Suppression of Rayleigh-Bénard convection and restratification by horizontal convection

This study demonstrates that sufficiently strong horizontal convection can suppress Rayleigh-Bénard convection and induce a restratified state in a fluid layer, identifying specific scaling laws for the onset of neutral and strong stratification regimes driven by the interplay between horizontal and vertical buoyancy fluxes.

The Gist

The Big Picture: A Pot of Soup with a Twist

Imagine you have a giant pot of soup sitting on a stove.

• The Stove (Bottom Heating): The stove is turned on, heating the soup from the bottom. In physics, this is called Rayleigh-Bénard Convection (RBC). Normally, this makes the soup boil violently. Hot bubbles rise from the bottom, cool down at the top, and sink back down, creating a chaotic, churning mess.
• The Lid (Top Cooling): Now, imagine the lid of the pot isn't just a flat lid. It's a "smart lid" that is cold in the middle and warm at the edges. This creates a sideways temperature difference. In physics, this is Horizontal Convection (HC). This sideways difference tries to make the soup flow in a giant, slow circle: sinking in the cold middle and rising at the warm edges.

The Question: What happens when you have a boiling stove and a smart lid at the same time? Does the soup keep boiling chaotically, or does the lid's gentle push calm it down and make the soup stable?

The Discovery: The "Re-stratification" Magic

The authors of this paper found that if the "smart lid" effect (Horizontal Convection) is strong enough, it can completely shut down the chaotic boiling from the stove.

They call this process "Re-stratification."

• Before: The soup is unstable. Hot stuff is at the bottom, cold stuff is at the top, but it's churning so much that the layers are mixed up.
• After: The soup becomes stable. The heavy, cold stuff settles at the bottom, and the light, warm stuff floats at the top, but now it's calm. The layers are distinct and peaceful.

Think of it like a crowded dance floor.

• RBC (Boiling): Everyone is jumping up and down randomly. It's chaotic.
• HC (The Lid): A DJ starts playing a slow, organized waltz.
• Re-stratification: If the DJ is loud enough, everyone stops jumping and starts waltzing in neat, calm circles. The chaos is suppressed, and order is restored.

The Two "States" of the Soup

The researchers discovered two specific moments where the balance shifts:

1. The "Neutral" State: This is the tipping point. The chaotic boiling from the bottom is exactly canceled out by the calm waltzing from the top. The soup isn't boiling, but it isn't perfectly calm yet. It's like a tug-of-war where both teams are pulling with equal strength.
2. The "Strong" State: The "smart lid" wins. The horizontal flow is so strong that it completely crushes the bottom heating. The soup becomes perfectly stable and layered. The bottom heating is still there, but it's too weak to cause any bubbles.

How They Figured It Out

The scientists didn't just guess; they used two methods:

1. Math Magic (Scaling Laws): They used equations to predict exactly how strong the "lid" needs to be to stop the "stove." They found a rule of thumb: If you increase the bottom heat, you need to increase the top "smartness" by a specific amount (roughly to the power of 4/5) to keep the soup calm.
2. Computer Simulations: They built a virtual pot of soup in a computer and cranked the numbers up to extreme levels (simulating deep oceans or icy moons) to watch the transition happen in real-time.

Why Should You Care? (The Real World Connection)

You might think, "I don't have a pot of soup with a smart lid." But this happens in nature!

• Earth's Deep Ocean: The Earth's core heats the bottom of the ocean (geothermal heat). Normally, this should make the deep ocean boil. But it doesn't. Why? Because the surface of the ocean is colder at the poles and warmer at the equator. This surface difference acts like our "smart lid," keeping the deep ocean calm and stable so fish and currents can exist.
• Icy Moons (Europa & Enceladus): These moons have oceans under miles of ice. The ice is thicker in some places than others, creating a "smart lid" effect. The moon's core heats the bottom. This paper helps us understand if those hidden oceans are churning violently or if they are calm, layered lakes. This is crucial for figuring out if life could exist there.
• Snowball Earth: Thousands of years ago, Earth was covered in ice. The ocean was heated from below by the core, but the ice surface had temperature variations. This paper helps us understand how the oceans survived that frozen era.

The Takeaway

The main lesson is that horizontal differences (side-to-side) can be more powerful than vertical differences (top-to-bottom).

Even if you are heating a fluid from the bottom (trying to make it boil), a strong enough temperature difference from side-to-side can force the fluid to sit still and organize itself into neat, stable layers. It's a reminder that in the fluid world, sometimes the gentle, sideways push is stronger than the violent, upward shove.

Technical Summary

1. Problem Statement and Motivation

The paper investigates the competition between two distinct modes of convection in a fluid layer:

• Rayleigh-Bénard Convection (RBC): Driven by a uniform, destabilizing buoyancy flux at the bottom (e.g., geothermal heating), which typically creates unstable, overturning convection cells.
• Horizontal Convection (HC): Driven by a horizontally varying buoyancy distribution at the top surface (e.g., due to variable ice thickness or surface temperature gradients), which typically creates a stable, overturning circulation.

The Core Question: In geophysical systems like subglacial lakes, the oceans of "Snowball Earth," and icy moons (Europa, Enceladus), both bottom heating (RBC) and horizontal surface variations (HC) exist simultaneously. The authors ask: Under what conditions does HC suppress the destabilizing RBC to produce a stably stratified interior?

Definition of Restratification: The authors define "restratification" as the state where the volume-averaged vertical buoyancy gradient, $\langle b_z \rangle$, becomes positive ($\langle b_z \rangle > 0$), despite the presence of destabilizing bottom heating.

2. Methodology

The study employs a combination of theoretical scaling analysis and high-resolution numerical simulations.

• Governing Equations: The Boussinesq equations for momentum and buoyancy are used in a 2D fluid layer of depth $h$.
• Boundary Conditions:
  • Bottom ($z=0$): Uniform diffusive buoyancy flux ($b_z = -\beta$), representing RBC.
  • Top ($z=h$): Prescribed sinusoidal buoyancy profile ($b = -b_\star \cos kx$), representing HC.
  • Sides: No-slip and insulated.
• Control Parameters:
  • Vertical Flux Rayleigh Number ($Ra_V$): Quantifies the strength of the bottom heating.
  • Horizontal Rayleigh Number ($Ra_H$): Quantifies the strength of the top surface buoyancy variation (defined using layer depth $h$ rather than horizontal length to simplify scaling).
  • Prandtl Number ($Pr$): Ratio of viscous to thermal diffusivity.
  • Aspect Ratio ($\Gamma$): Ratio of domain width to depth.
• Numerical Approach:
  • Direct Numerical Simulations (DNS) using the spectral-element code Nek5000.
  • Simulations cover Rayleigh numbers up to $Ra_V \sim 10^{10}$.
  • Both 2D and limited 3D simulations (to assess dimensionality effects) were performed.
  • Power Integrals: Theoretical constraints derived from energy and buoyancy variance budgets were used to define regimes and derive scaling laws.

3. Key Contributions and Regimes

The authors identify three distinct regimes based on the sign and magnitude of the volume-averaged vertical buoyancy gradient $\langle b_z \rangle$:

1. RBC-Dominated Regime: $\langle b_z \rangle < 0$. The fluid is unstable; bottom heating drives cellular convection.
2. Neutral Stratification: $\langle b_z \rangle = 0$. The destabilizing effect of RBC is exactly offset by the stabilizing effect of HC.
3. Strong Stratification (Restratification): $\langle b_z \rangle > 0$ (specifically $\langle b_z \rangle \ge \beta$). HC dominates, suppressing RBC and creating a globally stable interior.

4. Key Results and Scaling Laws

A. Scaling Laws for Transitions

Using power integral arguments and numerical data, the authors derive scaling laws for the critical horizontal Rayleigh numbers required to reach the neutral and strong stratification states as a function of the vertical Rayleigh number ($Ra_V$) in the high-Rayleigh regime ($Ra_V \gtrsim 10^4$):

• Onset of Neutral Stratification ($Ra_H^N$):
  $$Ra_H^N \sim Ra_V^{4/5}$$
  This indicates that to achieve a neutral state, the horizontal forcing must scale slightly less than linearly with the bottom heating.

• Onset of Strong Stratification ($Ra_H^S$):
  $$Ra_H^S \sim Ra_V$$
  To fully suppress RBC and achieve strong restratification, the horizontal forcing must scale linearly with the bottom heating.

• Low Rayleigh Regime: For $Ra_V \lesssim 10^4$, the scalings follow a different power law: $Ra_H \sim Ra_V^{1/2}\Gamma$.

B. Boundary Layer Dynamics

The study highlights the top boundary layer as the controlling region for the global stratification:

• In the Neutral State, the top boundary layer thickness ($\delta$) follows the RBC scaling ($\delta \sim h Ra_V^{-1/5}$), but the buoyancy jump across it follows the HC scaling.
• In the Strong Stratification State, the flow is dominated by HC overturning cells. The bottom plumes are intermittent and weak, while the top boundary layer structure dictates the global mean stratification.

C. Sensitivity to Prandtl Number ($Pr$)

• Increasing $Pr$ (relevant for icy moon oceans where $Pr \approx 10-13$) enhances restratification.
• Higher $Pr$ lowers the critical $Ra_H$ required for both neutral and strong stratification, though the effect is modest (approx. 10% reduction in $Ra_H^N$ between $Pr=1$ and $Pr=13$).
• Physically, higher $Pr$ thins the buoyancy boundary layer relative to the viscous layer, reducing the horizontal forcing needed to stabilize the column.

D. Three-Dimensionality

• 3D simulations ($\ell_y = h$) showed that while 3D turbulence increases viscous dissipation and decreases kinetic energy by ~40% compared to 2D, the mechanism of restratification remains unchanged.
• The mean stratification $\langle b_z \rangle$ increases slightly in 3D, confirming that the 2D results are robust qualitative guides for 3D geophysical systems.

E. Comparison with Previous Work (CNF)

The authors compare their findings with Couston, Nandaha, and Favier (2022). They note that while CNF observed "bursting" (transient HC events interrupting RBC) as a transition, their own analysis shows that bursting occurs before true restratification ($\langle b_z \rangle < 0$ during bursts). True restratification requires a higher $Ra_H$ where the time-averaged stratification becomes positive.

5. Significance and Implications

• Geophysical Relevance: The results provide a framework for understanding the stability of oceans in environments where wind stress is absent or negligible, such as:
  • Subglacial Lakes: Where geothermal heating competes with variable ice thickness.
  • Snowball Earth: Where the ocean is covered by ice, and circulation is driven by geothermal flux and lateral variations in ice thickness.
  • Icy Moons (Europa, Enceladus): Determining whether their subsurface oceans are well-mixed (RBC dominated) or stably stratified (HC dominated) is crucial for modeling heat transport, chemical mixing, and potential habitability.
• Modeling Implications: If $\langle b_z \rangle > 0$, standard ocean models (using the "traditional approximation" and baroclinic instability theories) are valid. If $\langle b_z \rangle < 0$, the system behaves like a stellar convection zone, requiring fundamentally different modeling approaches.
• Theoretical Insight: The paper resolves the competition between vertical and horizontal forcing by identifying the top boundary layer as the "bottleneck" that determines the global stratification state, offering precise scaling laws for predicting these transitions.

In summary, the paper demonstrates that horizontal convection is a potent mechanism capable of suppressing geothermal Rayleigh-Bénard convection, provided the horizontal forcing exceeds a specific threshold relative to the bottom heating. This finding is critical for accurately modeling the internal dynamics of ice-covered planetary bodies.