Geometric mechanics is a framework for the derivation and analysis of dynamical systems based on the Lie group transformation properties of the variational principles from which they arise. For example, geometric mechanics preserves the properties of energy conservation, Kelvin-Noether circulation dynamics, and the conservation laws resulting from the particle relabelling symmetry of ideal geophysical fluid dynamics (GFD). GFD models of real-world dynamics such as climate change involve interactions among disparate scales in space and time whose approximate modelling can introduce uncertainty. To meet this type of challenge, this thesis focuses on developing modelling schemes for introducing noise and coupling among multiscale multiphysics dynamics within the geometric mechanics framework.

First, we develop a structure-preserving stochastic perturbation framework known as the Stochastic Forcing by Lie Transport (SFLT) using the reduced Lagrange-d'Alembert Pontryagin principle from geometric mechanics. When applied to GFD, SFLT preserves the conservation of energy and modifies the Kelvin-Noether circulation dynamics. Physically, the stochastic forces introduced by SFLT encompass stochastic variants of Craik-Leibovich forces and material entrainment effects. The Eulerian Average SFLT (EA SFLT) framework is also proposed as a modification of the SFLT framework where one replaces the fluid momentum and advected quantities with their expectations over the underlying probability space.

Then, we formulate a class of wave-current interaction systems where the wave dynamics are assumed to be symmetry-breaking features of the current. The resulting equation contains the wave momentum, which links the tangled and untangled Poisson structure of the system. We show that this modelling approach is flexible enough to readily incorporate the stochastic perturbation frameworks of both SALT (Stochastic Transport by Lie Transport) and SFLT (Stochastic Forcing by Lie Transport). Lastly, we exemplify the wave-current interaction modelling approach by constructing three two-dimensional models to study satellite imagery of the ocean.