Planetary magnetic fields.

Observations, theory, models.

Julien Aubert

1– Intro

• Magnetic field as a signal carrying information about

constitution

dynamics of the interior

thermal history of a planetary system.

• Planetary dynamo modelling aims at

retrieving information by confronting numerical models and ob-

servations

investigating the theoretical difficulties of the dynamo problem.

2– Plan

• The geodynamo: where it all started

• Mars: mysteries of the lost dynamo

• Uranus/Neptune: remote is exotic

• Jupiter/Saturn: zonal flows and dynamos

3– Earth: the most comprehensive dataset

Finlay & Jackson

4– Modelling

• Spherical shell geometry, rotation and magnetic induction are

the 3 essential ingredients. One source of kinetic energy (most

likely convection) is required.

solarviews.com

5– Solving the equations

• Navier Stokes (Boussinesq) equations + Maxwell equations =

magnetohydrodynamic model

6– Success

• explains the large scale space (> 1000 km) and time (>100 yrs)

features of the geodynamo

Model gufm1 for 1990 Glatzmaier-Roberts simulation

7– Invert for innner structure?

Hulot et al.

−100

−50

0

20

−60

−30

0

30

a. Pm = 0 b. Pm = 2

0 45 90 135 180−250

−200

−150

−100

−50

0

50

tangent cylin

der

tangent cylin

der

colatitude (degrees)

zonalflow

vel

oci

tynea

rsu

rface

Pm = 0, Λ = 0Pm = 0.5, Λ = 0.4Pm = 1, Λ = 1.2Pm = 2, Λ = 2.3Pm = 6, Λ = 6.0Pm = 10, Λ = 11.5

c.

• Surface measurement allows to infer inner magnetic structure

and confirm dynamo mechanism.

8– Challenges for future modelling

• Are models wrong or lowpass-filtering reality? Need to have accurate ratio

between viscous, thermal, compositional, magnetic time scales.

local (box) modelglobal model with 2D approximation

60

80

100

120

140

160

180

200

220

240

0 0.1 0.2 0.3 0.4 0.5 0.61e−07

1e−06

1e−05

0.0001

0.001

0.01

0.1

1

10

mag

net

ic R

eyn

old

s n

um

ber

time

magnetic/kinetic energy ratio

magnetic Reynolds numberElsasser number

ener

gy

rat

io,

Els

asse

r nu

mb

er

Stellmach et al.

9– Mars: an ancient, short lived dynamo?

Mars Global Surveyor at 200 km altitude, Purucker et al.

10– Interior of Mars

• Mars has half the Earth’s radius and an iron core which is at least

partially liquid.

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11– Scaling of dynamo constraints: velocity amplitude

• Magnetic Reynolds number Rem = UD/λ has to exceed 100

for a dynamo to work.

• convective velocities U have been scaled with heat flux Q in lab

experiments:

U =

(

αg

ρCpΩD3

)2/5

Q2/5

0.1 1.0 10.0 100.0

1

10

Q

U

• dynamo condition implies Q > 1 GW which is very little above

adiabatic, much less than typical adiabatic heat flux (∼ 1 TW).

• So if there is core convection, a dynamo is extremely likely!

12– Scaling of dynamo constraints: Ohmic losses

• A working dynamo produces a lot of ohmic heat.

• Joule heating power QJ has been scaled in numerical models:

τ =Em

QJ

∝

1

Rem

An asymptotic regime is reached

(independant of the kine-

matic/magnetic diffusivity ratio):

small-scale eddies dissipate a

large-scale field.30 50 100 200 300 500 1000

0.2

0.3

0.5 1

2

3 5

10

20

Rm

τ /1

0−3

Magnetic dissipation time

a

Christensen 2004

• QJ = 1 − 10 TW has to be extracted from the energy sources.

The onset of a dynamo is coincident with a sudden “power

hunger” (cf Karlsruhe experiment).

13– Possibilities?

• Why hasn’t Mars a dynamo today while Earth has one?

• Death of convection? Possible if plate tectonics stopped, if inner

core formation was delayed in comparison to the Earth.

• Possible later turn-on of the dynamo, stopped due to a frozen

core.

• In any case the thermal history of the core needs to be clarified.

QJ is a central unknown for this history.

• Scaling will help to better estimate dissipation from surface ob-

servations.

• Present models have a high potential because they show unex-

pected asymptotic convergence, a hint of the closeness with the

physics of real systems.

14– Remote & exotic: Uranus and Neptune

Uranus Neptune

surface radial fields in gauss, Holme and Bloxham, Voyager II.

• Surface fields have a singular equatorial dipole + multipoles con-

tent.

• two instances mean that we are not “catching” a reversal.

15– Interior of Uranus/Neptune

7

6

5

4

3

2

1

5

4

3

2

1

100

100

P (GPa)

P (GPa)200

200

300

300

T (

10

3 K

)T

(1

03 K

)

Melting exp.

solarviews.com Cavazzoni et al.

• Dynamo action could take place in the middle conducting fluid

ice layer.

16– The Elsasser Number mystery

• Planetary dynamos are believed to work in a

Coriolis force ∼ Lorentz force

equilibrium regime.

• The Elsasser number

Λ =σB2

ρΩ

measures the ratio of the two. This should always be of order

1 and provide a convenient way of scaling planetary magnetic

fields...

• ... but Λ ∼ 1 in the Earth, Λ ∼ 0.01 in Uranus/Neptune. How

can this be? More generally, how does an equatorial dipole dy-

namo work?

17– Equatorial dipole dynamo model

Magnetic tension in the

normal/tangential vector

basis

T =B2

Rc

en +∂

∂s

(

B2

2

)

es

In this representation

field line thickness is

weighted with B2.

• Where thickness varies, or where thick lines are curved, work is

done against magnetic tension.

18– Model predictions

• This model can be perturbed to yield either an axial, or an equa-

torial dipole.

50 52 54 56 58 60 62 64

magnetic diffusion time

103

104

tota

l M

E

0306090

120150180

dip

ole

tilt

(d

eg

)

Ra=6.2e4

Ra=7e4

Ra=6.2e4

a

b

Rayleigh number

is increased is decreased

an equatorialdipole field

is added

Ra=6.2e4

• Dynamo is not efficient when dipole axis and rotation axis

are orthogonal because of field line shear by vortices. Hence

Λe/Λa = 0.05.

19– Constraint on the size of the dynamo region?

no dynamo

axial dipoledynamo

equatorialdipole

dynamo

0.4 0.5 0.6 0.7

aspect ratio

1.0

1.1

1.2

1.3

1.4

1.5

1.6

Ra/R

ac

• Exotic dynamos with equatorial dipoles and multipoles show

when the dynamo shell is small. This could be a constraint for

Uranus and Neptune.

20– Zonal flows in the gas giants

• The surface, and probably the deeper dynamo region of gas gi-

ants is swept by strong zonal flows.

21– The origin of zonal flows

Potential vorticity

q =ω + 2Ω

H

q is materially conserved and

therefore develops regions with

flat profiles. This means nonzero

ω and therefore zonal flow.

• Zonal flows are the result of potential vorticity mixing by turbu-

lence.

• This simply describes exchanges of angular momentum between

the rotating frame and the fluid within.

22– Dynamos with zonal flows

• The magnetic field of Saturn is highly axisymmetric, but Cowling

theorem prevents dynamos producing axisymmetric fields.

• The surface field looks

axisymmetric on surface,

• ...but is not deeper in the

shell where dynamo ac-

tion takes place.

• This dynamo requires

free-slip boundaries and

has a different mecha-

nism from the previous

models.

23– Finally...

• Other magnetic analyses have increased the payload of space

missions, for instance the discovery of liquid water oceans under

the surface of jovian satellites.

• Present numerical modelling has a large potential, especially with

the upcoming planetary magnetic observations.

• The knowledge of the Earth dynamo benefits from this of other

natural dynamos.

• Dynamo processes are intimely connected to surface and mantle

geodynamics.