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

This paper uses scaling analysis and numerical simulations to demonstrate how the competition between bottom-driven convection and baroclinic eddies governs the persistence of upper-ocean stratification and vertical heat transport in icy moon oceans, ultimately proposing a scaling law for meridional buoyancy transport.

The Gist

The Big Picture: A Tug-of-War in the Deep Ocean

Imagine the subsurface oceans of icy moons (like Europa or Enceladus) as a giant, rotating bathtub. Inside this bathtub, two powerful forces are fighting a tug-of-war to decide how heat moves from the bottom to the top.

1. The Bottom Heater (Convection): Deep down, the rocky core of the moon is hot. It acts like a stove, sending heat upward. This heat wants to rise straight up, creating "thermal plumes" (like bubbles rising in boiling water) that punch through the water to reach the ice ceiling.
2. The Side Cooler (Baroclinic Eddies): The ice shell on top isn't the same thickness everywhere. Some parts are thick, some are thin. This creates a temperature difference across the ocean's surface. This difference acts like a "slant" or a ramp. It creates swirling currents (eddies) that want to move heat sideways, along the slant, rather than letting it go straight up.

The Question: Who wins? Does the heat from the bottom shoot straight up to melt the thin ice, or does it get swept sideways and dumped under the thick ice?

The Three Stages of the Battle

The researchers used supercomputers to simulate this battle and found three distinct outcomes, depending on how strong the bottom heater is compared to the side slope.

1. The "Sideways Sweep" (Weak Bottom Heat)

Analogy: Imagine a strong wind blowing across a calm lake. If you try to throw a leaf straight up, the wind catches it and blows it sideways before it can rise.

• What happens: When the heat from the bottom is weak, the "side slope" (created by the ice thickness) wins. The water forms a stable, layered "lid" near the top.
• The Result: The heat from the bottom tries to rise, hits this lid, and gets deflected. Instead of warming the ice directly above it, the heat is carried sideways by swirling eddies to the colder, thicker parts of the ice shell. The bottom heat is completely "deflected."

2. The "Cracking the Lid" (Medium Bottom Heat)

Analogy: Now imagine you turn up the stove. The bubbles (plumes) get bigger and more energetic. They start punching holes in the wind-blown lid, but they can't break it all the way through yet.

• What happens: The rising heat is strong enough to erode the stable layer in some spots. You get a mix of swirling eddies and rising plumes.
• The Result: Some heat still gets swept sideways, but some of it manages to punch through the lid and reach the ice directly. It's a messy transition zone.

3. The "Total Breakthrough" (Strong Bottom Heat)

Analogy: The stove is now on "High." The bubbles are so powerful and fast that they smash through the wind and the lid entirely.

• What happens: The rising heat is so strong that it destroys the stable layers. The water becomes turbulent and mixed from bottom to top.
• The Result: The heat shoots straight up. The bottom heating pattern is projected directly onto the ice ceiling. The "sideways sweep" is defeated.

The "Magic Formula"

The authors discovered a mathematical rule (a scaling law) that predicts exactly when the switch happens.

• If the bottom heat is below a certain threshold, the heat gets swept sideways.
• If the bottom heat is above that threshold, the heat shoots straight up.

They found that the threshold depends on how steep the "side slope" is. The steeper the slope (the bigger the difference in ice thickness), the harder it is for the bottom heat to break through.

Why This Matters for Icy Moons

This isn't just about math; it changes how we understand the solar system.

• The Mystery of the Ice: We know that moons like Europa and Enceladus have ice shells that are much thicker in some places than others.
• The Problem: If the heat from the rocky core just shot straight up, it would melt the thin ice and leave the thick ice alone, eventually making the shell uniform (flat). But the shells aren't flat; they have huge variations.
• The Solution: This paper confirms that the "Sideways Sweep" is likely happening. The heat from the core is being swept away to the thick ice, keeping the thin ice cold and the thick ice warm.
• The Implication: This means the core's heat alone cannot explain the current shape of the ice. There must be other heat sources (like friction or tidal flexing) happening inside the ice shell itself to keep the thick ice from melting away.

Summary

Think of the ocean as a battlefield.

• Convection is the army trying to march straight up.
• Baroclinic Eddies are the terrain (hills and valleys) trying to force the army to march sideways.
• The Paper tells us: If the army is weak, the terrain wins, and the heat goes sideways. If the army is strong enough, it conquers the terrain and goes straight up.

For our icy moons, the "army" (core heat) is currently too weak to conquer the "terrain" (ice thickness differences), so the heat is being diverted, helping to maintain the strange, lumpy shapes of their frozen surfaces.

Technical Summary

1. Problem Statement

In geophysical and planetary environments, such as Earth's oceans and the subsurface oceans of icy moons (e.g., Europa, Enceladus, Titan), two competing physical processes often coexist:

• Bottom Heating: Geothermal activity at the seafloor drives Rayleigh-Bénard convection, which destabilizes the fluid interior and attempts to transport heat vertically to the surface.
• Horizontal Gradients: Variations in overlying ice thickness (on icy moons) or solar radiation (on Earth) create lateral temperature/buoyancy gradients. These drive baroclinic eddies, which transport denser fluid beneath lighter fluid, creating a stable, stratified layer near the surface.

The central problem is determining the conditions under which these processes compete. Specifically, does the bottom heat flux reach the surface directly via convection, or is it laterally deflected by the stratified layer and transported by baroclinic eddies to colder regions? Previous studies (e.g., Kang 2023) proposed scaling laws but were limited by computational constraints that prevented the adequate resolution of convective plumes, leaving uncertainty about the transition thresholds.

2. Methodology

The authors employed three-dimensional numerical simulations using the GPU-accelerated non-hydrostatic Boussinesq solver Oceananigans.jl.

• Domain Setup: A rotating rectangular domain ($L_x, L_y, L_z$) on a $\beta$-plane (Coriolis parameter $f$ varies linearly with latitude $y$).
• Boundary Conditions:
  • Bottom: Uniform heat flux ($Q_0$) driving convection.
  • Top: A sinusoidal temperature profile ($T_{top}$) representing lateral buoyancy gradients, relaxed toward the imposed profile at a rate $\gamma_T$.
  • Lateral: Periodic in the zonal ($x$) direction; free-slip and no-flux in the meridional ($y$) direction.
• Governing Equations: Non-hydrostatic Boussinesq equations including Coriolis force, viscosity, and thermal diffusion.
• Key Dimensionless Parameters:
  • $Ra_h$ (Horizontal Rayleigh Number): Measures the imposed upper-surface buoyancy contrast ($\Delta b_0$). It relates to the Rossby deformation radius.
  • $Ra_v$ (Vertical Rayleigh Number): Measures the bottom buoyancy flux ($B_0$). It relates to the characteristic size of rotating convective plumes.
  • $Ek$ (Ekman Number) & $Pr$ (Prandtl Number): Fixed or varied to explore the parameter space.
• Scope: The study explored a broad parameter space where both baroclinic eddies and convective plumes are adequately resolved, addressing the under-resolution issues in prior work.

3. Key Contributions

1. High-Resolution Simulation: The authors conducted simulations that fully resolve the interaction between small-scale convective plumes and large-scale baroclinic eddies, validating and extending previous analytical scaling laws.
2. Identification of Regimes: They clearly delineated three distinct dynamical regimes based on the ratio of bottom heating to surface stratification ($Ra_v$ vs. $Ra_h$).
3. Refined Scaling Laws: They derived and validated analytical scaling laws for:
   • The critical threshold ($Ra_{vc}$) where convection overcomes stratification.
   • The meridional heat transport in the diffusion-dominated regime.
   • A correction factor for finite surface relaxation rates ($\gamma_T$).
4. Planetary Application: They applied these findings to icy moons, providing a robust framework to interpret heat transport and ice shell dynamics.

4. Key Results

A. Dynamical Regimes

The system transitions through three states as the bottom heating strength ($Ra_v$) increases relative to the surface stratification ($Ra_h$):

1. Baroclinic-Eddy Dominated ($Ra_v < Ra_{vc}$):
   • Convection is suppressed by a stable stratified layer near the surface.
   • Bottom heat is transported upward by convection only until it hits the stratified layer.
   • Deflection: Baroclinic eddies take over, transporting heat along tilted isopycnals toward the colder side of the domain. The bottom heat is completely deflected laterally; it does not reach the surface directly beneath the heat source.
2. Transition Regime ($Ra_v \approx Ra_{vc}$):
   • Convective plumes begin to erode the stratified layer.
   • Some isopycnals originating from the surface are drawn down to the bottom, allowing partial direct heat transport.
3. Convection-Dominated ($Ra_v > Ra_{vc}$):
   • Convective plumes penetrate the entire depth, destroying the surface stratification.
   • Bottom heat is transported vertically with negligible lateral deflection, projecting the bottom heating pattern directly onto the surface.

B. Scaling Laws

• Critical Threshold ($Ra_{vc}$): The transition occurs when the bottom buoyancy flux is strong enough to overcome the stratification. The authors confirm the scaling:
  $$Ra_{vc} \propto Ra_h^{5/2}$$
  Specifically, $Ra_{vc} = k \Gamma_y^{-3} Ra_h^{5/2}$, where $k \approx 0.25$.
• Correction for Surface Relaxation: When the surface temperature is relaxed at a finite rate ($\gamma_T$), the effective horizontal gradient is reduced. The authors propose a modified critical value $\tilde{Ra}_{vc}$ that accounts for this, showing $\tilde{Ra}_{vc} < Ra_{vc}$.
• Heat Transport Scaling:
  • In the diffusion-dominated regime (very low $Ra_v$), horizontal heat flux is finite and set by diffusion, scaling as $Ra_h^{10/7}$.
  • In the transition regime, meridional heat flux scales linearly with $Ra_v$ until the convection-dominated regime is reached.

C. Validation

The numerical results collapse onto a single curve when normalized by the modified critical Rayleigh number ($Ra_v / \tilde{Ra}_{vc}$). The data aligns well with the proposed $Ra_h^{5/2}$ scaling and previous studies (Kang 2023; Kvorka & Čadek 2024), though the latter's extrapolation to Titan was found to overestimate the relevant parameter range by orders of magnitude.

5. Significance and Implications for Icy Moons

The study has profound implications for understanding the subsurface oceans of Europa, Enceladus, and Titan:

• Stratified Upper Oceans: Based on estimated parameters (ice thickness variations and maximum conductive heat flux $\approx 40$ mW m$^{-2}$), all three moons fall into the baroclinic-eddy dominated regime ($Ra_v \ll Ra_{vc}$).
• Heat Deflection: Consequently, geothermal heat from the silicate core is completely deflected laterally by the stratified ocean. It is transported toward latitudes with thicker ice shells (colder surface temperatures) rather than reaching the thinner ice directly.
• Ice Shell Dynamics: This implies that the observed large-scale variations in ice shell thickness on these moons cannot be sustained solely by core heating. Instead, the heat distribution required to maintain these thickness variations must involve spatially heterogeneous heating within the ice shell itself (e.g., tidal dissipation within the ice).
• Broader Applicability: The framework also applies to gas giant atmospheres, helping explain latitudinal distributions of outgoing longwave radiation driven by internal heating and heterogeneous solar forcing.

In summary, this paper resolves the "tug-of-war" between convection and baroclinic eddies, establishing that for icy moons, the ocean is likely stratified enough to prevent core heat from directly melting thin ice, necessitating internal ice heating to explain observed surface features.