II' Plasma Physics Fundamentals

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II. Plasma Physics Fundamentals

• 4. The Particle Picture
• 5. The Kinetic Theory
• 6. The Fluid Description of Plasmas

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6. The Fluid Description of Plasmas

• 6.1 The Fluid Equations for a Plasma
• 6.2 Plasma Diffusion
• 6.3 Fluid Model of Fully Ionized Plasmas

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6.3 Fluid Model of Fully Ionized Plasmas

• 6.3.1 The Magnetohydrodynamic Equations
• 6.3.2 Diffusion in Fully Ionized Plasmas
• 6.3.3 Hydromagnetic Equilibrium
• 6.3.4 Plasma b
• 6.3.5 Diffusion of Magnetic Field in a Plasma

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6.3.1 Magnetohydrodynamic Equations

• Goal to derive a single fluid description for a
  fully ionized plasma
• Single-fluid quantities define mass density,
  fluid velocity and current density from the same
  quantities referred to electrons and ions

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Magnetohydrodynamic Equations (II)

• Equation of motion for electron and ions with
  Coulomb collisions, neni and a gravitational
  term (that can be used to represent any
  additional non e.m. force)

• Approximation 1 the viscosity tensor has been
  neglected, acceptable for Larmor radius small
  w.r.t. the scale length of variations of the
  fluid quantities.

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Magnetohydrodynamic Equations (III)

• Approximation 2 neglect the convective term,
  acceptable when the changes produced by the fluid
  convective motion are relatively small

• These equation can be added and by setting
  ppepi, -qiqee and Pei-Pie obtaining

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Magnetohydrodynamic Equations (IV)

• By substituting the definition of the single
  fluid variables r, u and j the equation

can be written as
that is the single fluid equation of motion for
the mass flow. There is no electric force because
the fluid is globally neutral (neni).
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Magnetohydrodynamic Equations (V)

• To characterize the electrical properties of the
  single-fluid it is necessary to derive an
  equation that retains the electric field
• By multiplying the ion eq. of motion by me, the
  electron one by mi, by subtracting them and
  taking the limit me/ migt0, d/dtgt0 it is obtained

that is the generalized Ohms law that includes
the Hall term (jxB) and the pressure effects
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Magnetohydrodynamic Equations (VI)

• Analogous procedures applied to the ion and
  electron continuity equations (multiplying by the
  masses, adding or subtracting the equations) lead
  to the continuity for the mass density rm or for
  the charge density r

• The single-fluid equations of continuity and
  motion and the Ohms law constitute the set of
  magnetohydrodynamic (MHD) equations.

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6.3.2 Diffusion in Fully Ionized Plasmas

• The MHD equations, in absence of gravity and for
  steady-state conditions, with a simplified
  version of the Ohms law, are

• The parallel (to B) component of the last
  equation reduce simply to the ordinary Ohms law

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Diffusion in Fully Ionized Plasmas (II)

• The component perpendicular to B is found by
  taking the the cross product with B

that is
and finally

• The first term is the usual ExB drift (for both
  species together), the second is a diffusion
  driven by the gradient of the pressure

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Diffusion in Fully Ionized Plasmas (III)

• The diffusion in the direction of -grad p
  produces a flux

• For isothermal, ideal gas-type plasma the
  perpendicular flux can be written as

that is a Ficks law with diffusion coefficient
named classical diffusion coefficient
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Diffusion in Fully Ionized Plasmas (IV)

• The classical diffusion coefficient is
  proportional to 1/B2 as in the case of weakly
  ionized plasmas it is typical of a random-walk
  type of process with characteristic step length
  equal to the Larmor radius
• The classical diffusion coefficient is
  proportional to n, not constant, because does not
  describe the scattering with a fixed neutral
  background
• Because the resistivity decreases with T3/2 so
  does the classical diffusion coefficient (the
  opposite of a partially ionized plasma)

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Diffusion in Fully Ionized Plasmas (IV)

• The classical diffusion is automatically
  ambipolar, as it was derived for a single fluid
  (both species are diffusing at the same rate)
• Since the equation for the perpendicular velocity
  does not contain any term along E that depend on
  E itself, it can be concluded that there is no
  perpendicular mobility an electric field
  perpendicular to B produces just a ExB drift.

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Diffusion in Fully Ionized Plasmas (V)

• Experiments with magnetically confined plasmas
  showed a diffusion rate much higher than the one
  predicted by the classical diffusion
• A semiempirical formula was devised this is the
  Bohm diffusion coefficient that goes like 1/B and
  increases with the temperature

• Bohm diffusion ultimately makes more difficult to
  reach fusion conditions in magnetically confined
  plasma

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6.3.3 Hydromagnetic Equilibrium

• The MHD momentum equation, in absence of gravity
  and for steady-state conditions is considered to
  describe an equilibrium condition for a plasma in
  a magnetic field.

• The momentum equation expresses the force balance
  between the pressure gradient and the Lorentz
  force
• In force balance both j and B must be
  perpendicular to grad p j and B must then lie on
  constant p surfaces

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Hydromagnetic Equilibrium (II)
j
B
grad p

• For an axial magnetic field in a cylindrical
  configuration with radial pressure gradient, the
  current must be azimuthal
• The momentum equation in the perpendicular plane
  (w.r.t. B) will then give an expression for j

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Hydromagnetic Equilibrium (II)

• The cross product of the momentum with B yields

and, in the usual approximations, solving for j
yield again the expression for the diamagnetic
current

• From the MHD point of view the diamagnetic
  current is generated by the grad p force that
  interacts (via a cross product) with B

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Hydromagnetic Equilibrium (IV)

• The connection between the fluid and the particle
  point of view was previously discussed the
  diamagnetic current arises from an unbalance of
  the Larmor gyration velocities in a fluid element
• From a strict particle point of view the
  confinement of the plasma with a gradient of
  pressure occurs because each particle guiding
  center is tight to a line of force and diffusion
  is not permitted (in absence of collisions)

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Hydromagnetic Equilibrium (V)

• For the equilibrium case under consideration, the
  momentum equation in the direction parallel to B
  will be simply

where s is a generalized coordinate along the
lines of force.

• For isothermal plasma it will be

then the density is constant along the lines of
force

• This condition is valid only for a static case
  (u0).
• For example in a magnetic mirror there are more
  particles trapped at the midplane (lower line of
  force density) than at the mirror end sections

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6.3.4 Plasma b

• By considering the equation for the force balance
  along with the Amperes law (without displacement
  current)

the current density j can be eliminated as

• The r.h.s. can be neglected in many cases like a
  straight magnetic field or when B varies slowly
  along B itself

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Plasma b (II)

• The condition

implies
that is the sum of the thermal pressure and of
the magnetic pressure

• The quantity b is defined as the ratio between
  the thermal (kinetic) pressure and the magnetic
  pressure

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Plasma b (III)
high B low p
jD
B
low B high p

• The diamagnetic current jD decreases the magnetic
  field inside the plasma keeping the sum of
  thermal and magnetic pressures constant
  everywhere in the cylinder

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Plasma b (IV)

• In a fusion reactor the energy produced is
  proportional to n2 while cost increases with the
  magnetic field
• For magnetically confined plasma experiments the
  plasma b is defined as the ratio between the
  maximum kinetic pressure and the maximum magnetic
  pressure (non local b)
• For economically viable fusion machines is then
  necessary to keep b as high as possible (ideally,
  close to one, or 100)
• Current fusion experiments typically reach bs in
  the order of few percents

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6.3.5 Diffusion of Magnetic Field in a Plasma

• If a plasma has no resistivity (idealized case)
  it will behave like a superconductor
• If a magnetic field is applied to the plasma (or
  the plasma is moving towards a region of higher
  magnetic field) a variation of magnetic field in
  the plasma would produce (Faradays law) an
  electric field
• Electric fields are not allowed in a perfect
  conductor a they would generate infinite currents
• A surface current (on infinitely thin surface
  layer) is then generated to repel the magnetic
  field from the plasma

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Diffusion of Magnetic Field in a Plasma (II)

• In plasma with zero resistivity the magnetic
  field is frozen to its value changes are
  prevented and the plasma carries the field with
  itself
• If a perfectly conducting plasma approaches a
  region of higher magnetic field it will distort
  the lines of forces while repelling them
• When the plasma has a finite resistivity the
  magnetic field will penetrate the plasma and
  cause dissipation via Joule effect

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Diffusion of Magnetic Field in a Plasma (III)

• The MHD equations along with Maxwell equations
  can be used to analyze the problem of magnetic
  field diffusion in a plasma.
• The Ohms law is

• If the plasma is at rest and the magnetic field
  is moving through it u0. Substituting E in the
  Faradays law

it is found
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Diffusion of Magnetic Field in a Plasma (IV)

• By eliminating j with Amperes law (neglecting
  the displacement current, assuming slow
  variations of the electric field)

it is found

• Since div B0, a diffusion equation for B is
  obtained as

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Diffusion of Magnetic Field in a Plasma (V)

• A rough estimate of the solution for the magnetic
  field diffusion equation can be found by taking

where L is the scale length of the spatial
variation of B

• The diffusion equation for B becomes then

where t is the time scale for the field
penetration into the plasma
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Diffusion of Magnetic Field in a Plasma (VI)

• When the field lines penetrate the plasma the
  induced currents cause dissipation. The power
  density dissipated in the plasma is

• If t is diffusion time for B the energy
  dissipated in the plasma will be t pdiss
• This energy can be estimated by substituting the
  expression for pdiss and by approximating j from

where L is the scale length of the spatial
variation of B, as discussed before
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Diffusion of Magnetic Field in a Plasma (VII)

• The energy dissipated in the plasma during the
  timet m0L2/h is then

that t is in the order of the time required to
dissipate the magnetic field energy in the plasma