This chapter is devoted to the analysis of MHD equilibria and stability. By equilibria, we mean a plasma state that is time-independent. Such states may or may not have equilibrium flows. When the states do not have equilibrium flows, that is, U = 0 in some appropriate frame of reference, the equilibria are called magnetostatic equlibria. When the states have flows that cannot be simply eliminated by a Galilean transformation, the equilbria are called magnetohydrodynamic equilibria. When we introduce small perturbations in a particular equilibrium which is itself time-independent, the time dependence of the perturbations determines the stability of the system. If an equilibrium is unstable, the instability typically grows exponentially in time. The mathematical problem for the stability of magnetostatic equilibria is made tractable due to the formulation of the so-called energy principle. It turns out that when MHD equilibria contain flows that are spatially dependent, the power of the energy principle is weakened significantly, and there has been a general tendency to rely on the normal mode method, for which we provide simple examples.

In nature and in the laboratory, plasmas can be stable according to the equations of ideal MHD. However, even ideally stable plasmas can become unstable in the presence of small departures from idealness, such as a small amount of resistivity. This may appear counter-intuitive upon first glance unless one takes into account the fact that in the presence of even small dissipation the frozen field theorem discussed in Chapter 6 is violated, which enables the plasma to access states of lower potential energy through motions that would be forbidden for ideal plasmas, i.e., by allowing magnetic field lines to slip with respect to the plasma fluid. Such instabilities are called resistive instabilities. These instabilities are part of a general class of phenomena called magnetic reconnection, which is a subject of great interest for space, laboratory, and astrophysical plasmas.

Most energy generated in the gravitational collapse to a supernova is radiated in neutrinos, hence the role of these particles in a supernova explosion is crucial. Current models of core collapse supernovae focus on multidimensional hydrodynamics and nuclear burning and treat neutrino transport in a simplified manner. In this last chapter an example of accurate neutrino treatment in a spherically symmetric collapse is given. The role of multidimensional effects is discussed. These results are of interest for the multidimensional models with large-scale convection as well as for the ongoing experimental search for neutrinos from supernovae.

Supernova Models and Neutrinos

Supernovae (SNe) are produced by stars that end their late evolution in a catastrophic explosive process. The name supernova was introduced and the difference between SNe and novae in terms of their estimated explosive powers was described in [515]. The luminosity of a SN at its maximum, which lasts for several days, is comparable to the total luminosity of its host galaxy. It was first hypothesized in [516] that SN explosions should be accompanied by the formation of neutron stars; the neutron had been discovered just two years earlier. The total energy involved into explosion is approximately 1053 erg, mostly released in the form of neutrinos. Approximately 1% of this energy, namely, 1051 erg, is released in the form of kinetic energy of the SN ejecta. Only approximately 1% of that kinetic energy, i.e., 1049 erg, is emitted in the form of photons, which are detected as the SN event [517, 518]. The relevant timescales are as follows: collapse of the core occurs on a ≤ 0.1 s timescale, the SW propagation inside the collapsing core takes ~10 ms, and the neutrino cooling time of the hot neutron star is 10 s.

There are two general types of a SN, classified according to the presence of hydrogen absorption lines in observed spectra. Absorption lines are present in the spectra of Type II SNe and absent in the spectra of Type I SNe. In addition, Type II SNe normally contain compact remnants, although such a remnant was not found in the nearby SN 1987A [519]. In this chapter,mainly Type II SNe, or core-collapse SNe, are discussed.