Turbulence geometry and Navier-Stokes equations

It is proposed to consider turbulent media and, in particular, random vector fields from a geometric point of view. This leads to a geometry similar to, but not identical to, Finsler's.
We show that a turbulence generates a horizontal differential symmetric 2-form on the tangent bundle, which we call the Mahalanobis metric.
Thus, vector fields on the underlying manifold generate Riemannian structures on it by the restriction of the Mahalanobis metric on the graphs of vector fields.
In the case of so-called Gaussian turbulences, these Riemannian structures coincide and generate a unique Riemannian structure.
Moreover, similar to Finsler geometry, turbulence generates a nonlinear connection in the tangent bundle but the Mahalanobis metric generates an affine connection in the tangent bundle.
This affine connection and the Mahalanobis metric give us everything we need to construct the Navier-Stokes equations for turbulent media.
We will present two implementations of this scheme: for the flow of ideal gases and plasma, where turbulence is described by the Maxwell-Boltzmann distribution law, and compare them with the standard Navier-Stokes equations.