Upward Overshooting in Turbulent Compressible Convection. II. Simulations at Large Relative Stability Parameters

An earlier study on upward overshooting demonstrated that the first and second zero-points of velocity correlation (with the velocity at the interface) are good indicators for measuring overshooting distance (Chan et al. 2010). In the previous simulations on upward overshooting at small S (Cai 2020), based on these indicators, we have partitioned the convectively stable zone into three layers: the thermal adjustment layer (mixing both entropy and material), the turbulent dissipation layer (mixing material but not entropy), and the thermal dissipation layer (mixing neither entropy nor material). This partition is still valid in our simulations on upward overshooting at large S. A comparison of layer structures among Zahn (1991), Zhang & Li (2012), and Cai (2020) is given in the Appendix. A detailed discussion on how to define the extent of overshooting distance is reported in Cai (2020). Figure 1 shows the snapshots of the contours of the temperature perturbation and vertical velocity for the case D1. The interface between the convectively unstable and stable zones (z = 1.0), the first zero of velocity correlation (z = 1.105), and the second zero of velocity correlation (z = 1.33) are illustrated by dashed lines. From the contours, we see that strong and narrow downward drafts are formed at the top of the convection zone. On the contrary, the upward drafts are generally weak and broad when they arrive at the top of the convection zone. Despite the weakness of the upward drafts, the figure clearly shows that they can pass through the interface, penetrate into the convectively stable layer, and turn around after reaching the first zero-point. The vertical velocity decays rapidly after the upward draft penetrates into the convective stable zone. The temperature perturbation, on the other hand, becomes negative before the upward draft reaches the first zero-point. Unlike the vertical velocity, the magnitude of the temperature perturbation in this penetrative layer is comparable to that in the convectively unstable layer (see Figure 2(a)). Since the temperature perturbation is anticorrelated with vertical velocity, the enthalpy flux F_e ($=\overline{\rho {c}_{p}{v}_{z}{T}_{1}}$, where overbar denotes spatial and temporal averages) is negative in this region (see Figure 2(c)). To balance the negative transport of F_e, the thermal structure is adjusted accordingly so that the temperature gradient ∇ exceeds the radiative temperature gradient ∇_rad (see Figure 3(b)). The upward drafts can hardly cross the first zero-point. Although the vertical velocity is small above the first zero-point, the temperature perturbation and the horizontal velocity are nonnegligible, at least in the region below the second zero-point (see Figure 2(a)). The enthalpy flux is negligible in this region (see Figure 2(c)), but the mixing is still active by the turbulent dissipation. Above the second zero-point, the Péclet number Pe_Hp ≲ 1 (see Figure 2(e)), where Pe_Hp ≲ 1 uses local pressure scale height as the characteristic length) and the fluid motions are dominated by thermal diffusion.

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Figure 2 shows the statistical results of case D1 (S = 250 on the left panel) and case D7 (S = 16000 in the right panel). From the asymptotic solution of a Reynolds stress model, Zhang & Li (2012) suggested to use the location of the peak of T₁'' (the rms turbulent temperature perturbation) as the boundary of the penetrative layer (or the thermal adjustment layer in our paper). In our simulations, the peak of T₁'' is close to the first zero-point for the case S = 250 (Figure 2(a)). However, the peak of T₁'' seems closer to the interface, rather than the first zero-point, for the case S = 16,000 (Figure 2(b)). Hence, the peak of T₁'' might not be the best proxy for identifying the boundary of the thermal adjustment layer. From Figures 2(a) and 2(b), we find that the first zero-point is almost located at the inflection point of T₁''. Thus we suggest to use the inflection point of T₁'' as the indicator for the boundary of the thermal adjustment layer. It has to be mentioned that the inflection point of T₁'' works better at high S cases. Zhang & Li (2012) have made the assumption that the diffusive term of T₁'' is unimportant in the overshooting zone. It would be more consistent with their assumption if we choose the inflection point of T₁'' as the proxy. Figures 2(c) and 2(d) show the enthalpy flux F_e, the diffusive flux F_d − F_ad (where ${F}_{d}=-\kappa \partial \overline{T}/\partial z$ and ${F}_{\mathrm{ad}}=-\kappa {(\partial \overline{T}/\partial z)}_{\mathrm{ad}}$), and the kinetic fluxes ${F}_{k}=0.5\overline{\rho {v}_{z}({v}_{h}^{2}+{v}_{z}^{2})}$ for D1 and D7, respectively. As mentioned before, there is a dip of F_e in the thermal adjustment layer. From these two subfigures, we see that the absolute value of the dip minimum increases with S. Thus, it is expected that the thermal structure in this layer will deviate more from the initial static state for the case D7 than D1. However, compared with D1, Figure 3(b) clearly shows that D7 deviates less from the static state (see the wiggles above the interface in Figure 3(b)). The reason for this is that the thermal diffusivity ${\kappa }_{2}={F}_{\mathrm{tot}}[S({m}_{\mathrm{ad}}-{m}_{1})+{m}_{\mathrm{ad}}+1]/g$ is much larger in case D7. Thus a small adjustment of thermal structure can lead to significant change in fluxes. Figures 2(e) and 2(f) plot both Pe and Pe_Hp. Apparently, the second zero-point approximately locates at the position of Pe_Hp ≈ 1 (the variation is within a factor of 0.5 ∼ 2.0). It is consistent with the assumption made on the turbulent dissipation layer in the Reynolds stress model (Zhang & Li 2012). The second zero is located a little bit away from the location Pe_Hp = 1 (being above for the low S case and below for the high S case). There are two possible reasons for this. One is that the characteristic length scale of Pe should be associated with the length scale of turbulent motions. It can be different for different cases. The second reason is that the characteristic velocity of Pe is evaluated by rms velocity, while the second zero tends to capture the extreme events.

The last two columns of Table 1 give the overshooting distances measured by the first (δ₁) and second (δ₂) zero-points, respectively. To better illustrate the result, we plot log(δ₁) and $\mathrm{log}({\delta }_{2})$ against $\mathrm{log}(S)$ in Figure 3(a). From the figure, we see that δ₁ or δ₂ first decreases, and then increases with S (U-shape). It is different from our previous simulation result at small S, where only a decreasing trend was discovered (Cai 2020). It is counterintuitive that the overshooting distance can increase with S. Here we discuss possible explanations by considering first the thermal adjustment layer. In our previous simulation (Cai 2020), we deduced a relationship between δ₁ and W_* (the exit vertical velocity at the interface):

$$\begin{eqnarray}\begin{array}{rcl}{W}_{* }^{2} & \approx & \displaystyle \frac{{g}^{2}}{c}({\rm{\nabla }}-{{\rm{\nabla }}}_{\mathrm{ad}}){\delta }_{1}^{2}\\ & = & \displaystyle \frac{{g}^{2}}{c}\displaystyle \frac{{m}_{\mathrm{ad}}-{m}_{1}}{{m}_{\mathrm{ad}}+1}\displaystyle \frac{S}{S({m}_{\mathrm{ad}}-{m}_{1})+({m}_{\mathrm{ad}}+1)}{\delta }_{1}^{2}\,,\end{array}\end{eqnarray} \tag{ 1 }$$

where c is the asymmetry coefficient. The exit velocity W_* is calculated as the rms vertical velocity of upward drafts at the interface. To obtain this relationship, we have made the assumption that ${\rm{\nabla }}-{{\rm{\nabla }}}_{\mathrm{ad}}$ is almost a constant in the thermal adjustment layer. It can be seen from Figure 3(b) that this assumption is reasonable in our simulations, especially when S is large. Equation (1) infers that δ₁ can be affected by the thermal structure (${\rm{\nabla }}-{{\rm{\nabla }}}_{\mathrm{ad}}$), the asymmetry coefficient (c), and the exit vertical velocity at the interface (W_*). First, we consider the effect of the thermal structure. From Figure 3(b) we notice that the absolute value of ${\rm{\nabla }}-{{\rm{\nabla }}}_{\mathrm{ad}}$ increases with S. However, Equation (1) implies that ${\delta }_{1}$ decreases with $| {\rm{\nabla }}-{{\rm{\nabla }}}_{\mathrm{ad}}|$. Thus the increase of ${\delta }_{1}$ is not due to the effect of thermal structure. Second, we consider the effect of the asymmetry coefficient. According to the definition in Zahn (1991), the asymmetry coefficient c is related to the triple moment of the vertical velocity. In our calculation, we find that c varies too much in the overshooting zone. However, the filling factor is more stable. Thus it is more appropriate to compare filling factors, instead of asymmetry coefficients among different cases. Figure 3(c) plots the filling factors at the interface for different cases. The filling factor first decreases, and then increases with S. Despite that the filling factor shows a similar trend with δ₁, its variation (max/min ∼1.1) is too small to explain the significant change of δ₁ across different cases. Third, we consider the effect of the exit velocity at the interface. Although the root mean average velocity $\langle v^{\prime\prime} {\rangle }_{{cz}}$ in the convection zone is almost the same (see Table 1), the exit vertical velocity at the interface is quite different (Figure 3(d)). It has a U-shape, and its variation (max/min ∼1.7) is comparable to that of δ₁ (max/min ∼ 2.6). Thus we conclude that the exit vertical velocity is the key factor affecting δ₁.

Now we discuss the extent of the turbulent dissipation layer. As shown in Figures 2(e) and 2(f), the boundary of the turbulent dissipation layer is located at the position of Pe_Hp ∼ 1. From our previous work (Cai 2020), Pe_Hp is of the following form:

$$\begin{eqnarray}&&{\mathrm{Pe}}_{\mathrm{Hp}}=\displaystyle \frac{{c}_{p}{pv}^{\prime\prime} }{{F}_{\mathrm{tot}}({m}_{2}+1)}\,.\end{eqnarray} \tag{ 2 }$$

Let Pe_Hp ∼ 1, the boundary of the turbulent dissipation layer (or δ₂) can be inferred from the value of p(z). The above equation indicates that δ₂ can be affected by m₂ and v''. In our simulation, m₂ varies by two orders of magnitude from 1.875 to 151.5. It is easy to find that δ₂ decreases with m₂ from the above equation. Thus the trend of δ₂ seen in Figure 3(a) cannot be explained only by m₂. Figures 3(e) and (f) plot the averaged vertical and horizontal velocities in the convectively stable zone, respectively. We see that both the averaged vertical and horizontal velocities have U-shapes, similar to the shape of δ₂. Since the horizontal velocity is several times larger than the vertical velocity, δ₂ is mainly affected by the horizontal velocity. It is interesting to note that the fluid motions can be more active in the stable layer when S is large enough. In the simulations of downward overshooting (Brummell et al. 2002; Rogers & Glatzmaier 2005), the case without the stable zone is assumed to be equivalent to $S\to \infty$. For the upward overshooting, however, they are completely different.

To better understand why the overshooting distance increases with S when S is large, we consider two possible mechanism: the shear instability and the counter cell. Figures 4(a) and 4(b) plot the vertical and horizontal velocities of the cases D4–D8 in the convectively stable zone. The velocity profiles are quite similar in the cases D4–D7. However, both the horizontal and vertical velocities are stronger in the convectively stable zone for case D8. In stars, it has been proposed that the material mixing could be induced by the shear instability (Zahn 1992; Maeder 1995). Linear stability analysis has shown that the shear instability occurs when the Richardson number Ri is smaller than a critical value Ri_c = 1/4 (Howard 1961; Miles 1961). Here we define the Richardson number Ri as:

$$\begin{eqnarray}&&\mathrm{Ri}(z)={N}^{2}/[\overline{{\left(\partial {v}_{x}/\partial z\right)}^{2}+{\left(\partial {v}_{y}/\partial z\right)}^{2}}],\end{eqnarray} \tag{ 3 }$$

where $N=\sqrt{g({{\rm{\nabla }}}_{\mathrm{ad}}-{\rm{\nabla }})/{H}_{p}}$ is the Brunt–Väisälä frequency, and the overline represents the average value taken both temporally and horizontally. Figure 4(c) shows Ri for the cases D4–D8. From the figure, we see that all the Richardson numbers are above the critical value. The situation does not change even when the effects of the Péclet number (Zahn 1992; Maeder 1995) and the horizontal integral scale of turbulence (Prat & Lignières 2014) are taken into account. Considering the effect of thermal dissipation, Zahn (1992) suggested to use $\mathrm{RiPe}\lt 1/4$ as the instability criteria. Maeder (1995) improved the result and suggested to use $\mathrm{RiPe}/(6+\mathrm{Pe})\lt 1/4$ as the instability criteria. In the overshooting zone, the Péclet number is greater than 1. It can be easily shown that the shear instability could take place in the overshooting zone if $\mathrm{Ri}\lt 7/4$. Figure 4(c) shows that Ri is much larger than this value. Prat & Lignières (2014) used a horizontal turbulent length scale ${\ell }=2\pi {\int }_{0}^{\infty }E(k){k}^{-1}{dk}/{\int }_{0}^{\infty }E(k){dk}$ to calculate the Péclet number. Figures 4(e)–(f) present the result on the evaluated horizontal turbulent length scale. ℓ is much larger than H_p. The Péclet number would be even higher if ℓ is used as the characteristic length. As a result, it is unlikely for the shear instability to take place in the convectively stable zone. Now we consider the possibility of the counter cell. Figure 4(d) shows the anisotropic degree ($\omega ={v}_{z}^{2}/({v}_{x}^{2}+{v}_{y}^{2}+{v}_{z}^{2})$) in the convectively stable zone. If the counter cell is important, we would expect that the local minimum points of ω are correlated to the cell nodes. The cells nodes are the barriers to prevent the material from mixing. From Figure 4(d), we can infer that there is one counter cell in the thermal adjustment layer for all the cases. The case is different above the thermal adjustment layer. The figure clearly shows that two counter cells (one extends to the top boundary of the box) exist above this layer in case D4. The cell node between these two cells is approximately at the boundary of the turbulent dissipation layer. Similar counter cells are shown in the cases D5 and D6, but the location of the cell node is subtle. We expect that these two cells would merge when S further increases. For cases D7 and D8, the profile of ω decreases all the way to the top boundary. This indicates that only one counter cell exists and extends all the way to the upper boundary of the box. Once the materials cross the upper boundary of the thermal adjustment layer, they can reach much deeper places through the motion of this large counter cell. The mixing process will be slowed down when the materials enter into the thermal dissipation layer. The counter cells are also confirmed in the profile of velocity field (Figure 5). As a result, we conclude that the counter cells have a significant impact on the extent of the turbulent dissipation layer.

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The recent work of Korre et al. (2019) performed 3D simulations on downward overshooting for Boussinesq flow with the stability parameter in the range of S ∈ [2, 100]. Apart from the different settings on geometry and background structure, their work shares some similarities with ours. First, both works use the correlations of vertical velocities as the proxy of the penetrative distance. The zeros of the velocity correlation function tend to pick out the effect of the strongest plume rather than the average effect. In their work, they have reported that the depth measured by the first zero-point of the velocity correlation is close to the depth of thermal mixing. It agrees well with the result of our simulations. Second, their estimated overshooting distance follows scaling laws with δ ∼ S^−0.5 when the transition region between the unstable and stable zone is thin. In our simulations, the estimated scaling is δ₁ ∼ S^−0.47 if only cases D1–D3 with relatively smaller S are considered. It is close to their estimations. Third, both the studies use the Péclet number as the proxy to define the boundary of the overshooting layer. Despite these similarities, there are also some differences. For example, in our simulations the turbulent dissipation layer is much wider than the thermal adjustment layer, while the opposite result was reported in their simulations. This is probably because different velocity and length scales are used in the calculation of the Péclet number. It is anticipated that this discrepancy would be diminished if the same scales are used. It seems that the Peclet number should be quoted using a length scale associated with turbulent motions. A calibration of parameters on turbulent Reynolds stress models shows that the vertical turbulent length scale is about 1.2H_p (the parameter is calibrated by the data of auto-correlation of vertical velocity in the convection zone; the detailed calibration method can be found in Cai 2018). The vertical turbulent length scale is very close to the H_p. Thus it is reasonable to use H_p as characteristic length scale when evaluating the P'eclet number.