O’Brien and McPherron (2000).

The contribution from the dawn-to-dusk directed tail current to the Dst index is more difficult to estimate. During strong activity the cross-tail current intensifies and moves closer to the Earth, enhancing the nightside contribution to Dst. The estimates of this effect on Dst vary in the range 25–50% (e.g., Turner et al. 2000; Alexeev et al. 1996). Furthermore, fast temporal changes in the ionospheric currents induce strong localized currents in the ground, which may contribute up to 25% to the Dst index (Langel and Estes 1985; Häkkinen et al. 2002).

Another widely used index is the planetary K index, Kp. Each magnetic observatory has its own K index and Kp is an average of K indices from 13 mid-latitude stations. It is a quasi-logarithmic range index expressed in a scale of one-thirds: 0, 0+, 1−, 1, 1+, …, 8+, 9−, 9. Kp is based on mid-latitude observations and thus more sensitive to high-latitude auroral current systems and to substorm activity than the Dst index. Kp is a 3-h index and does not reflect rapid changes in the magnetospheric currents.

The fastest variations in the current systems take place at auroral latitudes. To describe the strength of the auroral currents the auroral electrojet indices (AE) are commonly used. The standard AE index is calculated from 11 or 12 magnetometer stations located under the average auroral oval in the northern hemisphere. It is derived from the magnetic north component at each station by determining the envelope of the largest negative deviation from the quiet time background, called the AL index, and the largest positive deviation, called the AU index. The AE index itself is AE = AU − AL (all in nT). Thus AL is the measure of the strongest westward current in the auroral oval, AU is the measure of the strongest eastward current, and AE characterizes the total electrojet activity. AE, AU, AL are typically given with 1-min time resolution.

As the auroral electrojets flow at the altitude of about 100 km, their magnetic deviations on the ground are much larger than those caused by the ring current. For example, during typical substorm activations AE is in the range 200–400 nT and can during strong storms exceed 2000 nT, whereas the equatorial Dst perturbations exceed − 200 nT only during the strongest storms.

1.3 Magnetospheric Particles and Plasmas

The magnetosphere is a vast domain with a wide range of relevant physical parameters. The energies, temperatures and densities vary by several orders of magnitude and change also significantly as response to variable solar wind conditions. The inner magnetosphere consists of three main particle domains; the cold and relatively dense plasmasphere, the more energetic ring current and the high-energy radiation belts. They are not spatially distinct regions, but partially overlap and their mutual interactions are critical to the physics of radiation belts. The plasma sheet in the outer magnetosphere acts as the source of suprathermal particles that are injected into the inner magnetosphere during periods of magnetospheric activity.

This introductory discussion remains at a very general level. We introduce the details of individual particle motion in Chap. 2 and the basic plasma concepts in Chap. 3.

1.3.1 Outer Magnetosphere

The outer magnetosphere can be considered to begin at distances of about 7–8 R _E where the nightside magnetic field becomes increasingly stretched. Table 1.1 summarizes typical plasma parameters in the mid-tail region, at about X = −20 R _E from the Earth. Here X is the Earth-centered coordinate along the Earth–Sun line, positive toward the Sun. The tail lobes are almost empty, particle number densities being of the order of 0.01 cm⁻³. The central plasma sheet where the cross-tail current is embedded (Fig. 1.3) is, in turn, a region of hot high-density plasma. It is surrounded by the plasma sheet boundary layer with density and temperature intermediate to values in the central plasma sheet and tail lobes. The field lines of the boundary layer connect to the poleward edge of the auroral oval. The actual numbers differ considerably from the typical values under changing solar wind conditions and, in particular, during strong magnetospheric disturbances.

Table 1.1 Typical values of plasma parameters in the mid-tail. Plasma beta (β) is the ratio between kinetic and magnetic pressures (Eq. 3.28)

Table 1.1 also includes typical parameters in the magnetosheath at the same X-coordinate. The magnetosheath consists of solar wind plasma that has been compressed and heated by the Earth’s bow shock. It has higher density and lower temperature than observed in the outer magnetosphere. Typical densities of the unperturbed solar wind at 1 AU extend from about 3 cm⁻³ in the fast (∼ 750 km s⁻¹) to about 10 cm⁻³ in the slow (∼ 350 km s⁻¹) solar wind, again with large deviations. Table 1.1 shows that, while the magnetic field magnitude is rather similar in all regions shown, plasma beta (the ratio between the kinetic and magnetic field pressures), is a useful parameter to distinguish between different regions.

1.3.2 Inner Magnetosphere

The inner magnetosphere is the region where the magnetic field is quasi-dipolar. It is populated by different spatially overlapping particle species with different origins and widely different energies: the ring current, the radiation belts and the plasmasphere. The ring current and radiation belts consist mainly of trapped particles in the quasi-dipolar field drifting due to magnetic field gradient and curvature effects around the Earth, whereas the motion and spatial extent of plasmaspheric plasma is mostly influenced by the corotation and convection electric fields (Chap. 2).

The ring current arises from the azimuthal drift of energetic charged particles around the Earth; positively charged particles drifting toward the west and electrons toward the east. Basically all drifting particles contribute to the ring current. The drift currents are proportional to the energy density of the particles and the main ring current carriers are positive ions in the energy range 10–200 keV, whose fluxes are much larger than those of the higher-energy radiation belt particles. The ring current flows at geocentric distances 3–8 R _E, and peaks at about 3–4 R _E. At the earthward edge of the ring current the negative pressure gradient introduces a local eastward diamagnetic current, but the net current remains westward.

During magnetospheric activity the role of the ionosphere as the plasma source of ring current enhances, increasing the relative abundance of oxygen (O⁺) and helium (He⁺) ions in the magnetosphere (to be discussed in Sect. 6.3.1). As a result a significant fraction of ring current can at times be carried by oxygen ions of atmospheric origin. The heavy-ion content furthermore modifies the properties of plasma waves in the inner magnetosphere, which has consequences on the wave–particle interactions with the radiation belt electrons, as will be discussed from Chap. 4 onward.

The plasmasphere is the innermost part of the magnetosphere. It consists of cold (∼1 eV) and dense (≳ 10³ cm⁻³) plasma of ionospheric origin. The existence of the plasmasphere was already known before the spaceflight era based on the propagation characteristics of lightning-generated and man-made very low-frequency (VLF) waves. The plasmasphere has a relatively clear outer edge, the plasmapause , where the proton density drops several orders of magnitude. The location and structure of the plasmapause vary considerably as a function of magnetic activity (Fig. 1.4). During magnetospheric quiescence the density decreases smoothly at distances from 4–6 R _E, whereas during strong activity the plasmapause is steeper and pushed closer to the Earth.

Fig. 1.4

Plasma density in the night sector organized by the activity index Kp. Kp < 1 +  corresponds to a very quiet magnetosphere, whereas Kp = 4 − 5 indicates a significant activity level, although not yet a big magnetic storm. The L-shell is defined in Sect. 2.6. It corresponds to the magnetic field lines of a given L-parameter. (Adapted from Chappell (1972), reprinted by permission from American Geophysical Union)

The location of the plasmapause is determined by the interplay between the sunward convection of plasma sheet particles and the plasmaspheric plasma corotating with the Earth. In Sect. 2.3 we add the convective and corotational electric fields to the guiding center motion of charged particles and find that an outward bulge called plasmaspheric plume develops on the duskside around 18 h magnetic local time (MLT). Plasmaspheric plumes are most common and pronounced during geomagnetic storms and substorms, but they can exist also during quiet conditions (e.g., Moldwin et al. 2016, and references therein). During geomagnetic storms the plume can expand out to geostationary orbit and bend toward earlier MLT.

Figure 1.5 shows global observations of the plasmasphere taken by the EUV instrument onboard the IMAGE satellite before and after a moderate geomagnetic storm in June 2000. Before the storm the plasmasphere was more or less symmetric. After the storm the plasmasphere was significantly eroded leaving a plume extending from the dusk toward the dayside magnetopause. When traversing the plume. the trapped radiation belt electrons, otherwise outside the plasmapause, encounter a colder and higher-density plasma with plasma wave environment similar to the plasmasphere proper. Consequently, the influence of the plasmasphere on radiation belt particles extends beyond its nominal boundary depicted in Fig. 1.4.

Fig. 1.5

Plasmapheric plume and plasmaspheric erosion as observed by the IMAGE EUV instrument. The picture is taken from above the northern hemisphere and the Sun is to the right. (Figure courtesy: Jerry Goldstein, Southwest Research Institute, for more information of this particular storm see Goldstein et al. 2004)

The plasma parameters in the plasmasphere, in the plume and at the plasmapause are critical to the generation and propagation of plasma waves that, in turn, interact with the energetic particles in the ring current and radiation belts. Thus, the coldest and the hottest components of the inner magnetosphere are intimately coupled to each other through wave–particle interactions.

1.3.3 Cosmic Rays

In addition to ion and electron radiation belts another important component of corpuscular radiation in the near-Earth space consists of cosmic rays. The kinetic energies of a large fraction of cosmic ray particles are so large that the geomagnetic field cannot trap them. Instead, the particles traverse through the Earth’s magnetosphere without much deflection of their trajectories. Some of them hit the atmosphere interacting with nuclei of atmospheric atoms and molecules causing showers of elementary particles being possible to detect on ground. Those with highest energies can penetrate all the way to the ground.

The spectrum of cosmic ray ions at energies below about 10¹⁵ eV per nucleon in the near-Earth space has three main components:

• Galactic cosmic rays (GCR), whose spectrum peaks at energies above 100 MeV per nucleon, are most likely accelerated by supernova remnant shock waves in our galaxy.

• Solar cosmic rays (SCR) are accelerated by coronal and interplanetary shocks related to solar eruptions. Their energies are mostly below 100 MeV per nucleon and a fraction of them can become trapped in the inner radiation belt.

• Anomalous cosmic rays (ACR) are ions of solar origin captured and accelerated by the heliospheric termination shock, where the supersonic solar wind becomes subsonic before encountering the interstellar plasma, or in the heliosheath outside the heliopause. Some of the ions are injected back toward the Sun. Near the Earth the ACR spectrum peaks at about 10 MeV per nucleon and thus the particles can become trapped in the geomagnetic field.

Although the galactic cosmic rays cannot directly be trapped into the radiation belts, they contribute indirectly to the inner belt composition through the Cosmic Ray Albedo Neutron Decay (GRAND) mechanism. The cosmic ray bombardment of the atmosphere produces neutrons that move in all directions. Although the average neutron lifetime is 14 min 38 s, during which a multi-MeV neutron either hits the Earth or escapes far away from the magnetosphere, a small fraction of them decay to protons while still in the inner magnetosphere and may become trapped in the inner radiation belt (to be discussed in Sect. 6.3.3).

Below about 10 GeV GCR and ACR fluxes are modulated by the 11- and 22-year solar cycles, so they provide quasi-stationary background radiation in the timescales of radiation belt observations. The arrivals of SCRs are, in turn, transient phenomena related to solar flares and coronal mass ejections.

The cosmic ray electrons also have galactic and solar components. Furthermore, the magnetosphere of Jupiter accelerates high-energy electrons escaping to the interplanetary space. These Jovian electrons can be observed near the Earth at intervals of about 13 months when the Earth and Jupiter are connected by the IMF.

Supernova shock waves are the most likely sources of the accelerated GCR electrons, whereas in the acceleration of SCR and Jovian electrons also other mechanisms besides shock acceleration are important, in particular inductive electric fields associated with magnetic reconnection in solar flares and the Jovian magnetosphere.

The acceleration and identity of the observed very highest-energy cosmic rays up to about 3 × 10²⁰ eV remain enigmatic. It should not be possible to observe protons with energies higher than 6 × 10¹⁹ eV, known as the Greisen–Zatsepin–Kuzmin cut-off, unless they are accelerated not too far from the observing site. Above the cut-off the interaction of protons with the blue-shifted cosmic microwave background produces pions that carry away the excessive energy. It is possible that the highest-energy particles are nuclei of heavier elements. This is, for the time being, an open question.

1.4 Magnetospheric Dynamics

Strong solar wind forcing drives storms and more intermittent substorms in the magnetosphere. Both are critical dynamical elements in the temporal and spatial evolution of the radiation belts. They are primarily caused by various large-scale heliospheric structures such as interplanetary counterparts of coronal mass ejections (CMEs/ICMEs),¹⁰ stream interaction regions (SIRs) of slow and fast solar wind flows, and fast solar wind supporting Alfvénic fluctuations (to be discussed more in detail in Sect. 7.3.1). ICMEs are often preceded by interplanetary fast forward shocks and turbulent sheath regions between the shock and the ejecta, which all create their distinct responses in the magnetosphere and radiation belts. Because fast solar wind streams originate from coronal holes, which can persist over several solar rotations, the slow and fast stream pattern repeats in 27-day intervals and SIRs are often called co-rotating interaction regions (CIRs). However, stream interaction region is a physically more descriptive term. SIRs may gradually evolve to become bounded by shocks, but fully developed SIR shocks are only seldom observed sunward of the Earth’s orbit. The duration of these large-scale heliospheric structures near the orbit of the Earth varies from a few hours to days. On average, the passage of a sheath region past the Earth takes 8–9 h and the passage of an ICME or SIR about 1 day. The fast streams typically influence the Earth’s environment for several days.

1.4.1 Magnetospheric Convection

Magnetospheric plasma is in a continuous large-scale advective motion, which in this context is, somewhat inaccurately, called magnetospheric convection (for a thorough introduction, see Kennel 1995). The convection is most directly observable in the polar ionosphere, where the plasma flows from the dayside across the polar cap to the nightside and turns back to the dayside through the morning and evening sector auroral region. The non-resistive ideal magnetohydrodynamics (MHD, Sect. 3.2.3) is a fairly accurate description of the large-scale plasma motion above the resistive ionosphere. In ideal MHD the magnetic field lines are electric equipotentials and the electric field E and plasma velocity V are related to each other through the simple relation

$$\displaystyle \begin{aligned} \mathbf{E} = -\mathbf{V}\times\mathbf{B}\;. \end{aligned} $$

(1.18)

Consequently, the observable convective motion, or alternatively the electric potential, in the ionosphere can be mapped along the magnetic field lines to plasma motion in the tail lobes and the plasma sheet. As the electric field in the tail plasma sheet points from dawn to dusk and the magnetic field to the north, the convection brings plasma particles from the nightside plasma sheet toward the Earth where a fraction of them become carriers of the ring current and form the source population for the radiation belts.

In ideal MHD the plasma and the magnetic field lines are said to be frozen-in to each other. This means that two plasma elements that are connected by a magnetic field line remain so when plasma flows from one place to another (the proof of this statement can be found in most plasma physics textbooks, e.g., Koskinen 2011). It is convenient to illustrate the motion with moving field lines, although the magnetic field lines are not physical entities and their motion is just a convenient metaphor. A more physical description is that the magnetic field evolves in space and time such that the plasma elements maintain their magnetic connection.

The convection is sustained by solar wind energy input into the magnetosphere. The input is weakest, but yet finite, when the interplanetary magnetic field (IMF) points toward the north, and is enhanced during southward pointing IMF. If the magnetopause were fully closed, plasma would circulate inside the magnetosphere so that the magnetic flux tubes crossing the polar cap from dayside to nightside would reach to the outer boundary of the magnetosphere where some type of viscous interaction with the anti-sunward solar wind flow would be needed to maintain the circulation. This was the mechanism proposed by Axford and Hines (1961) to explain the convection. The classical (collisional) viscosity on the magnetopause is vanishingly small, but finite gyro radius effects and wave–particle interactions give rise to some level of anomalous viscosity.¹¹ It is estimated to provide about 10% of the momentum transfer from the solar wind to the magnetosphere.

The magnetosphere is, however, not fully closed. In the same year, when Axford and Hines presented their viscous interaction model, Dungey (1961) explained the convection in terms of magnetic reconnection . The Dungey cycle begins with a violation of the frozen-in condition at the dayside magnetopause current sheet. A magnetic field line in the solar wind is cut and reconnected with a terrestrial field line. Reconnection is most efficient for oppositely directed magnetic fields, as is the case in the dayside equatorial plane when the IMF points southward, but remains finite under other orientations. Subsequent to the dayside reconnection the solar wind flow drags the newly-connected field line to the nightside and the part of the field line that is inside the magnetosphere becomes a tail lobe field line. Consequently, an increasing amount of magnetic flux is piling up in the lobes. At some distance far in the tail the oppositely directed field lines in the northern and southern lobes reconnect again across the cross-tail current layer. At this point the ionospheric end of the field line has reached the auroral oval near local midnight. Now the earthward outflow from the reconnection site in the tail drags the newly-closed field line toward the Earth. The return flow cannot penetrate to the plasmasphere corotating with the Earth and the convective flow must proceed via the dawn and dusk sectors around the Earth to the dayside. In the ionosphere the flow returns toward the dayside along the dawnside and duskside auroral oval. Once approaching the dayside magnetopause, the magnetospheric plasma provides the inflow to the dayside reconnection inside of the magnetopause. Note that the resistive ionosphere breaks the frozen-in condition of ideal MHD and it is not reasonable to use the picture of moving field lines in the atmosphere.

The increase in the tail lobe magnetic flux and strengthening of plasma convection inside the magnetosphere during southward IMF have a strong observational basis. Calculating the east-west component of the motion-induced solar wind electric field (E = V B _south) incident on the magnetopause and estimating the corresponding potential drop over the magnetosphere, some 10% of the solar wind electric field is estimated to “penetrate” into the magnetosphere as the dawn-to-dusk directed convection electric field. Note that E = −V ×B is not a causal relationship indicating whether it is the electric field that drives the magnetospheric convection, or convection that gives rise to the motion-induced electric field. The ultimate driver of the circulation is the solar wind forcing on the magnetosphere.

The plasma circulation is not as smooth as the above discussion may suggest. If the reconnection rates at the dayside magnetopause and nightside current sheet balance each other, a steady-state convection can, indeed, arise. This is, however, seldom the case since the changes in the driving solar wind and in the magnetospheric response are faster than the magnetospheric circulation timescale of a few hours. Reconnection may cause significant erosion of the dayside magnetospheric magnetic field placing the magnetopause closer to the Earth than a simple pressure balance consideration would indicate. The changing magnetic flux in the tail lobes causes expansion and contraction of the polar caps affecting the size and shape of the auroral ovals.

Furthermore, the convection in the plasma sheet has been found to consist of intermittent high-speed bursty bulk flows (BBF) with almost stagnant plasma in between (Angelopoulos et al. 1992, and references therein). It is noteworthy that while BBFs are more frequent during high auroral activity, they also appear during auroral quiescence. BBFs have been estimated to be the primary mechanism of earthward mass and energy transport in regions where they have been observed (Angelopoulos et al. 1994). Thus the high-latitude convection observed in the ionosphere corresponds to an average of the BBFs and slower background flows in the outer magnetosphere.

1.4.2 Geomagnetic Storms

Strong perturbations of the geomagnetic field known as geomagnetic (or magnetic) storms have been known since the nineteenth century. Because we look at the storms in this book mostly from the magnetospheric viewpoint, we call them also magnetospheric storms. As illustrated in Fig. 1.6, the storms are periods of most dynamic evolution of radiation belts. They often, but not always, commence with a significant positive deviation in the horizontal component of the magnetic field (H) measured on the ground (Fig. 1.7), called storm sudden commencement (SSC) . An SSC is a signature of an ICME-driven shock and the associated pressure pulse arriving at the Earth’s magnetopause. SSCs are also observed during pressure pulses related to SIRs or to ICMEs that are not sufficiently fast to drive a shock in the solar wind but still disturb and pile-up the solar wind ahead of them. If the solar wind parameters are known, the pressure effect can be removed from the Dst index as discussed in Sect. 1.2.3.

Fig. 1.6

Outer radiation belt response to solar and magnetospheric activity from the SAMPEX satellite and Van Allen Probes observations over a period of more than two solar cycles. The uppermost panel shows 27-day window-averaged relativistic (>2 MeV) electron fluxes at geostationary orbit, the second panel the monthly minimum of the Dst index, and the third panel the yearly window-averaged sunspot number (black) and weekly window-averaged solar wind speed (red). The spectrogram in the lowest panel is a composite of 27-day window-averaged SAMPEX observations of relativistic (∼2 MeV) electron fluxes until September 2012 and Van Allen Probes REPT observations of (∼2.1 MeV) electron fluxes after 5 September 2012. The shift from SAMPEX to Van Allen Probes is visible in the change of sensitivity to particle flux in the slot region (From Li et al. 2017, Creative Commons Attribution-NonCommercial-NoDerivs License)

Fig. 1.7

The horizontal component (H) of the magnetic field measured at four low-latitude stations during a magnetic storm on 15 May 1997. An ICME-driven solar wind shock hit the magnetosphere on 15 May at about 02 UT causing the storm sudden commencement which is indicated by a sudden positive jump of the H component at all stations (thick blue line). The main phase of the storm started after 06 UT as indicated by the strong negative deviation in the H component. The solid vertical lines give the UT midnight and the tick-marks on the horizontal axis are given for each 3 h. (Figure courtesy: L. Häkkinen, adapted from Koskinen 2011, reprinted by permission from SpringerNature)

Storms in the magnetosphere can also be driven by low-speed ICMEs and SIRs without a significant pressure pulse. SIR-driven storms occur if the field fluctuations have sufficiently long periods of strong enough southward magnetic field to sustain global convention electric field to enhance the ring current. Thus there are storms without a clear SSC signature in the Dst index. On the other hand, a shock wave hitting the magnetopause is not always followed by a geomagnetic storm, in particular, if the IMF points dominantly toward the north during the following solar wind structure. In such cases the positive deviation in the magnetograms is called a sudden impulse (SI) , after which the Dst index returns close to its background level with small temporal variations only. If the dynamic pressure remains at enhanced level, Dst can maintain positive deviation for some period.

After the SSC an initial phase of the storm begins. It is characterized by a positive deviation of Dst, typically a few tens of nT. The initial phase is caused by a combination of predominantly northward IMF and high dynamic pressure. The phase can have very different durations depending on the type and structure of the solar wind driver. It can be very brief if the storm is driven by an ICME with a southward magnetic field following immediately a sheath with predominantly southward magnetic field. In such a case the storm main phase , which is a period characterized by a rapid decrease of the H component of the equatorial magnetic field, starts as soon as the energy transfer into the magnetosphere has become strong enough. If the sheath has a predominantly northward IMF, the main phase will not begin until a southward field of the ejecta enhances reconnection on the dayside magnetopause.

If there is no southward IMF either in the sheath or in the ICME, no regular global storm is expected to take place. However, pressure pulses/shocks followed by northward IMF can cause significant consequences to the radiation belt environment, as they can shake and compress the magnetosphere strongly and trigger a sequence of substorms (Sect. 1.4.3).

During the storm main phase, the enhanced energy input from the solar wind leads to energization and increase of the number of ring current carriers in the inner magnetosphere, as the enhanced magnetospheric convection transports an increasing amount of charged particles from the tail to the ring current region. Here substorms, discussed below, have important contribution, as they inject fresh particles from the near-Earth tail. The ring current enhancement is typically asymmetric because not all current carrying ions are on closed drift paths but a significant fraction of them passes the Earth on the evening side and continue toward the dayside magnetopause. This is illustrated in Fig. 1.7 where the Honolulu and Kakioka magnetometers show the steepest main phase development when these stations were in the dusk side of the globe.

When energy input from the solar wind ceases, the energetic ring current ions are lost faster than fresh ones are supplemented from the tail. The Dst index starts to return toward the background level. This phase is called the recovery phase. It is usually much longer than the main phase, because the dominating loss processes of the ring current carriers: charge exchange with the low-energy neutral atoms of the Earth’s exosphere, wave–particle interactions, and Coulomb collisions (Sect. 6.3.2), are slower than the rapid increase of the current during the main phase. As ICMEs last typically 1 day, storms driven by ICMEs trailed by a slow wind tend to have relatively short recovery phases, whereas storms driven by SIRs and ICMEs followed by a fast stream can have much longer recovery phases. This is because Alfvénic fluctuations, i.e., large-amplitude MHD Alfvén waves (Sect. 4.4), in fast streams interacting with the magnetospheric boundary lead to triggering substorms, which inject particles to the inner magnetosphere. This can keep keep the ring current populated with fresh particles up to or longer than a week. The ring current development can also be more complex, often resulting in multi-step enhancement of Dst or events where Dst does not recover to quiet-time level between relatively closely-spaced intensifications. This typically occurs when both sheath and ICME ejecta carry southward field or when the Earth is impacted by multiple interacting ICMEs.

1.4.3 Substorms

From the radiation belt viewpoint the key significance of magnetospheric substorms is their ability to inject fresh particles in the energy range from tens to a few hundred keV from the tail plasma sheet into the inner magnetosphere. After being injected to the quasi-dipolar magnetosphere, charged particles start to drift around the Earth, contributing to the ring current and radiation belt populations. The injections have a twofold role: They provide particles to be accelerated to high energies. Simultaneously the injected electrons and protons drive waves that can lead to both acceleration and loss of radiation belt electrons and ring current carriers.

Magnetospheric substorms result from piling of tail lobe magnetic flux in the near-to-mid-tail region during enhanced convection. The details of the substorm cycle are still debated after more than half a century of research. Observationally it is clear that substorms encompass global configurational changes in the magnetosphere, namely the stretching of the near-Earth nightside magnetic field and related thinning of the plasma sheet during the flux pile-up (substorm growth phase), followed by a relatively rapid return of the near-Earth field toward a dipolar shape (expansion phase), and a slower return to a quiet-time stretched configuration associated with thickening of the plasma sheet (recovery phase). A substorm cycle typically lasts 2–3 h. The strongest activity occurs following the onset of the expansion phase: The cross-tail current in the near-Earth tail disrupts and couples to the polar region ionospheric currents through magnetic field-aligned currents forming the so-called substorm current wedge. This leads to intense precipitation of magnetospheric particles causing the most fascinating auroral displays. During geomagnetic storms the substorm cycle may not be equally well-defined. For example, a new growth phase may begin and the onset of the next expansion may follow soon after the previous expansion phase.

A widely used, though not the only, description of the substorm cycle is the so-called near-Earth neutral line model (NENL model, for a review, see Baker et al. 1996). In the model the current sheet is pinched off by a new magnetic reconnection neutral line once enough flux has piled up in the tail. The new neutral line forms somewhere at distances of 8–30 R _E from the Earth, which is much closer to the Earth than the far-tail neutral line of the Dungey cycle (Sect. 1.4.1). Earthward of the neutral line plasma is pushed rapidly toward the Earth. Tailward of the neutral line plasma flows tailward, and together with the far-tail neutral line, a tailward moving structure called plasmoid forms. Sometimes recurrent substorm onsets can create a chain of plasmoids. While it is common to illustrate the plasmoid formation using two-dimensional cartoons in the noon–midnight meridional plane, the three-dimensional evolution of the substorm process in the magnetotail is far more complex. In reality a plasmoid is a magnetic flux rope whose two-dimensional cut looks like a closed loop of magnetic field around a magnetic null point.

As pointed out in Sect. 1.4.1, the plasma flow in the central plasma sheet is not quite smooth and a significant fraction of energy and mass transport takes place as bursty bulk flows (BBFs). The BBFs are thought to be associated with localized reconnection events in the plasma sheet roughly at the same distances from the Earth as the reconnection line of the NENL model. They create small flux tubes called dipolarizing flux bundles (DFBs). The name derives from their enhanced northward magnetic field component B _Z corresponding to a more dipole-like state of the geomagnetic field compared to a more stretched configuration. Once created, DFBs surge toward the Earth due to the force caused by magnetic curvature tension in the fluid picture. They are preceded by sharp increases of B _Z called dipolarization fronts. DFBs are also associated with large azimuthal electric fields, up to several mV m⁻¹, which are capable of accelerating charged particles to high energies. Whether the braking of the bursty bulk flows and coalescence of dipolarization fronts closer to the Earth cause the formation of the substorm current wedge, or not, is a controversial issue.

The NENL model has been challenged by the common observation that the auroral substorm activation starts at the most equatorward arc and expands thereafter poleward. Whether the NENL model or some of the competing approaches (for a discussion, see e.g., Koskinen 2011) is the most appropriate substorm description, is not relevant to our discussion of radiation belts. What is essential is that the substorm expansions dipolarize the tail magnetic field configuration having been stretched during the growth phase and inject fresh particles into the inner magnetosphere. The particle injections can be observed as dispersionless, meaning that injected particles arrive to the observing spacecraft simultaneously at all energies, or dispersive when particles of higher energies arrive before those of lower energies. Because the dispersion arises from energy-dependent gradient and curvature drifts of the particles (Sect. 2.2.2), a dispersionless injection suggests that the acceleration occurs relatively close to the observing spacecraft, whereas dispersive arrival indicates acceleration further away from the observation when the particle distribution has had time to develop dispersion due to energy-dependent drift motion.

Dispersionless substorm injections are typically observed close to the midnight sector at geostationary orbit (6.6 R _E) and beyond, but have been found all the way down to about 4 R _E (Friedel et al. 1996). The injection sites move earthward as the substorm progresses and are also controlled by geomagnetic activity, although the extent of the dispersionless region is unclear, both in local time and radial directions. Neither have the details of acceleration of the injected particles been fully resolved. It has been suggested to be related both to betatron and Fermi acceleration (Sect. 2.4.4) associated with earthward moving dipolarization fronts. Another important aspect of dipolarization fronts for radiation belts is their braking close to Earth, which can launch magnetosonic waves that can effectively interact with radiation belt electrons.