### 5.2 Dispersion relations and Landau and cyclotron resonance

Usually, a (warm and multi-component) plasma is analysed for stability by Fourier decomposing the linear fluctuations of the electromagnetic fields into plane waves with a given wave vector . Each species contributes a distinct dispersion branch of kinetic waves, which are weakly damped normal modes or unstable in the presence of free energy. For the general dispersion relation we refer to the standard text book literature, e.g., Stix (1992), Melrose and McPhedran (1991), or Baumjohann and Treumann (1996). As an important example, we present the dispersion equation for parallel propagating left ( sign) and right (+ sign) handed circularly polarised electromagnetic waves with wave vector component along the background magnetic field, . This dispersion equation can be written as:
where the fractional mass density, , has been used. For the low-frequency ion wave modes below the displacement current term can be neglected, since usually . On the other hand without plasma, i.e., for , we obtain the free-space electromagnetic wave. In the gyrofrequency domain, i.e. for , the typical wave length is just of the order of the gyroradius, , of the species that dominates in mass. The gyroradius is based on the Alfvén speed as defined by the total mass density, , which is most appropriate to normalise any phase speed.

For a given parallel wave vector, , solutions of Equation (41) in terms of the complex frequency are sought, whereby a positive signifies the growth of a plasma microinstability. The dielectric constant involves a resonance integral over the velocity-space derivative of the VDF, corresponding to the pitch-angle gradient of a particle in the wave frame as defined by the phase speed, and reads as follows:

The symbols refer to the random velocity components perpendicular and parallel to in the species proper frame, whereby . The prime at indicates a Doppler shift into the rest frame of species , drifting at bulk speed , i.e. . The VDF is here understood to be normalised to the particle number density . Numerical investigations of Equation (41) have been carried out in the literature under various conditions. The monograph on space plasma instabilities Gary (1993) contains many relevant results for the solar wind and other space plasmas. The multi-component solar wind plasma is usually stable but sometimes close to the margin of a microscopic instability.

When interacting with a wave, a particle sees a stationary electric field if its velocity meets the condition for cyclotron resonance, where . Energy and momentum between particle and wave are exchanged as a result of this wave-particle interaction. The velocity distributions are reshaped, until the free energy in the form of temperature anisotropy, beam drift or ion differential motion, and skewness or heat flux is reduced or removed. These processes are analytically described by quasilinear theory, based on Equation (9) with the fields being decomposed into means and fluctuations, or fully calculated and visualised by direct numerical simulations.

In addition to electromagnetic modes, electrostatic waves frequently occur with variable intensity in different regions of the heliosphere (see the reviews by Gurnett, 1991 and MacDowall and Kellog, 2001). Below the proton plasma frequency one has the ion acoustic modes and above the electron plasma frequency the electrostatic Langmuir oscillations and the free-space electromagnetic waves. These prevailing waves are derived from the simple electrostatic dispersion equation:

where is the Debye wave number of species , given by . We recall that the thermal speed is . The dielectric function involves the Landau-resonance integral over the derivative of the VDF in the -direction. The dispersion relation (43) gives the wave vector in dependence on the propagation direction and phase speed . Here is the complex frequency as a function of . The wavelength may vary between infinity and the very short electron Debye length . Observationally, the occurrence of ion acoustic waves was found (Gurnett, 1991) to be correlated with the electron to proton temperature ratio, , and the magnetic field direction. Highest wave intensities are observed around the heliospheric current sheet (MacDowall and Kellog, 2001), where usually , implying weak Landau damping.