  We develop a stochastic model for the velocity gradients dynamics along a
Lagrangian trajectory. Comparing with different attempts proposed in the
literature, the present model, at the cost of introducing a free parameter
known in turbulence phenomenology as the intermittency coefficient, gives a
realistic picture of velocity gradient statistics at any Reynolds number. To
achieve this level of accuracy, we use as a first modelling step a regularized
self-stretching term in the framework of the Recent Fluid Deformation (RFD)
approximation that was shown to give a realistic picture of small scales
statistics of turbulence only up to moderate Reynolds numbers. As a second
step, we constrain the dynamics, in the spirit of Girimaji & Pope (1990), in
order to impose a peculiar statistical structure to the dissipation seen by the
Lagrangian particle. This probabilistic closure uses as a building block a
random field that fulfils the statistical description of the intermittency,
i.e. multifractal, phenomenon. To do so, we define and generalize to a
statistically stationary framework a proposition made by Schmitt (2003). These
considerations lead us to propose a non-linear and non-Markovian closed
dynamics for the elements of the velocity gradient tensor. We numerically
integrate this dynamics and observe that a stationary regime is indeed reached,
in which (i) the gradients variance is proportional to the Reynolds number,
(ii) gradients are typically correlated over the (small) Kolmogorov time scale
and gradients norms over the (large) integral time scale (iii) the joint
probability distribution function of the two non vanishing invariants $Q$ and
$R$ reproduces the characteristic teardrop shape, (iv) vorticity gets
preferentially aligned with the intermediate eigendirection of the deformation
tensor and (v) gradients are strongly non-Gaussian and intermittent.
