### There are 28 modules to our tutorial at present.

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.

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.

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.

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