From classical to statistical ocean dynamics

Traditional ocean modeling treats fields resolved oil the model grid according to the classical dynamics of continua. Variability on smaller scales is included through sundry "eddy viscosities", "mixing coefficients" and other schemes. In this paper we develop an alternative approach based oil statistical dynamics. First, we recognize that we treat probabilities of flows, not the flows themselves. Modeled dependent variables Lire the moments (expectations) of the probabilities of possible flows. Second, we address the challenge to obtain file equations of motion for the moments of probable flows rather than file (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 "eddy viscosity", 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 arc also evident; here, two phenomena arc described where the results differ. We consider eddies interacting with bottom topography. It is seen that traditional "eddy viscosity" and/or "topographic drag", which would reduce large scale flows toward rest, Lire wrong. The second law of thermodynamics is violated; the "arrow of time" is running backwards! From statistical dynamics, approximate corrections Lire 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 Lire stably stratifying, their rates of turbulent transfer should differ. This suggests a further model improvement.