On the Dynamics of Overshooting Convection in Spherical Shells: Effect of Density Stratification and Rotation

Abstract

Overshooting of turbulent motions from convective regions into adjacent stably stratified zones plays a significant role in stellar interior dynamics, as this process may lead to mixing of chemical species and contribute to the transport of angular momentum and magnetic fields. We present a series of fully nonlinear, three-dimensional (3D) anelastic simulations of overshooting convection in a spherical shell that are focused on the dependence of the overshooting dynamics on the density stratification and the rotation, both key ingredients in stars that however have not been studied systematically together via global simulations. We demonstrate that the overshoot lengthscale is not simply a monotonic function of the density stratification in the convective region, but instead it depends on the ratio of the density stratifications in the two zones. Additionally, we find that the overshoot lengthscale decreases with decreasing Rossby number Ro and scales as Ro^0.23 while it also depends on latitude with higher Rossby cases leading to a weaker latitudinal variation. We examine the mean flows arising due to rotation and find that they extend beyond the base of the convection zone into the stable region. Our findings may provide a better understanding of the dynamical interaction between stellar convective and radiative regions, and motivate future studies particularly related to the solar tachocline and the implications of its overlapping with the overshoot region.

Keywords: Solar convective zone (1998); Solar interior (1500); Solar radiative zone (1999); Solar rotation (1524); Stellar convective zones (301); Stellar rotation (1629).

Figure 1.

Profile of the nondimensional heating function Q_nd versus r/r_o for the case with N_ρ = 3. The black dashed vertical line corresponds to the base of the convective region.

Figure 2.

Profile of the nondimensional background entropy gradient $d\overline{S}∕dr$ versus r/r_o. The black dashed vertical line corresponds to the base of the convective region.

Figure 3.

Snapshots of meridional slices of the radial velocity u_r at a selected longitude for different values of N_ρ (indicated above each panel) at Ra = 10⁵ for the nonrotating runs. The inner black line marks the bottom of the convective region. The convective motions overshoot into the stable region in all N_ρ cases and the amount of overshooting depends on N_ρ.

Figure 4.

Time- and spherically-averaged kinetic energy profile ${\tilde{E}}_k(r)$ versus r/r_o at Ra = 10⁵. The black dashed vertical line corresponds to the base of the convective region. (a) The profiles of ${\tilde{E}}_k(r)$ are shown for all of the different N_ρ cases. ${\tilde{E}}_k(r)$ decays as a half-Gaussian below the base of the CZ. (b) Profile of ${\tilde{E}}_k(r)$ along with the fitted Gaussian function f_G(r) (see Equation (19)) below the base of the CZ for the case with N_ρ = 3.

Figure 5.

Dependence of the computed overshoot lengthscale δ_G on the density stratification in the CZ N_ρ at Ra = 10⁵. The lengthscale δ_G increases for N_ρ ≲ 0.7 and decreases for N_ρ > 0.7, indicating a nonmonotonic dependence on N_ρ.

Figure 6.

(a) Dependence of δ_G on the ratio of the density stratification in the RZ over the one in the CZ given by the parameter R_ρ. The overshoot lengthscale is ${\delta }_G=0.078R_{\rho }^{0.36}$. (b) Snapshot of the radial velocity u_r in r/r_o and θ at a fixed longitude for the Ra = 10⁵ and N_ρ = 3 run. The black dashed line indicates the bottom of the CZ, and the the blue solid line indicates the depth down to r_c – δ_G. The overshoot lengthscale δ_G characterizes remarkably well the depth in the RZ down to which the convective motions overshoot on average.

Figure 7.

Time- and spherically-averaged total adjusted entropy gradient profile $d{\tilde{S}}_T∕dr$ against r/r_o for four N_ρ cases compared with the background $d\overline{S}∕dr$. The black dashed vertical line corresponds to the base of the convective region. The weak partial thermal mixing in the RZ has a slight dependence on N_ρ.

Figure 8.

Snapshots of shell slices of the radial velocity u_r at r = 0.9r_o, i.e., close to the surface of the spherical shell for (a) Ro ≈ 0.007 at Ra = 10⁵ and Ek = 0.001, (b) Ro ≈ 0.043 at Ra = 10⁶ and Ek = 0.001, (c) Ro ≈ 0.23 at Ra = 10⁵ and Ek = 0.01, and (d) Ro ≈ 1.7 at Ra = 10⁵ and Ek = 0.1. The flow becomes more aligned with the axis of rotation with decreasing values of Ro.

Figure 9.

Snapshots of meridional slices of the radial velocity u_r varying in r and θ at a selected longitude for four different values of Ro with (a) Ro≈0.007, Ra=10⁵, and Ek=0.001, (b) Ro≈0.043, Ra=10⁶, and Ek=0.001, (c) Ro≈0.23, Ra=10⁵, and Ek=0.01, and (d) Ro≈1.7, Ra=10⁵, and Ek=0.1. The amount of overshooting below the base of the convection zone seems to be decreasing with decreasing Ro.

Figure 10.

Dependence of the computed overshoot lengthscale δ_G on Rossby number Ro. Error bars associated with uncertainty in the fit are indicated. The black line is the fitted function that describes how δ_G scales with respect to Ro, namely δ_G ∝ Ro^0.23±0.072. The different colors correspond to different Ra and N_ρ indicated in the upper left corner of the plot, while the different shapes correspond to the different values of Ek (shown in the lower left corner of the plot).

Figure 11.

Computed overshoot lengthscale δ_G as a function of latitude θ for four different Ro cases. For larger values of Ro, the latitudinal dependence of δ_G is weaker.

Figure 12.

Profiles of the time- and spherically-averaged nondimensional radial fluxes multiplied by the surface area 4πr² against the radius r/r_o with (a) Ro ≈ 0.007 at Ra = 10⁵ and Ek = 0.001, (b) Ro ≈ 0.043 at Ra = 10⁶ and Ek = 0.001, (c) Ro ≈ 0.23 at Ra = 10⁵ and Ek = 0.01, and (d) Ro ≈ 1.7 at Ra = 10⁵ and Ek = 0.1. The black vertical line corresponds to the base of the convective region.

Figure 13.

Meridional slices illustrating the meridional circulation streamlines with underlying contours of the mass flux $\pm \sqrt{\langle \overline{\rho }{u}_r^2\rangle +\langle \overline{\rho }{u}_{\theta }^2\rangle }$ (blue color corresponds to clockwise motion, and red color to counterclockwise) and the differential rotation profiles (⟨u_ϕ⟩/(r sin θ)), respectively for (a), (b) Ro ≈ 0.007 at Ra = 10⁵ and Ek = 0.001; (c), (d) Ro ≈ 0.043 at Ra = 10⁶ and Ek = 0.001; (e), (f) Ro ≈ 0.23 at Ra = 10⁵ and Ek = 0.01; and (g), (h) Ro ≈ 1.7 at Ra = 10⁵ and Ek = 0.1.

Figure 14.

Time- and spherically-averaged nondimensional kinetic energy in the mean flows. The black dashed vertical line corresponds to the base of the convective region. (a) Profile of the kinetic energy related to the differential rotation ${\tilde{E}}_{\text{DR}}(r)$. ${\tilde{E}}_{\text{DR}}(r)$ is largest for the solar-like and the anti-solar cases in the CZ and the RZ. (b) Profile of the kinetic energy associated with the meridional circulation ${\tilde{E}}_{\text{MC}}(r)$ for four different values of Ro. For the three highest Ro cases, ${\tilde{E}}_{\text{MC}}(r)$ is substantially large below the base of the CZ indicating that the mean flows propagate into the RZ.

Figure 15.

Time- and azimuthally-averaged latitudinal and radial fluxes contributing to the angular momentum transport for the solar-like case with Ro ≈ 0.043. The fluxes associated with the Reynolds stresses are shown in panels (a) and (e), the ones related to the mean flows are shown in (b) and (f), the ones related to the Coriolis term are shown in (c) and (g), and the ones associated with the viscous stresses are shown in panels (d) and (h). The fluxes corresponding to the action of the Coriolis force on the mean flows are dominant and as such we have divided them by 10 (panels (c) and (g)) in order to highlight the profiles of the other weaker fluxes and examine all of them under the same color bar.

Figure 16.

Time- and azimuthally-averaged latitudinal and radial fluxes contributing to the angular momentum transport for the anti-solar case with Ro ≈ 0.23. The fluxes associated with the Reynolds stresses are shown in panels (a) and (e), the ones related to the mean flows are shown in (b) and (f), the ones related to the Coriolis term are shown in (c) and (g), and the ones associated with the viscous stresses are shown in panels (d) and (h). The fluxes corresponding to the action of the Coriolis force on the mean flows are dominant and as such we have divided them by 10 (panels (c) and (g)) in order to highlight the profiles of the other weaker fluxes and examine all of them under the same color bar.

Figure 17.

Thermal-wind balance for the solar-like and the anti-solar cases. (a) Profile of the LHS part and the RHS part of Equation (42) versus latitude θ at different radii within the overshoot region for the solar-like case with Ro ≈ 0.043 (similarly for panel (c), but for the anti-solar case with Ro ≈ 0.23). (b) Meridional slices of the LHS and RHS terms illustrating the thermal-wind balance across the whole shell for the solar-like case (similarly for panel (d), but for the anti-solar case). The thermal-wind balance is mostly satisfied across the shell for the solar-like case, but is somewhat sustained only in the RZ for the anti-solar case.

Figure 18.

Gyroscopic pumping balances for the solar-like and the anti-solar cases. Profile of the LHS and RHS terms of Equation (43) versus latitude θ at different radii within the overshoot region for the solar-like case with Ro ≈ 0.043 (similarly for panel (c), but for the anti-solar case with Ro ≈ 0.23). (b) Meridional slices of the LHS and RHS terms illustrating the gyroscopic pumping balance across the whole shell for the solar-like case (similarly for panel (d), but for the anti-solar case). The gyroscopic pumping balance is overall achieved within the CZ and the overshoot region for both the solar-like case and the anti-solar case.

Figure 19.

Time- and spherically-averaged total adjusted entropy gradient profile $d{\tilde{S}}_T∕dr$ against r/r_o at N_ρ = 3 for three different Rayleigh numbers compared with the background $d\overline{S}∕dr$. The black vertical line corresponds to the base of the convection region. Larger values of Ra lead to stronger partial thermal mixing in the RZ and a smaller subadiabatic layer close to the bottom of the CZ.

Figure 20.

Latitudinal dependence of the time- and azimuthally-averaged entropy perturbations (with their spherically symmetric mean subtracted) volume averaged within the overshoot region ⟨S⟩_ov for the solar-like case with Ro ≈ 0.043. ⟨S⟩_ov does not vary substantially with latitude and exhibits a nonmonotonic profile.