In Chapter 3, we introduced the eikonal approximation for vector wave equations, and considered its application in various settings. Up to now we have postponed a discussion of situations where the approximation breaks down. The most common problems are caustics, tunneling, and mode conversions. These breakdowns are all local in x-space, but they have very different characteristics. Resolution of these three distinct problems require three different strategies, but all benefit from a phase space perspective. In this chapter, we consider caustics, and defer the discussion of tunneling and mode conversion to Chapter 6.

Caustics are associated with singularities that can arise locally when we project the Lagrange manifold of rays from its natural home in ray phase space down to x-space. This leads to a loss of good behavior of the eikonal phase in x-space, and a resulting nonphysical (infinite) prediction for the action density at the caustic. Caustics can result from focusing/refraction of neighboring rays, or their reflection near the plasma boundary, for example. This projection singularity leads to arbitrarily large components of ∂∂θ, and unphysical singularities of the wave amplitude along the ray through Eq. (3.57e). Near caustics, the eikonal approximation is no longer valid in x-space. A local asymptotic solution must be found and matched to the incoming and outgoing rays. This is most often done by transforming to k-space, where the solution is well-behaved.

The theory of caustics is intimately entwined with the stationary phase approximation. Away from caustics, the Fourier transform of an eikonal field is also an eikonal field, when the Fourier integral is evaluated using the stationary phase approximation.

It is a fact of immediate importance to our everyday experience that light nearly always travels in straight lines from the source to our eyes, perhaps scattering off some object along the way. Without the ability to assume this as a fact about the world around us, our extraordinary talent for instinctively comprehending spatial relationships in everyday life would be severely compromised. Consider how much computer power must be expended to disentangle the multiple images of distant galaxies to map the dark matter distribution in the visible universe [MRE+07]. Imagine what life would be like if we had to do similar mental computations just to navigate around the furniture in our living room.

How do we build upon this insight that light nearly always travels in straight lines in order to develop a theory with predictive power?

Many aspects of the properties and behavior of matter at extreme conditions are adequately described by simple fluid theory. Indeed, most of the rest of this text is devoted to developing mathematical models for the properties and behavior of matter at extreme conditions, models that are based on treating the matter as a fluid of interacting ions, electrons, and radiation. The fluid description is adequate largely because the spatial scales and time scales over which the interacting particles, via collisions, establish an equilibrium distribution of particle energies is – as we shall see in this chapter – very short compared to typical time scales over which equilibrium conditions change as a result of the fluid motion, for example. Thus, we do not, in general, have to follow the motion of each individual particle of which the plasma is comprised, but only the bulk motion of larger-scale volume or mass elements of the plasma. This is what is done in a radiation-hydrodynamics simulation code. In subsequent chapters we will learn the mathematical models from which radiation-hydrodynamics simulation codes are constructed, how the codes are constructed, and how to do simulations with a radiation-hydrodynamics code.

There are some phenomena, however, for which the fluid treatment is inadequate, and the macroscopic fluid picture must be replaced by a microscopic kinetic theory. Such situations arise, for example, when and where the three fluids are not in equilibrium with each other, or when multiple ion species are present in the plasma. In those situations where it is necessary to follow the motions and interactions of the individual particles on the very short interaction spatial scales and time scales, we need to use a simulation code that is based on kinetic models rather than fluid models. There are a number of such codes, called “particle-in-cell” (PIC) codes, but we will not discuss these codes in this text. There are also a number of hybrid codes available, that treat the electrons as a fluid and the ions kinetically. We also do not discuss hybrid codes in this text.

We learned in Chapter 1 that there are two principal ways to add energy to matter to bring the matter up to high temperature and pressure: via absorption of energy from an incident particle or photon beam, or via high-velocity collision with other matter. As for the first mechanism – absorption of energy from an incident particle or photon beam – we also learned in Chapter 1 that it is much easier to focus a photon beam to very high energy density since photons, being electrically neutral, are not subject to the Coulomb forces that act to push apart the charged particles making up an electron or ion beam. Accordingly, lasers are commonly used to create extreme conditions in matter. Indeed, lasers can create a very wide range of extreme conditions. They can also create extreme conditions which cannot be created any other way outside astrophysical objects, except by nuclear detonations.

In this chapter we discuss the physical mechanisms by which the energy in a laser beam is absorbed in matter. The physical mechanisms are different for different laser intensities. We also discuss how the absorbed energy gets converted into material pressure. It is the pressure gradient created in the material by the laser energy absorption that drives the material motions that we discuss in much more detail in the next two chapters.

Magnetic fields are often given a secondary position of importance in the study of dense plasma. Indeed, many aspects of plasma may be adequately described without including magnetic fields. There are some physical processes, however, that require consideration of the effects of magnetic fields, and in certain circumstances their effects may dominate.

Magnetic fields are found nearly everywhere there is plasma, from the farthest regions of deep space to that of our closer terrestrial environment. The magnetosphere surrounding the Earth has an induction of order 10–5 G, and the stream of charged particles from the Sun (the solar wind) interacts with it to create the Van Allen radiation belts. As the charged particles spiral down the magnetic field lines toward the Earth's surface, their collisions with the atoms and molecules of the upper atmosphere create the polar aurorae.

A bit farther from Earth, the magnetic fields are responsible for the existence of solar flares. Flares occur around sunspots where intense magnetic fields penetrate the photosphere and link the corona to the solar interior. These flares are powered by the sudden release of magnetic energy stored in the corona and can extend far from the surface of the Sun. One also observes coronal mass ejections, in the form of charged particles, which occur during magnetic reconnection events.

In Section 2.2 we discussed the nature of collisions between electrons and ions, suggesting that we might view plasma as two separate fluids, an electron fluid and an ion fluid, as the electrons and ions have substantial mass differences. Yet the two fluids are not independent since their constituents undergo collisions and thus exchange momentum and energy. There is, however, a third fluid to be considered, namely radiation. In Chapter 3 we found that we could describe radiation as a propagating electromagnetic wave. Since the radiation, though, undergoes interactions with the electron fluid involving exchanges of momentum and energy, the properties of radiation may also be treated with methods not much different from those for particles. In treating the radiation as a fluid of point, massless particles – which are called photons – we can derive an equilibrium distribution function for radiation in much the same way we did in Chapter 2 for ions and electrons.

Radiation as a fluid and the Planck distribution function

Let us begin our discussion of radiation transport by treating the radiation as a photon fluid. Each of these photons carries an energy E with associated frequency ν, where E = hν and ν = ω/2π. These massless particles – the photons – also carry momentum p = E/c = hν/c. In the absence of any matter the photons travel in straight lines at the speed of light, c. Hence, they can transport energy and momentum long distances, very rapidly. Even in the presence of matter, the mean free path of the photons may be much longer than the collisional mean free path of the particles with mass; the speed of photons is much greater than thermal velocities of the massive particles. Radiation also exhibits pressure, and is thus capable of doing work. As we shall see, the interaction of radiation with matter has a special importance. Since the radiation energy is propagated at the speed of light, relativistic effects must be considered, although they are important only in certain respects.

Continuum flux

Our goal in this chapter is to provide a complete mathematical description of plasmas in motion, and to describe how the motion changes with time in response to applied forces. The equations of motion are the central and most essential part of all simulation computer codes. All the other physical processes operative in matter at extreme conditions – thermal energy transport, radiation energy transport, ionization – affect the motion, and in the simulation code are added as source terms to the basic hydrodynamic equations.

We begin by considering motion in one dimension only. We also start with the simplest case of plasma that is a non-viscous and non-conducting fluid. Of course, real plasmas are both viscous and conducting, and we will learn how to add these descriptions to our mathematical modeling in later chapters. We are concerned with describing the bulk motion of the fluid rather than the thermal motion of the constituent particles, so a continuum fluid description will be adequate. We also assume that the bulk material velocities are much less than the velocity of light, so we can treat the fluid as non-relativistic.