Convective overshooting on the Sun: radiative effects

We calculate solar models with convective overshooting at the top and at the base of the outer convection zone, and test the models by comparing their eigenfrequencies to the observed solar p-mode frequencies. Radiative temperature relaxation is included in form of a characteristic time that describes both optically thick and thin cases, and a modified mixing-length formalism is used, with gas parcels traveling varying path lengths. These modifications to the common mixing-length theory generally change the efficiency of the convective energy transport, and therefore the stratification at and immediately below the surface of the Sun. Radiative relaxation lowers the convective efficiency and so leads to a steeper temperature gradient, with the consequence that the temperature becomes somewhat larger in the near-surface layer, but slightly lower in the upper convection zone; due to the latter effect there is a negative correction to eigenfrequencies above approximate to 2 mHz. The effect of convective parcels with varying path lengths is opposite. In the solar interior, radiative relaxation is in the diffusion limit and therefore has no immediate effect at the base of I-he convection zone. However, the larger mixing-length to scale-height ratio caused by the near-surface effect leads to farther overshooting at the base. The effect of the multiple-path models is in the same direction. For most of our models the extent of the overshooting is larger than permitted by the helioseismic constraint 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 the temperature gradient. Both the radiation and the multiple-path effects lead to more extended overshooting. The models reach approximate to 200 km of overshooting, with temperature fluctuations of up to several hundred Kelvin. We compare the results with spectroscopic investigations, and with recent three-dimensional hydrodynamic numerical simulations. A general result is that mixing-length theory appears unable to reproduce in detail the properties of solar convection that are directly observed at the surface or inferred by helioseismology. The improvements based on even sophisticated modifications remain limited.