Chapter 3 Plasma as fluids 3.1 Introduction 3.2 Relation of plasma physics to ordinary electromagnetics 3.3 The fluid equation for a plasma 3.4 Fluid drifts perpendicular to B 3.5 Fluid drifts parallel to B

3.1 Introduction In a plasma the situation is much more complicated than single particle motion. One must solve a self-consistent (自恰）problem. About 80% 0f plasma phenomena can be explained by a crude model which was used in fluid mechanics. A more refined theory for plasma is kinetic theory of plasma. some plasma problem neither fluid theory nor kinetic theory. In this time, we have to use the computer to simulation.

3.2 Relation of plasma physics to ordinary electromagnetics Maxwell’s equation In vacuum: In a medium: Permeability 磁导率

In plasma, we only need use the Maxwell’s equation in vacuum, but we have to note that the current j must include the free and magnetization current, charge density has to include the polarization charge density. In general,

However, in Plasma The relation between M and H is no longer linear. The dielectric constant of a plasma we have known that a fluctuating E field gives rise to a polarization current . This leads to a polarization charge given by the equation of continuity: if we consider the effect of , we do not need consider .

This is the low frequency dielectric constant for transverse motions. If and for hydrogen, then This is the low frequency dielectric constant for transverse motions. But we have to note that: the expression for is valid only for and for E perpendicular to B.

3.3 The fluid equation for a plasma In the fluid approximation, we consider the plasma to be composed of two or more interpenetrating fluids. Continuity equation consider a cube element of volume dxdydz, density the rate of change of number of particle in the cube is

define flow velocity of fluid element u as the average over particle velocity: Momentum equation Lorentz force, ignore convection

Convection: momentum change due to particle motion define pressure tensor:

Use Combine Lorentz force and convection: Define the convective derivative (total derivative)

Consider G(x,t) to be any property of a fluid in one-dimensional x space. The change of G with time in a fame moving with the fluid is the sum of two terms: In three dimensions: For example, take G to be the density of cars near a freeway entrance at rush hour. A driver will see the density around him increasing as he approaches the crowed freeway. At the same time, the local streets may be filling with cars that enter from driveways, so that the density will increase even if the observer does not move.

Equation of state Evolution equation for pressure tensor involves heat-flux tensor, a 3rd order velocity moment. To close the system of fluid equations, we need equation of state for the pressure. In the present of strong magnetic field, a useful approximation is the adiabatic invariants of particle motion can be used to derive equation of state for , perpendicular:

second adiabatic invariant parallel: second adiabatic invariant L is the length of plasma along the fieldline take A the cross section area of plasma volume V=AL, nV=const. BA=const. We obtain then These are so-called double adiabatic equation of state. Electric charge and current density : Complete set of fluid equation contain Maxwell’s equation, momentum equation, equation of continuity, equation of state and left two equations.

Diamagnetic drift and diamagnetic current Momentum equation left hand side term is smaller than RHS terms In the lowest order of cross product with

perpendicular component drift Diamagnetic drift No counter part in guiding center drift. Diamagnetic current The induced magnetic field reduces the strength of the confining external field. The diamagnetic current is just the magnetization current.

Parallel pressure balance For isothermal compression Parallel pressure balance Assume scalar pressure, ignore inertial terms in the momentum equation, the parallel direction: assume Assume electron parallel thermal conductivity is large,

Te is constant along field line, This is the Boltzmann relation for electrons. Ion cannot be in Boltzmann equilibrium, because quasi-neutrality require Ion parallel force balance equation :

Electron contribution to ion parallel force balance like an effective ion pressure. Microscopic guiding Macroscopic center drift flow current and curvature