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Earth Rotation changes due to the Core Dynamics

Whereas the variation of the LOD at short timescale (seasonnal, annual) is
mainly due to the atmosphere, the variation at decadal timescale seems
mainly due to the liquid core.

I. Decadal changes in Earth rotation and role of the liquid core

On decadal time scales, the variations in the LOD have been shown
to be correlated with the secular variation of the magnetic field.
This suggests that the core plays an important role in these LOD changes.
These LOD variations are thus believed to be associated with the changes
in the core angular momentum (CAM). See II and III of this web page.
There, we show that different dynamics for the core with different CMB
velocity fields can lead to CAM explaining the observed scaled decadal
variation of the LOD.

II. Short timescale changes in Earth rotation

A close correlation between the variation of the length of day (LOD)
and the total atmospheric angular momentum (AAM) has been shown
on annual and subannual time scales.

On subannual time scales down to periods of about 30 days, the phase of
the LOD variations leads that of the AAM. For sub-seasonal time scales,
the discrepancy can be due to either core-mantle coupling or to the action
of the oceans (perhaps also due to hydrology). The mechanisms by which
the core might cause a phase lead of the LOD variations is different from
that by which the oceans might do so for at least two reasons.
The first one is due to the fact that the moment of inertia of the ocean is
much smaller than that of the core. The second is a more geographic one.
The ocean is between the atmosphere and the mantle whereas the core
lies underneath the mantle. The core can only react to atmospheric forcing
of LOD variations through its response to mantle motion, while the ocean
may directly interfere with the transmission of angular momentum between
atmosphere and mantle. A simple three layer model of the Earth
(atmosphere, mantle and core) treating the core as a rotating body coupled
to the mantle can cause a phase lead of the LOD variations of the correct
magnitude (see Zatman and Bloxham, 1997, GRL, 24, (14), 1799-1802).

Correlation studies :
- core decoupled for period as 9 days, coupled for semi-annual
- core decoupled for period less than 30 days, coupled at semi-annual

Phase studies :
period > 30 days : LOD phase lead with respect to AAM phase
=> significant core-mantle coupling
=> can be explained by a simple model of core-mantle coupling (with
core = rotating solid body coupled to the mantle).

Core dynamics :
axisymmetric inertial waves + core behaves as global rigid body (global motion);
the axisymmetric inertial waves may become important at short timescales
-> axial core-mantle coupling more effective;
the assumption of core as a global rigid body breaks down at short timescales
=> core might be a possible explanation of observed phase difference between
LOD and AAM;
(Zatman and Bloxham, 1997, GRL,24, 14, 1799-1802).

III. How to compute the Earth rotation changes due to the core?

1. computation of flow at CMB
   
   -> surface magnetic field
   
   -> at CMB, the induction equation for the magnetic field relates the time
   
   variation of the magnetic field to the difffusion and the advection terms
   
   -> for time scales of a few decades, the diffusion (the dissipation part of the
   
   induction equation) can be considered as negligible with respect to the
   
   advection term - Frozen flux approximation (Alfvèn)
   -> at the CMB, for the frozen flux approximation, the induction equation links
   
   the radial magnetic field with the tangential flux
   
   = 2 unknowns, 1 equation
   
   => need for approximation
   
   - from physical assumptions, the flow at the CMB can be obtained
   

2. computation of other torques on mantle (see point IV) or core angular
   momentum (CAM) (see point V)
   

IV. Torques at the CMB

• topographic torque: effect of pressure on boundary topography
  -> possibly explains observed lod changes at decadal time-scale
  The topographic torque can be either weak or large depending on
  the topography and the method of calculation
  If it is large, the topographic torque (pressure torque arising from core
  
  flow against bumps in the CMB) can explain the observed decadal
  
  variations of the lod.

  Note :The pressure torque is proportional to the square of the boundary
  amplitude; a boundary topography of the order of 3 km in
  magnitude would then be insufficient to cause the observed
  decadal LOD variation. However, the pressure torque may be
  responsible for the excitation of the observed 30-year Markowitz
  wobble in the polar motion.

• viscous torque: effect of viscosity in core (viscous drag between core flow and CMB)
  -> too small to explain the observed decadal changes in Earth's rotation with
  present day estimates of core viscosity

• electromagnetic torque: (see geomagnetism)
  Magnetic field at CMB will induce electric currents in an electrically
  
  conducting mantle -> poloidal torque and advective torque.
  
  Also a leakage torque arises from currents in mantle caused by toroidal
  field that diffuses from core to mantle. This torque can not be calculated
  from observations
  ~ due to non-uniqueness, consider inverse problem: look for flows that
  - are consistent with observed secular variations of magnetic field
  - explain decadal lod variations by electromagnetic coupling alone
  -> can be found for reasonable mantle conductivity profiles (10⁸ S)
  For moderate mantle conductance, electromagnetic coupling can explain
  
  the observed decadal variations of the lod.
  

• gravitational torque:
  - effect of non-radial gravity in the core arising from mantle heterogeneity
  convection models using seismic tomography data in the mantle
  must be used to compute that torque
  -> unlikely to be important
  -indirect interaction through inner core
  see point VI

Note:

- Depending on the choice of parameters, the electromagnetic torque, the topographic
torque and the gravitational torque can be large and can explain, each one separately,
the total LOD variation.

- Fluctuations in the length of day (LOD) at decade periods can be attributed to
exchanges of angular momentum between the core and the mantle. It is assumed
that the changes in angular momentum in the fluid are carried by simple flows of
which the characteristics are described here.

V. Core angular momentum computation

principles:

-The radial magnetic field at CMB and the flow at CMB can be


obtained from surface magnetic field data and core dynamics approximations.


-The core angular momentum, the flow in the whole core must be computed


in order to obtain results concerning the core angular momentum.


-Simple motions mainly due to the buoyancy and Coriolis forces are usually


considered within the liquid core. These motions are uniform on cylinders


coaxials with the rotation axis. So-called torsional oscillations relate the


amplitudes of the zonal velocity field on the different cylinders rotating


with the same frequency. Usually two torsional oscillations of frequencies


around 60 years are required to explain the LOD variations.


results concerning the core angular momentum and its ability to fit the scaled LOD

- toroidal flux

- magnetostrophic, geostrophic + torsional oscillations

- steady flow in a drifting frame

- very good fit to scaled LOD

=> many different assumptions on the flow give rise to a very good correlation


between the scaled decadal variation of the LOD and the core angular momentum


=> LOD provides no constraint on the core dynamics

Rem.: only problem for the fit before 1900 due to low quality of magnetic


data (poor distribution, fewer data)

VI. Role of the inner core in LOD variation

-The core toroidal motions together with the important axial magnetic field at Inner Core
Boundary (ICB) are able to induce very important electromagnetic torque at that boundary.
This torque is able to induce a differential rotation of the inner core. But there is also an
important restoring gravitational coupling between the mantle and the inner core.
This torque locks the inner core in the mantle. This strong coupling might be sufficient to explain
the LOD variations as shown by Buffett, showing the important role of the inner core in the coupling
mechanisms.
-For a reconciliation with the inner core differential rotation possibly observed by the seismologists,
see section on the inner core differential rotation on this website.

VII. Present situation / Conclusion

1. robust correlation between decadal variations of the lod and modelled
   core angular momentum;
2. various coupling mechanisms; a mixture of electromagnetic, topographic
   and gravitational most likely;
3. no constraint on core dynamics from studies of lod variations but study of dynamics
   provides the best chance of progress.