IAS Turbulence: ‘Big whirls have little whirls’

by Arthur Petersen

The phenomenon of ‘turbulence’, in the sense of turbulent flow, is omnipresent in the natural and engineering sciences. Turbulent flow, as opposed to smooth (called ‘laminar’) flow, is characterized by its irregular, unpredictable behaviour. Still, some important physical regularities can be observed, such as the scaling laws captured in verse by Lewis F. Richardson: ‘big whirls have little whirls that feed on their velocity, and little whirls have lesser whirls and so on to viscosity’.[i]

Scientists can experience beauty and intuit transcendence, that is, they can wonder. The emotion of wonder can make one appreciate what is great in science, and depends on an aesthetic sense of intellectual beauty. It comes with joy, delight, and pleasure. This can happen when we study the equations that describe the early universe, but also when we study a non-esoteric phenomenon such as turbulence. I illustrate this here with two examples from my own research, the first relating to equations, the second to patterns.[ii]

The Navier–Stokes equations, which were derived in the middle of the nineteenth century, account for all fluid dynamical phenomena, including turbulence. It is the ‘nonlinearity’ of these equations, the u_ju_i terms (multiplicative terms including velocities in all three directions, represented by the subscripts, which have values 1, 2, and 3) in the equations shown, that make them both analytically intractable and phenomenologically featuring the turbulence.

Since the middle of the twentieth century, numerical mathematics and computer science have made it possible to simulate turbulent flow using models that approximate the Navier–Stokes equations. What is remarkable, indeed astounding, is that all turbulent flows can be simulated on the basis of these equations.

The swirling patterns that arise in turbulent flow can be enjoyed both for the understanding of complex phenomena they make possible and for their beauty. Studies of the history of science document scientists talking about this visual beauty of fluid dynamics.[iii] The American Physical Society even awards an annual prize for the most beautiful videos or poster of fluid motion.[iv] Let me here give an example from my own work. Using the Cray supercomputer in the Netherlands at the end of the 1990s, I simulated turbulence in the convective atmospheric boundary layer (for instance, the bottom one and a half kilometres of the atmosphere on a sunny day in summer). In the figure [below/above],  the vertical movements of the air at 1300 metres are shown in a section of a convective atmosphere boundary layer six by six kilometres wide and one and a half kilometres high (discretised using a three-dimensional grid of 128 by 128 by 66 boxes). Light and dark shades correspond to upward and downward velocities, respectively. The figure therefore makes visible a large-scale organisation of air moving upwards, in ‘updrafts’, and air moving downwards, in ‘downdrafts’. What I find particularly striking about the behaviour of updrafts and downdrafts in convective turbulent flow is that their behaviour features universal regularities. Apparently, in terms of their physical behaviour and resulting patterns, it does not fundamentally matter whether the Navier–Stokes equations are describing convection of warm air in the earth’s atmosphere or of extremely hot hydrogen and helium gas in the sun. Of course, there are differences between different types of turbulent flow and there will always remain uncertainties in representing turbulent flow in simplified mathematical models. But fundamentally, the simplified equations that I developed for the earth’s atmosphere can also be used to describe convection in stars!