Abstract: Scalar Turbulence and Heat Transport Scaling in High-Rayleigh Number Convection

This structured literature review synthesizes the theoretical and experimental foundations concerning scalar turbulence and heat transport scaling in high-Rayleigh number (Ra) Rayleigh–Bénard convection (RBC), focusing on the critical role of thermal boundary layers (TBLs). The primary objective is to critically assess the evolution, central tenets, and ongoing debates surrounding the dominant heat flux scaling laws, particularly the predicted $\text{Nu} \propto \text{Ra}^{2/7}$ relationship.

The review traces the evolution of understanding from the classical $\text{Nu} \propto \text{Ra}^{1/3}$ prediction to the foundational Shraiman and Siggia (SS) $\text{Ra}^{2/7}$ scaling, which hinges on a passive scalar approximation for temperature advection within the turbulent bulk and distinct boundary layer turbulence dynamics. Key themes explored include the interplay between bulk turbulence (often described by Kolmogorov scaling) and the unique characteristics of the TBLs, the mechanism of plume dynamics (the primary mode of heat transport), and the theoretical structure proposed by Grossmann and Lohse (GL), which attempts a unified description of the Nusselt ($\text{Nu}$) and Reynolds ($\text{Re}$) numbers across various Ra and Prandtl ($\text{Pr}$) regimes.

A central finding is the persistent, yet increasingly constrained, debate between the $\text{Ra}^{2/7}$ and $\text{Ra}^{1/3}$ exponents, with modern high-Ra experiments and Direct Numerical Simulations (DNS) often yielding exponents that cluster near $0.28$ ($\approx 2/7$), especially in the proposed ‘soft’ or ‘intermediate’ turbulent regime. The primary conflicting viewpoint remains the validity of the passive scalar assumption in the bulk of this active, buoyancy-driven flow, which directly influences the predicted boundary layer velocity and temperature profiles.

Significant gaps remain in fully characterizing the flow structure in the proposed ‘ultimate’ regime ($\text{Ra} > 10^{14}$), particularly regarding the detailed scaling of the TBL velocity and the definitive role of the Large Scale Circulation (LSC). Future research should focus on high-fidelity, high-Ra DNS with sufficient resolution to resolve TBL microstructure, novel experimental techniques to directly measure logarithmic velocity profiles within the TBL, and advanced theoretical modeling that incorporates the full non-passive nature of the temperature field to reconcile observed scaling with theoretical predictions across all relevant Ra numbers.

The core findings and theoretical frameworks within the literature review on Scalar Turbulence and Heat Transport Scaling in High-Rayleigh Number Convection offer significant conceptual and structural insights that could be useful for addressing the Navier-Stokes Millennium Problem (specifically, the question of existence and smoothness of solutions for the 3D incompressible Navier-Stokes equations).

The utility stems from the literature’s focus on:

Scaling Laws and Singularities (The Core Problem)

The Millennium Problem is fundamentally about understanding whether the Navier-Stokes equations can lead to a singularity (infinite energy dissipation or velocity) in finite time.

• Turbulence Scaling ($\text{Nu} \propto \text{Ra}^{2/7}$): The literature review details how the $\text{Nu} \propto \text{Ra}^{2/7}$ and similar scaling laws are derived from assumptions about the structure of turbulence (like Kolmogorov $\text{K41}$ scaling) and the balance of energy/fluxes in the governing equations. These scaling relations are empirical and theoretical efforts to characterize the behavior of solutions at extreme parameters ($\text{Ra} \to \infty$).
• Analogy to Singularities: The theoretical debate between $\text{Nu} \propto \text{Ra}^{2/7}$ and the classical $\text{Ra}^{1/3}$ is essentially a debate over how energy dissipates as the system becomes more turbulent. A singularity in the Navier-Stokes equations would represent a point of infinite energy/vorticity dissipation. The scaling laws, while not proving or disproving singularities, provide a quantitative framework for how solutions should behave in the limit of infinite driving force ($\text{Ra}$), forcing theorists to identify the critical physical mechanism (e.g., the thermal boundary layer dynamics in the Shraiman & Siggia theory) that controls the flow.

Boundary Layers and Energy Dissipation

The literature emphasizes the crucial distinction between bulk turbulence and thermal boundary layers (TBLs).

• Dissipation Localization: In high-$\text{Ra}$ convection, a significant portion of the total energy (both kinetic and thermal) dissipation is confined to the thin TBLs. The Grossmann-Lohse (GL) theory explicitly formalizes this by partitioning the total dissipation into contributions from the bulk and the boundary layers.
• Relevance to Navier-Stokes: The Millennium Problem requires understanding if localized regions of extreme energy concentration can form. The $\text{RBC}$ studies show a physical mechanism for concentrating dissipation (the TBLs). Mathematical analysis of the Navier-Stokes equations could draw on this by investigating if the boundary layer structure provides a natural “regularizing” mechanism or, conversely, a prime location for the growth of potentially singular gradients.

Passive vs. Active Scalar Turbulence

The conflicting viewpoints section is highly relevant.

• Passive Scalar Approximation: The $\text{Nu} \propto \text{Ra}^{2/7}$ scaling relies on the assumption of passive scalar turbulence in the bulk (where temperature acts as a passive tracer). This is a simplification that allows for cleaner mathematical analysis.
• Mathematical Simplification: For the Navier-Stokes Problem, a common approach is to study simplified, related equations (like the Euler equations or 2D Navier-Stokes) that do have global smooth solutions. The $\text{RBC}$ literature demonstrates how the results change fundamentally when the active nature of the temperature field (buoyancy, the driving force) is correctly accounted for, moving beyond the passive scalar simplification. This provides a test case: any proposed proof for 3D Navier-Stokes must hold for the full, non-simplified equations where buoyancy is active.

In summary, the $\text{RBC}$ literature provides a mathematically tractable, physically realized system of equations closely related to Navier-Stokes ($\text{RBC}$ is $\text{Navier-Stokes} + \text{Temperature Field} + \text{Boussinesq}$ approximation). The efforts to derive and validate scaling exponents force a deep confrontation with the structure of solutions at high Reynolds numbers, which is the exact regime where the existence and smoothness of the pure Navier-Stokes solutions are questioned.