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?
20 30
0
0
−50 −30
−60 −100 a. Pm = 0 b. Pm =2
50
0
−50
−100 Pm =0, Λ=0 Pm =0.5, Λ=0.4 −150 Pm =1, Λ=1.2 Pm =2, Λ=2.3 Pm =6, Λ=6.0 −200 Pm = 10, Λ=11.5 zonal flow velocity near surface
tangent cylinder −250 tangent cylinder Hulot et al. 0 45 90 135 180 colatitude (degrees) 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) model global model with 2D approximation
240 10 magnetic/kinetic energy ratio 220 Elsasser number 1 magnetic Reynolds number 200 0.1 180 0.01 160 0.001 140 0.0001 120 1e−05 magnetic Reynolds number 100 energy ratio, Elsasser number
80 1e−06
60 1e−07 0 0.1 0.2 0.3 0.4 0.5 0.6 time 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:
10 2/5
αg U = 2/5 1 U 3 Q µρCpΩD ¶
0.1 1.0 10.0 100.0 Q • 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: Magnetic dissipation time 20
10 a 1 Em ∝ τ = 5 QJ Rem 3 −3 2 /10 An asymptotic regime is reached τ 1
(independant of the kine-
0.5 matic/magnetic diffusivity ratio): small-scale eddies dissipate a 0.3 0.2 30 50 100 200 300 500 1000 large-scale field. Rm 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 )
K 5
3 4 10 (
T 3 2 1 Melting exp. 100 200 300 P (GPa)
5
) 4 K
3
10 3 (
T 2 1
100 200 300 P (GPa)
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 σB2 Λ = 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 B2 ∂ B2 T = en + es Rc ∂s µ 2 ¶ 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. an equatorial Rayleigh number dipole field is increased is decreased is added
180 150 a 120 Ra=7e4 90 60 Ra=6.2e4
dipole tilt (deg) 30 Ra=6.2e4 Ra=6.2e4 0
104 b total ME 103
50 52 54 56 58 60 62 64 magnetic diffusion time
• 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?
1.6 axial dipole 1.5 dynamo
1.4 equatorial dipole 1.3
Ra/Rac dynamo
1.2
1.1 no dynamo
1.0 0.4 0.5 0.6 0.7 aspect ratio
• 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 ω + 2Ω q = 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.