Lesson 3: MHD reconnec.on, MHD currents

Transcription

1 Lesson3:MHDreconnec.on, MHDcurrents AGF 351 Op.calmethodsinauroralphysicsresearch UNIS, AnitaAikio UniversityofOulu Finland Photo:J.Jussila MHDbasics MHD cannot address discrete or single particle effects such as gyro motion and small-scale effects (smaller than the ion gyroradius r i ). MHD equations are valid for much lower frequencies than the plasma frequency ω pe. In Maxwell s equations the displacement current ɛ 0 E/ t has been neglected by assuming that there are no electromagnetic waves propagating at the speed of light. Assuming electrons and single charged ions and neutral plasma n e = n i = n, the total current density j, the total mass density ρ m, effective mass M, and total mass velocity flux ρ m v are given by j = en(v i v e ) (5.17) ρ m = n(m i + m e ) (5.18) M = m i + m e (5.19) ρ m v = n(m i v i + m e v e ) (5.20) 1

2 MHDequa.ons One fluidresis5vemhdequa5ons: ρ m + (ρ m v) = 0 (mass continuity equation) (5.26) ( t ) v t + v v ρ m = j B p (momentum equation) (5.27) E + v B = j σ (generalized Ohm s law) (5.28) d dt (pρ γ m ) = 0 (equation of state) (5.29) B = µ o j (Ampère s law) (5.30) B = 0 (5.31) E = B t (Faraday s law) (5.32) E =0 (Gauss law) (5.33) Whenplasmaconduc5vityσ=ne 2 /m e ν e >, (5.28)=>IdealMHD Induc.onequa.on Let s start from the resistive MHD Ohm s law in eq. (5.28) E + v B = j σ (5.35) and take the curl and use Faraday s law in eq. (5.32) to get B t = (v B j σ ). (5.36) We insert j from Ampère s law in eq. (5.30) and use the identity (see Appendix A) ( B) = ( B) 2 B (5.37) together with eq. (5.31). The result is the induction equation B t = (v B)+ 1 µ 0 σ 2 B. (5.38) convec5ondiffusion 2

3 Convec.on Ifσ > ineq.(5.38),wegettheconvec5onequa5on: B t = (v B) Ifplasmamoves,themagne5cfieldlinesmustfollow,becausetheycan'tdiffuse acrossplasma.itissaidthatthemagne5cisfrozenintheplasma. ByapplyingFaraday slawonthelevhandsideoftheeq.,weimmediatelysee that E = v B whichgivesanequa5onforplasmavelocity v = E B B 2. Thisapproxima5onisvalidinmostpartsofthemagnetosphere,butcanbeviolated e.g. at boundaries and in the reconnec5on (magne5c merging) regions. Even thoughconduc5vityisnotinfiniteintheionosphere,collisionsareinfrequentinthe Fregionandtheequa5onaboveisvalidalsoforplasmaintheFregion. Diffusion Ifplasmaisatrest(v=0),theinduc5onequa5onsimplifiestodiffusionequa5on B t = D m 2 B, wherethediffusioncoefficientd m is D m = 1 µ 0 σ. Thecharacteris5c5meofmagne5cdiffusionisfoundbyreplacingV 2 by1/l B2, wherel B isthecharacteris5cgradientlengthoftheinhomogeneityinthe magne5cfield.thenbcanbesolvedas B o = B 0 exp(±t/τ d ), wherethemagne5cdiffusion5meτ d isgivenby τ d = µ 0 σl 2 B = L 2 B/D m. If σ > (orl B isverylarge),thediffusion5mebecomesverylongandmagne5c fieldisnotabletodiffuseacrossplasma. 3

4 Magne.cmerging If the magnetic induction eq. (5.38) is written in a simple dimensional form B τ = vb L B + B τ d. (5.46) The ratio of the first and second term give the magnetic Reynolds number R m = µ 0 σl B v. (5.47) If R m 1 convection dominates and diffusion can be neglected. For example, the solar wind magnetic Reynolds number is about R m However,ifplasmavelocityvorthegradientscalelengthL B orconduc5vityσ decreases,magne5cfieldstartstodiffuse.thismayoccurwithinaverylimitedregion, e.g.atthesubsolarmagnetopauseorinthemagnetotail. X typeneutralline Magne5cfielddiffusion(R m <1)occursinalimitedregion. Magne5cfieldiszeroonlyatasingleline,attheneutralline(inY direc5on). ConstantE y (reconnec5onelectricfield)insteady state. 4

5 X typeneutralline:realityismorecomplicated Pritchett, 2001 Whenthediffusionregionwidthbecomessmallerthantheioniner5allengthδ i = c/ω pi, ionsstarttodiffusefromthemagne5cfield(toppanel),whereaselectronss5llfollowthe ExB driv(bocompanel). X typeneutralline:realityismorecomplicated Mozer et al., 2002 Ionsdiffusefromthemagne5cfieldintheiondiffusionregion,whereasmagne5cfield remainsfrozenintothemo5onoftheelectronsanddiffuselaterinsidethesmaller electrondiffusionregion.separa5onofionsandelectronssetsupthehallcurrentsystem. 5

6 Plasmaconvec.on Plasmaconvec5oninthe ionosphereisdueto reconnec5onoftheimfand thegeomagne5cfieldatthe daysidemagnetopaseandin themagnetotail. Theresultisthe2 cell convec5onpacerninthe ionosphereduringsouthward IMFcondi5ons. DuringnorthwardIMF, reconnec5onmaytakeplace polewardofthecusp. MHDperpendicularcurrents We start from the momentum equation (5.27) (5.22) and take the cross product with B j = 1 ( ) dv ρ B 2 m dt B + p B. (5.62) Thesecondtermgivesthediamagne:ccurrent(perpendiculartoB) j = B p B 2 andthefirsttermgivesthepolariza:oncurrent,akainer:alcurrent,(also perpendiculartob).thesecondformcomesbyusinge= vxb. j = ρ m B 2 dv dt B <=> j = ρ m E B 2 t 6

7 [ ] MHDFACfromdiamagne.ccurrent Byusingthecondi5onofcurrentcon5nuity, j = j itcanbeshown (Vasyliunas,1970)thatweget yields / ion j eq B = B eq p Beq 2 eq V, (5.65) the so called Vasyliunas equation. Here V is the differential flux tube volume (i.e. the volume of a magnetic flux tube of unit magnetic flux). This volume is given by ion ds V = eq B, (5.66) Diversionofthediamagne5ccurrentatthemagne5cequatorplanetoproduceFAC. MHDFACfromtheiner.alcurrent ( ) ( ) Similarly,itcanbeshownthatfortheiner5alcurrentwegetthefollowingFAC / ion eq j ion B = ρ m dω eq B 2 dt ds wherethefield alignedcomponentofplasmavor5cityisgivenby Ω = v Ω = ˆb v + dawn + + noon + noon dusk dawn dusk midnight Plasmaflowintheeq.plane(leV)andFACsbyvor5city(right):solidlinesatdusk correspondtoupwardfacfromtheionsophereanddashedlinesinthedawnto downwardfac. 7

8 Grouptask3:WhatkindofhorizontalandF A currentsareflowinginthepolarionosphere? Exercise: Add EF and currents in the three panels. Conductivity Electric field (arrows) Pedersen current and FACs (FAC: dot=up, cross=down) Hall current and FACs Grouptask3:WhatkindofhorizontalandF A currentsareflowinginthepolarionosphere? Solu5onontheblackboard. 8