Double averaging methodology and double‐decomposition: description of intermediate scales between a fluid particle and a catchment

Copyright © 2014 John Wiley & Sons, Ltd. HOW TO AVERAGE AND WHY As an observer zooms out from a fluid particle to a river depth to a catchment scale, and also from a split second to aminute, a day and a hydrological season, the small-scale details disappear, and the large-scale patterns emerge. One of the ways to describe the process of ‘zooming-out’ in mathematical terms is averaging. Averaging over time or over an ensemble of statistical realizations was first proposed by Osborn Reynolds in 1895. It is suitable for studying turbulent flows because it filters out the rapidly changing turbulent fluctuations from the slowly changing or stationary flow, resulting in a simpler flow description. For similar reasons, the description of spatially heterogeneous flows becomes simpler when the small-scale spatial disturbances are filtered out by spatial averaging. The methodology that combines time (or ensemble) averaging with spatial averaging is called the double-averaging methodology. The theoretical background for spatial averaging was first developed in the area of porous media flows, motivated by the need to derive various macroscopic laws such as equations for single-phase and multi-phase flows in porous media. The main research groups that developed the spatial averaging methodology were led by W.G. Gray (e.g. Gray, 1975), S. Whitaker (e.g. Whitaker, 1999) and J. Bear (e.g. Bear and Bachmat, 1990). As a rational tool for relating the fundamental laws at a range of spatial scales, spatial averaging methodology is applicable not only to porous media flows but also to a wide range of physical systems (Gray et al., 1993). One of its key concepts is the so-called representative elementary volume (REV), which is sufficiently large to capture a statistically significant sample of small-scale processes and sufficiently small to not smooth the large-scale heterogeneities. For a given flow domain, a REV, and a set of small-scale (microscopic) balance equations formulated in terms of smallscale variables, spatial averaging is performed by moving the REV over the flow domain, averaging microscopic variables and equations over the fluid phase within the REV, and assigning the result of averaging to the REV centre. The flow domain at the macroscopic scale therefore becomes continuous. The macroscopic balance equations are usually expressed as partial differential equations, which are very similar to microscopic equations, except for the additional terms which quantify large-scale effects of small-scale processes. Reggiani et al. (1998, 1999) have used the main concepts of spatial averaging to propose a unifying framework for catchment scale hydrological models. The framework is based on the concepts of a representative elementary watershed (REW), defined as a sub-watershed of a larger watershed being modelled. A REW consists of the following sub-regions: unsaturated region, saturated region, overland flow region, main channel region and concentrated overland flow regions. The fundamental equations are expressed in integral form applicable to a