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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.
solarviews.com
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.
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