Is space time? A spatiotemporal theory of transitional turbulence

Recent advances in fluid dynamics reveal that the recurrent flows observed in moderate Reynolds number turbulence result from close passes to unstable invariant solutions of Navier-Stokes equations. By now hundreds of such solutions been computed for a variety of flow geometries, but always confined to small computational domains (minimal cells).

However, it is possible to determine such solutions on spatially infinite domains. Flows of interest (pipe, channel flows) often come equipped with D continuous spatial symmetries. If the theory is recast as a (D+1)-dimensional space-time theory, the space-time invariant solutions are (D+1)-tori (and not the 1-dimensional periodic orbits of the traditional periodic orbit theory). The symbolic dynamics is likewise (D+1)-dimensional (rather than a single temporal string of symbols), and the corresponding zeta functions should be sums over tori, rather than 1-dimensional periodic orbits. In this theory there is no time, there is only a repertoire of admissible spatiotemporal patterns.