Plasma Waves in the Inner Magnetosphere

Abstract

Understanding the role of plasma waves, extending from magnetohydrodynamic (MHD) waves at ultra-low-frequency (ULF) oscillations in the millihertz range to very-low-frequency (VLF) whistler-mode emissions at frequencies of a few kHz, is necessary in studies of sources and losses of radiation belt particles. In order to make this theoretically heavy part of the book accessible to a reader, who is not familiar with wave–particle interactions, we have divided the treatise into three chapters. In the present chapter we introduce the most important wave modes that are critical to the dynamics of radiation belts. The drivers of these waves are discussed in Chap. 5 and the roles of the wave modes as sources and losses of radiation belt particles are dealt with in Chap. 6.

Understanding the role of plasma waves, extending from magnetohydrodynamic (MHD) waves at ultra-low-frequency (ULF) oscillations in the millihertz range to very-low-frequency (VLF) whistler-mode emissions at frequencies of a few kHz, is necessary in studies of sources and losses of radiation belt particles. In order to make this theoretically heavy part of the book accessible to a reader, who is not familiar with wave–particle interactions, we have divided the treatise into three chapters. In the present chapter we introduce the most important wave modes that are critical to the dynamics of radiation belts. The drivers of these waves are discussed in Chap. 5 and the roles of the wave modes as sources and losses of radiation belt particles are dealt with in Chap. 6.

Basic plasma wave concepts such as dispersion equation, wave vector, index of refraction, phase and group velocities, etc., are summarized in Appendices A.2 and A.3.

4.1 Wave Environment of Radiation Belts

We begin with ULF waves. They can be observed directly in space and as geomagnetic pulsations on the ground. The ground-based observations are particularly useful when local space observations are not available or the low frequency of the waves makes them difficult to identify by using instruments onboard fast moving satellites. Ground-based magnetometers can also capture ULF waves more globally through the wide longitudinal and latitudinal coverage of magnetometer stations. On the other hand, not all ULF wave modes reach the ground and those that do so may become distorted in the ionosphere.

In studies of geomagnetic pulsations the ULF waves are traditionally grouped as irregular (Pi) and continuous (Pc) pulsations and further according to the observed periods. Table 4.1 summarizes the pulsation periods most frequently encountered in magnetospheric physics.

Table 4.1 Periods and frequencies of Pc1–Pc5 and Pi1 and Pi2 pulsations (for further details, see Jacobs et al. 1964)

In the radiation belt context the most important ULF waves have frequencies in the ranges of Pc1, Pc4, and Pc5 pulsations. Pc4 and Pc5 waves are global-scale magnetohydrodynamic waves (Sect. 4.4). Their role is particularly important in radial diffusion and transport of radiation belt electrons (Chap. 6). The Pc1 range includes electromagnetic ion cyclotron (EMIC) waves, also known as Alfvén ion cyclotron waves, whose frequencies are below the local ion gyro frequency but higher than those of Pc4 and Pc5 waves. The dispersion equation of EMIC waves can be found by solving the cold plasma dispersion equation (Sect. 4.3), although determining their growth and decay rates requires calculation based on Vlasov theory. EMIC waves play an important role in the ring current and in the loss of ultra-relativistic radiation belt electrons.

Figure 4.1 illustrates the most common equatorial domains of waves whose interactions with charged particles can lead to acceleration, transport and loss of radiation belt electrons. EMIC waves are predominantly observed in the afternoon sector close to the plasmapause and beyond. The next wave mode in the order of increasing frequency is the equatorial magnetosonic noise , observed from a few Hz to a few hundreds of Hz. Magnetosonic noise is found both inside and outside the dayside plasmapause. The plasmaspheric hiss can be found all over the plasmasphere with highest occurrence rates on the dayside as indicated in Fig. 4.1. The frequency of hiss emissions extends to several kHz. However, their interaction with radiation belt electrons is most efficient at frequencies below 100 Hz. The highest-frequency waves in Fig. 4.1 are the VLF whistler-mode chorus emissions from about 0.5 kHz to 10 kHz. They are observed outside the plasmasphere, most commonly from the dawn sector to the dayside.

Fig. 4.1

Schematic map of the equatorial occurrence of the wave modes that are most important to the radiation belt electrons. Note that the occurrence of different modes varies depending on the magnetospheric activity and availability of free energy to drive the waves, and, e.g., chorus waves and EMIC waves can be observed at all local times, although less frequently than in the domains indicated here. More detailed empirical maps are presented in Chap. 5

4.2 Waves in Vlasov Description

The basic characteristics of the most important wave modes in radiation belt physics can be found from reduced plasma descriptions, such as cold plasma theory (EMIC, whistler-mode chorus, plasmaspheric hiss) or magnetohydrodynamics (ULF waves). However, these theories are not sufficient to describe how the waves are driven nor how the waves accelerate, scatter and transport plasma particles. To understand the source and loss mechanisms of energetic particles in radiation belts a more detailed treatment is needed. For this reason we start our discussion of plasma waves from the elements of Vlasov theory and move thereafter to the cold plasma and MHD descriptions.

4.2.1 Landau’s Solution of the Vlasov Equation

The Vlasov equation for particle species α (3.14)

$$\displaystyle \begin{aligned} \frac{\partial f_\alpha}{\partial t} + \mathbf{v}\cdot\frac{\partial f_\alpha}{\partial\mathbf{r}} + \frac{q_\alpha}{m_\alpha}(\mathbf{E}+\mathbf{v}\times\mathbf{B})\cdot\frac{\partial f_\alpha}{\partial\mathbf{v}} = 0 \end{aligned}$$

is not easy to solve. It has to be done under the constraint that the electromagnetic field fulfils Maxwell’s equations, whose source terms (ρ, J) are determined by the distribution function, which, in turn, evolves according to the Vlasov equation. When looking for analytical solutions the background plasma and magnetic field must in practice be assumed homogeneous. In space physics this is a problem at various boundary layers, where the wavelengths become comparable to the thickness of the boundary. Furthermore, the force term in the Vlasov equation is nonlinear and the Vlasov equation can be solved analytically only for small perturbations when linearization is possible. This is sometimes a serious limitation in radiation belts where the wave amplitudes are known to grow to the nonlinear regime as will be discussed in the subsequent chapters.

We start by writing the distribution function for plasma species α and the electromagnetic field as sums of equilibrium solutions (subscript 0) and small perturbations (subscript 1)

$$\displaystyle \begin{aligned} \begin{array}{rcl} f_\alpha & =&\displaystyle f_{\alpha0} +f_{\alpha1} \\ \mathbf{E} & =&\displaystyle {\mathbf{E}}_0 + {\mathbf{E}}_1 \\ \mathbf{B} & =&\displaystyle {\mathbf{B}}_0 + {\mathbf{B}}_1 \end{array} \end{aligned} $$

and linearize the Vlasov equation by considering only the first-order terms in perturbations. The problem still remains difficult. For example, the general linearized solution for homogeneous plasma in a homogeneous background magnetic field was not presented until late 1950s by Bernstein (1958). Inclusion of spatial inhomogeneities rapidly leads to problems that require numerical methods.

Lev Landau (1946) found the solution to the Vlasov equation in the absence of background fields. At the first sight, this may seem irrelevant in the context of radiation belts where particle dynamics is controlled by the magnetospheric magnetic field. However, the wave–particle interactions described by Landau’s solution are important also in magnetized plasma and lay the foundation for the transfer of energy from plasma waves to charged particles, and vice versa.

Let us consider homogeneous plasma without ambient electromagnetic fields (E ₀ = B ₀ = 0) in the electrostatic approximation, in which the electric field perturbation is given as the gradient of a scalar potential E ₁ = −∇φ ₁ and the magnetic field perturbation B ₁ = 0. The linearized Vlasov equation is now

$$\displaystyle \begin{aligned} \frac{\partial f_{\alpha1}}{\partial t} + \mathbf{v}\cdot\frac{\partial f_{\alpha1}}{\partial\mathbf{r}} - \frac{q_\alpha}{m_\alpha}\frac{\partial\varphi_1}{\partial\mathbf{r}} \cdot\frac{\partial f_{\alpha0}}{\partial\mathbf{v}} = 0\;, \end{aligned} $$

(4.1)

where

$$\displaystyle \begin{aligned} \nabla^2\varphi_1= -\frac{1}{\epsilon_0}\sum_\alpha n_\alpha q_\alpha\int f_{\alpha1}\,\mathrm{d}^3v\;. \end{aligned} $$

(4.2)

Here it is convenient to normalize the distribution function to 1. As we assume the plasma being homogeneous, the constant background density n _α has in (4.2) been moved outside the integral.

Vlasov tried to solve these equations at the end of the 1930s using Fourier transformations in space and time. He ended up with the integral

$$\displaystyle \begin{aligned} \int_{-\infty}^\infty\frac{\partial f_{\alpha0}/\partial v}{\omega -kv}\,\mathrm{d} v\;, \end{aligned}$$

which has a singularity along the path of integration. Vlasov did not find the way how to deal with the singularity.

Landau realized that, because the perturbation must begin at some point in time, the problem should be treated as an initial value problem and, instead of a Fourier transform, a Laplace transform is to be applied in the time domain. In this approach the initial perturbations turned out to be transients that fade away with time and the asymptotic solution gives the intrinsic properties of the plasma, i.e., the dispersion equation between the frequency and the wave number. The Laplace transformation makes the frequency a complex quantity ω = ω _r + iω _i, which, when inserted in the plane wave expression $\exp (\mathrm{i} (\mathbf {k\cdot r}-\omega t))$, leads to a term proportional to $\exp (\omega _i t)$ that either grows (ω _i > 0) or decays (ω _i < 0) exponentially as a function of time.

After Fourier transforming in space and Laplace transforming in time the perturbations f _α1 and φ ₁, the asymptotic solution leads to the dispersion equation

$$\displaystyle \begin{aligned} K(\omega,\mathbf{k}) = 0\,, \end{aligned} $$

(4.3)

where

$$\displaystyle \begin{aligned} K(\omega,\mathbf{k}) = 1 + \frac{1}{\epsilon_0}\sum_\alpha\frac{n_\alpha q_\alpha^2} {m_\alpha}\frac{1}{k^2}\int\frac{\mathbf{k}\cdot\partial f_{\alpha0}/\partial\mathbf{v}}{\omega-\mathbf{k}\cdot\mathbf{v}}\,\mathrm{d}^3v\,. \end{aligned} $$

(4.4)

K is called the dielectric function because it describes the dielectric behavior of the plasma, i.e., it formally relates the electric field to the electric displacement D = K𝜖 ₀ E . Now the frequency is ω = ip, where p is the coordinate in Laplace transformed time domain $\exp (-pt)$.

Because K(ω, k) contains the information of the relation between frequency and wave vector, we do not usually need to make the inverse transformations back to the (t, r)-space. However, it is important to know, how the inverse Laplace transformation is to be done in order to correctly treat the pole in (4.4). This is a non-trivial exercise in complex integration. The procedure can be found in advanced plasma physics textbooks (e.g., Koskinen 2011). Here we skip the technical details.

Non-magnetized homogeneous plasma is essentially one-dimensional. We can simplify the notation by selecting one of the coordinate axes in the direction of k and write the one-dimensional distribution function as

$$\displaystyle \begin{aligned} F_{\alpha0}(u) \equiv \int f_{\alpha0}(\mathbf{v})\, \delta\left(u-\frac{\mathbf{k}\cdot\mathbf{v}}{\vert k\vert}\right)\mathrm{d}^3v \,, \end{aligned} $$

(4.5)

where δ(x) is Dirac’s delta.

Careful analysis of the inverse Laplace transform indicates that the integral in (4.4) must be calculated along a contour that is closed in the upper half of the complex plane and passes below the pole. The integration path is called the Landau contour , denoted by ∫_L, and the dispersion equation is

$$\displaystyle \begin{aligned} K(\omega,k)\equiv 1-\sum_\alpha\frac{\omega_{p\alpha}^2}{k^2} \int_L\frac{\partial F_{\alpha0}(u)/\partial u}{u-\omega/|k|}\,\mathrm{d} u = 0\;. \end{aligned} $$

(4.6)

The pole in the integral leads to a complex solution of (4.6)

$$\displaystyle \begin{aligned} \omega(k) = \omega_r(k) + \mathrm{i} \omega_i(k)\,. \end{aligned} $$

(4.7)

If ω _i < 0, the electrostatic potential φ ₁ is damped and the distribution function is stable. If ω _i > 0, φ ₁ grows corresponding to an instability.

Recall that this analysis is done by assuming small perturbations and the result is valid at the asymptotic limit. Consequently, the solution is valid when |ω _i|≪|ω _r|. Such solutions are called normal modes . Larger |ω _i| leads either to an overdamped wave or to a perturbation growing to the nonlinear regime.

4.2.2 Landau Damping of the Langmuir Wave

The Landau integration can be performed analytically for some specific distribution functions only. Already the Maxwellian distribution leads to technical complications.

Assume again E ₀ = B ₀ = 0 and consider the one-dimensional Maxwellian

$$\displaystyle \begin{aligned} F_{\alpha0}(u)=\sqrt{\frac{m_\alpha}{2\pi k_BT_\alpha}}\exp(-u^2/v_{th,\alpha}^2)\;, \end{aligned} $$

(4.8)

where the thermal speed v _th,α is defined as

$$\displaystyle \begin{aligned} v_{th,\alpha} = \sqrt{\frac{2k_BT_\alpha}{m_\alpha}}\;. \end{aligned} $$

(4.9)

A difficulty, although manageable, with the Landau contour is the calculation of the closure of the integration path of

$$\displaystyle \begin{aligned} \int\frac{\partial F_{\alpha0}/\partial u}{u-\omega/|k|}\,\mathrm{d} u \propto \int\frac{uF_{\alpha0}}{u-\omega/|k|}\,\mathrm{d} u \end{aligned}$$

in the complex plane when u →∞. The result is commonly expressed in terms of the plasma dispersion function

(4.10)

and its derivatives. Z(ζ) is related to the error function of mathematics and must, in practise, be computed numerically.

Considering the electron oscillations only, similar to the case of cold plasma oscillation (Sect. 3.1.2) but assuming now a finite temperature, the dispersion equation turns out to be

$$\displaystyle \begin{aligned} 1-\frac{\omega_{pe}^2}{k^2v_{th,e}^2}\mathrm{Z}'\left(\frac{\omega}{kv_{th,e}}\right) = 0\,, \end{aligned} $$

(4.11)

where Z′ denotes the derivative of the plasma dispersion function with respect to its argument.

For normal modes (|ω _i|≪ ω _r) the dispersion equation can be expanded around ω = ω _r as

$$\displaystyle \begin{aligned} \begin{array}{rcl} 1-\sum_\alpha\frac{\omega_{p\alpha}^2}{k^2} \left(1+\mathrm{i} \omega_i\frac{\partial}{\partial\omega_i}\right) \left[\mathrm{P}\int\frac{\partial F_{\alpha0}/\partial u}{u-\omega_r/|k|}\,\mathrm{d} u +\pi \mathrm{i}\left(\frac{\partial F_{\alpha0}}{\partial u}\right)_{u=\omega_r/|k|} \right]=0\;.\\ {} \end{array} \end{aligned} $$

(4.12)

Here P indicates the Cauchy principal value. The second term in the brackets comes from the residue at the pole. Because the pole in this case is on the real axis, the residue is multiplied by πi instead of 2πi. Using this expression we can find solutions for the dispersion equation at long and short wavelengths. These correspond to series expansions of the dispersion function Z for large and small arguments, respectively. At intermediate wavelengths numerical computation of Z cannot be avoided.

The most fundamental normal mode is the propagating variant of the fundamental plasma oscillation (Sect. 3.1.2), known as the Langmuir wave . It can be found as the long wavelength (ω∕k ≫ v _th) solution of (4.12). At this limit

$$\displaystyle \begin{aligned} \begin{array}{rcl} -\mathrm{P}\int\frac{\partial F_{\alpha0}/\partial u}{u-\omega_r/|k|}\,\mathrm{d} u =\int\frac{\partial F_{\alpha0}}{\partial u} \left(\frac{1}{\omega/|k|}+\frac{u}{(\omega/|k|)^2}+\frac{u^2}{(\omega/|k|)^3} +...\right)\,\mathrm{d} u\;.\\ \end{array} \end{aligned} $$

(4.13)

By using this expansion, considering electron dynamics only, and inserting a Maxwellian electron distribution function, we find the dispersion equation for the Langmuir wave. The real part of the frequency is

$$\displaystyle \begin{aligned} \omega_r\approx\omega_{pe}(1+3k^2\lambda_{De}^2)^{1/2} \approx\omega_{pe}\left(1+\frac{3}{2}k^2\lambda_{De}^2\right) \end{aligned} $$

(4.14)

and the imaginary part

$$\displaystyle \begin{aligned} \omega_i\approx -\sqrt{\frac{\pi}{8}}\frac{\omega_{pe}}{|k^3\lambda_{De}^3|} \exp\left(-\frac{1}{2k^2\lambda_{De}^2}-\frac{3}{2}\right)\;. \end{aligned} $$

(4.15)

The finite temperature of the Maxwellian distribution makes the standing cold plasma oscillation to propagate. Furthermore, the negative imaginary part of the frequency indicates that the wave is damped. This phenomenon is known as Landau damping.

Landau damping is not limited to electrostatic waves. As will be discussed in Chap. 6, it is also important in the context of electromagnetic waves causing resonant scattering of electrons with pitch angles close to 90^∘.

4.2.3 Physical Interpretation of Landau Damping

Landau’s solution was met with scepticism until it was experimentally verified in laboratory experiments in the 1960s (Malmberg and Wharton 1964). The problem was that the Vlasov equation conserves entropy, whereas the Landau solution does not seem to do so. The electric field of the Langmuir wave interacts with electrons accelerating those whose velocity is slightly less than the phase velocity of the wave, and decelerating those that move a little faster. In a Maxwellian distribution ∂f∕∂v < 0, meaning that there are more slower than faster electrons around the phase velocity (Fig. 4.2). Figuratively speaking the wave forces the particles near the phase velocity to “glide down” along the slope of the distribution function until the population is warm enough to damp the oscillation below the observable level. Thus, there is a net energy transfer from the wave to the particles.

Fig. 4.2

In a Maxwellian plasma ∂f∕∂v < 0 and there are more particles that are accelerated by the Langmuir wave in the vicinity of the phase velocity v _ph than those that lose energy to the wave. Thus the wave is damped and the electron population is heated

Although Landau damping looks like a dissipative process, the entropy is conserved in Vlasov theory and no information must be lost from the combined system consisting of both the distribution function and the electrostatic potential. The apparent contradiction can be resolved by carefully considering what happens to the distribution function in the damping process (for a detailed discussion, see, e.g., Krall and Trivelpiece 1973). At the time-asymptotic limit an extra term appears to the distribution function in the Fourier space

$$\displaystyle \begin{aligned} f_{\alpha\mathbf{k}}=\hat{f}_{\alpha b}\exp(-\mathrm{i} \mathbf{k}\cdot\mathbf{v}t) +\sum_{\omega_{\mathbf{k}}}\hat{f}_{\alpha\mathbf{k}}\exp(-\mathrm{i} \omega_{\mathbf{k}}t)\;, {} \end{aligned} $$

(4.16)

where ω _k are the solutions of the dispersion equation and $\hat {f}_{\alpha b}$ and $\hat {f}_{\alpha \mathbf {k}}$ are time-independent amplitudes. The terms in the sum over ω _k are damped at the same rate as the perturbed potential φ _k(t). In the first term on the RHS of (4.16) the subscript b stands for ballistic. The ballistic term is a consequence of the Liouville theorem, according to which the Vlasov equation conserves entropy. As the system is deterministic, every particle “remembers” its initial perturbation wherever it moves in the phase space.

When t increases, the ballistic term becomes increasingly oscillatory in the velocity space and its contribution to φ _k(t) behaves at the limit t →∞ as

$$\displaystyle \begin{aligned} k^2\varphi_{\mathbf{k}} = \frac{1}{\epsilon_0} \sum_\alpha q_\alpha n_\alpha\int\hat{f}_{\alpha b}\exp(-\mathrm{i} \mathbf{k}\cdot\mathbf{v}t) \mathrm{d}^3v\ \rightarrow\ 0\;. \end{aligned} $$

(4.17)

That is, at the time-asymptotic limit the ballistic terms of each particle species contain the information of the initial perturbation but they do not contribute to the observable electric field.

The existence of ballistic terms leads to an observable nonlinear phenomenon called the Landau echo . Assume that an initial perturbation took place at time t ₁ and its spectrum was narrow near wave number k ₁. Wait until the perturbation has been damped below the observable limit and only the ballistic term superposed on the equilibrium distribution remains. Then launch another narrow-band wave near k ₂ at time t ₂ and wait until it also is damped. At time t = t ₃ defined by

$$\displaystyle \begin{aligned} k_1(t_3-t_1) - k_2(t_3-t_2) = 0 {} \end{aligned} $$

(4.18)

the oscillations in the ballistic terms interfere positively. This beating of the ballistic terms of the first two perturbations produces a new observable fluctuation, which is the Landau echo. Also the echo is transient because the condition (4.18) is satisfied only for a short while and Landau damping acts on this fluctuation as well. The effect has been verified in laboratories and shows that the Landau damping does not violate the conservation of entropy in the timescale shorter than the collisional time.

As collisional timescales in tenuous space plasmas often are very long compared to the relevant timescales of interesting plasma phenomena, the existence of Landau echoes indicates that even in the case of small-amplitude perturbations there can be nonlinear mixing of wave modes at the microscopic level. This is one viewpoint to plasma turbulence.

4.2.4 Solution of the Vlasov Equation in Magnetized Plasma

Magnetospheric plasma is embedded in a background magnetic field and we need to look for a more general description including the background fields E ₀(r, t) and B ₀(r, t). The linearized Vlasov equation is written as

$$\displaystyle \begin{aligned} \left[\frac{\partial}{\partial t} + \mathbf{v}\cdot\frac{\partial}{\partial\mathbf{r}} +\frac{q_\alpha}{m_\alpha}({\mathbf{E}}_0+\mathbf{v}\times {\mathbf{B}}_0)\cdot\frac{\partial}{\partial\mathbf{v}}\right] f_{\alpha1} = - \frac{q_\alpha}{m_\alpha}({\mathbf{E}}_1+\mathbf{v}\times{\mathbf{B}}_1)\cdot\frac{\partial f_{\alpha0}}{\partial\mathbf{v}} {}\;. \end{aligned} $$

(4.19)

This is possible to solve by employing the method of characteristics, which can be described as “integration over unperturbed orbits”. Define new variables (r ′, v ′, t′) as

$$\displaystyle \begin{aligned} \frac{\mathrm{d}\mathbf{r}'}{\mathrm{d} t'}=\mathbf{v}'\ ;\ \frac{\mathrm{d}\mathbf{v}'}{\mathrm{d} t'} = \frac{q_\alpha}{m_\alpha} \left[{\mathbf{E}}_0(\mathbf{r}',t')+ \mathbf{v}'\times{\mathbf{B}}_0(\mathbf{r}',t')\right]\,, \end{aligned} $$

(4.20)

where the acceleration is determined by the background fields and the boundary conditions are

$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathbf{r}'(t'=t)& =&\displaystyle \mathbf{r}\\ \mathbf{v}'(t'=t)& =&\displaystyle \mathbf{v}\;.{} \end{array} \end{aligned} $$

(4.21)

Consider f _α1(r ′, v ′, t′) and use (4.19) to calculate its total time derivative

$$\displaystyle \begin{aligned} \begin{array}{rcl} \frac{\mathrm{d} f_{\alpha1}(\mathbf{r}',\mathbf{v}',t')}{\mathrm{d} t'} & =&\displaystyle \frac{\partial f_{\alpha1}(\mathbf{r}',\mathbf{v}',t')}{\partial t'} + \frac{\mathrm{d} \mathbf{r}'}{\mathrm{d} t'}\cdot \frac{\partial f_{\alpha1}(\mathbf{r}',\mathbf{v}',t')}{\partial \mathbf{r}'} + \frac{\mathrm{d}\mathbf{v}'}{\mathrm{d} t'}\cdot \frac{\partial f_{\alpha1}(\mathbf{r}',\mathbf{v}',t')}{\partial \mathbf{v}'} \\ & =&\displaystyle -\frac{q_\alpha}{m_\alpha} \left[{\mathbf{E}}_1(\mathbf{r}',t')+ \mathbf{v}'\times{\mathbf{B}}_1(\mathbf{r}',t')\right]\cdot \frac{\partial f_{\alpha0}(\mathbf{r}',\mathbf{v}')}{\partial \mathbf{v}'}\;. {} \end{array} \end{aligned} $$

(4.22)

The boundary conditions (4.21) imply that f _α1(r ′, v ′, t′) = f _α1(r, v, t) at time t′ = t. Thus the solution of (4.22) at t′ = t is a solution of the Vlasov equation. The point of this procedure is that (4.22) can be calculated by a direct integration because its LHS is an exact differential. The formal solution is

$$\displaystyle \begin{aligned} \begin{array}{rcl} f_{\alpha1}(\mathbf{r},\mathbf{v},t) & =&\displaystyle -\frac{q_\alpha}{m_\alpha}\int\limits_{-\infty}^t \left[{\mathbf{E}}_1(\mathbf{r}',t')+ \mathbf{v}'\times{\mathbf{B}}_1(\mathbf{r}',t')\right]\cdot \frac{\partial f_{\alpha0}(\mathbf{r}',\mathbf{v}')}{\partial \mathbf{v}'}\,\mathrm{d} t'\\ & +&\displaystyle f_{\alpha1}(\mathbf{r}'(-\infty),\mathbf{v}'(-\infty),t'(-\infty))\;. \end{array} \end{aligned} $$

(4.23)

The procedure can be interpreted in the following way: The perturbation of the distribution function f _α1 has been found by integrating the Vlasov equation from −∞ to t along the path in the (r, v)-space that at each individual time coincides with the orbit of a charged particle in the background fields E ₀ and B ₀. This, of course, requires that the deviation from the background orbit at each step in the integration is small. Consequently, the method is limited to linear perturbations.

From f _α1 we can calculate n _α1(r, t) and V _α1(r, t) and insert these in Maxwell’s equations

$$\displaystyle \begin{aligned} \begin{array}{rcl} \nabla\times{\mathbf{E}}_1 & =&\displaystyle -\frac{\partial{\mathbf{B}}_1}{\partial t} \end{array} \end{aligned} $$

(4.24)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \nabla\cdot{\mathbf{E}}_1 & =&\displaystyle \frac{1}{\epsilon_0}\sum_\alpha q_{\alpha}n_{\alpha1} \end{array} \end{aligned} $$

(4.25)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \nabla\times{\mathbf{B}}_1 & =&\displaystyle \frac{1}{c^2}\frac{\partial{\mathbf{E}}_1}{\partial t} +\mu_0\sum_\alpha q_{\alpha}(n_{\alpha}{\mathbf{V}}_{\alpha})_1\;. \end{array} \end{aligned} $$

(4.26)

This set of equations can now (in principle) be solved as an initial value problem in the same way as the Landau solution. Accepting that the Landau contour is the correct way to deal with the resonant integrals, assuming that the waves are plane waves ${\mathbf {E}}_1(\mathbf {r}, t) = {\mathbf {E}}_{\mathbf {k}\omega }\exp (i\mathbf {k}\cdot \mathbf {r}-\mathrm{i} \omega t)$, and f _α1(r ′, v ′, t →−∞) → 0, the growing solutions (Im(ω) > 0) are found to be

$$\displaystyle \begin{aligned} f_{\alpha\mathbf{k}} = -\frac{q_\alpha}{m_\alpha}\int\limits_{-\infty}^0 ({\mathbf{E}}_{\mathbf{k}\omega}+ \mathbf{v}'\times{\mathbf{B}}_{\mathbf{k}\omega})\cdot \frac{\partial f_{\alpha0}(\mathbf{v}')}{\partial \mathbf{v}'} \exp[\mathrm{i}(\mathbf{k}\cdot\mathbf{R}-\omega\tau)]\,\mathrm{d} \tau\;, {} \end{aligned} $$

(4.27)

where τ = t′− t , R = r ′−r. The damped solutions (Im(ω) < 0) are found by analytic continuation of f _αk to the lower half-plane. By inserting this into Maxwell’s equations in the (ω, k) space and eliminating B _kω we get the wave equation

$$\displaystyle \begin{aligned} {\mathsf K}\cdot\mathbf{E} = 0\;. \end{aligned} $$

(4.28)

Now the dielectric function is the dielectric tensor or dispersion tensor K. It is even in a homogeneous background magnetic field a complicated function. Let us start by considering the field-free isotropic case (E ₀ = B ₀ = 0 and f ₀ = f ₀(v ²)). Define $F_{\alpha 0}(u)=\int f_{\alpha 0}\,\delta (u-\mathbf {k}\cdot \mathbf {v}/|k|)\,\mathrm{d} ^3v$ and denote the component of the wave electric field in the direction of wave propagation by E _k = (k ⋅E)∕|k| and the transverse component by E _⊥ = (k ×E)∕|k| . The wave equation now becomes

$$\displaystyle \begin{aligned} \left[ \begin{array}{ccc} K_\perp & 0 & 0 \\ 0 & K_\perp & 0 \\ 0 & 0 & K_{\mathbf{k}} \\ \end{array} \right]\left[ \begin{array}{c} E_{\perp1}\\E_{\perp2}\\E_{\mathbf{k}} \end{array} \right]=0\;, \end{aligned} $$

(4.29)

where

$$\displaystyle \begin{aligned} \begin{array}{rcl} K_\perp & =&\displaystyle 1-\frac{k^2c^2}{\omega^2} -\sum_\alpha\frac{\omega_{p\alpha}^2}{\omega} \int\frac{F_{\alpha0}}{\omega-|k|u}\,\mathrm{d} u \end{array} \end{aligned} $$

(4.30)

$$\displaystyle \begin{aligned} \begin{array}{rcl} K_{\mathbf{k}} & =&\displaystyle 1 + \sum_\alpha\frac{\omega_{p\alpha}^2}{\omega} \int_L\frac{F_{\alpha0}/\partial u}{\omega/|k|-u}\,\mathrm{d} u\;. \end{array} \end{aligned} $$

(4.31)

These give

$$\displaystyle \begin{aligned} \begin{array}{lll} \mbox{electrostatic}\ \mbox{modes}: & K_{\mathbf{k}}=0 & ({\mathbf{E}}_\perp=0) \\ \mbox{electromagnetic}\ \mbox{modes}: & K_\perp=0 & ({\mathbf{E}}_{\mathbf{k}}=0)\;. \end{array} \end{aligned}$$

The electrostatic solution is Landau’s solution familiar from above. The dispersion equation for the electromagnetic modes is

$$\displaystyle \begin{aligned} \omega^2=k^2c^2+\sum_\alpha\omega_{p\alpha}^2 \int\limits_{-\infty}^\infty\frac{\omega F_{\alpha0}}{\omega-|k|u}\,\mathrm{d} u\;. \end{aligned} $$

(4.32)

This has propagating solutions if ω ≫ kv _th,e and we find the electromagnetic wave in non-magnetized cold plasma

$$\displaystyle \begin{aligned} \omega^2\approx k^2c^2+\omega_{pe}^2\;. \end{aligned} $$

(4.33)

The propagation is limited to frequencies higher than ω _pe.

Include next a homogeneous background magnetic field B ₀ = B ₀ e _z , but keep the background electric field E ₀ zero. Assume that the background particle distribution function is gyrotropic but may be anisotropic $f_{\alpha 0} = f_{\alpha 0}(v_\perp ^2,v_\parallel )$. Already in this very symmetric configuration the derivation of the dielectric tensor is a tedious procedure. A lengthy calculation, first presented by Bernstein (1958), leads to the dielectric tensor in the form

$$\displaystyle \begin{aligned} \begin{array}{rcl} {\mathsf K}(\omega,\mathbf{k}) & =&\displaystyle \left(1-\sum_\alpha\frac{\omega_{p\alpha}^2}{\omega^2}\right){\mathsf I} -\sum_\alpha\sum_{n=-\infty}^\infty\frac{2\pi\omega_{p\alpha}^2} {n_{\alpha0}\omega^2}\ \times {}\\ & &\displaystyle \int\limits_0^\infty\int\limits_{-\infty}^\infty v_\perp \mathrm{d} v_\perp \mathrm{d} v_\parallel\left( k_\parallel\frac{\partial f_{\alpha0}}{\partial v_\parallel} +\frac{n\omega_{c\alpha}}{v_\perp}\frac{\partial f_{\alpha0}}{\partial v_\perp}\right) \frac{{\mathsf S}_{n\alpha}(v_\parallel,v_\perp)} {k_\parallel v_\parallel+n\omega_{c\alpha}-\omega}\;. \end{array} \end{aligned} $$

(4.34)

I is the unit tensor and the tensor S _nα is

$$\displaystyle \begin{aligned} {\mathsf S}_{n\alpha}(v_\parallel,v_\perp) = \left[ \begin{array}{ccc} {\displaystyle\frac{n^2\omega_{c\alpha}^2}{k_\perp^2}J_n^2} & {\displaystyle\frac{inv_\perp\omega_{c\alpha}}{k_\perp}J_nJ_n^{\prime}} & {\displaystyle\frac{nv_\parallel\omega_{c\alpha}}{k_\perp}J_n^2} \\ -{\displaystyle\frac{inv_\perp\omega_{c\alpha}}{k_\perp}J_nJ_n^{\prime}} & v_\perp^2J_n^{\prime2} & -iv_\parallel v_\perp J_nJ_n^{\prime} \\ {\displaystyle\frac{nv_\parallel\omega_{c\alpha}}{k_\perp}J_n^2} & iv_\parallel v_\perp J_nJ_n^{\prime} & v_\parallel^2J_n^2 \end{array} \right]\, . \end{aligned} $$

(4.35)

Here J _n are the ordinary Bessel functions of the first kind with the argument k _⊥ v _⊥∕ω _cα, and $J_n^{\prime }=\mathrm{d} J_n/\mathrm{d} (k_\perp v_\perp /\omega _{c\alpha })$ .

Finite B ₀ makes the plasma behavior anisotropic. The temperature may now be different in parallel and perpendicular directions as, e.g., in the case of a bi-Maxwellian distribution

$$\displaystyle \begin{aligned} f_{\alpha0} = \frac{m_\alpha}{2\pi k_BT_{\alpha\perp}} \sqrt{\frac{m_\alpha}{2\pi k_BT_{\alpha\parallel}}} \exp\left[-\frac{m_\alpha}{2k_B}\left( \frac{v_\perp^2}{T_{\alpha_\perp}}+\frac{v_\parallel^2}{T_{\alpha_\parallel}} \right)\right]\;. \end{aligned} $$

(4.36)

When this is inserted into the elements of K, the resonant integrals in the direction of v _∥ can be expressed in terms of the plasma dispersion function Z (4.10).

The wave modes are the non-trivial solutions of

$$\displaystyle \begin{aligned} {\mathsf K}\cdot\mathbf{E}=0 \,. \end{aligned} $$

(4.37)

The mode structure is now more complex than in non-magnetized plasma:

• The distinction between electrostatic and electromagnetic modes is no more exact; there still are electrostatic modes fulfilling E ∥k as an approximation but also the electromagnetic modes may have an electric field component along k.

• The Bessel functions introduce harmonic mode structure organized according to ω = nω _cα for each particle species α.

• The Landau resonance ω = k ⋅v of the isotropic plasma is replaced by

  $$\displaystyle \begin{aligned} \omega-n\omega_{c\alpha}=k_\parallel v_\parallel\,. {} \end{aligned} $$

  (4.38)

  Thus only the velocity component along B ₀ is associated with Landau damping (n = 0) and only for waves with k _∥≠ 0 .

4.2.4.1 Parallel Propagation

Let us first look at the solutions for wave modes propagating parallel to the background magnetic field (k _⊥ = 0). At the lowest frequencies (ω ≪ ω _ci) we find the parallel propagating Alfvén wave

$$\displaystyle \begin{aligned} \omega_r = \frac{k_\parallel v_A}{\sqrt{1+v_A^2/c^2}}\,, \end{aligned} $$

(4.39)

where $v_A = B_0/\sqrt {\rho _m\mu _0}$ is the Alfvén speed . This is an MHD mode to be discussed further in Sect. 4.4. Note that (4.39) contains a “cold plasma correction” ($v_A^2/c^2$) in the denominator, which is not found in MHD. This due to the inclusion of the displacement current into Ampère’s law in Vlasov and cold plasma descriptions, in contrast to standard MHD. When the Vlasov equation is solved together with the full set of Maxwell’s equations, the solutions include the cold plasma and MHD approximations as limiting cases.

In Vlasov theory Alfvén waves are damped, which is not found in ideal MHD. The damping rate is very small at low frequencies. When ω → ω _ci, the mode approaches the ion gyro resonance (see Fig. 4.4 in the discussion of cold plasma waves, Sect. 4.3), and the damping rate increases. At this limit the mode is the left-hand (L) circularly polarized electromagnetic ion cyclotron (EMIC) wave, which is damped not only by the resonant ions but also by relativistic electrons with sufficiently large Lorentz factor γ and Doppler shift k _∥ v _∥ of the frequency. This is an important loss mechanism of ultra-relativistic radiation belt electrons (Sect. 6.5.4).

We return to the right-hand (R) and left-hand (L) circularly polarized electromagnetic modes in cold plasma theory (Sect. 4.3), where they can be described in a more transparent manner. The most important right-hand polarized wave mode in radiation belts is the whistler mode . Again Vlasov theory is needed to describe the damping and growth of the whistler-mode waves leading to acceleration and pitch-angle scattering of radiation belt electrons as discussed in Chap. 6. Near the electron gyro frequency the whistler mode goes over to the electromagnetic electron cyclotron wave.

In the linear approximation the parallel propagating electromagnetic waves do not have harmonic structure. However, if the amplitude grows to nonlinear regime, the representation of the wave, e.g., as a Fourier series contains higher harmonics.

4.2.4.2 Perpendicular Propagation

For perpendicular propagation (k _∥ = 0) the wave equation reduces to

$$\displaystyle \begin{aligned} \left[ \begin{array}{ccc} K_{xx} & K_{xy} & 0 \\ K_{yx} & K_{yy} & 0 \\ 0 & 0 & K_{zz} \end{array} \right] \cdot \left[ \begin{array}{c} E_x \\ E_y \\ E_z \end{array} \right] = 0\;, \end{aligned} $$

(4.40)

where the z-axis is along the background magnetic field.

Assuming an isotropic background distribution function one component of the dispersion equation is

$$\displaystyle \begin{aligned} K_{zz} = 1 -\frac{k^2c^2}{\omega^2} - \frac{2\pi}{\omega}\sum_\alpha\sum_n \omega_{p\alpha}^2\int\limits_{-\infty}^{\infty}\mathrm{d} v_\parallel \int\limits_0^\infty\frac{J_n^2f_{\alpha0}v_\perp} {\omega-n\omega_{c\alpha}}\,\mathrm{d} v_\perp = 0\;. \end{aligned} $$

(4.41)

One solution of this equation is the so-called ordinary mode (O-mode), which is also found in cold plasma approximation. K _zz = 0 furthermore gives a series of modes with narrow bands slightly above the harmonics of the cyclotron frequency

$$\displaystyle \begin{aligned} \omega = n\omega_{c\alpha}\left\lbrace 1+ \mathcal{O}\left[ \frac{\omega_{p\alpha}^2}{k^2c^2}(kr_{L\alpha})^{2n}\right]\right\rbrace\,, \end{aligned} $$

(4.42)

where r _Lα is the gyro radius of species α and $\mathcal O$ indicates terms of the order of its argument, in this case small as compared to 1. These modes are electrostatic cyclotron waves . Both electrons and all ion species have their own families of electrostatic cyclotron modes.

The remaining perpendicular propagating modes are found from the determinant

$$\displaystyle \begin{aligned} \left\vert \begin{array}{cc} K_{xx} & K_{xy} \\ -K_{xy} & K_{yy} \end{array} \right\vert = 0\;. \end{aligned} $$

(4.43)

This equation covers the electromagnetic modes for which |E ⋅k|≪|E ×k|. They are called extraordinary modes (X-modes), which are also found in cold plasma theory.

At frequencies below the cold plasma lower hybrid resonance frequency ω _LHR (defined by Eq. 4.70 below) the X-mode is often called the magnetosonic mode . It is an extension of the perpendicular propagating magnetosonic mode of MHD (Sect. 4.4) from frequencies below the ion gyro frequency to higher frequencies. In the finite-temperature Vlasov theory the X-mode has quasi-resonances, where the group velocity of the wave (A.28) ∂ω∕∂ k → 0, at the multiples of the ion gyro frequency nω _ci for n ≥ 1 up to ω _LHR. This gives to the X-mode wave an observable banded structure, an example of which is shown in Fig. 5.14. These bands were first identified in Bernstein’s dielectric tensor (4.34) and they are, consequently, known as Bernstein modes.

Another set of Bernstein-mode solutions of (4.43) is found at short wavelengths. These modes are quasi-electrostatic (|E ⋅k|≫|E ×k|) and they are found both for electrons and all ion species. The exactly perpendicular modes are not Landau damped. If the modes have finite k _∥, they experience cyclotron damping when n ≠ 0.

4.2.4.3 Propagation to Arbitrary Directions

A convenient way to illustrate the wave solutions at arbitrary directions of the wave vector ω = ω(k _∥, k _⊥) is to represent them as dispersion surfaces in three-dimensional (ω, k _∥, k _⊥)-space. An example of dispersion surfaces is given in Fig. 4.3. The surface has been calculated by solving Eq. (4.37) for plasma parameters corresponding to the inner magnetosphere slightly outside the plasmapause using the numerical dispersion equation solver WHAMP (Waves in homogeneous anisotropic magnetized plasmas) originally written by Kjell Rönnmark.¹ Figure 4.3 illustrates how the parallel propagating right-hand polarized whistler mode joins the perpendicular propagating X-mode when the direction of the wave vector is rotated from parallel toward perpendicular direction, For further examples of dispersion surfaces, see, e.g., André (1985) or Koskinen (2011).

Fig. 4.3

The dispersion surface that contains the parallel propagating right-hand polarized whistler mode and perpendicular propagating X-mode. The frequency on the vertical axis is normalized to the local proton gyro frequency and the parallel and perpendicular wave numbers are normalized to the proton gyro radius (Figure courtesy: Yann Pfau-Kempf)

Fig. 4.4

Parallel propagation in high plasma density approximation, which is a good approximation in the radiation belts. The red lines indicate the R-modes and the blue lines the L-modes. Cut-offs are found where the dispersion curve meets the frequency axis (k = 0) and resonances at the limit k →∞

The growth/damping rate ω _i varies from one point to another on a dispersion surface and the solution may in some domains of the surfaces be strongly damped. Depending on the local plasma parameters and characteristics of the particle populations, there may be free energy to drive the instabilities leading to growing solutions of the dispersion equation. These are discussed using practical examples in Chap. 5.

4.3 Cold Plasma Waves

While the Vlasov theory is necessary for the treatment of the growth and damping of plasma waves, the real part of the dispersion equation for several of the most important linear wave modes in radiation belt physics can be derived from the much simpler cold plasma theory.

4.3.1 Dispersion Equation for Cold Plasma Waves in Magnetized Plasma

From Maxwell’s equations and Ohm’s law J = σ ⋅E, where σ is generally a tensor, it is straightforward to derive a wave equation in the form

$$\displaystyle \begin{aligned} \mathbf{k}\times(\mathbf{k}\times\mathbf{E}) + \frac{\omega^2}{c^2}\,{\mathsf K}\cdot\mathbf{E} = 0\,, \end{aligned} $$

(4.44)

where

$$\displaystyle \begin{aligned} {\mathsf K} = {\mathsf I} + \frac{\mathrm{i}}{\omega\epsilon_0}\sigma \end{aligned} $$

(4.45)

is the dielectric tensor and I the unit tensor. K is a dimensionless quantity and we can relate it to the electric permittivity of the dielectric medium familiar from classical electrodynamics as

$$\displaystyle \begin{aligned} \mathbf{D} = \epsilon\cdot\mathbf{E} = \epsilon_0\,{\mathsf K}\cdot\mathbf{E}\,. \end{aligned} $$

(4.46)

In case of no background fields (E ₀ = B ₀ = 0) K reduces to a scalar

$$\displaystyle \begin{aligned} K = 1 - \frac{\omega_{pe}^2}{\omega^2} \equiv n^2\,, \end{aligned} $$

(4.47)

i.e., K is the square root of the refractive index n defined in Appendix A (A.24). The wave equation has the already familiar solutions

$$\displaystyle \begin{aligned} \begin{array}{ccll} \mathbf{k}\parallel\mathbf{E} &\Rightarrow& \omega^2=\omega_{pe}^2&\ \ \mbox{longitudinal}\ \mbox{standing}\ \mbox{plasma}\ \mbox{oscillation} \\ \mathbf{k}\perp\mathbf{E} &\Rightarrow& \omega^2=k^2c^2+\omega_{pe}^2&\ \ \mbox{electromagnetic}\ \mbox{wave}\ \mbox{in}\ \mbox{plasma} \end{array} \end{aligned}$$

Consider small perturbations B ₁ to a homogeneous background magnetic field B ₀ (B ₁ ≪ B ₀). In the cold plasma approximation all particles of species α are assumed to move at their macroscopic fluid velocity V _α(r, t). Thus the total plasma current is

$$\displaystyle \begin{aligned} \mathbf{J} = \sum_\alpha n_\alpha q_\alpha{\mathbf{V}}_\alpha\,. \end{aligned} $$

(4.48)

Assuming that V _α(r, t) oscillates sinusoidally $\propto \exp (-\mathrm{i} \omega t)$ the first-order macroscopic equation of motion is

$$\displaystyle \begin{aligned} -\mathrm{i} \omega{\mathbf{V}}_\alpha = q_\alpha(\mathbf{E} + {\mathbf{V}}_\alpha\times{\mathbf{B}}_0)\,. \end{aligned} $$

(4.49)

It is convenient to consider the plane perpendicular to B ₀ as a complex plane and use the basis of unit vectors $\lbrace \sqrt {1/2}({\mathbf {e}}_x+\mathrm{i} {\mathbf {e}}_y),\sqrt {1/2}({\mathbf {e}}_x-\mathrm{i} {\mathbf {e}}_y), {\mathbf {e}}_z\rbrace$ where B ₀ ∥e _z. Denote the components in this basis by integers d = {−1, 1, 0} and express the plasma and gyro frequencies as

$$\displaystyle \begin{aligned} X_\alpha = \frac{\omega_{p\alpha}^2}{\omega^2}\ ,\ Y_\alpha = \frac{s_\alpha\omega_{c\alpha}}{\omega}\,. \end{aligned} $$

(4.50)

Note that ω _cα is an unsigned quantity and the sign of the charge is indicated by s _α. In this basis the components of the current are

$$\displaystyle \begin{aligned} J_{d,\alpha}=\mathrm{i}\epsilon_0\omega\,\frac{X_\alpha}{1-dY_\alpha}\,E_d \end{aligned} $$

(4.51)

and the dielectric tensor (4.45) is diagonal

$$\displaystyle \begin{aligned} {\mathsf K} = \left\lbrack \begin{array}{ccc} 1-\sum_\alpha\displaystyle{\frac{X_\alpha}{1-Y_\alpha}} & 0 & 0\\ 0 &1-\sum_\alpha\displaystyle{\frac{X_\alpha}{1+Y_\alpha}}& 0\\ 0&0&1-\sum_\alpha X_\alpha \end{array} \right\rbrack\,. \end{aligned} $$

(4.52)

The components of the tensor are denoted by letters R, L and P:

$$\displaystyle \begin{aligned} \begin{array}{rcl} R & =&\displaystyle 1-\sum_\alpha\frac{\omega_{p\alpha}^2}{\omega^2} \left(\frac{\omega}{\omega+s_\alpha\omega_{c\alpha}}\right) \end{array} \end{aligned} $$

(4.53)

$$\displaystyle \begin{aligned} \begin{array}{rcl} L & =&\displaystyle 1-\sum_\alpha\frac{\omega_{p\alpha}^2}{\omega^2} \left(\frac{\omega}{\omega-s_\alpha\omega_{c\alpha}}\right) \end{array} \end{aligned} $$

(4.54)

$$\displaystyle \begin{aligned} \begin{array}{rcl} P & =&\displaystyle 1-\sum_\alpha\frac{\omega_{p\alpha}^2}{\omega^2}\,. \end{array} \end{aligned} $$

(4.55)

The component R has a singularity when ω = ω _ce and s _α = −1. At this frequency the wave is in resonance with the gyro motion of the electrons. Thus R corresponds to the right-hand circularly polarized wave. Similarly L has a resonance with positive ions and corresponds to the left-hand circularly polarized wave. Recall that different from optics, where the handedness is given by the sense of the rotation of the wave electric field approaching the observer, the left- and right-handedness in magnetized plasmas correspond to the sense of the gyro motion of charged particles around the background magnetic field in the frame of reference of the guiding center. If the observer looks into the direction to which magnetic field points, the left-hand polarized wave rotates in the same sense as the gyro motion of positively charged particle. If the observer looks against the magnetic field, the rotation of the wave appears right-handed.

The component P corresponds to a standing plasma oscillation in the cold plasma approximation. As discussed in the context of Vlasov theory above, a finite temperature makes the plasma oscillation a propagating Langmuir wave.

Transforming K back to the {x, y, z}-basis we get

$$\displaystyle \begin{aligned} {\mathsf K} = \left\lbrack \begin{array}{ccc} S & -\mathrm{i} D & 0\\ \mathrm{i} D & S & 0\\ 0 & 0 & P \end{array} \right\rbrack\,, \end{aligned} $$

(4.56)

where S = (R + L)∕2 and D = (R − L)∕2.

The wave equation can be written in terms of the wave normal vector n = c k∕ω as

$$\displaystyle \begin{aligned} \mathbf{n}\times(\mathbf{n}\times\mathbf{E}) + {\mathsf K}\cdot\mathbf{E} = 0\,. \end{aligned} $$

(4.57)

Recall that B ₀ is in the z-direction. Select the x-axis so that n is in the xz-plane. The angle θ between n and B ₀ is the wave normal angle (WNA). In these coordinates the wave equation is

$$\displaystyle \begin{aligned} \left\lbrack \begin{array}{ccc} S-n^2\cos^2\theta & -\mathrm{i} D & n^2\cos\theta\sin\theta\\ \mathrm{i} D & S-n^2 & 0\\ n^2\cos\theta\sin\theta & 0 & P-n^2\sin^2\theta \end{array} \right\rbrack \left\lbrack \begin{array}{c} E_x\\ E_y\\ E_z \end{array} \right\rbrack =0\,. \end{aligned} $$

(4.58)

The solutions of the wave equation are the non-trivial roots of the dispersion equation

$$\displaystyle \begin{aligned} An^4 - Bn^2 + C = 0\,, \end{aligned} $$

(4.59)

where

$$\displaystyle \begin{aligned} \begin{array}{rcl} A & =&\displaystyle S\sin^2\theta + P\cos^2\theta \\ B & =&\displaystyle RL\sin^2\theta + PS(1+\cos^2\theta) \\ C & =&\displaystyle PRL\,. \end{array} \end{aligned} $$

(4.60)

It is convenient to solve the dispersion equation (4.59) for $\tan ^2\theta$ as

$$\displaystyle \begin{aligned} \tan^2\theta = \frac{-P(n^2-R)(n^2-L)}{(Sn^2-RL)(n^2-P)}\,. \end{aligned} $$

(4.61)

With this equation it is straightforward to discuss the propagation of the waves to different directions with respect to the background magnetic field. The modes propagating in the direction of the magnetic field (θ = 0) and perpendicular to it (θ = π∕2) are called the principal modes

$$\displaystyle \begin{aligned} \begin{array}{ll} \theta = 0: &\ \ P=0,\ n^2=R,\ n^2=L \\ \theta = \pi/2: &\ \ n^2=RL/S,\ n^2=P\,. \end{array} \end{aligned}$$

These modes have cut-offs

$$\displaystyle \begin{aligned} \begin{array}{l} n^2\rightarrow 0\ \ (v_p\rightarrow\infty,\ k\rightarrow 0,\ \lambda\rightarrow\infty) \\ P=0,\ R=0,\ \mathrm{or}\ \ L=0 \end{array} \end{aligned}$$

and resonances

$$\displaystyle \begin{aligned} \begin{array}{l} n^2\rightarrow\infty\ \ (v_p\rightarrow 0,\ k\rightarrow \infty,\ \lambda\rightarrow 0) \\ \tan^2\theta = -P/S\ \ (\mbox{under}\ \mbox{the}\ \mbox{condition}\ P\ne 0)\,. \end{array} \end{aligned}$$

When the wave approaches a region where it has a cut-off (n ² → 0), it cannot propagate further and is reflected. At a resonance the wave energy is absorbed by the plasma.

4.3.2 Parallel Propagation (θ = 0)

Figure 4.4 presents the solutions of the cold plasma dispersion equation for parallel propagation.

The resonance frequency of the right-hand polarized mode

$$\displaystyle \begin{aligned} n_R^2 = R = 1 -\sum_i\frac{\omega_{pi}^2}{\omega(\omega + \omega_{ci})} - \frac{\omega_{pe}^2}{\omega(\omega - \omega_{ce})} \end{aligned} $$

(4.62)

is ω = ω _ce. The left-hand polarized mode

$$\displaystyle \begin{aligned} n_L^2 = L = 1 -\sum_i\frac{\omega_{pi}^2}{\omega(\omega - \omega_{ci})} - \frac{\omega_{pe}^2}{\omega(\omega + \omega_{ce})} \end{aligned} $$

(4.63)

has resonances ω = ω _ci for each ion species of different masses.

The low-frequency branches of the left- and right-hand modes propagating below their respective cyclotron frequencies are of particular importance in radiation belt physics. At the low frequency limit $(\omega \rightarrow 0)\ \ n^2\rightarrow c^2/v_A^2$ and L- and R-modes merge to parallel propagating MHD waves at the Alfvén speed v _A = ω∕k (Sect. 4.4).

With increasing k the phase velocities of the L- and R-modes become different. As a linearly polarized wave can be expressed as a sum of left- and right-hand polarized components, this leads to the Faraday rotation of the polarization of linearly polarized waves.

4.3.2.1 Electromagnetic Ion Cyclotron Wave

The parallel propagating left-hand polarized waves below ω _ci of each ion species are the electromagnetic ion cyclotron (EMIC) waves. In the inner magnetosphere the most important ion species are protons and singly charged helium and oxygen ions, the last two being of ionospheric origin. Figure 4.5 is an example of simultaneous observation of hydrogen and helium ion cyclotron waves in the dayside magnetosphere.

Fig. 4.5

Multi-band EMIC wave observation by Van Allen Probe A over a period of 30 min on 14 April 2014, in the noon sector (MLT ≈ 11 and L ≈ 5.7). The uppermost panel shows the magnetic power spectrum in H⁺ and He⁺ bands. In the panel the He⁺ gyro frequency is indicated by the red line. The middle and lower panels indicate that the waves are circularly polarized (ellipticity close to 0) and propagating along the magnetic field (small WNA) (From Fu et al. 2018, reprinted by permission from COSPAR)

4.3.2.2 Whistler Mode

The R-mode propagating at frequencies between ω _ci and ω _ce is known as the whistler mode. If ω _ci ≪ ω ≪ ω _ce the dispersion equation can be approximated by

$$\displaystyle \begin{aligned} k = {\omega_{pe}\over c}\sqrt{\frac{\omega}{\omega_{ce}}} \end{aligned} $$

(4.64)

giving the phase and group velocities

$$\displaystyle \begin{aligned} \begin{array}{rcl} v_p\,\ =\,\ {\omega\over k} & =&\displaystyle \frac{c\sqrt{\omega_{ce}}}{\omega_{pe}}\ \sqrt{\omega} \end{array} \end{aligned} $$

(4.65)

$$\displaystyle \begin{aligned} \begin{array}{rcl} v_g\ =\ \frac{\partial\omega}{\partial k} & =&\displaystyle \frac{2c\sqrt{\omega_{ce}}}{\omega_{pe}}\ \sqrt{\omega}\,. \end{array} \end{aligned} $$

(4.66)

The dispersive whistler mode was identified for the first time during the First World War as descending whistling tones induced into telecommunication cables in the frequency band around 10 kHz. The origin of the signals was not understood until Storey (1953) suggested that the waves originated from wide-band electromagnetic emissions of lightning strokes. A fraction of the wave energy is ducted along the magnetic field as a whistler wave to the other hemisphere. The time of arrival depends on the frequency as

$$\displaystyle \begin{aligned} t(\omega) = \int\frac{\mathrm{d} s}{v_g} = \int\frac{\omega_{pe}(s)}{2c\sqrt{\omega\,\omega_{ce}(s)}}\,\mathrm{d} s\ \propto\frac{1}{\sqrt{\omega}} \end{aligned} $$

(4.67)

implying that the higher frequencies arrive before the lower tones, resulting in the whistling sound when replayed as an audio signal. This explanation was not accepted immediately because it requires a higher plasma density in the inner magnetosphere than was known at the time. Storey actually found the plasmasphere, which has thereafter been thoroughly studied using radio wave propagation experiments and in situ satellite observations.

There are all the time thunderstorms somewhere in the atmosphere and thus the lightning-generated whistlers are continuously observed in the recordings of ground-based VLF receivers. To avoid confusion, sometimes also misunderstanding, it is advisable to dedicate the term “whistler” to the descending-tone lightning-generated signals and call all right-hand polarized waves in this frequency range generally “whistler-mode waves”. For example, the man-made VLF signals from naval communication transmitters, which are known to affect the radiation belts, do not whistle because they are narrow-band signals from the beginning. Also the whistler-mode chorus (Sect. 5.2) and plasmaspheric hiss (Sect. 5.3) waves, which are most important in radiation belt physics, are different from lightning-generated whistlers. For example, the chorus is composed of rising tones, reflecting the local nonlinear physics in the inner magnetosphere rather than long-distance propagation.

4.3.3 Perpendicular Propagation (θ = π∕2)

The perpendicular propagating ordinary and extraordinary electromagnetic waves were already introduced in Vlasov theory (Sect. 4.2.4).² Figure 4.6 shows their dispersion curves.

Fig. 4.6

Sketch of perpendicular propagating waves in cold plasma approximation. The red curve is the O-mode, which is the same as the electromagnetic wave in cold non-magnetized plasma. The X-mode has three different branches of which the lowermost is the most important in radiation belt context

The ordinary (O) mode is the mode whose index of refraction is

$$\displaystyle \begin{aligned} n_O^2 = P = 1 - \frac{\omega_{pi}^2}{\omega^2} - \frac{\omega_{pe}^2}{\omega^2} \approx 1- \frac{\omega_{pe}^2}{\omega^2}\,. \end{aligned} $$

(4.68)

This is the same as the refractive index of an electromagnetic wave in isotropic plasma (4.47). Its electric field is linearly polarized in the direction of the background magnetic field (E ∥B ₀). For exactly perpendicular propagation the dispersion equation does not contain the magnetic field. The mode has a cut-off at ω = ω _pe (Fig. 4.6).

The extraordinary (X) mode is the solution of $n_X^2 = RL/S$ . With the trivial approximation ω _ce ≫ ω _ci two hybrid resonances are found (Fig. 4.6). The upper hybrid resonance is

$$\displaystyle \begin{aligned} \omega_{UHR}^2 \approx \omega_{pe}^2 + \omega_{ce}^2 \end{aligned} $$

(4.69)

and the lower , written here for one ion species,

$$\displaystyle \begin{aligned} \omega_{LHR}^2 \approx\frac{\omega_{ci}^2+\omega_{pi}^2}{ 1+(\omega_{pe}^2/\omega_{ce}^2)} \approx \omega_{ce}\omega_{ci}\left(\frac{\omega_{pe}^2 + \omega_{ce}\omega_{ci}}{\omega_{pe}^2 + \omega_{pi}^2}\right)\,. \end{aligned} $$

(4.70)

The upper hybrid resonance can be used to determine the plasma density from wave observations if there is an independent way to determine the local magnetic field. The waves propagating close to the lower hybrid resonance frequency are important because they can resonate with both electrons and ions. In low-density plasma (ω _pe ≪ ω _ce) ω _LHR → ω _ci. When ω _pe > ω _ce, as is the case close to the equator within radiation belts, $\omega _{LHR} \approx \sqrt {\omega _{ce}\omega _{ci}}$ is a good approximation.

At the limit of low frequency

$$\displaystyle \begin{aligned} n_X^2\rightarrow 1+\frac{\omega_{pi}^2}{\omega_{ci}^2} = 1+\frac{c^2}{v_A^2}\,. \end{aligned} $$

(4.71)

This is the cold plasma representation of the MHD magnetosonic mode . In MHD (Sect. 4.4) its dispersion equation is found to be

$$\displaystyle \begin{aligned} \frac{\omega^2}{k^2} = v_s^2 +v_A^2\,, \end{aligned} $$

(4.72)

where v _s is the speed of sound. In cold plasma v _s is neglected (→ 0), whereas in MHD the displacement current is neglected corresponding to the limit c →∞. In tenuous space plasmas v _A can, however, be a considerable fraction of c. Combining finite v _s and the cold plasma solution the dispersion equation is

$$\displaystyle \begin{aligned} \frac{\omega^2}{k^2} = \frac{v_s^2 +v_A^2}{1 + v_A^2/c^2}\,. \end{aligned} $$

(4.73)

For increasing k _⊥ the magnetosonic/X-mode branch approaches the lower hybrid resonance.

4.3.4 Propagation at Arbitrary Wave Normal Angles

The propagation of plasma waves at wave normal angles between 0^∘ and 90^∘ depends on the local plasma parameters. In Chaps. 5 and 6 we present several examples of obliquely propagating whistler- and X-mode waves in observations and numerical analyses. As noted at the end of Sect. 4.2.4 the solutions of the dispersion equation propagating at arbitrary WNAs can be represent as dispersion surfaces in three-dimensional (ω, k _∥, k _⊥)-space.

Figure 4.3 was an example of the surface containing the parallel and obliquely propagating right-hand polarized whistler mode and the perpendicular propagating linearly polarized magnetosonic/X-mode below the lower hybrid resonance frequency. While both the whistler mode and the X-mode are observable at frequencies below the lower hybrid resonance frequency, they can be distinguished if enough components of the wave fields are measured to determine the wave polarization.

4.4 Magnetohydrodynamic Waves

The ULF Pc4 and Pc5 waves well below the ion gyro frequency in the magnetosphere belong to the family of magnetohydrodynamic or Alfvén waves. Their wavelengths are comparable to the Earth’s radius and thus the dipole geometry constrains the modes that can propagate in the inner magnetosphere. As will be discussed in Chap. 6, these waves play a major role in the diffusive transport of charged particles in the inner magnetosphere.

4.4.1 Dispersion Equation for Alfvén Waves

We start the discussion by introducing the linearized dispersion equation for Alfvén waves in a homogeneous ambient magnetic field. Consider a compressible, non-viscous, perfectly conductive fluid in a magnetic field described by the MHD equations

$$\displaystyle \begin{aligned} \begin{array}{rcl} \frac{\partial\rho_m}{\partial t} + \nabla\cdot(\rho_m\mathbf{V}) & =&\displaystyle 0 \end{array} \end{aligned} $$

(4.74)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \rho_m\frac{\partial\mathbf{V}}{\partial t} + \rho_m(\mathbf{V}\cdot\nabla)\mathbf{V} & =&\displaystyle -\nabla P + \mathbf{J}\times\mathbf{B} \end{array} \end{aligned} $$

(4.75)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \nabla P & =&\displaystyle v_s^2\nabla\rho_m {} \end{array} \end{aligned} $$

(4.76)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \nabla\times\mathbf{B} & =&\displaystyle \mu_0\mathbf{J} \end{array} \end{aligned} $$

(4.77)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \nabla\times\mathbf{E} & =&\displaystyle -\frac{\partial\mathbf{B}}{\partial t} \end{array} \end{aligned} $$

(4.78)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathbf{E} + \mathbf{V}\times\mathbf{B} & =&\displaystyle 0\;. \end{array} \end{aligned} $$

(4.79)

In Eq. (4.76) we have taken the gradient of the equation of state and introduced the speed of sound

$$\displaystyle \begin{aligned} v_s = \sqrt{\gamma_p P/\rho_m} = \sqrt{\gamma_p k_B/m}\,, \end{aligned} $$

(4.80)

where γ _p is the polytropic index and k _B the Boltzmann constant.

From this set of equations we can eliminate J, E, and P

$$\displaystyle \begin{aligned} \begin{array}{rcl} \frac{\partial\rho_m}{\partial t} + \nabla\cdot(\rho_m\mathbf{V}) & =&\displaystyle 0 \end{array} \end{aligned} $$

(4.81)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \rho_m\frac{\partial\mathbf{V}}{\partial t} + \rho_m(\mathbf{V}\cdot\nabla)\mathbf{V} & =&\displaystyle -v_s^2\nabla\rho_m + (\nabla\times\mathbf{B})\times\mathbf{B}/\mu_0 \end{array} \end{aligned} $$

(4.82)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \nabla\times(\mathbf{V}\times\mathbf{B}) & =&\displaystyle \frac{\partial\mathbf{B}}{\partial t}\;. \end{array} \end{aligned} $$

(4.83)

Assume that in equilibrium the density ρ _m0 is constant and look for the solution in the rest frame of the plasma where V = 0. Furthermore, let the background magnetic field B ₀ be uniform. By considering small perturbations to the variables

$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathbf{B}(\mathbf{r},t) & =&\displaystyle {\mathbf{B}}_0 + {\mathbf{B}}_1(\mathbf{r},t) \end{array} \end{aligned} $$

(4.84)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \rho_m(\mathbf{r},t) & =&\displaystyle \rho_{m0} + \rho_{m1}(\mathbf{r},t) \end{array} \end{aligned} $$

(4.85)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathbf{V}(\mathbf{r},t) & =&\displaystyle {\mathbf{V}}_1(\mathbf{r},t) \end{array} \end{aligned} $$

(4.86)

we can linearize the equations by picking up the first-order terms

$$\displaystyle \begin{aligned} \begin{array}{rcl} \frac{\partial\rho_{m1}}{\partial t} + \rho_{m0}(\nabla\cdot{\mathbf{V}}_1) & =&\displaystyle 0 \end{array} \end{aligned} $$

(4.87)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \rho_{m0}\frac{\partial{\mathbf{V}}_1}{\partial t} + v_s^2\nabla\rho_{m1} + {\mathbf{B}}_0\times(\nabla\times{\mathbf{B}}_1)/\mu_0 & =&\displaystyle 0 \end{array} \end{aligned} $$

(4.88)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \frac{\partial{\mathbf{B}}_1}{\partial t}-\nabla\times({\mathbf{V}}_1\times{\mathbf{B}}_0) & =&\displaystyle 0\;.{} \end{array} \end{aligned} $$

(4.89)

From these we find an equation for the velocity perturbation V ₁

$$\displaystyle \begin{aligned} \frac{\partial^2{\mathbf{V}}_1}{\partial t^2} - v_s^2\nabla(\nabla\cdot{\mathbf{V}}_1) +{\mathbf{v}}_A\times\lbrace\nabla\times\lbrack\nabla\times({\mathbf{V}}_1\times{\mathbf{v}}_A)\rbrack\rbrace = 0\;, \end{aligned} $$

(4.90)

where we have introduced the Alfvén velocity as a vector

$$\displaystyle \begin{aligned} {\mathbf{v}}_A = \frac{{\mathbf{B}}_0}{\sqrt{\mu_0\rho_{m0}}}\;. \end{aligned} $$

(4.91)

By looking for plane wave solutions ${\mathbf {V}}_1(\mathbf {r},t) = {\mathbf {V}}_1\exp \lbrack \mathrm{i} (\mathbf {k}\cdot \mathbf {r}-\omega t)\rbrack$ we get an algebraic equation

$$\displaystyle \begin{aligned} -\omega^2{\mathbf{V}}_1 + v_s^2(\mathbf{k}\cdot{\mathbf{V}}_1)\mathbf{k} -{\mathbf{v}}_A\times\lbrace\mathbf{k}\times\lbrack\mathbf{k}\times({\mathbf{V}}_1\times{\mathbf{v}}_A)\rbrack\rbrace = 0\,. \end{aligned} $$

(4.92)

After straightforward vector manipulation this leads to the dispersion equation for ideal MHD waves

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle -\omega^2{\mathbf{V}}_1 + (v_s^2+v_A^2)(\mathbf{k}\cdot{\mathbf{V}}_1)\mathbf{k}+ \\ & &\displaystyle +(\mathbf{k}\cdot{\mathbf{v}}_A)\lbrack((\mathbf{k}\cdot{\mathbf{v}}_A){\mathbf{V}}_1 - ({\mathbf{v}}_A\cdot{\mathbf{V}}_1)\mathbf{k} -(\mathbf{k}\cdot{\mathbf{V}}_1){\mathbf{v}}_A)\rbrack = 0\;. {} \end{array} \end{aligned} $$

(4.93)

4.4.1.1 Parallel Propagation

For k ∥B ₀, the dispersion equation reduces to

$$\displaystyle \begin{aligned} (k^2v_A^2-\omega^2){\mathbf{V}}_1 +\left(\frac{v_s^2}{ v_A^2}-1\right)k^2({\mathbf{V}}_1\cdot{\mathbf{v}}_A){\mathbf{v}}_A =0\;. \end{aligned} $$

(4.94)

This describes two different wave modes. V ₁ ∥B ₀ ∥k yields the sound wave

$$\displaystyle \begin{aligned} \frac{\omega}{k} = v_s\;. \end{aligned} $$

(4.95)

The second solution is a linearly polarized transverse wave with V ₁ ⊥B ₀ ∥k. Now V ₁ ⋅v _A = 0 and we find the Alfvén wave

$$\displaystyle \begin{aligned} \frac{\omega}{k} = v_A\;. \end{aligned} $$

(4.96)

The magnetic field of the Alfvén wave is

$$\displaystyle \begin{aligned} {\mathbf{B}}_1 = -\,\frac{{\mathbf{V}}_1}{\omega/k}B_0\,. \end{aligned} $$

(4.97)

The wave magnetic and electric fields are perpendicular to the background field. This mode does not perturb the density or pressure but causes shear stress on the magnetic field (∇⋅ (BB)∕μ ₀). Consequently, it is also called the shear Alfvén wave.

Parallel propagating linearly polarized waves can be decomposed to left- and right-handed circularly polarized components. With increasing k the circularly polarized components of the Alfvén wave split to two branches found in cold plasma theory: the left-hand polarized electromagnetic ion cyclotron wave approaching the ion cyclotron frequency from below and the right-hand polarized whistler mode (Fig. 4.4). Physically, this splitting is due to the decoupling of the electron and ion motions through the Hall effect (3.41).

4.4.1.2 Perpendicular Propagation

Perpendicular propagation (k ⊥B ₀) implies k ⋅v _A = 0 , and the dispersion equation (4.93) reduces to

$$\displaystyle \begin{aligned} {\mathbf{V}}_1 = (v_s^2+v_A^2)(\mathbf{k}\cdot{\mathbf{V}}_1)\mathbf{k}/\omega^2\;. \end{aligned} $$

(4.98)

Clearly k ∥V ₁, and we have found the magnetosonic wave in the MHD approximation.

$$\displaystyle \begin{aligned} \frac{\omega}{k} = \sqrt{v_s^2+v_A^2}\;. \end{aligned} $$

(4.99)

For a plane wave the linearized convection equation (4.89) becomes

$$\displaystyle \begin{aligned} \omega\,{\mathbf{B}}_1 + \mathbf{k}\times({\mathbf{V}}_1\times{\mathbf{B}}_0) = 0\,, \end{aligned} $$

(4.100)

which yields the magnetic field of the wave

$$\displaystyle \begin{aligned} {\mathbf{B}}_1 = \frac{V_1}{\omega/k}{\mathbf{B}}_0\,. \end{aligned} $$

(4.101)

The wave magnetic field is in the direction of the background magnetic field B ₀. The wave electric field is obtained from the ideal MHD Ohm’s law E ₁ = −V ₁ ×B ₀ and is perpendicular to B ₀ and we have obtained the same polarization as in the cold plasma description. In MHD the wave is known as the compressional (or fast) Alfvén (or MHD) wave.

4.4.1.3 Propagation at Oblique Angles

To find the dispersion equation at arbitrary wave normal angles insert θ into the dot products of the dispersion equation. Selecting the z-axis parallel to B ₀ and the x-axis so that k is in the xz-plane, the components of the dispersion equation are

$$\displaystyle \begin{aligned} \begin{array}{rcl} V_{1x}(-\omega^2+k^2v_A^2 + k^2v_s^2\sin^2\theta) + V_{1z}(k^2v_s^2\sin\theta\cos\theta) & =&\displaystyle 0 \end{array} \end{aligned} $$

(4.102)

$$\displaystyle \begin{aligned} \begin{array}{rcl} V_{1y}(-\omega^2+k^2v_A^2\cos^2\theta) & =&\displaystyle 0 \end{array} \end{aligned} $$

(4.103)

$$\displaystyle \begin{aligned} \begin{array}{rcl} V_{1x}(k^2v_s^2\sin\theta\cos\theta) + V_{1z}(-\omega^2 + k^2v_s^2\cos^2\theta) & =&\displaystyle 0\;. \end{array} \end{aligned} $$

(4.104)

The y-component yields a linearly polarized mode with the phase velocity

$$\displaystyle \begin{aligned} \frac{\omega}{k} = v_A\cos\theta\;. \end{aligned} $$

(4.105)

This is the extension of the shear Alfvén wave to oblique directions. It does not propagate perpendicular to the magnetic field, as there ω∕k → 0.

The non-trivial solutions of the remaining pair of equations are found by setting the determinant of the coefficients of V _1x and V _1z equal to zero

$$\displaystyle \begin{aligned} \left(\frac{\omega}{k}\right)^2 = {1\over 2}(v_s^2+v_A^2) \pm {1\over 2}\lbrack(v_s^2+v_A^2)^2 - 4v_s^2v_A^2\cos^2\theta\rbrack^{1/2}\;. \end{aligned} $$

(4.106)

These modes are compressional. The plus sign gives the generalization of the fast MHD mode. It can propagate to all directions with respect to the background magnetic field. The magnetic field and density compressions of the fast mode oscillate in the same phase. The solution with the minus sign is the slow MHD mode . Its density and magnetic perturbations oscillate in opposite phases. The slow mode is strongly damped through the Landau mechanism, the calculation of which requires a kinetic approach.

The discussion above assumes homogeneous magnetic field and isotropic plasma pressure. The simplest extension of MHD into anisotropic plasma is the double adiabatic theory (Chew et al. 1956) with separate equations of state for parallel and perpendicular pressures. This leads to the firehose mode in the direction parallel and the mirror mode perpendicular to the background magnetic field (see, e.g., Koskinen 2011). The mirror mode is of interest in the inner equatorial magnetosphere with anisotropic plasma pressure (P _⊥ > P _∥). Its density and magnetic field oscillate in opposite phases³ similar to the slow-mode wave. Note that the mirror mode propagates perpendicular to the background field, whereas the phase velocity of the slow mode goes to zero when θ → 90^∘. The lowest-frequency long-wavelength ULF oscillations observed in the inhomogeneous magnetosphere can be either slow-mode or mirror-mode waves (e.g., Southwood and Hughes 1983; Chen and Hasegawa 1991).

The compressional MHD waves can steepen to shocks. Fast-mode shocks are ubiquitous in the solar wind. They form when an obstacle moves faster than the local magnetosonic speed, e.g., in front of planetary magnetospheres or when an ICME is fast enough relative to the background flow. Also SIRs gradually develop fast forward and fast reverse shocks, although mostly beyond the Earth orbit. Compressional shocks hitting the magnetopause can launch ULF waves inside the magnetosphere (Sect. 5.4).

Slow-mode shocks are strongly damped and thus difficult to observe. In the magnetosphere they have been found in association with magnetic reconnection where they have an important role decoupling the ion motion from the electron plasma flow and accelerating inflowing ions to the outflow velocities.

4.4.2 MHD Pc4–Pc5 ULF Waves

The wavelengths of magnetospheric MHD waves with periods in the Pc4–Pc5 range (45–600 s, or 1.7–22 mHz) are very long. For example, assuming equatorial Alfvén speed of 300 km s⁻¹, a Pc5 wave with f = 2 mHz has the wavelength of about 10 R _E, which is comparable to the size of the inner magnetosphere. In fact, the frequency of about 1 mHz is in practice the lowest for which the oscillation can still be described as a wave in the inner magnetosphere. At such long wavelengths the assumption of a homogeneous background magnetic field B ₀ assumed in Sect. 4.4.1 is no more valid, nor can the fluctuations be considered as plane waves. The solutions to the full set of coupled nonlinear hydromagnetic equations must be sought using numerical methods. The boundary conditions are usually given at the magnetopause and in the ionosphere.

The ULF waves in the quasi-dipolar inner magnetosphere retain the mode structure of the MHD waves in a homogeneous magnetic field: the shear Alfvén wave with the wave vector along the background magnetic field and the fast compressional mode wave that can propagate to all directions. Because the Alfvén speed in the inner magnetosphere is much larger than the sound speed, the phase speed of the perpendicular propagating fast mode (4.106) can be approximated by the Alfvén speed $v_A = B/\sqrt {\mu _0\rho _m}$.

However, the polarization of the ULF waves becomes more complicated and depends on the background field geometry. In the nearly dipolar inner magnetospheric field the electric and magnetic components of the ULF waves are useful to give in local magnetic field-aligned coordinates. In the literature several different notations are used. A well-motivated convention is to use the right-handed set of unit vectors {e _ν, e _ϕ, e _μ}, where e _μ is along the background magnetic field line, e _ϕ is in the azimuthal direction (eastward) and e _ν = e _ϕ ×e _μ points radially outward in the equatorial plane. The electric and magnetic field components in these directions are often indicated using other subscripts, e.g., {r, a, p} for radial, azimuthal and parallel.

The wave electric field E ₁ of the MHD waves is always perpendicular to the background magnetic field and can thus have only two components δE _ν and δE _ϕ. The wave magnetic field B ₁ is perpendicular to E ₁ and can point to all directions. The different polarizations are characterized according to the appearance of the magnetic field fluctuations. The the wave with the magnetic field in the azimuthal direction, i.e., B ₁ = δB _ϕ is called the toroidal mode corresponding to the shear Alfvén wave. The associated wave electric field must be in the radial direction E ₁ = δE _ν. The poloidal mode , in turn, refers to the fast mode, which can propagate at all wave normal angles. The perpendicular propagating (compressional) mode has the wave magnetic field B ₁ = δB _μ and the parallel propagating mode B ₁ = δB _ν. The associated wave electric field is in both cases azimuthal E ₁ = δE _ϕ. The observed ULF waves usually contain a mixture of the different polarizations.

To keep the discussion simple, let us consider the different polarizations on the dipole equator where e _ν is the radial unit vector e _r and we can use cylindrical coordinates. In the cylindrically symmetric geometry, the total electric field can be expanded in cylindrical harmonics as

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \mathbf{E}(r,\phi,t) = {\mathbf{E}}_0(r,\phi) & +&\displaystyle \sum_{m=0}^\infty\delta E_{rm}\sin{}(m\phi\pm\omega t + \xi_{rm})\,{\mathbf{e}}_r \\ & +&\displaystyle \sum_{m=0}^\infty\delta E_{\phi m}\sin{}(m\phi\pm\omega t + \xi_{\phi m})\,{\mathbf{e}}_\phi\,. \end{array} \end{aligned} $$

(4.107)

Here E ₀(r, ϕ) is the time-independent convection electric field, m is the azimuthal mode number, δE _rm are the amplitudes of the toroidal modes and δE _ϕm of the poloidal modes, and ξ _rm and ξ _ϕm represent their respective phase lags.

Note that the wave field is often expanded in terms of exponential basis functions $\exp (\mathrm{i} (m\phi - \omega t))$. In such an expansion the azimuthal mode number m is an integer from −∞ to ∞. Since ϕ increases eastward, negative m corresponds to a westward and positive m to an eastward propagating wave phase. In the expansion (4.107) the opposite propagation directions correspond to the two signs of ± ωt.

The terminology varies in the literature. Often only the division to toroidal and poloidal modes according to the electric field components is used and the poloidal mode includes both compressional and non-compressional polarizations. Sometimes the parallel propagating (non-compressional) poloidal mode is called the poloidal Alfvén mode to distinguish it from the compressional oscillation. Furthermore, since the observed oscillation typically has both toroidal and poloidal electric field components and all three magnetic field components simultaneously, the combination of toroidal electric field component δE _ν and compressional magnetic field component δB _μ is sometimes considered as another compressional mode. It is, however, a redundant combination of toroidal and compressional polarizations.

Figure 4.7 shows two examples of THEMIS-A satellite observations of Pc5 waves exhibiting all polarizations simultaneously. In both cases the fluctuations took place following solar wind pressure enhancements. During Event A the spacecraft was in the post-midnight sector (about 03 MLT) at L ≈ 10, during Event B on the duskside flank (about 18 MLT) at L ≈ 9. Both events feature clear ULF range fluctuations with all three polarizations superposed. The amplitude of the toroidal mode was the largest during both events. This is consistent with the statistical result of Hudson et al. (2004a), that the Pc5 waves in the dawn and dusk sectors are preferentially toroidal.

Fig. 4.7

Magnetic (blue) and electric field (red) components in magnetic field-aligned coordinates. The data have been band-pass filtered to match the observed ULF wave frequencies in the Pc5 range, 0.9–2.7 mHz (Event A) and 1.8–2.5 mHz (Event B). Here the components are: p is directed along the background magnetic field, r points (nearly) radially outward and a is directed azimuthally eastward (From Shen et al. 2015, reprinted by permission from American Geophysical Union)

ULF waves with small azimuthal mode number m have a predominantly toroidal polarization, whereas waves with large m are primarily poloidal. For limiting cases m = 0 and |m|→∞, the ideal-MHD solution in a dipole field configuration yields purely toroidal and purely poloidal modes, respectively. For finite m, different polarizations are coupled and toroidal and poloidal waves can have both small and large m.

The division of the ULF waves to toroidal, poloidal and compressional waves is further complicated by the fact that the magnetic field and density oscillations are often found to be in opposite phases (e.g., Zhang et al. 2019, and references therein) suggesting that they would be slow-mode waves or, in the case of anisotropic pressure, mirror-mode waves as noted in Sect. 4.4.1. Empirical determination between these is challenging because the direction and velocity of the wave propagation are difficult to observe.

The magnetospheric magnetic field lines are connected to the ionosphere where toroidal Alfvén waves are partially reflected and partially transmitted through the neutral atmosphere to the ground. This allows for remote observations of magnetospheric ULF oscillations using ground-based magnetometers with appropriate sampling rates. The ground-based measurements provide a wider latitudinal and longitudinal coverage of a given wave event than single-point space observations. For example, longitudinally separated magnetometers can be used to determine the azimuthal wave number (m), provided that their distance is smaller than half of the azimuthal wavelength of the oscillation. The waves are, however, attenuated when propagating from the ionosphere to the ground, which obscures their properties.

Let us assume, for illustration, that the ionosphere is a perfectly conductive boundary at both ends of the dipole field flux tubes. Such a flux tube is a wave guide for parallel propagating waves with conductive end plates known in electrodynamics as a resonance cavity. The perfect conductivity implies that the wave electric field vanishes at the end plates and thus only selected wavelengths fulfil Maxwell’s equations. If the length of the field line from one hemisphere to the other is l, the allowed wavelengths are λ _∥ = 2l∕n , where n is an integer. Thus the eigenfrequencies of these field line resonances (FLRs) are

$$\displaystyle \begin{aligned} f = \frac{nv_A}{2l}\,. \end{aligned} $$

(4.108)

The lowest frequency (n = 1) corresponds to a half-wave that has maximum amplitude at the dipole equator, as do the odd harmonics (n = 3, 5, …) as well. The even harmonics (n = 2, 4, …) in turn have minima at the equator. Having observations of the magnetic field and plasma density the eigenfrequencies can be estimated and related to the observed frequency of the oscillation. The 90-degree time lag between the radial electric component and azimuthal magnetic component of the toroidal mode in Fig. 4.7 is an indication that the observed toroidal oscillation was a standing field line resonance, as expected from the theory of standing electromagnetic waves in resonant cavities.

Another resonance cavity may form for perpendicular propagating waves between the dayside magnetopause and the near-equatorial ionosphere leading to standing cavity mode oscillations (CMOs), again with vanishing electric field at the boundaries. When compressional waves launched externally at the dayside magnetopause propagate inward, they are attenuated. The cavity modes peak at resonant L–shells where the frequency matches with the field line resonance of the toroidal (shear) wave, and the shear mode is amplified at the expense of the compressional mode (Kivelson and Southwood 1986).

4.5 Summary of Wave Modes

To keep track of the multitude of wave modes in the radiation belts, the most important wave modes for our treatise are briefly summarized in Table 4.2.

Table 4.2 Key wave modes in the inner magnetosphere relevant for dynamics of radiation belt particles, their frequencies, polarization and dominant wave normal angles