Magnetic Diffusion using Lattice-Boltzmann

Revsta Mexcana de Físca S 58 2 188 14 DICIEMBRE 212 Magnetc Dffuson usng Lattce-Boltzmann F. Fonseca Physcs Department, Unversdad Naconal de Colomba, Bogotá Colomba. e-mal: frfonsecaf@unal.edu.co Recbdo el 25 de juno de 21; aceptado el 22 de octubre de 21 We have mplemented a lattce-boltzmann model LBM to smulate the magnetc dffuson MD phenomena. An error n our model for the equlbrum dstrbuton functon s up to order Ou 2. The magnetc feld n our model s consdered as a vector valued magnetc dstrbuton functon whch follows a vector Boltzmann equaton. We dscuss the dffuson of magnetc feld trough plasma n one or two dmensonal confguratons. Also we make a comparson between the analytcal and smulaton confguratons fndng a good agreement. Keywords: Magnetc dffuson; lattce Boltzmann; magnetc smulaton. Se ha desarrollado un modelo de dfusón magnétco MD, aplcando la técnca de lattce-boltzmann LBM. El error en nuestro modelo, asocado a la funcón de dstrbucón de equlbro, es de orden Ou2. El modelo de campo magnétco se consdera una funcón de dstrbucón asocada a cada una de las componentes del campo, de tal forma, que se cumple una ecuacón vectoral de Boltzmann. Se dscute la dfusón de plasma magnétco en con guracones de una y dos dmensones. De la msma forma, se realza una comparacón entre lo resultados analítcos y los dados por la smulacón, encontrándose buena concordanca. Descrptores: Dfusón magnétca; lattce Boltzmann; Smulacón Magnétca. PACS: 75.7.Cn; 75.78.Cd 1. Introducton Lattce Boltzmann has been appled wth success to many problems n Physcs. The method s sutable to smulate hydrodynamc systems [1], multphase fluds [2], charge dstrbuton n electrolytes [3], chemcal-reactve flows and the flow n porous meda [4]. One of the frst applcatons of the method comes from the poneer work of Wolfram on lattcegas automata [4-11]. There have been many attempts to buld models of magneto-hydrodynamcs MHD usng Lattce-Boltzmann LB. Ones of the frst was a 2D model based on lattce-gas automata [12]-[13], where a basc automata model called FHP, Frsch, Hasslacher and Pomeau, s extended to nclude other degrees of freedom n order to gve justfcaton to the vector magnetc potental. Also a work done by Succ et.al, [14] shows that the lattce-boltzmann scheme for the Naver- Stokes equatons can be extended to nclude the effects of a two-dmensonal magnetc feld. Chen et.al., [15],[16] and [17], n 11, propose a LB equaton model, for 2D and 3D, that gves rse for moments as mass densty, magnetc feld, momentum for ncompressble MHD smulatons, and ntroduces a unque relaxaton tme that admts easy handlng transport coefcents. Fogacca, et.al, [18] n 16, mplement an algorthm usng LBE, n the electrostatc lmt studyng 2D turbulence. Also, n ref. [1] a BGK Bhatnagar- Gross-Krook scheme models the collson term recoverng the macroscopc dsspatve MHD equatons. In 28, [2], Muñoz & Mendoza proposed a 3D lattce-boltzmann model n a cubc lattce wth 1 veloctes D3Q1, that recuperates MHD equatons and reproduces the Hartmann flow and magnetc reconnecton. Based on the lattce-boltzmann technque we fnd a novel soluton to obtan the magneto-hydrodynamcs equatons. Ths paper ams to use the anzatz hypothess [21], a soluton to the magnetc dffuson equaton usng a defnton of the tensor Π, n the Chapman-Enskog expanson [21]. In secton II, we begn presentng a short revew of magnetc dffuson equatons. In secton III, we present the basc set of equatons of Boltzmann technque and the equlbrum functons for flud and the magnetc feld, based on the 2dq scheme [21]. After, we dervate the magnetc dffuson equatons usng the re-defnton of Π. In secton IV, we obtan the equlbrum functon whch we use on the lattce for the mplementaton n the computatonal scheme. In secton V, we compare our results wth the theoretcal approach gven by [22], n one and two dmensons for magnetc dffuson, and also we present the vortcty structure of the magnetc feld startng from a random ntal confguratons. At the end of the secton V, we present conclusons. 2. Magnetc dffuson equatons The equatons governng the magneto-hydrodynamc phenomena are: u ρ + u u = p + j B + µ f 2 u + ρ g 1 ρ + ρ u = 2 E = B 3 B = µ m j 4

Where we have the momentum, the equaton of mass contnuty, Maxwell s equatons and Ohm s law, respectvely. The dsplacement current n Ampere s law, has not been taken nto account reasons by t s vald for a non-relatvstc approach for an nertal flud. Assumed n the above equatons are the followng relatons: B = 6 j = 7 If we take the curl of Ohm s law, we get: j = σe + u E Usng Ampere and faraday s laws and the dentty: B = B 2 B We obtan MAGNETIC DIFFUSION USING LATTICE-BOLTZMANN 18 j = σ E + u B 5 where and m are local collson operators that gve local nteracton rules among partcle collsons. Usng BGK approxmaton [23], the collson operators could be approxmated by a sngle tme relaxaton process that mght happen for a gven partcle probablty dstrbuton at constant rate. These 8 dstrbutons are called f eq gven by: x, t and g eq f x, t f eq x, t Ω = Ω m = g,j x, t g eq,j τ m x, t and they are x, t 15 16 Here and τ m measure the approachng rate of the system to the statstcal equlbrum. The physcal condtons to ths equlbrum are that the momentum and energy relatons are conserved. Also, the analytcal form of the equlbrum dstrbuton has to ensure the sotropc and Gallean nvarance. Expandng the dstrbuton functons and the tme and space dervatves, we obtan: B = u B + 1 µ m σ 2 B 1 Equaton 1 s called the nducton equaton and descrbes temporal evoluton of the magnetc feld n terms of two effects, namely the advecton of magnetc feld wth the plasma and ts dffuson through the plasma. Assumng the flud n rest, eq. 11 t becomes a pure dffuson equaton: B = D m 2 B 11 Wth D m = 1/µ m σ. In general we can suppose that D m s non-unform, then we get: B = 2 D m B Lattce veloctes of the D2Q scheme. 3. The lattce Boltzmann model 12 The Boltzmann equaton gves temporal evoluton of a sngle partcle probablty dstrbuton functon f x, t, whch n the lattce-boltzmann method s transform nto a dscrete functon. In the model we use f x, t to justfy the flud and g x, t to the magnetc feld. Both of them, at the ste become as: f x + e, t + 1 = f x, t + Ω f x, t 13 g,j x + e, t + 1 = g,j x, t + Ω m,j g,j x, t 14 Where f f = f + εf 1 + ε 2 f 2 + g,α = g,α + εg1,α + ε2 g 2,α + t = ε t1 + ε 2 t2 + = ε 1 + 17 = f eq and g,α = geq,α, and the parameter ε whch s assumed small. Also, t s supossed that t 2 s smaller than the tme scale t 1 and t s assocated wth dffuson phenomena. Replacng equatons 17 nto eqs. 15-16 we obtan at frst order: ε : t1 + e 1 f = 1 f 1 And second order: t1 + e 1 g,j = 1 g 1,j τ m ε 2 : t2 f + 1 2 t 1 + e 1 2 f + t1 + e 1 f 1 = 1 f 2 t2 g,j + 1 2 t 1 + e 1 2 g,j + t1 + e 1 g 1,j = 1 g 2,j τ m 18 1 Rev. Mex. Fs. S 58 2 212 188 14

1 F. FONSECA Usng some algebra, we have: 1 f 2 = t2 f + 1 1 t1 + e 1 f 1 2 1 g 2,j τ = t 2 g,j m + 1 1 t1 + e 1 g 1,j 2τ m Proposng the next defntons: 1 1 2 1 1 2τ m 1 1 2τ m N N N f = ρ g,α = B α e,α f = µ α e,α e,β f e,α e,β f 1 = Π α,β = Π 1 α,β e,α g,β = Γ α,β e,α g 1,β = Γ 1 α,β e,α e,β g,γ = Λ α,β,γ e,α e,β g 1,γ = Λ 1 α,β,γ 2 21 Agan, dong some algebra on eqs. 1 and eqs. 2, we obtan: t ρ + ν µ ν = t µ ν + µ Π µ,ν + επ 1 µ,ν = t B ν + µ Γ µ,ν + εγ 1 µ,ν = 22 or Replacng eq. 25 n equatons 23 and 24, we obtan: t Γ µ,ν + Dδ α,µ α t B ν = 26 Applyng the ν operator: Assumng ν t Γ µ,ν + D ν µ t B ν = 27 t δ µ,ν µ Γ µ,ν + Dδ µ,ν t µ µ B ν = 28 t δ µ,ν µ Γ µ,ν + D µ µ B ν = 2 Usng eq. 23 n eq. 3 Whch s the same as µ Γ µ,ν + D µ µ B ν = 3 t B ν + D µ µ B ν = 31 t B ν = +D 2 B ν = D m 2 B ν 32 4. The dstrbuton functon We use the d2q scheme shown n Fg. 1, for drectons e and weghts w on each cell: w = 4 1 1 36 f = f = 1, 2, 3, 4 f = 5, 6, 7, 8 33 t Γ µ,ν + α Λ α,µ,ν + ελ 1 α,µ,ν = If we choose τ m = 1/2, then Γ 1 = and Λ 1 =, we fnd: t B ν + µ Γ µ,ν = 23 t Γ µ,ν + α Λ α,µ,ν = 24 We wll use the tensor Γ as a dagonal matrx, where we defne the dagonal components as the temporal dervatve of the feld B ν, and the D s a factor that balances dmensons n the system: Λ B α,µ,ν = Dδ α,µ t B ν D ν 25 t B ν FIGURE 1. Lattce veloctes of the D2Q scheme. Rev. Mex. Fs. S 58 2 212 188 14

MAGNETIC DIFFUSION USING LATTICE-BOLTZMANN 11 Both, drectons v and weghts w, follow the next relatons: w e,α = 34 w e,α e,β = 1 3 δ α,β 35 w e,α e,β e,γ = 36 The dstrbuton functon that we assume for g eq,α g eq,α = { w Ae,β u β B α +CB α f > w EB α otherwse = s: 37 Where A, C and E are quanttes proportonal to the magnetc feld B α an ts temporal dervatve δ t B α, and should be determned. Usng eq. 25, the tensoral relatons eqs. 34-34 and the defnton of g eq,α, eq.37, we obtan: D m δ α,µ B ν Λ α,µ,ν = D m δ α,µ B ν Λ α,µ,ν = e,αe,β g,γ = C 1 3 δ α,µb ν = C 1 3 δ α,µb ν B ν 3D m = CB ν Usng Assumng 38 e,β g,γ = Γ β,γ Γ β,γ = A α u α 1 3 δ α,β Γ β,γ = A 3 u βb γ 3 Γ β,γ = u β Bγ 4 And equatng eq.3 and eq.4, we have: Fnally A = 3 41 g,α = B α 42 Usng the tensoral relatons eqs. 34-36 and the defnton of g eq,α, eq.37, we fnd: g 5,γ = CB γ + EB 4 γ 5 B γ = CB α + EB 4 α Replacng eq. 38 n eq. 43, we obtan: B γ 5 B γ = 3D m + EB 4 γ EB γ = 4 B γ 15 4 D B γ m 43 44 Collectng the results gven n eq. 38, eq. 41 and eq. 44 and replacng them n eq. 37, we obtan: B α w 3e,β u β B α +3D m f > g eq,α = w 4 B α 15 45 4 D B α m otherwse = we assume the expres- For the dstrbuton functon, f eq son gven n ref. [24], whch s: f eq = 5. Results ρw 1 + 3 e u + 2 e u 2 3 2 u2 1 ρw e 2 B 2 3 B 2 2 f > ρw 1 3 2 u2 otherwse = 46 In the work of Wlmot, [22], the dffuson of a magnetc feld has been consdered. There have been obtaned results for confguratons n one, two and three dmensons. In order to compare and valdate our smulaton results n one and two dmensons we use the analytcal results gven n Wlmot, [22]. 5.1. Results for unform dffusvty Takng the one-dmensonal magnetc dffuson equaton B = D 2 B x 2 47 and the magnetc feld fxed at two ponts ±l, then we get: B l, t = B l, t = B 48 D s assumed constant, eqs. 12 and 32, and the magnetc ntal profle s: { +B, f x > B x, = 4 B, f x < The soluton s x B x, t = B l + 2B π n=1 1 n exp n 2 π 2 Dt l 2 nπx sn l 5 Fgure 2 shows the analytcal results, equaton 5, for dfferent values of τ h. In order to compare the analytcal result gven by equaton 5, wth the result provded by the Lattce-Boltzmann smulaton, sectons III and IV, we purport the two curves n Fg. 3. The one-dmensonal smulaton result was obtaned by projectng the result of a twodmensonal Lattce-Boltzmann smulaton, Fg. 4, on the B x plane. The two curves concde qute well, obtanng a good agreement between the theory and smulaton. Rev. Mex. Fs. S 58 2 212 188 14

12 F. FONSECA 5.2. Results for a magnetc feld wth crcular feld lnes Takng the radal 2D-dmensonal magnetc dffuson equaton B 2 = D B r 2 + 1 B r r B r 2 51 For an ntal magnetc dffuson flux of radus a, we have: B r, = F δ r a 52 FIGURE 2. Analytc Soluton, ref. [22], wth dfferent τ th, whch s defned as τ th =Dt/l 2. The soluton, [22], s: F B r, t = r 4D 2 t 2 + F 4D 2 t 2 a a r 2 +s 2 rs s exp I ds 4Dt 2Dt r s 2 2 + s 2 rs exp I 1 ds 53 4Dt 2Dt Where I and I 1 are the hyperbolc Bessel functon of zero and frst order, respectvely. In the same way we compare the results for the radal analytcal soluton, eq.53, whch s shown n Fg. 5, wth that obtaned from the Lattce-Boltzmann smulaton, Fg. 4. The two results have the same overall behavor. 5.3. Results for a random ntal confguraton FIGURE 3. Analytc one-dmensonal soluton, dash lne, supermposed over the smulaton result, contnuous lne, wth τ th =,225. We performed smulatons on a 2 2 spatal grd wth perodc boundary condtons. We start from an ntal random confguraton, for veloctes and magnetc felds. Fgures 6, 7 and 8 show the magnetc feld confguraton for dfferent tme steps, such as 1, 4 and 4, respectvely. Clearly, we can see the vortcty structure for magnetc feld. FIGURE 4. The two-dmensonal Lattce-Boltzmann smulaton result, for a system sze L = 2 2, wth τ m = 1/2. FIGURE 5. The two-dmensonal analytc result, Eq. 53 for a system sze L = 2 2. Rev. Mex. Fs. S 58 2 212 188 14

MAGNETIC DIFFUSION USING LATTICE-BOLTZMANN 13 FIGURE 6. Magnetc smulaton result for a system of spatal grd of 2 2, for a smulaton tme t = 1. FIGURE 8. Magnetc smulaton result for a system of spatal grd of 2 2, for a smulaton tme t = 4. 6. Conclusons We have mplemented a new strategy usng the lattce- Boltzmann technque to obtan the magneto-hydrodynamc equatons and n partcular the magnetc dffuson equaton usng the anzatz hypothess [21], usng a defnton of the tensor Π, extracted from the Chapman-Enskog expanson. The comparson between the theory and the lattce-boltzmann smulaton agrees qute well for the one and two-dmensonal case. As a future work we wll extend the method to reproduce equaton 1 and to study magnetc reconnecton phenomena. Acknowledgements FIGURE 7. Magnetc smulaton result for a system of spatal grd of 2 2, for a smulaton tme t = 4. Ths work was supported by Unversdad Naconal de Colomba DIB-83355. 1. R. Benz, S. Succ, and Vergassola, Phys. Rep. 222 12 145. 2. X. Chan and H. Chen, Phys. Rev. E. 47 13 1815. 3. F. Fonseca and A. Franco, Mcroelectronc Journal 3 28 1224. 4. S. Chen et al., J. Stat. Phys. 45 12 471. 5. S. Wolfram, J. Stat. Phys. 45 12 471. 6. E. N. Parker, J. Astrophys. Astr. 17 16 147. 7. M. Pattson, K. Premnath, N. Morley, and M. Abdou, Fuson Engneerng and Desgn 83 28 557U572. 8. Y.S. Cha, Chnese Journal of Physcs 43 25 681.62.. N. Kleeorn, D. Moss, I. Rogachevsk1, and D. Sokolo, Astronomy & Astrophyscs 4 23 U18. 1. J.M.V.A. Koelman, Europhys. Lett. 15 11 63U67. 11. X. He1 and L.-S. Luo, Journal of Statstcal Physcs 88 17 27U44. 12. D. Montgomery and G. Doolen, Phys. Lett. A 12 187 22. 13. D. Montgomery and G. Doolen, Complex Systems 1 187 831. 14. S. Succ, M. Vergassola, and R. Benz, Phys. Rev. A 43 11 4521. 15. H. Chen and W.H. Matthaeus, Phys. Rev. Lett. 58 187 1845. 16. H. Chen, W.H. Matthaeus, and L.W. Klen, Phys. Fluds 31 188 143. 17. S. Chen, H. Chen, D. Martnez, and W. Matthaeus, Phys. Rev. Lett. 67 11 3776. 18. G. Fogacca, R. Benz, and F. Romanell, Phys. Rev. E. 54 16 4384. Rev. Mex. Fs. S 58 2 212 188 14

14 F. FONSECA 1. G. Breyanns and D. Valougeorgs, Computers & Fluds 35 26 2. 2. M. Mendoza and J. Muñoz, Computers & Fluds 77 28 26713. 21. D.A. Wolf-Gladrow, Lattce-Gas Cellular Automata and Lattce Boltzmann Models - An Introducton Sprnger, Berln, 2. 22. A. Wlmot-Smth, E. Prest, and G. Horng, Geophyscal and Astrophyscal Flud Dynamcs 25 177. 23. P. Bhatnagar, E. Gross, and M. Krook, Physcal Revew 4 154 511. 24. P.J. Dellar, Journal of Computatonal Physcs 17 22 5. Rev. Mex. Fs. S 58 2 212 188 14