The well-known Knudsen relations and the total exchange flow (TEF) analysis framework provide quantifications of exchange flow across an open boundary to the adjacent ocean in terms of bulk values (Knudsen theory: inflow and outflow volume or salinity) or with resolution in salinity space (TEF: profiles of volume and salt flux in salinity coordinates). In the present study, these theories are extended toward mixing of salinity, defined as the decay of salinity variance due to turbulent mixing. In addition to the advective fluxes, diffusive fluxes across the boundary are also considered now. These new Knudsen and TEF relations for mixing are derived by applying Gauss’s theorem to the salinity square and salinity variance equations. As a result of the analysis, four different Knudsen relations for the mixing in estuaries are derived. The first one is exact and considers nonperiodicity as well as nonconstancy of the inflow and outflow salinities. The other three formulations are approximate only, in the sense that either nonperiodicity or nonconstancy or both are relaxed. The simplest of those formulations has recently been derived by MacCready et al. and estimates the estuarine mixing as the product of inflow salinity, outflow salinity, and time-averaged river runoff. These four mixing estimates are systematically assessed by means of a number of idealized estuarine test cases. For periodic tidal flow, the simplest estimate still predicts the effective (physical plus numerical) mixing within an error of about 10%.
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