511th WE-Heraeus-Seminar

From the into the –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. A&B (1974-85) "- 2 books on (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 (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<

€

€ (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 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 <

• (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 ) • €(iii) Friedberg collisionless MHD€ = = 0, ( ρ = −∇p + j × B+ , € 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<

• 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