Karhunen-Loeve Basis Functions of Kolmogorov Turbulence in the Sphere

TL;DR

This paper derives the Karhunen-Loeve basis functions for Kolmogorov turbulence in a sphere using 3D Zernike functions, providing a mathematical framework for modeling isotropic turbulence effects.

Contribution

It introduces a method to compute the Karhunen-Loeve modes of isotropic Kolmogorov turbulence in spherical domains using 3D Zernike functions and Fourier-based diagonalization techniques.

Findings

Derived explicit forms of Karhunen-Loeve modes for Kolmogorov turbulence in spheres.

Established an efficient diagonalization method in wavenumber space.

Extended symmetry arguments from 2D to 3D turbulence modeling.

Abstract

The statistically independent Karhunen-Loeve modes of refractive indices with isotropic Kolmogorov spectrum of the covariance are calculated in a sphere of given radius, rendered as series of 3D Zernike functions. Many of the symmetry arguments of the associated 2D problem for the circular input pupil remain valid. The technique of efficient diagonalization of the eigenvalue problem in wavenumber space is founded on the Fourier representation of the 3D Zernike basis.