Accepted Manuscript - TARA

where g1 1 and h1 1 are the equatorial dipole Gauss coefficients. From the x-component of (6), the
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rate of change of m x is
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m˙
x = 3 [A x + D rx + D tx ]dS (16)
2µ 0
∫S
where the contribution by tangential advection is
A x = (u θ cosθ cosφ − u φ sin φ)B r (17)
Similarly, the y-component of (6) yields a corresponding expression for the rate of change of
m y ,
m˙
y = 3
with the advective contribution being
2µ 0
∫S
In terms of m x and m y , the longitude of the dipole is
and the azimuthal angular velocity of the dipole axis is
[A y + D ry + D ty ]dS (18)
A y = (u θ cosθ sin φ + u φ cosφ)B r (19)
φ 0 = tan −1 ( m y
) = tan −1 ( h1 1
) (20)
m x
Accepted Manuscript
g 1 1
φ˙
0 = m˙
ym x − m˙
x m y h˙
1
=
1g1 1 − g ˙ 1h 1 1 1
m x2 + m
2 y g 12 1 + h 1 2
. (21)
1
We now define the equatorial component of the dipole moment as
m e = 4πa3 √ ∫
g 12 1 + h 1 12 = ρ e dS (22)
µ 0 S
in terms of the equatorial dipole moment density ρ e on the CMB,
ρ e = 3r c
2µ 0
B r sin θ cosφ ′ , (23)
where φ ′ = φ−φ 0 is the longitude relative to the magnetic pole and φ 0 (t) is the time-dependent
longitude of the magnetic pole. The equatorial component of (6) yields an expression for the
rate of change of the equatorial dipole moment in terms of three contributions,
m˙
e = 3 [A e + D re + D te ]dS . (24)
2µ 0
∫S
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Note that the equatorial unit vector ê is time-dependent, therefore in general, ê· ˙⃗m = m˙
e − ⃗m· ˙ê,
but since ⃗m · ˙ê = 0, we obtain (24). The contribution to m˙
e from tangential advection is
A e = (u θ cos θ cos φ ′ − u φ sin φ ′ )B r (25)
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