Abstract :
Traditional ocean modeling treats fields resolved on the model grid according to the classical dynamics of continua. Variability on smaller scales is included through sundry ldquoeddy viscositiesrdquo, ldquomixing coefficientsrdquo and other schemes. In this paper we develop an alternative approach based on statistical dynamics. First, we recognize that we treat probabilities of flows, not the flows themselves. Modeled dependent variables are the moments (expectations) of the probabilities of possible flows. Second, we address the challenge to obtain the equations of motion for the moments of probable flows rather than the (traditional) equations for explicit flows. For linear terms and on larger resolved scales, the statistical equations agree with classical dynamics where those of traditional modeling works well. Differences arise where traditional modeling would relegate unresolved motion to ldquoeddy viscosityrdquo, etc.. Instead, changes of entropy (<-log P> over the probability distribution of possible flows) with respect to the modeled moments act as forcings upon those moments. In this way we obtain a consistent framework for specifying the terms which, traditionally, represent subgridscale effects. Although these statistical equations are close to the classical equations in many ways, important differences are also evident; here, two phenomena are described where the results differ. We consider eddies interacting with bottom topography. It is seen that traditional ldquoeddy viscosityrdquo and/or ldquotopographic dragrdquo, which would reduce large scale flows toward rest, are wrong. The second law of thermodynamics is violated; the ldquoarrow of timerdquo is running backwards! From statistical dynamics, approximate corrections are obtained, yielding a practical improvement to the fidelity of ocean models. Another phenomenon occurs at much smaller scales in the turbulent mixing of heat and salt. Even when both heat and salt are stably stratifying, their rates of turbulent transfer should differ. This suggests a further model improvement.
