Properties of High-Energy-Density Plasmas

Abstract

The discussion of energy in Sect. 2.1 was entirely based on the notion of a polytropic gas. The speed of sound waves, which we found by examining fluctuations in density and velocity, was found to depend upon the variation of pressure with density. These observations reveal the tip of an iceberg, and the iceberg is known as the closure problem. The fluid equations can be derived by taking moments of the velocity distribution of the particles, as is done for example in graduate courses in plasma physics. Here we designate particle velocities by v and fluid velocities by u. Thus the continuity equation is the moment taken with v⁰, the momentum equation is the moment with v, the energy equation is the moment with v², the heat transport equation is the moment with v³, and one can keep going. The closure problem arises because every moment equation contains terms involving the next higher moment. Equation (2.1) involves the momentum (ρu), (2.2) involves the energy density (as p), and (2.3) would involve the heat flux had we not assumed it to be zero. Because we assumed the heat flux to be zero, (2.1) to (2.3) form a closed system of equations. In general, one obtains a closed system of fluid equations by assuming that some moment of the velocity distribution is either zero or a known function of lower moments. As another example, sometimes the energy equation is expressed as an equation for temperature and the heat flux is written as —κ_th∇T, which also produces a closed system of three equations.