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Lecture 2 Basic MHD Equations MHD = Magnetohydrodynamics Lecture 2 Basic MHD Equations MHD = Magnetohydrodynamics Maxwell’s Equation: v 1 ∂E v v v ρ : charge density ∇ × B = µj + 2 c ∂t v v v j : current density ∇ ⋅ B = 0 v ε = 8.854×10−12 Fm−1 magnetic permittivity v v ∂B 0 ∇ × E = − µ = 4π ×10−7 Hm−1 magnetic permeability ∂t 0 ∗ 1 v v ρ c = = 3×108 m / s speed of light ∇ ⋅ E = ε µ0ε 0 v v B H = µ If v << c non-relativistic v v D = εE Electric displaceme nt v v ∇× B = µvj kT 1 λ = ( ) 2 << l in plasma d 4πne2 scale v v OHM' s law : vj = σ (E + vv × B) v E in the frame of reference moving with plasma σ electric conductivity ohm m-1 A generalization of Ohm’s law e− p + 3-fluid model for , and neutral atom n ne n p v v v ∇pe me ∂j v vv vv 1 1 v v v nee(E0 + ) = ( + ∇ ⋅ (vj + jv) + + )Bj + j × B nee c ∂t Ωτ ei Ωτ en 2 f Ωτ in v v v v v + []∇Pe × B − ()j × B × B B v v v v E0 = E + v × B eB Ω = electron gyration frequency me 2 v v σ v v σf Ωτ in v v v If ignore electron inertia & σE0 = j + j × B − ( j × B)× B nee neeB pressure gradient: 2 −1 nee me σ = −1 −1 electron conductivity at B = 0 τ ei +τ en v v j // B: σ 0 E0 = j if v v v v σ 3 v v j ⊥ B: σ 3E0 = j + j × B nee σ σ = cowling conductivity 3 τ 1+ f 2 BΩ in nee v v v B× E vj = σ E +σ 0 Solve above equation 1 0 2 B σ σ B σ = 3 σ = 3 σ 1 B 2 1 2 nee 1+ (σ 3 ) nee If gas is fully ionized n =0 σ n e2τ a σ = σ = Ωτ σ σ = σ = e ei 1 2 2 2 ei 1 3 1+ Ω τ ei me v Compare to vj = σE v v There is an additional term normal to B and E , it is called Hall current Induction Equation v v Ohm’s Law: ∇× B = µvj v v v ∂B v v j v j v = −∇×(−vv × B + ) ⇐ E = − vv × B ∂t σ σ v ∂B v v = −∇× E ∂t v v v v v v v v v v v v v = ∇×(vv × B) − ∇×(η∇× B) ⇐ ∇×(∇× B) = ∇(∇ ⋅ B) − ∇ ⋅∇B v ∂B v 2 v 1 = ∇×(vv × B) +η∇ B η = magnetic diffusivity ∂t µσ v v v ∇× B v v j j = E = vv × B + µ σ r v For condition of the sun : E ≈ vv × B 3 A typical active region, B=100G, v=10 m/s, E0=v0 B0=10 V/m If no motion, 1 j = = 8×10−4 Am−2 l =107 m µl σ =103 ohmm−1 −7 −1 E0 = j0 /σ = 8×10 Vm Electric conductivity Fully ionized collision- dominated plasma n e2τ σ = e ei me 3/ 2 6 T Coulomb logarithm lnΛ is Collision time τ ei = 0.266×10 ne ln Λ between 5 and 20 T 3/ 2 σ =1.53×10−2 ohmm−1 ln Λ Diffusion constant η = 5.2×107 (ln Λ)T −3/ 2 m2s−1 typical value in solar chromosphere or corona : η =109T −3/ 2 nk T for turbulent plasma, anomalous collision time :τ * = w −1 B pe w wpe : plasma frequency n e2τ * anomalous conductivity : σ * = e me Condition for this is electron conduction speed >thermal speed Plasma equations ∂ρ continuity equation : + ∇ ⋅(ρvv) = 0 ∂t Dvv ∂vv v v equation of motion : ρ = ρ( + vv ⋅∇vv) = −∇P + vj × B + F Dt ∂t D : derivative of following the motion Dt v vj × B : Lorentz force v v v F = Fg + Fv viscosity ν : coefficient viscosity v 2 v Fv = ρν∇ v T 5/ 2 ρν = 2.21×10−16 kgm−1s−1 ln Λ P = nkT n = np + ne Perfect gas law Energy equation: • Thermal conduction : or Ds ρT = −ζ v v Dt q = −K∇T v De p Dρ K thermal conduction tensor ρ = − = −ζ Dt ρ Dt • Radiation : l = ∇⋅q = −K ∇2T e: internal energy, s: entropy r r r Kr heat diffusivity optical thick, this is valid; Ideal gas: optical thin, l = n n Q(T ) e = c T r e H υ • Heating : K cρ N + 2 c = c + γ = = H = ρε + H r + H w ρ υ m c N υ nuclear + viscous disipatation + waves N: # of degrees of freedom • Energetics : j 2 v v Different terms for ζ E ⋅ j = + vv ⋅ j × B σ A few dimension less parameters l V Reynolds number : R = 0 0 e γ l V Magnetic Reynolds number : R = 0 0 coupling between flow and magnetic fields m η Solar atmosphere: Rm >>1 coupling is strong V0 rp0 Mach number : M = Cs = speed of sound wave Cs ρ0 V0 β0 Alfven mach number : M A = VA = 1/ 2 Alfven speed VA (µρ0 ) 2µp0 Gas plasma β : β = 2 pressure B0 Magnetic β <<1 low β magnetic dominant - corona V0 inertial possby number : R0 = l0Ω Coriolis Rm γ Viscoucity Prandtl number : pm = = Re η diffusion Example. Sunspot 7 3 -6 -1 l0 =10 m v0 =10 m/s Ω = 10 s 4 20 -3 3 T0 =10 k n0 =10 m β0 =10 G 7 4 Rm = 3×10 , Cs = 2×10 m/s (M = 0.05) 5 12 VA = 3×10 m/s Re = 10 R0 = 100 More about Induction Equation v ∂B v 2 = ∇×(vv × B) +η∇ B ∂t l V v 0 0 1st term can be dropped, ∂B 2 •if Rm = <<1 =η∇ B Ædiffusion equation η ∂t 3 2 2 2 diffusion time scale l0 −8 l0 T τ d = =1.9×10 η lnΛ 6 15 -3 7 14 T =10 K η =10 m l0 =10 m τ d =10 S For solar flare τ d =100 to 1000S l0 =100 to 1000 m v •if Rm >>1 frozen-in condition, B Moves with flow -- generally true in solar condition v v D B B Dρ ( ) = ( ⋅∇)vv is similar to fluid motion = −ρ∇ ⋅v prove: ∂t ρ ρ Dt 2 ηt1 2 ηt2 A diffusing current sheet: (a): the variation with time of the magnetic field strength; (b): a sketch of the magnetic field lines at three times Lorentz Force v 2 v 1 v v v v B B F = vj × B = (∇× B)× B = (B ⋅∇) − ∇( ) L µ µ 2µ magnetic tention magnetic pressure Example (1) uniform field v D× B v vj = = 0 F = 0 µ L 2 v v B e v v B e2x (2) B = B e× yv j = 0 × zv j × B = − 0 xv 0 µ µ pressure term v zˆ (3) B = −yxˆ + yˆ vj = µ v xˆ ( vj × B) = − tension term y=0 µ (4) X-type neutral point 2 2 B0 = yxˆ + xyˆ (solution y − x = const) T balances with P, but unstable. (a) (b) The magnetic pressure P and tension T forces due to: (a) A uniform field; (b) a unidirectional field whose strength increases along the x-axis. The magnetic field lines near an X- type neutral point in equilibrium with no current The resultant magnetic force (R) due to a symmetrically curved field (Equation 2.59) Flux tubes and current sheets Flux tubes F = ∫ B ⋅ds =constant B2 P + = constant (dark sunspots) 2µ current sheet : B varies a lot in a very short spatical scale L B current density j = µL if v = 0, current sheet diffuses at a speed of η/l, η : magnetic diffusivity enhanced plasma pressure in the center of the sheet expels material away at a speed of VA ,reconnection A section of a magnetic flux tube (a): a current sheet in the yz plane across which the magnetic field rotates from B1 to B2; (b): a section across a neutral current sheet in the centre of which the magnetic field vanishes and the plasma pressure is P0; ©: the reconnection of magnetic field lines by their passage through a current sheet. The (a) central sheet bifurcates into two (b) (c) pairs of slow shocks. Homework 1. Using reasonable parameters in solar photosphere, chromosphere, corona and in sunspots, calculate the following parameters: Alfven speed, sound speed, and plasma ß. 2. Refer to Equation 2.3, assume a reasonable diffusion constant, and magnetic field strength, compute magnetic topology as a function of time using the induction equation (ignore the velocity term)..
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