Waves in plasma. Denis Gialis

Transcription

1 Waves in plasma Denis Gialis This is a short introduction on waves in a non-relativistic plasma. We will consider a plasma of electrons and protons which is fully ionized, nonrelativistic and homogeneous. 1 Preliminaries 1.1 Maxwell s equations The electromagnetic field ( E, B, in such a plasma, is governed by Maxwell s equations: E = e (n p n e ɛ 0, (1 B = 0, ( E = B t, (3 B = µ 0 J + 1 c E t, (4 where n p and n e are respectively the local proton density and the local electron density. In our case, we can add the definition of the total current density vector J J e + J p = e (n p Vp n e Ve, (5 where V p and V e are respectively the local proton fluid velocity and the local electron fluid velocity (see Sec. 1.. Ph. D. in Astrophysics - University J. Fourier, Grenoble, FRANCE. 1

2 Waves in plasma - Denis Gialis c Fluid reduction For a distribution function f( r, v, t (= f( x, t, the Fokker-Planck equation is such that f t = < xi > f ( ] < xi x j > f. (6 x i t x j t For a process on a time t much shorter than the characteristic colliding time, we can neglect the diffusion terms, and we obtain f t = ( < r > f ( < v > f. (7 r t v t The plasma dynamics is dominated by the electromagnetic field and we have < r > t < v > t = v, (8 = q ( E( r, t + v B( r, t. m (9 We then deduce the Vlasov s equation ( v v = 0 f t + v f r + q m ( E( r, t + v B( r, t f v = 0, (10 and the hydrodynamics quantities (d v = dv 1 dv dv 3 : n( r, t f( r, v, t d v, (11 V ( r, t P ( r, t v f( r, v, t d v, n( r, t (1 m ( v V ( v V f( r, v, t d v, (13 respectively the density, the Euler s velocity (or fluid velocity and the kinetic pressure tensor. The integration of the Vlasov s equation yields the continuity equation n t = (n V, (14

3 Waves in plasma - Denis Gialis c because the velocity distribution at the infinity equals to 0, ( E + v B f ( ] v d v = E + v B f d v = 0. (15 v The Euler s equation is obtained by the first order moment of the Vlasov s equation, ( f f v + v t r + q ( E + v B f d v = 0, (16 m v and, because v v = ( v V ( v V V V + v V + V v, ( v v f ( d v = ( v v f r r d v = P m + n V V. (17 Also, we have, by integration parts, v ( ] ( E + v B f d v = n E + V B. (18 v Then, the Euler s equation is V t + ( V V = P n m + q m ( E + V B. (19 With an isotropic and adiabatic assumption: P = P I and P n γ = constant, where γ = c p /c v is the adiabatic index. In general anisotropic case, by writing P = B P I + (P P B, and in B adiabatic transformations (of duration t 1/ ce, we obtain P = constant, n B (0 P B = constant. n 3 (1 In Eq. (19, for a two fluid model (p-e plasma, we could have to add a term of momentum exchange which results from taking account a collision term in the Fokker-Planck equation (6: it can be approximated by ν v, where ν 3

4 Waves in plasma - Denis Gialis c is the colliding frequency or the momentum exchange rate per collision 1. It yields two coupled Euler s equations: V ( e t + Ve Ve = Pe e ( E + Ve n e m e m B ν ep ( V e V p, e ( V ( p t + Vp Vp = Pp + e ( E + Vp n p m p m B ν pe ( V p V e, p (3 where ν ep n p σ ep v e v p and ν pe n e σ pe v p v e. Magneto-Hydro-Dynamics.1 MHD model assumptions We want to construct a single fluid theory: (1 We consider phenomena that occur on a scale time τ 0 such that τ 0 cp 1, where the Larmor s pulsation cp = e B/m p, or on a lenght scale l 0 such that l 0 pp c, where the plasma pulsation pp = n p e /ɛ 0 m p. We will define the Alfvén velocity as V A = ( cp / pp c c. ( We assume the local thermodynamics equilibrium (T e T p, and a quasi-neutral p-e plasma: n p n e n p + n e. (3 P = P I and P n γ = constant (isotropic and adiabatic assumption. (4 We define the following quantities: ρ n p m p + n e m e n p m p, (4 V n p m p Vp + n e m e Ve n p m p + n e m e V p, (5 P P p + P e, (6 J e (n p Vp n e Ve. (7 With these assumptions, the Eq. ( and (3 lead to V ρ ] ( t + V V = P +(n p n e e E+ J B+(ν ep n e m e ν pe n p m p ( V e V p, 1 It depends from the interaction modelling between the two particle species. We have the approximation < v e > / < v p > m p /m e, but V e V p 0. 4

5 Waves in plasma - Denis Gialis c or, taking into account the quasi-neutrality assumption and equality in momentum exchange (n e m e ν ep = n p m p ν pe or ν ep = (m p /m e ν pe, V ρ ] ( t + V V = P + J B. (8 Also, because V e = V p J/e n e V J/e n e, we deduce from Eq. ( that E + V B m e ν n e e J J B n e e = 0, (9 by neglecting dv e /dt and P e in the fluid model, and with ν = ν ep. With η n e e (Spitzer conductivity, m e ν ce e B/m e and b = B/B, the generalized Ohm law is written J + ( ( ce J b = η E + V B. (30 ν The current density vector, with E = ( E b b and E = b ( E b, can be easily expressed J = η E ( + η E + V B ( + η b E + V B, (31 where η = η/(1 + c /ν and η = η/(ν/ c + c /ν. There are two asymptotic limits: (1 When ce ν, the Ohm s law becomes isotropic and J ( = η E + V B. This is the resistive MHD regime. ( When the plasma is collisionless, i.e ν 0 (or η +, with a finite ratio ce /ν, we have the approximation of the ideal MHD or the non-resistive MHD regime: E + V B = 0. But, when J B /e ne V B 1, we have also J ( = η E + V B : this is the case when l 0 pp c (first MHD assumption and V 0 l 0 /τ 0 0. Otherwise it is necessary to have ce ν. When the electromagnetic field is slightly variable (over scales l 0 and τ 0, E E 0 /l 0 B 0 /τ 0, E/ t E 0 /τ 0 and B B 0 /l 0, where E 0 and B 0 are the characteristic values of E and B. Thus we have (1/c E/ t B (l 0/τ 0 c 1, (3 5

6 Waves in plasma - Denis Gialis c in the non-relativistic MHD regime, and the Eq. (4 becomes B = µ 0 J. Assuming an isotropic Ohm s law, we deduce from Eq. (3 and (4, B t ( V B = ν m B, (33 where ν m = 1/µ 0 η is the magnetic diffusivity. We can form the ratio between the advective and the diffusive terms ( V B ν m B V 0 B 0 /l 0 ν m B 0 /l 0 = V 0 l 0 ν m, (34 where V 0 ( c is the characteristic velocity ( V A of the MHD fluid. This ratio is defined as the magnetic Reynold number R m. When R m 1, the plasma dynamics is dominated by its resistivity and its diffusion. When R m 1, this is the ideal MHD regime: the diffusion can be neglected and, the magnetic field and the MHD fluid are frozen together.. Ideal MHD and waves According to the previous section, the ideal MHD equations are: ρ = t ( ρ V, (35 B t = ( V B, (36 d ( P ρ γ dt = 0, (37 ρ V t + ( V V ] = P + B µ 0 B. (38 In the Eq. (38, the term ( B B can be written ( B B (1/ B. Thus, we have V ρ ] ( t + V V = P P m + T m, (39 where the magnetic pressure P m B /µ 0, and the magnetic tension, T m, is such that T m ( B B = d ( B B b b +, (40 µ 0 ds µ 0 µ 0 R c 6

7 Waves in plasma - Denis Gialis c where s is the curvilinear abscissa, b is the unit vector perpendicular to b = B/B, and Rc, the curvature radius of the magnetic field line. Thus, the Laplace s force (per unit of volum can be written f L = J B = P m + P m R c b. (41 If P/P m < 1, the plasma undergoes the magnetic dynamics, and if P/P m > 1, the magnetic field undergoes the plasma dynamics. At last, the Eq. (37 can be transformed P t = v P γ P v. (4 Let us study the effects of a weak perturbation. The linearization process is the following one: considering a Fourier decomposition, all complex quantities X will be such that X = X 0 +x where X 0 is a real time invariant quantity and x/x 0 1. The perturbative quantity x will be expi ( t k r] (where i = 1. The only condition on the pulsation is that cp. Thus, we will have / t i, i k, i k and k, when one derive these perturbative quantities (only. Assuming that V 0 = 0 and writing ρ = ρ 0 + δρ, the Eqs. (35 to (38 yield δρ = ρ 0 k v, (43 1 b = k ( v B 0, (44 i p = v P 0 + iγ P 0 k v, (45 i ρ 0 v = p + B 0 b µ 0 ] + 1 µ 0 ( B 0 b + ( b B 0 ]. (46 In fact, the Eq. (46 is obtained because, following the Eq. (39, we have ( P 0 = B 0 + ( B 0 B 0. (47 µ 0 µ 0 7

8 Waves in plasma - Denis Gialis c Example: MHD waves in homogeneous medium We consider here a simple case: the medium is homogeneous i.e the spatial variation of quantities X 0 are negligible. The Eqs. (45 and (46 yield With the Eq. (44, we obtain p = γ P 0 k v, (48 ρ 0 v = p k + 1 ( B0 k b. (49 µ 0 ( ρ 0 v = γ P 0 k v k 1 ( B0 k k ( v µ ] B 0. (50 0 Let us define the basis vectors e = B 0 / B 0, e = k / k when k = 0 (otherwise e is an arbitrary perpendicular unit vector, and e = e e, with k = k e + k e and v = v e + v e + v e. We will have 1 ( B0 k k ( v µ ] B 0 = 1 ( B0 v k e + v k e. (51 0 µ 0 Thus, the Eq. (50 leads to the system ( Cs k v Cs k k v = 0, (5 ( Cs k VA k v Cs k k v = 0, (53 ( VA k v = 0, (54 with the sound velocity C s = γ P 0 /ρ 0 and the Alfvén velocity V A = B 0 /(ρ 0 µ 0 for (B 0/µ 0 /ρ 0 c 1. In its matricial form, this system becomes ( Cs k C s k k 0 Cs k k ( Cs k V A k ( VA k v v v = 0. (55 The solutions of this system (proper modes are determined by det = 0. The Alfvén waves are then defined by the following dispersion relation, = V A k, (56 8

9 Waves in plasma - Denis Gialis c and the fast (+ and slow ( magnetosonic waves are determined by the dispersion relation, ( f,s = k (C s + V k A ± (Cs + VA 4 Cs VA. (57 k 3 Electromagnetic waves in fluid model 3.1 Assumptions In the Sec. 3, we will consider the following assumptions: General assumption of adiabaticity: (1 The phase velocity of waves, v φ = /k, is supposed to be v th, the thermal velocity of charged particle species. ( The energy density of the EM waves is the thermal energy density. (3 The damping of EM waves is neglected. (4 The collision effects are neglected: ν 0. Additional assumption: the plasma is homogeneous. We will define: The plasma pulsation of the species a: pa = n a q a/ɛ 0 m a, where n a, q a and m a, are respectively the density, the charge and the mass of a. The cyclotronic or Larmor pulsation of the species a: ca = q a B 0 /m a, where B 0 is the intensity of the magnetic field. 3. Electromagnetic mode The plasma is assumed to be isotropic and unmagnetized (B 0 = 0. We first neglect the ionic response ( J p and the temperature (T e = T p 0. With the previous assumptions, the Eq. ( leads to V ( e t + Ve Ve = e ( E + Ve B m. (58 e 9

10 Waves in plasma - Denis Gialis c The fields E and B are considered as perturbations. Following the linearization process, as explained in Sec.., the Eq. (58 simply becomes and, because J e = n e e V e, we obtain Thus, the Eqs. (3 and (4 could be written and we deduce V e t = e m e E, (59 J e = i pe ɛ 0 E. (60 k E = B, (61 k B = i Je ɛ 0 E, (6 ( k E k k E = pe c E. (63 For the electromagnetic mode ( k E = 0, this equation yields the following dispersion relation: = pe + k c. (64 For > pe, the phase velocity /k is superluminal and the EM wave well verifies the assumption of adiabaticity. For < pe, the EM wave is evanescent. For k E = 0, we obtain the plasmon mode = pe, but with no energy transport. It corresponds to Langmuir s oscillations. 3.3 Bohm-Gross and ionic acoustic modes The plasma is assumed to be isotropic and unmagnetized (B 0 = 0. We neglect the ionic response ( J p 0 and the ionic temperature (T p 0. With these assumptions, the Eq. ( leads to V ( e t + Ve Ve P = e e ( E + Ve n e m e m B, (65 e 10

11 Waves in plasma - Denis Gialis c and we suppose the isentropic state equation (37. According to the linearization process and the continuity equation (giving the Eq. (4, we obtain first p = γ P e k V e, (66 with P e = n e k B T e, where k B is the Boltzmann constant. Thus, the Eq. (65 can be written J e = i pe ɛ 0 E + γ k B T e m e ( k Je k. (67 For k E = 0, we obtain the Bohm-Gross mode: indeed, because of the previous equation, k J e = 0, and we have J e = i pe The dispersion relation of the Bohm-Gross mode is thus ɛ 0 E 1 γ k B T e. (68 k m e = pe + γ k B T e m e k. (69 For < pe, there is no pure electronic mode. We have to take into account the ionic dynamics. We first assume low frequencies such that k B T p m p < k < k B T e m e. (70 This is at the limit of the first adiabatic assumption (see Sec We neglect the electronic inertia. The Eq. (68 gives J e i pe m e ɛ 0 E γ k B T e k. (71 Concerning the ionic current, the Eq. (3 (with T p 0 and the linearization process lead to J p = i pp ɛ 0 E. (7 For low frequencies pp, one can assume the quasi-neutrality of current: J e + J p 0. Indeed, because B = 0, we have J e + J p + i ɛ 0 E = 0 and ɛ 0 E J e + J p pp 1 m e pe ( ] + o. (73 γ k B T e k pp pp 11

12 Waves in plasma - Denis Gialis c We then deduce the following dispersion relation for the phonon mode; with C s = γ k B T e /m p. = C s k, (74 For higher frequencies such that pp > > k k B T e /m e, we cannot assume the quasi-neutrality. The Eqs. (68 and (7 yield pp + pp (m e /m p k C s = 1, (75 with pp / pe = m e /m p. By neglecting the term (m e /m p, we obtain the dispersion relation of the ionic acoustic mode; pp C s k pp + C s k. (76 In this mode, we cannot have pp C s k which leads to pp, because the condition /k k B T e /m p would not be satisfied and we would have to take into account Landau s damping. 3.4 Magnetized electronic and ionic modes The plasma is assumed to be magnetized. { e, e, e } as described in Sec Electronic modes We will define the basis We first neglect the ionic response ( J p 0 and the temperature (T e = T p 0. With these assumptions, the Eq. ( leads to V ( e t + Ve Ve = e ( E + Ve B m, (77 e and the linearization process yields J e = i pe ɛ 0 E + i ce J e e. (78 1

13 Waves in plasma - Denis Gialis c Because we deduce J e e = i pe ɛ 0 E e, (79 J e e = i pe ɛ 0 E e i ce J e + i ( ce Je e e, (80 pe J e = i ce The Eqs. (3 and (4 gives k ( k E with the coefficients, ɛ 0 E ce + i ɛ ( ] 0 E e ce ɛ 0 E e e. (81 + ɛ c ( E + ( ce ɛ ( E e e i ɛ ( E ] e = 0 (8 ɛ ( = 1 pe, (83 ce ɛ ( = ce pe. (84 ce By defining E = E + E with E = E e and E e = 0, we obtain k ( k E with the last coefficient, + c ɛ ( E + ɛ ( E i ɛ ( E e ] = 0, (85 ɛ ( = 1 pe. (86 (1 For a parallel propagation i.e k = k e, the Eq. (85 is equivalent to ɛ c ɛ c k i ɛ c 0 i ɛ c ɛ c k E E y E z = 0, (87 13

14 Waves in plasma - Denis Gialis c in the basis { e, e, e }. For E = E, we recognize the plasmon mode (see Sec. 3. with = pe. For E E, the other proper modes are obtained for det = 0 i.e k c = (ɛ ± ɛ. The two dispersion relations correspond to the following modes: (1-a The left mode (L-mode: k = c with the cut-off frequency (k = 0 such that L = ] pe 1, (88 ( + ce (1-b The right mode (R-mode: k = c with the cut-off frequency (k = 0 such that R = pe + ce 4 ce. (89 ] pe 1, (90 ( ce pe + ce 4 + ce, (91 and the resonance frequency (k + for ce. We can plot two different Brillouin diagrams (-k plane for the R-mode: for a strong magnetized plasma, ce > pe, and for a dense plasma, ce < pe. The different phase velocity between the R-mode and the L-mode is at the origin of the Faraday effect, for a rectilinear polarized wave, which is the rotation of the rectilinear polarization plane. For ce, the R-mode is named whistler mode and the dispersion relation is simply: = k c ce. (9 pe The group velocity is thus v g = / k. 14

15 Waves in plasma - Denis Gialis c ( For a perpendicular propagation i.e, for example, k = k e, the Eq. (85 is equivalent to ɛ c k ɛ c i ɛ c 0 i ɛ c ɛ c k E E y E z = 0, (93 in the basis { e, e, e }. For E = 0, we obtain the ordinary mode with the dispersion relation: = pe + k c. (94 This is similar to the electromagnetic mode previously obtained in Sec. 3.. For E 0, we deduce from det = 0, the extraordinary mode with the following dispersion relation: which can be written k c = (ɛ ɛ /ɛ, (95 k c = ( R ( L UH, (96 where the frequency of the high hybrid resonance, corresponding to the Langmuir electronic oscillation, is such that UH = 1 ] ce + pe + ce + pe 4 ce (cp + pp. (97 For pe pp and, because ce cp, UH ce + pe. (3 For an oblique propagation, k = k e + k e, with k = k sin θ and k = k cos θ. The Eq. (85 is equivalent to ɛ c k 0 k k 0 ɛ c k k i ɛ c i ɛ c ɛ c k k k in the basis { e, e, e }. By defining, E E y E z = 0, (98 a 1 (θ = ɛ sin θ + ɛ cos θ, (99 a (θ = ɛ ɛ (1 + cos θ + (ɛ ɛ sin θ, (100 a 3 = ɛ (ɛ ɛ, (101 15

16 Waves in plasma - Denis Gialis c we obtain the proper modes (det = 0 with the equation a 1 ( k c 4 ( k c a + a 3 = 0. (10 Thus, we have the dispersion relation k = a ± a 4a 1 a 3. (103 c a 1 There are resonances (k + for a 1 0 and cut-off (k 0 for a Ionic modes We also consider a cold plasma by neglecting the temperature (T e = T p 0, but we introduce the ionic current ( J p 0. With these assumptions, the Eqs. ( and (3, and the linearization process, yield pe J e = i ce pp J p = i cp ɛ 0 E + i ce ɛ ( ] 0 E e ce ɛ 0 E e e, (104 ( ] E e e, (105 ɛ 0 E i cp ɛ 0 E e cp ɛ 0 and we obtain the same relation than in the previous section, k ( k E + c ɛ ( E + ɛ ( E i ɛ ( E e ] = 0, (106 but with the following new definitions for coefficients ɛ ( = 1 pe ce pp, (107 cp ɛ ( = ce pe ce cp pp, (108 cp ɛ ( = 1 pe pp. (109 For high frequencies ( cp, the situation is the same one than in the Sec

17 Waves in plasma - Denis Gialis c For very low frequencies ( 0, because ce cp, we have ɛ pe/, ɛ 1 + c /V A and, ɛ (c /V A / cp. (1 For a parallel propagation i.e k = k e, and for E E, the Eq. (106 leads to k c = (ɛ ± ɛ. For low frequencies ( pp, and for n e n p (i.e pp/ cp pe/ ce, we deduce = k VA 1 ± ( ] + o. (110 cp We thus define again the following modes: (1-a The left mode (L-mode: (1-b The right mode (R-mode: cp k V A 1 cp. (111 k V A 1 + cp. (11 For 0, these two modes blend into the Alfvén mode, as previously described in Sec..3: k V A. ( For a perpendicular propagation i.e, for example, k = k e, and for E 0, we deduce from the Eq. (106 the dispersion relation for the extraordinary mode: k c = (ɛ ɛ /ɛ, (113 which leads to k c = ( R ( L ( LH ( UH, (114 where the frequency of the low hybrid resonance, corresponding to the Langmuir oscillation, is such that LH = 1 ] ce + pe ce + pe 4 ce (cp + pp. (115 For pe pp and, because ce cp, we have LH ce cp + pp ce + pe 17 ce cp. (116

18 Waves in plasma - Denis Gialis c For 0, we recognize again the Alfvén mode because k V A / 1 + V A /c k V A (with V A c. (3 If we slightly modify the assumptions by taking into account the electronic compressibility effects (T e 0, and if we consider pulsations such that cp < < ce, the modification of the Eq. (104 with the pressure term, and the Eq. (105 give J e i pe ɛ 0 E + γ k B T e k Je, (117 m e pp J p i ɛ cp 0 E, (118 with ce/ 1 and, by neglecting terms in cp /. Because /k γ kb T e /m e, the Eq. (117 can be written J e i pe m e γ k B T e k ɛ 0 E. (119 The quasi-neutrality of current, J e + J p = 0, leads to dispersion relation of the ionic cyclotron mode; = cp + C s k, (10 with C s = γ k B T e /m p, and because pp/ pe = m e /m p. 18

19 Waves in plasma - Denis Gialis c Summary Let us summarize the previous results concerning waves propagating in an isotropic p e plasma. I - For an unmagnetized plasma i.e B 0 = 0 I-1 The ionic response ( J p and the temperature (T e = T p 0 are neglected: - The electromagnetic mode ( k E = 0: = pe + k c. (11 For > pe, the phase velocity /k is superluminal. For < pe, the EM wave is evanescent (k = k r + i k i, with k i 0. - The plasmon mode or Langmuir s oscillations ( k E = 0: = pe. No energy transport. I- The ionic response ( J p and the ionic temperature (T p 0 are first neglected: - The Bohm-Gross mode ( k E = 0: = pe + γ k B T e m e k. (1 For < pe, there is no pure electronic mode. We have to take into account the ionic dynamics ( J p. - The phonon mode ( pp and J e + J p 0: with C s = γ k B T e /m p. = C s k, (13 - The ionic acoustic mode (k k B T e /m e < < pp : pp C s k pp + C s k. (14 19

20 Waves in plasma - Denis Gialis c II - For a magnetized plasma i.e B 0 0 II-1 The electronic modes ( J p 0 and T e = T p 0: - The L-mode ( k B: k = c ] pe 1, (15 ( + ce with the cut-off frequency (k = 0 such that - The R-mode ( k B: L = k = c with the cut-off frequency (k = 0 such that R = pe + ce 4 ce. (16 ] pe 1, (17 ( ce pe + ce 4 + ce, (18 and the resonance frequency (k + for ce. For ce, the R-mode is named whistler mode and the dispersion relation is simply: = k c ce. (19 pe - The ordinary mode equivalent to the electromagnetic mode ( E = 0 and k B: = pe + k c. (130 - The extraordinary mode ( E 0 and k B: k c = ( R ( L UH, (131 with the frequency of the high hybrid resonance, UH = 1 ] ce + pe + ce + pe 4 ce (cp + pp 0. (13

21 Waves in plasma - Denis Gialis c For pe pp, UH ce + pe. II- The ionic modes ( J p 0 and T e = T p 0: - The L-mode ( k B: - The R-mode ( k B: k V A 1 cp. (133 k V A 1 + cp. (134 For 0, these two modes blend into the Alfvén mode: k V A. - The extraordinary mode ( E 0 and k B: k c = ( R ( L ( LH ( UH, (135 with the frequency of the low hybrid resonance, LH = 1 ] ce + pe ce + pe 4 ce (cp + pp For pe pp and, because ce cp, we have. (136 LH ce cp + pp ce + pe ce cp. (137 For 0 (or cp, we obtain the Alfvén mode because k V A / 1 + V A /c k V A (with V A c. - The slow (- and fast (+ magnetosonic modes ( cp : ( f,s = k (C s + V k A ± (Cs + VA 4 Cs VA. (138 k 1