In this thesis I study analytical properties and applications to data assimilation for nonlinear
stochastic transport partial differential equations which originate in fluid dynamics. The thesis
has two parts.
In the first part (Chapters 3-5) I focus on theoretical results for three two-dimensional stochastic
transport models. I prove analytical properties (existence, uniqueness, continuity with respect
to initial conditions) for the corresponding classes of stochastic partial differential equations
(SPDEs) driven by transport noise: the stochastic Euler equation (SE), the stochastic great lake
equation (SGLE), and the stochastic rotating shallow water (SRSW) model. The well-posedness
strategy is based on constructing an approximating sequence of solutions which is proven to be
relatively compact and to converge to the solution of the original equation in a suitable sense.
The solution is global for the 2D SE equation and local for the SGLE and for the SRSW model.
In the second part (Chapter 6) I prove the applicability of one of these models (the SRSW
model) in a data assimilation setting. I implement a data assimilation methodology based on
a bootstrap particle filter combined with two additional procedures (tempering and jittering).
The methodology is tested first on the Lorenz ’63 model and then applied to the SRSW model.