14. Energy transport.

Transcription

1 Phys780: Plasma Physics Lecture 14. Energy transport Energy transport. Chapman-Enskog theory. ([8], p.51-75) We derive macroscopic properties of plasma by calculating moments of the kinetic equation for particles of type a: f a t + (v β f a )+ ( ) Faβ f a = St aa +St ab x β v β m a where St aa and St ab are collision terms for a a and a b collisions. Some properties of the collision terms result from the conservations laws: St ab d 3 v = 0 because there is no transformation from particles of one type into

2 Phys780: Plasma Physics Lecture 14. Energy transport. another; m a vst aa d 3 v = 0 because the momentum is conserved for collisions of particles of the same type; ma v St aa d 3 v = 0 because of the conservation of energy for collisions of the same type particles; m a vst ab d 3 v + m b vst ba d 3 v = 0 because the total momentum in a b and b a collisions is conserved; ma v St ab d 3 mb v v + St bad 3 v = 0 because the total energy is conserved. We separate particle s velocity v into the mean macroscopic velocity V

3 Phys780: Plasma Physics Lecture 14. Energy transport. 3 and fluctuation component v : v = V + v, where V = 1 n vfd 3 v v. We defined the friction force for particles a colliding with particles b as: R ab = m a v St ab d 3 v Then for the conservation of the total momentum we have: R ab = R ba From the conservation of the total energy we have: Q ab + R abva +Q ba + R bavb = 0

4 Phys780: Plasma Physics Lecture 14. Energy transport. 4 where Q ab = and similarly for Q ba. Then, ma v St ab d 3 v Q ab +Q ba = R ab ( V a V b ). We consider the energy equation obtained by integrating the kinetic equation with a factor mv /: ( mn v ) + ( mn v ) v β ene t x β V = mv = St abd 3 v. In the local thermodynamic equilibrium, for Maxwellian distribution function: v = V + v = V + 3T m.

5 Phys780: Plasma Physics Lecture 14. Energy transport. 5 Now, we consider the second term: v v β = (Vα +v α) α = α = V αv β +V α V β v α }{{} 0 + +V α v α v β + }{{} (pδ αβ +π αβ )/mn ( V + 5T m ) v α }{{ } 3T/m (V β +v β) v α v β V β + V V β + V απ αβ mn + = v v β = v β + }{{} 0 where we used relation p = nt which follows from the definition of p, π αβ is the viscous stress tensor.,

6 Phys780: Plasma Physics Lecture 14. Energy transport. 6 The collision term is: where = V β m( V + v ) St ab d 3 v = mv mv βst ab d 3 v + St abd 3 v = V R+Q, Q = m v Stab d 3 v. Finally, the energy equation is written as ( nmv + 3nT ) + t + [( nmv + 5nT ] )V β +π αβ V α +q β x β = = en E V + R V +Q,

7 Phys780: Plasma Physics Lecture 14. Energy transport. 7 where v v β q β = nm = m v v β fd 3 v. Using the continuity and momentum equations we get: ( ) nmv + ( ) nmv V β = t x β ( ) Vα V α = mnv α +V β = t x β ( = V α p π αβ +en[ E x α x + 1 ) V β c B] α +R α Substituting this into the energy equation we get: ( ) 3nT + ( ) 5nT t x β V β +π αβ V α +q β V α p x α V α π αβ x β +env α E α +V α R α =.

8 Phys780: Plasma Physics Lecture 14. Energy transport. 8 or 3n t = env α E α +R α V α +Q ( 3nT ) +V β x β ( 3nT ) + + 5nT V β V α +π αβ + q = Q; x β x β x β ( ) T t +V T β + 3T ( n x β t + nv ) β + x β }{{} 0 +p V β V α +π αβ + q = Q, x β x β x β where we used the continuity equation and p = nt. Finally, using the definition of the full derivative we get in the case of the isotropic pressure: 3 ndt dt +p V + q = Q,

9 Phys780: Plasma Physics Lecture 14. Energy transport. 9 where is the heat flux. q = nm v v = mv v fd 3 v So far we assumed that the local distribution function was Maxwellian. However, if the plasma is not uniform then the distribution function deviates from the Maxwellian because of mass, momentum and energy transport processes: diffusion, viscosity and heat conduction, which lead the system to the complete thermodynamic equilibrium. The heat and viscous fluxes depend on first-order non-maxwellian corrections to the distribution function.

10 Phys780: Plasma Physics Lecture 14. Energy transport. 10 Chapman-Enskog theory The transport processes in gases were studied by Chapman and Enskog in , and were applied to plasma by Chapman and Cowling in The final classical transport theory was formulated by Braginskii (S.I. Braginskii, Transport processes in a plasma, in Reviews of Plasma Physics, Consultants Bureau, New York NY, 1965, Vol. 1, p.05). This theory considers Coulomb collisions. However, particles also interact with waves and turbulence. These are considered in so-called neoclassical transport theory. Here we consider the classical theory. The plasma fluid equations that we derived are incomplete. The viscosity tensor, π αβ, heat flux, q, and collision terms, R,Q, can be described by equation for higher moments of the kinetic equation. However, in general this is an infinite series of moment equations because each of the moments is coupled to higher moments. Therefore, we need an additional information to close the system.

11 Phys780: Plasma Physics Lecture 14. Energy transport. 11 There are two basic closure schemes. In truncation schemes, higher order moments are assumed to vanish. For instance, one classical scheme is a thirteen moment approximation. Another approach is to exploit a small parameter in the equations (asymptotic schemes). A classical example of these is the Chapman-Enskog theory, in which the small parameter is the ratio of the mean-free path between collisions to the scale length of macroscopic variations. Consider a plasma in the local thermodynamic equilibrium described by the Maxwellian distribution function, with slowly varying macroscopic properties: n( r,t),t( r,t), V( r,t). Because of these variations the distribution function is slightly different from Maxwellian. Consider a small perturbation to the distribution function: f = f M +δf where δf f M, and solve the Fokker-Planck equation for non-uniform plasma.

12 Phys780: Plasma Physics Lecture 14. Energy transport. 1 Suppose the plasma has a electron temperature gradient: T e = e z dt dz, and V = 0 and n =const, and assume that the ions are stationary because of their high mass, that there is no magnetic field, and that the plasma is isolated so that there is no current. The temperature gradient results in a pressure gradient: p e = n e T e which is balanced by electric field created by charges on the plasma surface (this is called thermoelectric or Seeback effect). The Fokker-Planck equation in this stationary case ( f/ t = 0) has the form: f v z z e m E f z = C 1 v z v 3 sinθ θ sinθ f θ,

13 Phys780: Plasma Physics Lecture 14. Energy transport. 13 where C = 4πZ e 4 n i Λ C m Recall that C/v 3 = ν ei is the collision frequency of electrons with ions with charge Ze. We substitute f = f M +δf into this equation. In the right-hand side, the term with f M vanishes because when V = 0, ( ) 3/ ) m f M = n e exp( mv πt e T e does not depend on angle θ in the velocity space. In the left-hand side we can neglect δf compared to f M. Thus, we get the following equation for δf: v z f M z e m E f M v z = C v 3 1 sinθ θ sinθ δf θ

14 Phys780: Plasma Physics Lecture 14. Energy transport. 14 Calculating the derivatives: we get f M = mv z f M = mvcosθ f M v z T e T e f M z = 3 T e T e z f M + mv vcosθ T e T e z = 1 T e T e z ( ) mv 3 T e T e ( ) mv 3 f M, T e T e z f M = f M + eevcosθ T e f M = = C 1 v 3 sinθ θ sinθ δf θ. We seek the solution of this equation in the form: δf = Φ(v)cosθ

15 Phys780: Plasma Physics Lecture 14. Energy transport. 15 and get: Φ(v) = v3 C [ v T e T e z ( ) mv 3 T e + eev T e ] f M. The distribution function must satisfy the stationary condition (mean velocity equals zero): vδfd 3 v = 0 that is or Φ(v)vcos θd 3 v = 0, Φ(v)v = 0. This gives equation: 1 T e z ( m v 7 3 v 5 ) T e ee v 5 = 0,

16 Phys780: Plasma Physics Lecture 14. Energy transport. 16 where v 5 = ( ) 3/ m = 4πn e πt e 0 ) 3/ ( Te = 4πn e ( m πt e (we used 0 exp( x )x k+1 dx = k!/) Similarly, Then, we have v 7 = 1 f M v 5 d 3 v = ) exp ( mv v 7 dv = T e ) 4 e x x 7 dx = m 0 ( ) 5/ = 48 π n Te e. m f M v 7 d 3 v = 384 π n e T e z ( ) = 48eE. ( ) 7/ Te. m

17 Phys780: Plasma Physics Lecture 14. Energy transport. 17 This gives a relation for the electric field strength which provides the plasma equilibrium ee = 5 T e z. Now, we can consider the momentum equation for electrons and find the force R ei that acts on electron from ions: Since we get p e +n e ee +R ei = 0 p e = n e T e z R ei = 3 n T e e z. This is called thermoforce. It appears because the collision frequency depends on velocity. Electrons which come from higher temperature regions experience less friction than electrons which come from lower temperature regions. The net result is a macroscopic friction force

18 Phys780: Plasma Physics Lecture 14. Energy transport. 18 directed against the temperature gradient. Using the solution for δf we can calculate the heat flux q: q = mv vδfd3 v mv 3 q z = π Φ(v)cos θsinθdθv dv Using the expression for Φ(v) with already determined E, we get where Φ(v) = v4 CT e q z = m T e 4CT e z v 9 = [ ] mv Te +8 T e z f M, 1 3 f M v 9 d 3 v = 3840 π ( m v 9 8 v 7 ) T e ( ) 9/ Te n e. m

19 Phys780: Plasma Physics Lecture 14. Energy transport. 19 Thus, q z = m T e 1CT e z ( ) π ( ) 7/ Te n e = m ( ) 5/ n e Te T e = 64 π C m z = Te 5/ T e = 8 π m 1/ e 4 ZΛ C z, where we used the expression for the collision parameter C and n e = Zn i. Generally, the heat flux equation is q = κ e T e, where Te 5/ κ e = 8 π m 1/ e 4 ZΛ C is the thermal conductivity for electrons. It depends only on

20 Phys780: Plasma Physics Lecture 14. Energy transport. 0 temperature. Consider an order-of-magnitude estimate: κ n ( ) 5/ e Te n ( ) 5/ e Te C m ν ei v 3 n e τ ei m where τ ei = 1/ν ei is the characteristic collision time. In the case of the stationary plasma we have the heat conduction equation: 3 n T e e = q = (κ T e ) t In 1D case, T e = χ T e t z, where χ = κ T e 3/n e m τ ei vtτ ei λ ei v T, is the temperature diffusion coefficient, λ ei is the mean-free path, v T is the electron thermal velocity. T e m,

21 Phys780: Plasma Physics Lecture 14. Energy transport. 1 The order-of-magnitude estimate for the deviation from the Maxwellian distribution: δf/f M Φ(v)/f M v T lnt e ν ei z λ ei /L, where L = ( lnt e / z) 1 is the characteristic length scale of plasma temperature variations. Therefore, the Chapman-Enskog theory is valid when λ L.