Applying the parameterising manifold approach to the wind-driven circulation

Abstract

Recently Chekroun et al. developed a method based on so called ‘parameterising manifolds’ to study stochastic (partial) differential equations. This method is promising for obtaining stochastic reduced models. We apply their method to a model of the wind-driven ocean ocean circulation at midlatitudes. To model the stream function ψ we use the quasi-geostrophic barotropic potential vorticity equation. The Reynolds number (R) is used as bifurcation parameter. ψ is susceptible to several types of bifurcations. We study the pitchfork bifurcation which occurs for R ≈ 36.6.

We apply reduction methods to model perturbations around the stationary state. First the deterministic case is considered and the parameterising manifold approach is compared to the Galerkin type of method used in Van der Vaart et al. It is shown that the ‘parameterising manifold’ approach may be seen as an extension of the Galerkin method. To include also baroclinic effects, we model baroclinic instabilities by noise on the perturbations. The stochastic reduced model we obtain from the parameterising manifold approach captures the dynamics of the full stochastic model well.