Predrag Cvitanović: Turbulence in spacetime

Abstract

For two centuries we have had the equations that describe the motion of fluids, but we cannot solve them where we need them. For pipe, channel and plane flows for long time intervals, on large spatial domains, turbulent instabilities make any accurate numerical time integration difficult. However, recent progress in ‘compressing’ turbulence data by equation-assisted thinking, in terms of so-called ‘exact coherent structures’ suggests a radically different approach. The way we perceive turbulence – the mere fact one can identify a cloud in a snapshot – suggests these terabytes should be zipped into small labelled files, a label for each pattern explored by turbulence, and a graph of transitions among them. This pattern recognition problem is exceptionally constrained by the exact differential equations that the data must respect. Here the Navier-Stokes equations are recast as a space-time theory, with both space and time taken to infinity, the traditional Direct Numerical Simulation codes have to be abandoned. In this theory there is no time, there is only a repertoire of admissible spatiotemporal patterns. To determine these, radically different kinds of codes will have to be written, with space and time treated on equal footing.