Q: Why do the Sun and planets have magnetic fields?

Transcription

1 Q: Why do the Sun and planets have magnetic fields? Dana Longcope Montana State University w/ liberal borrowing from Bagenal, Stanley, Christensen, Schrijver, Charbonneau,

2 Q: Why do the Sun and planets have magnetic fields? A: They all have dynamos

3 Plasma (stars) Liquid iron (terrestrial planets) Metallic hydrogen (gas giants) Ionized water (ice giants) Complex turbulent flows Rotation: breaks mirror-symmetry not required, but needed for largescale, organized fields Figure of merit: Magnetic Reynold s # Rm = velocity size conductivity From Stanley 2013

4 A Toy w/ all ingredients conducting fluid multi-part conductor 1 V disk = vbdr = ( Ωr)Bdr = 0 0 2π ΩΦ disk v E E = v B No magnets No batteries No (net) charge lack of mirror symmetry complex motions differential motion of parts

5 A Toy w/ all ingredients conducting fluid multi-part conductor lack of mirror symmetry complex motions differential motion of parts 1 V disk = vbdr = ( Ωr)Bdr = 0 0 2π ΩΦ disk IR = Ω 2π Φ L di disk dt = Ω 2π M I L di w,d dt di dt = motional EMF # % $ Growth: Growth: Ω 2π M w,d L back EMF R & (I = γi L ' generation dissipation γ > 0 Ω > 2π v Rm = µ 0 σ v > 2π I(t) = I 0 e γt R M w,d > 2π 1/σ µ 0 = 2π µ 0 σ 2

6 Toy dynamo amplifies fields of either sign: two attracting states v v I(t) = I 0 e γt I 0 > 0 I 0 < 0 Reverse velocity AND reflect in mirror è still amplifies Do one and not the other è no amplification

7 Will I grow forever? Torque on disk carrying current: τ = Fr dr = IBr dr = 0 1 2π IΦ disk 0 Power needed to turn disk: = M w,d 2π I 2 F = I B F I P Ω = Ωτ = Ω 2π M w,d I 2 Subtracting Ohmic losses W W cr I # P Ω I 2 R = % $ = d dt Ω 2π M R & w,d (I 2 = γli 2 ' 1 ( 2 LI 2 ) Stored in energy of B

8 Fluid dynamics Reality: Conducting fluid MHD Faraday s + Ohm s laws ρ t = (ρv) $ ρc v & % T t + v T Effect of B on conducting fluid ρ v t +ρ(v )v = - p + ρg +! σ + J B Lorentz force ' ) = 2 3 ( T v + v : σ! + Q " + 1 σ J 2 Ohmic heat B t = E = % v B - 1 & ' σ J ( ) * J = 1 µ 0 B $ 1 % & σ J ' ( ) = $ 1 ' & B) = η 2 B % σµ 0 ( η = 1 µ 0 σ magnetic diffusivity

9 Reality: Conducting fluid MHD Fluid dynamics ρ t = (ρv) $ ρc v & % T t + v T If B is weak: kinematic equations ρ v t +ρ(v )v = - p + ρg +! σ + J B ' ) = 2 3 ( T v + v : σ! + Q " + 1 σ J 2 Traditional (neutral) fluid solve first Faraday s + Ohm s laws B t + (v )B = (B )v B( v)+η 2 B Linear equation for B(x,t) solve w/ known v(x,t)

10 Dynamo action in MHD DB Dt B [ v I( v)] = B M B t + (v )B = (B )v B( v)+η 2 B If M has a positive eigenvalue l > 0 B can grow exponentially: DYNAMO ACTION λ η 2 γ ~ v η = v " 2 1 η % $ ' # v & B à -B : same e-vector è same l Reverse velocity AND reflect in mirror è l à l Do one and not the other è l à -l Growth: Rm = v η = µ 0 vσ > 1

11 Q: What kind of flow has l > 0? y v(x, y) = v 0 x ( xˆx yŷ) " M = v x / x v x / y $ $ v y / x v y / y # % ' ' = v 0 & " $ # % ' & " M $ # $ B 0 0 % ' &' = v " 0 B $ 0 # $ 0 % ' &' λ = + v 0

12 Q: What kind of flow has l > 0? y v(x, y) = v 0 x ( xˆx yŷ) " M = v x / x v x / y $ $ v y / x v y / y # % ' ' = v 0 & " $ # % ' & B = B 0 ˆx " M $ # $ B 0 0 % ' &' = v " 0 B $ 0 # $ 0 % ' &' λ = + v 0 A: stretching flow

13 Q: What kind of flow has l > 0? y v(x, y) = v 0 x ( xˆx yŷ) " M = v x / x v x / y $ $ v y / x v y / y # % ' ' = v 0 & " $ # % ' & B = B 0 ˆx " M $ # $ B 0 0 % ' &' = v " 0 B $ 0 # $ 0 % ' &' λ = + v 0 A: stretching flow Aspect ratio growth ~ Lyapunov exponent

14 Q: What kind of flow has l > 0? Turbulent flows have pos. Lyapunov exponent: l > 0 tend to stretch balls into strands tend to amplify fields Conditions for turbulence: driving: e.g. Rayleigh-Taylor instability viscosity fights driving must be small Rotation can organize turbulence: align stretching direction à azimuthal (toroidal) known as W-effect must be significant w.r.t. fluid motion Re = v υ >>1 Ro = v Ω <<1 h [m 2 /s] u [m 2 /s] L [m] v [m/s] W [rad/s] Rm Re Ro Sun (CZ) Earth (core)

15 Plasma (stars) Liquid iron (terrestrial planets) Metallic hydrogen (gas giants) Ionized water (ice giants) Complex turbulent flows Rotation: breaks mirror-symmetry not required, but needed for largescale, organized fields Figure of merit: Magnetic Reynold s # Rm = velocity size conductivity From Stanley 2013

16 How this works for Earth Non-conducting mantle B = µ 0 J = 0 B = χ B = 2 χ = 0 χ(r,θ,φ) =,m B r (r,θ,φ) = χ r = g,m,m # Y m (θ,φ) % $ ( +1) g,m R r & ( ' +1 % Y m (θ,φ) ' & simplifies w/ increasing r R r ( * ) +2 Christensen Turbulent conducting fluid: DYNAMO B = µ 0 J 0 Complex flows complex field

17 A Spherical Harmonic Refresher l = 1 dipole Y 1 0 (θ,ϕ) ~ cosθ Y 1 ±1 (θ,ϕ) ~ sinθ e ±iϕ!g 1,±1 ~ g 1,1 ih 1,1 (g 1,0, g 1,1, h 1,1 )! µ dipole moment l = 2 quadrupole Y 2 0 (θ,ϕ) ~ 3 4 cos2θ Y 2 ±2 (θ,ϕ) ~ sinθ e ±2iϕ (g 2,0, g 2,1, h 2,1, g 2,2, h 2,2 )! Q quadrupole tensor higher l: Y l 0 (θ,ϕ) ~ coslθ +" Y l ±l (θ,ϕ) ~ sinθ e ±liϕ Finer scale: l periods around circle More components: 2l +1 real coefficients

18 B r (r,θ,φ) = l,m (l +1) "g l,m # Y m l (θ,φ) % $ R r & ( ' l+2 Simplifies w/ surface dipole!g l,m More even distribution over l 2 l+2 "g l,m e l Dominated by low l core-mantle boundary

19 Observation: what B r looks like today core-mantle boundary: lower boundary of potential region

20 Simplifies w/ increasing r B r (r,θ,φ) = χ r =,m ( +1) g,m % Y m (θ,φ) ' & R r ( * ) +2

21 Evolution of surface for 100 years

22 Evolution of core-mantle boundary for 100 years

23 Use evolution to infer fluid velocity v ~ 3 X 10-4 m/s è Dx = 1,000 km è in 100 years Subsonic flow, Ignore stratification Christensen v = 0 Ignore diffusion B t + (v )B = (B )v B( v)+η 2 B known B r t + (vb r ) = 0

24 Longer-term evolution Christensen normal reversed Like toy dynamo, Earth works in 2 modes. Flips between them seemingly at random

25 Model geodynamo Glatzmaier & Roberts 1995 Numerical solution of MHD Toroidal structure inside convecting core

26 Other planets Christensen

27 Other planets Christensen

28 What that means Christensen

29 Stanley

30 Stanley

31 Level of saturation from Christensen R c = 0.4 R R c = 0.6 R B saturates (exp growth ends) when driving power thermal conduction q 0 balances Ohmic dissipation dimensionless factors

32 How it works in the Sun Entire Star: H/He plasma Convection Zone (CZ) Outer 200,000 km Turbulence: Re = Thermally driven Good conductor Rm = 10 8 Rotation effective Ro = 10-2 Corona conductive but tenuous: J smaller (~0?)

33 Evidence of the dynamo Magnetic field where there are sunspots Field outside sunspots and elsewhere too

34 Evidence of the dynamo Field is fibril

35 Evidence of the dynamo Field orientation: mostly toroidal + (-) + (-) (-) +

36 Synoptic plot: unwrapped view built up over time

37 Rm = 10 8 Ro = 10-2 Dynamo comparison: Sun vs. Earth Rm = 10 2 Ro = 10-6

38 Assume corona has small (negligible) current: B = µ 0 J = 0 B = χ,m B = 2 χ = 0 χ(r,θ,φ) = g,m # Y m (θ,φ) % $ R r & ( ' +1

39 r = R r = 2.5 R

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45 Cliver et. al 2013

46 Stanley

47 Convection zones 2 R R * R Fully radiative d ce F0 F5 G0 G5 K0 K5

48 Other stars Evidence of magnetic activity Activity on main sequence: types F à M B-V > 0.4 (From Linsky 1985)

49 Explaining Activity Levels Individual Variation Variance within class

50 The Dynamo Number Parker s Dynamo # N D = a dyn W' d 2 h 4 Dynamo is linear instability for N D > N crit Dynamo a-effect: a dyn ( Ñ v) Wd º t v ~ h = h turb ~ d t 2 c dw W ' = ~ dr W d N D ~ ( W t c ) 2 ~ ( P / t ) obs c - 2 = Ro -2

51 Activity vs. Rossby Number 41 Local stars P obs from S(t) young stars old stars (from Noyes et al. 1984)

52 Activity vs. Rossby Number (from Patten and Simon 1996) Stars in open cluster 2391 (30My old) R X from ROSAT observations Rotation periods P obs from optical photometry N R = Ro = P obs /t c

53 Summary Magnetic fields all from dynamos Conducting fluid Complex motions w/ enough umph Create complex fields Fields evolve in time reverse occasionally Differences from different parameters: Rm, Re, Ro