Entropy, Closures and Subgrid Modeling

Abstract

Maximum entropy states or statistical mechanical equilibrium solutions have played an important role in the development of a fundamental understanding of turbulence and its role in geophysical flows. In modern general circulation models of the earth's atmosphere and oceans most parameterizations of the subgrid-scale energy and enstrophy transfers are based on ad hoc methods or ideas developed from equilibrium statistical mechanics or entropy production hypotheses. In this paper we review recent developments in nonequilibrium statistical dynamical closure theory, its application to subgrid-scale modeling of eddy-eddy, eddy-mean field and eddy-topographic interactions and the relationship to minimum enstrophy, maximum entropy and entropy production arguments.

Statistical Dynamical Closures and Subgrid Modeling for Inhomogeneous QG and 3D Turbulence

“…Frederiksen [4] derived general expressions for the eddy-topographic force, eddy viscosity and stochastic backscatter based on the QDIA closure. O'Kane and Frederiksen [18] and Frederiksen and O'Kane [1] calculated and analysed inhomogeneous subgrid-scale parameterizations for observed atmospheric flows over global topography and compared the strengths of the subgrid-scale eddy-topographic, eddy-mean field, quadratic mean and mean field-topographic terms.…”

“…In recent years there have been significant advances in the formulation and applications of subgrid-scale parameterizations to quasigeostrophic (QG) and three-dimensional (3D) turbulence as reviewed by Frederiksen and O'Kane [1]. Statistical dynamical closure theory or stochastic modeling methods form the basis of much of this work.…”

“…They also applied the ȕ-plane QDIA closure to the issue of predictability and this was further pursued in a detailed examination of ensemble prediction during blocking regime transitions by O'Kane and Frederiksen [16]. O'Kane and Frederiksen [17] applied the ȕ-plane QDIA closure to data assimilation while O'Kane and Frederiksen [18] and Frederiksen and O'Kane [1] examined the numerical evaluation of the QDIA subgrid model of Frederiksen [4] for equilibrium and non-equilibrium turbulent flows on f-and ȕ-planes.…”

Entropy production and self-organized (sub)criticality in earthquake dynamics

“…Explicitly, Martyushev & Seleznev (2006) state that 'attempts to derive the maximum entropy principle have so far been unconvincing because they often involve additional hypotheses'. Frederiksen & O'Kane (2008) suggest that this is not surprising, as far-from-equilibrium phenomena are often characterized by multiple stationary states and hysteresis. Martyushev & Seleznev (2006) also explain the apparent paradox that minimum entropy production can be viewed as the maximum possible entropy production rate for a system otherwise strongly constrained to be near equilibrium, for example, heat conduction in a solid.…”

Stochastic subgrid-scale modelling for non-equilibrium geophysical flows

“…Most parametrizations have represented the unresolved processes through heuristic specifications of eddy diffusion (Frederiksen et al 1996), or through arguments based on baroclinic instability (Gent et al 1995) or ideas based on equilibrium statistical mechanics or entropy production hypotheses (Frederiksen & O'Kane 2008). Leith (1971) and Kraichnan (1976) developed a theoretical foundation for subgrid-scale modelling based on closure theory for homogeneous turbulence.…”