  A new phenomenological model of turbulent fluctuations is constructed by
considering the Lagrangian dynamics of 4 points (the tetrad). The closure of
the equations of motion is achieved by postulating an anisotropic, i.e. tetrad
shape dependent, relation of the local pressure and the velocity gradient
defined on the tetrad. The non-local contribution to the pressure and the
incoherent small scale fluctuations are modeled as Gaussian white "noise". The
resulting stochastic model for the coarse-grained velocity gradient is analyzed
approximately, yielding predictions for the probability distribution functions
of different 2nd and 3d order invariants. The results are compared with the
direct numerical simulation of the Navier-Stokes. The model provides a
reasonable representation of the non-linear dynamics involved in energy
transfer and vortex stretching and allows to study interesting aspects of the
statistical geometry of turbulence, e.g. vorticity/strain alignment. In a state
with a constant energy flux (and K41 power spectrum) it exhibits the anomalous
scaling of high moments associated with formation of high gradient sheets -
events associated with large energy transfer. An approach to the more complete
analysis of the stochastic model, properly including the effect of
fluctuations, is outlined and will enable further quantitative juxtaposition of
the model with the results of the DNS.
