EART 162 PLANETARY INTERIORS This week Observations of

EART 162: PLANETARY INTERIORS

This week • Observations of planetary magnetism • Mechanisms to explain magnetism • Dynamo concepts (self-exciting dynamo, convection dynamo) • Some examples • Some material taken from: – MIT 12. 501 Open. Course. Ware Ch. 3

Solar system fields

Earth’s field in space domain Credit: Gary Glatzmaier • How can we describe the structure of these fields?

Review: magnetic dipole • Two representations: m – Two fictitious magnetic monopoles. – Or, current carrying loop. • Magnetic moment has units of Am 2 • Earth’s field is obviously not purely dipolar. m m = i. A Current carrying loop with area A, current i

Review: ferromagnetism • Permanent magnetization of a crystal below a temperature known as the Curie temperature. • Rocks contain ferromagnetic minerals. • When cooled in the presence of a dynamo (or other) field, they become coherently magnetized in the direction of the field. • Hence, magnetized rocks are evidence for past dynamos on rocky planets.

The magnetic potential • Analogous to gravity potential, you can define the magnetic scalar potential V such that: • (Different from the magnetic vector potential B = curl(A))

Potential and field for a dipole m m = IA

In Obtaining B from V • In spherical coordinates:

Complex arrangements Magnetic quadrupole How can we handle arbitrarily complex sums of pairs of monopoles?

Magnetic spherical harmonics • Laplace’s equation for magnetism: • Solutions are spherical harmonics: • Here g’s and h’s are “Gauss coefficients. ” – Why no degree-0 term? – What does degree-1 represent? – How fast does a dipole potential (field? ) decay with r?

Dipole components MIT Open. Course. Ware

Earth’s field in frequency domain What’s this?

Moon’s field in space domain Earth for comparison • Exercise: sketch the Moon’s power spectrum

Moon’s field in frequency domain • Why no low degree strength? • What happens at degree ~15? • How are such high degree measurements possible on Moon but not Earth? Purucker et al. 2010

Lunar dipole field? • Which body would have a stronger dipole field at the surface, the Moon or the Earth, if they both had similar dynamos?

Sources of fields • Permanently magnetized core? • Magnetized mantle and/or core? • East-west current in the liquid part of the core? – Require continuous generation of the field.

Dynamo concepts • 1. Power source required. • 2. Frozen flux theorem. • 3. Field geometry (alpha and omega effects).

Dynamo concepts: power input • Assume there exists an external field. – Can we generate field from this field? • Examine the Lorentz force on electrons in this system. Lorentz force for positive charges

Dynamo Concepts (2) i B • Now, let’s harvest the current generated. • Then, turn it into a loop. What is the field direction in this loop? • Self exciting dynamo.

Dynamo concepts (3) i B • Once the external field is turned off, what will happen over time?

Dynamo concepts (3) i B • Once the external field is turned off, what will happen over time? – Current gets weaker due to resistance in the disk (Ohmic dissipation). – When current drops, the field drops. – When the field drops, the induced current drops some more, until the field disappears.

Dynamo concepts (4) • Dynamos require power in order to overcome Ohmic dissipation. • Recall the adiabatic heat flux: • How does this compare with that required for convection? • How does this compare with that required for a dynamo?

Convection, the dynamo and the adiabat • For a thermal convection dynamo, the core-mantle boundary heat flux must be greater than the heat flux that would be conducted down an adiabat – the requirement on convection. • Imagine cooling the overlying mantle of an adiabatic outer core. • Venus dynamo paper. T 2 Adiabat for instantly cooled CMB T 1 Adiabat New, disturbed temperature profile To

Sources of dynamo power • Convection driven by thermal gradient or crystallization of dense material at inner core boundary (leaving behind buoyant liquid – see next slide). – Applies to Earth, Jupiter, Saturn, Sun, Uranus, Neptune, probably Ganymede and Mercury – May have strong non-dipolar terms in some dynamos. – Vesta and asteroids? • Mechanical stirring – more exotic dynamo?

Origins of convection/power • d Credit: Catherine Johnson

Lunar paleomagnetic record Modern Earth field (~ 50 μT) Spike due to late heavy bombardment? Impact stirring?

Tidally stirred dynamo? • Differential rotation of the liquid core and mantle results in a frictional power input to the core. – Takes place after roughly 30 Earth radii. • Can this stirring be used to power a dynamo? Dwyer, Nimmo, and Stevenson (2011)

Impact stirred dynamo? (1) • Some lunar craters are associated with magnetic fields. • Could a transient core dynamo be associated with the impact event, and tidal unlocking? Mare Crisium, from Le Bars et al. (2011)

Impact stirred dynamo (2) • Moon is tidally unlocked due to an impact • Torques cause the A-axis to fall back into place, but overshoot, and oscillate until locking (after damping): Slightly slower than synchronous: Empty focus Planet Overshoots, now slightly super synchronous: Empty focus Planet

Frozen flux theorem • Faraday’s law of induction: • Currents are generated to oppose changes in magnetic field (blue line) B Copper disk (bad drawing) B

Field diffusion timescale • Field will “diffuse away” on a timescale that is determined by the conductivity of the conductor. – Like the currents in the disk in the previous slide. • In a sphere of radius a, conductivity σ: (see Stacey, Physics of the Earth) • About 15, 000 years for the Earth.

Dynamo equation: First term looks familiar? Second term? Magnetic diffusivity: Second term is the induction field, similar to curl of “v cross B”. Can derive by eliminating E and j from equations below: Ohm’s law Ampere’s law Faraday’s law (previous slide) u is permeability of free space σ is electrical conductivity

Magnetic Reynolds number Rm = VL/λ • V and L are the characteristic length scales of the fluid flow. • Ratio represents the competition of the velocity/length scales of the problem, and the characteristic timescale for the field to diffuse away. • Rm > 10 could yield a dynamo.

Field geometry: Initial poloidal field a toroidal one • Omega effect • Rotation angular speed is higher at low radius (conserve angular momentum of downwelling). – Shears the poloidal field into a toroidal one. In a frame rotating with the planet:

Omega effect

Toroidal field back into poloidal • Alpha effect • Helical flow, via the Coriolis effect. • Loops coalesce into poloidal field. Coriolis-induced flow patterns: High altitude winds Surface winds Coriolis-induced field lines: First look at side view Similar to atmospheric circulation patterns / cyclone directions

Dynamos on various bodies • • Earth Moon Asteroids Mercury Ganymede Venus and Mars Jupiter and Saturn Uranus and Neptune

Magnetic fields

Mars and the Moon

Neptune and Uranus • Voyager 2 flyby, that’s it! • Tilted dipoles • Strong non-dipolar terms • Pressures too low for liquid metallic H • Convection is ice shell?

Neptune flyby • Octupole term is required

Models • Numerical dynamo models for Uranus and Neptune Stanley and Bloxham (2006)

Summary • Planetary magnetism can be described by spherical harmonics • Dynamos are self-exciting. • Dynamos must be powered by some convective or other mechanical force. • Frozen flux theorem and considerations of geometry of the field can explain the spinaxis aligned geometry of most dynamos.