Fast spectral solutions of the double-gyre problem in a turbulent flow regime

•Developing a simplified model of the double-gyre problem based on layer-averaged Navier–Stokes equations.•Presenting multi-scale spectral solutions to the simplified double-gyre model.•Verification of the resulting spectral solutions with CABARET solutions of a quasi-geostrophic double-gyre model.•Validation of the spectral solutions for integral dynamic quantities and zonally averaged flow profiles. Several semi-analytical models are considered for a double-gyre problem in a turbulent flow regime for which a reference fully numerical eddy-resolving solution is obtained. The semi-analytical models correspond to solving the depth-averaged Navier–Stokes equations using the spectral Galerkin approach. The robustness of the linear and Smagorinsky eddy-viscosity models for turbulent diffusion approximation is investigated. To capture essential properties of the double-gyre configuration, such as the integral kinetic energy, the integral angular momentum, and the jet mean-flow distribution, an improved semi-analytical model is suggested that is inspired by the idea of scale decomposition between the jet and the surrounding flow.