The concept of water renewal refers to the processes by which water initially located in the domain of interest (the “original water”) is progressively replaced by water originating from its environment (the “renewing water”). Determining the rate at which water renewal is achieved is of use in many hydrodynamical, pollution and ecological studies.Over the last few decades, many methods were suggested for quantifying water renewal rates. Most of them, be they based on in situ measurements or numerical results, aimed at estimating relevant timescales. Herein, a generic approach to such timescales is presented that relies on the Constituent-oriented Age and Residence time Theory (CART, www.climate.be/CART). Accordingly, by solving a partial differential problem in a backward mode, the residence time of the original water may be calculated at any time and position, i.e. the time needed for every particle of this water type to reach for the first time an open boundary of the domain. Additional information may be obtained by estimating the age of the renewing water, which is defined at any time and location as the time that has elapsed since entering the domain. To obtain this timescale, one must solve a partial differential problem in a forward mode. Clearly, the age of the renewing water and the residence time of the original water are complementary, diagnostic quantities, whose domain averaged values are seen to be equal at a steady state. Upon estimating its residence time, an original water particle is discarded as soon as it hits one of the open boundaries of the domain of interest, thereby ignoring the possibility that this water particle may re-enter the domain at a later time. As this is unlikely to be acceptable in all water renewal studies, an alternative approach is suggested, that consists in estimating the exposure time, i.e. the time spent in the domain of interest. This timescale may be estimated in the domain and its environment — while the residence time of the original water and the age of the renewing water are defined only in the domain of interest. If this alternative strategy is selected, there is no need to split the water into two categories, original water and renewing water.In the domain of interest, the exposure time is always larger than the residence time, and the relative difference between them is termed the “return coefficient”. The latter measures the propensity for a water parcel to re-enter the domain of interest after leaving it for the first time.All of the above timescales may be calculated in a Lagrangian framework, by having recourse to relevant random walk algorithms. However, in the present study, most of the developments are performed in a Eulerian mode, making it possible to establish rather easily properties that the timescales under consideration satisfy at any time and position. The physical meaning and the usefulness — as diagnostic quantities — of these timescales is illustrated by tackling a couple of idealised, zero- or one-dimensional flow configurations that allow analytical solutions to be obtained.Using the unstructured-mesh, finite-element Second-generation Louvain-la-Neuve Iceocean Model (SLIM, www.climate.be/slim), the flows in the upper layer of Lake Tanganyika and the Scheldt Estuary are simulated along with the abovementioned timescales. By doing so, a picture of the long-term transport properties is obtained that is of use to gain insight into the fate of contaminants or the ecology of the domains of interest. |
