  This paper is an attempt to introduce methods and concepts of the
Riemann-Cartan geometry largely used in such physical theories as general
relativity, gauge theories, solid dynamics, etc. to fluid dynamics in general
and to studying and modeling turbulence in particular. Thus, in order to
account for the rotational degrees of freedom of the irregular dynamics of
small scale vortexes, we further generalize our unified first-order hyperbolic
formulation of continuum fluid and solid mechanics which treats the flowing
medium as a Riemann-Cartan manifold with zero curvature but non-vanishing
torsion. We associate the rotational degrees of freedom of the main field of
our theory, the distortion field, to the dynamics of microscopic (unresolved)
vortexes. The distortion field characterizes the deformation and rotation of
the material elements and can be viewed as anholonomic basis triad with
non-vanishing torsion. The torsion tensor is then used to characterize
distortion's spin and is treated as an independent field with its own time
evolution equation. This new governing equation has essentially the structure
of the non-linear electrodynamics in a moving medium and can be viewed as a
Yang-Mills-type gauge theory. The system is closed by providing an example of
the total energy potential. The extended system describes not only irreversible
dynamics (which raises the entropy) due to the viscosity or plasticity effect
but it also has dispersive features which are due to the reversible energy
exchange (which conserves the entropy) between micro and macro scales. Both the
irreversible and dispersive processes are represented by relaxation-type
algebraic source terms so that the overall system remains first-order
hyperbolic. The turbulent state is then treated as an excitation of the
equilibrium laminar state due to the non-linear interplay between dissipation
and dispersion.
