Convective overshooting on the Sun : radiative effects

We calculate solar models with convective overshooting at the top and at th e base of the outer convection zone, and test the models by comparing their eigenfrequencies to the observed solar p-mode frequencies. Radiative tempe rature relaxation is included in form of a characteristic time that describ es both optically thick and thin cases, and a modified mixing-length formal ism is used, with gas parcels traveling varying path lengths. These modific ations to the common mixing-length theory generally change the efficiency o f the convective energy transport, and therefore the stratification at and immediately below the surface of the Sun. Radiative relaxation lowers the c onvective efficiency and so leads to a steeper temperature gradient, with t he consequence that the temperature becomes somewhat larger in the near-sur face layer, but slightly lower in the upper convection zone; due to the lat ter effect there is a negative correction to eigenfrequencies above approxi mate to 2 mHz. The effect of convective parcels with varying path lengths i s opposite. In the solar interior, radiative relaxation is in the diffusion limit and t herefore has no immediate effect at the base of I-he convection zone. Howev er, the larger mixing-length to scale-height ratio caused by the near-surfa ce effect leads to farther overshooting at the base. The effect of the mult iple-path models is in the same direction. For most of our models the exten t of the overshooting is larger than permitted by the helioseismic constrai nt of approximate to 0.1 pressure scale heights, but for some models it is marginal. At the surface the efficient optically thin radiative relaxation smoothes t he temperature gradient. Both the radiation and the multiple-path effects l ead to more extended overshooting. The models reach approximate to 200 km o f overshooting, with temperature fluctuations of up to several hundred Kelv in. We compare the results with spectroscopic investigations, and with rece nt three-dimensional hydrodynamic numerical simulations. A general result is that mixing-length theory appears unable to reproduce i n detail the properties of solar convection that are directly observed at t he surface or inferred by helioseismology. The improvements based on even s ophisticated modifications remain limited.