  We consider a non-linear Bayesian data assimilation model for the periodic
two-dimensional Navier-Stokes equations with initial condition modelled by a
Gaussian process prior. We show that if the system is updated with sufficiently
many discrete noisy measurements of the velocity field, then the posterior
distribution eventually concentrates near the ground truth solution of the time
evolution equation, and in particular that the initial condition is recovered
consistently by the posterior mean vector field. We further show that the
convergence rate can in general not be faster than inverse logarithmic in
sample size, but describe specific conditions on the initial conditions when
faster rates are possible. In the proofs we provide an explicit quantitative
estimate for backward uniqueness of solutions of the two-dimensional
Navier-Stokes equations.
