Chapter 3. Transport Equations

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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

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3.1 Boltzmann Equation (1)

• Boltzmanns probability density fs for each
  species s.
• The number of particles in a phase space
  volume element d3r d3v is fs d3r d3vs.

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3.1 Boltzmann Equation (2)

• If there are collisions, the densities in phase
  space will change, dfs/dt ?fs/?t (Boltzmann
  collision integral), resulting in the Boltzmann
  equation

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3.2 Velocity Moments of Distribution Functions (1)
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Velocity Moments of Distribution Functions (2)
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Moments of Distribution Functions (3)
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Moments of Distribution Functions (4)
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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

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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

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3.3 Transport Equations (3)

• To obtain the momentum equation we multiply the
  BE with mscs and integrate over all velocities.

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3.3 Transport Equations (4)
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3.3 Transport Equations (5)
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3.3 Transport Equations (6)

• This leaves

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3.3 Transport Equations (7)
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Discussion of Transport Equations (8)

• The momentum equation describes the evolution of
  the first-order velocity momentum us in terms of
  the second-order momentum Ps. Similarly the
  continuity equation describes the evolution of
  the density ns (zero-order momentum) using the
  first-order velocity momentum us, etc. This
  means, the transport equations are not a closed
  system.
• To close the system we need an approximate
  distribution function fs(r,vs,t). We will use the
  local drifting Maxwellian.

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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.

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Maxwellian Velocity Distribution Function (2)
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Maxwellian Velocity Distribution Function (3)
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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

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3.5 Closing the System of Transport Equations (2)
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3.6 Maxwell Equations

• The electric and magnetic fields E and B in a
  medium are related by Maxwells equations

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