  Insufficient numerical resolution of grid-based, direct numerical simulations
(DNS) hampers the development of instabilitydriven turbulence at small
(unresolved) scales. As an alternative to DNS, sub-grid models can potentially
reproduce the effects of turbulence at small scales in terms of the resolved
scales, and hence can capture physical effects with less computational
resources. We present a new sub-grid model, the
MHD-instability-induced-turbulence (MInIT) mean-field model. MInIT is a
physically motivated model based on the evolution of the turbulent (Maxwell,
Reynolds, and Faraday) stress tensors and their relation with the turbulent
energy densities of the magneto-rotational (MRI) and parasitic instabilities,
modeled with two partial differential evolution equations with stiff source
terms. Their solution allows obtaining the turbulent stress tensors through the
constant coefficients that link them to the energy densities. The model is
assessed using data from MRI in-box DNS and applying a filtering operation to
compare the filtered data with that from the model. Using the $L_2$-norm as the
metric for the comparison, we find less than one order-of-magnitude difference
between the two sets of data. No dependence on filter size or length scale of
unresolved scales is found, as opposed to results using the gradient model
(which we also use to contrast our model) in which the $L_2$-norm of some of
the stresses increases with filter size. We conclude that MInIT can help DNS by
properly capturing small-scale turbulent stresses which has potential
implications on the dynamics of highly-magnetized rotating compact objects,
such as those formed during binary neutron star mergers.
