Scale-dependence of Error Growth in Turbulence

We examine the rate of error growth in turbulence simulations as a function of the length scales at which the uncertainties in the initial fluid state reside.To this end, we carry out large-eddy and direct numerical simulations of the sinusoidally-driven three-dimensional fluid under periodic boundary conditions in all directions. In several settings, we estimate the rate of divergence of fluid states that are initially close to one another with deviations confined to a subset of length scales in the system. We find that smaller-scale fluctuations in the initial condition translate to faster-growing errors, resulting in finite-time predictions that deteriorate quicker. We discuss the implications of our results for weather forecasting, where it is well known that atmospheric phenomena change the faster the smaller are their spatial scales.