A Tug-of-War Between Baroclinic Eddies and Convection: Implications for Icy Moon Oceans

Abstract

In many geophysical and planetary environments, such as Earth's ocean and atmosphere as well as subsurface oceans of icy satellites, convection driven by bottom geothermal heating usually coexists with baroclinic eddies driven by lateral buoyancy/temperature gradients. These processes compete against each other, with convection destabilizing the stratification and baroclinic eddies re-stabilizing it, thereby controlling whether the bottom heat flux is significantly redistributed as it is transmitted to the upper surface. Using scaling analysis and numerical simulations, we show that a stratified layer persists near the upper surface up to ${\rm Ra}_{v}\sim {\rm Ra}_h^{5/2}$, where ${\rm Ra}_h\equiv Δb_0/(L_zf^2)$ measures the imposed upper-surface buoyancy contrast $Δb_0$ and ${\rm Ra}_v\equiv B_0/(L_z^2f^3)$ measures the strength of the bottom buoyancy flux $B_0$, $L_z$ is the domain depth and $f$ is the Coriolis parameter. For ${\rm Ra}_v<{\rm Ra}_h^{5/2}$, baroclinic eddies dominate over convection, maintain the upper stratified layer, and completely deflect the bottom buoyancy/heat input into meridional transport. In contrast, when ${\rm Ra}_v>{\rm Ra}_h^{5/2}$, convective plumes penetrate the stratification and transport buoyancy/heat vertically with negligible deflection. Building on these results, we further propose a scaling law for the meridional buoyancy/heat transport in this system. Applications to icy satellites are discussed.

Figures (3)

• Figure 1: Sketch of the model setup. The domain has zonal, meridional, and vertical extents of $L_x$, $L_y$, and $L_z$, respectively, and rotates along $z-$axis, yielding the Coriolis parameter of $f+\beta y$. A uniform heat flux $Q_0$ (red spirals) is imposed at the bottom plate; at the top, the fluid is relaxed (blue arrow) toward an imposed sinusoidal temperature profile shown in the right panel. The lateral boundary conditions are displayed on the left.
• Figure 2: Three-dimensional fields of nondimensional temperature $T^*$ (left column) and corresponding anomaly $T_e^*$ after subtracting the zonal mean (middle column), in the Group-1 simulations with ${\rm Ra}_h=0.04$ and (a) ${\rm Ra}_v=4\times10^{-8}$, (b) ${\rm Ra}_v=4\times10^{-7}$, (c) ${\rm Ra}_v=4\times10^{-6}$, and (d) ${\rm Ra}_v=4\times10^{-5}$. The right column shows schematic diagrams of the shape of isopycnals (black solid curves) and heat transport components, including downward diffusion (green vectors), baroclinic transport (yellow spirals), and convective transport (red spirals).
• Figure 3: (a) Meridional heat flux $Q_h^*=2\overline{|\iint (vT)^*\mathrm{d}x^*\mathrm{d}z^*|}/(\Gamma_x\Gamma_y)$ against the vertical Rayleigh number ${\rm Ra}_{v}$. (b) Similar to (a), but both $Q_h^*$ and ${\rm Ra}_v$ are rescaled by $\widetilde{\rm Ra}_{vc}$. (c) ${\rm Ra}_{vc}$ against ${\rm Ra}_h$. In all panels, black squares represent simulations in Group-1; gray triangles mark $Q_h^*$ without bottom heating; red circles represent simulations in Group-2; blue dots represent results from Kang_2023_HC_and_RBC_heatdeflection, labeled as Kang23; and yellow stars represent results from Kvorka_and_Cadek_2024_DALeffect, labeled as K&C24. In panels (a) and (b), gray solid and dashed lines represent the one-to-one line and the best-fit relation $y=x^{1/2}$, respectively. In panel (c), solid lines represent predicted ${\rm Ra}_{vc}$ from equation (\ref{['eq:Ravc']}) and dashed curves represent $\widetilde{\rm Ra}_{vc}$ from equation (\ref{['eq:Ravc_modified']}). For Group-1 (black), ${\rm Ra}_{vc}$ and $\widetilde{\rm Ra}_{vc}$ overlap.