By Bo Lehnert

ISBN-10: 9185917001

ISBN-13: 9789185917006

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59); • The new values of A¯q and A¯Γ are further inserted into the last member of Eq. 53), but with the flux factor f¯Γq being replaced by f¯Γf . This gives a new value of q ∗ to be inserted into a second step of iteration. The obtained values of A¯m and A¯s also define a new value of h0 through Eq. 56), to be used in the second iteration; • The process is repeated, until the two flux factors of Eq. 59) converge towards the same value. 2). 01. We now proceed to a larger number of nonzero amplitudes as follows: • A detailed plateau analysis is first performed on the four-amplitude case, with its special sub-case of three amplitudes represented by the line a4 = 0 in Fig.

18) Is = ρ(sin θ)f g . 20) 0 where ρk are small radii of circles centered around the origin ρ = 0 when G is divergent there, and ρk = 0 when G is convergent at ρ = 0. The case ρk = 0 will later be treated in detail. 20) have been obtained from integration of the local contributions dq0 , dM0 , dm0 , and ds0 to the total charge, magnetic moment, mass, and angular momentum. With the volume element dV = 2πρ2 (sin θ)dρ dθ these contributions become dq0 = ρ¯ dV , dM0 = 21 (sin θ) C ρ dq0 , dm0 = cw2 dV , and ds0 = Cρ (sin θ)dm0 .

In the analysis of the contribution from the magnetic islands we introduce the normalized flux function defined by Eq. 31) in the upper half-plane of the sphere ρ = ε 1. It becomes ˜ Ψ≡ Γ (ρ = ε, θ) C εδ/3 = (sin3 θ)(Dθ T − 2T ) . 31) and the corresponding vector potential both consist of the product of two separate functions of ρ and θ. This also applies to the components of the magnetic field given by Eq. 8). The polar component Bθ is then seen to be proportional to the function Ψ of Eq. 34), whereas the radial component Bρ becomes proportional to dΨ .

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