Derivation and Numerical Assessment of a Stochastic Large–Scale Hydrostatic Primitive Equations Model

Planetary flows are shaped by interactions at scales much smaller than the flows themselves, with mesoscale and sub–mesoscale eddies playing key roles in mixing, particle transport and tracer dispersion. To capture these effects, we introduce a stochastic formulation of the primitive equations within the Location Uncertainty (LU) framework. Derived from conservation principles via a stochastic Reynolds transport theorem, this approach decomposes velocity into a smooth–in–time large–scale component and a random–in–time field representing unresolved scales effects. To model the velocity noise term, we develop two data–driven methods based on Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD) and extend this to hybrid approaches combining model– and data–driven constraints. Simulations show that the LU framework enhances gyre flow predictions, improving mixing, jet structure, and tracer transport while revealing the interplay between small– and large–scale dynamics.