  This article is concerned with the problem of determining an unknown source
of non-potential, external time-dependent perturbations of an incompressible
fluid from large-scale observations on the flow field. A relaxation-based
approach is proposed for accomplishing this, which leverages a nonlinear
property of the equations of motions to asymptotically enslave small scales to
large scales. In particular, an algorithm is introduced that systematically
produces approximations of the flow field on the unobserved scales in order to
generate an approximation to the unknown force; the process is then repeated to
generate an improved approximation of the unobserved scales, and so on. A
mathematical proof of convergence of this algorithm is established in the
context of the two-dimensional Navier-Stokes equations with periodic boundary
conditions under the assumption that the force belongs to the observational
subspace of phase space; at each stage in the algorithm, it is shown that the
model error, represented as the difference between the approximating and true
force, asymptotically decrements to zero in a geometric fashion provided that
sufficiently many scales are observed and certain parameters of the algorithm
are appropriately tuned; the issue of the sharpness of the assumptions, among
other practical considerations such as the transient periods between updates,
are also discussed.
