Fluid equations for mass, momentum and energy

Abstract

In Chapter I have made it clear that the emphasis of this course is on Magnetohydrodynamics (MHD for short). In the previous Chapter we have seen that plasmas are most accurately described by particle distribution functions in phase space. The spatial and temporal evolution of these distribution functions are governed by the Boltzmann equation (2.9). This is a partial differential equation in a 7-dimensional space. Chapman and Enskog were the first to obtain a solution to the Boltzmann equation in 1916 for a mono-atomic gas close to thermodynamic equilibrium. However, in plasma physics we are confronted with a far more difficult situation. The full set of Boltzmann-Maxwell equations (2.7) – (2.9) provides a very detailed and complete description of plasma behaviour. At one end of the spectrum, it contains microscopic information about the orbits of the individual charged particles on the very short cyclotron time scale and the gyro-radius length scale. At the other end, it accurately describes the macroscopic behaviour of large astrophysical and fusion plasmas. The complexity arising from this wide range of plasma physics information makes it virtually impossible to solve the Boltzmann-Maxwell equations (2.7) – (2.9) in any nontrivial situation. The collision term makes life even more complicated. Usually this collision term is written as an integral involving the distribution function itself, so that Boltzmann equation (2.9) is really an integro-differential equation. You do not have to be a big genius to understand that it is horrendously difficult to obtain a solution to the Vlasov equation for collisionlcss plasmas or even worse to the Boltzmann equation under realistic plasma physics conditions. This realization has led to the development of simpler mathematical models with a narrower range of applicability. Magnetohydrodynamics is such a, model. Psychology has probably a name for it; if you cannot get it, you pretend you do not want it. Actually, it often makes sense to not want to solve the Boltzmann equation (2.9) since its solution would give too much and too detailed information. In many cases our interest lies in macroscopic quantities like density, temperature and pressure and how these macroscopic quantities vary in space and time. These macroscopic quantities arc obtained as moments of the distribution function. Clearly, it will be simpler to investigate their evolution than that of the distribution function.