## Chapter 1:

A plasma can be either collisionless or collisional. In a collisionless plasma, the motion of electrons and ions is determined by the electromagnetic forces via the Lorentz force equation. In general one can adopt two approaches to finding out how the particles move in space as a function of time. In one approach, it is assumed that the electromagnetic fields are constant in time (but not necessarily in space). This is also known as the single particle picture of the plasma. In the other approach, one allows for the fact that the electric and magnetic fields in the plasma are not necessarily constant in time. Due to the continuous motion of the electrons and ions in the plasma, it is possible for the current and charge densities to change with time. This, in turn, leads to the variation of electric and magnetic fields in space and time as prescribed by Maxwell's equations. Depending on the nature of the problem at hand, one or the other approach is most useful. For example, if one is interested in how electrons or ions move in the Earth's magnetic field, the single particle approach provides a good description. On the other hand, there are many instances where the single particle method does not do a good enough job and one must allow for the time variations of the electromagnetic fields. In this module, the way in which such situations are handled is described.

## Chapter 2:

In general, one can study the time evolution of the plasma by finding the solution to the Lorentz-Maxwell equations. This has to be done in a self-consistent manner meaning that the particle positions and velocities (and therefore charge and current densities) at any given time must be consistent with the electromagnetic fields. Thus, as the particles move around in a plasma, the electromagnetic fields change with time. One of the fascinating aspects of plasmas is the fact that under certain circumstances these changes are very small and can be viewed as noise, while at other times the fields change drastically. In the former case, the plasma is said to be in a stable configuration where as in the latter case the plasma is unstable. One way in which one can determine if the plasma is stable or not is through the use of velocity distribution function. When the plasma has an unstable velocity distribution function, some of the existing noise in the system begin to grow in amplitude. In this respect, unstable plasmas behave very much like an LRC circuit where an initial signal with a characteristic frequency is amplified to larger amplitudes. In the initial stages of wave growth, in an unstable plasma, the amplitude of the oscillating electromagnetic fields is small enough that one can ignore their effect on the motion of the particles. This early phase of the instability is referred to as the linear stage. Ignoring the effects of the oscillating electromagnetic fields on the particle motion, allows for a number of mathematical simplifications which in turn make it possible to determine which waves are likely to grow in the plasma. In other words, there are mathematical tools which allow us to determine what waves (i.e. wavelength and frequency) will experience growth for a given plasma environment. This mathematical tool is also referred to as linear theory signifying the fact that it is most useful when the plasma is in a linear stage, i.e. the wave amplitudes are small.

## Chapter 3:

As the waves grow, their influence on particle motion becomes larger and one can no longer use linear theory. The exact mathematical treatment of the plasma in this nonlinear stage is much more difficult. In general, the waves interact with particles in such a way that the velocity distribution function evolves toward a more stable configuration. This in turn limits the growth of the waves and eventually the plasma reaches a new equilibrium state. In this state, the velocity distribution function is no longer unstable and the electromagnetic fields in the plasma exhibit some level of wave activity. To quantify and understand this process in more detail Numerical Simulations are used. As a way of demonstrating the nonlinear stage of wave-particle interactions, we use the excitation of Ion-Acoustic waves by an ion beam instability. As will be shown, the growth of ion-acoustic waves results in the trapping of the ions in the plasma which in turn leads to the saturation of the instability. This nonlinear process is demonstrated by numerical simulation of the ion-acoustic instability.