Prandtl number dependence of rotating internally heated convection

Abstract

We investigate the influence of the Prandtl number ($Pr$) on penetrative internally heated convection (IHC) in both non-rotating and rotating regimes using three-dimensional direct numerical simulations. By varying $Pr$ between 0.1 and 100, we show that the global mean temperature $\langle \overline{T} \rangle$ is not very sensitive to $Pr$, and is primarily controlled by the dynamics of the unstably stratified top boundary layer. In contrast, the Prandtl number dictates the behavior of the lower, stably stratified region and affects the vertical convective heat flux $\langle \overline{wT} \rangle$. In the non-rotating case, low $Pr$ fluids exhibit a ``symmetry recovery'' where turbulent stirring agitates the stable layer, whereas high $Pr$ fluids transition toward a ``dead zone'' of suppressed fluctuations. Under rotation, we find that $\langle \overline{wT} \rangle$ is enhanced across all Prandtl numbers, though global cooling efficiency, measured by the reduction in $\langle \overline{T} \rangle$, is only improved for $Pr\ge1$ due to the emergence of Ekman pumping. These results demonstrate that while IHC shares some scaling similarities with Rayleigh-Bénard convection at the top boundary, the internal stratification creates a unique sensitivity to $Pr$ that is critical for understanding heat transport in planetary and stellar interiors.

Figures (16)

• Figure 1: A non-dimensional schematic diagram for rotating uniform internally heated convection. The upper and lower plates are at the same temperature, and the domain is periodic in the $x$ and $y$ directions and rotates about the $z$ axis. $\mathcal{F}_B$ and $\mathcal{F}_T$ are the mean heat fluxes out the bottom and top plates, $\overline{\langle T \rangle}$ the mean temperature, and $g$ is the acceleration due to gravity.
• Figure 2: Volumetric visualisation of the instantaneous temperature field without rotation for $R=10^{10}$ and from left to right: $Pr=0.1$, $Pr=1$, $Pr=10$ and $Pr=100$.
• Figure 3: Global responses. $\mathcal{F}_B$ (top), $\overline{\langle T \rangle}$, (middle) and $Re_w$ (bottom) against $R$ (left) and $Pr$ (right). For clarity, only selected values of $Pr$ are shown on the left column plots.
• Figure 4: Plots of the horizontally averaged temperature and temperature fluctuation against $z$ (top), and scaled horizontally averaged vertical velocity and horizontal velocity fluctuations against $z$ (bottom). All plots are for $R=10^9$ and varying Prandtl numbers.
• Figure 5: Thermal and viscous boundary layer sizes for $R=10^9$ and all values of $Pr$ simulated.