Third order correlations and skewness in convection. I. A new approach suitable for three-equation non-local models

Abstract

Non-local models of stellar convection can account for mixing effects in regions adjacent to convectively unstable layers and for changes to the mean temperature structure caused by free, buoyancy driven convection. The physical completeness of such models, however, depends on how third order correlations, which characterize the non-local transport processes, are expressed in terms of second order correlations and the stellar mean structure. Physical arguments and 3D hydrodynamical simulations were used to develop and test new closure relations for the skewness of the vertical velocity and temperature fields and third order cross-correlations to improve the predictive capabilities of non-local models of convection used in stellar astrophysics and in other disciplines such as meteorology. The structural form of the closure correlations was developed by a series of physical arguments and their accuracy was evaluated through self-consistency tests based on 3D hydrodynamical simulations for the Sun and a DA type white dwarf. The new closure relations derived for the skewness of vertical velocity and temperature fields provided improvements of up to an order of magnitude compared to previous models. This allows releasing the full potential of closure relations for the vertical velocity and temperature cross-correlations previously proposed in meteorology as well as the construction of new, more reliable models for the third order moments of vertical velocity and temperature in non-local models of turbulent convection. The new models for the skewness and third order cross-correlations of vertical velocity and temperature permit the construction of non-local models of turbulent convection which remove, among others, several major short-comings of three equation non-local convection models that are based on the downgradient approximation.

Figures (9)

• Figure 1: Skewness of vertical velocity $S_w$ in 3D RHD simulations of solar granulation. Upper panel: open vertical boundary conditions at top and bottom (using the code of Muthsam10a). Lower panel: closed vertical boundary conditions (using the code of robinson03b). Data directly computed from the 3D RHD numerical simulations (purple line with crosses) are compared with the new TOM model and the DGA. For the new TOM model the green line with x-shaped points shows the no damping case ($c_6=0$). The full, new model ($c_6=0.1$) is indicated by a dark blue line with circles. The light brown line with asterisks denotes the DGA ($d_3=-0.1$). The light blue line with squares shows the DGA with ten times larger turbulent diffusivity ($d_3=-1$).
• Figure 2: Skewness of temperature $S_{\theta}$ (upper panel) and the TOM $\overline{w^3}$ (lower panel). The group of models displayed is the same as in the upper panel of Fig. \ref{['Fig1']}. Data were taken from the 3D RHD simulation with open vertical boundary conditions as in the upper panel of Fig. \ref{['Fig1']}. The same colour scheme was used as in Fig. \ref{['Fig1']} except for the product of the cross-correlation $C_{w,\theta}$ with $S_w$ (dark blue with filled squares as points).
• Figure 3: Comparisons for the TOMs $\overline{w^2\theta}$ (upper row of panels) and $\overline{w\theta^2}$ (lower row of panels). Simulation data and models are the same as those studied in Fig. \ref{['Fig1']} (upper panel) and Fig. \ref{['Fig2']} (lower panel) for $S_w$ and $\overline{w^3}$, respectively: each panel displays results for the solar surface 3D RHD simulation with open vertical boundary conditions. The full vertical range is displayed in the left column of panels. The right column of panels zooms into the upper 40% of the simulation domain. The same colour coding was used as for Fig. \ref{['Fig1']}. In addition, the results obtained for the TSMF closure with DGA (dark red line with asterisks) and without (dark green line with crosses) are shown. For both variants of the TSMF model the quantity $S_w$ was computed directly from 3D RHD data, as explained in the main text.
• Figure 4: The skewnesses $S_w$ and $S_{\theta}$ and the TOM $\overline{w^3}$ in a 3D RHD simulation of convection in the upper envelope of a DA white dwarf. The upper row of panels compares $S_w$ (left panel) and $S_{\theta}$ (right panel) for the closure models discussed in Fig. \ref{['Fig1']} with input data taken from the 3D RHD simulation. The colour coding is the same as for Fig. \ref{['Fig1']} and \ref{['Fig2']}. The lower row of panels displays a test of these closure models for the TOM $\overline{w^3}$ (left panel) alongside a zoom into the upper half of the simulation domain (right panel). A detailed discussion is given in the main text.
• Figure 5: Testing closure models for the TOMs $\overline{w^2\theta}$ and $\overline{w\theta^2}$ with data of a 3D RHD simulation of envelope convection in a DA white dwarf. The same models as discussed in Figs. \ref{['Fig1']} and \ref{['Fig3']} were investigated. The case of the TOM $\overline{w^2\theta}$ is shown in the upper row for the full vertical range (left panel) and for the upper half of the domain (right panel). For line style and colour coding see Fig. \ref{['Fig3']}. The lower row repeats this comparison for $\overline{w\theta^2}$. A discussion of the results is given in the main text.