A Lévy Based Approach to Random Vector Fields: With a View Towards Turbulence

Abstract

Using integration of deterministic, matrix-valued functions with respect to vector-valued, volatility modulated Lévy bases, we construct random vector fields on Rⁿ. In the statistically homogeneous case, the vector field is given as a convolution. With applications to turbulence in mind, the convolution kernel is expressed in terms of the energy spectrum. The theory is applied to atmospheric boundary layer turbulence where, in particular, the Shkarofsky correlation family (a generalisation of the Matérn correlation family) is shown to fit the data well. A modification of the Shkarofsky correlation family, which reproduces a more rigorously derived result on the small-scale behaviour of fully developed turbulence, is introduced. Since turbulence possesses structure across a wide range of length scales, simulation is non-trivial. Using a smooth partition of unity, a simple algorithm is derived to decompose the simulation problem into computationally tractable subproblems. Applications within the wind energy industry are suggested.