Magnetized plasma : About the Braginskii s 1 macroscopic model 2
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1 Magnetzed plasma : About the Bragnsk s 1 macroscopc model 2 B. Nkonga JAD Unv. Nce/INRIA Sopha-Antpols 1 S. I. Bragnsk, n Revews of Plasma Physcs, edted by M. A. Leontovch Consultants Bureau, New York, 1965, Vol. I, p Talk H. Gullard, 1st summer school of the Large Scale Intatve FUSION : September 15-18, 2009 n Strasbourg. fuson/ B. Nkonga. Flud Theory 1 / 56
2 Overvew 1 Knetc and macroscopc equatons for Smple plasma 2 Flud Theory : Scalng and dmensonal analyss 3 Flud Theory : Hlbert s expanson and asymptotc analyss 4 Flud Theory : Frst order correcton of Bragnsk. 5 Bragnsk transport Coeffcents B. Nkonga. Flud Theory 2 / 56
3 State of the matter : Plasma Temperature versus Number of charged partcles/m 3 B. Nkonga. Flud Theory 3 / 56
4 Models + Maxwell s Equatons for E and B 1 N-body : x k t : R R 3, k = O /m 3 800m 3 Newton Equaton for each charged partcle dx k dt = v k and m k dv k dt = q k m k E + v k B + l C kl 2 Knetc : f k t, x, v : R 7 R, k = O10 t f k + v x f k + q k m k E + v B v f k = l C kl 3 Flud :ω k t, x : R 4 R N k, k = O10 t ω k + L, ω k, B, E = S k B. Nkonga. Flud Theory 4 / 56
5 Knetc equaton for Smple plasma D v t f e + qe m e L v v f e = C ee f e, f e + C e f e, f D v t f + q m L v v f = C e f, f e + C f, f where for electrons k = e and ons k = Moreover f k f k t, x, v s the dstrbuton functon. m k s the mass q k s the charge C kl are collsons operators. D v t = t + v x s the materal dervatve at the velocty v, D v t s the materal dervatve at the velocty v, L v = E + v B s the Lorenz force E and B are govern by Maxwell equatons. B. Nkonga. Flud Theory 5 / 56
6 Coulomb bnary scatterng law The Landau form of the Coulomb collson s Eq. 4.3 of Bragnsk: C kl f k, f l = Γ kl 2 [ ] v O kl f k, f l where, wth a = v v we have [ O kl f k, f l = dv mk B a f k v f l v R 3 m l v f l v ] f k v v where, for rgd spheres approxmaton, Cut-offs estmaton gves Γ kl = 4πq2 k q2 l ln Λ m 2 k and for any vector a B a = a 2 I a a a 3 ln Λ s the Coulomb logarthm. B. Nkonga. Flud Theory 6 / 56
7 Propertes of the scatterng tensor B = a 2 I a a a 3 a 1 B a s symmetrc B a T = B a and even B a = B a a 1 2 B a derve from a potental B a = a = a a a a 3 a t s n the kernel of B a B a a = a B a = a = 2a 2 and Tr [B a ] = a a 3 a 5 v B a = 2a a 3 = v B a [Tr a a B] = a 2 I 3a a a 5 7 In sphercal coordnatesvelocty space a B a 2 a 3 B θ, φ B θ, φ = 1 1 sn θ sn θ θ θ sn 2 θ θ 2 = 1 1 µ µ µ s the angular part of a 2 and s often wrtten n terms of the ptch angle varable : µ = cos θ µ 2 θ 2 B. Nkonga. Flud Theory 7 / 56
8 Other formulatons of Coulomb bnary scatterng law O kl f k, f l = dv R 3 [ mk B a f k v f l v m l v 1 Fokker-Planck form, wth D l v = O kl f k, f l = f l v ] f k v v [ dv fl v ] B a and P5 R 3 f k v D l v 1 + m k f k v m l v D l v v 2 Rosenbluth57-Trubnkov58 form : O kl f k, f l = m k m k f k v H l v v v f k v v v G l v Rosenbluth potentals H l v and G l v: H l v = dv f l v wth G l v = dv a f l v R 3 a R 3 B. Nkonga. Flud Theory 8 / 56
9 Other formulatons of Coulomb bnary scatterng law [ 1 Fokker-Planck form, wth D l v = dv fl v ] B a and P5 R 3 O kl f k, f l = 1 + m k f k v m l v D l v v f k v D l v O kl f k, f l = D l v f k v v 2 Rosenbluth57-Trubnkov58 form : O kl f k, f l = 1 + m k m k f k v H l v v v + m k m l f k v v D l v f k v H l v v D l v = H l v v D l v s the dffuson tensor and H l v = 1 2 v v G l v B. Nkonga. Flud Theory 8 / 56
10 Macroscopc equatons : D u t = t + u x D u k t n k + n k x u k = 0 m k n k D u k t u k + x p k q k n k E + u k B = x π k + R k n k D u k t p k γ k p k x u k γ k 1 = x q k π k : x u k + Q k t B + x E = 0 q k and γ k 5 3 are constants parameters. T k = n k p k Statc constrant : t x B s constant. Addtonal relaton to defne E V V = x B = µ 0 J = µ 0 q k n k u k k n k u k p k B B. Nkonga. Flud Theory 9 / 56
11 Macroscopc equatons : Transport π k = q k = R k = Q k = dv [m k f k v u k v u k v u k 2 ] I R 3 3 v u k dv [m 2 ] k f k v R 3 2 [ dv m k v u k ] C kl R 3 l [ ] v u k 2 dv m k C kl R 3 2 l Scalng and asymptotc expansons of knetc equatons : Defne π k, R k, q k and Q k as functons of V B. Nkonga. Flud Theory 10 / 56
12 Knetc Transport theory : Strategy 1 Defne an approprate frame and scalng. 2 Evaluate non-dmensonal coeffcent n term of a small parameter. 3 Proceed to an expansons accordng to these terms 4 Obtaned approxmatons of probablty densty functons. 5 Use these approxmatons to evaluate transport terms. B. Nkonga. Flud Theory 11 / 56
13 Overvew 1 Knetc and macroscopc equatons for Smple plasma 2 Flud Theory : Scalng and dmensonal analyss 3 Flud Theory : Hlbert s expanson and asymptotc analyss 4 Flud Theory : Frst order correcton of Bragnsk. 5 Bragnsk transport Coeffcents B. Nkonga. Flud Theory 12 / 56
14 Knetc equaton n a non nertal frames Coordnate transformaton : κ t, x, v = v u t, x and κ t, x, v = v u t, x Let us defne f k t, x, κ = f k t, x, v v=κ+u and g k t, x, κ = f k t, x, v v=κ+ue There are 4 possbles formulatons for smple plasma knetc equatons: Electrons and ons n mean electrons velocty frame [ ] D u t g e + κ x g e + qe m e E + u + κ B D u t u κ x u κ g e = C ee g e, g e + C e g e, g [ ] D u t g + κ x g + q m E + u + κ B D u t u κ x u κ g = C e g, g e + C g, g The Coulomb collson operator s nvarant under Gallean transformaton. κ = v, κ = v, B v v = B κ κ B. Nkonga. Flud Theory 13 / 56
15 Knetc equaton n a non nertal frames Coordnate transformaton : κ t, x, v = v u t, x and κ t, x, v = v u t, x Let us defne f k t, x, κ = f k t, x, v v=κ+u and g k t, x, κ = f k t, x, v v=κ+ue There are 4 possbles formulatons for smple plasma knetc equatons: Electrons and ons n the opposte mean velocty frame D u t fe + κ f [ ] x e + qe m e E + u + κ B D u t u κ x u f κ e = C ee fe, f e + C e fe, f [ ] D u t g + κ x g + q m E + u + κ B D u t u κ x u κ g = C e g, g e + C g, g The Coulomb collson operator s nvarant under Gallean transformaton. κ = v, κ = v, B v v = B κ κ B. Nkonga. Flud Theory 13 / 56
16 Knetc equaton n a non nertal frames Coordnate transformaton : κ t, x, v = v u t, x and κ t, x, v = v u t, x Let us defne f k t, x, κ = f k t, x, v v=κ+u and g k t, x, κ = f k t, x, v v=κ+ue There are 4 possbles formulatons for smple plasma knetc equatons: Electrons and ons n ther mean velocty frame Bragnsk... [ ] D u t g e + κ x g e + qe m e E + u + κ B D u t u κ x u κ g e = C ee g e, g e + C e g e, g D u t f + κ f [ ] x + q m E + u + κ B D u t u κ x u f κ = C e f, f e + C f, f The Coulomb collson operator s nvarant under Gallean transformaton. κ = v, κ = v, B v v = B κ κ B. Nkonga. Flud Theory 13 / 56
17 Knetc equaton n a non nertal frames Coordnate transformaton : κ t, x, v = v u t, x and κ t, x, v = v u t, x Let us defne f k t, x, κ = f k t, x, v v=κ+u and g k t, x, κ = f k t, x, v v=κ+ue There are 4 possbles formulatons for smple plasma knetc equatons: Electrons and ons n mean ons velocty frame Gralle... D u t fe + κ f [ ] x e + qe m e E + u + κ B D u t u κ x u f κ e = C ee fe, f e + C e fe, f D u t f + κ f [ ] x + q m E + u + κ B D u t u κ x u f κ = C e f, f e + C f, f The Coulomb collson operator s nvarant under Gallean transformaton. κ = v, κ = v, B v v = B κ κ B. Nkonga. Flud Theory 13 / 56
18 Dmensonless equatons D u t fe + κ f [ ] x e + qe m e E + u + κ B D u t u κ x u f κ e = C ee fe, f e + C e fe, f Dmensonless where D u t fk + [t 0] [κ k ] κ x [x 0 ] f k [u k ] [κ k ] + q k [t 0 ] [E 0 ] m k [κ k ] E + [B 0 ] [u k ] [E 0 ] = [t 0] [C kk ] C kk fk, [f k ] f k D u t u + [t 0] [κ k ] [x 0 ] u + [κ k ] [u k ] κ D u t = t + [t 0] [u k ] u x [x 0 ] κ x u f κ k B f κ k + [t 0] [C kl ] C kl fk, [f k ] f l B. Nkonga. Flud Theory 14 / 56
19 Scalng Hypotheses [ρ] s the on Larmor radus, [r] s characterstc short length ɛ = [ρ] [r] me m Ions and electrons are of the same scale for 1 Denstes n 0 2 Temperatures T 0 3 Cross-sectons σ 0 4 Macroscopc veloctes u 0 [κ k ] = k B [T 0 ] m k = [κ ] [κ e ] = ɛ [l k ] [τ k ] [κ k ] = 1 [σ 0 ] [n 0 ] = [τ e] [τ ] = ɛ [u k ] [u 0 ] = [u e ] [u ] = ɛ0 1 Note that u e u 3 P. Degond, B. Lucqun-Desreux, Transport coeffcents of plasmas and dsparate mass bnary gases. Transp. Theory and Stat. Phys. 25 pp , J.J. Ramos, Flud Theory of Magnetzed Plasma Dynamcs at Low Collsonalty. Physcs of plasmas vol MIT Report PSFC/JA B. Gralle, T. Mangn, and M. Massot. Knetc theory of plasmas: translatonal energy. Math. Models Methods Appl. Sc. M3AS , B. Nkonga. Flud Theory 15 / 56
20 Scalng Hypotheses Other mportant parameters are : 1 The collsonalty [ν ] = [R] [l k ] 2 The pressure rato [β] = 2µ 0 [p] [B B] For ITER we have [T 0 ] 10keV, [n 0 ] m 3 and [l k ] 100m a. a R.V. Budny, Fuson alpha parameters n tokamak wth hgh DT fuson rates Nucl. Fuson Therefore [κ e ] ms 1, [κ ] ms 1 [τ e ] 10 4 s, [τ ] s B. Nkonga. Flud Theory 16 / 56
21 Scalng Hypotheses :: ɛ me m [τ e] [τ ] 1 1 Collsons scales [C ee ] [C e ] [f e] [τ e ], [C e] m e [f ] m [τ e ] [f ] [τ ] ɛ3 and [C ] [f ] [τ ] Indeed m e m = ɛ 2 and [τ e ] [τ ] = ɛ B. Nkonga. Flud Theory 17 / 56
22 Veloctes dstrbutons for onsred and electrons blue. [u ] [κ ] [u e ] [κ e ] 1 [u e ] = [u 0 ] 2 [u ] = [u 0 ] 3 M = [u ] [κ ] 1 Therefore M e = [u e ] [κ e ] = [u ] [κ ] wth ε = ɛm Indeed [κ ] [κ e ] = ε [κ e ] = 1 ɛ [κ ] 1 ɛ [u 0 ] B. Nkonga. Flud Theory 18 / 56
23 Veloctes dstrbutons for onsred and electrons blue. 1 [u e ] = [u 0 ] [u ] [κ ] [u e ] [κ e ] 2 [u ] = [u 0 ] 3 M = [u ] [κ ] ɛ < 1 Therefore M e = [u e ] [κ e ] = [u ] [κ ] [κ ] [κ e ] = ε wth ε = ɛm Indeed [κ e ] = 1 ɛ [κ ] 1 ɛ [u 0 ] B. Nkonga. Flud Theory 18 / 56
24 Scalng Hypotheses :: ɛ = me m 1, ε = ɛm 2 Large observaton tme and space length scales : Hydrodynamc [t 0 ] = [τ ] ε = M [τ e ] ε 2 and [x 0 ] = [l ] ε = [l e] ε = [l 0] ε = [t 0] [x 0 ] = 1 [κ ] Electrcal and thermal energes are of the same scale q e [x 0 ] [E 0 ] = m [κ ] 2 = m e [κ e ] 2 Strongly magnetzed plasma [B 0 ] [u ] [E 0 ] = 1 B. Nkonga. Flud Theory 19 / 56
25 Electrons : ε = ɛm, u u e, κ κ e D u t = t + [t 0] [u e ] u x, [x 0 ] [t 0 ] [u e ] [x 0 ] = 1, D u t g e + [t 0] [κ e ] κ x g e [u e ] [x 0 ] [κ e ] + q e [t 0 ] [E 0 ] m e [κ e ] E + [B 0 ] [u e ] [E 0 ] = [t 0] [C ee ] [f e ] D u t u + [t 0] [κ e ] [x 0 ] u + [κ e ] [u e ] κ κ x u κ g e B κ g e C ee g e, g e + [t 0] [C e ] C e g e, g [f e ] B. Nkonga. Flud Theory 20 / 56
26 Electrons : ε = ɛm, u u e, κ κ e D u t = t + u x, [t 0 ] [κ e ] [x 0 ] = [κ e ] [u e ] = 1 ɛm D u t g e + [t 0] [κ e ] κ x g e [u e ] [x 0 ] [κ e ] + q e [t 0 ] [E 0 ] m e [κ e ] E + [B 0 ] [u e ] [E 0 ] = [t 0] [C ee ] [f e ] D u t u + [t 0] [κ e ] [x 0 ] u + [κ e ] [u e ] κ κ x u κ g e B κ g e C ee g e, g e + [t 0] [C e ] C e g e, g [f e ] B. Nkonga. Flud Theory 20 / 56
27 Electrons : ε = ɛm, u u e, κ κ e D u t = t + u x, D u t g e + 1 ε κ x g e ε q e [t 0 ] [E 0 ] m e [κ e ] q e [t 0 ] [E 0 ] = [t 0] [κ e ] = 1 m e [κ e ] [x 0 ] ɛm D u t u + 1 ε κ x u κ g e E + [B 0 ] [u e ] u + 1 [E 0 ] ε κ B κ g e = [t 0] [C ee ] [f e ] C ee g e, g e + [t 0] [C e ] C e g e, g [f e ] B. Nkonga. Flud Theory 20 / 56
28 Electrons : ε = ɛm, u u e, κ κ e D u [B 0 ] [u e ] t = t + u x, = [B 0 ] [u ] = 1 [E 0 ] [E 0 ] D u t g e + 1 ε κ x g e ε D u t u + 1 ε κ x u κ g e 1 E + [B 0 ] [u e ] u + 1 ε [E 0 ] ε κ B κ g e = [t 0] [C ee ] [f e ] C ee g e, g e + [t 0] [C e ] C e g e, g [f e ] B. Nkonga. Flud Theory 20 / 56
29 Electrons : ε = ɛm, u u e, κ κ e D u t = t + u x, D u t g e + 1 ε κ x g e ε 1 ε = [t 0] [C ee ] [f e ] [t 0 ] [C ee ] [ g e ] = [t 0] [C e ] [ g e ] D u t u + 1 ε κ x u = [t 0] [τ e ] = M ε 2 κ g e E + u + 1 ε κ B κ g e C ee g e, g e + [t 0] [C e ] C e g e, g [f e ] B. Nkonga. Flud Theory 20 / 56
30 Electrons : ε = ɛm, u u e, κ κ e D u t = t + u x, D u t g e + 1 ε κ x g e ε D u t u + 1 ε κ x u κ g e 1 E + u + 1 ε ε κ B κ g e = M ε 2 C ee g e, g e + M ε 2 C e g e, g B. Nkonga. Flud Theory 20 / 56
31 Ions : ε = ɛm D u t = t + [t 0] [u ] u x, [x 0 ] D u t f + [t 0] [κ ] κ x [x 0 ] f [u ] [κ ] + q [t 0 ] [E 0 ] m [κ ] E + [B 0 ] [u ] [E 0 ] [t 0 ] [u ] [x 0 ] = 1, D u t u + [t 0] [κ ] [x 0 ] u + [κ ] [u ] κ = [t 0] [C e ] C e f, [f ] f e κ x u f κ B f κ + [t 0] [C ] C f, [f ] f B. Nkonga. Flud Theory 21 / 56
32 Ions : ε = ɛm D u [t 0 ] [κ ] t = t + u x, = [κ ] [x 0 ] [u ] = 1 M D u t f + [t 0] [κ ] κ x [x 0 ] f [u ] D u [κ ] t u + [t 0] [κ ] κ x u κ [x 0 ] f + q [t 0 ] [E 0 ] E + [B 0 ] [u ] u + [κ ] m [κ ] [E 0 ] [u ] κ B f κ = [t 0] [C e ] C e f, [f ] f e + [t 0] [C ] C f, [f ] f B. Nkonga. Flud Theory 21 / 56
33 Ions : ε = ɛm + q [t 0 ] [E 0 ] m [κ ] q [t 0 ] [E 0 ] Z m [κ ] = [t 0] [κ ] [x 0 ] = Z M D u t f + 1 κ M f x M D u t u + 1 κ x u M f κ E + [B 0 ] [u ] u + 1 κ B [E 0 ] M f κ = [t 0] [C e ] C e f, [f ] f e + [t 0] [C ] C f, [f ] f B. Nkonga. Flud Theory 21 / 56
34 Ions : ε = ɛm [B 0 ] [u ] [E 0 ] = [B 0 ] [u ] [E 0 ] = 1 D u t f + 1 κ M f x M D u t u + 1 κ x u M f κ + Z E + [B 0 ] [u ] u + 1 κ B M [E 0 ] M f κ = [t 0] [C e ] C e f, [f ] f e + [t 0] [C ] C f, [f ] f B. Nkonga. Flud Theory 21 / 56
35 Ions : ε = ɛm [t 0 ] [C e ] = [t 0] m e = 1 [t 0 ] [C ] and = [t 0] [f ] [τ e ] m M [f ] [τ ] = 1 ε D u t f + 1 κ M f x M D u t u + 1 κ x u M f κ + Z E + u + 1 κ B M M f κ = [t 0] [C e ] C e f, [f ] f e + [t 0] [C ] C f, [f ] f B. Nkonga. Flud Theory 21 / 56
36 Ions : ε = ɛm D u t f + 1 κ M f x M D u t u + 1 κ x u M f κ + Z E + u + 1 κ B M M f κ = 1 C e f, M f e + 1 ε C f, f B. Nkonga. Flud Theory 21 / 56
37 Dmensonless smple plasma system M < 1 Electrons : u u e, κ κ e ε2 M [ t u + u x u κ g e ] + ε M [ t g e + u x g e κ x u κ g e ] + 1 M [κ x g e E + u B κ g e ] = 1 ε [ κ B κ g e + C ee g e, g e + C e g e, g ] Ions: u u, κ κ [ M t u + u x u f ] κ + [ t f + u f x κ x u f ] κ + 1 [ C e f, M f e + κ f x + Z E + u B f ] κ + Z M 2 κ B κ f = 1 ε [C f, f ] B. Nkonga. Flud Theory 22 / 56
38 Fast dynamcs M 1 and ε = ɛ : Sonc Electrons : u u e, κ κ e ε 2 [ t u + u x u κ g e ] + ε [ t g e + u x g e κ x u κ g e ] + [κ x g e E + u B κ g e ] = 1 ε [ κ B κ g e + C ee g e, g e + C e g e, g ] Ions: u u, κ κ [ t u + u x u f ] κ + [ t f + u f x κ x u f ] κ + [ C e f, f e + κ f x + Z E + u B f ] κ + Z κ B f κ = 1 [C f, ε f ] B. Nkonga. Flud Theory 22 / 56
39 Slow dynamcs M ɛ and ε = ɛ 2 : Drft Electrons : u u e, κ κ e ε ε [ t u + u x u κ g e ] + ε [ t g e + u x g e κ x u κ g e ] + 1 ε [κ x g e E + u B κ g e ] = 1 ε [ κ B κ g e + C ee g e, g e + C e g e, g ] Ions: u u, κ κ ε [ t u + u x u f ] κ + [ t f + u f x κ x u f ] κ + 1 [ C e f, f e + κ f x + Z E + u B f ] κ ε = 1 ε [ Z κ B κ f + C f, f ] B. Nkonga. Flud Theory 22 / 56
40 Slow dynamcs of Bragnsk. Electrons : u u e, κ κ e [ t u + u x u κ g e ] + [ t g e + u x g e κ x u κ g e ] + [κ x g e E + u B κ g e ] = 1 ε [ κ B κ g e + C ee g e, g e + C e g e, g ] Ions: u u, κ κ [ t u + u x u f ] κ + [ t f + u f x κ x u f ] κ + [ C e f, f e + κ f x + Z E + u B f ] κ = 1 [ Z κ B κ ε f + C f, f ] B. Nkonga. Flud Theory 22 / 56
41 Relatons δu = u e u g e κ = f e κ + u e = f e κ + u + u e u = f e κ + δu g κ = f κ + u e = f κ + u + u e u = f κ + δu B. Nkonga. Flud Theory 23 / 56
42 Overvew 1 Knetc and macroscopc equatons for Smple plasma 2 Flud Theory : Scalng and dmensonal analyss 3 Flud Theory : Hlbert s expanson and asymptotc analyss 4 Flud Theory : Frst order correcton of Bragnsk. 5 Bragnsk transport Coeffcents B. Nkonga. Flud Theory 24 / 56
43 Taylor s and Hlbert s expansons : ε = ɛm B κ ɛκ B κ ɛκ κ B κ + ɛ2 2 κ κ : κ κ B κ + ɛ 3 Then, wth σ = n T I τ n T I. In the ons frame we have : [ D κ = dκ f κ B κ ɛκ ] R 3 and = n B κ ɛ 0 + ɛ 2 n T 3κ κ κ 2 I κ 5 + ɛ 3 O e fe, f = D κ f e κ + ɛ 2 m e fe κ κ m κ D κ = n B κ f e κ 0 ɛ m e 2κ ɛ2 m κ f 3κ κ κ 2 I fe 3 e κ + n T κ 5 + ɛ 3 κ B. Nkonga. Flud Theory 25 / 56
44 m 2 e Note that we have Γ e = Γ e m 2 B. Nkonga. Flud Theory 26 / 56 Taylor s and Hlbert s expansons : ε = ɛm B κ e ɛκ B κ e ɛκ κ B κ ɛ e e κ κ : κ e κ e B κ e +ɛ 3 κ κ B κ e e = κ κ e κ κ e 5 e 2 I 3κ e κ κ κ e e + κ e κ κ e 3 In the electrons frame we have D e ɛκ = dκ e [ ge κ e B κ e ɛκ ] R 3 = dκ e [ ge κ e B κ ] e + ɛ dκ e [ ge κ e κ κ B κ ] R 3 R 3 e e + ɛ 2 Case of g e κ e M e κ e O e g, g e 4n e 3 me 2πT e g κ κ + m κ g κ T e
45 Taylor s and Hlbert s expansons : ε = ɛm g e = g e 0 + ε g e 1 + ε2 M M 2 g e 2 + f = f 0 + ε f 1 M + ε2 f 2 + M 2 B e = B 0 e + ε B 1 e + ε2 M M 2 B 2 e + B e = B 0 e + ε B 1 e + ε2 M M 2 B 2 e + C e = Ce 0 + ε Ce 1 + ε2 Ce 2 + M M 2 C e = C 0 e + ε M C 1 e + B. Nkonga. Flud Theory 27 / 56
46 Expanson of electron-on collsons wth C e fe, f = Γ e 2 κ O e = Ce 0 fe, f + ε Ce 1 fe, M f + ε2 M 2 O e = D κ κ f e + m e m fe κ κ D Wtchng the ons frame we have [ D κ = dκ f κ B κ κ ] = n B κ ɛ 2 R 3 Therefore C 0 e fe, f = n Γ e 2 v B κ f κ e B. Nkonga. Flud Theory 28 / 56
47 Thermalzaton of dstrbutons functons { κ B κ g e C ee g e, g e 0 + C 0 0 e g e, g 0 = 0 κ B f 0 κ + C f 0, f 0 = 0 g e 0 0 = M e κ and f = M κ where M e κ = M e,0 exp m e κ 2 and M κ = M,0 exp m κ 2 2T e 2T For any change of varable κ = κ ± ɛδu :: g e 0 κ M e κ ± ɛδu κ B κ g0 0 e + C ee g e, g e 0 + C 0 0 e g e, g 0 ε = 0 + M 1 Whch thermalzaton s consstent wth physcal applcatons? 2 What s the defnton of g 0? Is g0 κ = f 0 κ + δu? B. Nkonga. Flud Theory 29 / 56
48 Frst order correcton : ε = M ɛ g e κ = M e κ 1 + Φ 1 e κ + ɛ 2 f κ = M e κ 1 + Φ 1 κ + ɛ 2 Expanson of the collsons C e = Γ e 2 v O e wth n ons frame O e = D κ f κ e + m e fe κ κ D = n B κ m f κ e + ɛ 2 where, wth σ = [ n T I + τ, D κ = dκ f κ B κ κ ] 1 = n B κ + R 3 2 [σ : κ κ B κ ] Then, as δu = v e v ɛ, we have the followng estmaton f e κ = f e κ + v = f e κ δu + v e = g e κ δu = M e κ δu 1 + Φ 1 e κ δu + ɛ 2 = M e κ 1 + m e δu κ + T Φ 1 e κ + ɛ 2 e B. Nkonga. Flud Theory 30 / 56
49 Frst order correcton : ε = M ɛ g e κ = M e κ f e κ = M e κ 1 + Φ 1 e κ 1 + m e δu κ + T Φ 1 e κ e + ɛ 2, f κ = M e κ + ɛ Φ 1 κ + ɛ 2, O e = n B κ f κ e + ɛ 2 Expanson of the collsons B κ [ κ M e κ ] = βb κ κ = 0 C e = Γ e 2 κ O e = n Γ e 2 κ B κ f κ k + ɛ 2 = n Γ e 2 me κ M e κ B κ δu + B κ T Φ 1 κ e κ e [ me + ɛ 2 B κ κ Φ 1 e κ ] + ɛ 2 = n Γ e 2 M e κ κ B κ δu + κ T e = n [ Γ e 2 M e κ 2m e T e κ 3 κ δu + κ B κ Φ ] 1 κ e κ + ɛ 2 = 0 + M e κ C e Φ1 e, f 0 2n Γ e m e 2T e κ 3 κ δu + ɛ 2 B. Nkonga. Flud Theory 30 / 56
50 Frst order correcton : ε = M ɛ g e κ = M e κ f e κ = M e κ Expanson of the collsons 1 + Φ 1 e κ 1 + m e δu κ + T Φ 1 e κ e + ɛ 2, f κ = M e κ + ɛ Φ 1 κ + ɛ 2, O e = n B κ f κ e + ɛ 2 C e = 0 ɛ 0 +C e g e 0 Φ 1 e, f 0 g e 0 2n Γ e m e 2T e κ 3 κ δu +ɛ 2 0 C ee = C ee g e, g e 0 +C ee g e 0 Φ 1 e, g e 0 + C ee g e, 0 g e 0 Φ 1 e +ɛ 2 C = C f 0, f 0 +C f 0 Φ1, f 0 + C f 0, f 0 Φ 1 +ɛ 2 C e = 0 ɛ 0 +ɛ B. Nkonga. Flud Theory 30 / 56
51 Overvew 1 Knetc and macroscopc equatons for Smple plasma 2 Flud Theory : Scalng and dmensonal analyss 3 Flud Theory : Hlbert s expanson and asymptotc analyss 4 Flud Theory : Frst order correcton of Bragnsk. 5 Bragnsk transport Coeffcents B. Nkonga. Flud Theory 31 / 56
52 Frst order correcton for Slow dynamcs of Bragnsk. Electrons : u u e, κ κ e and g 1 e g 1 e κ = Φ 1 e κ M e κ t u + u x u κ g e 0 + t g e 0 + u x g e 0 κ x u κ g e 0 +κ x g e 0 E + u B κ g e 0 = κ B κ g 1 e + C ee g 1 e, g e 0 + Cee g 0 e, g 1 e +C e g 1 e, f 0 2n Γ e m e 2T e κ 3 κ δum e κ Ions: u u, κ κ and f 1 f 1 κ = Φ 1 κ M κ t u + u x u f 0 κ + t f 0 + u f 0 x κ x u f 0 κ C e f 0, f e 0 + κ f 0 x + Z E + u B f 0 κ = Z κ B f 1 κ + C f 1, f 0 + C f 0, f 1 B. Nkonga. Flud Theory 32 / 56
53 Transport contrbuton for frst order approxmaton C e = Γ e 2 κ O 1 e + ɛ 2 where O 1 me e κ = n M e κ B κ δu + B κ κ T Φ 1 e κ e Frcton contrbuton s R 1 e = Γ e m e κ δu κ O 1 Γ e e dκ = m e O 1 2 R 3 2 edκ R 3 Accordng to ntegraton formulas of polynomals functons over balls 6 R 1 e = m en e τ e δu + R e where τ e = 3 m e T 3 2 e 4n 2πq 2 e q 2 ln Λ and R e R e Φ1 e = n Γ e M e κ B κ 2 Φ 1 κ e κdκ R 3 6 John A. Baker. Integraton Over Spheres and the Dvergence Theorem for Balls. The Amercan Mathematcal Monthly, Vol. 104, No. 1. Jan., 1997, pp B. Nkonga. Flud Theory 33 / 56
54 Transport contrbuton for frst order approxmaton Frcton contrbuton s R 1 e = m en e δu + R e Heat τ e Q 1 e = Γ R3 e κ δu 2 m e κ O e dκ = Γ e m e κ δu O 1 2 edκ R 3 Γ e = m e B 2 T κ κ ÕO 1 e dκ δu R 1 e R 3 = 0 + Q 2δu e = ɛ 2 B. Nkonga. Flud Theory 33 / 56
55 Transport : second order contrbutons There s also an other second order term assocated to O 2 e κ = n m e m g 0 e κ κ B κ 1 2m [σ : κ κ B κ ] κ g 0 e κ where σ = n T I τ n T I Indeed, we have R 2 e = Γ e 2 R 3 m e O 2 edκ = 0 κ B = 2κ κ 3, 1 2 [ κ κ B] = κ 2 I 3κ κ κ 5 and S 2 [ σ : s s s s ] 3 I sds = 0 B. Nkonga. Flud Theory 34 / 56
56 Transport contrbuton for frst order approxmaton Q 2δT e Q 2δT e = m eγ e 2 = m eγ e 2 = m eγ e 2 = m eγ e 2 m eγ e 2 κ δu 2 κ O 2 e κ dκ = m eγ e dκ κ δu O 2 e κ dκ 2 2 R 3 R3 κ δu 2 κ O 2 2 edκ = m eγ e κ δu O 2 2 edκ [ R 3 n m e κ g R m e 0 κ B 1 ] [σ : κ κ B] κ g e 0 dκ 3 2m [ 2 κ 2 n m e R κ 3 g e 0 1 ] κ [σ : κ κ B] κ g e 0 dκ m 3 [ 2m 2 n m e + 2n ] m e T g edκ 0 m2 en Γ e T R3 dκ R 3 m2 en Γ e m 3m en e m τ e κ 1 T T e m m T e κ 4πTe m e n e T e T = 3m en e δt m τ e Q 2 e = Q 2δT e + Q 2δu e π 2T e m e 3 1 m T e 2 as g 0 e = n e π 2T e m e = δu R 1 e 3m en e m τ e δt 3 g e 0 R 3 2 e x κ dκ B. Nkonga. Flud Theory 35 / 56
57 Transport frst and second order contrbutons 1 Frcton 2 Heat R e R 1 e + R 2 e = m en e τ e δu + R e and R e = R e Q e Q 1 e + Q 2 e = δu R 1 e 3m en e m τ e δt = m en e δu δu δu R e 3m en e δt τ e m τ e and Q e = 3m en e m τ e δt Where δu = u e u and δt = T e T B. Nkonga. Flud Theory 36 / 56
58 Solublty condtons for L k, Φ 1 k = b k For example, wth f 1 = M κ Φ 1 κ = g 0 f 1, we have L, Φ = κ B f 1 κ + C f 1, f 0 + C f 0, f 1 Note that γ 0 + γ 2 κ 2 s always n the kernel of L. The requrement that correcton must not change macroscopc parameters : 1 κ M k κ Φ 1 k κ dκ = 0 R 3 κ 2 contans also the assumed requrement for exstence and unqueness of the soluton B. Nkonga. Flud Theory 37 / 56
59 Therefore D u k t n k = n k x u k m e n e D u e t u e = x p e E + u e B + R e x π k = x p + Z E + u B R e D u k t T k = 2 3 T k x u k x q k π k : x u k + Q kl m n D u t u Dervatves wth respect to tme and space of the Maxwellan are [ g e 0 ne 3 = n e 2 m ] eκ κ Te g e 0 2T e T e Then the left hand sde of electrons correcton equaton can be estmated [ me κ 2 2T e 5 2 x T e κ T e [ = D u e t u e κ g e 0 + t g e 0 + u e x g e 0 κ x u e κ g e 0 +κ x g e 0 E + u e B κ g e 0 = + R1 e κ + m e m e T e T e L e κ κ + R e κ L e m e T e ] κ [ x u e ] T κ κ 2 3 x u e + L e κ : κ κ κ 2 3 g 0 e ] g 0 e B. Nkonga. Flud Theory 38 / 56
60 Equaton for frst order correctons g 1 e Φ 1 e κ M e κ and f 1 Φ 1 κ M κ [ meκ κ 2T e 5 2 xt e κ T e + R 1 e ] κ + me ] T κ [ κ 2 xu e κ m et e T e 3 x u e M e κ = κ B κ g 1 e g + Cee 1 e, g0 e + C ee g e 0, g1 e + C e g 1 e, f 0 2n Γ e m e 2T e κ 3 κ δum e κ Integro-dfferental lnear equaton for Φ 1 e usng L e κ B κ g 1 e + C ee g 1 e, g e 0 + Cee g 0 e, g 1 e + C e = g el 0 L e κ κ + g el 0 L e Lnear partal dfferental equaton for Φ 1 κ B f 1 κ + C f 1, f 0 + C f 0, f 1 L e = L e + 2n Γ e me δu: 2Te κ 3 L e κ : g 1 e, f 0 = f 0 L κ κ + f 0 L κ : g e 0 R e κ m e T e κ κ κ 2 3 κ κ κ 2 3 B. Nkonga. Flud Theory 39 / 56
61 Resoluton of frst order correctons equatons Accordng to symmetres of the RHS, Φ 1 e κ and Φ 1 κ are found under the followng form : Φ 1 k κ = P k κ κ + P k κ : κ κ κ 2 3 Moreover, RHS operators L k κ and L k κ can be expanded wth Laguerre-Sonne polynomals. For example, let us denote by x = m e κ 2 me κ 2 L e κ = 5 x T e + 2n Γ e m e 2T e 2 T e [ 2T e κ 3 δu ] = xt e Y δt T e,1l x + δu Y δu e,l L 3 2 l x e L 3 2 l x functons gves very smple expanson for the frst term : Y δt e,1 = 1. B. Nkonga. Flud Theory 40 / 56 l>0 2T e
62 Vector splttng n strongly magnetzed plasma In strongly magnetzed plasma, macroscopc vectors are often splt nto parallel, perpendcular and components. For example : x T e = M x T e + M x T e + M x T e = xt e + x T e + x T e where M = b b, M = I b b, M x T e = b x T e 0 b z b y M = b z 0 b x b y b x 0 These matrces are lnearly ndependent when b 0 stable under multplcaton. What about tensors? B. Nkonga. Flud Theory 41 / 56
63 Tensor splttng n strongly magnetzed plasma We have L k κ L k = m k [ x u k ] + [ x u k ] T 2 2T k 3 x u k I For ths symmetrc tensor, Bragnsk propose the followng splttng: 4 L k = Π l b : L k l=0 Π 0 = M 1 2 M 2 3 M 1 3 M Π 1 = M M 1 2 M M Π 2 = M M M M Π 3 = 1 2 M M [M ] T M Π 4 = M M + [M ] T M [A B : W ] j = A j B kl W kl k l [A B : W ] j = A k B jl W kl k l B. Nkonga. Flud Theory 42 / 56
64 Splttng of the frst order approxmaton Accordng to prevous splttng n strongly magnetzed plasma 1 The vector P k κ s found under the form [ ] P k κ = L 3 2 l x X δt k,l M + X δt k,l M + X δt k,l M x T k T k l>0[ ] + L 3 2 l x X δu k,l M + X k,l δu M + X δu k,l M δu l>0 wth the constran that 1 R 3 κ κ 2 M k κ Φ 1 k κ dκ = 0 = l > 0 2 and P k κ under the form P k κ = 4 L 3 2 l x X δτζ Π l u k, b : L k l>0 ζ=0 k,l Π l B. Nkonga. Flud Theory 43 / 56
65 Systems to be solved Φ 1 k κ = P k κ κ + We have κ B Φ 1 κ e = κ B P k κ + x = B B 2 B x x = x x B x = x B, and κ B x = 0 κ B x = B κ x κ B x = B κ x ndeed κ B B x = κ B B x B B κ x. We have P k κ = L 3 2 l x X δt xt e k,l + X δt x T e k,l + X δt x T e T k T k,l + k T k l>0 and therefore, as κ B Φ 1 κ e = κ BPP k κ + κ B Φ 1 κ e = B L 3 2 l x X δt k,l κ x T e X δt T k,l κ x T e + k T k l>0 Systems for X δt k,l and X δt k,l are coupled B. Nkonga. Flud Theory 44 / 56 P k,
66 Systems to be solved :: Φ 1 k κ = P k κ κ + L e κ κ = Y δt 3 e,1 L 21 x x κ Te Y δt 3 e,1 T L 21 x x κ Te k T k κ B Φ 1 κ e = 3 L 2 l l>0 x X δt k,l B κ x Te 3 L 2 l T k l>0 x B X δt k,l κ x Te + T k Compacted form. Usng dot product wth κ and the relaton κ κ = x 2T e m e l>0 where xe x L 3 2 l x X δt e,l X θ k,l = X θ θ k,l + ıx k,l Y θ k,l = Yθ k,l + ıyθ k,l ı B + C x : [s s] xc C x s a tensor assocated to lnearzed collsons. = Y δt e,1xe x L x + B. Nkonga. Flud Theory 45 / 56
67 Systems to be solved :: Φ 1 k κ = P k κ κ + Compacted form. Usng dot product wth κ and the relaton κ κ = x 2T e m e l>0 xe x L 3 2 l x X δt e,l ı B + C x : [s s] = Y δt e,1xe x L x + Varatonal prncples Onsager symmetry formulated as L 2 -projecton for any q > 0. + [ X δt e,l x 3 2 e x L 3 2 q x L 3 2 l x ı B + C x : s s ds ] 8dx l>0 0 S 2 4π 15 π + [ ] = Y δt e,1 x 3 2 e x L 3 2 q x L 3 2 8dx 1 x 15 π + 15 π 8 = ! = ! = 5 3 π B. Nkonga. Flud Theory 45 / 56
68 Systems to be solved :: Φ 1 k κ = P k κ κ + Compacted form. Usng dot product wth κ and the relaton κ κ = x 2T e m e l>0 xe x L 3 2 l x X δt e,l ı B + C x : [s s] = Y δt e,1xe x L x + Varatonal prncples as L 2 -projecton for any q > 0. + [ X δt e,l x 3 2 e x L 3 2 q x L 3 2 l x ı B + 23 ] Tr 8dx C x 15 π = YδT e,1δ q,1 + l>0 0 X are solutons of a lnear system of the followng form : A θ ex θ e = C θ ey θ e and A θ X θ = C θ Y θ B. Nkonga. Flud Theory 45 / 56
69 Fnal When prevous systems are solved, we obtan analytcal formula for g e κ M e κ 1 + P e κ κ + P e κ : κ κ κ 2 3 f κ M e κ 1 + P κ κ + P κ : κ κ κ 2 3 Then they are used to compute transport contrbutons. B. Nkonga. Flud Theory 46 / 56
70 Overvew 1 Knetc and macroscopc equatons for Smple plasma 2 Flud Theory : Scalng and dmensonal analyss 3 Flud Theory : Hlbert s expanson and asymptotc analyss 4 Flud Theory : Frst order correcton of Bragnsk. 5 Bragnsk transport Coeffcents B. Nkonga. Flud Theory 47 / 56
71 Bragnsk Transport Coeffcents : An example R 1 e = m en e τ e P k κ = Therefore α m en e τ e δu n Γ e R 3 2 M e κ B κ Φ 1 κ e κdκ = m en e n Γ e δu τ e R 3 2 M e κ B κ κ [P P k κ κ] = m en e n Γ e δu τ e R 3 2 M e κ B κp P k κ dκ [ ] L 3 2 l x X δt k,l M + X δt k,l M + X δt k,l M x T k T k l>0[ ] + L 3 2 l x X δu k,l M + X k,l δu M + X δu k,l M δu l>0 + n Γ e 2 N l l=1 X δu e,l dκ R 3 [M e κ L 3 2l me κ 2 2T e ] B κ B. Nkonga. Flud Theory 48 / 56
72 Bragnsk Transport Coeffcents : An example α m en e τ e + n Γ e 2 N l l=1 X δu e,l dκ R 3 B κ = 1 κ B s where r = κ, s = r κ The ntegral part of α can be computed as R 3 dκ [G r B κ ] = + 0 l=1 [M e κ L 3 2l me κ 2 and [ r 2 G r 1 ] dr B s ds = 8π r S T e ] B κ S 2 B s ds = 8π 3 I + Therefore α s equvalent to a scalar. α m en e + n Γ e 8π N l + X δu τ e 2 3 e,l r [M e r L 3 me r 2l 2 ] dr 2T e 0 rg rdr B. Nkonga. Flud Theory 49 / 56
73 Bragnsk transport closure : quas neutral plasma n = n e Electrons β t R e = en e αj β t x T e Q e = Q + J R e n e e Q e = κ e x T e + en e β j ej 4 η el Π l u e, b π e where l=0 Ions R = R Q = 3m en e T e T m τ e Q = κ e x T + en e β j J 4 η l Π l u, b α = α M +α M α M κ k = κ k M +κ k M +κ k M β t = β t M +βt M +β M t β j k = β j k M +βj k M +β j k M See [Bragnsk, 1965] for numercal values of theses parameters. Some numercal examples α = β j π m e e 2 n e τ e, α = 1.96α, β t = l=0 3 2ω coll e τ e, β t = 0.71 B. Nkonga. Flud Theory 50 / 56
74 Appendx κ κ = r 2 = x 2T e m e α m en e τ e m en e τ e m en e τ e and rdr = T e m e dx + n Γ e 2 + n Γ e 2 + m en e τ e 8π 3 8π 3 N l l=1 X δu e,l T e M e 0 m e T e N l m 2 e l=1 X δu e,l r [M e r L 2l 3 me r 2 ] dr 0 2T e N l + [ exp x L 3 2 l x + l=1 + [ 0 X δu e,l 0 exp x L 3 2 l x ] dx ] dx Γ kl = 4πq2 k q2 l ln Λ m 2 k and τ e = 3 m e T 3 2 e 4n 2πq 2 e q 2 ln Λ τ e = 3π m e T 3 2 e = n 2πm 2 e Γ e 3 4πn Γ e 2πTe m e 3 2 = 3n e 4πn Γ e M e 0 B. Nkonga. Flud Theory 51 / 56
75 Appendx As s 2 k s homogeneous of degree two. Then corollary 1 page 39 6 S 2 s 2 k 1 ds = x 2 k dx k dx l dx p = = 10π 3 1 = 10π = 4π 3 Therefore, as B s = I s s, we have B s ds = 4π 4π S 2 3 and S I = 8π 3 I s s s s 3 I ds = 0 x 2 k π1 x2 k dx k 6 John A. Baker. Integraton Over Spheres and the Dvergence Theorem for Balls. The Amercan Mathematcal Monthly, Vol. 104, No. 1. Jan., 1997, pp B. Nkonga. Flud Theory 52 / 56
76 Appendx :: fk κ M k κ 1 + P e κ κ r = x 2T 1 2 k T k, rdr = dx, m k m k M k r = n k π 2T 3 2 k e x m k [ v u k 2 ] [ κ q k = m k v f k v 2 ] dv = m k R 3 2 R 3 2 κ f k κ dκ [ κ 2 [ ] κ ] 0 + m k R 3 2 M k κ κ κ P k dκ + + r 2 r [m 2 [ k 0 2 M k r r 2 ] r ] + r 6 4π r s s dsp P k dr m k [M k r ] S P k dr + 4πm kn k + x 2T 5 k 2 π 2T 3 k 2 x e Pk P k x T k dx m k m k m k 4n k 3 T 2 + [ k x 2 3 xe x P k x ] dx 4n k π m k 0 3 T 2 [ + k x π m k 0 2 L 20 + L 2 1 x e x x ] P k dx + 4n k 3 T 2 k 15 π X δt xt e + X δt x Te k,1 k,1 + X δt x Te k,1 π m k 8 T k T k T k 4n k 3 T 2 k 15 π X δu k,1 π m k 8 xδu + X δu k,1 δu x δu + X k,1 x δu + 5n ktk 2 X δt xt e + X δt x Te k,1 k,1 + X δt x Te k,1 + X δu k,1 2m k T k T k T δu xδu + X k,1 δu x δu + X k,1 x δu + k B. Nkonga. Flud Theory 53 / 56
77 Appendx :: fk κ M k κ 1 + P e κ κ We have + [ [κ κ]p P k κ M k κ = r 2 [ P k r M k r r 2 s s ] ] ds dr R 3 0 S 2 = 4π + r 3 P k r M k r rdr = 4π 3 2Tk 2 T + k x 3 2 e x P k x dx m k m k 0 = 4π 3 2Tk 2 T k + P kl x 3 2 e x L m k m 0 x L 3 2 l x dx k l 0 1 Therefore the constran κ M k κ Φ 1 k κ dκ = 0 s acheved when R 3 κ 2 l > 0, accordng to orthogonalty of Laguerre-Sonne polynomals and zero ntegral on sphere for monomal wth an odd component of the mult-ndex : 1 + κ M k κ Φ rs 1 R 3 r 2 k κ dκ = r 2 M k rp P k r r 2 s s ds dr 0 S 2 r 2 rs + + r 2 M k rp P k r : 1 rs r 2 s s s s 3 I ds dr = 0 0 S 2 r 2 B. Nkonga. Flud Theory 54 / 56
78 Appendx :: D e ɛκ = [ = dκ e fe κ e B κ e ɛκ ] R[ 3 dκ e fe κ e B κ ] e + ɛ R 3 Case of f e κ e = g 0 e κ e R 3 dκ e [ fe κ e κ κ B κ ] e e + ɛ 2 D e ɛκ [ = dκ e fe κ e B κ e ɛκ ] = 8π R r g 0 e r dr + ɛ Case of f e κ e = δu κ e g 0 e κ e D e ɛκ [ = dκ e fe κ e B κ e ɛκ ] = 8π R r g 0 e r dr + ɛ 2 B. Nkonga. Flud Theory 55 / 56
79 Appendx :: [ D e ɛκ = dκ e fe κ e B κ e ɛκ ] R[ 3 = dκ e fe κ e B κ ] [ e + ɛ dκ e fe κ e κ κ B κ ] R 3 R 3 e e + ɛ 2 Case of f e κ e = δu κ e g 0 e κ e D e ɛκ = ɛ dκ [ e δu κ e g 0 e κ e κ κ B κ ] 8π R 3 e e = r g 0 e r dr + B. Nkonga. Flud Theory 56 / 56
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