In this paper, we consider a phenomenological model, incorporating the main features of hydrodynamic fluid turbulence, aimed at predicting the structure of the velocity gradient tensor, M, coarse-grained at a spatial scale r. This model (M. Chertkov, A. Pumir and B.I. Shraiman, Phys. Fluids 11, 2394 (1999)) is formulated as a set of stochastic ordinary differential equations depending on three dimensionless parameters. The joint probability distribution functions of the second and third invariants of M, as well as the scaling laws of the average enstrophy, strain and energy transfer are computed by using a semi-classical method of resolution of the model. These results are compared with direct numerical simulations (DNS) data. The semi-classical solutions correctly reproduce the DNS data behavior provided the parameter that controls nonlinearity reduction induced by pressure is finely tuned.
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