chapter  8
44 Pages

## Geomagnetism

We revisit both the Helmholtz and the Mie decomposition from Section 5.2 in a spherical framework. While the Helmholtz decomposition can be transferred to a representation intrinsic to spherical vector ﬁelds, the Mie decomposition remains a decomposition in the Euclidean space R3. Yet, the poloidal and toroidal scalars allow a representation in terms of spherical tools like the fundamental solution for the Beltrami operator. Furthermore, we distinguish the two cases of global decompositions on the entire sphere Ω and spherical shells BR0,R1(0), respectively, as well as local decompositions on regular surfaces Γ ⊂ Ω and conical shells CR0,R1(Γ) ⊂ BR0,R1(0), respectively. With regard to the good data coverage by satellites, the global decompositions are often suﬃcient for the modeling aspects under consideration. The local framework is of interest whenever we are concerned with a small modeling region and/or if data is only locally available. One should be aware of the fact that the local decompositions usually require the availability of boundary data.