We consider a non-linear filtering problem, whereby the signal obeys the
stochastic Navier-Stokes equations and is observed through a linear mapping
with additive noise. The setup is relevant to data assimilation for numerical
weather prediction and climate modelling, where similar models are used for
unknown ocean or wind velocities. We present a particle filtering methodology
that uses likelihood informed importance proposals, adaptive tempering, and a
small number of appropriate Markov Chain Monte Carlo steps. We provide a
detailed design for each of these steps and show in our numerical examples that
they are all crucial in terms of achieving good performance and efficiency.