This paper describes a series of three‐dimensional simulations of shallow inefficient convection in the outer layers of the Sun. The computational domain is a closed box containing the convection–radiation transition layer, located at the top of the solar convection zone. The most salient features of the simulations are that: (i) the position of the lower boundary can have a major effect on the characteristics of solar surface convection (thermal structure, kinetic energy and turbulent pressure); (ii) the width of the box has only a minor effect on the thermal structure, but a more significant effect on the dynamics (rms velocities); (iii) between the surface and a depth of 1 Mm, even though the density and pressure increase by an order of magnitude, the vertical correlation length of vertical velocity is always close to 600 km; (iv) in this region the vertical velocity cannot be scaled by the pressure or the density scaleheight; this casts doubt on the applicability of the mixing length theory, not only in the superadiabatic layer, but also in the adjacent underlying layers; (v) the final statistically steady state is not strictly dependent on the initial atmospheric stratification.

We construct models of the structure and evolution of the Sun which include variable magnetic fields and turbulence. The magnetic effects are (1) magnetic pressure, (2) magnetic energy, and (3) magnetic modulation to turbulence. The effects of turbulence are (1) turbulent pressure, (2) turbulent kinetic energy, and (3) turbulent inhibition of the radiative energy loss of a convective eddy, and (4) turbulent generation of magnetic fields. Using these ingredients we construct five types of solar variability models (including the standard solar model) with magnetic effects. These models are in part based on three-dimensional numerical simulations of the superadiabatic layers near the surface of the Sun. The models are tested with several sets of observational data, namely, the changes of (1) the total solar irradiance, (2) the photospheric temperature, (3) radius, (4) the position of the convection zone base, and (5) low-and medium-degree solar oscillation frequencies. We find that turbulence plays a major role in solar variability, and only a model that includes a magnetically modulated turbulent mechanism can agree with all the current available observational data. We find that because of the somewhat poor quality of all observations (other than the helioseismological ones), we need all data sets in order to restrict the range of models.

The development of 2D and 3D simulations of solar convection has lead to a picture of convection quite unlike the usually assumed Kolmogorov spectrum turbulent flow. We investigate the impact of this changed structure on the dissipation properties of the convection zone, parametrized by an effective viscosity coefficient. We use an expansion treatment developed by Goodman & Oh 1997, applied to a numerical model of solar convection (Robinson et al. 2003) to calculate an effective viscosity as a function of frequency and compare this to currently existing prescriptions based on the assumption of Kolmogorov turbulence (Zahn 1966, Goldreich & Keeley 1977. The results match quite closely a linear scaling with period, even though this same formalism applied to a Kolmogorov spectrum of eddies gives a scaling with power-law index of 5 3 .

The general consensus is that in order to reproduce the observed solar p-mode oscillation frequencies, turbulence should be included in solar models. However, until now there has not been any well-tested efficient method to incorporate turbulence into solar modeling. We present here two methods to include turbulence in solar modeling within the framework of the mixing length theory, using the turbulent velocity obtained from numerical simulations of the highly superadiabatic layer of the sun at three stages of its evolution. The first approach is to include the turbulent pressure alone, and the second is to include both the turbulent pressure and the turbulent kinetic energy. The latter is achieved by introducing two variables: the turbulent kinetic energy per unit mass, and the effective ratio of specific heats due to the turbulent perturbation. These are treated as additions to the standard thermodynamic coordinates (e.g. pressure and temperature). We investigate the effects of both treatments of turbulence on the structure variables, the adiabatic sound speed, the structure of the highly superadiabatic layer, and the p-mode frequencies. We find that the second method reproduces the SAL structure obtained in 3D simulations, and produces a p-mode frequency correction an order of magnitude better than the first method.

This paper describes three‐dimensional (3D) large eddy simulations of stellar surface convection using realistic model physics. The simulations include the present Sun, a subgiant of one solar mass and a lower‐gravity subgiant, also of one solar mass. We examine the thermal structure (superadiabaticity) after modification by 3D turbulence, the overshoot of convective motions into the radiative atmosphere and the range of convection cell sizes. Differences between models based on the mixing length theory (MLT) and the simulations are found to increase significantly in the more evolved stages as the surface gravity decreases. We find that the full width at half maximum (FWHM) of the turbulent vertical velocity correlation provides a good objective measure of the vertical size of the convective cells. Just below the convection surface, the FWHM is close to the mean vertical size of the granules and 2 × FWHM is close to the mean horizontal diameter of the granules. For the Sun, 2 × FWHM = 1200 km, a value close to the observed mean granule size. For all the simulations, the mean horizontal diameter is close to 10 times the pressure scaleheight at the photospheric surface, in agreement with previous work.