Both geophysical fluid-dynamical (GFD) systems and magnetohydrodynamical (MHD) systems are governed by symmetries that manifest as conservation laws, such as those for energy, momentum, and circulation. Preserving these conservation laws in numerical simulations improves long-time stability and physical fidelity. Simultaneously, fully resolving the wide range of scales motion in these dynamical systems is computationally prohibitive, necessitating complexity reduction that fulfils the constraints imposed by the conservation laws.

We will highlight the applicability of circulation-preserving noise (stochastic advection by Lie transport, SALT) and energy-preserving noise (stochastic forcing by Lie transport, SFLT) as possible structure-preserving approaches for reduced-complexity modelling of GFD and MHD systems. These methods introduce carefully designed stochastic terms to account for the influence of unresolved dynamics while maintaining key physical invariants. We will present ongoing work focused on understanding the qualitative dynamics of these stochastic methods, which serves as a preliminary step in the development of reduced-complexity models for, e.g., uncertainty quantification and data assimilation.