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This volume is an illuminating demonstration of a broad range of mathematical modeling strategies and analytical techniques as well as their relevance to studies of geophysical flows. A well-known reduced model for large-scale flows in the atmosphere and oceans, the barotropic (two-dimensional) quasi-geostrophic (QG) equations, and some close relatives are used to create a showcase of methodologies. The overarching goal of the volume is to explain and demonstrate how statistical mechanics can be exploited to construct efficient representations of the effects of small-scale fluctuations on large-scale geophysical mean fields and coherent structures without having to resort to detailed simulations of the small scales. Within the rich collection of concrete examples, the reader will find both simplified set-ups specifically selected as demonstrators for a particular method or technique and much more-complex cases that are of interest as demonstrators as well as in terms of their geophysical fluid dynamics content. [ABSTRACT FROM AUTHOR]