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 is oriented towards the direction of the first basis vector . 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 . This can be done by using the randomization method we already know, but adding a projector to every mode being summed:
with the projector
By calculating , it can be verified, that the resulting field is indeed incompressible.