Abstract
We parameterize the error in the time-relaxation model of turbulence by means of a deterministic corrector with stochastic noise term that represents the unresolved effects of the small scales on the large scales. Our method uses the large scales from a reference solution of the Navier–Stokes equations to compute the corrections needed to ensure the large scales from the turbulence model are the same as those in the reference solution over time. In the absence of model error, the theory of determining modes implies the size of the corrections needed over time decrease to zero provided enough Fourier modes were observed from the reference solution. In the presence of model error, the corrections do not decrease to zero. In this case, we call them the unresolved tendencies. After a theoretical analysis on the size of the these tendencies, our main result parameterizes them using a least-squares fit and characterizes the resulting residual as a time-correlated Gaussian noise processes. In doing so, we not only obtain an improved turbulence model, but a concrete representation of the error in the time-relaxation model of turbulence.