Chapter 3. Transport Equations The plasma flows in planetary ionospheres can be in equilibrium (d/dt = 0), like the midlatitude terrestrial ionosphere, or in noneqilibrium (d/dt  0). Different temperatures of the interacting species, or flow speeds in excess of the thermal speed can cause nonequilibrium flow. Transport equations are required to describe the flow. Reasonable simplifying assumptions are usually made in applications

3.1 Boltzmann Equation (1) Boltzmann’s probability density f s for each species s. The number of particles in a “phase space” volume element d 3 r d 3 v is f s d 3 r d 3 v s.

3.1 Boltzmann Equation (2) If there are collisions, the densities in phase space will change, df s /dt =  f s /  t (Boltzmann collision integral), resulting in the Boltzmann equation:

3.2 Velocity Moments of Distribution Functions (1)

Velocity Moments of Distribution Functions (2)

Moments of Distribution Functions (3)

Moments of Distribution Functions (4)

3.3 Transport Equations (1) We can use the Boltzmann equation to describe the evolution in space and time of the physically important velocity moments. Use the following relation to rewrite the Boltzmann equation:

3.3 Transport Equations (2) Miraculously we can use the Boltzmann equation (BE) to derive the continuity and momentum equations. Integrating BE over all velocities gives

3.3 Transport Equations (3) To obtain the momentum equation we multiply the BE with m s c s and integrate over all velocities.

3.3 Transport Equations (4)

3.3 Transport Equations (5)

3.3 Transport Equations (6) This leaves

3.3 Transport Equations (7)

Discussion of Transport Equations (8) The momentum equation describes the evolution of the first-order velocity momentum u s in terms of the second-order momentum P s. Similarly the continuity equation describes the evolution of the density n s (zero-order momentum) using the first-order velocity momentum u s, etc. This means, the transport equations are not a closed system. To “close” the system we need an approximate distribution function f s (r,v s,t). We will use the local drifting Maxwellian.

3.4 Maxwellian Velocity Distribution Function (1) When collisions dominate, the distribution function becomes Maxwellian (no prove given here). In the following I suppress the species subscript s.

Maxwellian Velocity Distribution Function (2)

Maxwellian Velocity Distribution Function (3)

3.5 Closing the System of Transport Equations (1) To close the system of transport equations we need a function f. One generally uses an orthogonal series expansion for f as shown below:

3.5 Closing the System of Transport Equations (2)

3.6 Maxwell Equations The electric and magnetic fields E and B in a medium are related by Maxwell’s equations: