Magnetized plasma : About the Braginskii’s 1 - Laboratoire … · 2017-05-09 · Magnetized...

Magnetized plasma : About the Braginskii’s 1

macroscopic model 2

B. Nkonga

JAD Univ. Nice/INRIA Sophia-Antipolis

1S. I. Braginskii, in Reviews of Plasma Physics, edited by M. A. Leontovich(Consultants Bureau, New York, 1965), Vol. I, p. 205.

2Talk H. Guillard, 1st summer school of the Large Scale Initiative ”FUSION”:September 15-18, 2009 in Strasbourg. http://www-math.u-strasbg.fr/ae fusion/

B. Nkonga . Fluid Theory 1 / 56

Overview

1 Kinetic and macroscopic equations for Simple plasma

2 Fluid Theory : Scaling and dimensional analysis

3 Fluid Theory : Hilbert’s expansion and asymptotic analysis

4 Fluid Theory : First order correction of Braginskii.

5 Braginskii transport Coefficients


State of the matter : Plasma

Temperature versus (Number of charged particles)/m3


Models + Maxwell’s Equations for EEE and BBB

1 N-body : xxxk (t) : R −→ R3, k = O((1020/m3) ∗ 800m3

)Newton Equation for each charged particle

dxxxkdt

= vvvk and mkdvvvkdt

=qkmk

(EEE + vvvk ×BBB) +∑`

Ck`

2 Kinetic : fk (t,xxx,vvv) : R7 −→ R, k = O(10)

∂tfk + vvv · ∂∂∂xxxfk +qkmk

(EEE + vvv ×BBB) · ∂∂∂vvvfk =∑`

Ck`

3 Fluid :ωωωk (t,xxx) : R4 −→ RNk , k = O(10)

∂tωωωk +LLL (∂∂∂,ωωωk,BBB,EEE) = Sk


Kinetic equation for Simple plasma

DDDvvvt fe + qe

meLLL (vvv) · ∂∂∂vvvfe = Cee (fe, fe) + Cei (fe, fi)

DDDvvvt fi + qi

miLLL (vvv) · ∂∂∂vvvfi = Cie (fi, fe) + Cii (fi, fi)

where for electrons (k = e) and ions (k = i)

fk ≡ fk (t,xxx,vvv) is the distribution function.

mk is the mass

qk is the charge

Ck` are collisions operators.

Moreover

DDDvvvt = ∂t + vvv · ∂∂∂xxx is the material derivative at the velocity vvv,

DDDvvvt is the material derivative at the velocity vvv,

LLL (vvv) = EEE + vvv ×BBB is the Lorenz force

EEE and BBB are govern by Maxwell equations.


Coulomb binary scattering law

The Landau form of the Coulomb collision is ( Eq. 4.3 of Braginskii):

Ck` (fk, f`) = −Γk`2

∂

∂vvv·[OOOk` (fk, f`)

]where, with aaa = vvv − vvv′ we have

OOOk` (fk, f`) =

∫R3

dvvv′(BBB (aaa)

[mk

m`fk (vvv)

∂f` (vvv′)

∂vvv′− f`

(vvv′) ∂fk (vvv)

∂vvv

])where, for rigid spheres approximation, Cut-offs estimation gives

Γk` =4πq2kq

2` ln Λ

m2k

and for any vector aaa BBB (aaa) =|aaa|2III− aaa⊗ aaa|aaa|3

ln Λ is the Coulomb logarithm.


Properties of the scattering tensor BBB =|aaa|2III− aaa⊗ aaa|aaa|3

∀aaa

1 BBB (aaa) is symmetric BBB (aaa)T = BBB (aaa) and even BBB (−aaa) = BBB (aaa)

2 BBB (aaa) derive from a potential BBB (aaa) = ∂∂∂aaa

(aaa

|aaa|

)= −∂∂∂aaa

(∂∂∂aaa

1

|aaa|

)3 ∀aaa it is in the kernel of BBB (aaa) ⇐⇒ BBB (aaa)aaa = 0

4 ∂∂∂aaa ·BBB (aaa) = ∂∂∂aaa

(2

|aaa|

)= − 2aaa

|aaa|3and Tr [BBB (aaa) ] =

2

|aaa|5 ∂∂∂vvv ·BBB (aaa) = − 2aaa

|aaa|3= −∂∂∂vvv′ ·BBB (aaa)

61

2[Tr(∂∂∂aaa ⊗ ∂∂∂aaa)BBB] = −|a

aa|2III− 3aaa⊗ aaa|aaa|5

7 In spherical coordinates(velocity space) ∂∂∂aaa · (BBB∂∂∂aaa) ≡2

|aaa|3B (∂∂∂θ, ∂∂∂φ)

B(∂∂∂θ, ∂∂∂φ

)=

1

2

(1

sin θ

∂

∂θ

(sin θ

∂

∂θ

)+

1

sin2 θ

∂2

∂θ2

)=

1

2

(∂

∂µ

((1− µ2

)∂

∂µ

)+

1

1− µ2

∂2

∂θ2

)

is the angular part of1

2

∂2

∂aaa2and is often written in terms of the pitch angle variable : µ = cos θ


Other formulations of Coulomb binary scattering law

OOOk` (fk, f`) =

∫R3

dvvv′(BBB (aaa)

[mk

m`fk (vvv)

∂f` (vvv′)

∂vvv′− f`

(vvv′) ∂fk (vvv)

∂vvv

])1 Fokker-Planck form, with DDD` (vvv) =

∫R3

dvvv′[f`(vvv′)BBB (aaa)

]and (P5)

OOOk` (fk, f`) =

(1 +

mk

m`

)fk (vvv)

∂

∂vvv·DDD` (vvv)− ∂

∂vvv·(fk (vvv)DDD` (vvv)

)2 Rosenbluth(57)-Trubnikov(58) form :

OOOk` (fk, f`) = 2(

1 + mkmk

)fk (vvv)

∂H` (vvv)

∂vvv− ∂

∂vvv·(fk (vvv)

∂

∂vvv· ∂∂vvvG` (vvv)

)Rosenbluth potentials H` (vvv) and G` (vvv):

H` (vvv) =

∫R3

dvvv′f` (vvv′)

|aaa|with G` (vvv) =

∫R3

dvvv′|aaa|f`(vvv′)

DDD` (vvv) is the diffusion tensor


Other formulations of Coulomb binary scattering law

1 Fokker-Planck form, with DDD` (vvv) =

∫R3

dvvv′[f`(vvv′)BBB (aaa)

]and (P5)

OOOk` (fk, f`) =

(1 +

mk

m`

)fk (vvv)

∂

∂vvv·DDD` (vvv)− ∂

∂vvv·(fk (vvv)DDD` (vvv)

)

OOOk` (fk, f`) = −DDD` (vvv)∂fk (vvv)

∂vvv+mk

m`fk (vvv)

∂

∂vvv·DDD` (vvv)

2 Rosenbluth(57)-Trubnikov(58) form :

OOOk` (fk, f`) =

(1 +

mk

mk

)fk (vvv)

∂H` (vvv)

∂vvv− ∂

∂vvv·(fk (vvv)H` (vvv)

)∂

∂vvv·DDD` (vvv) =

∂H` (vvv)

∂vvvand H` (vvv) =

1

2

∂

∂vvv· ∂∂vvvG` (vvv)

DDD` (vvv) is the diffusion tensor


Macroscopic equations : DDDuuut = ∂t + uuu · ∂∂∂xxx

DDDuuukt nk + nk∂∂∂xxx · uuuk = 0

mknkDDDuuukt uuuk + ∂∂∂xxxpk − qknk (EEE + uuuk ×BBB) = −∂∂∂xxx · πππk +RRRk

nkγk − 1

(DDDuuukt pk − γkpk∂∂∂xxx · uuuk

)= −∂∂∂xxx · qqqk − πππk : ∂∂∂xxxuuuk +Qk

∂tBBB + ∂∂∂xxx ×EEE = 0

qk and γk ≡5

3are constants parameters. Tk = nkpk

Static constraint : ∂t (∂∂∂xxx ·BBB) is constant.

Additional relation to “define” EEE (VVV )

∂∂∂xxx ×BBB = −µ0JJJ = −µ0∑k

qknkuuuk

VVV =

nkuuukpkBBB


Macroscopic equations : Transport

πππk =

∫R3

dvvv

[mkfk

((vvv − uuuk)⊗ (vvv − uuuk)−

|vvv − uuuk|2

3III)]

qqqk =

∫R3

dvvv

[mkfk

|vvv − uuuk|2

2vvv

]RRRk =

∫R3

dvvv

[mk (vvv − uuuk)

∑`

Ck`

]

Qk =

∫R3

dvvv

[mk|vvv − uuuk|2

2

∑`

Ck`

]

Scaling and asymptotic expansions of kinetic equations :

Define πππk, RRRk, qqqk and Qk as functions of VVV


Kinetic Transport theory : Strategy

1 Define an appropriate frame and scaling.

2 Evaluate non-dimensional coefficient in term of a small parameter.

3 Proceed to an expansions according to these terms

4 Obtained approximations of probability density functions.

5 Use these approximations to evaluate transport terms.


Overview







Kinetic equation in a non inertial framesCoordinate transformation : κκκ (t,xxx,vvv) = vvv − uuu (t,xxx) and κκκ′ (t,xxx,vvv) = vvv′ − uuu (t,xxx)

Let us define

fk (t,xxx,κκκ) = fk (t,xxx,vvv) |vvv=κκκ+uuui and gk (t,xxx,κκκ) = fk (t,xxx,vvv) |vvv=κκκ+uuueThere are 4 possibles formulations for simple plasma kinetic equations:Electrons and ions in mean electrons velocity frame

DDDuuut ge + κκκ · ∂∂∂xxxge +

[qeme

(EEE + (uuu+ κκκ)×BBB)−DDDuuutuuu− κκκ · ∂∂∂xxxuuu

]· ∂∂∂κκκge

= Cee (ge, ge) + Cei (ge, gi)

DDDuuut gi + κκκ · ∂∂∂xxxgi +

[qimi


]· ∂∂∂κκκgi

= Cie (gi, ge) + Cii (gi, gi)

The Coulomb collision operator is invariant under Galilean transformation.

∂∂∂κκκ = ∂∂∂vvv, ∂∂∂κκκ′ = ∂∂∂vvv′ , BBB(vvv − vvv′

)= BBB

(κκκ− κκκ′

)B. Nkonga . Fluid Theory 13 / 56


Let us define

fk (t,xxx,κκκ) = fk (t,xxx,vvv) |vvv=κκκ+uuui and gk (t,xxx,κκκ) = fk (t,xxx,vvv) |vvv=κκκ+uuueThere are 4 possibles formulations for simple plasma kinetic equations:Electrons and ions in the “opposite” mean velocity frame

DDDuuut fe + κκκ · ∂∂∂xxxfe +

[qeme


]· ∂∂∂κκκfe

= Cee(fe, fe

)+ Cei

(fe, fi

)DDDuuut gi + κκκ · ∂∂∂xxxgi +

[qimi


]· ∂∂∂κκκgi

= Cie (gi, ge) + Cii (gi, gi)

The Coulomb collision operator is invariant under Galilean transformation.


)= BBB




Let us define

fk (t,xxx,κκκ) = fk (t,xxx,vvv) |vvv=κκκ+uuui and gk (t,xxx,κκκ) = fk (t,xxx,vvv) |vvv=κκκ+uuueThere are 4 possibles formulations for simple plasma kinetic equations:Electrons and ions in their mean velocity frame (Braginskii ...)

DDDuuut ge + κκκ · ∂∂∂xxxge +

[qeme


]· ∂∂∂κκκge

= Cee (ge, ge) + Cei (ge, gi)

DDDuuut fi + κκκ · ∂∂∂xxxfi +

[qimi


]· ∂∂∂κκκfi

= Cie(fi, fe

)+ Cii

(fi, fi

)The Coulomb collision operator is invariant under Galilean transformation.


)= BBB




Let us define

fk (t,xxx,κκκ) = fk (t,xxx,vvv) |vvv=κκκ+uuui and gk (t,xxx,κκκ) = fk (t,xxx,vvv) |vvv=κκκ+uuueThere are 4 possibles formulations for simple plasma kinetic equations:Electrons and ions in mean ions velocity frame (Graille ...)


[qeme



= Cee(fe, fe

)+ Cei

(fe, fi

)DDDuuut fi + κκκ · ∂∂∂xxxfi +

[qimi


]· ∂∂∂κκκfi

= Cie(fi, fe

)+ Cii

(fi, fi

)The Coulomb collision operator is invariant under Galilean transformation.


)= BBB



Dimensionless equations


[qeme



= Cee(fe, fe

)+ Cei

(fe, fi

)Dimensionless

DDDuuut fk +[t0] [κκκk]

[xxx0]κκκ · ∂∂∂xxxfk −

[uuuk]

[κκκk]

(DDDuuutuuu+

[t0] [κκκk]

[xxx0]κκκ · ∂∂∂xxxuuu

)· ∂∂∂κκκfk

+qk [t0] [EEE0]

mk [κκκk]

(EEE +

[BBB0] [uuuk]

[EEE0]

(uuu+

[κκκk]

[uuuk]κκκ

)×BBB

)· ∂∂∂κκκfk

=[t0] [Ckk]

[fk]Ckk

(fk, fk

)+

[t0] [Ck`][fk]

Ck`(fk, f`

)where

DDDuuut = ∂t +[t0] [uuuk]

[xxx0]uuu · ∂∂∂xxx


Scaling Hypotheses 3 4 5

[ρ]i is the ion Larmor radius,[r] is characteristic short length

ε =[ρ]i[r] '

√memi' 2 10−2 � 1

Ions and electrons areof the same scale for

1 Densities≡ n02 Temperatures≡ T03 Cross-sections≡ σ04 Macroscopic

velocities ≡ uuu0

[κκκk] =

√kB [T0]

mk=⇒ [κκκi]

[κκκe]= ε

[`k] ≡ [τk] [κκκk] =1

[σ0] [n0]=⇒ [τe]

[τi]= ε

[uuuk] ≡ [uuu0] =⇒ [uuue]

[uuui]= ε0 ≡ 1

Note that uuue 6= uuui3P. Degond, B. Lucquin-Desreux, Transport coefficients of plasmas and disparate

mass binary gases. Transp. Theory and Stat. Phys. 25 pp. 595-633, (1996).4J.J. Ramos, Fluid Theory of Magnetized Plasma Dynamics at Low Collisionality.

Physics of plasmas vol. 14 (1) 2007. MIT Report PSFC/JA-06-295B. Graille, T. Mangin, and M. Massot. Kinetic theory of plasmas: translational

energy. Math. Models Methods Appl. Sc. (M3AS) 527-599, 19(4) (2009).B. Nkonga . Fluid Theory 15 / 56

Scaling Hypotheses

Other important parameters are :

1 The collisionality [ν?] =[R]

[`k]

2 The pressure ratio [β] =2µ0 [p]

[BBB ·BBB]

For ITER we have [T0] ' 10keV , [n0] ' 1020m−3 and [`k] ' 100m a.

aR.V. Budny, Fusion alpha parameters in tokamak with high DT fusion ratesNucl. Fusion 42 (2002) 1382-1392

Therefore[κκκe] ' 1.3 106ms−1, [κκκi] ' 3.1 104ms−1

[τe] ' 10−4s, [τi] ' 0.8 10−2s


Scaling Hypotheses :: ε '√

me

mi' [τe]

[τi]� 1 (1)

Collisions scales

[Cee] ≡ [Cei] ≡[fe]

[τe], [Cie] ≡

me

mi

[fi]

[τe]≡ [fi]

[τi]ε3 and [Cii] ≡

[fi]

[τi]

Indeedme

mi= ε2 and

[τe]

[τi]= ε


Velocities distributions for ions(red) and electrons (blue).

[κκκe]

[κκκi]

[uuue]

[uuui]

1 [uuue] = [uuu0]

2 [uuui] = [uuu0]

3 Mi =[uuui]

[κκκi]' 1

Therefore

Me =[uuue]

[κκκe]=

[uuui]

[κκκi]

[κκκi]

[κκκe]= ε

with ε = εMi

Indeed

[κκκe] =1

ε[κκκi] '

1

ε[uuu0]


Velocities distributions for ions(red) and electrons (blue).

[κκκe]

[κκκi]

[uuue]

[uuui]

1 [uuue] = [uuu0]

2 [uuui] = [uuu0]

3 Mi =[uuui]

[κκκi]' ε ∼< 1

Therefore

Me =[uuue]

[κκκe]=

[uuui]

[κκκi]

[κκκi]

[κκκe]= ε

with ε = εMi

Indeed

[κκκe] =1

ε[κκκi] '

1

ε[uuu0]


Scaling Hypotheses :: ε =√

me

mi� 1, ε = εMi (2)

Large observation time and space length scales : “Hydrodynamic”

[t0] =[τi]

ε=

Mi [τe]

ε2and [xxx0] =

[`i]

ε=

[`e]

ε=

[`0]

ε=⇒ [t0]

[xxx0]=

1

[κκκi]

Electrical and thermal energies are of the same scale

|qe| [xxx0] [EEE0] = mi [κκκi]2 = me [κκκe]

2

Strongly magnetized plasma

[BBB0] [uuui]

[EEE0]= 1


Electrons : ε = εMi, uuu ≡ uuue, κκκ ≡ κκκe

DDDuuut = ∂t +[t0] [uuue]

[xxx0]uuu · ∂∂∂xxx,

[t0] [uuue]

[xxx0]= 1,

DDDuuut ge +[t0] [κκκe]

[xxx0]κκκ · ∂∂∂xxxge −

[uuue]

[κκκe]

(DDDuuutuuu+

[t0] [κκκe]


)· ∂∂∂κκκge

+qe [t0] [EEE0]

me [κκκe]

(EEE +

[BBB0] [uuue]

[EEE0]

(uuu+

[κκκe]

[uuue]κκκ

)×BBB


=[t0] [Cee]

[fe]Cee (ge, ge) +

[t0] [Cei][fe]

Cei (ge, gi)



DDDuuut = ∂t + uuu · ∂∂∂xxx,[t0] [κκκe]

[xxx0]= ‘

[κκκe]

[uuue]=

1

εMi

DDDuuut ge +[t0] [κκκe]

[xxx0]κκκ · ∂∂∂xxxge −

[uuue]

[κκκe]

(DDDuuutuuu+

[t0] [κκκe]



+qe [t0] [EEE0]

me [κκκe]

(EEE +

[BBB0] [uuue]

[EEE0]

(uuu+

[κκκe]

[uuue]κκκ

)×BBB


=[t0] [Cee]

[fe]Cee (ge, ge) +

[t0] [Cei][fe]

Cei (ge, gi)



DDDuuut = ∂t + uuu · ∂∂∂xxx,|qe| [t0] [EEE0]

me [κκκe]=

[t0] [κκκe]

[xxx0]=

1

εMi

DDDuuut ge +1

εκκκ · ∂∂∂xxxge − ε

(DDDuuutuuu+

1

εκκκ · ∂∂∂xxxuuu


− |qe| [t0] [EEE0]

me [κκκe]

(EEE +

[BBB0] [uuue]

[EEE0]

(uuu+

1

εκκκ

)×BBB


=[t0] [Cee]

[fe]Cee (ge, ge) +

[t0] [Cei][fe]

Cei (ge, gi)



DDDuuut = ∂t + uuu · ∂∂∂xxx,[BBB0] [uuue]

[EEE0]=

[BBB0] [uuui]

[EEE0]= 1

DDDuuut ge +1


(DDDuuutuuu+

1



− 1

ε

(EEE +

[BBB0] [uuue]

[EEE0]

(uuu+

1

εκκκ

)×BBB


=[t0] [Cee]

[fe]Cee (ge, ge) +

[t0] [Cei][fe]

Cei (ge, gi)



DDDuuut = ∂t + uuu · ∂∂∂xxx,[t0] [Cee]

[ge]=

[t0] [Cei][ge]

=[t0]

[τe]=

Mi

ε2

DDDuuut ge +1


(DDDuuutuuu+

1



− 1

ε

(EEE +

(uuu+

1

εκκκ

)×BBB


=[t0] [Cee]

[fe]Cee (ge, ge) +

[t0] [Cei][fe]

Cei (ge, gi)



DDDuuut = ∂t + uuu · ∂∂∂xxx,

DDDuuut ge +1


(DDDuuutuuu+

1



− 1

ε

(EEE +

(uuu+

1

εκκκ

)×BBB


=Mi

ε2Cee (ge, ge) +

Mi

ε2Cei (ge, gi)


Ions : ε = εMi

DDDuuut = ∂t +[t0] [uuui]

[xxx0]uuu · ∂∂∂xxx,

[t0] [uuui]

[xxx0]= 1,

DDDuuut fi +[t0] [κκκi]

[xxx0]κκκ · ∂∂∂xxxfi −

[uuui]

[κκκi]

(DDDuuutuuu+

[t0] [κκκi]


)· ∂∂∂κκκfi

+qi [t0] [EEE0]

mi [κκκi]

(EEE +

[BBB0] [uuui]

[EEE0]

(uuu+

[κκκi]

[uuui]κκκ

)×BBB


=[t0] [Cie]

[fi]Cie(fi, fe

)+

[t0] [Cii][fi]

Cii(fi, fi

)


Ions : ε = εMi

DDDuuut = ∂t + uuu · ∂∂∂xxx,[t0] [κκκi]

[xxx0]=

[κκκi]

[uuui]=

1

Mi

DDDuuut fi +[t0] [κκκi]

[xxx0]κκκ · ∂∂∂xxxfi −

[uuui]

[κκκi]

(DDDuuutuuu+

[t0] [κκκi]



+qi [t0] [EEE0]

mi [κκκi]

(EEE +

[BBB0] [uuui]

[EEE0]

(uuu+

[κκκi]

[uuui]κκκ

)×BBB


=[t0] [Cie]

[fi]Cie(fi, fe

)+

[t0] [Cii][fi]

Cii(fi, fi

)


Ions : ε = εMi

qi [t0] [EEE0]

Zimi [κκκi]=

[t0] [κκκi]

[xxx0]=ZiMi

DDDuuut fi +1

Miκκκ · ∂∂∂xxxfi −Mi

(DDDuuutuuu+

1

Miκκκ · ∂∂∂xxxuuu


+qi [t0] [EEE0]

mi [κκκi]

(EEE +

[BBB0] [uuui]

[EEE0]

(uuu+

1

Miκκκ

)×BBB


=[t0] [Cie]

[fi]Cie(fi, fe

)+

[t0] [Cii][fi]

Cii(fi, fi

)


Ions : ε = εMi

[BBB0] [uuui]

[EEE0]=

[BBB0] [uuui]

[EEE0]= 1

DDDuuut fi +1


(DDDuuutuuu+

1



+ZiMi

(EEE +

[BBB0] [uuui]

[EEE0]

(uuu+

1

Miκκκ

)×BBB


=[t0] [Cie]

[fi]Cie(fi, fe

)+

[t0] [Cii][fi]

Cii(fi, fi

)


Ions : ε = εMi

[t0] [Cie][fi]

=[t0]

[τe]

me

mi=

1

Miand

[t0] [Cii][fi]

=[t0]

[τi]=

1

ε

DDDuuut fi +1


(DDDuuutuuu+

1



+ZiMi

(EEE +

(uuu+

1

Miκκκ

)×BBB


=[t0] [Cie]

[fi]Cie(fi, fe

)+

[t0] [Cii][fi]

Cii(fi, fi

)


Ions : ε = εMi

DDDuuut fi +1


(DDDuuutuuu+

1



+ZiMi

(EEE +

(uuu+

1

Miκκκ

)×BBB


=1

MiCie(fi, fe

)+

1

εCii(fi, fi

)


Dimensionless “simple plasma” system ∀ Mi ∼< 1

Electrons : uuu ≡ uuue, κκκ ≡ κκκe

− ε2

Mi[(∂tuuu+ uuu · ∂∂∂xxxuuu) · ∂∂∂κκκge] +

ε

Mi[∂tge + uuu · ∂∂∂xxxge − (κκκ · ∂∂∂xxxuuu) · ∂∂∂κκκge]

+1

Mi[κκκ · ∂∂∂xxxge − (EEE + uuu×BBB) · ∂∂∂κκκge]

=1

ε[− (κκκ×BBB) · ∂∂∂κκκge + Cee (ge, ge) + Cei (ge, gi) ]

Ions: uuu ≡ uuui, κκκ ≡ κκκi

−Mi

[(∂tuuu+ uuu · ∂∂∂xxxuuu) · ∂∂∂κκκfi

]+[∂tfi + uuu · ∂∂∂xxxfi − (κκκ · ∂∂∂xxxuuu) · ∂∂∂κκκfi

]+

1

Mi

[−Cie

(fi, fe

)+ κκκ · ∂∂∂xxxfi + Zi (EEE + uuu×BBB) · ∂∂∂κκκfi

]+Zi

M2i

(κκκ×BBB) · ∂∂∂κκκfi =1

ε

[Cii(fi, fi

) ]B. Nkonga . Fluid Theory 22 / 56

Fast dynamics Mi ≡ 1 and ε = ε : Sonic

Electrons : uuu ≡ uuue, κκκ ≡ κκκe−ε2 [(∂tuuu+ uuu · ∂∂∂xxxuuu) · ∂∂∂κκκge] + ε [∂tge + uuu · ∂∂∂xxxge − (κκκ · ∂∂∂xxxuuu) · ∂∂∂κκκge]

+ [κκκ · ∂∂∂xxxge − (EEE + uuu×BBB) · ∂∂∂κκκge]=

1



−[(∂tuuu+ uuu · ∂∂∂xxxuuu) · ∂∂∂κκκfi


]+[−Cie

(fi, fe


]+ Zi (κκκ×BBB) · ∂∂∂κκκfi =

1

ε

[Cii(fi, fi

) ]


Slow dynamics Mi ≡ ε and ε = ε2 : Drift

Electrons : uuu ≡ uuue, κκκ ≡ κκκe

−ε√ε [(∂tuuu+ uuu · ∂∂∂xxxuuu) · ∂∂∂κκκge] +

√ε [∂tge + uuu · ∂∂∂xxxge − (κκκ · ∂∂∂xxxuuu) · ∂∂∂κκκge]

+1√ε

[κκκ · ∂∂∂xxxge − (EEE + uuu×BBB) · ∂∂∂κκκge]

=1



−√ε[(∂tuuu+ uuu · ∂∂∂xxxuuu) · ∂∂∂κκκfi


]+

1√ε

[−Cie

(fi, fe


]=

1

ε

[− Zi (κκκ×BBB) · ∂∂∂κκκfi + Cii

(fi, fi

) ]B. Nkonga . Fluid Theory 22 / 56

Slow dynamics of Braginskii.

Electrons : uuu ≡ uuue, κκκ ≡ κκκe− [(∂tuuu+ uuu · ∂∂∂xxxuuu) · ∂∂∂κκκge] + [∂tge + uuu · ∂∂∂xxxge − (κκκ · ∂∂∂xxxuuu) · ∂∂∂κκκge]

+ [κκκ · ∂∂∂xxxge − (EEE + uuu×BBB) · ∂∂∂κκκge]=

1



−[(∂tuuu+ uuu · ∂∂∂xxxuuu) · ∂∂∂κκκfi


]+[−Cie

(fi, fe


]=

1

ε

[− Zi (κκκ×BBB) · ∂∂∂κκκfi + Cii

(fi, fi

) ]


Relations

ge (κκκ) = fe (κκκ+ uuue) = fe (κκκ+ uuui + (uuue − uuui) ) = fe (κκκ+ δuuu)

gi (κκκ) = fi (κκκ+ uuue) = fi (κκκ+ uuui + (uuue − uuui) ) = fi (κκκ+ δuuu)

δuuu = uuue − uuui


Overview







Taylor’s and Hilbert’s expansions : ε = εMi

BBB(κκκ− εκκκ′i

)' BBB (κκκ)− εκκκ′i · ∂∂∂κκκBBB (κκκ) +

ε2

2

(κκκ′i ⊗ κκκ′i

): (∂∂∂κκκ ⊗ ∂∂∂κκκ)BBB (κκκ) + ε3

Then, with σσσi = niTiIII− τττ i ' niTiIII. In the ions frame we have :

DDDi (κκκ) =

∫R3

dκκκ′i

[fi(κκκ′i)BBB(κκκ− εκκκ′i

) ]= niBBB (κκκ)− ε ∗ 0 + ε2niTi

(3κκκ⊗ κκκ− |κκκ|2III

|κκκ|5

)+ ε3

and

OOOei(fe, fi

)= −DDDi (κκκ)

∂fe (κκκ)

∂κκκ+ ε2

me

mife (κκκ)

∂

∂κκκ·DDDi (κκκ)

= −niBBB (κκκ)∂fe∂κκκ− 0 ∗ ε− ε2

(me

mi

2κκκ

|κκκ|3fe (κκκ) + niTi

(3κκκ⊗ κκκ− |κκκ|2III

|κκκ|5

)∂fe∂κκκ

)+ ε3



BBB(κκκ′e − εκκκ

)' BBB

(κκκ′e)−εκκκ·∂∂∂κκκ′eBBB

(κκκ′e)+ε2

2(κκκ⊗ κκκ) :

(∂∂∂κκκ′e ⊗ ∂∂∂κκκ′e

)BBB(κκκ′e)+ε3

κκκ · ∂∂∂κκκ′eBBB (κκκ′e) = −κκκ · κκκ′e|κκκ′e|5

(|κκκ′e|2III− 3κκκ′e ⊗ κκκ′e

)− κκκ⊗ κκκ′e + κκκ′e ⊗ κκκ

|κκκ′e|3In the electrons frame we have

DDDe (εκκκ) = −∫R3

dκκκ′e[ge(κκκ′e)BBB(κκκ′e − εκκκ

) ]= −

∫R3

dκκκ′e[ge(κκκ′e)BBB(κκκ′e) ]

+ ε

∫R3

dκκκ′e[ge(κκκ′e)κκκ · ∂∂∂κκκ′eBBB

(κκκ′e) ]

+ ε2

Case of ge (κκκ′e) 'Me (|κκκ′e|)

OOOie (gi, ge) ' −4ne3

√me

2πTe

(∂gi (κκκ)

∂κκκ+mi

Teκκκgi (κκκ)

)

Note that we have Γie = Γeim2e

m2i



ge = g0e +ε

Mig1e +

ε2

M2i

g2e + · · ·

fi = f0i +ε

Mif1i +

ε2

M2i

f2i + · · ·

BBBei = BBB0ei +

ε

MiBBB1ei +

ε2

M2i

BBB2ei + · · ·

BBBie = BBB0ie +

ε

MiBBB1ie +

ε2

M2i

BBB2ie + · · ·

Cei = C0ei +ε

MiC1ei +

ε2

M2i

C2ei + · · ·

Cie = C0ie +ε

MiC1ie + · · ·


Expansion of electron-ion collisions

Cei(fe, fi

)= −Γei

2∂∂∂κκκ · OOOei = C0ei

(fe, fi

)+

ε

MiC1ei(fe, fi

)+

ε2

M2i

withOOOei = −DDDi (κκκ)∂∂∂κκκfe +

me

mife (κκκ)∂∂∂κκκ · (DDDi)

Witching the ions frame we have

DDDi (κκκ) =

∫R3

dκκκ′[fi(κκκ′)BBB(κκκ− κκκ′

) ]= niBBB (κκκ) + 0 + ε2

Therefore

C0ei(fe, fi

)=niΓei

2∂∂∂vvv ·

(BBB (κκκ)∂∂∂κκκfe

)


Thermalization of distributions functions

{− (κκκ×BBB) · ∂∂∂κκκg0e + Cee

(g0e , g

0e

)+ C0ei

(g0e , g

0i

)= 0

− (κκκ×BBB) · ∂∂∂κκκf0i + Cii(f0i , f

0i

)= 0

g0e =Me (|κκκ|) and f0i =Mi (|κκκ|) where

Me (|κκκ|) =Me,0 exp

(−me|κκκ|2

2Te

)and Mi (|κκκ|) =Mi,0 exp

(−mi|κκκ|2

2Ti

)

For any change of variable κκκ∗ = κκκ± εδuuu :: g0∗e (κκκ∗) ≡Me (|κκκ± εδuuu|)

− (κκκ∗ ×BBB) · ∂∂∂κκκ∗ g0∗e + Cee(g0∗e , g

0∗e

)+ C0ei

(g0∗e , g

0∗i

)= 0 +

ε

Mi· · ·

1 Which thermalization is consistent with physical applications?

2 What is the definition of g0i ? Is g0i (κκκ) = f0i (κκκ+ δuuu)?


First order correction : ε = Miε

ge (κκκ) = Me (|κκκ| )(

1 + Φ1e (κκκ)

)+ ε2

fi (κκκ) = Me (|κκκ| )(

1 + Φ1i (κκκ)

)+ ε2

Expansion of the collisions Cei = −Γei2∂∂∂vvv · OOOei with (in ions frame)

OOOei = −DDDi (κκκ)∂∂∂κκκfe +me

mife (κκκ)∂∂∂κκκ · (DDDi) = −niBBB (κκκ)∂∂∂κκκfe + ε2

where, with σσσi = niTiIII + τττ i,

DDDi (κκκ) =

∫R3

dκκκ′[fi(κκκ′)BBB(κκκ− κκκ′

) ]= niBBB (κκκ) +

��

��1

2[σσσi : (∂∂∂κκκ ⊗ ∂∂∂κκκ)BBB (κκκ) ]

Then, as ‖δuuu‖ = ‖vvve − vvvi‖ ' ε, we have the following estimation

fe (κκκ) = fe (κκκ+ vvvi) = fe (κκκ− δuuu+ vvve) = ge (κκκ− δuuu)

= Me (|κκκ− δuuu| )(

1 + Φ1e (κκκ− δuuu)

)+ ε2

= Me (|κκκ| )(

1 +me

Teδuuu · κκκ+ Φ1

e (κκκ)

)+ ε2



ge (κκκ) =Me (|κκκ| )(

1 + Φ1e (κκκ)

)+ ε2, fi (κκκ) =Me (|κκκ| )

(1 + Φ1

i (κκκ))

+ ε2

fe (κκκ) =Me (|κκκ| )(

1 +me


e (κκκ)

)+ ε2, OOOei = −niBBB (κκκ)∂∂∂κκκfe + ε2

Expansion of the collisions BBB (κκκ) [∂∂∂κκκMe (|κκκ| )] = βBBB (κκκ)κκκ = 0

Cei = −Γei2∂∂∂κκκ · OOOei =

niΓei2

∂∂∂κκκ ·(BBB (κκκ)∂∂∂κκκfk

)+ ε2

=niΓei

2∂∂∂κκκ ·

(Me (|κκκ| )

(me

TeBBB (κκκ) δuuu+BBB (κκκ)∂∂∂κκκΦ1

e (κκκ)

))+ ε2

=niΓei

2Me (|κκκ| )

[me

Te(∂∂∂κκκ ·BBB (κκκ) ) · δuuu + ∂∂∂κκκ ·

(BBB (κκκ)∂∂∂κκκΦ1

e (κκκ))]

+ ε2

=niΓei

2Me (|κκκ| )

[− 2me

Te|κκκ|3κκκ · δuuu + ∂∂∂κκκ ·

(BBB (κκκ)∂∂∂κκκΦ1

e (κκκ))]

+ ε2

= 0 +Me (|κκκ| )(C′ei(

Φ1e, f

0i

)− 2niΓeime

2Te|κκκ|3κκκ · δuuu

)+ ε2



ge (κκκ) =Me (|κκκ| )(

1 + Φ1e (κκκ)

)+ ε2, fi (κκκ) =Me (|κκκ| )

(1 + Φ1

i (κκκ))

+ ε2

fe (κκκ) =Me (|κκκ| )(

1 +me


e (κκκ)

)+ ε2, OOOei = −niBBB (κκκ)∂∂∂κκκfe + ε2

Expansion of the collisions

Cei = 0 ∗ ε0 +C′ei(g0e Φ

1e, f

0i

)− g0e

2niΓeime

2Te|κκκ|3κκκ · δuuu +ε2

Cee = Cee(g0e , g

0e

)+Cee

(g0e Φ

1e, g

0e

)+ Cee

(g0e , g

0e Φ

1e

)+ε2

Cii = Cii(f0i , f

0i

)+Cii

(f0i Φ1

i , f0i

)+ Cii

(f0i , f

0i Φ1

i

)+ε2

Cie = 0 ∗ ε0 +ε


Overview







First order correction for Slow dynamics of Braginskii.

Electrons : uuu ≡ uuue, κκκ ≡ κκκe and g1e ≡ g1e (κκκ) = Φ1e (κκκ)Me (|κκκ| )

− (∂tuuu+ uuu · ∂∂∂xxxuuu) · ∂∂∂κκκg0e + ∂tg0e + uuu · ∂∂∂xxxg0e − (κκκ · ∂∂∂xxxuuu) · ∂∂∂κκκg0e

+κκκ · ∂∂∂xxxg0e − (EEE + uuu×BBB) · ∂∂∂κκκg0e= − (κκκ×BBB) · ∂∂∂κκκg1e + Cee

(g1e, g

0e

)+ Cee

(g0e , g

1e

)+C′ei

(g1e, f

0i

)− 2niΓeime

2Te|κκκ|3κκκ · δuuuMe (|κκκ| )

Ions: uuu ≡ uuui, κκκ ≡ κκκi and f 1i ≡ f 1

i (κκκ) = Φ1i (κκκ)Mi (|κκκ| )

− (∂tuuu+ uuu · ∂∂∂xxxuuu) · ∂∂∂κκκf0i + ∂tf0i + uuu · ∂∂∂xxxf0i − (κκκ · ∂∂∂xxxuuu) · ∂∂∂κκκf0i

��

��−Cie(f0i , f

0e

)+ κκκ · ∂∂∂xxxf0i + Zi (EEE + uuu×BBB) · ∂∂∂κκκf0i

= −Zi (κκκ×BBB) · ∂∂∂κκκf1i + Cii(f1i , f

0i

)+ Cii

(f0i , f

1i

)


Transport contribution for first order approximation

Cei = −Γei2∂∂∂κκκ ·

(OOO1ei

)+ ε2

where OOO1ei (κκκ) = −niMe (|κκκ| )

(me

TeBBB (κκκ) δuuu+BBB (κκκ)∂∂∂κκκΦ1

e (κκκ)

)Friction contribution is

RRR1ei = −Γei

2

∫R3

me (κκκ− δuuu)∂∂∂κκκ ·(OOO1ei

)dκκκ =

Γei2

∫R3

meOOO1eidκκκ

According to integration formulas of polynomials functions over balls 6

RRR1ei = −mene

τeδuuu+RRR′ei where τe =

3√meT

32e

4ni√

2πq2eq2i ln Λ

and RRR′ei ≡ RRR′ei(

Φ1e

)= −niΓei

2

∫R3

Me (|κκκ| )BBB (κκκ)∂∂∂κκκΦ1e (κκκ)dκκκ

6John A. Baker. Integration Over Spheres and the Divergence Theorem for Balls.The American Mathematical Monthly, Vol. 104, No. 1. (Jan., 1997), pp. 36-47.



Friction contribution is RRR1ei = −mene

τeδuuu+RRR′ei

Heat

Q1ei = −Γei

2

∫R3

me|κκκ− δuuu|2

2∂∂∂κκκ ·

(OOO1ei

)dκκκ

=Γei2

∫R3

me (κκκ− δuuu) · OOO1eidκκκ

=(((

(((((((

((((Γei2

∫R3

me

(BBBT (κκκ)κκκ

)· OOO1

eidκκκ − δuuu ·RRR1ei

= 0 +Q2δuuuei = ε2


Transport : second order contributions

There is also an other second order term associated to

OOO2ei (κκκ) =

nime

mig0e (|κκκ|)∂∂∂κκκ ·BBB (κκκ)− 1

2mi[σσσi : (∂∂∂κκκ ⊗ ∂∂∂κκκ)BBB (κκκ) ]∂∂∂κκκg

0e (|κκκ|)

where σσσi = niTiIII− τττ i ' niTiIII

RRR2ei =

Γei2

∫R3

meOOO2eidκκκ = 0

Indeed, we have

∂∂∂κκκ ·BBB = − 2κκκ

|κκκ|3,

1

2[(∂∂∂κκκ ⊗ ∂∂∂κκκ)BBB] = −|κ

κκ|2III− 3κκκ⊗ κκκ|κκκ|5

and ∫S2

[σσσi :

(sss⊗ sss− sss · sss

3III) ]

sssdsss = 0



Q2δTei = −meΓei

2

∫R3

dκκκ|κκκ− δuuu|2

2∂∂∂κκκ·OOO2

ei (κκκ) dκκκ =meΓei

2

∫R3

dκκκ (κκκ− δuuu)·OOO2ei (κκκ) dκκκ

Q2δTei = −meΓei

2

∫R3

|κκκ− δuuu|2

2∂∂∂κκκ · OOO2

eidκκκ =meΓei

2

∫R3

(κκκ− δuuu) · OOO2eidκκκ

=meΓei

2

∫R3

κκκ ·[nime

mig0e∂∂∂κκκ ·BBB−

1

2mi[σσσi : (∂∂∂κκκ ⊗ ∂∂∂κκκ)BBB]∂∂∂κκκg

0e

]dκκκ

=meΓei

2

∫R3

[−2|κκκ|2

|κκκ|3nime

mig0e −

1

2miκκκ · [σσσi : (∂∂∂κκκ ⊗ ∂∂∂κκκ)BBB]∂∂∂κκκg

0e

]dκκκ

' meΓei2

∫R3

[− 2

|κκκ|nime

mi+

2nimeTimiTe|κκκ|

]g0edκκκ '

m2eniΓeimi

(TiTe− 1

)∫R3

g0e|κκκ|dκκκ

' −m2eniΓeimi

(1− Ti

Te

)4πTeme

ne

(π

2Teme

)− 32

as g0e = ne

(π

2Teme

)− 32

e−x

' −3menemiτe

(Te − Ti) = −3menemiτe

δT

Q2ei = Q2δT

ei +Q2δuuuei = −δuuu ·RRR1

ei −3menemiτe

δT


Transport first and second order contributions

1 Friction

RRRei ≡ RRR1ei +RRR2

ei = −meneτe

δuuu+RRR′ei and RRRie = −RRRei

2 Heat

Qei ≡ Q1ei +Q2

ei = −δuuu ·RRR1ei −

3menemiτe

δT

=meneτe

δuuu · δuuu− δuuu ·RRR′ei −3menemiτe

δT

and Qie =3menemiτe

δT

Whereδuuu = uuue − uuui and δT = Te − Ti


Solubility conditions for Lk(∂, Φ1

k

)= bbbk

For example, with f1i =Mi (|κκκ|) Φ1i (κκκ) = g0i f

1i , we have

Li(∂, Φi

)= − (κκκ×BBB) · ∂∂∂κκκf1i + Cii

(f1i , f

0i

)+ Cii

(f0i , f

1i

)Note that γ0 + γ2|κκκ|2 is always in the kernel of Li.The requirement that correction must not change macroscopic parameters: ∫

R3

1κκκ|κκκ|2

Mk (|κκκ|) Φ1k (κκκ) dκκκ = 0

contains also the assumed requirement “for existence and uniqueness ofthe solution”


ThereforeDDDuuukt nk = −nk∂∂∂xxx · uuuk

meneDDDuuuet uuue = −∂∂∂xxxpe − (EEE + uuue ×BBB) +RRRei��−∂∂∂xxx · πππk

miniDDDuuuit uuui = −∂∂∂xxxpi + Zi (EEE + uuui ×BBB) ��−RRRei

DDDuuukt Tk = −2

3Tk∂∂∂xxx · uuuk

(((((((

(((((((

((

+23 (−∂∂∂xxx · qqqk − πππk : ∂∂∂xxxuuuk +Qk`)

Derivatives with respect to time and space of the Maxwellian are

∂∂∂g0e =

[∂∂∂nene−(

3

2− meκκκ · κκκ

2Te

)∂∂∂TeTe

]g0e

Then the left hand side of electrons correction equation can be estimated

−DDDuuuet uuue · ∂∂∂κκκg0e + ∂tg0e + uuue · ∂∂∂xxxg0e − (κκκ · ∂∂∂xxxuuue) · ∂∂∂κκκg0e

+κκκ · ∂∂∂xxxg0e − (EEE + uuue ×BBB) · ∂∂∂κκκg0e =[(me|κκκ|2

2Te− 5

2

)∂∂∂xxxTe · κκκTe

+RRR1ei · κκκmeTe

+me

Te

(κκκ · [∂∂∂xxxuuue]T κκκ−

|κκκ|2

3∂∂∂xxx · uuue

)]g0e

=

[L′eL′eL′e (|κκκ|) · κκκ+

RRR′ei · κκκmeTe

+LeLeLe (|κκκ|) :

(κκκ⊗ κκκ− |κ

κκ|2

3

)]g0e


Equation for first order correctionsg1e ≡ Φ1

e (κκκ)Me (|κκκ| ) and f1i ≡ Φ1

i (κκκ)Mi (|κκκ| )

[(meκκκ · κκκ

2Te−

5

2

)∂∂∂xxxTe · κκκTe

+RRR1ei · κκκmeTe

+me

Te

(κκκ ·[∂∂∂xxxuuue

]Tκκκ−

|κκκ|2

3∂∂∂xxx · uuue

)]Me

(|κκκ|)

= −(κκκ×BBB

)· ∂∂∂κκκg1

e + Cee(g1e, g

0e

)+ Cee

(g0e , g

1e

)+ C′ei

(g1e, f

0i

)−

2niΓeime

2Te|κκκ|3κκκ · δuuuMe

(|κκκ|)

Integro-differential “linear” equation for Φ1e using LeLeLe = L′eL

′eL′e +

2niΓeime2Te|κκκ|3

δuuu:

− (κκκ×BBB) · ∂∂∂κκκg1e + Cee(g1e, g

0e

)+ Cee

(g0e , g

1e

)+ C′ei

(g1e, f

0i

)− g0e

RRR′ei · κκκmeTe

= g0eLeLeLe (|κκκ|) · κκκ+ g0eLeLeLe (|κκκ|) :


κκ|2

3

)Linear partial differential equation for Φ1

i

− (κκκ×BBB) · ∂∂∂κκκf1i + Cii(f1i , f

0i

)+ Cii

(f0i , f

1i

)= f0i LiLiLi (|κκκ|) · κκκ+ f0i LiLiLi (|κκκ|) :


κκ|2

3


Resolution of first order corrections equations

According to symmetries of the RHS, Φ1e (κκκ) and Φ1

i (κκκ) are found underthe following form :

Φ1k (κκκ) = PkPkPk (|κκκ|) · κκκ+PkPkPk (|κκκ|) :


κκ|2

3

)Moreover, RHS operators LkLkLk (|κκκ|) and LkLkLk (|κκκ|) can be expanded with

Laguerre-Sonine polynomials. For example, let us denote by x =me|κκκ|2

2Te

LeLeLe (|κκκ|) =

(me|κκκ|2

2Te− 5

2

)∂∂∂xxxTeTe

+2niΓeime

2Te|κκκ|3δuuu

= −∂∂∂xxxTeTeYδTe,1L

321 (x) + δuuu

[∑`>0

Yδuuue,`L32` (x)

]

L32` (x) functions gives very simple expansion for the first term : YδTe,1 = 1.


Vector splitting in strongly magnetized plasma

In strongly magnetized plasma, macroscopic vectors are often split intoparallel, perpendicular and () components. For example :

∂∂∂xxxTe = MMM∂∂∂xxxTe +MMM⊥∂∂∂xxxTe +MMM×∂∂∂xxxTe = ∂∂∂‖xxxTe + ∂∂∂⊥xxx Te + ∂∂∂×xxx Te

where MMM = bbb⊗ bbb, MMM⊥ = III− bbb⊗ bbb, MMM×∂∂∂xxxTe = bbb× ∂∂∂xxxTe

MMM× =

0 −bz bybz 0 −bx−by bx 0

These matrices are linearly independent when ‖bbb‖ 6= 0 stable undermultiplication.What about tensors?


Tensor splitting in strongly magnetized plasma

We have

LkLkLk (|κκκ|) ≡ LkLkLk =mk

2Tk

([∂∂∂xxxuuuk] + [∂∂∂xxxuuuk]

T − 2

3∂∂∂xxx · uuukIII

)For this symmetric tensor, Braginskii propose the following splitting:

LkLkLk =

4∑`=0

ΠΠΠ`ΠΠΠ`ΠΠΠ` (bbb) : LkLkLk

ΠΠΠ0ΠΠΠ0ΠΠΠ0 = −(MMM− 1

2MMM⊥)⊗ (23MMM− 1

3MMM⊥)

ΠΠΠ1ΠΠΠ1ΠΠΠ1 = −MMM⊥ �MMM⊥ − 1

2MMM⊥ ⊗MMM

ΠΠΠ3ΠΠΠ3ΠΠΠ3 = 12MMM⊥ �MMM× + 1

2 [MMM×]T �MMM⊥

∥∥∥∥∥∥ ΠΠΠ2ΠΠΠ2ΠΠΠ2 = −MMM⊥ �MMM−MMM�MMM⊥

ΠΠΠ4ΠΠΠ4ΠΠΠ4 = MMM�MMM× + [MMM×]T �MMM

[(AAA⊗BBB) : WWW ]ij =∑k

∑l

AijBklWkl

[(AAA�BBB) : WWW ]ij =∑k

∑l

AikBjlWkl


Splitting of the first order approximation

According to previous splitting in strongly magnetized plasma1 The vector PkPkPk (|κκκ|) is found under the form

PkPkPk (|κκκ|) =

[∑`>0

L32` (x)

(X δT‖k,` MMM + X δT⊥k,` MMM⊥ + X δT×k,` MMM×

)] ∂∂∂xxxTkTk

+

[∑`>0

L32` (x)

(X δuuu‖k,` MMM + X δuuu⊥k,` MMM⊥ + X δuuu×k,` MMM×

)]δuuu

with the constrain that∫R3

1κκκ|κκκ|2

Mk (|κκκ|) Φ1k (κκκ) dκκκ = 0 =⇒ ` > 0

2 and PkPkPk (|κκκ|) under the form

PkPkPk (|κκκ|) =∑`>0

L32` (x)

4∑ζ=0

X δτζk,`ΠΠΠ`ΠΠΠ`ΠΠΠ` (uuuk, bbb)

: LkLkLk


Systems to be solved Φ1k (κκκ) = PkPkPk (|κκκ|) · κκκ+ · · ·

We have (κκκ×BBB) · ∂∂∂κκκΦ1e = (κκκ×BBB) ·PkPkPk (|κκκ|) + · · ·

∂∂∂‖xxx =

BBB

|BBB|2(BBB · ∂∂∂xxx)

∂∂∂⊥xxx = ∂∂∂xxx − ∂∂∂‖xxx

∂∂∂×xxx =BBB × ∂∂∂xxx|BBB|

, and(κκκ×BBB) · ∂∂∂‖xxx = 0(κκκ×BBB) · ∂∂∂⊥xxx = |BBB|κκκ · ∂∂∂×xxx(κκκ×BBB) · ∂∂∂×xxx = −|BBB|κκκ · ∂∂∂⊥xxx

,

indeed (κκκ×BBB) · (BBB × ∂∂∂xxx) = (κκκ ·BBB) (BBB · ∂∂∂xxx)− (BBB ·BBB) (κκκ · ∂∂∂xxx). We have

PkPkPk (|κκκ|) =∑`>0

L32` (x)

(X δT‖k,`

∂∂∂‖xxxTeTk

+ X δT⊥k,`

∂∂∂⊥xxx TeTk

+ X δT×k,`

∂∂∂×xxx TeTk

)+ · · ·

and therefore, as (κκκ×BBB) · ∂∂∂κκκΦ1e = (κκκ×BBB)PkPkPk (|κκκ|) + · · ·

(κκκ×BBB) · ∂∂∂κκκΦ1e = |BBB|

∑`>0

L32` (x)

(X δT⊥k,` κκκ ·

∂∂∂×xxx TeTk−X δT×k,` κκκ ·

∂∂∂⊥xxx TeTk

)+ · · ·

Systems for X δT⊥k,` and X δT×k,` are coupled


Systems to be solved :: Φ1k (κκκ) = PkPkPk (|κκκ|) · κκκ+ · · ·

LeLeLe(|κκκ|)· κκκ = −

(YδTe,1L

321

(x)κκκ

)·∂∂∂⊥xxx Te

Tk−(YδTe,1L

321

(x)κκκ

)·∂∂∂×xxx Te

Tk

−(κκκ×BBB

)· ∂∂∂κκκΦ

1e =

∑`>0

L32`

(x)XδT×k,` |BBB|κκκ

· ∂∂∂⊥xxx TeTk

−

∑`>0

L32`

(x)|BBB|XδT⊥k,` κκκ

· ∂∂∂×xxx TeTk

+

Compacted form . Using dot product with κκκ and the relation κκκ ·κκκ = x2Teme∑

`>0

xe−xL32` (x)X δTe,`

(ı|BBB|+C (x)C (x)C (x) : [sss⊗ sss]

)= YδTe,1xe−xL

321 (x) + · · ·

where

X θk,` = X θ⊥k,` + ıX θ×k,`Yθk,l = Yθk,l + ıYθk,lxC (x)C (x)C (x) is a tensor associated to linearized collisions.




`>0



)= YδTe,1xe−xL

321 (x) + · · ·

Variational principles (Onsager symmetry)formulated as L2-projection for any q > 0.∑

`>0

X δTe,`∫ +∞

0

[x

32 e−xL

32q (x)L

32` (x)

(ı|BBB|+C (x)C (x)C (x) :

∫S2

sss⊗ sss dsss

4π

)]8dx

15√π

= YδTe,1∫ +∞

0

[x

32 e−xL

32q (x)L

321 (x)

]8dx

15√π

+ · · ·

15√π

8=

(3

2+ 1

)! =

5

2

3

2

(1

2

)! =

5

2

3

2

√π

2




`>0



)= YδTe,1xe−xL

321 (x) + · · ·

Variational principles as L2-projection for any q > 0.

∑`>0

X δTe,`∫ +∞

0

[x

32 e−xL

32q (x)L

32

` (x)

(ı|BBB|+ 2

3Tr(C (x)C (x)C (x)

))] 8dx

15√π

= YδTe,1δq,1 + · · ·

X are solutions of a linear system of the following form :

AAAθeX θe = CCCθeYθe and AAAθiX θi = CCCθiYθi


Final

When previous systems are solved, we obtain analytical formula for

ge (κκκ) ' Me (|κκκ| )(

1 +PePePe (|κκκ|) · κκκ+PePePe (|κκκ|) :


κκ|2

3

))fi (κκκ) ' Me (|κκκ| )

(1 +PiPiPi (|κκκ|) · κκκ+PiPiPi (|κκκ|) :


κκ|2

3

))Then they are used to compute transport contributions.


Overview







Braginskii Transport Coefficients : An example

RRR1ei = −mene

τeδuuu−

∫R3

niΓei2Me (|κκκ| )BBB (κκκ)∂∂∂κκκΦ1

e (κκκ)dκκκ

= −meneτe

δuuu−∫R3

niΓei2Me (|κκκ| )BBB (κκκ)∂∂∂κκκ [PkPkPk (|κκκ|) · κκκ]

= −meneτe

δuuu−∫R3

niΓei2Me (|κκκ| )BBB (κκκ)PkPkPk (|κκκ|)dκκκ

PkPkPk (|κκκ|) =

[∑`>0

L32` (x)

(X δT‖k,` MMM + X δT⊥k,` MMM⊥ + X δT×k,` MMM×

)] ∂∂∂xxxTkTk

+

[∑`>0

L32` (x)

(X δuuu‖k,` MMM + X δuuu⊥k,` MMM⊥ + X δuuu×k,` MMM×

)]δuuu

Therefore

α‖ 'meneτe

+niΓei

2

N∑`=1

X δuuu‖e,`∫R3

dκκκ

[Me (|κκκ| )L

32`

(me|κκκ|2

2Te

)BBB (κκκ)

]B. Nkonga . Fluid Theory 48 / 56

Braginskii Transport Coefficients : An example

α‖ 'meneτe

+niΓei

2

N∑`=1

X δuuu‖e,`∫R3

dκκκ

[Me (|κκκ| )L

32`

(me|κκκ|2

2Te

)BBB (κκκ)

]

BBB (κκκ) =1

rBBB (sss) where r = |κκκ|, sss =

κκκ

|κκκ|and

∫S2

BBB (sss) dsss =8π

3III

The integral part of α‖ can be computed as∫R3

dκκκ [G (r)BBB (κκκ) ] =

∫ +∞

0

r2[G (r)

1

r

]dr

∫S2

BBB (sss) dsss =8π

3

∫ +∞

0

rG (r)dr

Therefore α‖ is equivalent to a scalar.

α‖ 'meneτe

+niΓei

2

8π

3

N∑`=1

X δuuu‖e,`∫ +∞

0r

[Me (r)L

32`

(mer

2

2Te

)]dr


Braginskii transport closure : quasi neutral plasma ni = ne

Electrons

RRRe = −eneαααααααααJJJ −βtβtβtβtβtβtβtβtβt∂∂∂xxxTeQe = −Qi +

JJJ ·RRRenee

Qe = −κκκe∂∂∂xxxTe + eneβjβjβjeJJJ

πππe ≡4∑`=0

ηe`ΠΠΠ` (uuue, bbb)

Ions

RRRi = −RRRiQi =

3menemiτe

(Te − Ti)

Qi = −κκκe∂∂∂xxxTi + eneβjβjβj iJJJ

πππi ≡4∑`=0

ηi`ΠΠΠ` (uuui, bbb)

where

ααααααααα = α‖MMM +α⊥MMM⊥ −α×MMM×βtβtβtβtβtβtβtβtβt = βt‖MMM +βt⊥MMM⊥ +βt×MMM×

∣∣∣∣ κκκk = κ‖kMMM +κ⊥kMMM⊥ +κ×kMMM×βjβjβjk = βj‖kMMM +βj⊥kMMM⊥ +βj×kMMM×

See [Braginskii, 1965] for numerical values of theses parameters. Somenumerical examples

α⊥ =me

e2neτe, α‖ = 1.96α⊥, βt× =

3

2ωcolle τe, βt‖ = 0.71


Appendix

κκκ · κκκ = r2 = x2Teme

and rdr =Teme

dx

α‖ 'meneτe

+niΓei

2

8π

3

N∑`=1


0r

[Me (r)L

32`

(mer

2

2Te

)]dr

' meneτe

+niΓei

2

8π

3

TeMe (0)

me

N∑`=1


0

[exp (−x)L

32` (x)

]dx

' meneτe

+meneτe

Tem2e

N∑`=1


0

[exp (−x)L

32` (x)

]dx

Γk` =4πq2kq

2` ln Λ

m2k

and τe =3√meT

32e

4ni√

2πq2eq2i ln Λ

τe =3π√meT

32e

ni√

2πm2eΓei

=3

4πniΓei

(2πTeme

) 32

=3ne

4πniΓeiMe (0)


Appendix

As s2k is homogeneous of degree two. Then (corollary 1 page 39 6)∫S2

s2kdsss = (3 + 2)

∫ 1

−1x2kdxk

∫ ∫dxldxp = 10

∫ 1

0x2kπ(1− x2k)dxk

= 10π

(1

3− 1

5

)= 10π

2

15=

4π

3

Therefore, as BBB (sss) = III− sss⊗ sss, we have∫S2

BBB (sss) dsss =

(4π − 4π

3

)III =

8π

3III

and ∫S2

(sss⊗ sss− sss · sss

3III)dsss = 0

6John A. Baker. Integration Over Spheres and the Divergence Theorem for Balls.The American Mathematical Monthly, Vol. 104, No. 1. (Jan., 1997), pp. 36-47.


Appendix :: fk (κκκ) 'Mk (|κκκ| ) (1 +PePePe (|κκκ|) · κκκ · · · )

r =

(x

2Tkmk

) 12

, rdr =Tkmk

dx, Mk (r) = nk

(π

2Tkmk

)− 32

e−x

qqqk =

∫R3

[mk|vvv − uuuk|2

2vvv

]fk

(vvv)dvvv =

∫R3

[mk|κκκ|2

2κκκ

]fk

(κκκ)dκκκ

' 0 +

∫R3

[mk|κκκ|2

2Mk

(|κκκ|) [κκκ⊗ κκκ

]PkPkPk(|κκκ|) ]

dκκκ + · · ·

'∫ +∞

0r2

[mk

r2

2Mk

(r) ∫

S2

[r2sss⊗ sss

]dsssPkPkPk

(r) ]

dr '∫ +∞

0mk

r6

2

[Mk

(r) 4π

3PkPkPk(r) ]

dr + · · ·

'4πmknk

6

∫ +∞

0

(x 2Tk

mk

) 52(π

2Tk

mk

)− 32e−xPkPkPk

(x) Tk

mkdx + · · ·

'4nk

3√π

T2k

mk

∫ +∞

0

[x

32 xe−xPkPkPk

(x) ]

dx ' −4nk

3√π

T2k

mk

∫ +∞

0

[x

32

(−

5

2L

320 + L

321

(x))

e−xPkPkPk

(x) ]

dx + · · ·

' −4nk

3√π

T2k

mk

15√π

8

XδT‖k,1

∂∂∂‖xxxTe

Tk+ XδT⊥k,1

∂∂∂⊥xxx Te

Tk+ XδT×k,1

∂∂∂×xxx Te

Tk

−

4nk

3√π

T2k

mk

15√π

8

(Xδuuu‖k,1

∂∂∂‖xxxδuuu + Xδuuu⊥k,1 ∂∂∂⊥xxx δuuu + Xδuuu×

k,1∂∂∂×xxx δuuu

)+ · · ·

'−5nkT

2k

2mk

XδT‖k,1

∂∂∂‖xxxTe

Tk+ XδT⊥k,1

∂∂∂⊥xxx Te

Tk+ XδT×k,1

∂∂∂×xxx Te

Tk+ Xδuuu‖

k,1∂∂∂‖xxxδuuu + Xδuuu⊥k,1 ∂∂∂

⊥xxx δuuu + Xδuuu×k,1 ∂∂∂

×xxx δuuu

+ · · ·


Appendix :: fk (κκκ) 'Mk (|κκκ| ) (1 +PePePe (|κκκ|) · κκκ · · · )We have∫

R3

[κκκ⊗ κκκ]PkPkPk (|κκκ|)Mk (|κκκ|) =

∫ +∞

0

r2PkPkPk (r)Mk (r)

[∫S2

[r2sss⊗ sss

]dsss

]dr

=4π

3

∫ +∞

0

r3PkPkPk (r)Mk (r) rdr =4π

3

(2Tkmk

) 32 Tkmk

∫ +∞

0

x32 e−xPkPkPk (x) dx

=4π

3

(2Tkmk

) 32 Tkmk

∑`

Pk`Pk`Pk`(∫ +∞

0

x32 e−xL

320 (x)L

32

` (x) dx

)

Therefore the constrain

∫R3

(1κκκ

|κκκ|2

)Mk (|κκκ|) Φ1

k (κκκ) dκκκ = 0 is achieved when

` > 0, according to orthogonality of Laguerre-Sonine polynomials and zerointegral on sphere for monomial with an odd component of the multi-index :∫

R3

1κκκr2

Mk (|κκκ|) Φ1k (κκκ) dκκκ =

∫ +∞

0

r2Mk (r)PkPkPk (r) ·

∫S2

rsssr2sss⊗ sssr2rsss

dsss

dr+

∫ +∞

0

r2Mk (r)PkPkPk (r) :

∫S2

1rsssr2

r2(sss⊗ sss− sss · sss

3III)dsss

dr = 0


Appendix ::


dκκκ′e

[fe(κκκ′e)BBB(κκκ′e − εκκκ

) ]= −

∫R3

dκκκ′e

[fe(κκκ′e)BBB(κκκ′e) ]

+ ε

∫R3

dκκκ′e

[fe(κκκ′e)κκκ · ∂∂∂κκκ′eBBB

(κκκ′e) ]

+ ε2

Case of fe (κκκe) = g0e (|κκκe|)


dκκκ′e


) ]= −8π

3

∫ +∞

0rg0e (r) dr + ε2 @

Case of fe (κκκ′e) = δuuu · κκκ′eg0e (|κκκ′e|)


dκκκ′e


) ]= −8π

3

∫ +∞

0rg0e (r) dr + ε2


Appendix ::


dκκκ′e


) ]= −

∫R3

dκκκ′e

[fe(κκκ′e)BBB(κκκ′e) ]

+ ε

∫R3

dκκκ′e

[fe(κκκ′e)κκκ · ∂∂∂κκκ′eBBB

(κκκ′e) ]

+ ε2

Case of fe (κκκ′e) = δuuu · κκκ′eg0e (|κκκ′e|)

DDDe (εκκκ) = ε

∫R3

dκκκ′e[δuuu · κκκ′eg0e

(|κκκ′e|)κκκ · ∂∂∂κκκ′eBBB

(κκκ′e) ]

= −8π

3

∫ +∞

0rg0e (r) dr + ε2


Magnetized plasma : About the Braginskii’s 1 - Laboratoire … · 2017-05-09 · Magnetized...

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Transcript of Magnetized plasma : About the Braginskii’s 1 - Laboratoire … · 2017-05-09 · Magnetized...