Ultimate regimes in horizontal and internally heated convection

This paper derives asymptotic models for the ultimate regimes of horizontal and internally heated convection by combining turbulent boundary-layer relations with exact dissipation balances, revealing that their kinetic-energy scaling exponents are 1/3—distinct from the 1/2 exponent in Rayleigh-Bénard convection—due to the absence of a specific response factor in their global kinetic-energy balances.

The Gist

Imagine you are trying to understand how heat moves through a fluid, like water in a pot or air in the atmosphere. Scientists call this convection. Usually, we think of heat moving because the bottom of a pot is hot and the top is cold (like boiling water). This is called Rayleigh–Bénard convection (RBC).

But in the real world, heat often moves in more complicated ways. Sometimes, the heat source is spread out inside the fluid itself (like the Earth's core generating heat). Other times, the top and bottom are the same temperature, but one side of a container is hot and the other is cold (like ocean currents driven by the sun warming one side of the sea). These are called Horizontal Convection (HC) and Internal Heating (IHC).

For decades, scientists have been arguing about what happens when the heat gets really intense. They call this the "Ultimate Regime." It's the point where the smooth, laminar flow breaks down into chaotic, turbulent swirls, and the rules for how fast heat moves change.

This paper by Olga Shishkina and Detlef Lohse solves a long-standing puzzle: What are the exact rules for heat transport in these "Ultimate" states for Horizontal and Internal heating?

Here is the breakdown in simple terms:

1. The Old Story vs. The New Story

• The Old Story (RBC): For standard boiling water (RBC), the leading theory suggested that when things get super turbulent, the heat transport speed increases with the square root of the driving force. Think of it like a car: if you press the gas pedal twice as hard, you go $\sqrt{2}$ times faster.
• The New Discovery: The authors realized that for Horizontal and Internal heating, the "engine" works differently. They found that in these specific setups, the heat transport speed increases much more slowly. If you double the heat, the transport only goes up by the cube root (roughly 1.26 times), not the square root.

2. The "Traffic Jam" Analogy

To understand why the rules change, imagine a highway:

• Rayleigh–Bénard (Standard Boiling): Imagine a highway where the traffic (heat) is pushed by a giant wind blowing from the bottom to the top. The wind pushes the cars, and the cars push the wind back. It's a two-way street where the cars and the wind help each other go faster. This allows for a faster speed limit (the 1/2 exponent).
• Horizontal/Internal Heating: Now, imagine the highway is on a flat plain. The cars (heat) are generated inside the cars themselves, or the wind is blowing sideways. The cars can't get a "boost" from the wind in the same way. They have to rely entirely on their own internal engines to get through the traffic jams near the walls (the boundaries).
  • Because they don't get that extra "push" from the global flow, they hit a bottleneck sooner. The speed limit drops. The math changes from a square root ($\sqrt{x}$) to a cube root ($\sqrt[3]{x}$).

3. The "Boundary Layer" Bottleneck

The paper focuses on what happens right next to the walls of the container.

• In the "Ultimate Regime," the thin layer of fluid touching the wall becomes turbulent.
• The authors combined two things:
  1. How turbulence behaves near a wall (like how water swirls around a rock in a river).
  2. The exact energy budget (how much energy is created vs. how much is lost to friction).

They found that in Horizontal and Internal heating, the "energy budget" equation is missing a key ingredient that exists in standard boiling. Because that ingredient is missing, the fluid can't accelerate as fast as we thought.

4. Why Does This Matter?

You might ask, "Who cares about the difference between a square root and a cube root?"

• Predicting the Future: This math helps us predict how heat moves in massive systems like the Earth's oceans (Horizontal Convection) or the Earth's core (Internal Heating).
• Climate Models: If we use the wrong math (the old 1/2 rule) to model ocean currents, we might think the ocean mixes heat much faster than it actually does. This could throw off our climate change predictions.
• Engineering: It helps engineers design better cooling systems for nuclear reactors or electronics, where heat is generated inside the material, not just at the surface.

The Takeaway

The authors didn't just guess; they derived a new "recipe" for these extreme conditions.

• For Standard Boiling (RBC): The ultimate speed limit is 1/2.
• For Horizontal & Internal Heating: The ultimate speed limit is 1/3.

They proved that while the turbulence looks similar in all these systems, the global rules of how energy is balanced are different. By fixing the energy balance equation, they corrected the speed limit for the most extreme heat scenarios, ensuring our models of the Earth and stars are as accurate as possible.

In short: They found that when heat is generated inside a fluid or moves sideways, the system hits a "speed bump" earlier than we thought, forcing the heat to move more slowly than in a standard pot of boiling water.

Technical Summary

1. Problem Statement

The paper addresses the theoretical prediction of the ultimate regime of thermal convection for two specific configurations: Horizontal Convection (HC) and Pure Internally Heated Convection (IHC).

• Context: In Rayleigh–Bénard Convection (RBC), the "ultimate regime" occurs at very high Rayleigh numbers ($Ra$) where boundary layers transition from laminar to turbulent, altering global heat transport scaling laws. While recent work by Shishkina and Lohse (2024) established asymptotic models for RBC (predicting a scaling exponent of $1/2$), the corresponding models for HC and IHC remained unresolved.
• The Gap: Existing models for HC and IHC lacked a rigorous asymptotic derivation consistent with mathematical upper bounds. Specifically, there was a need to determine if the scaling exponents for these systems would follow the RBC $1/2$ law or adhere to the stricter $1/3$ bounds derived from rigorous mathematical analysis (e.g., Siggers et al., 2004 for HC).
• Objective: To derive asymptotic scaling models for the ultimate regimes of HC and pure IHC, determine the scaling exponents for the Nusselt number ($Nu$) or dimensionless bulk temperature ($\tilde{\Delta}$), and Reynolds number ($Re$), and verify consistency with rigorous transport bounds.

2. Methodology

The authors employ a Grossmann–Lohse (GL) type scaling theory, adapted from their recent RBC work, by combining:

1. Turbulent Boundary Layer Relations: They retain the standard relations for turbulent wall layers derived by Shishkina and Lohse (2024). These relate the friction velocity ($u_\tau$) and Nusselt number to the Reynolds number ($Re_\tau$) and Prandtl number ($Pr$), incorporating logarithmic corrections.
2. Exact Global Dissipation Balances: This is the critical modification. Instead of using the RBC kinetic energy balance, they substitute the exact global kinetic energy dissipation balances specific to HC and IHC.

Key Mathematical Steps:

• Boundary Layer Closure: They assume Landau-type closures for eddy viscosity and thermal diffusivity outside the viscous sublayer, leading to generic relations for $Nu$ and $Re$ in terms of $Re_\tau$ and $Pr$.
• System-Specific Balances:
  • RBC: $\epsilon_u \sim (Nu - 1) Ra Pr^{-2}$.
  • HC: $\epsilon_u \sim Ra Pr^{-2}$ (Note: The factor $Nu$ is absent).
  • Pure IHC: $\epsilon_u \sim Rr Pr^{-2} \Phi$, where $Rr$ is the Rayleigh–Roberts number and $\Phi$ is the convective flux (treated as $O(1)$).
• Asymptotic Matching: By equating the boundary layer estimates with the global balances, they derive scaling laws for different Prandtl number regimes (low $Pr$ and high $Pr$) and identify transition slopes between subregimes.

3. Key Contributions

• Derivation of Ultimate Regime Models: The paper provides the first asymptotic models for the ultimate regimes of HC and pure IHC, extending the GL framework beyond RBC.
• Identification of the Scaling Exponent: The authors demonstrate that for fixed $Pr$, the scaling exponent for the heat transport (or inverse bulk temperature) in the ultimate regime is $1/3$ for both HC and IHC. This contrasts with the $1/2$ exponent found in RBC.
• Explanation of the Physical Mechanism: The paper clarifies that the difference in exponents arises because the global kinetic energy balance in HC and IHC lacks the additional response factor (the Nusselt number $Nu$ in HC or the inverse bulk temperature in IHC) present in the RBC balance.
• Consistency with Rigorous Bounds: The derived $1/3$ exponent for HC is shown to be exactly consistent with the rigorous upper bound derived by Siggers et al. (2004). For IHC, the results align with the exact balance analogy between HC and IHC established by Wang et al. (2021).
• Mapping of Subregimes: The authors map out the complete phase diagram for the ultimate regimes, identifying four distinct subregimes ($II'_\ell, IV'_\ell, IV'_u, III'_\infty$) and the transition slopes between them.

4. Key Results

A. Horizontal Convection (HC)

• Scaling Laws: In the ultimate regime, the Nusselt number ($Nu$) and Reynolds number ($Re$) scale as:
  • $Nu \sim Pr^{\gamma} Ra^{1/3} (\log Ra)^{-4/3}$
  • $Re \sim Pr^{\delta} Ra^{1/3} (\log Ra)^{2/3}$
  • (Where exponents $\gamma$ and $\delta$ depend on whether $Pr \lesssim 1$ or $Pr \gtrsim 1$).
• Consistency: The $Ra^{1/3}$ exponent matches the rigorous upper bound $Nu \lesssim Ra^{1/3}$.
• Subregimes:
  • Low $Pr$($Pr \lesssim 1$): Branch $IV'_\ell$ with $Nu \sim Pr^{1/3} Ra^{1/3}$.
  • High $Pr$($Pr \gtrsim 1$): Branch $IV'_u$ with $Nu \sim Pr^{-1/3} Ra^{1/3}$.
  • Transitions: The model defines transition lines (e.g., $Pr \sim Ra^{-1}$ and $Pr \sim Ra^{1/4}$) connecting the ultimate branches to classical laminar/turbulent branches.

B. Pure Internally Heated Convection (IHC)

• Analogy: Pure IHC with equal-temperature plates is formally analogous to HC via the mapping $Ra \leftrightarrow Rr$ (Rayleigh–Roberts number) and $Nu \leftrightarrow \tilde{\Delta}^{-1}$ (inverse dimensionless bulk temperature).
• Scaling Laws:
  • $\tilde{\Delta}^{-1} \sim Pr^{\gamma} Rr^{1/3} (\log Rr)^{-4/3}$
  • $Re \sim Pr^{\delta} Rr^{1/3} (\log Rr)^{2/3}$
• Result: The inverse mean bulk temperature grows as $Rr^{1/3}$ at fixed $Pr$.
• Distinction: The authors explicitly distinguish "pure" IHC (where heat must cross boundary layers) from source-sink systems where heat is absorbed in the bulk. In the latter, $1/2$ exponents are possible, but in pure IHC, the $1/3$ exponent holds.

C. Comparison with RBC

• RBC: Ultimate exponent = $1/2$.
• HC/IHC: Ultimate exponent = $1/3$.
• Reason: The absence of the $Nu$ (or $\tilde{\Delta}^{-1}$) factor in the kinetic energy balance of HC/IHC reduces the scaling power.

5. Significance

• Theoretical Unification: The work unifies the understanding of turbulent convection across different geometries (RBC, HC, IHC). It establishes that the "recipe" for deriving ultimate regimes is universal: combine turbulent wall laws with the specific global kinetic energy balance of the system.
• Validation of Rigorous Bounds: The theoretical derivation confirms that the mathematically rigorous upper bounds (specifically the $1/3$ limit for HC) are not just theoretical limits but are likely the actual asymptotic scaling laws realized in the ultimate turbulent regime.
• Geophysical and Astrophysical Relevance: Since HC models large-scale geophysical flows (e.g., ocean circulation driven by surface heating/cooling) and IHC models stellar interiors or planetary cores, these corrected scaling laws ($1/3$ vs $1/2$) are crucial for accurate extrapolations of heat transport in these natural systems.
• Clarification of IHC Physics: The paper resolves potential confusion regarding IHC scaling by distinguishing between pure volumetric heating (boundary-limited, $1/3$) and source-sink mixing (bulk-limited, potentially $1/2$).

In conclusion, Shishkina and Lohse provide a robust theoretical framework demonstrating that the ultimate regime scaling for horizontal and internally heated convection is governed by a $1/3$ power law, driven by the specific structure of their global energy balances, distinct from the $1/2$ law of Rayleigh–Bénard convection.