Differential rotation of the inner core

=> variation in seismic wave travel time
rem: an anisotropy axis might be a too simple picture.

Differential rotation from observations:

What do seismic data tell us
systematic differences in the differential travel times at different epochs
=> motion of the anisotropy axis of the inner core by about :
- 1.1 degree/yr eastwards -> Song and Richards, 1996, Nature, 382 , 221-224
- 3.0 degree/yr -> Su et al. , 1996, Science, 274, 1883-1887
- 0.2 degree/yr -> Creager, 1997, Science, 278, 1248-1288
- at the limit of data possibilities, questionable
-> Souriau et al., 1997, GRL, 24, 2103-2106
- 0.2 degree/yr -> Souriau and Poupinet ., 2000
- 0.3 - 1.1 degree/yr -> Song, 2000
- 0.0 degree/yr -> Isse and Nakanishi, 2000
- 0.1 degree/yr -> Vidale et al., 2000

by analysing seismic free oscillations
for estimating differential rotation, free-oscillation "splitting functions"
are good candidates because they are insensitive to local structure.
The inner core differential rotation is essentially zero over the last
20 years implying that the inner core is most likely gravitationnaly
locked to the mantle.
- 0.0 degree/yr -> Laske and Masters, 1999, Nature, 402, 66-69
- essentially zero over the last 20 years
-> Laske, Masters and Gilbert, 2000

<=> westward motion of outer part of core (angular momentum balance)
<=> westward drift of magnetic field

=> disrupts the axisymmetric flow centred around cylinders

=> rotating core undergoes torsional oscillations (periods of several decades)
large electromagnetic torque on inner core the cylinder tangential to the inner core rotates westward inducing eastward differential rotation of the inner core => consistent with the eastward inner core superrotation (Pais and Hulot, 2000, PEPI, 118, 291-316). misalignement of inner core and mantle => gravitational restoring torque on inner core => inner core remains aligned with mantle = inner core locked to mantle => inconsistency with inner core differential rotation (Buffett, 1996, GRL, 23, (17), 2279-2282 and (25), 3803-3806) but can be accommodated by allowing the inner core shape to adjust as it rotates to the gravitational potential imposed by the overlying mantle (Buffett, 1997, Nature, 388, 571-573) Inner core differential rotation and length of day (LOD) Buffett and Creager, 1999, GRL, 26,10,1509-1512. inner core rotates faster than mantle => large gravitational torques; but LOD data : at least one order of magnitude lower than predicted with that torque => torque must be opposed by another torque; at CMB, electromagnetic torque due to a westward flow at the top of the core is compatible and consistent with the required torque. gravitational forces on the aspherical inner core may be strong enough to lock the rotation of the inner core to that of the mantle. However, relative motion is permitted if the boundary topography adjusts during rotation to remain nearly fixed with respect to the mantle. In this model, gravitational forces on the inner core are included and the relaxation of the inner core shape is allowed assuming an effective viscosity 5 10¹⁶ Pa s for the inner core. The predicted rotation rate of the inner core relative to the mantle is then 0.02 degree/yr. Buffet, 2000
differential rotation of the inner core
=> consistent with the eastward inner core superrotation
(Pais and Hulot, 2000, PEPI, 118, 291-316). misalignement of inner core and mantle => gravitational restoring torque on inner core => inner core remains aligned with mantle = inner core locked to mantle => inconsistency with inner core differential rotation (Buffett, 1996, GRL, 23, (17), 2279-2282 and (25), 3803-3806) but can be accommodated by allowing the inner core shape to adjust as it rotates to the gravitational potential imposed by the overlying mantle (Buffett, 1997, Nature, 388, 571-573)
=> gravitational restoring torque on inner core
=> inner core remains aligned with mantle = inner core locked to mantle
=> inconsistency with inner core differential rotation
(Buffett, 1996, GRL, 23, (17), 2279-2282 and (25), 3803-3806)
but can be accommodated by allowing the inner core shape to adjust as it rotates
to the gravitational potential imposed by the overlying mantle
(Buffett, 1997, Nature, 388, 571-573)
Buffett and Creager, 1999, GRL, 26,10,1509-1512.
inner core rotates faster than mantle
=> large gravitational torques;
but LOD data : at least one order of magnitude lower than predicted with
that torque
=> torque must be opposed by another torque;
at CMB, electromagnetic torque due to a westward flow
at the top of the core is compatible and consistent with the required
torque.
gravitational forces on the aspherical inner core may be strong enough
to lock the rotation of the inner core to that of the mantle. However,
relative motion is permitted if the boundary topography adjusts during
rotation to remain nearly fixed with respect to the mantle. In this model,
gravitational forces on the inner core are included and the relaxation
of the inner core shape is allowed assuming an effective viscosity
5 10¹⁶ Pa s for the inner core. The predicted rotation rate of the
inner core relative to the mantle is then 0.02 degree/yr.
Buffet, 2000

Convection + dynamo action in outer core
=> excess of temperature inside tangent cylinder surrounding inner core
=> temperature difference with the liquid outside this cylinder
=> prograde thermal wind and strong azimuthal magnetic field inside the cylinder
=> electromagnetic torque at ICB -> consistency with the differential rotation
(a temperature difference of 0.001 K is sufficient)
(Aurnou et al., 1996, GRL, 23, 3401-3404)

The combination of toroidal and poloidal magnetic fields exerts electromagnetic
torques on the solid inner core, resulting in its anomalous spin. Aurnau et al.
(1988) have shown that there is a close relation between the inner core rotation
and the generation of the toroidal magnetic field (which is induced by the
poloidal magnetic field). If the inner core is believed to be rotating in the prograde sense,
columnar flow following geostrophic contours outside the tangent cylinder of the
inner core may be eliminated (induces retrograde inner core rotation). However,
thermal wind azimuthal flow resulting from a positive temperature anomaly whinin
the inner core is compatible with a prograde inner core rotation.