Some mathematical models generalizing the model of homogeneous and isotropic turbulence

Homogeneous and isotropic turbulence is an example of a system of random fields invariant with respect to a group of motions. Along with homogeneous and isotropic fields, locally homogeneous and locally isotropic ones play an important role in turbulence theory; such local fields may also have an accurate mathematical definition. The random fields invariant with respect to groups of transformations different from a group of Euclidean motions can also be considered; the ‘spectral representation’ of such a field and of a corresponding correlation function often has an unusual form, although its sense remains the same. The algebraic theory of group representations gives the general method of obtaining the spectral representation for the fields. The examples of homogeneous random fields on a sphere and fields in a semiplane invariant with respect to all similarity transformations present interesting examples of random fields invariant with respect to ‘motions’ of special type which might be of some importance for turbulence theory.