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II. Plasma Physics Fundamentals

II. Plasma Physics Fundamentals. 4. The Particle Picture 5. The Kinetic Theory 6. The Fluid Description of Plasmas. 6. The Fluid Description of Plasmas. 6.1 The Fluid Equations for a Plasma. 6.1 The Fluid Equations for a Plasma. 6.1.1 Plasmas as Fluids: Introduction

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II. Plasma Physics Fundamentals

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  1. II. Plasma Physics Fundamentals 4. The Particle Picture 5. The Kinetic Theory 6. The Fluid Description of Plasmas

  2. 6. The Fluid Description of Plasmas 6.1 The Fluid Equations for a Plasma

  3. 6.1 The Fluid Equations for a Plasma 6.1.1 Plasmas as Fluids: Introduction 6.1.2 The Continuity Equation 6.1.3 The Equation of State 6.1.4 The Equation of State for Adiabatic Conditions 6.1.5 The Momentum Equation - Cold Plasma 6.1.6 The Momentum Equation - Warm Plasma 6.1.7 The Momentum Equation - Collisional Plasma 6.1.8 The Set of Fluid Equations 6.1.9 Fluid Drifts: Diamagnetic Drift 6.1.10 Fluid Drifts: Curvature Drift 6.1.11 Fluid Drifts: grad B Drift 6.1.12 The Plasma Approximation

  4. 6.1.1 Plasmas as Fluids: Introduction • The particle description of a plasma was based on trajectories for given electric and magnetic fields • Computational particle models allow in principle to obtain a microscopic description of the plasma with its self-consistent electric and magnetic fields • The kinetic theory yields also a microscopic, self-consistent description of the plasma based on the evolution of a “continuum” distribution function • Most practical applications of the kinetic theory rely also on numerical implementation of the kinetic equations

  5. Plasmas as Fluids: Introduction (II) • The analysis of several important plasma phenomena does not require the resolution of a microscopic approach • The plasma behavior can be often well represented by a macroscopic description as in a fluid model • Unlike neutral fluids, plasmas respond to electric and magnetic fields • The fluidodynamics of plasmas is then expected to show additional phenomena than ordinary hydro, or gasdynamics

  6. Plasmas as Fluids: Introduction (III) • The “continuum” or “fluid-like” character of ordinary fluids is essentially due to the frequent (short-range) collisions among the neutral particles that neutralize most of the microscopic patterns • Plasmas are, in general, less subject to short-range collisions and properties like collective effects and quasi-neutrality are responsible for the fluid-like behavior

  7. Plasmas as Fluids: Introduction (IV) • Plasmas can be considered as composed of interpenetrating fluids (one for each particle species) • A typical case is a two-fluid model: an electron and an ion fluids interacting with each other and subject to e.m. forces • A neutral fluid component can also be added, as well as other ion fluids (for different ion species or ionization levels)

  8. 6.1.1 The Continuity Equation dS dS • The number of particles N in a volume V changes only if these is a net flux of particles in the volume or if there are particle sources or “sinks” (conservation of mass) • The flux per volume unit (flux density) is nu, where u is the fluid velocity

  9. The Continuity Equation (II) • By applying the divergence theorem then and since the volume V is arbitrary that is the equation of continuity. Source and sink terms should be added to the r.h.s.

  10. 6.1.2 The Equation of State • Ideal gas: a gas where the interaction among the different molecules is negligible • An ideal gas is described by the Boltzmann distribution of energies for each energy state Wn: • The normalization condition

  11. The Equation of State (II) • By taking the log of it is found • The sum over the energy statesWn can be expressed in terms of the energiesek of each of the N molecules of the gas as

  12. The Equation of State (III) • The expression becomes then • By using the approximation the ideal gas statistics can be written as

  13. The Equation of State (IV) • For a classical (non quantum) description of the ideal gas the sum in the expression can be reduced to an integral over the volume V of the gas itself. The energy of a state characterized by a temperature T can be written as where f(T) is a given function of the temperature

  14. The Equation of State (V) • Since the pressurep is defined as it is found or (equation of state of an ideal gas) • For T expressed in kelvin and observing that the density is just n=N/V

  15. 6.1.3 The Equation of State for Adiabatic Conditions • For an ideal gas in adiabatic conditions the entropy is constant and a relationship links volume, temperature and pressure with the specific heats • The specific heat at constant volumecv is the variation of the total energy with respect the temperature • The specific heat at constant pressurecp is the variation of the free (thermal) energy (W) with respect the temperature

  16. The Equation of State for Adiabatic Conditions (II) • The ratio cp / cv is a constant that can be expressed in terms of the number of degrees of freedomNdf of the system • For an ideal gas in adiabatic conditions the following relation holds: • By using the ideal gas equation of statep=nkBT it can also be written, for adiabatic conditions:

  17. 6.1.4 The Momentum Equation - Cold Plasma • For a charged particle in an e.m. field the equation of motion is • If there are many particles, without collisions and superimposed thermal motions, they will all obey to the same equation of motion and will have the same fluid velocityu • In the fluid approximation then the variation of momentum density will be

  18. The Momentum Equation - Cold Plasma (II) • The time derivative dv/dt is taken at the position of each particle, that is in a moving frame • In general, for any function f(x,t) where the first term of the r.h.s. is the variation of f at a fixed point and the second term is the change of f seen by an observer moving with velocity ux in a region where f is different

  19. The Momentum Equation - Cold Plasma (III) • In the most general, three dimensional, case: that is called convective derivative • The operator is a scalar differential operator

  20. The Momentum Equation - Cold Plasma (IV) • Example. • f is a temperature of a fluid, the fluid is heated and pumped towards a region of higher a temperature. • df/dt>0 is the variation of temperature when the heater is on • if the fluid is moving towards a region of higher temperature then u>0 and • the temperature change in a fixed element of fluid is a balance of different terms:

  21. The Momentum Equation - Cold Plasma (V) • By using the convective derivative the variation of momentum density in the fluid will be that is the momentum equation or equation of motion for a fluid • Because the assumptions of particles without thermal motions and collsions, this equation applies only to the cold plasma case without collisions

  22. 6.1.5 The Momentum Equation - Warm Plasma dS B A x • If the particles have thermal motions the fluid description must include a (thermal) pressure term • The flux along x across a fluid element due to the particle motion is the difference of the fluxes through each face A and B of area dS

  23. The Momentum Equation - Warm Plasma (II) • The number of particles per time unit crossing A with velocity between vx and vx+Dvx is where • Each particle carries a momentummvx. The total momentum carried per time unit across A is where the average is taken over the distribution and n/2 is the density of particles going toward A (the others are going away from A)

  24. The Momentum Equation - Warm Plasma (III) • By repeating the same estimate for the face B the net change of momentum of the fluid element can be expressed as • By decomposing the velocity of each particle as the average (fluid) velocity plus the thermal component for a 1D maxwellian distribution it will be

  25. The Momentum Equation - Warm Plasma (IV) • By using the continuity equation and the equation of state a final expression for the change of momentum density in a fluid element due to the thermal motions can be written in term of the pressure gradient

  26. The Momentum Equation - Warm Plasma (V) • The momentum equation in 3D, including e.m. and pressure effects, will be then • The case studied is still not completely general as the transfer of momentum due to the thermal motions was considered only in the direction of the motion itself (isotropic case) • For anisotropic cases a stress tensor Pij=mn<vivj>, instead of a scalar pressure p, should be considered. The momentum equation is then:

  27. 6.1.6 The Momentum Equation - Collisional Plasma • In presence of neutrals the charged fluid will affected also by collisions that cause change in momentum • Momentum exchange with neutrals will be proportional to the relative velocity between the charged fluid u and the neutral fluid u0. • The momentum density variation due to the collisional interaction with neutrals can be estimated from the mean free time between collisions t (assumed constant). Then

  28. The Momentum Equation - Collisional Plasma (II) • The reciprocal of the mean free time between collisions is the collision frequencyn • Collisions among charged particles will cause additional change in momentum • There will be in general a different collision frequency for different types of collisions (ion-electron, electron-ion, ion-ion, electron-electron) • For example, for electron-ion and electron neutrals the momentum equation will be

  29. 6.1.7 Comparison with the Navier-Stokes Equation • Navier-Stokes equation for ordinary fluid: where r=mn and n is the kinematic viscosity that includes the anisotropic effects (as in the tensor P) • The Navier-Stokes equation resembles the plasma fluid momentum equation, without the e.m. term • The N-S equation represent a (neutral) fluid dominated by collisions • The plasma fluid momentum equation was derived under the assumption of a maxwellian distribution function (to compute <v2>)

  30. Comparison with the Navier-Stokes Equation (II) • The maxwellian distribution function is typically the result of a collisional process that reaches an equilibrium • The same derivation would hold for other distributions as long as they yield the same average square velocity • The fluid theory is therefore not very sensitive to deviations from the maxwellian distribution

  31. 6.1.8 Fluid Equation Set • The fluid equations for a plasma, along with the Maxwell equations, constitute a self-consistent set of equations for the plasma in an e.m. field • Maxwell Equations • Fluid equations (continuity, momentum) for electron and ion species (j=i,e):

  32. Fluid Equation Set (II) • Equation of state (adiabatic closure) • The set written for the electron and ion fluids consists of 16 scalar equations with 16 scalar unknowns

  33. 6.1.9 Fluid Drifts: Diamagnetic Drift • Fluid momentum equation: • For slow motions w.r.t. the Larmor frequency and E in the same direction as grad p it can be shown that the plasma motion perpendicular to B can be approximated by the fluid momentum equation with the l.h.s. set to zero:

  34. Fluid Drifts: Diamagnetic Drift (II) • Uniform E and B are considered, while n and p have a gradient • To study the motion perpendicular to B the cross product of the momentum equation with B is taken: that yields • The first term is the usual ExB drift, as in the particle description, the second term is called diamagnetic drift

  35. Fluid Drifts: Diamagnetic Drift (III) • The diamagnetic drift is originated by the pressure gradient and therefore is an effect that cannot be described in the single particle picture B grad n vD

  36. Fluid Drifts: Diamagnetic Drift (IV) • The diamagnetic drift occurs even the guiding centers are stationary • The diamagnetic drift does not depend on the mass but changes sign with the charge: this causes a diamagnetic current since electrons and ions drift in opposite directions

  37. 6.1.10 Fluid Drifts: Curvature Drift • In a bend magnetic field all the particles in a fluid element are subjected to the centrifugal force • The r.h.s. of the fluid momentum equation should also include a centrifugal force density term as where the r.h.s. is obtained for a maxwellian distribution with one degree of freedom • Analogously to the diamagnetic drift caused by grad p, the centrifugal force drift will be

  38. 6.1.11 Fluid Drifts: grad B Drift • A fluid element will not show any grad B drift, even if the guiding centers are: the particle drifts in any fixed fluid element cancel out. • In absence of electric forces a particle will change its Larmor radius because the grad B but the energy of the particle is constant (there is no work done on the particle) • Inside the same fluid element two particles with the same energy will “see” the same magnetic field and therefore will have, locally, the same Larmor radius

  39. Fluid Drifts: grad B Drift (II) • In fluid element the particle drifts are canceling out • The fluid theory is “filtering out” the grad B drift shown in the particle trajectory analisis Fluid Element grad |B| B

  40. 6.1.12 The Plasma Approximation • In a plasma, for low-frequency motions and when the electron inertia can be neglected, the electrons will follow the ion motions ensuring, on average, equal ion and electron densities • At the same time electric field are considered in the plasma, for example computed from the fluid equation on motion, then • This is the plasma approximation: the Poisson equation will be not used to compute electric fields in the plasma and quasineutrality is assumed

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