The theory is in striking agreement with pioneering experimental results in graphite published last year

Fourier's well-known heat equation, introduced in 1822, describes how temperature changes in space and time when heat flows through a material. In general, this formulation works well to describe heat conduction in objects that are macroscopic (typically, a millimeter or larger), and at high temperatures. It fails, however, in describing so-called "hydrodynamic heat phenomena."

One such phenomenon is Poiseuille heat flow, where the heat flux becomes similar to the flow of a fluid in a pipe: it has a maximum in the center and minima at the boundaries, suggesting that heat propagates as a viscous-fluid flow. Another, called "second sound" takes place when heat propagation in a crystal is akin to that of sound in air: given portions of the crystal oscillates quickly between being hot and cold, instead of following the gentle temperature variation observed in the usual (diffusive) propagation.

Neither of these phenomena is described by Fourier's equation. Until now, researchers have only been able to analyze these phenomena using microscopic models, whose complexity and high computational cost have hindered both understanding and application to anything but the simplest geometries. In contrast, in developing the novel viscous heat equations, MARVEL researchers have condensed all the relevant physics underlying heat conduction into accurate and easily-solvable equations. This introduces a novel basic research tool for the design of electronic devices, especially those integrating diamond, graphene or other low-dimensional or layered materials where hydrodynamic phenomena are now understood to be prevalent. {module INSIDE STORY}

The work is particularly timely. While these heat hydrodynamic phenomena have been observed since the 1960s, they were only seen at cryogenic temperatures (around -260 degrees C) and therefore thought to be irrelevant for everyday applications. These beliefs suddenly changed last March with the publication in Science of pioneering experiments that found second-sound (or wavelike) heat propagation in graphite employed in several engineering devices and promising material for next-generation electronics at the record temperature of -170 degrees C.

The novel formulation presented in the paper Generalization of Fourier's law into viscous heat equations yields results for graphite that are in striking agreement with these experiments and also predicts that this hydrodynamic heat propagation can be observed in diamond even at room temperature. This prediction is awaiting experimental confirmation, which would establish a new record for the maximum temperature at which hydrodynamic heat transfer is observed.

The work is very relevant for applications since such hydrodynamic heat propagation can emerge in materials for next-generation electronic devices, where overheating is the main limiting factor for miniaturization and efficiency. Knowing how to handle the heat generated in these devices is critical to understanding how to maximize their efficiency, or even predict if they will work or just melt because of overheating. The paper provides new and original insights into transport theories and also paves the way towards the understanding of shape and size effects, e.g., new-generation electronic devices and so-called "phononic" devices that control cooling and heating. Finally, this novel formulation can be adapted to describe viscous phenomena involving electricity, discovered by Philip Moll in 2017, now a professor at the Institute of Materials at EPFL.

For the mathematically inclined

In this work, MARVEL researchers have coarse-grained the microscopic integrodifferential phonon Boltzmann transport equation into mesoscopic (simpler) differential equations, which they have called "viscous heat equations". These viscous heat equations capture the regime where the atomic vibrations in a solid ("phonons") assume a collective ("drift") velocity akin to that of a fluid. They have shown how thermal conductivity and viscosity can be determined exactly and in closed-form as a sum over the eigenvectors of the scattering matrix (the "relaxons", a concept introduced in 2016 by Cepellotti, for which he was awarded the IBM Research Prize and the Metropolis Prize of the American Physical Society). Relaxons have well-defined parities, with even relaxons determining the thermal viscosity and odd relaxons determining the thermal conductivity, and thermal conductivity and viscosity govern the evolution of the temperature and drift-velocity fields in these two coupled viscous heat equations.

In the paper, the scientists also introduced a Fourier deviation number (FDN), a dimensionless parameter that quantifies the deviation from Fourier's law due to hydrodynamic effects. The FDN is a scalar descriptor that captures the deviations from Fourier's law due to viscous effects, playing a role analogous to the Reynolds number for fluids, which is a parameter that engineers use to distinguish the different possible behaviors of the solutions to the Navier-Stokes equations.

Climate costs are likely smallest if global warming is limited to 2 degrees Celsius; the politically negotiated Paris Agreement is thus also the economically sensible one, Potsdam researchers find in a new study

"To secure economic welfare for all people in these times of global warming, we need to balance the costs of climate change damages and those of climate change mitigation. Now our team found what we should aim for," says Anders Levermann from the Potsdam Institute for Climate Impact Research (PIK) and Columbia University's LDEO, New York, head of the team conducting the study. "We did a lot of thorough testing with our computers. And we have been amazed to find that limiting the global temperature increase to 2°C, as agreed in the science-based but the highly political process leading towards the 2015 Paris Agreement, indeed emerges as economically optimal."

Striving for economic growth

Climate policies such as for instance the replacement of coal-fired power plants by windmills and solar energy, or the introduction of CO2 pricing, have some economic costs. The same is true for climate damages. Cutting greenhouse gas emissions clearly reduces the damages, but so far observed temperature-induced losses in economic production have not really been accounted for in computations of economically optimal policy pathways. The researchers now did just that. They fed up-to-date research on economic damages driven by climate change effects into one of the most renowned supercomputer simulation systems, the Dynamic Integrated Climate-Economy model developed by the Nobel Laureate of Economics, William Nordhaus, and used in the past for US policy advice. The supercomputer simulation is trained to strive for economic growth. 

"It is remarkable how robustly reasonable the temperature limit of more or less 2°C is, standing out in almost all the cost-curves we've produced," says Sven Willner, also from PIK and an author of the study. The researchers tested a number of uncertainties in their study. For instance, they accounted for people's preference for consumption today instead of consumption tomorrow versus the notion that tomorrow's generations should not have fewer consumption means. The result that the 2°C limit is the economically most cost-efficient one was also true for the full range of possible climate sensitivities, hence the amount of warming that results from a doubling of CO2 in the atmosphere.

"The world is running out of excuses for doing nothing"

"Since we have already increased the temperature of the planet by more than one degree, 2°C requires fast and fundamental global action," says Levermann. "Our analysis is based on the observed relationship between temperature and economic growth, but there could be other effects that we cannot anticipate yet." Changes in the response of societies to climate stress - especially a violent flare-up of smoldering conflicts -, or the crossing of tipping points for critical elements in the Earth system could shift the cost-benefit analysis towards even more urgent action. {module INSIDE STORY}

"The world is running out of excuses to justify sitting back and doing nothing - all those who have been saying that climate stabilization would be nice but is too costly can see now that it is really unmitigated global warming that is too expensive," Levermann concludes. "Business, as usual, is clearly not a viable economic option anymore. We either decarbonize our economies or we let global warming fire up costs for businesses and societies worldwide."