511th WE-Heraeus-Seminar From the Heliosphere into the Sun –SailingagainsttheWind– Collection of presentations Edited by Hardi Peter ([email protected]) Physikzentrum Bad Honnef, Germany January 31 – February 3, 2012 http://www.mps.mpg.de/meetings/heliocorona/ Where to Go Beyond MHD! (Bad Hanof, Jan, 2012)! Eric Priest (St Andrews)! 1. INTRODUCTION! Eckart Marsch! towering figure in heliospheric physics:! "- pioneering work w. Helios A&B (1974-85)! "- 2 books on solar wind (90,91) with Rainer Schwenn! "- breakthro’s in theory …..! • 90’s: chair ESA S. Phys. Plan. Gp – recommend future mission! • 98 Tenerife Conference “Future of S. Phys.” ! "-- what after SoHO ??? ! • quality science, deep understanding,Recommendations fine! human qualities, ! •(i) delight/pleasureESA choose Solar! Orbiter as next solar/helio mission! To d a y: " "(i) MHD successes! (ii) European contributions to NASA Stereo & Probe Missions! (iii)"""" ESA discuss(ii) w NASAAlternative & other agencies models to &cooordinate assumptions future missions! ! 2. Eqns of Magnetohydrodynamics(MHD)! Unification of Eqns of:! (i) Maxwell! ∇ × B / µ = j + ∂ D / ∂ t, ∇.B = 0, ∇ × E = −∂ B / ∂ t, ∇.D = ρc, where B = µH, D = ε E, E = j / σ . where assume v<<c and so neglect displacement current! and in frame moving with plasma: ! E + v × B = j/σ € € (ii) Fluid Mechanics! dv Motion ρ = − ∇p, dt dρ Continuity + ρ∇.v = 0, dt Perfect gas p = R ρ T, Energy eqn. ............. where d / dt = ∂ / ∂t + v.∇ Add magnetic force! j × B € € € MHD Equations! dv ρ = −∇p + j × B equation of motion! dt Eliminate j, E from Maxwell’s eqns:! ∂B 1 € = ∇ × (v × B) +η∇2B induction equation! η = ∂t µσ magnetic diffusivity! Other equations are:! ∂ρ + ∇ • (ρ v) = 0 mass continuity€ ! € ∂t p = RρT perfect gas Law! € d $ p ' j 2 & ) = + ......... energy equation! dt % ργ ( σ € € 3. APPLICATIONS of MHD! (i) MHD Models of Global Corona! Progressed from early potential models! Wiegelmann – sophisticated nonlinear fff! Mikic et al ! – cf full MHD model with eclipse structures:! Evolution global corona thro’ nonlinear fff! modelled for several solar cycles! "emergence new flux, diffusion, meridional flow, ! differential rotation! (Mackay & Van Ballegooijen) ! • deduce polar flux! • predict locations of most ! € "prominences as twisted flux ropes ! • locations of most ! € """prominence eruptions & cme’s! 3. APPLICATIONS of MHD! (ii) Models of Solar Wind! Progressed from early spherically symmetric ! """""& coronal hole models! Fast Solar Wind! – self-consistent wave turbulence (Cranmer07)! (cf evolution of MHD turbulence,! Marsch90, Tu & Marsch95)! Slow Solar Wind - ? Reconnection! e.g. S-web model: reconnection in ! a complex web of separatrices & QSL’s ! in coronal streamer belt (Titov et al, 11)! Eckart Marsch! Aspects of his approach (Liv. Rev, Marsch06):! (i)"Use key space missions (Helios, SUMER, Ulysses)! (ii) Coronal heating/solar wind acceleration a unity! (iii)"Take us out of MHD comfort zone! Science Highlights: ! • 3D proton distribution functions ! & turbulent spectra, 0.3-1 AU" (Marsch 82,90)! • 2-fluid/kinetic models of solar wind ! "(Tu & Marsch 97, Vocks & Marsch 02)! • Discover/model source fast wind (funnels) ! " (Wilhelm et al 98,00; Tu et al 05)! • Heating/accelerating corona by ! "cyclotron resce/re co n n waves (Marsch82,98; ! "" " Marsch & Tu97,01) Tperp (Araneda08, Bourouaine10)! 4. PLASMA MODELS! – govern struc./t-evol. of plasma:! (i) (Macroscopic) Fluid - motion of plasma element ! " " "- -> multi-fluid " """""""of velocity v(r,t) ! (ii) "(Microscopic" """""""""""") Kinetic – describes distribution! ! " " """"""""""of pcle velys within element! (iii)" Hybrid" """""""""""" (mixture)! ! 4.1. Motion Single-Particle in prescribed E(r,t), B(r,t)! du dr m = e(E + u × B), = u dt dt In uniform B, particle gyrates: ! mu⊥ eB """""helix gyroradius rg" = "", frequency"ω! g = eB m € E × B u = Add constant E, "helix has drift velocity! drift B2 € t-variation/gradients ⇒ other drifts! € € 4.2 Collisionless Kinetic Model [Microscopic]! Distribution function "fs(""r,u,t )! "is density of particles of species s in r,u space at time t! Suppose no collisions, no p’cles created/destroyed &! "interact only€ thro’ E(r,t), B(r,t)! Then Liouville’s theorem:! dfs ∂fs dr ∂fs du ∂fs ≡ + • + • = 0 dt ∂t dt ∂r dt ∂u ∂fs ∂fs es ∂fs or! + u • + (E + u × B) • = 0 ∂t ∂r ms ∂u € NB "– characs. are eqns p’cle motion! (Vlasov equation)! ""- but E,B determined by Maxwell’s eqns:! ∇ × E = −∂B/∂t, ∇ • εE = q, € where! q = ∑es ∫ f s du, j = ∑es ∫ f su du , −2 S S ∇ × B = j+ c ∂E/∂t, ∇ • B = 0, """charge density! set of nonlinear integro-differential eqns! € € 4.3 Add Collision Term to Kinetic Model! ∂fs ∂fs es ∂fs $ ∂fs ' + u • + (E + u × B) • = & ) ∂t ∂r ms ∂u % ∂t ( c Boltzmann eqn! NB:! €• In general involves 2-particle distrib fns" ⇒"" inf. hierarchy! • If no Coulomb collisions but turbce "⇒" effective colln term """"""" ! • Lorentz gas (scatter off heavy static ions) "! [RHS = −νc ( f S − f MAXn )] ⇒ BGK classical transport€ coeffs! (σ, κ, µ) """"""or neoclassical in inhomogeneous B! € • Fokker-Planck eqn (effect€ many long-range collisions)! τ (τ ) ⇒ electrons (ions) relax to€ Maxwellians in few ! ee ii """&! " Te"""⇒ Ti after ! (mi /me )τ ee € € € € € 4.4 Fluid Models (Macroscopic)! • Plasma – better to regard it as at least 2 fluids: electrons, ions! • ? Derive from distrib. fns "fs("""r,u,t) of collisionless plasma! nS = ∫ fS du density! vS = ∫ u fS du/nS bulk velocity! Take moments of Vlasov: ! ∂fs ∂fs es ∂fs € + u • + (E + u × B) • = 0 t r m u n ∂ ∂ s ∂ € × u and ∫ du € ∂ρS (i) n=0! du + ∇ • (ρS vS ) = 0 mass continuity -> v ?! ∫ ∂t s € € dvS (ii) n=1! ∫ mSu du ρ = −∇ • P + j × B + q E Momentum -> Ps? ! dt s s € pressure€ tensor! Ps = ms ∫ (u − vS )(u − vS ) f S (r,u,t)du ""- arises spatial element contains particles w. many different u ‘s! € € (iii)! ∫ mS (u − vS )(u − vS ) du eqn for Ps includes heat flow tensor ! ⇒ rd € """"""""Qs "(3 order)! € € € …… ⇒ infinite hierarchy of coupled eqns! To form set of fluid eqns – truncate hierarchy:! Often assume quasi-neutral(ne=ni), heat flux (Qs=0), isotropic (Ps=psI)! e.g. Consider plasma of two fluids (electrons & hydrogen ions):! € dv Sum of momm eqs: "ρ """"= −∇p + j × B , where "! dt ρ = ρe + ρi, p = pe + pi, v = (ρeve + ρivi )/ρ m ! Electron mom 1 generalised ! € E + v × B = (−∇pe + j × B) +ηj " "(m <<m ):! ene e i € Ohm’s law! el. pressure! Hall term! ηcollisional t a (imp for phen ηmicroturbulent € on ion scales)! • For this MHD form of gend Ohm law, derivation€ is rigorous ! € n • For more general forms (inertial terms), no rigorous deriv ! " " " " " " " "- so status of a “model eqn” "[Balescu,1988]! Several “Fluid” models of a plasma:! • (i) Ideal single-fluid MHD! d % p ( E + v × B = 0, ' * = 0 dt ' γ * when L0>> η / v 0 and L0>> ion scales ( r g!,λi ) & ρ ) • €(ii) In two-fluid collisionless MHD! # 2 & € d p B d # p & B compensates for lack of €collisions: ! % ll ( = 0, % ⊥ ( = 0 if collns weak! (r << λ ), dt % 3 ( dt % ρB( g C $ ρ ' $ ' gyro-radius restricts flow p’cles & ! p ≠ p⊥ ll (double-adiabatic CGL model)! € dρ dp & dv ) € = = 0, ( ρ = −∇p + j × B+ , • €(iii) Friedberg collisionless MHD! € dt dt ' dt *⊥ from guiding-centre theory! E + v × B = 0, ∇ • v = 0. ⊥ • (iv) Interaction plasma & electromagc waves! ""- -> collisionless fluid eqns (exact for cold plasma) ! € • (v) Electron MHD – describes electrons in the whistler regime Le<<L0<<Li! 4.5 Validity of MHD ! • At 1st, MHD no use for collisionless" "not treat particles! • But MHD describes large-scale dynamics with(out) collisions! less e.g. beyond 10R0 solar wind coll ,! """but MHD models describe global "T, v"", ρ, B quite well,! """incl. shocks, stream€ interactions, turbulence, …! • Vlasov Kinetic theory-most complete description colllessplasma! € """but mathy intractable/ rarely used for global models! """instead, often model shock structure/ p’cle accn/ """"evolution of distrib fns by wave-p’cle interactions! Why MHD so successful?! (i) ideal MHD embodies conservn mass, momn, energy! " " """"""""universal in colll/co ll less plasma! (ii) Gyro-motion prevents particles travelling ⊥ B ! ⇒ net effect described by MHD-like eqns! (iii) Often wave-p’cle interactions impede€ p’cle motion along B! (iv) In collless plasma with " " ! E × B u = ⇒ E + v × B = 0 € "E"× B"- drift dominant! drift B2 ⊥ (v)" Adding Lorentz force on individual p’cles !⇒ € F = qE + j × B, € where qE << j × B when udrift << c and qE << ∇p when λDebye << L0 € € ? Important modificns to MHD in Collisionless Plasma! • p not isotropic so need! p and p ⊥ ll • Viscous stress tensor anisotropic & depends on B ! • Ohm’s law generalises! % ( €m e ∂j 1 j E + v × B = 2 ' + ∇ • (vj+ jv)* + (j × B − ∇ • pe ) + e ne & ∂t ) ene σ -- all terms allow B to slip thro’ plasma,! """but el. inertia doesn’t produce dissipation! € """and Hall term frozen to els & = 0 at null! -- el inertia important if! L0 < λe = c /ω pe (electron inertial length) -- Hall term if! L0 < λi / MA = c /(ω pi MA ) 1/ 2 -- electron stress if! L0 < β rgi / MA € 1/ 2 -- collision term if! L0 < (β / MA )(λeλi /λmfp ) • But need€ kinetic theory to describe p’cle accn &! """modifications€ to distrib fn by wave-p’cle interactions! € Compare parameters in Corona & Magnetosphere! Parameter " "Corona (above a.r.) " "Magnetsph(plasma sheet)! n (m-3)" """"1015 " " """""105! T (K) " """"106" """ """ """ "107! B (tesla) " """10-2 " " """ """ "10-8! -1 5 Rgi (m) " """10 " " """ """ "10 ! 1 6 λion"inertial " (m) " ""10 " """""""10 ! 4" """ """ """ " 16 λmpf" (m) " """10 10 ! 8 7 € Global L0"(m)" ""10 " """ """ """ "10 ! -1 4 Lelectron inertia (m) " "10 " " """ """ "10 ! € 1 6 LHall (m) " """10 " """ """ """ "10 ! -3 5 Lelectron stress (m) " "10 " " """ """ "10 ! -7 -7! Lcollision (m)" """10 " " """ """ "10 On global scale ideal MHD valid,! As decrease L, Hall comes in 1st! 5. CONCLUSIONS! • Need range models - different problems! """""""- appreciate different assumptions! • MHD "- better than seems at 1st! " " " "– need understand validity of different versions! " " " "- need go beyond single-fluid routinely! • Kinetic – use insights of MHD –! """"- importance of bc’s/ macroscopic evolution! Eckart – thanks ! - prodding us go beyond MHD! - now time is ripe to do just that!.