# Heat Rises...But How Fast?

Everyone knows that hot fluid rises and cold fluid sinks. These buoyancy driven flows, known as thermal convection, are physical phenomena of great importance for engineering applications and throughout the geophysical and astrophysical sciences. Heat transfer by natural convection is a basic consideration for insulation, ventilation, and climate control in buildings, in automotive engineering and for the thermal management of electronic devices and systems. Thermal convection occurs naturally in the earth’s atmosphere and oceans and plays a fundamental role in the global heat transfer processes determining the weather and controlling the climate. On geological time scales, convection in the earth’s mantle is the driving force behind continental drift. On the largest scales, convective heat transport in stars is a primary factor in the balance between ongoing thermonuclear explosions in the core and gravitational attraction that determines their structure.

Despite its ubiquity in everyday life, convection is one of the most challenging and–––especially when the fluid flow is turbulent–––largely unsolved problems in nonlinear physics and mathematics. Now U-M Physicist and Mathematician Charles Doering and U-M Applied & Interdisciplinary Mathematics Graduate Student Jared Whitehead have proven that turbulent convection is much more subtle than previously expected and that a popular theoretical prediction for the magnitude of the heat transport is, at least in some cases, a significant overestimate.

Ever since laboratory experiments by Henri Bénard at the turn of the 20th century and a mathematical model proposed by Lord Rayleigh in 1916, scientists have been trying to understand the spontaneous flows and resulting heat transfer enhancement that occurs when a layer of fluid is heated from below and cooled from above. Extreme temperature differences, indicated by large values of the dimensionless Rayleigh number (Ra), produce turbulent flows that actively transport much more heat across the layer than the fluid would conduct via thermal diffusion (diffusion is the purely molecular process of heat transfer if the fluid remained at rest). The convective heat transport enhancement factor is called the Nusselt number (Nu) and for many decades, experimental, computational, and theoretical physicists have sought to discern how Nu depends on Ra. It is generally expected that the Nusselt number increases as a power of the Rayleigh number, i.e., Nu~Ra^{ß} in the turbulent regime. The scaling exponent ß of the power law characterizes the transport process.

For nearly fifty years, one popular theory has held that molecular processes should become negligible in the "ultimate" high-Ra regime of turbulent convection. That theory implies that ß=1/2 so that Nu would be proportional to the square root of Ra. In this paper, Whitehead and Doering prove that for Lord Rayleigh’s original model the heat transport scaling exponent ß = 5/12 = .41666 ... < 1/2. In geophysical and astrophysical applications, the Rayleigh number can be as large as 10^{20} in which case the over-estimate of the conventional theory is significant. Moreover, this new rigorous mathematical result may help to explain why current experimental attempts to observe the conventional “ultimate” regime have been so difficult.

VIDEO CAPTION: *Direct numerical simulation of turbulent Rayleigh-Bénard convection. The color indicates temperature: red is hot, blue is cold, and green is average. In this simulation the fluid sticks to the bottom and top boundaries and the heat flux through the boundaries is fixed. Although these boundary conditions produce a quantitatively different heat transport scaling law, the fluid dynamics are qualitatively similar to the model studied by Whitehead & Doering.*