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Convolutions on the sphere: commutation with differential operators

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Abstract

We generalize the definition of convolution of vectors and tensors on the 2-sphere, and prove that it commutes with differential operators. Moreover, vectors and tensors that are normal/tangent to the spherical surface remain so after the convolution. These properties make the new filtering operation particularly useful to analyzing and modeling nonlinear dynamics in spherical systems, such as in geophysics, astrophysics, and in inertial confinement fusion applications. An essential tool we use is the theory of scalar, vector, and tensor spherical harmonics. We then show that our generalized filtering operation is equivalent to the (traditional) convolution of scalar fields of the Helmholtz decomposition of vectors and tensors.

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Notes

  1. That operation (3) also commutes with time derivatives is trivial.

  2. Of course, a Helmholtz decomposition may be accomplished by several methods, including SFTs.

  3. Spectral truncation is a specific choice of the filtering kernel, G.

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Acknowledgements

I thank Matthew Hecht and Geoffrey Vallis for valuable discussions on oceanic flows that spurred questions leading to this paper.

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  1. Department of Mechanical Engineering, University of Rochester, Rochester, NY, 14627, USA

    Hussein Aluie

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This work was supported by NASA Grant 80NSSC18K0772 and DOE Office of Fusion Energy Sciences Grant DE-SC0014318. Partial support was also provided by an initial seed Grant from the Institute of Geophysics, Planetary Physics, and Signatures (IGPPS) at Los Alamos National Laboratory (LANL), NSF Grant OCE-1259794, and DOE NNSA Award DE-NA0001944.

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Aluie, H. Convolutions on the sphere: commutation with differential operators. Int J Geomath 10, 9 (2019). https://doi.org/10.1007/s13137-019-0123-9

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