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Planetary Magnetic Fields 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..
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