Reduction of deterministic coupled atmosphere–ocean models to stochastic ocean models: a numerical case study of the Lorenz–Maas system

Abstract

Our starting point is the Lorenz–Maas coupled atmosphere–ocean model which was proposed by van der Schrier and van Veen and couples the ocean model by Maas with the Lorenz-84 model of the atmosphere. This 6-dimensional model (3-dimensional slow ocean and 3-dimensional fast atmosphere) is, to the knowledge of the authors, the simplest atmosphere–ocean model presently available which is derived from first principles in a controlled manner. This paper is an extensive numerical case study of the model, thereby implementing ‘Hasselmann's program,’ i.e. applying the various mathematical techniques of reducing the fully coupled deterministic model to a deterministic or stochastic model for the ocean alone, namely to: • the (deterministic) ‘statistical model,’ using the method of averaging, • the ‘linear stochastic model,’ based on the central limit theorem for the error in averaging, • the ‘nonlinear stochastic model,’ also known as ‘Hasselmann's equation’. The long-term and bifurcation behaviour of these models are studied and compared. The general result is that in most situations the nonlinear stochastic model outperforms the other ones.