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

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

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

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

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

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

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

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

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

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)

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

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 & " $ # 1 0 0 1 % ' & " M $ # $ B 0 0 % ' &' = v " 0 B $ 0 # $ 0 % ' &' λ = + v 0

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 & " $ # 1 0 0 1 % ' & B = B 0 ˆx " M $ # $ B 0 0 % ' &' = v " 0 B $ 0 # $ 0 % ' &' λ = + v 0 A: stretching flow

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 & " $ # 1 0 0 1 % ' & B = B 0 ˆx " M $ # $ B 0 0 % ' &' = v " 0 B $ 0 # $ 0 % ' &' λ = + v 0 A: stretching flow Aspect ratio growth ~ Lyapunov exponent

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) 1 10-2 10 8 1 10-6 10 8 10 10 10-2 Earth (core) 1 10-5 10 6 10-4 10-4 10 2 10 7 10-6

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

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

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 + 1 4 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

B r (r,θ,φ) = l,m (l +1) "g l,m # Y m l (θ,φ) % $ R r & ( ' l+2 Simplifies w/ radius @ surface dipole!g l,m More even distribution over l 2 l+2 "g l,m e -1.26 l Dominated by low l Christensen @ core-mantle boundary

Observation: what B r looks like today <14 @ core-mantle boundary: lower boundary of potential region

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

Evolution of field @ surface for 100 years

Evolution of field @ core-mantle boundary for 100 years

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

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

Model geodynamo http://www.es.ucsc.edu/~glatz/geodynamo.html Glatzmaier & Roberts 1995 Numerical solution of MHD Toroidal structure inside convecting core

Other planets Christensen

Other planets Christensen

What that means Christensen

Stanley

Stanley

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

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

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

Evidence of the dynamo Field is fibril

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

Synoptic plot: unwrapped view built up over time

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

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

r = R r = 2.5 R

Cliver et. al 2013

Stanley

Convection zones 2 R R * R Fully radiative d ce 8000 7000 6000 5000 4000 F0 F5 G0 G5 K0 K5

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

Explaining Activity Levels Individual Variation Variance within class

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

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

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

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