Mean Field Dynamo Theory

Jiacheng Meng Adviser: Prof. Xuening Bai, Prof. Yi Mao 2019.5.17 Outline

1. Introduction

2. Cowling’s theorem and Parker’s model

3. Mean field dynamo theory

4. An example: galactic magnetic field 1. Introduction Magnetic fields are observed in almost all astronomical objects.

Earth Sun

ISM Galaxy Cluster of galaxies ∂ B ⃗ Induction equation = ▽ × ( u ⃗ × B )⃗ − ▽ × η( ▽ × B )⃗ ∂t

η : Resistivity u ⃗ Velocity field 2 Flux Freezing η is constant: −η ▽ B ⃗

Diffusing term

Magnetic Reynolds number 2 U*L* L Rm = T = * η decay η Tdecay for different objects

Earth Star Galactic disk ~105 yr ~1010 yr ~1026 yr

Problem: ? Decay time for earth ? Period of solar magnetic activity ? Take the same time to gain magnetic field …… Dynamo Theory

Convert the kinetic energy into magnetic energy

If there is a small initial magnetic field, dynamo action can enlarge the field and sustain it. 2. Cowling’s Theorem and Parker’s Model History

Joseph Larmar Find some motion Dynamo patterns to achieve 1919 dynamo.

Firstly, we might consider symmetric William Gilbert field…… Earth is permanent magnet 1600 ⃗ ⃗ ⃗ B = B P + B T

Poloidal Field Toroidal Field

Poloidal Field Toroidal Field

Assume axisymmetry

There is no source for poloidal field

Advection term Diffusion term s = rsinθ Cowling’s Theorem (1934)

Anti-dynamo theorem

An axisymmetric magnetic field vanishing at infinity cannot be maintained by dynamo action.

Eventually, ⃗ ⃗ B P → 0 B T → 0

Axial symmetry must be broken.

Non-axisymmetric field is too complex to solve. Proof: A: poloidal field; B: toroidal field

Multiply by s2A, integrate and eliminate the divergence term

< 0

Bp = 0

< 0 Have to find some non- axisymmetric velocity field Parker’s Model (1955) to generate BP from BT.

Consider cyclonic motion for fluid in the earth. Local velocity field u ⃗

along r-direction Toroidal Field Field line is frozen in the fluid.

Poloidal Field

Cyclonic motion

Break the axial symmetry. Generate BP from BT. Achieve a loop between BP and BT.

Weakness: qualitative approach and numerical complication. 3. Mean field Dynamo Theory Steenbeck, Krause, and Radler in 1966: Mean Field Dynamo theory.

Break the axial symmetry by the fluctuation of the field.

At large scale, the mean field can be axisymmetric.

• Support Parker’s model for earth’s field Successful • Explain the solar cycle • Explain the origin of the field in the galactic disk Assume

Average the induction equation:

= 0 = 0

Mean field induction equation Evaluate ε (mean electromotive force)

Diffusion term β = constant

Turbulent resistivity: convection cells mix up the field line and reduce the mean field. Decompose the axisymmetric mean field

A and B for mean magnetic field

αΩ dynamo α2 dynamo α effect

ε = αB¯ j = σε = σαB¯ j is parallel to the mean magnetic field and can generate the perpendicular Similar to Parker’s model magnetic field.

τ α = − ⟨ u ′⃗ ⋅ ( ▽ × u ′⃗ ) ⟩ 3

ω ⃗

Kinetic helicity Ω effect

Toroidal field is generated by stretching due to differential rotation.

Inside of the sphere rotates faster. 4. An Example: Galactic Disk Assume thin disk. Keep z-derivatives and drop Bz. z ∂B ∂ ∂2B Ω ⃗ αΩ dynamo r = − (αB ) + β r B = 0 ∂t ∂z θ ∂z2 in cylindrical ∂B ∂Ω ∂2B 2h coordinate θ = B r + β r ∂t r ∂r ∂z2

z B ⃗ = 0 Assume α = α β constant 0 h

∂A pt Br = − Bθ = B0cos(k0z)e π ∂z k0 = pt 2h A = A0sim(k0z)e 2 k0α0Ω 2 2 k0α0Ω p = − k β ± (p + k β) = 0 π 0 π

α Ωh3 Dynamo number D = 0 D > 12.2, p > 0 β2

Increase Supernovae For interstellar medium, D = 7.6 Superbubbles

1 6Gyr p = −18 −5 200Myr 10 G 10 G Comparable to the observational value 26 Tdecay ∼ 10 yr M51 Magnetic Field (Fletcher et al. 2011) 10−5G Summary

• From Cowling’s theorem, axial symmetry must be broken to achieve dynamo.

• Mean Field theory decompose the field into mean part and fluctuation part. Mean part is still axisymmetric.

• Poloidal and toroidal field regenerate each other. • Using mean field dynamo theory, we can explain the magnetic field in many astrophysical objects.