Geodesic
Poloidal-toroidal decomposition of solenoidal vector fields in the ball
Sibirskij žurnal industrialʹnoj matematiki, Tome 22 (2019) no. 3, pp. 74-95

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Under study is the polynomial orthogonal basis system of vector fields in the ball which corresponds to the Helmholtz decomposition and is divided into the three parts: potential, harmonic, and solenoidal. It is shown that the decomposition of a solenoidal vector field with respect to this basis is a poloidal-toroidal decomposition (the Mie representation). In this case, the toroidal potentials are Zernike polynomials, whereas the poloidal potentials are generalized Zernike polynomials. The polynomial system of toroidal and poloidal vector fields in a ball can be used for solving practical problems, in particular, to represent the geomagnetic field in the Earth's core.
Keywords: solenoidal, toroidal and poloidal vector fields, Mie representation, vector spherical harmonic, Zernike polynomial. 
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     title = {Poloidal-toroidal decomposition of solenoidal vector fields in the ball},
     journal = {Sibirskij \v{z}urnal industrialʹnoj matematiki},
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     url = {http://geodesic.mathdoc.fr/item/SJIM_2019_22_3_a6/}
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S. G. Kazantsev; V. B. Kardakov. Poloidal-toroidal decomposition of solenoidal vector fields in the ball. Sibirskij žurnal industrialʹnoj matematiki, Tome 22 (2019) no. 3, pp. 74-95. http://geodesic.mathdoc.fr/item/SJIM_2019_22_3_a6/