Fake turbulence*

Turbulence is a high-dimensional dynamical system with known equations of motion. It can be numerically integrated, but the simulation results are also high-dimensional and hard to interpret.Reduced models project the flow evolution onto a lower-dimensional subspace, but they are not dynamical systems because some dynamics is discarded in the projection. A stochastic transfer operator then substitutes the equations of motion. Using as an example a turbulent channel flow at moderate but non-trivial Reynolds number, Re_τ ≈ 1000, we show that some projections containparticularly deterministic subsets that can be interpreted as coherent structures. We also show that these can be used to construct data-driven `fake' Markovian models that reproduce many of the statistical characteristics of the flow, including some in which the operator has not apparently been trained. This can be traced to the underlying eigenstructure of the transfer operator.