A closure model with plumes

Context. Oscillations of stellar p modes, excited by turbulent convection, are investigated. In the uppermost part of the solar convection zone, radiative cooling is responsible for the formation of turbulent plumes, hence the medium is modelled with downdrafts and updrafts. Aims. We take into account the asymmetry of the upand downflows created by turbulent plumes through an adapted closure model. In a companion paper, we apply it to the formalism of excitation of solar p modes developed by Samadi & Goupil (2001). Methods. Using results from 3D numerical simulations of the uppermost part of the solar convection zone, we show that the two-scale mass-flux model (TFM) is valid only for quasi-laminar or highly skewed flows (Gryanik & Hartmann 2002) and does not reproduce turbulent properties of the medium such as velocity-correlation products. We build a generalized two-scale mass-flux Model (GTFM) model that takes both the skew introduced by the presence of two flows and the effects of turbulence in each flow into account. In order to apply the GTFM to the solar case, we introduce the plume dynamics as modelled by Rieutord & Zahn (1995) and construct a closure model with plumes (CMP). Results. The CMP enables expressing the thirdand fourth-order correlation products in terms of second-order ones. When compared with 3D simulation results, the CMP improves the agreement for the fourth-order moments by a factor of two approximately compared with the use of the quasi-normal approximation or a skewness computed with the classical TFM. Conclusions. The asymmetry of turbulent convection in the solar case has an important impact on the vertical-velocity fourth-order moment, which has to be accounted for by models. The CMP is a significant improvement and is expected to improve the modelling of solar p-mode excitation.