A brief note on the computation of silent from nonsilent contributions of spatially localized magnetizations on a sphere

Any square-integrable vector field $$\mathbf {f}$$ f over a sphere $$\mathbb {S}$$ S can be decomposed into three unique contributions: one being the gradient of a function harmonic inside the sphere (denoted by $$\mathbf {f}_+$$ f + ), one being the gradient of a function harmonic in the exterior of the sphere (denoted by $$\mathbf {f}_-$$ f - ), and one being tangential and divergence-free (denoted by $$\mathbf {f}_{df}$$ f df ). In geomagnetic applications this is of relevance because, if we consider $$\mathbf {f}$$ f to be identified with a magnetization, only the contribution $$\mathbf {f}_+$$ f + can generate a non-vanishing magnetic field in the exterior of the sphere. Thus, we call $$\mathbf {f}_-$$ f - and $$\mathbf {f}_{df}$$ f df “silent” and $$\mathbf {f}_+$$ f + “nonsilent”. If $$\mathbf {f}$$ f is known to be spatially localized in a subregion of the sphere, then $$\mathbf {f}_+$$ f + and $$\mathbf {f}_-$$ f - are coupled due to their potential field nature. In this short paper, we derive an approach that makes use of this coupling in order to compute the contribution $$\mathbf {f}_-$$ f - from knowledge of the contribution $$\mathbf {f}_+$$ f + .