Fundamentals of Plasma Physics
2 Derivation offluid equations: Vlasov, 2-fluid, MHD 2.1 Phase-space Consider a particle moving in a one-dimensional space and l ...
2.2 Distribution function and Vlasov equation 31 reverse direction is called a passing particle, while a particle confined to a ...
32 Chapter 2. Derivation offluid equations: Vlasov, 2-fluid, MHD ∂f(x,v,t) ∂t dxdv = −f(x+ dx,v,t)vdv+f(x,v,t)vdv −f(x,v+ dv,t)a ...
2.3 Moments of the distribution function 33 For example, if the energyEof particles is constant along their orbits thenf=f(E)is ...
34 Chapter 2. Derivation offluid equations: Vlasov, 2-fluid, MHD particles atx.Thus we see that ∫ f(x,v)dv=n(x);the transition f ...
2.3 Moments of the distribution function 35 cumulative effect of the more infrequently occurring large angle collisions. In orde ...
36 Chapter 2. Derivation offluid equations: Vlasov, 2-fluid, MHD while collisions between different species must conserve the to ...
2.4 Two-fluid equations 37 (i) “pulling” both the time and space derivatives out of the velocity integral, (ii) writingv=v′(x,t) ...
38 Chapter 2. Derivation offluid equations: Vlasov, 2-fluid, MHD Equation (2.25) defines pressure for a three-dimensional isotro ...
2.4 Two-fluid equations 39 Let us now take the second moment of the Vlasov equation. Unlike the zeroth and first moments, here t ...
40 Chapter 2. Derivation offluid equations: Vlasov, 2-fluid, MHD The second identity is obtained by dotting the equation of moti ...
2.4 Two-fluid equations 41 2.4.1 Entropy of a distribution function Collisions cause the distribution function to tend towards a ...
42 Chapter 2. Derivation offluid equations: Vlasov, 2-fluid, MHD subsequent holes, we see there areN!ways of putting all the peg ...
2.4 Two-fluid equations 43 By now, it is obvious thatfcould be the velocity distribution function in which casef(v)dv is just th ...
44 Chapter 2. Derivation offluid equations: Vlasov, 2-fluid, MHD is N=V ∫ fdv. (2.46) The energy of an individual particle isE=m ...
2.4 Two-fluid equations 45 2.4.3 Relation between pressure and Maxwellian The scalar pressure has a simple relation to the gener ...
46 Chapter 2. Derivation offluid equations: Vlasov, 2-fluid, MHD 2.5 Magnetohydrodynamic equations Particle motion in the two-fl ...
2.5 Magnetohydrodynamic equations 47 by ←→ PMHD= ∑ σ ∫ mσv′v′fσdv. (2.68) We insert Eqs.(2.67) and (2.68) in Eq.(2.66), integrat ...
48 Chapter 2. Derivation offluid equations: Vlasov, 2-fluid, MHD The pressure term in the MHD equation of motion, Eq.(2.71) is ...
2.5 Magnetohydrodynamic equations 49 A formal proof of this frozen-influx property will now be established by direct calcula- ti ...
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