# Radiative heating achieves the ultimate regime of thermal convection

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Edited by Isaac M. Held, Geophysical Fluid Dynamics Laboratory, National Oceanic and Atmospheric Administration, Princeton, NJ, and approved July 31, 2018 (received for review April 23, 2018)

## Significance

Turbulent convection is ubiquitous in geophysical and astrophysical contexts: It drives winds in the atmosphere and currents in the ocean, it generates magnetic fields inside planets and stars, and it triggers supernova explosions inside collapsing stellar cores. In many such natural flows, convection is driven by the absorption of incoming radiation (light or neutrinos). We designed an experiment to reproduce such radiatively driven convection in the laboratory. In contrast with convection driven by heating and cooling plates, our radiative heating setup achieves the “ultimate” regime of turbulent convection, which is the one relevant to many natural flows. Such experiments can yield the constitutive laws of turbulent convection, to be implemented into geophysical and astrophysical models.

## Abstract

The absorption of light or radiation drives turbulent convection inside stars, supernovae, frozen lakes, and Earth’s mantle. In these contexts, the goal of laboratory and numerical studies is to determine the relation between the internal temperature gradients and the heat flux transported by the turbulent flow. This is the constitutive law of turbulent convection, to be input into large-scale models of such natural flows. However, in contrast with the radiative heating of natural flows, laboratory experiments have focused on convection driven by heating and cooling plates; the heat transport is then severely restricted by boundary layers near the plates, which prevents the realization of the mixing length scaling law used in evolution models of geophysical and astrophysical flows. There is therefore an important discrepancy between the scaling laws measured in laboratory experiments and those used, e.g., in stellar evolution models. Here we provide experimental and numerical evidence that radiatively driven convection spontaneously achieves the mixing length scaling regime, also known as the “ultimate” regime of thermal convection. This constitutes a clear observation of this regime of turbulent convection. Our study therefore bridges the gap between models of natural flows and laboratory experiments. It opens an experimental avenue for a priori determinations of the constitutive laws to be implemented into models of geophysical and astrophysical flows, as opposed to empirical fits of these constitutive laws to the scarce observational data.

Thermal convection drives natural flows in the atmosphere, in the oceans, and in the interior of planets and stars. The resulting turbulence controls the convective heat transport, the typical wind speed in the atmosphere, the ability of planets and stars to produce magnetic fields, and the triggering of supernova explosions inside collapsing stellar cores. The cornerstone setup to study thermal convection is the Rayleigh–Bénard one (RB), in which fluid is heated from below by a hot plate and cooled from above by a cold one. One then relates the convective heat transport enhancement to the temperature difference between the plates: One seeks a power-law relation

However, despite decades of investigations of the RB setup, the asymptotic regime of turbulent heat transport remains strongly debated and an outstanding challenge of nonlinear physics and turbulence research (5⇓⇓⇓⇓–10). Indeed, although the interior flow is strongly turbulent and transports heat very efficiently, this heat first has to be diffused across the boundary layers near the top and bottom plates. The temperature gradient is then confined to these lazy boundary layers (1); standard dimensional analysis arguments give a heat transport exponent *Discussion*), and so is the heat flux; simple dimensional analysis then leads to

There is a considerable controversy over the possible experimental detection of the mixing length regime in the RB setup (5⇓⇓⇓⇓–10): Most experiments report an increase in the exponent γ at the highest achievable Rayleigh numbers, but a clear ultimate regime with

A clear observation of the mixing length regime is all the more desirable in that it is the scaling law currently used in astrophysical contexts. As an example, stellar evolution models are to be integrated over the lifespan of a star, which makes DNS of the fluid dynamics prohibitively expensive and requires a parametrization of the convective effects (4, 22⇓–24). Such parametrizations are based on the mixing length scaling law, and, in the absence of experimental data to back it up, one needs to fit the details of the convective model to the scarce observational data, which strongly restricts the predictive power of the whole approach. There is therefore an important research gap between laboratory experiments of turbulent convection, on the one side, and parametrizations of geophysical and astrophysical convection on the other.

The present study bridges this gap by proposing an innovative experimental strategy leading to the mixing length regime of thermal convection. It is based on the following observation: In contrast with RB convection, many natural flows are driven by a flux of light or radiation, instead of heating and cooling plates. A first example is the mixing of frozen lakes in the spring, due to solar heating of the near-surface water (25⇓–27). A second example is stellar interiors (28) and the sun in particular, inside of which heat radiated by the core is transferred through the radiative zone before entering the convective one (29⇓–31). A related third example is convection in Earth’s mantle, which may be internally driven by radioactive decay (32, 33). Finally, the powering of supernova explosions by neutrino absorption in the collapsing stellar core provides an additional astrophysical example (34⇓–36).

In this article, we report on an experimental setup showing that radiative heating spontaneously achieves the mixing length regime of thermal convection. Indeed, radiative heating differs drastically from the RB setup, with important consequences for the transported heat flux: Heat is input directly inside an absorption layer of finite extension ℓ. When this absorption length is thicker than the boundary layers, radiative heating allows us to bypass the boundary layers and heat up the bulk turbulent flow directly.

The combination of radiation and convection is also ubiquitous in atmospheric physics. Solar radiation heats up the ground, which, in turn, emits black-body radiation in the infrared. This outgoing radiation is absorbed by

Fig. 1*A* provides a sketch of the experimental setup: A high-throughput projector shines at a cylindrical cell which has a transparent bottom plate. The cell contains a homogeneous mixture of water and dye. Dye absorbs the incoming light over a typical height ℓ. Through Beer–Lambert law (41), this leads to a source of heat that decays exponentially away from the bottom boundary over a height ℓ: The local heating rate is proportional to *SI Appendix*, the practical implementation of radiative heating requires extreme care: First, the choice of the dye is critical, as it must have a uniform absorbance over the visible spectrum. Second, the powerful spotlight has an efficiency of roughly

Another key aspect of the experiment is to avoid boundary layers at the cooling side. Traditionally, studies of internally heated convection consider cooling at a solid isothermal boundary (42). The resulting boundary layers near the cold plate then control the heat transport efficiency and lead to scaling laws similar to those of standard RB convection (43). The approach developed herein deviates from standard experiments: We run the experiment in quasi-stationary state, the fluid being radiatively heated without a cooling mechanism. The resulting temperature field increases linearly with time. On top of this linear drift, the turbulent flow develops some stationary internal temperature gradients. At the mathematical level, this drifting situation is exactly equivalent to the fluid being radiatively heated and uniformly cooled at a rate equal and opposite to the heating power (see *SI Appendix*); the stationary internal temperature gradients measured experimentally are those of a fluid that is both internally heated and internally cooled, therefore bypassing both the heating and cooling boundary layers. This drift method is therefore a useful experimental tool to bypass the cooling boundary layers, but it also corresponds to practical situations: frozen lakes radiatively heating up in the spring and collapsing stellar cores heated by a flux of neutrinos. In a similar fashion, the secular cooling of planet interiors is routinely modeled as a uniform heating term in theoretical studies of planetary convection (44⇓–46).

## Results

We characterize the temperature field using two precision thermocouples, one touching the bottom sapphire plate and the second one at midheight, both being centered horizontally. Fig. 1*B* presents the typical time series recorded during an experimental run. The common drift of the two sensors gives access to the heat flux P transferred from the projector to the fluid:

Fig. 2 reports the experimental curves

To confirm the experimental data, we have performed DNS of radiatively driven convection. The domain is a 3D cube with stress-free sidewalls and no-slip boundary conditions at the top and bottom boundaries. We first fix the Prandtl number to the value of water at 20°C, *SI Appendix* for a proof of this exact equivalence). We extract the Rayleigh and Nusselt numbers once the simulations reach a statistically steady state. The corresponding data points are shown in Fig. 2; the agreement with the experimental data is excellent. Indeed, the numerical values of the exponent γ are 0.29 in case I and 0.55 in case II, confirming the experimental values with a precision of a few percent. The slight departure of the case II exponent from 0.5 may originate from a small finite-Rayleigh-number viscous correction. To test this hypothesis, we have performed additional DNS using *Bottom*, we show the compensated Nusselt number

## Discussion

Another open question concerning the ultimate regime of thermal convection is the behavior of the Nusselt number with respect to the Prandtl number. For RB convection, Kraichnan predicted that

Fig. 3 presents snapshots of the temperature field in the high-Rayleigh-number DNS. For case I, most of the temperature gradients are contained in a very thin boundary layer. These temperature gradients are strong, because only diffusion acts to evacuate the heat away from the heating region, where there is little or no advection. The high temperature values only seldom penetrate the bulk of the domain through the emission of narrow plumes of warm fluid. By contrast, in case II, the high-temperature regions extend significantly farther from the bottom plate, and penetrate the bulk turbulent flow through taller and wider thermal plumes. The horizontally averaged temperature profiles for these two snapshots are provided in *SI Appendix*, Fig. S1; for case I, the temperature is homogeneous in the bulk, with a sharp boundary layer near the bottom wall. For case II, the temperature decreases less rapidly with height, but we can still identify a small thermal boundary layer near the bottom wall. Defining its thickness

We conclude by stressing once again that the current theory of stellar evolution strongly relies on the mixing length scaling regime of turbulent convection (22, 23, 50); the simplest implementations of mixing length theory consist in fitting the mixing length to the scarce observational data, which limits the predictive power of the whole approach. Instead, one would like to determine the constitutive laws of turbulent convection a priori, based on experimental and numerical studies in idealized geometries. A prerequisite is that such experiments should achieve the mixing length regime of thermal convection. While the mixing length regime has yet to be clearly observed in the standard RB system, our radiative experiment provides a clear observation of this scaling regime. The present study therefore puts the mixing length scaling theory on a firmer footing by reconciling it with laboratory experiments. It paves the way for the a priori experimental determination of the convective parametrizations to be input into stellar evolution models, potentially including nonlocality and/or overshooting at an internal boundary (23, 24).

## Acknowledgments

The authors thank J. Guilet, T. Foglizzo, C. R. Doering, D. Goluskin, and V. Bouillaut for insightful discussions, and V. Padilla for building part of the experimental setup. This research is supported by the European Research Council under Grant Agreement FLAVE 757239, and by Agence Nationale de la Recherche Grant ANR-10-LABX-0039.

## Footnotes

↵

^{1}S.L., S.A., and B.G. contributed equally to this work.- ↵
^{2}To whom correspondence should be addressed. Email: basile.gallet{at}cea.fr.

Author contributions: S.L., S.A., and B.G. designed research, performed research, analyzed data, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1806823115/-/DCSupplemental.

Published under the PNAS license.

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