Spherical Ornstein-Uhlenbeck processes

TL;DR

This paper analyzes the motion of a point on a sphere driven by an Ornstein-Uhlenbeck process, providing exact solutions in 2D and asymptotic behaviors in 3D, with potential applications to tumbling objects in turbulence.

Contribution

It introduces a fully characterized model of spherical motion with Ornstein-Uhlenbeck angular velocity, including exact 2D solutions and asymptotic 3D analysis for different persistence angles.

Findings

Exact solution for 2D case.

Correlation function exhibits slow variation in series sum for large persistence angle.

Asymptotic correlation functions in 3D for small and large persistence angles.

Abstract

The paper considers random motion of a point on the surface of a sphere, in the case where the angular velocity is determined by an Ornstein-Uhlenbeck process. The solution is fully characterized by only one dimensionless number, the persistence angle, which is the typical angle of rotation of the object during the correlation time of the angular velocity. We first show that the two-dimensional case is exactly solvable. When the persistence angle is large, a series for the correlation function has the surprising property that its sum varies much more slowly than any of its individual terms. In three dimensions we obtain asymptotic forms for the correlation function, in the limits where the persistence angle is very small and very large. The latter case exhibits a complicated transient, followed by a much slower exponential decay. The decay rate is determined by the solution of a radial Schrodinger equation in which the angular momentum quantum number takes an irrational value, namely j=(sqrt{17}-1)/2. Possible applications of the model to objects tumbling in a turbulent environment are discussed.