Poloidal-Toroidal Decomposition of Solenoidal Vector Fields in the Ball
- Authors: Kazantsev S.G.1, Kardakov V.B.2
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Affiliations:
- Sobolev Institute of Mathematics
- Novosibirsk State University of Architecture and Civil Engineering
- Issue: Vol 13, No 3 (2019)
- Pages: 480-499
- Section: Article
- URL: https://journals.rcsi.science/1990-4789/article/view/213224
- DOI: https://doi.org/10.1134/S1990478919030098
- ID: 213224
Cite item
Abstract
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.
About the authors
S. G. Kazantsev
Sobolev Institute of Mathematics
Author for correspondence.
Email: kazan@math.nsc.ru
Russian Federation, pr. Akad. Koptyuga 4, Novosibirsk, 630090
V. B. Kardakov
Novosibirsk State University of Architecture and Civil Engineering
Email: kazan@math.nsc.ru
Russian Federation, ul. Leningradskaya 113, Novosibirsk, 630113