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.