Accepted Manuscript - TARA

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event involves a 13.5% drop in the equatorial dipole moment since 1960. This is roughly 50
times the free decay rate of a fundamental-mode dipole field in the core and it has overwhelmed
the better-known 3.0% drop in the axial component of the dipole moment over the same period
of time, resulting in the 1.2 ◦ northward motion of the NGP.
3 Kinematic theory for dipole moment change
The dipole moment vector ⃗m due to a distributed electric current vector ⃗ J is
⃗m = 1 2
∫
V
⃗r × ⃗ JdV , (1)
where ⃗r is the position vector and V the volume of the conductor. The dipole moment vector
can also be expressed in terms of the magnetic field B ⃗ in the same volume as
⃗m = 3
2µ 0
∫V
⃗BdV , (2)
where µ 0 = 4π · 10 −7 NA −2 is permeability of free space. The temporal rate of change of the
dipole moment vector is therefore
˙⃗m = 3
Using Faraday’s law, (3) can be rewritten as
2µ 0
3 ˙⃗m
∫
= −
V
2µ 0
∫V
∇ × EdV ⃗ ∫
= −
˙⃗BdV . (3)
Accepted Manuscript
S
ˆr × ⃗ EdS , (4)
where ⃗ E is the electric field, ˆr the unit vector normal to the conductor boundary and S the
conductor surface. The electric field can be expressed in terms of the magnetic and velocity
fields in the conductor using Ohm’s law,
⃗E = −⃗u × ⃗ B + λ∇ × ⃗ B , (5)
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where ⃗u is the velocity field and λ the magnetic diffusivity. Substituting (5) into (4) and as-
suming the normal component of the velocity vanishes on approach to the conductor boundary
gives the rate of change of the dipole moment vector in terms of the magnetic and velocity
fields just below the boundary (Moffatt, 1978; Davidson, 2001):
2µ 0
3 ˙⃗m
∫
= ⃗u( ⃗ ∫
B · ˆr)dS − λ ˆr × (∇ × B)dS ⃗ . (6)
S
S
5
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