Representation of the geomagnetic field:
The mantle of the Earth is an insulator, so the potential field is obeying
Laplace's equation

Solution in spherical harmonic expansion for the observed field at the surface
of the Earth,
(low degrees -> CMB field)
high degrees -> crustal field

Coefficients estimated from a least-squares fit on surface magnetic
field observations
-> model of Bloxham and Jackson (1992, J. Geophys. Res., 97, 19537 - 19563)

CMB field from downward continuation of surface field
Limitations linked to the precision in observations and
to the use of truncated spherical harmonics expansions

- in magnetic field
    results of K.A.Whaler, 1986, GJR, 86, pp 563-588.
    -the truncation of the spherical harmonics series followed by
    straightforward least squares inversion leads to solutions that
    are strongly dependent on the particular truncation level
    chosen. Changing the truncation level changes all the
    coefficients, due to spatial aliasing of the higher-order
    -problems arise when trying to test hypotheses using spherical
    harmonics models for the secular variation, rather than original
- in core flow at CMB
  1. results of Hulot et al., 1992, GJI, 108, pp 224-246.
    - no information derived for the components of the flow with degree
    larger than 12;
    - one may truncate the spherical harmonics expansion of the flow to
    degree 12 with only a small impact on the first degrees of the flow;
    - the components of the flow with degree less than 5 are fairly well
    determined from the data;
    - the components of the flow with degree greater than 8 are not
  2. results of Celaya and Wahr, 1996, GJI, 126, pp 447-469.
    - noise may aliase the inversion but damping is effective in removing
    most of the noise;
    - high signal to noise ratio may imply misleading results when
    inversions are not damped;
    - only flows with rapid energy decrease in space and time domains are
    immune to aliasing;
    - the spectrum of the true flow may fall as 1\l² (where l is the
    of the spherical harmonics); this bounds the relative energy

Eletrical conductivity:

large variations in electrical conductivity (sigma) with material phase
(expected to be 1 order of magnitude between sigma at the top and at the bottom
of the mantle, outside D" layer)
D" region exhibits substantial lateral heterogeneities in electrical conductivity
In particular, high-conductivity obstacles

=> deviation of the magnetic strength lines

=> significant electromagnetic coupling between core and mantle

=> significantly affects the fluid flow

ex : strong lateral variations in D" can be responsible for

the reversals of geomagnetic field (see "reversals in geomagnetism")

Dynamo models:

depend on :
-> time scale and space scale
-> ratio of the strengths of toroidal/poloidal magnetic fields in outer core
-> dynamo action concentrated in upper region or extended throughout
the whole outer core:
strong magnetic torque <=> dynamo action concentrated in upper region
-> hypothesis on the small scale turbulent flow
-> hydrodynamic parameter range used to approximate the core flow

Reversals in geomagnetism:

Flow modeling:

Flow can be
- associated with different dynamics
- associated with different time-scales and space-scales
Flow is derived from observed geomagnetic secular variation (radial magnetic field)
Non-uniqueness: many different flows generate the same secular variation
Also non-uniqueness from non-perfect data

Resolving the non-uniqueness by assuming:
  1. frozen-flux approximation -> advection dominates diffusion -> magnetic field frozen into fluid -> flow can be steady, steady in a drifting frame, tangentially geostrophic, purely toroidal
  2. geostrophic balance: first order Taylor approximation -> magnetic force, viscosity and advection << Coriolis -> zonal flow accelerated by pressure field = cylindrical annuli -> co-centric cylinders for velocity field -> deviation from equilibrium => acceleration/deceleration of the cylinders => torsional oscillations or
  3. (piecewise) steady flow Symetric zonal flow -> cylinders => torsional oscillations or
  4. purely toroidal flow: -> usually assumed when top of outer core stably stratified => no upwelling (but also tangentially geostrophic flows can be consistent with stable stratification)
For more details, see "Flow modeling possibilities"

Model of the flow in the core and at the CMB:

The flow contains

a steady poloidal part
a steady toroidal part
a varying poloidal part
a varying toroidal part
a time varying part related to the jerks

The constant (mean steady) part contains co-axial cylinders and

stationary patches. The flow at the CMB can be associated with
up-wellings and down-wellings which cannot be extrapolated down
to the inner core. Only the cylinder part can be extrappolated down
within the core. The time varying part is related to the time variations
of the torsional waves. These are sufficient to explain the LOD
variations and the secular variation of the magentic field.

Hydrodynamical stresses at CMB:

(I) viscous coupling
-> tangential stresses related to viscous forces
-> thin Ekman-Hartman boundary layer
-> negligible for present core viscosity values
(II) electromagnetic coupling
-> tangential stresses related to Lorentz forces and the currents at the CMB
(III) topograhic coupling
-> normal stresses produced by dynamic pressure forces on CMB topography

Electromagnetic coupling:

(1) Poloidal field:
-> electric currents induced by the poloidal field in the mantle
-> can be calculated from the observed surface field
(2) Advective torque:
-> electric currents (only poloidal when the conductivity profile
is radial)
generated by the differences in the electric potential
(poloidal field) at the CMB
induced by the motions at the CMB
-> can be calculated from surface core flow computed using simplified

model as the steady flow approximation

(3) Leakage torque:
-> electric currents induced by the diffusion of the core toroidal field
into the mantle
-> cannot be deduced from observations (surface poloidal field)

Constraining the electromagnetic core-mantle coupling:

J.Wicht and D.Jault, PEPI, 111, 1999, pp161-177.

1.Constraint on the electromagnetic coupling

On the decade to century timescale, changes in the mantle angular momentum associated with the observed changes in the length of day (LOD) can be balanced by changes in core angular momentum (Hide and Dickey, 1991). The secular variation of the geomagentic field and the LOD decadal variation have been indeed shown to be compatible (Jault et al., 1988; Jackson et al., 1993). On the other hand, it is also possible to estimate the core angular momentum changes from core flow models (the tiny changes in core angular momentum give a global measure of time changes in core motions).The agreement gives us some confidence in the assuptions (frozen-flux, tangential geostrophy) that have been used to calculate the core surface motions. Conversely, we can assume this agreement and use it as a constraint on core surface flow models (Holme, 1998).

The torque due to electric currents induced by time varying magnetic field is the poloidal torque and the torque due to electric currents set by electric potential differences is the toroidal torque (also called advective torque).

According to the hypothesis of frozen-flux, the role of diffusion is neglected (leakage torque neglected) and we study rapid changes in the magnetic field at the core surface. Then the lines where Br vanishes at the core surface are material curves and there is no change of magnetic flux through the surfaces comprised within the curves Br = 0.

2.Computation of the electromagnetic torque

Length of day variations are caused by the axial component of the torque acting on the mantle. For calculation of the torque, a simple power law is used to model the conductivity in the mantle (see e.g., Braginsky and Fishman, 1976; Stix and Roberts (1984, Phys. Earth planet. Inter., 36, 49-60). The conductivity decreases for larger radial distances and becomes zero in the main insulating part of the mantle.

There are two boundary conditions : the continuity (thus the vanishing) of the radial current in the insulating part of the mantle and the continuity of the horizontal electric field at the core-mantle boundary. Since there is no way of measuring the toroidal field in the Earth's core, the advective torque is calculated and the diffusive term is neglected.

3.Deriving the vector field UBr

The traditional way to calculate the toroidal and poloidal parts of the flow at the CMB is to solve the magnetic induction equation considering the frozen-flux approximation (no diffusion term) together with another physical approximation such as the tangential geostrophic hypothesis, for getting the CMB flow, and then extract the toroidal part of the flow.

The new method to calculate the two components of the flow is as follows : consider the vector UBr, product of the tangential velocity at the CMB and the poloidal magnetic field at that boundary, and which appears in the induction equation; consider then the curves where UBr is zero; UBr vanishes on every null flux line (no diffusion), which means that the toroidal part has to cancel the poloidal contribution there. This provides with a second condition to solve the induction equation.

Since using this method, the flow can only be solved using the magnetic field and the induction equation along the lines UBr =0, and since there are large areas of the CMB without any null flux line, some regularisation is necessary for solving the global flow map at the CMB : - the first one is based on minimizing the horizontal Laplacian of the two flow components; - the second one minimizes the kinetic energy. This is a kind of damping of the smaller length-scales at the time-scale considered here.