Using integration of deterministic, matrix-valued functions with respect to vector-valued, volatility modulated Lévy bases, we construct random vector fields on $\mathbb{R}^n$. In the statistically homogeneous case, the vector field is given as a convolution of a deterministic kernel with respect to a homogeneous, volatility modulated Lévy basis. With applications to turbulence in mind, the kernel is, in the isotropic and incompressible case, 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. 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.