Non-local models of stellar convection usually rely on the assumption that the transfer of convective heat flux, turbulent kinetic energy and related quantities can be described as a diffusion process or that the fourth-order moments of velocity and temperature fluctuations follow a Gaussian distribution (quasi-normal approximation). The latter is also assumed in models of solar p-mode excitation.

We have used realistic numerical simulations of solar granulation and of granulation in a K dwarf to test the quasi-normal approximation and several alternatives. For the superadiabatic layer of the Sun and for the quasi-adiabatic zone underneath, we find that the hypothesis of quasi-normality is a rather poor approximation. In the superadiabatic layer, it overestimates some of the fourth-order moments of vertical velocity and temperature by up to a factor of 2 while it underestimates them in the quasi-adiabatic layers by up to a factor of 3.5. The model proposed by Gryanik & Hartmann and Gryanik et al. reduces the discrepancies within the quasi-adiabatic zone to typically less than 30 per cent and is partially comparable and partially in better agreement with the simulation data than two earlier models by Grossman & Narayan. Simulation data for the K dwarf confirm these results. However, none of the proposed models works well in the superadiabatic layer nor in the photospheric layers above. For the Sun, we provide evidence that the fourth-order moments of horizontal velocity fields can be estimated to within about 30 per cent with the quasi-normal approximation despite the complexity of the horizontal flow. Comparing our results to those from solar simulations with idealized microphysics and with related studies of geophysical convection zones confirms our conclusions about the quasi-normal approximation and the new models.

The improvements come from including the effects of coherent structures (such as granules or plumes), while the limitations are tied to the transition regions or boundaries such as the rapid radiative cooling that occurs at the top of the convection zone. Incorporating the model of Gryanik & Hartmann and Gryanik et al. into non-local convection models may well produce a significant improvement in the modelling of convection or of solar-like p-mode excitation in the quasi-adiabatic part of convection zones. For application to entire convection zones, modifications are necessary which can account for the change in background properties of the convective medium near boundaries or transition regions.