EART162: 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 OpenCourseWare 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 Am2
• Earth’s field is obviously
not purely dipolar.
m
m = iA
Current carrying loop with area A, current i
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 OpenCourseWare
Earth’s field in frequency domain
What’s this?
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
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
• 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: q  k  dT 
 dz  adiabatic
• 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.
T2
Adiabat for
instantly
cooled CMB
T1
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)
t = mos a 2 / p 2
• About 15,000 years for the Earth.
Initial poloidal field a toroidal one
• Omega effect
• Rotation angular
speed is higher at low
radius (conserve
angular momentum
of upwelling).
– 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
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.