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Chapter 3. Deriving the Fluid Equations from the Vlasov Equation 25

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Chapter 3. Deriving the Fluid Equations from the Vlasov Equation 25 Chapter 3. Deriving the Fluid Equations From the Vlasov Equation 25 Chapter 3. Deriving the Fluid Equations From the Vlasov Equation Topics or concepts to learn in Chapter 3: 1. The basic equations for study kinetic plasma physics: The Vlasov-Maxwell equations 2. Definition of fluid variables: number density, mass density, average velocity, thermal pressure (including scalar pressure and pressure tensor), heat flux, and entropy function 3. Derivation of plasma fluid equations from Vlasov-Maxwell equations: (a) The ion-electron two-fluid equations (b) The one-fluid equations and the MHD (magnetohydrodynamic) equations (c) The continuity equations of the number density, the mass density, and the charge density (d) The momentum equation (e) The momentum of the plasma and the E-, B- fields (f) The momentum flux (the pressure tensor) of the plasma and the E-, B- fields (g) The energy of the plasma and the E-, B- fields (h) The energy flux of the plasma and the E-, B- fields (i) The energy equations and “the equations of state” (j) The MHD Ohm’s law and the generalized Ohm’s law Suggested Readings: (1) Section 7.1 in Nicholson (1983) (2) Chapter 3 in Krall and Trivelpiece (1973) (3) Chapter 3 in F. F. Chen (1984) 3.1. The Vlasov-Maxwell System Vlasov equation of the αth species, shown in Eq. (2.7), can be rewritten as ∂ fα (x,v,t) eα + v⋅ ∇fα (x,v,t) + [E(x,t) + v × B(x,t)]⋅ ∇ v fα (x,v,t) = 0 (3.1) ∂t mα or ∂ fα (x,v,t) eα + ∇ ⋅{vfα (x,v,t)} + ∇ v ⋅{[E(x,t) + v × B(x,t)] fα (x,v,t)} = 0 (3.1') ∂t mα where ∇ =∂/∂x and ∇ v = ∂/∂v. 26 Chapter 3. Deriving the Fluid Equations From the Vlasov Equation Maxwell’s equations in Table 1.1. can be rewritten as ρ (x,t) ∇ ⋅ E(x,t) = c ε0 1 = ∑eα nα (x,t) (3.2) ε0 α 1 3 = ∑eα ∫∫∫ fα (x,v,t)d v ε0 α ∇ ⋅ B(x,t) = 0 (3.3) ∂B(x,t) ∇ × E(x,t) = − (3.4) ∂ t ∂E(x,t) ∇ × B(x,t) = µ J(x,t) + µ ε 0 0 0 ∂ t ∂E(x,t) = µ0[∑eα nα (x,t)Vα (x,t)]+ µ0ε0 (3.5) α ∂ t 3 ∂E(x,t) = µ0[∑eα ∫∫∫ vfα (x,v,t)d v]+ µ0ε0 α ∂ t The Vlasov equations of ions and electrons and the Maxwell’s equations are the governing equations of the Vlasov-Maxwell system, which includes eight unknowns ( fi, fe, E, B) and eight independent equations in a six-dimensional phase space. Chapter 3. Deriving the Fluid Equations From the Vlasov Equation 27 3.2. The Fluid Variables Before introducing the fluid equations, we need to define fluid variables of plasma. The number density of the αth species, in Eq. (3.2), is defined by n (x,t) ≡ f (x,v,t)d 3v (3.6) α ∫∫∫ α The average velocity of the αth species, in Eq. (3.5), is defined by vf (x,v,t)d 3v ∫∫∫ α Vα (x,t) ≡ (3.7) nα (x,t) The particle flux of the αth species is n V = vf (x,v,t)d 3v (3.8) α α ∫∫∫ α The mass flux of the αth species is m n V = m vf (x,v,t)d 3v (3.9) α α α ∫∫∫ α α The charge flux of the αth species is e n V = e vf (x,v,t)d 3v (3.10) α α α ∫∫∫ α α The momentum flux, or the kinetic pressure, of the αth species is m n V V + P (x,t) = m vvf (x,v,t)d 3v (3.11) α α α α α ∫∫∫ α α where mαnα Vα Vα is the dynamic pressure, and Pα (x,t) is the thermal pressure tensor. The thermal pressure tensor Pα (x,t) in Eq. (3.11) is defined by 3 Pα (x,t) ≡ mα [v − Vα (x,t)][v − Vα (x,t)] fα (x,v,t)d v (3.12) ∫∫∫ Since Pα (x,t) is a second rank symmetric tensor, trace of Pα (x,t) is invariant after an orthonormal coordinate transformation. For an isotropic pressure, we have ⎛ pα (x,t) 0 0 ⎞ ⎜ ⎟ P (x,t) 1 p (x,t) 0 p (x,t) 0 α = α = ⎜ α ⎟ ⎜ ⎟ ⎝ 0 0 pα (x,t)⎠ Thus, in general we can define a scalar thermal pressure pα (x,t) 1 p (x,t) ≡ trace[P (x,t)] (3.13) α 3 α The flux of the total kinetic pressure of the αth species is m n V V V + (P V )S + Q (x,t) = m vvvf (x,v,t)d 3v (3.14) α α α α α α α α ∫∫∫ α α where the heat-flux tensor Qα(x,t) is a third rank tensor, which is defined by 28 Chapter 3. Deriving the Fluid Equations From the Vlasov Equation Q (x,t) ≡ m [v − V (x,t)][v − V (x,t)][v − V (x,t)] f (x,v,t)d 3v (3.15) α ∫∫∫ α α α α α S (PαVα ) in Eq. (3.14) is a symmetric third rank tensor, which is defined by (P V )S ≡ P V + V P + m (v − V )V (v − V ) f (x,v,t)d 3v (3.16) α α α α α α ∫∫∫ α α α α α The kinetic energy flux of the αth species is 1 3 1 m n V ⋅ V V + p V + P ⋅ V + q (x,t) = m v⋅ vvf (x,v,t)d 3v 2 α α α α α 2 α α α α α ∫∫∫ 2 α α or 1 2 3 1 2 3 ( mα nαVα + pα )Vα + Pα ⋅ Vα + qα (x,t) = mαv vfα (x,v,t)d v (3.17) 2 2 ∫∫∫ 2 where the heat-flux vector qα(x,t) is defined by 1 q (x,t) ≡ m [v − V (x,t)]⋅[v − V (x,t)][v − V (x,t)] f (x,v,t)d 3v (3.18) α ∫∫∫ 2 α α α α α For advanced study: If it is needed, we can define a fourth rank tensor Rα (x,t) R (x,t) ≡ m (v − V )(v − V )(v − V )(v − V ) f (x,v,t)d 3v (3.19) α ∫∫∫ α α α α α α and a fifth rank tensor Sα (x,t) S (x,t) ≡ m (v − V )(v − V )(v − V )(v − V )(v − V ) f (x,v,t)d 3v (3.20) α ∫∫∫ α α α α α α α so that m n V V V V + (P V V )S + (Q V )S + R = m vvvvf (x,v,t)d 3v (3.21) α α α α α α α α α α α α ∫∫∫ α α and S S S mα nα Vα Vα Vα Vα + (Pα Vα Vα Vα ) + (Qα Vα Vα ) + (Rα Vα ) + Sα (3.22) = m vvvvvf (x,v,t)d 3v ∫∫∫ α α where the superscript s denotes a higher-rank symmetric tensor. An example of the 2nd-rank symmetric tensor is shown in (3.16). Exercise 3.0.1 Let us define a generalized local entropy function Sα of the αth species: fα 3 Sα = − ∫ ln fα d v + constant nα Chapter 3. Deriving the Fluid Equations From the Vlasov Equation 29 Let fα be a normal distribution function with number density nα , temperature Tα , and zero average velocity. Determine how the entropy Sα varies with varying of the number density n and the thermal pressure p . α α In addition to the general definitions of the fluid variables, we shall encounter the following integrations in deriving the fluid equations in the next section. ∂ ⋅{[E(x,t) + v × B(x,t)] f (x,v,t)}d 3v = 0 (3.23) ∫∫∫ ∂v α ∂ v ⋅{[E(x,t) + v × B(x,t)] f (x,v,t)}d 3v = −n (E + V × B) (3.24) ∫∫∫ ∂v α α α ∂ vv ⋅{[E(x,t) + v × B(x,t)] f (x,v,t)}d 3v ∫∫∫ v α ∂ (3.25) S S S 3 = −nα (EVα ) − nα [Vα (Vα × B)] − ∫∫∫ {(v − Vα )[(v − Vα ) × B]} fα (x,v,t)d v The following two integrations are useful for the higher moment integrations of the Vlasov equation. ∂ vvv ⋅{[E(x,t) + v × B(x,t)] f (x,v,t)}d 3v ∫∫∫ ∂v α S nα S S nα S = −nα (EVα Vα ) − (EPα ) − nα [Vα Vα (Vα × B)] − [Pα (Vα × B)] (3.26) mα mα S 3 − ∫∫∫ {(v − Vα )(v − Vα )[(v − Vα ) × B]} fα (x,v,t)d v ∂ vvvv ⋅{[E(x,t) + v × B(x,t)] f (x,v,t)}d 3v ∫∫∫ ∂v α n n = −n (EV V V )S − α (EP V )S − α (EQ )S α α α α m α α m α α α (3.27) S nα S nα S − nα [Vα Vα Vα (Vα × B)] − [Pα Vα (Vα × B)] − [Qα (Vα × B)] mα mα S 3 − ∫∫∫ {(v − Vα )(v − Vα )(v − Vα )[(v − Vα ) × B]} fα (x,v,t)d v and so forth. 30 Chapter 3. Deriving the Fluid Equations From the Vlasov Equation Exercise 3.1. Verify Eqs. (3.23), (3.24), (3.25), and (3.26). Hint: There are three terms in the integration of Eq. (3.23), they are all in the following form: ∂ [ E f (x,v,t)dv ]d 2v = E [ f (x,v,t)]vx = +∞ d 2v = 0 ∫∫ ∫ x α x ∫∫ x α vx =−∞ ∂vx There are nine terms in the integration of Eq.
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