In this work, we apply a particle filter with three additional procedures
(model reduction, tempering and jittering) to a damped and forced
incompressible 2D Euler dynamics defined on a simply connected bounded domain.
We show that using the combined algorithm, we are able to successfully
assimilate data from a reference system state (the ``truth") modelled by a
highly resolved numerical solution of the flow that has roughly $3.1\times10^6$
degrees of freedom for $10$ eddy turnover times, using modest computational
hardware.
The model reduction is performed through the introduction of a stochastic
advection by Lie transport (SALT) model as the signal on a coarser resolution.
The SALT approach was introduced as a general theory using a geometric
mechanics framework from Holm, Proc. Roy. Soc. A (2015). This work follows on
the numerical implementation for SALT presented by Cotter et al, SIAM
Multiscale Model. Sim. (2019) for the flow in consideration. The model
reduction is substantial: The reduced SALT model has $4.9\times 10^4$ degrees
of freedom.
Forecast reliability and estimated asymptotic behaviour of the particle
filter are also presented.