Representation of the geomagnetic field:
=> 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
Reversals in geomagnetism:
Model of the flow in the core and at the CMB:
The flow contains
The constant (mean steady) part contains co-axial cylinders and
Hydrodynamical stresses at CMB:
model as the steady flow approximation
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.