Turbulent convection in a rotating body is a common but poorly understood phenomenon in astrophysical and geophysical settings. Consideration of boundary effects offers a fresh angle on this thorny problem.
Heat energy is transported by three basic processes: conduction, radiation and convection. But which of them is likely to dominate in any given circumstance? In particular, how can we quantify the relative efficiency of each process, not least of convection, which is inherently nonlinear and turbulent? This problem is especially challenging for understanding the evolution and interior structure of stars and planets — a task that is further complicated by the fact that most stars and planets are rotating, which may modify the style of convection and influence its efficiency.
In a study that combines laboratory measurements and numerical simulations over an unusually wide range of parameters, King and colleagues (page 301 of this issue)1 have systematically investigated the behaviour of convective heat transport in a rotating fluid. They focus on exploring the hypothesis that the type of convection that occurs is primarily governed by the ratio of the characteristic thicknesses of the principal (rotationally controlled) boundary layers (Fig. 1).
It is traditional in fluid mechanics to characterize quantities in terms of dimensionless ratios. So the total heat flux carried by a fluid can be quantified by comparing it with that obtained by conduction alone in a solid with the same thermal properties. The resulting ratio is known as the Nusselt number, Nu. This is convenient because, for many problems, the conductive heat flux is relatively easy to calculate from basic geometrical and physical properties of the fluid. Nu = 1 then corresponds to a thermal efficiency that is no better than conduction alone, whereas for convection on domestic or industrial scales, Nu takes typical values from about 10 up to several thousand.
But how should we extrapolate estimates of this convective efficiency to systems on the scale of planets or stars? From nearly 100 years of measurements and theory, several semi-empirical scaling laws have emerged that relate Nu to another dimensionless parameter, the Rayleigh number, Ra, which effectively measures the ratio of buoyancy to viscous forces in a fluid and is proportional to the imposed temperature contrast. In a non-rotating fluid, Nu has been measured2,3 to vary as ∼Raα, where α can take a range of values (2/7 ≤ α ≤ 0.31). The measurements cover an enormous range, including the extreme so-called 'hard' turbulence regime at values of Ra > 108. This might seem to provide a good indication of how to extrapolate estimates of Nu to values of Ra ∼ 1025–1030, typical of planets and stars.
However, if convection takes place in a rapidly rotating fluid, things are much more complicated. This is because rapid rotation imparts an extra 'rigidity' to a fluid, but only in a direction parallel to the axis of rotation (Fig. 1). This directional rigidity changes the turbulent motions in ways that are still poorly understood, but that may profoundly influence the efficiency of convective heat transport, suggesting a scaling law closer to ∼Ra6/5 under some conditions. Such a difference in scaling exponent can change the extrapolated Nusselt number by orders of magnitude, rendering estimates of planetary or stellar heat flow extremely uncertain.
According to conventional wisdom, the force balance between buoyancy effects in the interior of the convecting region and Coriolis forces (which, for example, cause the air flow around low-pressure centres in Earth's atmosphere to circulate in the same sense as Earth's rotation) is the main factor governing the influence of rotation on the convection — notably, its efficiency as measured by Nu. King et al.1 have examined whether this interior force balance provides a quantitative criterion for determining whether Nu scales either in the non-rotating limit as ∼Ra2/7 or in the rotating limit as ∼Ra6/5, but find that it doesn't predict the observed changeover between regimes.
In contrast, they concentrate on the role of the boundary layers in the problem. Typical convective flows are not generally compatible with the mechanical and thermal boundary conditions imposed at the edges of the convective region. The fluid generally gets around this by creating thin regions of adjustment, known as boundary layers, in which diffusive effects (viscous or thermal) become large enough to allow the interior flow to match the imposed boundary conditions. Two types of boundary layer seem to be of most significance: first, a thermal boundary layer, dominated by thermal conduction, whose thickness δT ≈ D/Nu, where D is a typical dimension of the convecting system, independent of rotation; and, second, the Ekman layer, within which viscous forces are comparable in size to Coriolis forces and whose thickness scales as δE ∼ (ν/Ω)½, where Ω is the rotation rate and ν the kinematic viscosity.
King et al. have accumulated impressive evidence from their laboratory and numerical modelling experiments to show that the scaling dependence of Nu on Ra depends on whether δE > δT, or vice versa. This relationship seems to apply over a wide range of conditions, and even seems to work for systems bounded by stress-free boundaries, representing an idealization of the top of the atmosphere of a planet or star.
The notion that the ratio δE/δT might determine the character and properties of rotating, stratified flows is not especially new. It was applied4 in the 1980s to such problems as models of the circulation in the ocean thermocline, which is currently the subject of a controversy concerning Sandström's theorem on the nature and energetics of the thermohaline ocean circulation5,6. In that problem, the (highly turbulent) Ekman and thermal boundary layers play a clear role in the oceanic meridional circulation, so their importance in governing convective heat flow is not surprising.
In the convective problem considered by King et al., however, it is less clear precisely what role the Ekman boundary layers play in the detailed transport of mass and heat. So the authors' results are all the more surprising and impressive. In the problems they have considered, however, heat enters or leaves the convecting region strictly by thermal conduction through the bounding walls, which is not exactly how heat is introduced or extracted in real planets or stars. So it remains to be seen to what extent their results can be applied to more general mechanisms for driving convection.