lecture 2: dynamo theory · 2018. 10. 24. · homopolar disc dynamo a solid electrically conducting...

Lecture 2: Dynamo theory

Celine Guervilly

School of Mathematics, Statistics and Physics, Newcastle University, UK

Basis of electromagnetism: Maxwell’s equations

Faraday’s law of induction: if a magnetic field B varies with time then an electricfield E is produced.

∇× E = −∂B∂t

Ampere’s law (velocity speed of light)

∇× B = µ0j

where j is the current density and µ0 is the vacuum magnetic permeability.

Gauss’s law (electric monopoles from which electric field originates)

∇ · E =ρ

ε0

with ρ the charge density and ε0 the dielectric constant.

No magnetic monopole (no particle from which magnetic field lines radiate)

∇ · B = 0

Ohm’s law

Relates current density j to electric field E.

In a material at rest, we assume the simple form

j = σE

with σ the electrical conductivity.

In the reference frame moving with the fluid, the electric, magnetic fields andcurrent become

E ′ = E + u× B, B ′ = B, j ′ = j

In the original reference, Ohm’s law is

j = σ(E + u× B)

Homopolar disc dynamo

A solid electrically conducting disk rotates about anaxis.

Uniform magnetic field B0 aligned with the rotationaxis.

The electromotive force q(u× B0) separates thecharges.

A conducting wire is wound around the disc andjoins the rim and the axis: electric current flows inthe wire and across the disc.

Winding is such that the induced magnetic field Breinforces the applied magnetic field B0.

Dynamo: conversion of kinetic energy intomagnetic energy.

If the disc rotation rate exceeds a critical value, B0can be switched off and the dynamo will continueto operate: the dynamo has become self-excited.

Magnetic induction equation

Taking the curl of Ohm’s law,

∇× jσ

= ∇× (E + u× B)

Now using Faraday’s and Ampere’s laws, we get

∂B∂t

= ∇× (u× B − η∇× B),

where u is the fluid velocity, and η = 1/(µ0σ) is the magnetic diffusivity.

If η is constant, then∂B∂t

= ∇× (u× B) + η∇2B

where we used ∇ · B = 0.

Magnetic induction equation and magnetic Reynolds number

For an incompressible fluid (∇ · u = 0) we can re-write the magnetic inductionequation as

DBDt

= (B ·∇)u + η∇2B

where the material derivative is

DDt≡ ∂

∂t+ u ·∇

Induction is produced by the shear of magnetic field lines.

The induction equation is linear in B and the relative importance of each term isindependent of the field strength.

Magnetic Reynolds number:

|∇× (u× B)||η∇2B|

∼ULη

(=

L2/η

L/U=

magnetic diffusion timescaleturnover timescale

)

Rm ≡ ULη

Magnetic diffusion is dominant on small scales (Rm 1) and negligible on largescales (Rm 1).

Frozen flux limit: Rm → ∞Neglecting diffusion on the large scales:

DBDt

= (B ·∇)u.

Magnetic field lines behave as material lines and move with the fluid.The magnetic flux through a material surface is conserved."

SB · dS = const.

This is Alfven’s theorem: Magnetic flux is “frozen” into the fluid.

If the surface expands then the field must become weaker, so that the total flux isunchanged.Alfven theorem no longer holds exactly if there is diffusion: reconnection ispossible and field lines are no longer material curves.

Poloidal-Toroidal decomposition

∇ · B = 0 implies that only two independent scalar fields are needed to specify B.

In spherical geometry:

B = BT + BP, BT = ∇× Tr, BP = ∇×∇× Pr.

T: toroidal component, P: poloidal component.

Br =1r

L2(P),

Bθ =1

sinθ∂T∂φ

+1r∂

∂θ

∂rP∂r

,

Bφ = −∂T∂θ

+1

r sinθ∂

∂φ

∂rP∂r

,

with

L2 =∂

∂rr2 ∂

∂r− r2∇2 = −

1sinθ

∂

∂θsinθ

∂

∂θ−

1sin2 θ

∂2

∂2φ.

Radial component of the induction equation gives the equation for P (L2P = r · B).

Curl of the induction equation gives the equation for T (L2T = r ·∇× B).

Radial component of the diffusion term and its curl can be separated into poloidaland toroidal parts. Only coupling arises from the induction term.

Free decay modes: Rm → 0

Consider a conducting sphere of radius a surrounded by an electrically insulatingregion.

For r < a,∂B∂t

= η∇2B,

We look for solutions of the form B = B0(r,θ,φ)e−σt where σ is the decay rate.

We must solved the problem: (∇2 +σ/η)B0 = 0.

Spherical harmonics expansion:

T =

∞∑l=0

l∑m=0

tml (r)Ym

l (θ,φ), P =

∞∑l=0

l∑m=0

pml (r)Ym

l (θ,φ).

This greatly simplifies the L2 operator: L2(Yml ) = l(l + 1)Ym

l .

For each spherical harmonics, we have to solve:

1r2∂

∂rr2 ∂pl

t∂r

+

(σ

η−

l(l + 1)r2

)pl

t = 0

The solution of this equation is a spherical Bessel function jl.

Free decay toroidal modes

Outside the sphere (r > a): j = 0.For the toroidal component at r > a:

jr =1µ0

r ·∇× B =1µ0

L2T = 0 → T = 0.

The toroidal decay solution for r < a is tml = jl(

√σ/ηr).

T is continuous at r = a so matching the solutions at r = a gives σl = ηx2l /a2, with

xl the lowest zero of jl (for l = 1, x1 = 4.493).

The decay time for thespherical harmonic of degreel is τl = a2/ηx2

l .

In the Earth’s core(a =3500 km, η = 1m2/s), thetoroidal mode l = 1 decays inabout 20000 yrs.

Toroidal modes of larger ldecay faster.

Free decay poloidal modes

For the poloidal component at r > a: jT = −∇2P/µ0 = 0 (jT : toroidal current).At r > a:

pml =

Arl+1 , A = const.

and so,∂pm

l∂r

+l + 1

rpm

l = 0.

The poloidal decay solution for r < a is pml = jl(

√σ/ηr).

P and ∂P/∂r are continuous at r = a so matching the solutions at r = a and usingthe Bessel functions recurrence relations gives σl = ηx2

l−1/a2.

For the dipole l = 1, thelowest zero is x0 = π so thedecay time for the dipole isτ = a2/ηπ2.

In the Earth’s core, the dipoledecays in about 40000 yrs.

Poloidal modes with larger ldecay faster.

Magnetic Reynolds number in the core

The turnover timescale is

τU =RU≈ 106m

5× 10−4mm/s= 60 yr

Rm =τη

τU≈ 40000 yr

60 yr≈ 600

Frozen flux approximation in Earth’s core?

Core flow inversion

Cowling’s anti-dynamo theorem

Cowling (1934)

Axisymmetric magnetic fields cannot be maintained by dynamo action.

Important note:

This theorem disallows axisymmetric B, but not axisymmetric u.

An axisymmetric u can create a non-axisymmetric B. See examples of thePonomarenko dynamo and Dudley & James dynamo.

What about Saturn’s magnetic field then?

Saturn’s magnetic field measured outside of the planet is remarkably axisymmetric...

Saturn’s magnetic field by NASA Goddard

Possible scenario: a stably-stratified layer at the top of the core screens thenon-axisymmetric field produced in the core?

Cowling’s anti-dynamo theorem – Proof

Assuming that the magnetic field and the velocity are axisymmetric (anon-axisymmetric flow always create a non-axisymmetric field):

u = uφeφ + up,

B = Bφeφ + Bp = Bφeφ +∇× (Aeφ),

where Aeφ is the vector potential.The induction equation now becomes

∂A∂t

+1s(up · ∇

)(sA) = η

(∇2 −

1s2

)A,

∂Bφ∂t

+ s(up · ∇

) Bφs

= η

(∇2 −

1s2

)Bφ + sBp · ∇

(uφs

),

where s = r sinθ.

Both equations have an advection term and a diffusion term.

The azimuthal field has a source term: shearing of the poloidal magnetic fieldlines by the gradients of the angular velocity uφ/s.

The poloidal field has no source term so it will just decay. A source term can onlybe provided by non-axisymmetric terms.

The kinematic dynamo problem

Kinematic problem:

The velocity u is a given function of space (and possibly time).

The problem is linear in B.

In the simple case where u is independent of time, we look for solutions

B = B0(x, y, z)ept,

p = σ+ iω with growth rate σ and frequencyω.

∂B∂t

= ∇× (u× B) +1

Rm∇2B,

pB0 = ∇× (u× B0) +1

Rm∇2B0.

We compute the growth rate as a function of the control parameter Rm: positivegrowth rate→ dynamo.

Dynamic (or self-consistent) problem:

u is solved using the Navier-Stokes equation.

B changes u through the Lorentz force: the dynamo stops to grow and theamplitude of the magnetic field saturates.

Kinematic dynamos: disc dynamo

An example of a kinematic dynamo: Ponomarenko dynamo

Ponomarenko (1973)

Conductor fills all space.

No flow outside a cylinder of radius a.

Inside the cylinder: helical flow

u = sΩeφ + Uez,

withΩ and U constant.

Note:

Anti-dynamo theorem: no dynamo can be maintained by a planar flow (i.e. a flowwith only 2 components) (Zeldovich 1957).

So if U = 0: the dynamo fails.

An example of a kinematic dynamo: Ponomarenko dynamo

The control parameters are the magnetic Reynolds number as

Rm =aumax

η, with umax =

√U2 +Ω2a2,

and the pitch angle of the spiral: χ = U/(aΩ).

u depends only on s, so the solution simplifies to

B = B0(s) exp(imφ+ ikz + pt)

Substitue in the induction equation to get the ordinary differential equations forB1(s) (can be solved analytically).

Leads to a dispersion relation f(p, m, k, Rm,χ) = 0.

The smallest value of Rm for which Re(p) = 0 is minimised over m and k for agiven χ→ critical Rm for the onset of dynamo action.

Rmc ≈ 18: moderate Rm that canbe reached in a laboratoryexperiment with liquid sodium.

Surfaces of constantmagnetic field amplitudeshowing the spirallingfield following the flowspiral.

A laboratory Ponomarenko dynamo: Riga (2000)

Gailitis et al. (2000)

Conceptual dynamo mechanisms

Ω-effect: Differential rotation shears poloidal magnetic field lines to generate a toroidalmagnetic field.


Parker loop mechanism: Magnetic field lines are bent and twisted by a cyclonic eddy.This creates an electric current parallel to the original field.Poloidal field can be created out of toroidal field with this mechanism.


Stretch-Twist-Fold:

Loop of flux is stretched twice its length.

Alfven theorem: cross-section of the tube is halved→ |B| doubles.

Twist the loop and fold it.

Large gradients at R leads to larger diffusion there and reconnection.

Each new loop has the same flux as the original tube: total flux has doubled.

Mean field dynamo theory

We partition the velocity and magnetic field into averaged and fluctuating parts:

u = u + u ′, B = B + B ′,

where · represents a time/space/ensemble average and B ′ = u ′ = 0.

We assume that averaging commutes with differentiating: ∇× B = ∇× B forinstance.

We average the induction equation:

∂B∂t

= ∇× (u× B) + η∇2B.

∂B∂t

= ∇× (u× B) +∇×E+ η∇2B.

where E = u ′ × B ′ is the mean e.m.f. (u× B ′ = u ′ × B = 0).

Cowling’s theorem is avoided if E , 0.

To make further progress, we must adopt a mean-field closure. The simplest is theαmodel: E = αB (Steenbeck, Krause & Radler, 1966).

The α-effect can be understood as an averaged Parker loop mechanism.

The α-effect is correlated to the helicity (u ′ · (∇× u ′)) for low-amplitude orhighly fluctuating turbulence.

Mean-field models can be tuned to match the observations...

Validity of mean field dynamo theory

In the αmodel, we assume that B ′ is proportional to B, and so B = 0 implies thatB ′ = 0. Is this always true?

Equation for the fluctuating field:

∂B ′

∂t=∇× (u× B ′) +∇× (u ′ × B) +∇× (u ′ × B ′) +∇× (u ′ × B ′) + η∇2B ′

The α-model assumes that there is no small-scale dynamo: Rm 1 at smallscales.

In the Solar convective zone, Rm is huge on the large scale (∼ 1011)!

The assumptions used for the averaging require a sufficient length scaleseparation between the mean and the fluctuating parts.

Another example of a kinematic dynamo: G.O. Roberts dynamo

Roberts (1972)

u = (cos y, sin x, sin y + cos x)

ω = ∇× u = u

H = u ·ω = |u|2 > 0

u is independent of z, but has 3 components (avoid the planar flow anti-dynamotheorem).

Ponomarenko dynamo has a single roll and B is on the scale of the roll: model fora small-scale dynamo where the lengthscale of B is comparable with thelengthscale of u.

G.O Roberts dynamo: collection of rolls so the field can be coherent across manyrolls: B can be large scale.

Another example of a kinematic dynamo: G.O. Roberts dynamo

Roberts (1972)

B = B0(x, y) exp(pt + ikz)

Anti-dynamo theorem implies that k , 0 (Cowling’s theorem in planar geometry:a field independent of z cannot be maintained by dynamo action).

A laboratory G.O. Roberts dynamo: Karlsruhe (2000)

Stieglitz & Muller (2001)

52 tubes inside a cylinder of diameter1.7 m and height 0.7 m.

The scale separation betweenindividual tubes and the wholeapparatus is about 10.

Produce a large-scale magnetic field.

Good agreement between the dynamoonset in the experiment and kinematiccalculations.

Dynamo experiments with less constrained flows

Von Karman Sodium experiment (Cadarache, France)Monchaux et al. (2007)

2 impellers (radius: 15 cm) positioned 37 cm apart.

50 L of liquid Sodium.

Dynamo for Rm ∼ 50, but only when the impellers are made of soft iron (highmagnetic permability).

α-effect from the spiraling flow produced between the blades of the impellers?(e.g. Laguerre et al., 2008)


Von Karman Sodium experiment (Cadarache, France)

Monchaux et al. (2009) (Θ = (F1 − F2)/(F1 + F2), blue: Bx, red: By and green: Bz)


Rm = RePm where the magnetic Prandtl number is Pm = ν/η (ν = fluidviscosity)Pm ≈ 5× 10−6 for liquid Sodium→ Re > 107 for dynamo action in experiment.Without a precise control of the flow, the effects of the turbulence can be toincrease the critical Rm of the dynamo onset.More turbulent experiment have not yet reached the critical Rm.

Spherical Couette flow (Lathrop et al.,Maryland)Rm ≈ 700, but no dynamo so far.

Self-consistent dynamo simulations

Dimensionless governing equations:

∂u∂t

+ u ·∇u +2

Ekez × u = −∇P +∇2u +

RaPr

Ter + (∇× B)× B,

∂B∂t

= ∇× (u× B) +1

Pm∇2B,

∂T∂t

+ u ·∇T =1Pr∇2T,

∇ · u = 0, ∇ · B = 0.

Coriolis force, Buoyancy driving, Lorentz force.

Ekman number: Ek = ν/(Ωd2),

Rayleigh number: Ra = αg∆Td3/(νκ),

magnetic Prandtl number: Pm = ν/η,

Prandtl number: Pr = ν/κ,

withΩ the rotation rate, ν the viscosity, d the size of the domain, κ the thermalconductivity, and η the magnetic diffusivity.

Self-consistent dynamo simulations – First success in the 90’s

Glatzmaier & Roberts (1995)

Dynamo benchmark

Christensen et al. (2001), Ek = 10−3

a) Br at the outer radius ro, b) ur at r = 0.8ro, c) axisymmetric B, (colour: toroidal field,field lines: poloidal field), d) axisymmetric u (colour: zonal flow, streamlines:

meridional flow).

Generation of a large-scale magnetic field by the convective flows

Aubert (2003) Christensen & Wicht (2015)

Exploration of the parameter space

Schaeffer et al. 2017

Exploration of the parameter space

Christensen & Aubert (2006)

Reversals in dynamo simulations

Christensen & Aubert (2006)fdip= relative dipole strength

Reversals in standard geodynamosimulations occurs whenRo` = U/(Ω`) ≈ 0.1

This implies an imporant role ofinertia at the scale where the dynamooperates.

In the Earth’s core: Ro` ≈ 0.1 for` ≈ 10m.

Rm` = U`/η 1 for ` ≈ 10m.

α-effect could operate at these smallscales but requires well correlatedflows...

Recent dynamo simulations: feedback of the magnetic field

Schaeffer et al. 2017

A =

(EkEm

)1/2


Yadav et al. (2016): Effect of the field on the lengthscale is more pronounced at low Ek


Movie: Schaeffer et al. (2017) Ek = 10−7

lecture 2: dynamo theory · 2018. 10. 24. · homopolar disc dynamo a solid electrically conducting...

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