Random Vector Field Generation

In 1970, Kraichnan was the first to suggest a randomization method. For studying the diffusion of single particles in a random incompressible velocity field, he came up with a randomization method which includes a projector which ensures the incompressibility of the vector field.

Without loss of generality we assume that the mean velocity \bar{U} is oriented towards the direction of the first basis vector \mathbf{e}_1. Our goal is now to generate random fluctuations with a given covariance model around this mean velocity. And at the same time, making sure that the velocity field remains incompressible or in other words, ensure \nabla \cdot \mathbf U = 0. This can be done by using the randomization method we already know, but adding a projector to every mode being summed:

\mathbf{U}(\mathbf{x}) = \bar{U} \mathbf{e}_1 - \sqrt{\frac{\sigma^{2}}{N}} \sum_{i=1}^{N} \mathbf{p}(\mathbf{k}_i) \left[ Z_{1,i} \cos\left( \langle \mathbf{k}_{i}, \mathbf{x} \rangle \right) + \sin\left( \langle \mathbf{k}_{i}, \mathbf{x} \rangle \right) \right]

with the projector

\mathbf{p}(\mathbf{k}_i) = \mathbf{e}_1 - \frac{\mathbf{k}_i k_1}{k^2} \; .

By calculating \nabla \cdot \mathbf U = 0, it can be verified, that the resulting field is indeed incompressible.