Lectures on basic plasma physics: Kinetic approach

Transcription

1 Lectures on basic plasma physics: Kinetic approach Department of applied physics, Aalto University April 30, 2014

2 Motivation Layout 1 Motivation 2 Boltzmann equation (a nasty bastard) 3 Vlasov equation 4 Kinetic effects: Oscillations I 5 Kinetic effects: Oscillations II 6 Kinetic effects: Introduction to Collisions 7 Summary

3 Motivation Why we need the statistical approach We cannot follow the trajectories of particles In MHD we lose the information of the velocity space details. In hot plasmas, the details of the particles velocity distribution can have a significant effect. The details may lead to, e.g., instabilities in the wave particle interaction. Development of the velocity space tells what happens to the energy and momentum of the plasma.

4 Boltzmann equation (a nasty bastard) Layout 1 Motivation 2 Boltzmann equation (a nasty bastard) 3 Vlasov equation 4 Kinetic effects: Oscillations I 5 Kinetic effects: Oscillations II 6 Kinetic effects: Introduction to Collisions 7 Summary

5 Boltzmann equation (a nasty bastard) Time evolution of the distribution function Quantum mechanics was ruled out (well, not totally, fusion reactions may still take place) The behaviour of charged particles is completely deterministic in the microscopic level and the phase-space conservation states F s t + v F s + a s v F s = 0, (1) However, the exact distribution function, F s = δ(r i r)δ(v i v) is quite useless. Also the acceleration a s = qs m s (E exact + v B exact ) is difficult to calculate, because the exact fields resulting from the point charges would be spiky.

6 Boltzmann equation (a nasty bastard) Time evolution of the distribution function Solution: take an ensemble average of the time evolution of the distribution function. Averaged over an ensemble, both the distribution function and the fields will be smooth. Define f s = F s ensemble (2) The acceleration term is problematic. The fields depend on the particle trajectories, and hence are not statistically independent of F s. a s v F s a s v F s (3)

7 Boltzmann equation (a nasty bastard) Time evolution of the distribution function The acceleration term involves correlations. The goal of kinetic theory is to analyze these, and produce useful approximations. In ionized plasmas, the most important correlations result from Coulomb collisions between the charged particles, thus define a s v F s = a s F s C s (f) (4) where the correlations are transferred into a collision term C s (f) often involving distributions of all species. In general C s (f) is a complicated operator. In weakly coupled plasmas, Debye length is assumed small and the collision operator approximated bilinearly C s (f) = s C ss (f s, f s )

8 Boltzmann equation (a nasty bastard) Time evolution of the distribution function Putting in the Lorentz force for the acceleration we obtain the Fokker-Planck equation for a charged particle with smooth fields f s t + v f s + q s m s (E + v B) v f s = C s (f) (5) The Coulomb collisions conserve particles, momentum and energy. The same is required from the operator Particle conservation: C ss dv = 0 Momentum conservation: m s vc ss dv = m s vc s sdv Energy conservation: 1 2 m sv 2 C ss dv = 1 2 m s v2 C s sdv We will discuss the operator and its properties more thoroughly in the exercises.

9 Vlasov equation Layout 1 Motivation 2 Boltzmann equation (a nasty bastard) 3 Vlasov equation 4 Kinetic effects: Oscillations I 5 Kinetic effects: Oscillations II 6 Kinetic effects: Introduction to Collisions 7 Summary

10 Vlasov equation Vlasov equation Investigate phenomena for which the characteristic frequency is a lot higher than the collision frequency Dropping the collision term yields the Vlasov equation f s t + v f s + q s m s (E + v B) v f s = 0 (6) This equation will reveal some fundamental properties of plasmas. Mathematics will be more challenging, and we can treat analytically only a couple of examples. Analytical investigations are limited to linear regime, numerically it will be then possible to go beyond linear behaviour.

11 Vlasov equation Simple solutions Jeans theorem: Any steady-state solution of the collisionless Boltzmann equation (the Vlasov equation) depends on the phase space coordinates only through integrals of motion in the given potential, and conversely any function of the integrals is a steady-state solution (Exercise). Thus, many solutions: e.g. if E = 0, B = 0: ( m ) 3/2 f = e mv 2 /2kT 2πkT f = v v 4 + v0 4 (7) (8)

12 Vlasov equation Simple solutions Many solutions: e.g. if E = 0, B = B 0 ẑ: One set of integrals of motion (constants) are ζ x = v x qb m y, ζ y = v y + qb m x (9) f = f(v 2, ζ x, ζ y ) is then a solution to the vlasov eqution With given force, Vlasov solution equivalent to single particle motion Unfortunately, we need also the Maxwell s equations, as the fields are coupled to the particles

13 Vlasov equation Simple solutions Lecture assignment: prove that Maxwell-Boltzman distribution f = e W/2kT, where W = 1 2 mv2 + qφ is a solution to the Vlasov equation Solution: df dt = f q kt v Φ + f q ( m (E + v B) m ) 2kT v (v v) (10) = f q kt v B v = 0 (11)

14 Kinetic effects: Oscillations I Layout 1 Motivation 2 Boltzmann equation (a nasty bastard) 3 Vlasov equation 4 Kinetic effects: Oscillations I 5 Kinetic effects: Oscillations II 6 Kinetic effects: Introduction to Collisions 7 Summary

15 Kinetic effects: Oscillations I Oscillations Initially we assume the plasma to be in thermodynamical equilibrium, often described with a Maxwellian distribution What will happen, if the equilibrium is perturbed, e.g, with an external electromagnetic wave? Here, we investigate the Vlasov equation under a small amplitude harmonic electric wave. The perturbation is of the form exp[i(k r ωt)], and the magnetic field is assumed vanishingly small, i.e., neglected from the calculations. Only electron motion is considered, and ions are assumed singly-charged fixed background.

16 Kinetic effects: Oscillations I Oscillations The Vlasov equation for the electrons in unmagnetized plasma f e t + v f e + e m e Φ v f e = 0 (12) Now take n as the ion density. The electric field is coupled to the particles via the Poisson equation 2 Φ = e ) (n f e dv (13) ɛ 0 Difficult system linearize, assuming that the perturbations cause small deviations: f e (r, v, t) = f 0 (v) + f 1 (r, v, t), Here f 0 is the equilibrium distribution (often assumed Maxwellian).

17 Kinetic effects: Oscillations I Oscillations The equilibrium distribution fulfills f 0 dv = n. Thus, we end up with linearized equations f 1 t + v f 1 + e m e Φ v f 0 = 0 (14) 2 Φ = e ɛ 0 f 1 dv (15) Then, make the harmonic assumption, that perturbation quantities Φ, and f 1, are proportional to exp[i(k r ωt)]

18 Kinetic effects: Oscillations I Oscillations Calculating the derivatives reduces the differential equations into algebraic ones i(ω k v)f 1 + i e m e Φk v f 0 = 0 (16) k 2 Φ = e ɛ 0 f 1 dv (17) Solving these, assuming that Φ is non-zero, leaves us with a dispersion relation 1 + e2 k v f 0 ɛ 0 m e k 2 dv = 0 (18) ω k v

19 Kinetic effects: Oscillations II Layout 1 Motivation 2 Boltzmann equation (a nasty bastard) 3 Vlasov equation 4 Kinetic effects: Oscillations I 5 Kinetic effects: Oscillations II 6 Kinetic effects: Introduction to Collisions 7 Summary

20 Kinetic effects: Oscillations II Oscillations: the proper way We have obtained a dispersion equation ɛ(k, ω) = 0 Problem: singularity at k v = ω, what to do? Lev Landau proposed a solution in 1946: Perturbation must start at some moment. Treat as an initial value problem and investigate at later times! The linearized vlasov equation is fourier transformed only in place: f k (v, t) = 1 (2π) 3 f 1 e ik r dr (19) and laplace transformed in time: f k (v, p) = t 0 f k e pt dt, Re(p) p 0 (20)

21 Kinetic effects: Oscillations II Oscillations: the proper way This gives the equations (p + ik v) f k = f k (v, t 0 ) e m e (ik v f 0 ) Φ k (21) k 2 Φk = e ɛ 0 f k dv (22) solving for the potential gives k 2 Φk = e fk (v,t 0 ) ɛ 0 p+ik v dv 1 + e2 1 k vf0 ɛ 0 m e k 2 ip k v dv (23) The denominator looks familiar if we replace p by iω...

22 Kinetic effects: Oscillations II Oscillations: the proper way Make the integrals one dimensional F 0 (u) = f 0 (v)δ(u k v/k)dv (24) F k (u) = f k (v)δ(u k v/k)dv (25) The potential is now given by the inverse laplace transformation along the Bromwich contour k 2 Φ k (t) = 1 p0 +i 2πi p 0 i e F k (u,t 0 ) ɛ e2 ɛ 0 m e 1 k 2 p+iku du F 0 (u)/ u ip/k u duept dp (26)

23 Kinetic effects: Oscillations II Oscillations: the proper way Finite number of poles for Re(p) > α, deform the contour so far to negative Re(p) (with aid of analytic continuation) that e pt guarantees the contribution of the vertical parts vanishes and only residues at the poles are left.

24 Kinetic effects: Oscillations II Oscillations: the proper way Using the deformed contour and Residues at the poles R j, the integration gives a time asymptotic solution Φ k = j R j e (p j(k)t) (27) where p j are the solutions of ɛ(k, p) = 1 + e2 1 ɛ 0 m e k 2 F 0 (u)/ u du = 0 (28) ip/k u Thus, the values p j determine the behaviour of the perturbation (growth, damping,...).

25 Kinetic effects: Oscillations II Oscillations: the proper way Starting from the df/dt = 0, adding Maxwells equations and Lorentz force, and investigating a small perturbation in linear regime, yielded a simple expression for the time behaviour of the perturbation.

26 Kinetic effects: Oscillations II Oscillations: the proper way Starting from the df/dt = 0, adding Maxwells equations and Lorentz force, and investigating a small perturbation in linear regime, yielded a simple expression for the time behaviour of the perturbation. The dispersion relation still looks the same, if we replace p = iω, and we get the singularity again?!?

27 Kinetic effects: Oscillations II Oscillations: the proper way Starting from the df/dt = 0, adding Maxwells equations and Lorentz force, and investigating a small perturbation in linear regime, yielded a simple expression for the time behaviour of the perturbation. The dispersion relation still looks the same, if we replace p = iω, and we get the singularity again?!? BUT: Landau s solution tells how to handle the singularity.

28 Kinetic effects: Oscillations II Oscillations: the proper way Starting from the df/dt = 0, adding Maxwells equations and Lorentz force, and investigating a small perturbation in linear regime, yielded a simple expression for the time behaviour of the perturbation. The dispersion relation still looks the same, if we replace p = iω, and we get the singularity again?!? BUT: Landau s solution tells how to handle the singularity. The analytic continuation of the functions of the type h(p) = g(u) u ip/k du to calculate the inverse laplace transformation requires that the path for u goes below the pole ip/k for both Re(p) > 0 and Re(p) 0, rather than just straight along the real axis.

29 Kinetic effects: Oscillations II Oscillations: the proper way The Landau contour: h(p) = = P = g(u) du, Re(p) > 0 (29) u ip/k g(u) du + πig(ip/k), Re(p) = 0 (30) u ip/k g(u) du + 2πig(ip/k), Re(p) < 0 (31) u ip/k

30 Kinetic effects: Oscillations II Landau damping Let us investigate the situation in the region of very long wavelengths, i.e., ω/k v th. Assume further that Im(ω) 0. If we replace p with iω, the dispersion relation reads ( ɛ(k, ω) = 1 e2 1 F 0 (u)/ u ɛ 0 m e k 2 P u ω/k du + iπ F ) 0 u (ω/k) (32) Approximate the principal value as a serie expansion F 0 (u)/ u k2 P du = u ω/k ω 2 F 0 [1 + 2 ku ω + 3(ku ω ) ]du (33)

31 Kinetic effects: Oscillations II Landau damping Assume F 0 (u) = n 1 πvth e u 2 v 2 th, i.e., a Maxwellian distribution Including only the first term from the principal value: ɛ(k, ω) 1 e2 n ɛ 0 m e ω 2 = 0 ω2 = ne2 ɛ 0 m e (34) Including also the second term gives ɛ(k, ω) 1 ω2 p ω 2 3k2 λ 2 D ω 4 = 0 (35) where λ 2 D = v2 th /ω2 p. The solution is ω 2 ω 2 p(1 + 3k 2 λ 2 D ) and is called the Langmuir wave, or Bohm-Gross relation. ω 4 p

32 Kinetic effects: Oscillations II Landau damping Now perturb the frequency slightly, i.e., ω = ω 0 + iγ where ω 0 is the solution for the Langmuir wave, and γ ω 0. Expand the terms including ω and include also the residue term to get ɛ(k, ω) 1 ω2 p ω 2 0 iπ ω2 p k 2 (1 2i γ ) 3k 2 λ 2 D ω 0 ( ) F0 (u) u u=ω o/k ω 4 p ω 4 0 (1 4i γ ω 0 ) (36) Demanding the imaginary part to vanish with ω 0 ω p gives π γ = 8 ω p k 3 λ 3 D (37) e 1 2k 2 λ 2 D (38)

33 Kinetic effects: Oscillations II Landau damping: a method for the damping rate Write ω = ω r + iω i and assume ω i ω r. Linearize ɛ(k, ω) around ω r ɛ(k, ω) ɛ(k, ω r ) + iω i ɛ(k, ω r ) ω r = 0 (39) Further divide ɛ(k, ω r ) = ɛ r (k, ω r ) + iɛ i (k, ω r ) and assume ɛ r (k, ω r ) = 0. Using Cauchy-Riemann equations we finally get ɛ r (k, ω r ) ɛ(k, ω) iɛ i (k, ω r ) + iω i = 0 ω r (40) ɛ i (k, ω r ) ω i = ɛ r (k, ω r )/ ω r (41)

34 Kinetic effects: Oscillations II Landau damping: what have we learnt First of all, we see that γ is negative. This accounts for a damping perturbation (remember Φ k = j R je iω j ). Plugging in the definitions for ω p and v th we see that the damping rate has a strong depency on temperature and density, γ n2 4k 2 ɛ 2 0 k B T. Thus the plasma conditions have a great impact on the damping rate. T 3/2 e ne 2 This is the simplest analytical example to demonstrate the kinetic damping of the waves launched to plasma. The expressions may vary slightly depending on the approximations. A derivation exploiting the properties of plasma dispersion function is a bit more rigorous (see exercises).

35 Kinetic effects: Oscillations II The plasma dispersion function Let us define a function Z(ζ) = 1 π If the initial state of the plasma is Maxwellian F 0α (u) = 1 πvthα e u 2 v thα 2 e x2 dx (42) x ζ we can write the dispersion relation as ɛ(k, ω) = 1 α ω 2 pα k 2 vthα 2 Z (ω/kv thα ) (43) Thus Z(ζ) is called the plasma dispersion function, defined for Im(ζ) > 0 and analytically continued for Im(ζ) 0.

36 Kinetic effects: Oscillations II The plasma dispersion function: lecture assignment Show that the dispersion relation can be written with the dispersion function as expressed in the previous slide.

37 Kinetic effects: Oscillations II The plasma dispersion function: lecture assignment Show that the dispersion relation can be written with the dispersion function as expressed in the previous slide. Solution: F 0 / u u ω/k du = 2 1 πv 2 th 1 = 2 πv 2 th u v th = Z (ω/(kv th )) v 2 th More about Z(ζ) in the exercises. u2 v e th 2 du u/v th ω/kv th x x ω/(kv th ) e x2 dx (44)

38 Kinetic effects: Oscillations II Microscopic stability conditions In MHD macroscopic details of the plasma, e.g., gradients could cause instabilities Thus far the kinetic approach has considered only maxwellian distribution. It seems to be stable for electric perturbations as the damping rate was negative. But γ was shown to be proportional to the slope of the distribution. What about other distribution? Bump on tail, double peaked, etc.

39 Kinetic effects: Oscillations II Microscopic stability conditions Let us investigate the dispersion relation, or more precisely, the function Z(ζ) = F0 / u k2 du = u ζ ωp 2 (45) First of all, because k 2 > 0 we are interested in real, positive values of Z(ζ) Secondly, we know that if we get Re{Z(ζ)} > 0 for some Im{ζ} > 0 we have encountered a growing instability.

40 Kinetic effects: Oscillations II Microscopic stability conditions The mapping of ζ = ω/k into complex Z(ζ)-plane is called the Nyquist diagram Remember: for Real ζ, Z(ζ) = P F (u) u ζ du + iπf (ζ) Example: in point B, I{Z} = 0 and it changes from negative to positive a local minimum in F 0.

41 Kinetic effects: Oscillations II Microscopic stability conditions In addition the real part of Z(ζ) has to be positive R{Z(ζ)} =P = lim ɛ 0 { = lim ɛ 0 { ζ ɛ F 0 (u) u ζ du ζ ɛ + ζ+ɛ } d (F 0(u) F 0 (ζ)) u ζ + } F 0(u) F 0 (ζ) ζ+ɛ (u ζ) 2 du [ F0 (ζ) F 0 (ζ ɛ) + lim ɛ 0 ɛ F 0 (u) F 0 (ζ) (u ζ) 2 du > 0 =P F 0(ζ + ɛ) F 0 (ζ) ɛ ] (46)

42 Kinetic effects: Oscillations II Microscopic stability conditions The Penrose criterion for an instability to grow demands a minimum for F o at u = ζ and that F 0 (u) F 0 (ζ) du > 0 (u ζ) 2 This means that the distribution function has to have two peaks separate enough.

43 Kinetic effects: Oscillations II Microscopic stability conditions Multiple peaks are common in Fusion applications: Heating of the plasma with external sources creates a local fast ion or electron populations. The fusion reaction itself creates energetic alpha particles.

44 Kinetic effects: Introduction to Collisions Layout 1 Motivation 2 Boltzmann equation (a nasty bastard) 3 Vlasov equation 4 Kinetic effects: Oscillations I 5 Kinetic effects: Oscillations II 6 Kinetic effects: Introduction to Collisions 7 Summary

45 Kinetic effects: Introduction to Collisions Particles interact via collisions and macroscopic fields Deflection of charged particle from static, locally fixed potential does not change the energy of the particle, only momentum Deflection from another particle of finite mass will change the momentum and energy of both particles. The finite particles may gain or lose energy/momentum as they interact with each other It is as if some sort of friction or force field appeared to change the particles velocity. (The force field is of course due to the electric potential caused by the two particles)

46 Kinetic effects: Introduction to Collisions Particles interact via collisions and macroscopic fields

47 Kinetic effects: Introduction to Collisions Particles interact via collisions and macroscopic fields Our systems of interest are often orders of magnitudes larger than the dimensions of the region where the interaction truly happens From our point of view, the interaction appears as a point-like collision between two billiard balls and then macroscopic fields take over the motion again. An observation: Heat transfer is carried by collisions between particles, and temperature obeys diffusion equation. Schrödinger equation for a free particle is a diffusion equation describing the evolution of probability density. Three words: collisions, diffusion, probability. What is the common factor and how is it related to Coulomb collisions?

48 Kinetic effects: Introduction to Collisions Stochastic motion and macroscopic interpretation Einstein was the first person to construct the connection between the statistical motion of microscopic particles and diffusion equation. A huge step as diffusion equation had been connected only to macroscopic quantities A formal argumentation for the connection is following: Let τ be a small period of time during which the particle location changes from x to x according to some transition probability given by W τ (x ; ), where the first argument is the particle location, and the second one is the change in location.

49 Kinetic effects: Introduction to Collisions Stochastic motion and macroscopic interpretation Then the probablity for finding particles at time t + τ between interval x and x + dx is f(x, t + τ)dx = f(x, t)w τ (x ; )d dx (47) Make expansions: f(x, t) f(x, t + τ) = f(x, t) + τ t f(x, t) f(x, t) = f(x, t) x W τ (x ; ) = W τ (x, ) W τ (x, ) x (48) 2 f(x, t) x 2 (49) W τ (x, ) 2 x 2 (50)

50 Kinetic effects: Introduction to Collisions Stochastic motion and macroscopic interpretation Now combine the equations to get f(x, t) f(x, t) + τ = f(x, t)w τ (x; )d t [ ] x (f(x, t)w τ (x; )) d [ 2 2 ] + 2 x 2 (f(x, t)w τ (x; )) d + O( 3 ) (51) Note that is independent of x and W τ (x; )d = 1

51 Kinetic effects: Introduction to Collisions Stochastic motion and macroscopic interpretation Thus we have f(x, t) t = ( f(x, t) x + 2 x 2 ( f(x, t) ) τ W τ (x; )d ) 2 2τ W τ (x; )d + O( 3 τ ) (52) Define diffusion and friction coefficients as expectation values with respect to the transition probability W τ (x, ) µ(x, t) = τ, D(x, t) = 2 2τ (53)

52 Kinetic effects: Introduction to Collisions Stochastic motion and macroscopic interpretation We end up with a partial differential equation f(x, t) t = 2 (µ(x, t)f) + (D(x, t)f) + O x x2 ( 3 ) Let now the transition probability be W τ (x, ) = δ( ), and independent of x. Thus µ is independent of x. Noting that = ẋτ, and taking the limit τ 0, we obtain EXACTLY f t = ẋ f x This looks like the Vlasov equation. What went wrong??? τ (54) (55)

53 Kinetic effects: Introduction to Collisions The answer: nothing, but now ẋ contains the whole detailed microscopic motion. Consider the friction term and let the coordinate x be velocity v, then = vτ = τ q m (E exact + v B exact ). Writing the fields as a sum of the smooth background fields, E and B, and strong spiky microscopic fields, Ẽ and B, the expectation value gives τ = q (E + v B) + A(x, v) (56) m Function A corresponds to the friction caused by interaction. If we neglect the friction and diffusion terms, we arrive at the smoothed Vlasov equation. Remember that we have shown x (ẋf) + v ( vf) = ẋ x f + v v f (57)

54 Kinetic effects: Introduction to Collisions Fokker-Planck equation (FPE) in a plasma The most common form for FPE you encounter in plasma physics is f s t + v x f s + q s (E + v B) m s v f s = [ v A s 1 ] 2 v (D sf s ) (58) The coefficients A s and D s are, (u = v v ) A s = 2 ( c sb m m ) s dv f b (v ) u b s m b u 3 (59) D s = 2 ( c sb I m 2 dv f b (v ) b s u uu ) u 3, (60)

55 Kinetic effects: Introduction to Collisions Fokker-Planck equation (FPE) in a plasma Even this is a grude approximation, in fact the coefficients A s and D s are valid for the case with E = B = 0 but often applied out of necessity. If the distribution f s is only a very small perturbation to the background we may assume the fields E and B to be independent of f s, and that the major component of the plasma is in thermal equilibrium. The result will be a partial differential equation and the solution can be obtained rather easily with Monte Carlo methods even for difficult geometries. If the distribution f s represents a larger population of the plasma then Maxwell s equations will be coupled to Fokker-Planck equation. The result is a system of intego-differential equations,very difficult to solve

56 Kinetic effects: Introduction to Collisions Fokker-Planck equation (FPE) in a plasma It is possible to show following properties of FPE: particles are conserved energy is conserved momentum is conserved In addition to these you will show in the exercises that the steady state solution is a Maxwellian FPE increases entropy the distribution function will always stay positive

57 Summary Layout 1 Motivation 2 Boltzmann equation (a nasty bastard) 3 Vlasov equation 4 Kinetic effects: Oscillations I 5 Kinetic effects: Oscillations II 6 Kinetic effects: Introduction to Collisions 7 Summary

58 Summary Summary In MHD, the spatial shape of the plasma density, pressure, and magnetic field (gradients etc.) are the dominating effects. The analyzation of ideal MHD revealed, e.g., Alfven waves Investigation of the warm plasmas and inclusion of the details of the particle velocity distribution function revealed how the waves behave in time, we observed damping or growth depending on the shape of the distribution function. Investigation of collisions as a stochastic process revealed diffusion and friction in the velocity space. These terms drive any distribution function towards a Maxwellian.