Physics of Fusion power

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Physics of Fusion power

• Lecture4 Quasi-neutrality Force on the plasma

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Quasi-neutrality

• Using the Poisson equation
• And a Boltzmann relation for the densities
• One arrives at an equation for the potential

Positive added charge Response of the
plasma
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Solution

• The solution of the Poisson equation is

Potential in vacuum Shielding due to the
charge screening
Vacuum and plasma solution
The length scale for shielding is the Debye
length which depends on both Temperature as well
as density. It is around 10-5 m for a fusion
plasma
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Quasi-neutrality

• For length scales larger than the Debye length
  the charge separation is close to zero. One can
  use the approximation of quasi-neutrality
• Note that this does not mean that there is no
  electric field in the plasma
• Under the quasi-neutrality approximation the
  Poisson equation can no longer be used to
  calculate the electric field

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Divergence free current

• Using the continuity of charge
• Where J is the current density
• One directly obtains that the current density
  must be divergence free

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Also the displacement current must be neglected

• From the Maxwell equation
• Taking the divergence and using that the current
  is divergence free one obtains
• The displacement current must therefore be
  neglected, and the relevant equation is

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Quasi-neutrality

• The charge density is defined to be equal to zero
  (but a finite electric field does exist)
• This equation replaces the Poisson equation. (we
  do not calculate the electric field from
  Poissons equation, which would give zero field)
• Additionally, the displacement current is
  neglected.
• Length scales of the phenomena are larger than
  the Debye length, time scales longer than the
  plasma frequency.
• The current is divergence free.

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Force on the plasma

• The force on an individual particle due to the
  electro-magnetic field (s is species index)
• Assume a small volume such that
• Then the force per unit of volume is

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Force on the plasma

• For the electric field
• Define an average velocity
• Then for the magnetic field

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Force on the plasma

• Averaged over all particles
• Now sum over all species
• The total force density therefore is

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Force on the plasma

• This force contains only the electro-magnetic
  part. For a fluid with a finite temperature one
  has to add the pressure force

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Reformulating the Lorentz force

• Using
• The force can be written as
• Then using the vector identity

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Force on the plasma

• One obtains
• Important parameter (also efficiency parameter)
  the plasma-beta

Magnetic field pressure Magnetic field tension
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Magnetic equilibria

• For stationary magnetic confinement device force
  balance implies than inside the plasma
• Unfortunately, an isolated region plasma cannot
  be confined by its own currents.
• Pressure balance requires external confining force

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Flux conservation

• When trying to change the magnetic flux through a
  metal ring an electric field is generated
  (Faraday) which drives a current such that it
  tries to conserve the flux
• The current eventually decays due to the
  resistivity
• A perfect conductor, however, would conserve the
  magnetic flux

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Flux conservation

• A plasma is like a metal (electrons are free)
• A hot plasma has a small resistivity
• As a first approximation it is perfectly
  conducting
• Flux is then conserved but the fluid can be
  moving
• Flux is transported along with the fluid