Spherical decompositions in a global and local framework: theory and an application to geomagnetic modeling

Abstract

This paper gives an overview on decompositions of vector fields on the sphere that are of importance in geoscientific modeling. Various versions of the Mie and Helmholtz decomposition are presented. A special emphasis is set to integral representations for the different contributions, which is of interest, e.g., in numerical applications. Furthermore, the decompositions are treated in a global framework on the entire sphere and in a local framework on regular subsurfaces. In the end, an application to the modeling of ionospheric currents is indicated.

On the unique reconstruction of induced spherical magnetizations

“…In order to be able to obtain a spherical version of Theorem 1.4(c), we reformulate the decomposition of Theorem 2.1 in terms of a set of pseudo-differential operatorsõ (1) ,õ (2) ,õ (3) as indicated in [8,10,11,12]. More precisely, 12) where D denotes the pseudo-differential operator…”

“…By dω we denote the surface element on the sphere Ω R . The non-uniqueness of recovering a vertically integrated magnetization m from the knowledge of V in Ω ext R can be characterized by a fairly well-known decomposition (see, e.g., [1,8,11,12,13,18,19,22]) m =m (1) +m (2) +m (3) , (1.3) which has the property that V ≡ 0 in Ω ext R if and only ifm (2) ≡ 0 (in other words, any magnetization of the form m =m (1) +m (3) produces no magnetic potential in the exterior Ω ext R ). We call such a decomposition a Hardy-Hodge decomposition (cf.…”

Multiscale Modeling of the Geomagnetic Field and Ionospheric Currents

“…4.2 requires G . I / to be three times continuously differentiable (for more details on higher-order regularizations, the reader is referred to Gerhards (2011a) and Freeden and Gerhards (2012)). At this point it should be emphasized that G. I / and its regularization only depend on the scalar product Á, so that they can be regarded as functions acting on the interval OE 1; 1/ and OE 1; 1, respectively.…”

“…In geomagnetic applications they have been used in Freeden and Gerhards (2010) and Gerhards (2011aGerhards ( , 2012.…”

“…Corresponding relations for further applications of differential operators and combinations of the single layer kernel and Green's function become necessary for some applications later on, but we skip them at this point due to the similar argumentation. More details can be found in Gerhards (2011aGerhards ( , 2012. In the different context of gravity disturbances, regularizations of the single layer kernel have been used as well in Freeden and Wolf (2008) and Wolf (2009).…”